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Full text of "Elements Of X Ray Diffraction"

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ELEMENTS OF 

X-RAY DIFFRACTION 



ADDISON-WESLEY METALLURGY SERIES 
MORRIS COHEN, Consulting Editor 

Cidlity ELEMENTS OF X-RAY DIFFRACTION 
Guy ELEMENTS OF PHYSICAL METALLURGY 
Norton ELEMENTS OF CERAMICS 

Schuhmann METALLURGICAL ENGINEERING 

VOL. I: ENGINEERING PRINCIPLES 

Wagner THERMODYNAMICS OF ALLOYS 



ELEMENTS OF 

X-RAY DIFFRACTION 



by 
B. D. CULLITY 

Associate Professor of Metallurgy 
University of Notre Dame 



ADDISON-WESLEY PUBLISHING COMPANY, INC. 

READING, MASSACHUSETTS 



Copyright 1956 
ADD1SON-WESLEY PUBLISHING COMPANY, Inc. 



Printed ni the United States of America 

ALL RIGHTS RESERVED. THIS BOOK, OR PARTS THERE- 
OF, MAY NOT BE REI'RODl CED IN ANY FORM WITHOUT 
WRITTEN PERMISSION OF THE PUBLISHERS 

Library of Congress Catalog No 56-10137 



PREFACE 

X-ray diffraction is a tool for the investigation of the fine structure of 
matter. This technique had its beginnings in von Laue's discovery in 1912 
that crystals diffract x-rays, the manner of the diffraction revealing the 
structure of the crystal. At first, x-ray diffraction was used only for the 
determination of crystal structure. Later on, however, other uses were 
developed, and today the method is applied, not only to structure deter- 
mination, but to such diverse problems as chemical analysis and stress 
measurement, to the study of phase equilibria and the measurement of 
particle size, to the determination of the orientation of one crystal or the 
ensemble of orientations in a polycrystalline aggregate. 

The purpose of this book is to acquaint the reader who has no previous 
knowledge of the subject with the theory of x-ray diffraction, the experi- 
mental methods involved, and the main applications. Because the author 
is a metallurgist, the majority of these applications are described in terms 
of metals and alloys. However, little or no modification of experimental 
method is required for the examinatiorrof nonmetallic materials, inasmuch 
as the physical principles involved do not depend on the material investi- 
gated. This book should therefore be useful to metallurgists, chemists, 
physicists, ceramists, mineralogists, etc., namely, to all who use x-ray diffrac- 
tion purely as a laboratory tool for the sort of problems already mentioned. 

Members of this group, unlike x-ray crystallographers, are not normally 
concerned with the determination of complex crystal structures. For this 
reason the rotating-crystal method and space-group theory, the two chief 
tools in the solution of such structures, are described only briefly. 

This is a book of principles and methods intended for the student, and 
not a reference book for the advanced research worker. Thus no metal- 
lurgical data are given beyond those necessary to illustrate the diffraction 
methods involved. For example, the theory and practice of determining 
preferred orientation are treated in detail, but the reasons for preferred 
orientation, the conditions affecting its development, and actual orien- 
tations found in specific metals and alloys are not described, because these 
topics are adequately covered in existing books. In short, x-ray diffrac- 
tion is stressed rather than metallurgy. 

The book is divided into three main parts: fundamentals, experimental 
methods, and applications. The subject of crystal structure is approached 
through, and based on, the concept of the point lattice (Bravais lattice), 
because the point lattice of a substance is so closely related to its diffrac- 



VI PREFACE 

tion pattern. The entire book is written in terms of the Bragg law and 
can be read without any knowledge of the reciprocal lattice. (However, a 
brief treatment of reciprocal-lattice theory is given in an appendix for those 
who wish to pursue the subject further.) The methods of calculating the 
intensities of diffracted beams are introduced early in the book and used 
throughout. Since a rigorous derivation of many of the equations for dif- 
fracted intensity is too lengthy and complex a matter for a book of this 
kind, I have preferred a semiquantitative approach which, although it does 
not furnish a rigorous proof of the final result, at least makes it physically 
reasonable. This preference is based on my conviction that it is better 
for a student to grasp the physical reality behind a mathematical equation 
than to be able to glibly reproduce an involved mathematical derivation 
of whose physical meaning he is only dimly aware. 

Chapters on chemical analysis by diffraction and fluorescence have been 
included because of the present industrial importance of these analytical 
methods. In Chapter 7 the diffractometer, the newest instrument for dif- 
fraction experiments, is described in some detail ; here the material on the 
various kinds of counters and their associated circuits should be useful, 
not only to those engaged in diffraction work, but also to those working 
with radioactive tracers or similar substances who wish to know how their 
measuring instruments operate. 

Each chapter includes a set of problems. Many of these have been 
chosen to amplify and extend particular topics discussed in the text, and 
as such they form an integral part of the book. 

Chapter 18 contains an annotated list of books suitable for further study. 
The reader should become familiar with at least a few of these, as he pro- 
gresses through this book, in order that he may know where to turn for 
additional information. 

Like any author of a technical book, I am greatly indebted to previous 
writers on this and allied subjects. I must also acknowledge my gratitude 
to two of my former teachers at the Massachusetts Institute of Technology, 
Professor B. E. Warren and Professor John T. Norton: they will find many 
an echo of their own lectures in these pages. Professor Warren has kindly 
allowed me to use many problems of his devising, and the advice and 
encouragement of Professor Norton has been invaluable. My colleague at 
Notre Dame, Professor G. C. Kuczynski, has read the entire book as it was 
written, and his constructive criticisms have been most helpful. I would 
also like to thank the following, each of whom has read one or more chap- 
ters and offered valuable suggestions: Paul A. Beck, Herbert Friedman, 
S. S. Hsu, Lawrence Lee, Walter C. Miller, William Parrish, Howard 
Pickett, and Bernard Waldman. I am also indebted to C. G. Dunn for 
the loan of illustrative material and to many graduate students, August 



PREFACE Vll 

Freda in particular, who have helped with the preparation of diffraction 
patterns. Finally but not perfunctorily, I wish to thank Miss Rose Kunkle 
for her patience and diligence in preparing the typed manuscript. 

B. D. CULLITY 
Notre Dame, Indiana 
March, 1956 



CONTENTS 

FUNDAMENTALS 

CHAPTER 1 PROPERTIES OF X-RAYS 1 

1-1 Introduction 1 

1-2 Electromagnetic radiation 1 

1-3 The continuous spectrum . 4 

1-4 The characteristic spectrum 6 

1-5 Absorption . 10 

1-6 Filters 16 

1-7 Production of x-rays 17 

1 -8 Detection of x-rays 23 

1 9 Safety precautions . 25 

CHAPTER 2 THE GEOMETRY OF CRYSTALS 29 

^2-1 Introduction . 29 

J2-2 Lattices . 29 

2-3 Crystal systems 30 

^2-4 Symmetry 34 

2-5 Primitive and nonprimitive cells 36 

2-6 Lattice directions and planes * . 37 

2-7 Crystal structure J 42 

2-8 Atom sizes and coordination 52 

2-9 Crystal shape 54 

2-10 Twinned crystals . 55 

2-11 The stereographic projection . . 60 

CHAPTER 3 DIFFRACTION I: THE DIRECTIONS OF DIFFRACTED BEAMS 78 

3-1 Introduction . .78 

3-2 Diffraction f . 79 

^3-3 The Bragg law * ' . 84 

3-4 X-ray spectroscopy 85 

3-5 Diffraction directions - 88 

3-6 Diffraction methods . 89 

3-7 Diffraction under nonideal conditions . 96 

CHAPTER 4 DIFFRACTION II: THE INTENSITIES OF DIFFRACTED BEAMS . 104 

4-1 Introduction 104 

4-2 Scattering by an electrons . . 105 

4-3 Scattering by an atom >, . / 108 

4-4 Scattering by a unit cell */ . Ill 



CONTENTS 

4-5 Some useful relations . 118 

4-6 Structure-factor calculations ^ 118 

4-7 Application to powder method ' 123 

4-8 Multiplicity factor 124 

4-9 Lorentz factor 124 

1-10 Absorption factor 129 

4-11 Temperature factor 130 

4-12 Intensities of powder pattern lines 132 

4-13 Examples of intensity calculations 132 

4-14 Measurement of x-ray intensity 136 

EXPERIMENTAL METHODS 

LPTER 5 LAUE PHOTOGRAPHS 138 

5-1 Introduction 138 

5-2 Cameras . 138 

5-3 Specimen holders 143 

5-4 Collimators . .144 

5-5 The shapes of Laue spots . 146 

kPTER 6 POWDER PHOTOGRAPHS . .149 

6-1 Introduction . 149 

6-2 Debye-Scherrer method . 149 

6-3 Specimen preparation .... 153 

6-4 Film loading . . 154 

6-5 Cameras for high and low temperatures . 156 

6-6 Focusing cameras ... . 156 

6-7 Seemann-Bohlin camera . 157 

6-8 Back-reflection focusing cameras . . .160 

6-9 Pinhole photographs . 163 

6-10 Choice of radiation . .165 

6-11 Background radiation . 166 

6-12 Crystal monochromators . 168 

6-13 Measurement of line position 173 

6-14 Measurement of line intensity . 173 

VPTER 7 DlFFRACTOMETER MEASUREMENTS 177 

7-1 Introduction . . . 177 

7-2 General features .... 177 

7-3 X-ray optics . . . - 184 

7-4 Intensity calculations . ... 188 

7-5 Proportional counters . . . . 190 

7-6 Geiger counters . . ... .193 

7-7 Scintillation counters . - 201 

7-8 Sealers . ... .... .202 

7-9 Ratemeters . - 206 

7-10 Use of monochromators 211 



CONTENTS XI 

APPLICATIONS 

CHAPTER 8 ORIENTATION OF SINGLE CRYSTALS . . . 215 

8-1 Introduction . . .... 215 

8-2 Back-reflection Laue method . . .215 

8-3 Transmission Laue method .... . 229 

8-4 Diffractometer method ' . ... 237 

8-5 Setting a crystal in a required orientation . 240 

8-6 Effect of plastic deformation . 242 

8-7 Relative orientation of twinned crystals 250 

8-8 Relative orientation of precipitate and matrix . . . 256 

CHAPTER 9 THE STRUCTURE OF POLYCRYSTALLINE AGGREGATES . 259 

9-1 Introduction . 259 

CRYSTAL SIZE 

9-2 Grain size 259 

9-3 Particle size . 261 

CRYSTAL PERFECTION 

9-4 Crystal perfection . .... 263 

9-5 Depth of x-ray penetration . . 269 

CRYSTAL ORIENTATION 

9-6 General . .272 

9-7 Texture of wire and rod (photographic method) . . . 276 

9-8 Texture of sheet (photographic method) 280 

9-9 Texture of sheet (diffractometer method) . . 285 

9-10 Summary . . 295 

CHAPTER 10 THE DETERMINATION OF CRYSTAL STRUCTURE . . . 297 

10-1 Introduction . . 297 

10-2 Preliminary treatment of data . . . 299 

10-3 Indexing patterns of cubic crystals 301 

10-4 Indexing patterns of noncubic crystals (graphical methods) 304 
10-5 Indexing patterns of noncubic crystals (analytical methods) . .311 

10-6 The effect of cell distortion on the powder pattern . . . 314 
10-7 Determination of the number of atoms in a unit cell . .316 

10-8 Determination of atom positions . 317 

10-9 Example of structure determination .... . 320 

CHAPTER 11 PRECISE PARAMETER MEASUREMENTS . ... 324 

11-1 Introduction .... 324 

11-2 Debye-Scherrer cameras .... .... 326 

1 1-3 Back-reflection focusing cameras 333 

11-4 Pinhole cameras 333 

11-5 Diffractometers 334 

11-6 Method of least squares .335 



Xll CONTENTS 

11-7 Cohen's method .... 338 

11-8 Calibration method . . 342 

CHAPTER 12 PHASE-DIAGRAM DETERMINATION . . . 345 

12-1 Introduction . 345 

12-2 General principles . . 346 

12-3 Solid solutions . 351 

12-4 Determination of solvus curves (disappearing-phase method) 354 

12-5 Determination of solvus curves (parametric method) 356 

12-6 Ternary systems 359 

CHAPTER 13 ORDER-DISORDER TRANSFORMATIONS 363 

13-1 Introduction . 363 

13-2 Long-range order in AuCus 363 

13-3 Other examples of long-range order 369 

13-4 Detection of superlattice lines 372 

13-5 Short-range order and clustering 375 

CHAPTER 14 CHEMICAL ANALYSIS BY DIFFRACTION 378 

14-1 Introduction 378 

QUALITATIVE ANALYSIS 

14-2 Basic principles 379 

14-3 Hanawait method 379 

14-4 Examples of qualitative analysis 383 

14-5 Practical difficulties 386 

14-6 Identification of surface deposits 387 

QUANTITATIVE ANALYSIS (SINGLE PHASE) 

14-7 Chemical analysis by parameter measurement 388 

QUANTITATIVE ANALYSIS (MULTIPHASE) 

14-8 Basic principles . . . 388 

14-9 Direct comparison method . . . 391 

14-10 Internal standard method . . . 396 

14-11 Practical difficulties . . . 398 

CHAPTER 15 CHEMICAL ANALYSIS BY FLUORESCENCE 402 

15-1 Introduction . ... 402 

15-2 General principles . . 404 

15-3 Spectrometers ... . 407 

15-4 Intensity and resolution . . . 410 

15-5 Counters .... . 414 

15-6 Qualitative analysis .... ... 414 

15-7 Quantitative analysis ... . . 415 

15-8 Automatic spectrometers . . 417 

15-9 Nondispersive analysis ..... . 419 

15-10 Measurement of coating thickness 421 



CONTENTS xiil 

CHAPTER 16 CHEMICAL ANALYSIS BY ABSORPTION . . . 423 

16-1 Introduction . . . ... 423 

16-2 Absorption-edge method . . ... 424 

16-3 Direct-absorption method (monochromatic beam) . 427 

16-4 Direct-absorption method (polychromatic beam) 429 

16-5 Applications . . 429 

CHAPTER 17 STRESS MEASUREMENT . ... 431 

17-1 Introduction . 431 

17-2 Applied stress and residual stress . . 431 

17-3 Uniaxial stress . . 434 

17-4 Biaxial stress . 436 

17-5 Experimental technique (pinhole camera) 441 

17-6 Experimental technique (diffractometer) 444 

17-7 Superimposed macrostress and microstress 447 

17-8 Calibration 449 

1 7-9 Applications 451 

CHAPTER 18 SUGGESTIONS FOR FURTHER STUDY . 454 

18-1 Introduction 454 

18-2 Textbooks . 454 

18-3 Reference books . 457 

18-4 Periodicals 458 



APPENDIXES 

APPENDIX 1 LATTICE GEOMETRY . 459 

Al-1 Plane spacings 459 

Al-2 Cell volumes . . 460 

Al-3 Interplanar angles . . . 460 

APPENDIX 2 THE RHOMBOHEDRAL-HEXAGONAL TRANSFORMATION 462 

APPENDIX 3 WAVELENGTHS (IN ANGSTROMS) OF SOME CHARACTERISTIC 

EMISSION LINES AND ABSORPTION EDGES . . . 464 

APPENDIX 4 MASS ABSORPTION COEFFICIENTS AND DENSITIES . 466 

APPENDIX 5 VALUES OF siN 2 8 . 469 

APPENDIX 6 QUADRATIC FORMS OF MILLER INDICES . . . 471 

APPENDIX 7 VALUES OF (SIN 0)/X . . . 472 

APPENDIX 8 ATOMIC SCATTERING FACTORS . 474 

APPENDIX 9 MULTIPLICITY FACTORS FOR POWDER PHOTOGRAPHS . * . 477 

APPENDIX 10 LORENTZ-POLARIZATION FACTOR 478 

APPENDIX 11 PHYSICAL CONSTANTS . 480 



XIV CONTENTS 

APPENDIX 12 INTERNATIONAL ATOMIC WEIGHTS, 1953 481 

APPENDIX 13 CRYSTAL STRUCTURE DATA 482 

APPENDIX 14 ELECTRON AND NEUTRON DIFFRACTION 486 

A14-1 Introduction . ... . . 486 

A14r-2 Electron diffraction ... . 486 

A14-3 Neutron diffraction .... . 487 

APPENDIX 15 THE RECIPROCAL LATTICE . . 490 

A15-1 Introduction . .... .490 

A15-2 Vector multiplication . ... 490 

A15-3 The reciprocal lattice . . ... 491 

A15-4 Diffraction and the reciprocal lattice . 496 

A15-5 The rotating-crystal method . 499 

A15-6 The powder method . 500 

A15-7 The Laue method . . 502 

ANSWERS TO SELECTED PROBLEMS . 506 

INDEX ... 509 



CHAPTER 1 
PROPERTIES OF X-RAYS 

1-1 Introduction. X-rays were discovered in 1895 by the German 
physicist Roentgen and were so named because their nature was unknown 
at the time. Unlike ordinary light, these rays were invisible, but they 
traveled in straight lines and affected photographic film in the same way 
as light. On the other hand, they were much more penetrating than light 
and could easily pass through the human body, wood, quite thick pieces of 
metal, and other "opaque" objects. 

It is not always necessary to understand a thing in order to use it, and 
x-rays were almost immediately put to use by physicians and, somewhat 
later, by engineers, who wished to study the internal structure of opaque 
objects. By placing a source of x-rays on one side of the object and photo- 
graphic film on the other, a shadow picture, or radiograph, could be made, 
the less dense portions of the object allowing a greater proportion of the 
x-radiation to pass through than the more dense. In this way the point 
of fracture in a broken bone or the position of a crack in a metal casting 
could be located. 

Radiography was thus initiated without any precise understanding of 
the radiation used, because it was not until 1912 that the exact nature of 
x-rays was established. In that year the phenomenon of x-ray diffraction 
by crystals was discovered, and this discovery simultaneously proved the 
wave nature of x-rays and provided a new method for investigating the 
fine structure of matter. Although radiography is a very important tool 
in itself and has a wide field of applicability, it is ordinarily limited in the 
internal detail it can resolve, or disclose, to sizes of the order of 10"" 1 cm. 
Diffraction, on the other hand, can indirectly reveal details of internal 
structure of the order of 10~~ 8 cm in size, and it is with this phenomenon, 
and its applications to metallurgical problems, that this book is concerned. 
The properties of x-rays and the internal structure of crystals are here 
described in the first two chapters as necessary preliminaries to the dis- 
cussion of the diffraction of x-rays by crystals which follows. 

1-2 Electromagnetic radiation. We know today that x-rays are elec- 
tromagnetic radiation of exactly the same nature as light but of very much 
shorter wavelength. The unit of measurement in the x-ray region is the 
angstrom (A), equal to 10~ 8 cm, and x-rays used in diffraction have wave- 
lengths lying approximately in the range 0.5-2.5A, whereas the wavelength 
of visible light is of the order of 6000A. X-rays therefore occupy the 

1 



PROPERTIES OF X-RAYS 



[CHAP. 1 



Frequency Wavelength 




(cycles/sec) in millimicrons 




10 23 












10 22 








_10- 5 




10 21 




Gamma-rays < 




_io- 4 i 


X unit 


10 20 


" 






_io~ 3 




10 19 








_10- 2 




10' 8 


_J 


x-rays 


.. 


__io- ] i 


angstrom 


10 17 




_1 1 


millimicron 


10* 


J\ Ultraviolet - 


*~ " 


_io 




ID 15 








_10 2 




10 77 


M 


WHj. Visible 1H 





_10 3 1 


micron 


ion 




> Infrared - 




"lo* 4 




10"" 


Short radio waves 


10 




10E 




_10 7 1 


centimeter 






_10 8 




iof^ 




_10 9 1 


meter 







_10 10 




10^ 


uttTiill 


~10 12 1 


kilometer 


10f_ 




10 13 




iol 


Long radio waves 


__10 14 




10 




10 15 




iol 




_10 16 





1 megacycle 10_ 



1 kilocycle IQl 



FIG. i-i. The electromagnetic spectrum. The boundaries between regions are 
arbitrary, since no sharp upper or lower limits can be assigned. (F. W. Sears, Optics, 
3rd ed., Addison- Wesley Publishing Company, Inc., Cambridge, Mass., 1949 ) 

region between gamma and ultraviolet rays in the complete electromag- 
netic spectrum (Fig. 1-1). Other units sometimes used to measure x-ray 
wavelength are the X unit (XU) and the kilo X unit (kX = 1000 XU).* 
The X unit is only slightly larger than the angstrom, the exact relation 

bemg lkX= 1.00202A. 

It is worth while to review briefly some properties of electromagnetic 
waves. Suppose a monochromatic beam of x-rays, i.e., x-rays of a single 
wavelength, is traveling in the x direction (Fig. 1-2). Then it has asso- 
ciated with it an electric field E in, say, the y direction and, at right angles 
to this, a magnetic field H in the z direction. If the electric field is con- 
fined to the xy-plane as the wave travels along, the wave is said to be plane- 
polarized. (In a completely unpolarized wave, the electric field vector E 
and hence the magnetic field vector H can assume all directions in the 

* For the origin of these units, see Sec. 3-4. 



1-2] 



ELECTROMAGNETIC RADIATION 




FIG. 1-2. Electric and magnetic 
fields associated with a wave moving 
in the j-direction. 



t/2-plane.) The magnetic field is of 
no concern to us here and we need 
not consider it further. 

In the plane-polarized wave con- 
sidered, E is not constant with time 
but varies from a maximum in the 
+y direction through zero to a maxi- 
mum in the y direction and back 
again, at any particular point in 
space, say x = 0. At any instant of 
time, say t = 0, E varies in the same 
fashion with distance along thex-axis. 
If both variations are assumed to be sinusoidal, they may be expressed in 
the one equation 

E = Asin27r(- - lA (1-1) 

where A = amplitude of the wave, X = wavelength, and v = frequency. 
The variation of E is not necessarily sinusoidal, but the exact form of the 
wave matters little; the important feature is its periodicity. Figure 1-3 
shows the variation of E graphically. The wavelength and frequency are 

connected by the relation c 

X - -. (1-2) 

V 

where c = velocity of light = 3.00 X 10 10 cm/sec. 

Electromagnetic radiation, such as a beam of x-rays, carries energy, and 
the rate of flow of this energy through unit area perpendicular to the direc- 
tion of motion of the wave is called the intensity I. The average value of 
the intensity is proportional to the square of the amplitude of the wave, 
i.e., proportional to A 2 . In absolute units, intensity is measured in 
ergs/cm 2 /sec, but this measurement is a difficult one and is seldom carried 
out; most x-ray intensity measurements are made on a relative basis in 



+E 


-E 





+E 
i 





(a) (b) 

FIG. 1-3. The variation of E, (a) with t at a fixed value of x and (b) with x at 
a fixed value of t. 



4 PKOPERTIES OF X-RAYS [CHAP. 1 

arbitrary units, such as the degree of blackening of a photographic film 
exposed to the x-ray beam. 

An accelerated electric charge radiates energy. The acceleration may, 
of course, be either positive or negative, and thus a charge continuously 
oscillating about some mean position acts as an excellent source of electro- 
magnetic radiation. Radio waves, for example, are produced by the oscil- 
lation of charge back and forth in the broadcasting antenna, and visible 
light by oscillating electrons in the atoms of the substance emitting the 
light. In each case, the frequency of the radiation is the same as the fre- 
quency of the oscillator which produces it. 

Up to now we have been considering electromagnetic radiation as wave 
motion in accordance with classical theory. According to the quantum 
theory, however, electromagnetic radiation can also be considered as a 
stream of particles called quanta or photons. Each photon has associated 
with it an amount of energy hv, where h is Planck's constant (6.62 X 10~ 27 
erg -sec). A link is thus provided between the two viewpoints, because 
we can use the frequency of the wave motion to calculate the energy of 
the photon. Radiation thus has a dual wave-particle character, and we 
will use sometimes one concept, sometimes the other, to explain various 
phenomena, giving preference in general to the classical wave theory when- 
ever it is applicable. 

1-3 The continuous spectrum. X-rays are produced when any electri- 
cally charged particle of sufficient kinetic energy is rapidly decelerated. 
Electrons are usually used for this purpose, the radiation being produced 
in an x-ray tube which contains a source of electrons and two metal elec- 
trodes. The high voltage maintained across these electrodes, some tens 
of thousands of volts, rapidly draws the electrons to the anode, or target, 
which they strike with very high velocity. X-rays are produced at the 
point of impact and radiate in all directions. If e is the charge on the elec- 
tron (4.80 X 10~ 10 esu) and 1) the voltage (in esu)* across the electrodes, 
then the kinetic energy (in ergs) of *the electrons on impact is given by the 
equation 

KE - eV = \mv*, (1-3) 



where m is the mass of the electron (9.11 X 10~ 28 gm) and v its velocity 
just before impact. At a tube voltage of 30,000 volts (practical units), 
this velocity is about one-third that of light. Most of the kinetic energy 
of the electrons striking the target is converted into heat, less than 1 percent 
being transformed into x-rays. 

When the rays coming from the target are analyzed, they are found to 
consist of a mixture of different wavelengths, and the variation of intensity 

* 1 volt (practical units) = ^fo volt (esu). 



1-3] 



THE CONTINUOUS SPECTRUM 




1.0 2.0 

WAVELENGTH (angstroms) 

FIG. 1-4. X-ray spectrum of molybdenum as a function of applied voltage (sche- 
matic). Line widths not to scale. 

with wavelength is found to depend on the tube voltage. Figure 1-4 
shows the kind of curves obtained. The intensity is zero up to a certain 
wavelength, called the short-wavelengthjimit (XSWL), increases rapidly to a 
maximum and then decreases, with no sharp limit on the long wavelength 
side. * When the tube voltage is raised, the intensity of all wavelengths 
increases, and both the short-wavelength limit and the position of the max- 
imum shift to shorter wavelengths. We are concerned now with the 
smooth curves in Fig. 1-4, those corresponding to applied voltages of 
20 kv or less in the case of a molybdenum target. The radiation repre- 
sented by such curves is called heterochromatic, continuous, or white radia- 
tion, since it is made up, like white light, of rays of many wavelengths. 

The continuous spectrum is due to the rapid deceleration of the electrons 
hitting the target since, as mentioned above, any decelerated charge emits 
energy. Not every electron is decelerated in the same way, however; some 
are stopped in one impact and give up all their energy at once, while others 
are deviated this way and that by the atoms of the target, successively 
losing fractions of their total kinetic energy until it is all spent. Those 
electrons which are stopped in one impact will give rise to photons of 
maximum energy, i.e., to x-rays of minimum wavelength. Such electrons 
transfer all their energy eV into photon energy and we may write 



PROPERTIES OF X-RAYS [CHAP. 1 

c he 



12,400 

(1-4) 



This equation gives the short-wavelength limit (in angstroms) as a func- 
tion of the applied voltage V (in practical units). If an electron is not 
completely stopped in one encounter but undergoes a glancing impact 
which only partially decreases its velocity, then only a fraction of its energy 
eV is emitted as radiation and the photon produced has energy less than 
hpmax- In terms of wave motion, the corresponding x-ray has a frequency 
lower than v max and a wavelength longer than XSWL- The totality of these 
wavelengths, ranging upward from ASWL, constitutes the continuous spec- 
trum. 

We now see why the curves of Fig. 1-4 become higher and shift to the 
left as the applied voltage is increased, since the number of photons pro- 
duced per second and the average energy per photon are both increasing. 
The total x-ray energy emitted per second, which is proportional to the 
area under one of the curves of Fig. 1-4, also depends on the atomic num- 
ber Z of the target and on the tube current i, the latter being a measure of 
the number of electrons per second striking the target. This total x-ray 
intensity is given by 

/cent spectrum = AlZV, (1-5) 

where A is a proportionality constant and m is a constant with a value of 
about 2. Where large amounts of white radiation are desired, it is there- 
fore necessary to use a heavy metal like tungsten (Z = 74) as a target and 
as high a voltage as possible. Note that the material of t the target affects 
the intensity but not thg. wftV dfin fi^h distribution O f t.hp..p.ont.iniiniia spec- 
trum, 

1-4 The characteristic spectrum. When the voltage on an x-ray tube 
is raised above a certain critical value, characteristic of the target metal, 
sharp intensity maxima appear at certain wavelengths, superimposed on 
the continuous spectrum. Since they are so narrow and since their wave- 
lengths are characteristic of the target metal used, they are called charac- 
teristic lines. These lines fall into several sets, referred to as K, L, M, 
etc., in the order of increasing wavelength, all the lines together forming 
the characteristic spectrum of the metal used as the target. For a molyb- 
denum target the K lines have wavelengths of about 0.7A, the L lines 
about 5A, and the M lines still higher wavelengths. Ordinarily only the 
K lines are useful in x-ray diffraction, the longer-wavelength lines being 
too easily absorbed. There are several lines in the K set, but only the 



1-4] THE CHARACTERISTIC SPECTRUM 7 

three strongest are observed in normal diffraction work. These are the 
ctz, and Kfa, and for molybdenum their wavelengths are: 



0.70926A, 
Ka 2 : 0.71354A, 

0.63225A. 



The i and 2 components have wavelengths so close together that they 
are not always resolved as separate lines; if resolved, they are called the 
Ka doublet and, if not resolved, simply the Ka line* Similarly, K&\ is 
usually referred to as the K@ line, with the subscript dropped. Ka\ is 
always about twice as strong as Ka%, while the intensity ratio of Ka\ to 
Kfli depends on atomic number but averages about 5/1. 

These characteristic lines may be seen in the uppermost curve of Fig. 
1-4. Since the critical K excitation voltage, i.e., the voltage necessary to 
excite K characteristic radiation, is 20.01 kv for molybdenum, the K lines 
do not appear in the lower curves of Fig. 1-4. An increase in voltage 
above the critical voltage increases the intensities of the characteristic 
lines relative to the continuous spectrum but does not change their wave- 
lengths. Figure 1-5 shows the spectrum of molybdenum at 35 kv on a 
compressed vertical scale relative to that of Fig. 1-4 ; the increased voltage 
has shifted the continuous spectrum to still shorter wavelengths and in- 
creased the intensities of the K lines relative to the continuous spectrum 
but has not changed their wavelengths. 

The intensity of any characteristic line, measured above the continuous 
spectrum, depends both on the tube current i and the amount by which 
the applied voltage V exceeds the critical excitation voltage for that line. 
For a K line, the intensity is given by 

IK line = Bi(V - V K ) n , (1-6) 

where B is a proportionality constant, VK the K excitation voltage, and 
n a constant with a value of about 1.5. The intensity of a characteristic 
line can be quite large: for example, in the radiation from a copper target 
operated at 30 kv, the Ka line has an intensity about 90 times that of the 
wavelengths immediately adjacent to it in the continuous spectrum. Be- 
sides being very intense, characteristic lines are also very narrow, most of 
them less than 0.001A wide measured at half their maximum intensity, 
as shown in Fig. 1-5. The existence of this strong sharp Ka. line is what 
makes a great deal of x-ray diffraction possible, since many diffraction 
experiments require the use of monochromatic or approximately mono- 
chromatic radiation. 



* The wavelength of an unresolved Ka doublet is usually taken as the weighted 
average of the wavelengths of its components, Kai being given twice the weight 
of Ka%, since it is twice as strong. Thus the wavelength of the unresolved Mo Ka 
line is J(2 X 0.70926 + 0.71354) = 0.71069A. 



PROPERTIES OF X-RAYS 



[CHAP. 1 



60 



50 



.5 40 






1 30 



20 



10 







Ka 



*-<0.001A 



0.2 



1.0 



0.4 0.6 0.8 

WAVELENGTH (angstroms) 

FIG. 1-5. Spectrum of Mo at 35 kv (schematic). Line widths not to scale. 

The characteristic x-ray lines were discovered by W. H. Bragg and 
systematized by H. G. Moseley. The latter found that the wavelength of 
any particular line decreased as the atomic number of the emitter increased. 
In particular, he found a linear relation (Moseley's law) between the 
square root of the line frequency v and the atomic number Z : 



= C(Z - er), 



(1-7) 



where C and <r are constants. This relation is plotted in Fig. 1-6 for the 
Kai and Lai lines, the latter being the strongest line in the L series. These 
curves show, incidentally, that L lines are not always of long wavelength : 
the Lai line of a heavy metal like tungsten, for example, has about the 
same wavelength as the Ka\ line of copper, namely about 1.5A. The 



1-4] 



THE CHARACTERISTIC SPECTRUM 



3.0 2.5 2.0 



X (angstroms) 
1.5 1.0 



0.8 0.7 



80 



70 



60 



I 
W 



w 



50 



u 

s 40 



30 



20 - 



10 



T 



I 



I 



I 



T 




I 



1.0 



1.2 



1.4 



1.6 



1.8 



2.0 2.2 X 10 9 



FIG. 1-6. Moseley's relation between \/v and Z for two characteristic lines. 

wavelengths of the characteristic x-ray lines of almost all the known ele- 
ments have been precisely measured, mainly by M. Siegbahn and his 
associates, and a tabulation of these wavelengths for the strongest lines 
of the K and L series will be found 
in Appendix 3. 

While the cQntinuoi^s_srjex;truri^js 
caused byjthe T^^^^dej^tignj)^ 
electrons by the targe t ; the origin of 
^ 



M shell 



atoms j3i_tl^_taj^J)_jrnaterial itself. 
To understand this phenomenon, it 
is enough to consider an atom as con- 
sisting of a central nucleus surrounded 
by electrons lying in various shells 
(Fig. 1-7). If one of the electrons 
bombarding the target has sufficient 
kinetic energy, it can knock an elec- 
tron out of the K shell, leaving the 
atom in an excited, high-energy state, 




FlG ^ Elec tronic transitions in 
an at0 m (schematic). Emission proc- 
esses indicated by arrows. 



10 PROPERTIES OF X-RAYS [CHAP. 1 

One of the outer electrons immediately falls into the vacancy in the K shell, 
emitting energy in the process, and the atom is once again in its normal 
energy state. The energy emitted is in the form of radiation of a definite 
wavelength and is, in fact, characteristic K radiation. 

The Jff-shell vacancy may be filled by an electron from any one of the 
outer shells, thus giving rise to a series of K lines; Ka and K& lines, for 
example, result from the filling of a K-shell vacancy by an electron from 
the LOT M shells, respectively. It is possible to fill a 7-shell vacancy either 
from the L or M shell, so that one atom of the target may be emitting Ka 
radiation while its neighbor is emitting Kfi\ however, it is more probable 
that a jf-shell vacancy will be filled by an L electron than by an M elec- 
tron, and the result is that the Ka line is stronger than the K$ line. It 
also follows that it is impossible to excite one K line without exciting all 
the others. L characteristic lines originate in a similar way: an electron 
is knocked out of the L shell and the vacancy is filled by an electron from 
some outer shell. 

We now see why there should be a critical excitation voltage for charac- 
teristic radiation. K radiation, for example, cannot be excited unless the 
tube voltage is such that the bombarding electrons have enough energy 
to knock an electron out of the K shell of a target atom. If WK is the 
work required to remove a K electron, then the necessary kinetic energy 

of the electrons is given by 

ynxr = WK- (1~8) 

It requires less energy to remove an L electron than a K electron, since 
the former is farther from the nucleus; it therefore follows that the L excita- 
tion voltage is less than the K and that K characteristic radiation cannot 
be produced without L, M, etc., radiation accompanying it. 

1-6 Absorption. Further understanding of the electronic transitions 
which can occur in atoms can be gained by considering not only the inter- 
action of electrons and atoms, but also the interaction of x-rays and atoms. 
When x-rays encounter any form of matter, they are partly transmitted 
and partly absorbed. Experiment shows that the fractional decrease in 
the intensity 7 of an x-ray beam as it passes through any homogeneous 
substance is proportional to the distance traversed, x. In differential form, 

-J-/.AC, (1-9) 

where the proportionality constant /u is called the linear absorption coeffi- 
cient and is dependent on the substance considered, its density, and the 
wavelength of the x-rays. Integration of Eq. (1-9) gives 

4- - /or**, (1-10) 

where /o = intensity of incident x-ray beam and I x = intensity of trans- 
mitted beam after passing through a thickness x. 



1-5] 



ABSORPTION 



11 



/* = 


Joe- """ 



The linear absorption coefficient /z is proportional to the density p, which 
means that the quantity M/P is a constant of the material and independent 
of its physical state (solid, liquid, or gas). This latter quantity, called the 
mass absorption coefficient, is the one usually tabulated. Equation (1-10) 
may then be rewritten in a more usable form : 



(1-11) 



Values of the mass absorption coefficient /i/p are given in Appendix 4 for 
various characteristic wavelengths used in diffraction. 

It is occasionally necessary to know the mass absorption coefficient of a 
substance containing more than one element. Whether the substance is a 
mechanical mixture, a solution, or a chemical compound, and whether it 
is in the solid, liquid, or gaseous state, its mass absorption coefficient is 
simply the weighted average of the mass absorption coefficients of its 
constituent elements. If Wi, w 2 , etc., are the weight fractions of elements 
1, 2, etc., in the substance and (M/P)I, (M/p)2j etc., their mass absorption 
coefficients, then the mass absorption coefficient of the substance is given 
by 

- = Wl ( -J + W2 ( -J + . . .. (1-12) 



The way in which the absorption 
coefficient varies with wavelength 
gives the clue to the interaction of 
x-rays and atoms. The lower curve 
of Fig. 1-8 shows this variation for a 
nickel absorber; it is typical of all 
materials. The curve consists of two 
similar branches separated by a sharp 
discontinuity called an absorption 
edge. Along each branch the absorp- 
tion coefficient varies with wave- 
length approximately according to a 
relation of the form 

M 
P 

where k = a constant, with a different 
value for each branch of the curve, 
and Z = atomic number of absorber. 
Short-wavelength x-rays are there- 
fore highly penetrating and are 



?-(n*,gm) ENERGY PER * 
^ QUANTUM (erg) * 

O 5 O COt ' tO C*3 C 




\ 


critica 
for e, 
electro 


1 energ 
ection 
u from 

/ 


yW K 
otK 
nickel 




\ 






\ 


/ 
^^ 





























/' 












K absorption 
^edge 






/ 




/ 




y 




<^i 


V 



0.5 1.0 1.5 2.0 2. 
X (angstroms) 

FIG. 1-8. Variation with wave- 
length of the energy per x-ray quantum 
and of the mass absorption coefficient 
of nickel. 



12 PROPERTIES OF X-RAYS [CHAP. 1 

termed hard, while long-wavelength x-rays are easily absorbed and are said 
to be soft. 

Matter absorbs x-rays in two distinct ways, by scattering and by true 
absorption, and these two processes together make up the total absorption 
measured by the quantity M/P- The scattering of x-rays by atoms is similar 
in many ways to the scattering of visible light by dust particles in the air. 
It takes place in all directions, and since the energy in the scattered beams 
does not appear in the transmitted beam, it is, so far as the transmitted 
beam is concerned, said to be absorbed. The phenomenon of scattering 
will be discussed in greater detail in Chap. 4; it is enough to note here 
that, except for the very light elements, it is responsible for only a small 
fraction of the total absorption. True absorption is caused by electronic 
transitions within the atom and is best considered from the viewpoint of 
the quantum theory of radiation. Just as an electron of sufficient energy 
can knock a K electron, for example, out of an atom and thus cause the 
emission of K characteristic radiation, so also can an incident quantum of 
x-rays, provided it has the same minimum amount of energy WK- In the 
latter case, the ejected electron is called a photoelectron and the emitted 
characteristic radiation is called fluorescent radiation. It radiates in all 
directions and has exactly the same wavelength as the characteristic radia- 
tion caused by electron bombardment of a metal target. (In effect, an 
atom with a #-shell vacancy always emits K radiation no matter how the 
vacancy was originally created.) This phenomenon is the x-ray counter- 
part of the photoelectric effect in the ultraviolet region of the spectrum; 
there, photoelectrons can be ejected from the outer shells of a metal atom 
by the action of ultraviolet radiation, provided the latter has a wavelength 
less than a certain critical value. 

To say that the energy of the incoming quanta must exceed a certain 
value WK is equivalent to saying that the wavelength must be less than a 
certain value X#, since the energy per quantum is hv and wavelength is 
inversely proportional to frequency. These relations may be written 

he 



where V K and \K are the frequency and wavelength, respectively, of the 
K absorption edge. Now consider the absorption curve of Fig. 1-8 in light 
of the above. Suppose that x-rays of wavelength 2.5A are incident on a 
sheet of nickel and that this wavelength is continuously decreased. At 
first the absorption coefficient is about 180 cm 2 /gm, but as the wavelength 
decreases, the frequency increases and so does the energy per quantum, 
as shown by the upper curve, thus causing the absorption coefficient to 
decrease, since the greater the energy of a quantum the more easily it 
passes through an absorber. When the wavelength is reduced just below 



1-5] ABSORPTION 13 

the critical value A#, which is 1.488A for nickel, the absorption coefficient 
suddenly increases about eightfold in value. True absorption is now oc- 
curring and a large fraction of the incident quanta simply disappear, their 
energy being converted into fluorescent radiation and the kinetic energy 
of ejected photoelectrons. Since energy must be conserved in the process, 
it follows that the energy per quantum of the fluorescent radiation must 
be less than that of the incident radiation, or that the wavelength \K of 
the K absorption edge must be shorter than that of any K characteristic 
line. 

As the wavelength of the incident beam is decreased below Xx, the ab- 
sorption coefficient begins to decrease again, even though the production 
of K fluorescent radiation and photoelectrons is still occurring. At a wave- 
length of l.OA, for example, the incident quanta have more than enough 
energy to remove an electron from the K shell of nickel. But the more 
energetic the quanta become, the greater is their probability of passing 
right through the absorber, with the result that less and less of them take 
part in the ejection of photoelectrons. 

If the absorption curve of nickel is plotted for longer wavelengths than 
2.5A, i.e., beyond the limit of Fig. 1-8, other sharp discontinuities will be 
found. These are the L, M, N, etc., absorption edges; in fact, there are 
three closely spaced L edges (Lj, Ln, and I/m), five M edges, etc. Each 
of these discontinuities marks the wavelength of the incident beam whose 
quanta have just sufficient energy to eject an L, M, N, etc., electron from 
the atom. The right-hand branch of the curve of Fig. 1-8, for example, 
lies between the K and L absorption edges; in this wavelength region inci- 
dent x-rays have enough energy to remove L, M, etc., electrons from nickel 
but not enough to remove K electrons. Absorption-edge wavelengths 
vary with the atomic number of the absorber in the same way, but not 
quite as exactly, as characteristic emission wavelengths, that is, according 
to Moseley's law. Values of the K and L absorption-edge wavelengths 
are given in Appendix 3. 

The measured values of the absorption edges can be used to construct 
an energy-level diagram for the atom, which in turn can be used in the 
calculation of characteristic-line wavelengths. For example, if we take 
the energy of the neutral atom as zero, then the energy of an ionized atom 
(an atom in an excited state) will be some positive quantity, since work 
must be done to pull an electron away from the positively charged nucleus. 
If a K electron is removed, work equal to WK must be done and the atom 
is said to be in the K energy state. The energy WK may be calculated 
from the wavelength of the K absorption edge by the use of Eq. (1-14). 
Similarly, the energies of the L, M, etc., states can be calculated from the 
wavelengths of the L, M, etc., absorption edges and the results plotted in 
the form of an energy-level diagram for the atom (Fig. 1-9). 



14 



" A 


i 




A' 




A r /3 emission 




"cfl 








1 






O 








H 








O 








>* ir 








L 




' 






w 








w 




cs 

o 








1 


La 















X 








0? 




H T u 






r 










M A/a 


H r v 










n 








"A T 











PROPERTIES OF X-RAYS [CHAP. 1 

K state (A' electron removed) 



L state (L electron removed) 



M state (M electron removed) 



N state (N electron removed) 
valence electron removed 
neutral atom 

FIG. 1-9. Atomic energy levels (schematic). Excitation and emission processes 
indicated by arrows. (From Structure of Metals, by C. S. Barrett, McGraw-Hill 
Book Company, Inc., 1952.) 

Although this diagram is simplified, in that the substructure of the L, 
M, etc., levels is not shown, it illustrates the main principles. The arrows 
show the transitions of the atom, and their directions are therefore just 
the opposite of the arrows in Fig. 1-7, which shows the transitions of the 
electron. Thus, if a K electron is removed from an atom (whether by an 
incident electron or x-ray), the atom is raised to the K state. If an elec- 
tron then moves from the L to the K level to fill the vacancy, the atom 
undergoes a transition from the K to the L state. This transition is accom- 
panied by the emission of Ka characteristic radiation and the arrow indi- 
cating Kot emission is accordingly drawn from the K state to the L state. 

Figure 1-9 shows clearly how the wavelengths of characteristic emission 
lines can be calculated, since the difference in energy between two states 
will equal hv, where v is the frequency of the radiation emitted when the 



1-5] 



ABSORPTION 



15 



atom goes from one state to the other. Consider the Kai characteristic 
line, for example. The "L level" of an atom is actually a group of three 
closely spaced levels (Li, Ln, and LIU), and the emission of the Kai line 
is due to a K > Lm transition. The frequency VK ai of this line is there- 
fore given by the equations 

hi> K<*I 



(1-15) 



1 

X/,111 



where the subscripts K and Lm refer to absorption edges and the subscript 
Kai to the emission line. 

Excitation voltages can be calculated by a relation similar to Eq. (1-4). 
To excite K radiation, for example, in the target of an x-ray tube, the bom- 
barding electrons must have energy equal to WK> Therefore 



= W K = 

i. 
he 

' 
e\ K 

12,400 



he 

. 

* 



(1-16) 



where VK is the K excitation voltage (in practical units) and \K is the K 
absorption edge wavelength (in angstroms). 

Figure 1-10 summarizes some of the relations developed above. This 
curve gives the short-wavelength limit of the continuous spectrum as a 
function of applied voltage. 
Because of the similarity be- 
tween Eqs. (1-4) and (1-16), 
the same curve also enables us 
to determine the critical exci- 
tation voltage from the wave- 
length of an absorption edge. 



FIG. 1-10. Relation between 
the voltage applied to an x-ray 
tube and the short-wavelength 
limit of the continuous spectrum, 
and between the critical excita- 
tion voltage of any metal and the 
wavelength of its absorption edge. 



30 
25 

920 

I- 
1, 

5 



\ 














\ 












\ 












\ 


\ 












x^ 


^ 


^5^ 


-__. 











0.5 



1.0 1.5 2.0 
X (angstroms) 



2.5 3.0 



16 



PROPERTIES OF X-RAYS 



[CHAP. 1 




A'a 




1.2 



1.4 1.6 

X (angstroms) 



1.8 



1.2 



1.4 1.6 

X (angstroms) 

(b) Nickel filter 



1.8 



(a) No filter 

FIG. 1-11. Comparison of the spectra of copper radiation (a) before and (b) 
after passage through a nickel filter (schematic). The dashed line is the mass ab- 
sorption coefficient of nickel. 

1-6 Filters. Many x-ray diffraction experiments require radiation 
which is as closely monochromatic as possible. However, the beam from 
an x-ray tube operated at a voltage above VK contains not only the strong 
Ka line but also the weaker Kft line and the continuous spectrum. The 
intensity of these undesirable components can be decreased relative to the 
intensity of the Ka line by passing the beam through a filter made of a 
material whose K absorption edge lies between the Ka and Kfl wave- 
lengths of the target metal. Such a material will have an atomic number 1 
or 2 less than that of the target metal. 

A filter so chosen will absorb the Kfi component much more strongly 
than the Ka component, because of the abrupt change in its absorption 
coefficient between these two wavelengths. The effect of filtration is shown 
in Fig. 1-11, in which the partial spectra of the unfiltered and filtered 
beams from a copper target (Z = 29) are shown superimposed on a plot 
of the mass absorption coefficient of the nickel filter (Z = 28). 

The thicker the filter the lower the ratio of intensity of Kft to Ka in the 
transmitted beam. But filtration is never perfect, of course, no matter 
how thick the filter, and one must compromise between reasonable sup- 
pression of the Kfi component and the inevitable weakening of the Ka 
component which accompanies it. In practice it is found that a reduction 



1-7] 



PRODUCTION OF X-RAYS 



17 



TABLE 1-1 
FILTERS FOR SUPPRESSION OF K/3 RADIATION 









Filter thickness for 








Incident beam 


I(Ka) 500 




Target 


Filter 


I(Ka) 


ivm ~ i 

in trans, beam 


I(K<x) trans. 


KKfi) 


I(Kot] incident 








2 

mg/cm 


in. 




Mo 


Zr 


3.9 


75 


0.0045 


0.27 


Cu 


Ni 


5.6 


19 


0.0008 


0.40 


Co 


Fe 


5.7 


14 


0.0007 


0.44 


Fe 


Mn 


5.7 


13 


0.0007 


0.43 


Cr 


V 


5.1 


11 


0.0007 


0.44 



in the intensity of the Ka line to about half its original value will decrease 
the ratio of intensity of K& to Ka from about ^ in the incident beam to 
about -gfa in the transmitted beam ; this level is sufficiently low for most 
purposes. Table 1-1 shows the filters used in conjunction with the com- 
mon target metals, the thicknesses required, and the transmission factors 
for the Ka line. Filter materials are usually used in the form of thin foils. 
If it is not possible to obtain a given metal in the form of a stable foil, the 
oxide of the metal may be used. The powdered oxide is mixed with a 
suitable binder and spread on a paper backing, the required mass of metal 
per unit area being given in Table 1-1. 

1-7 Production of x-rays. We have seen that x-rays are produced 
whenever high-speed electrons collide with a metal target. Any x-ray 
tube must therefore contain (a) a source of electrons, (6) a high acceler- 
ating voltage, and (c) a metal target. Furthermore, since most of the 
kinetic energy of the electrons is converted into heat in the target, the 
latter must be water-cooled to prevent its melting. 

All x-ray tubes contain two electrodes, an anode (the metal target) 
maintained, with few exceptions, at ground potential, and a cathode, 
maintained at a high negative potential, normally of the order of 30,000 
to 50,000 volts for diffraction work. X-ray tubes may be divided into two 
basic types, according to the way in which electrons are provided: filament 
tubes, in which the source of electrons is a hot filament, and gas tubes, in 
which electrons are produced by the ionization of a small quantity of gas 
in the tube. 

Filament tubes, invented by Coolidge in 1913, are by far the more 
widely used\ They consist of an evacuated glass envelope which insulates 
the anode at one end from the cathode at the other, the cathode being a 
tungsten filament and the anode a water-cooled block of copper con- 
taining the desired target metal as a small insert at one end. Figure 1-12 



18 



PROPERTIES OF X-RAYS 



[CHAP. 1 





1-7] 



PRODUCTION OF X-EAY8 



19 



is a photograph of such a tube, and Fig. 1-13 shows its internal construc- 
tion. One lead of the high-voltage transformer is connected to the fila- 
ment and the other to ground, the target being grounded by its own cooling- 
water connection. The filament is heated by a filament current of about 
3 amp and emits electrons which are rapidly drawn to the target by the 
high voltage across the tube. Surrounding the filament is a small metal 
cup maintained at the same high (negative) voltage as the filament: it 
therefore repels the electrons and tends to focus them into a narrow region 
of the target, called the focal spot. X-rays are emitted from the focal 
spot in all directions and escape from the tube through two or more win- 
dows in the tube housing. Since these windows must be vacuum tight 
and yet highly transparent to x-rays, they are usually made of beryllium, 
aluminum, or mica. 

Although one might think that an x-ray tube would operate only from 
a DC source, since the electron flow must occur only in one direction, it is 
actually possible to operate a tube from an AC source such as a transformer 
because of the rectifying properties of the tube itself. Current exists 
during the half-cycle in which the filament is negative with respect to the 
target; during the reverse half-cycle the filament is positive, but no elec- 
trons can flow since only the filament is hot enough to emit electrons. 
Thus a simple circuit such as shown in Fig. 1-14 suffices for many installa- 
tions, although more elaborate circuits, containing rectifying tubes, smooth- 
ing capacitors, and voltage stabilizers, are often used, particularly when 
the x-ray intensity must be kept constant within narrow limits. In Fig. 
1-14, the voltage applied to the tube is controlled by the autotransformer 
which controls the voltage applied to the primary of the high-voltage 
transformer. The voltmeter shown measures the input voltage but may 
be calibrated, if desired, to read the output voltage applied to the tube. 

\-ray tube 



ri'ISZil 

~ 



high-voltage transformer 

M AK Q-0-0-0 Q.ooo Q Q Q Q Q,Q Q .* 



ground 
autotransformer f 0000001)1)0 " 




filament 
rheostat 



000000000 



filament 
transformer 



110 volts AC 



110 volts AC 
FIG. 1-14. Wiring diagram for self-rectifying filament tube. 



20 



PROPERTIES OP X-RAYS 



[CHAP. 1 




c 
o 



1-8] 



DETECTION OF X-RAYS 



23 



electrons 



x-rays 



target 
metal 





anode 



FIG. 1-16. Reduction in apparent 
size of focal spot. 



FIG. 1-17. Schematic drawings of two 
types of rotating anode for high -power 
x-rav tubes. 



Since an x-ray tube is less than 1 percent efficient in producing x-rays 
and since the diffraction of x-rays by crystals is far less efficient than this, 
it follows that the intensities of diffracted x-ray beams are extremely low. 
In fact, it may require as much as several hours exposure to a photographic 
film in order to detect them at all. Constant efforts are therefore being 
made to increase the intensity of the x-ray source. One solution to this 
problem is the rotating-anodc tube, in which rotation of the anode con- 
tinuously brings fresh target metal into the focal-spot area and so allows 
a greater power input without excessive heating of the anode. Figure 1-17 
shows two designs that have been used successfully; the shafts rotate 
through vacuum-tight seals in the tube housing. Such tubes can operate 
at a power level 5 to 10 times higher than that of a fixed-focus tube, with 
corresponding reductions in exposure time. 

1-8 Detection of x-rays. The principal means used to detect x-ray 
beams are fluorescent screens, photographic film, and ionization devices. 

Fluorescent screens are made of a thin layer of zinc sulfide, containing 
a trace of nickel, mounted on a cardboard backing. Under the action of 
x-rays, this compound fluoresces in the visible region, i.e., emits visible 
light, in this case yellow light. Although most diffracted beams are too 
weak to be detected by this method, fluorescent screens are widely used 
in diffraction work to locate the position of the primary beam when adjust- 
ing apparatus. A fluorescing crystal may also be used in conjunction with 
a phototube; the combination, called a scintillation counter, is a very 
sensitive detector of x-rays. 



24 



PROPERTIES OF X-RAYS 



[CHAP. 1 



(a) 




(h) 



K edge of 
silver 

(0.48A). 




A' edge of 
bromine 
(0.92A) 

V 





1 1 5 

X (angstroms) 

FIG. 1-18. Relation between film 
sensitivity and effective shape of con- 
tinuous spectrum (schematic): (a) con- 
tinuous spectrum from a tungsten target 
at 40 kv; (b) film sensitivity; (c) black- 
ening curve for spectrum shown in (a). 



Photographic film is affected by 
x-rays in much the same way as by 
visible light, and film is the most 
widely used means of recording dif- 
fracted x-ray beams. However, the 
emulsion on ordinary film is too 
thin to absorb much of the incident 
x-radiation, and only absorbed x- 
rays can be effective in blackening 
the film. For this reason, x-ray films 
are made with rather thick layers of 
emulsion on both sides in order to 
increase the total absorption. The 
grain size is also made large for the 
same purpose: this has the unfor- 
tunate consequence that x-ray films 
are grainy, do not resolve fine de- 
tail, and cannot stand much enlarge- 
ment. 

Because the mass absorption co- 
efficient of any substance varies with 
wavelength, it follows that film sen- 
sitivity, i.e., the amount of blacken- 
ing caused by x-ray beams of the 
same intensity, depends on their 
wavelength. This should be borne 
lh mind whenever white radiation is 
recorded photographically; for one 
thing, this sensitivity variation al- 
ters the effective shape of the con- 
tinuous spectrum. Figure l-18(a) 
shows the intensity of the continu- 
ous spectrum as a function of wave- 
length and (b) the variation of film 
sensitivity. This latter curve is 
merely a plot of the mass absorp- 
tion coefficient of silver bromide, 
the active ingredient of the emul- 
sion, and is marked by discontinui- 
ties at the K absorption edges of 
silver and bromine. (Note, inciden- 
tally, how much more sensitive the 
film is to the A' radiation from cop- 



1-9] SAFETY PRECAUTIONS 25 

per than to the K radiation from molybdenum, other things being equal.) 
Curve (c) of Fig. 1-18 shows the net result, namely the amount of film 
blackening caused by the various wavelength components of the continu- 
ous spectrum, or what might be called the "effective photographic in- 
tensity" of the continuous spectrum. These curves are only approximate, 
however, and in practice it is almost impossible to measure photographi- 
cally the relative intensities of two beams of different wavelength. On the 
other hand, the relative intensities of beams of the same wavelength can 
be accurately measured by photographic means, and such measurements 
are described in Chap. 6. 

lonization devices measure the intensity of x-ray beams by the amount 
of ionization they produce in a gas. X-ray quanta can cause ionization 
just as high-speed electrons can, namely, by knocking an electron out of a 
gas molecule and leaving behind a positive ion. This phenomenon can be 
made the basis of intensity measurements by passing the x-ray beam 
through a chamber containing a suitable gas and two electrodes having a 
constant potential difference between them. The electrons are attracted 
to the anode and the positive ions to the cathode and a current is thus 
produced in an external circuit. In the ionization chamber, this current is 
constant for a constant x-ray intensity, and the magnitude of the current 
is a measure of the x-ray intensity. In the Geiger counter and proportional 
counter, this current pulsates, and the number of pulses per unit of time is 
proportional to the x-ray intensity. These devices are discussed more 
fully in Chap. 7. 

In general, fluorescent screens are used today only for the detection of 
x-ray beams, while photographic film and the various forms of counters 
permit both detection and measurement of intensity. Photographic film 
is the most widely used method of observing diffraction effects, because it 
can record a number of diffracted beams at one time and their relative 
positions in space and the film can be used as a basis for intensity measure- 
ments if desired. Intensities can be measured much more rapidly with 
counters, and these instruments are becoming more and more popular for 
quantitative work. However, they record only one diffracted beam at a 
time. 

1-9 Safety precautions. The operator of x-ray apparatus is exposed 
to two obvious dangers, electric shock and radiation injury, but both of 
these hazards can be reduced to negligible proportions by proper design of 
equipment and reasonable care on the part of the user. Nevertheless, it is 
only prudent for the x-ray worker to be continually aware of these hazards. 

The danger of electric shock is always present around high-voltage appa- 
ratus. The anode end of most x-ray tubes is usually grounded and there- 
fore safe, but the cathode end is a source of danger. Gas tubes and filament 



26 PROPERTIES OF X-RAYS [CHAP. 1 

tubes of the nonshockproof variety (such as the one shown in Fig. 1-12) 
must be so mounted that their cathode end is absolutely inaccessible to 
the user during operation; this may be accomplished by placing the cathode 
end below a table top, in a box, behind a screen, etc. The installation 
should be so contrived that it is impossible for the operator to touch the 
high-voltage parts without automatically disconnecting the high voltage. 
Shockproof sealed-off tubes are also available: these are encased in a 
grounded metal covering, and an insulated, shockproof cable connects the 
cathode end to the transformer. Being shockproof, such a tube has the 
advantage that it need not be permanently fixed in position but may be 
set up in various positions as required for particular experiments. 

The radiation hazard is due to the fact that x-rays can kill human tis- 
sue; in fact, it is precisely this property which is utilized in x-ray therapy 
for the killing of cancer cells. The biological effects of x-rays include burns 
(due to localized high-intensity beams), radiation sickness (due to radia- 
tion received generally by the whole body), and, at a lower level of radia- 
tion intensity, genetic mutations. The burns are painful and may be 
difficult, if not impossible, to heal. Slight exposures to x-rays are not 
cumulative, but above a certain level called the "tolerance dose," they 
do have a cumulative effect and can produce permanent injury. The 
x-rays used in diffraction are particularly harmful because they have rela- 
tively long wavelengths and are therefore easily absorbed by the body. 

There is no excuse today for receiving serious injuries as early x-ray 
workers did through ignorance. There would probably be no accidents if 
x-rays were visible and produced an immediate burning sensation, but 
they are invisible and burns may not be immediately felt. If the body 
has received general radiation above the tolerance dose, the first noticeable 
effect will be a lowering of the white-blood-cell count, so periodic blood 
counts are advisable if there is any doubt about the general level of in- 
tensity in the laboratory. 

The safest procedure for the experimenter to follow is: first, to locate 
the primary beam from the tube with a small fluorescent screen fixed to 
the end of a rod and thereafter avoid it; and second, to make sure that he 
is well shielded by lead or lead-glass screens from the radiation scattered 
by the camera or other apparatus which may be in the path of the primary 
beam. Strict and constant attention to these precautions will ensure 
safety. 

PROBLEMS 

1-1. What is the frequency (per second) and energy per quantum (in ergs) of 
x-ray beams of wavelength 0.71 A (Mo Ka) and 1.54A (Cu Ka)l 

1-2. Calculate the velocity and kinetic energy with which the electrons strike 
the target of an x-ray tube operated at 50,000 volts. What is the short-wavelength 



PROBLEMS 27 

limit of the continuous spectrum emitted and the maximum energy per quantum 
of radiation? 

1-3. Graphically verify Moseley's law for the K($\ lines of Cu, Mo, and W. 

1-4. Plot the ratio of transmitted to incident intensity vs. thickness of lead 
sheet for Mo Kot radiation and a thickness range of 0.00 to 0.02 mm. 

1-5. Graphically verify Eq. (1-13) for a lead absorber and Mo Kot, Rh Ka, and 
Ag Ka radiation. (The mass absorption coefficients of lead for these radiations 
are 141, 95.8, and 74.4, respectively.) From the curve, determine the mass ab- 
sorption coefficient of lead for the shortest wavelength radiation from a tube op- 
erated at 60,000 volts. 

1-6. Lead screens for the protection of personnel in x-ray diffraction laboratories 
are usually at least 1 mm thick. Calculate the "transmission factor" (/trans. //incident) 
of such a screen for Cu Kot, Mo Kot, and the shortest wavelength radiation from a 
tube operated at 60,000 volts. 

1-7. (a) Calculate the mass and linear absorption coefficients of air for Cr Ka 
radiation. Assume that air contains 80 percent nitrogen and 20 percent oxygen 
by weight, (b) Plot the transmission factor of air for Cr Ka radiation and a path 
length of to 20 cm. 

1-8. A sheet of aluminum 1 mm thick reduces the intensity of a monochromatic 
x-ray beam to 23.9 percent of its original value. What is the wavelength of the 
x-rays? 

1-9. Calculate the K excitation voltage of copper. 

1-10. Calculate the wavelength of the Lm absorption edge of molybdenum. 

1-11. Calculate the wavelength of the Cu Ka\ line. 

1-12. Plot the curve shown in Fig. 1-10 and save it for future reference. 

1-13. What voltage must be applied to a molybdenum-target tube in order 
that the emitted x-rays excite A' fluorescent radiation from a piece of copper placed 
in the x-ray beam? What is the wavelength of the fluorescent radiation? 

In Problems 14 and 15 take the intensity ratios of Ka to K@ in unfiltered radia- 
tion from Table 1-1. 

1-14. Suppose that a nickel filter is required to produce an intensity ratio of 
Cu Ka to Cu K/3 of 100/1 in the filtered beam. Calculate the thickness of the fil- 
ter and the transmission factor for the Cu Ka line. (JJL/P of nickel for Cu Kft ra- 
diation = 286 cm Y gin.) 

1-16. Filters for Co K radiation are usually made of iron oxide (Fe 2 03) powder 
rather than iron foil. If a filter contains 5 mg Fe 2 3 /cm 2 , what is the transmission 
factor for the Co Ka line? What is the intensity ratio of Co Ka to Co KQ in the 
filtered beam? (Density of Fe 2 3 = 5.24 gm/cm 3 , /i/P of iron for Co Ka radiation 
= 59.5 cm 2 /gm, M/P of oxygen for Co Ka radiation = 20.2, pt/P of iron for Co Kfi 
radiation = 371, JJL/P of oxygen for Co K0 radiation = 15.0.) 

1-16. What is the power input to an x-ray tube operating at 40,000 volts and 
a tube current of 25 ma? If the power cannot exceed this level, what is the maxi- 
mum allowable tube current at 50,000 volts? 

1-17, A copper-target x-ray tube is operated at 40,000 volts and 25 ma. The 
efficiency of an x-ray tube is so low that, for all practical purposes, one may as- 
sume that all the input energy goes into heating the target. If there were no dissi- 



28 PROPERTIES OF X-RAYS [CHAP. 1 

pation of heat by water-cooling, conduction, radiation, etc., how long would it 
take a 100-gm copper target to melt? (Melting point of copper = 1083C, mean 
specific heat = 6.65 cal/mole/C, latent heat of fusion = 3,220 cal/mole.) 

1-18. Assume that the sensitivity of x-ray film is proportional to the mass ab- 
sorption coefficient of the silver bromide in the emulsion for the particular wave- 
length involved. What, then, is the ratio of film sensitivities to Cu Ka and Mo Ka 
radiation? 



CHAPTER 2 
THE GEOMETRY OF CRYSTALS 

2-1 Introduction. Turning from the properties of x-rays, we must now 
consider the geometry and structure of crystals in order to discover what 
there is about crystals in general that enables them to diffract x-rays. We 
must also consider particular crystals of various kinds and how the very 
large number of crystals found in nature are classified into a relatively 
small number of groups. Finally, we will examine the ways in which the 
orientation of lines and planes in crystals can be represented in terms of 
symbols or in graphical form. 

A crystal may be defined as a solid composed of atoms arranged in a pat- 
tern periodic in three dimensions. As such, crystals differ in a fundamental 
way from gases and liquids because the atomic arrangements in the latter 
do not possess the essential requirement of periodicity. Not all solids are 
crystalline, however; some are amorphous, like glass, and do not have any 
regular interior arrangement of atoms. There is, in fact, no essential 
difference between an amorphous solid and a liquid, and the former is 
often referred to as an "undercooled liquid." 

2-2 Lattices. In thinking about crystals, it is often convenient to ig- 
nore the, actual atoms composing the crystal and their periodic arrange- 
ment in Space, and to think instead of a set of imaginary points which has 
a fixed relation in space to the atoms of the crystal and may be regarded 
as a sort of framework or skeleton on which the actual crystal is built up. 

This set of points can be formed as follows. Imagine space to be divided 
by three sets of planes, the planes in each set being parallel and equally 
spaced. This division of space will produce a set of cells each identical in 
size, shape, and orientation to its neighbors. Each cell is a parallelepiped, 
since its opposite faces are parallel and each face is a parallelogram.^ The 
space-dividing planes will intersect each other in a set of lines (Fig. 2-1), 
and these lines in turn intersect in the set of points referred to above. A 
set of points so formed has an important property: it constitutes a point 
lattice, which is defined as an array of points in space so arranged that each 
point has identical surroundings. By "identical surroundings*' we mean 
that the lattice of points, when viewed in a particular direction from one 
lattice point, would have exactly the same appearance when viewed in the 
same direction from any other lattice point. 

Since all the cells of the lattice shown in Fig. 2-1 are identical, we may 
choose any one, for example the heavily outlined one, as a unit cell. The 

29 



30 



THE GEOMETRY OF CRYSTALS 



[CHAP. 2 




FIG. 2-1. A point lattice. 

size and shape of the unit cell can in turn be described by the three vec- 
tors* a, b, and c drawn from one corner of the cell taken as origin (Fig. 
2-2). These vectors define the cell and are called the crystallographic axes 
of the cell. They may also be described in terms of their lengths (a, 6, c) 
and the angles between them (a, ft 7). These lengths and angles are the 
lattice constants or lattice parameters of the unit cell. 

Note that the vectors a, b, c define, not only the unit cell, but also the 
whole point lattice through the translations provided by these vectors. 
In other words, the whole set of points in the lattice can be produced by 
repeated action of the vectors a, b, c on one lattice point located at the 
origin, or, stated alternatively, the 
vector coordinates of any point in the 
lattice are Pa, Qb, and /fc, where 
P, Q, and R are whole numbers. It 
follows that the arrangement of 
points in a point lattice is absolutely 
periodic in three dimensions, points 
being repeated at regular intervals 
along any line one chooses to draw 
through the lattice. 




FIG. 2-2. A unit cell. 



2-3 Crystal systems, (jn dividing space by three sets of planes, we can 
of course produce unit cells of various shapes, depending on how we ar- 
range the planesT) For example, if the planes in the three sets are all equally 

* Vectors are here represented by boldface symbols. The same symbol in italics 
stands for the absolute value of the vector. 



2-3] 



CRYSTAL SYSTEMS 



31 



TABLE 2-1 
CRYSTAL SYSTEMS AND BRAVAIS LATTICES 

(The symbol ^ implies nonequality by reason of symmetry. Accidental equality 
may occur, as shown by an example in Sec. 2-4.) 



System 


Axials lengths and angles 


Bravais 
lattice 


Lattice 
symbol 


r 
Cubic 


Three equal axes at right angles 

a = /, = r , a = p = J = 90 


Simple 
Body-centered 
Face-centered 


P 
1 
F 


Tetragonal 


Three axes at right angles, two equal 

a = 6 ^ c , a = p = 7 = 90 


Simple 
Body-centered 


P 
I 


Orthorhombic 


Three unequal axes at right angles 

a ^ b i- c, a = p = 7 = 90 


Simple 
Body -centered 
Base-centered 
Face-centered 


.P 
I 
C 
F 


Rhombohedral 


Three equal axes, equally inclined 

a = b = c , a = P 7 * 90 


Simple 


P 


Hexagonal 


Two equal coplanar axes at 120, 
third axis at right angles 
a = b ? c, a = p = 90, 7 = 120 


Simple 


P 


Monoclinic 


Three unequal axes, 
one pair not at right angles 

a * b * c, a - y = 90 * P 


Simple 
Base -centered 


P 
C 


Triclinic 


Three unequal axes, unequally inclined 
and none at right angles 
a * b * c, a ^ p ^ X ^ 90 


Simple 


P 



* Also called trigonal. 

spaced and mutually perpendicular, the unit cell is cubic. In this case the 
vectors a, b, c are all equal and at right angles to one another, or a = b = c 
and a = = 7 = 90. By thus giving special values to the axial lengths 
and angles, we can produce unit cells of various shapes and therefore 
various kinds of point lattices, since the points of the lattice are located at 
the cell corners. It turns out that only seven different kinds of cells are 
necessary to include all the possible point lattices. These correspond to 
the seven crystal systems into which all crystals can be classified. These 
systems are listed in Table 2-1. 

Seven different point lattices can be obtained simply by putting points 
at the corners of the unit cells of the seven crystal systems. However, 
there are other arrangements of points which fulfill the requirements of a 
point lattice, namely, that each point have identical surroundings. The 
French crystallographer Bravais worked on this problem and in 1848 
demonstrated that there are fourteen possible point lattices and no more; 
this important result is commemorated by our use of the terms Bravais 



32 



THE GEOMETRY OF CRYSTALS 



[CHAP. 2 





SIMPLE 
CUBIC (P) 



BODY-CENTERED FACE-C 'ENTERED 
CUBIC (/) CUBIC 1 (F) 




SIMPLE BOD Y-( CENTERED SIMPLE BODY-CENTERED 

TETRAGONAL TETRAGONAL ORTHORHOMBIC ORTHORHOMBIC 

(P) (/) (P) (/) 





BASE-CENTERED FACE-CENTERED RHOMBOHEDRAL 
ORTHORHOMBIC 1 ORTHORHOMBIC (/?) 

(O (F) 



^^ 








c 




120 


a 


W J-** 


^^ 


HEXAGONAL 

(P) 




SIMPLE 
MONOCLINIC 





BASE-CENTERED TRICLINIC (P) 

(P) MONOCLINIC 1 (C) 

FIG. 2-3. The fourteen Bravais lattices. 



lattice and point lattice as synonymous. For example, if a point is placed 
at the center of each cell of a cubic point lattice, the new array of points 
also forms a point lattice. Similarly, another point lattice can be based 



2-3] 



CRYSTAL SYSTEMS 



33 



on a cubic unit cell having lattice points at each corner and in the center 
of each face. 

The fourteen Bravais lattices are described in Table 2-1 and illustrated 
in Fig. 2-3, where the symbols P, F, /, etc., have the following meanings. 
We must first distinguish between simple, or primitive, cells (symbol P 
or R) and nonprimitive cells (any other symbol): primitive cells have only 
one lattice point per cell while nonprimitive have more than one. A lattice 
point in the interior of a cell "belongs" to that cell, while one in a cell face 
is shared by two cells and one at a corner is shared by eight. The number 
of lattice points per cell is therefore given by 



N = 



N f 

-- 

2 



N c 

, 

8 



(2-1 ; 



where N t = number of interior points, N/ = number of points on faces, 
and N c = number of points on corners. Any cell containing lattice points 
on the corners only is therefore primitive, while one containing additional 
points in the interior or on faces is nonprimitive. The symbols F and / 
refer to face-centered and body-centered cells, respectively, while A, B, 
and C refer tqjmse-centered cells, centered on one pair of opposite faces 
A, B, or C. (The A face is the face defined by the b and c axes, etc.) The 
symbol R is used especially for the rhombohedral system. In Fig. 2-3, 
axes of equal length in a particular system are given the same symbol to 
indicate their equality, e.g., the cubic axes are all marked a, the two equal 
tetragonal axes are marked a and the third one c, etc. 

At first glance, the list of Bravais lattices in Table 2-1 appears incom- 
plete. Why not, for example, a base-centered tetragonal lattice? The 
full lines in Fig. 2-4 delineate such a cell, centered on the C face, but we 
see that the same array of lattice points can be referred to the simple 
tetragonal cell shown by dashed lines, so that the base-centered arrange- 
ment of points is not a new lattice. 




/ 



FIG. 2-4. Relation of tetragonal C FIG. 2-5. Extension of lattice points 
lattice (full lines) to tetragonal P iat- through space by the unit cell vectors 
tice (dashed lines). a, b, c. 



34 



THE GEOMETRY OF CRYSTALS 



[CHAP. 2 



The lattice points in a nonprimitive unit cell can be extended through 
space by repeated applications of the unit-cell vectors a, b, c just like those 
of a primitive cell. We may regard the lattice points associated with a 
unit cell as being translated one by one or as a group. In either case, equiv- 
alent lattice points in adjacent unit cells are separated by one of the vectors 
a, b, c, wherever these points happen to be located in the cell (Fig. 2-5). 

2-4 Symmetry, i Both Bravais lattices and the real crystals which are 
built up on them exhibit various kinds of symmetry. A body or structure 
is said to be symmetrical when its component parts are arranged in such 
balance, so to speak, that certain operations can be performed on the body 
which will bring it into coincidence with itself. These are termed symmetry 
operations. /For example, if a body is symmetrical with respect to a plane 
passing through it, then reflection of either half of the body in the plane 
as in a mirror will produce a body coinciding with the other half. Thus a 
cub has se ir -ral planes of symmetry, one of which is shown in Fig. 2-6(a). 

There are in all four macroscopic* symmetry operations or elements: 
reflection, rotation, inversion, and rotation-inversion. A body has n-fold 
rotational symmetry about an axis if a rotation of 360 /n brings it into 
self-coincidence. Thus a cube has a 4-fold rotation axis normal to each 
face, a 3-fold axis along each body diagonal, and 2-fold axes joining the 
centers of opposite edgesf Some of these are shown in Fig. 2-6 (b) where 
the small plane figures (square, triangle, and ellipse) designate the various 




\ 

X 


c 

p' 1 ' 


> 


\ 




r 


/ 


\ 


V 




\ 


X 




, 


z 

c. 


~7 
? 






r 


7 


A 


/ 






/ 



(b) 



(ci) 



FIG, 2-6. Some symmetry elements of a cube, (a) Reflection plane. AI be- 
comes A%. (b) Rotation axes. 4-fold axis: A\ becomes A^ 3-fold axis: A\ becomes 
AZ\ 2-fold axis: AI becomes A*, (c) Inversion center. AI becomes A%. (d) Rota- 
tion-inversion axis. 4-fold axis: AI becomes A\\ inversion center: A\ becomes A*. 



* So called to distinguish them from certain microscopic symmetry operations 
with which we are not concerned here. The macrosopic elements can be deduced 
from the angles between the faces of a well-developed crystal, without any knowl- 
edge of the atom arrangement inside the crystal. The microscopic symmetry ele- 
ments, on the other hand, depend entirely on atom arrangement, and their pres- 
ence cannot be inferred from the external development of the crystal. 



2-4] SYMMETRY 35 

kinds of axes. In general, rotation axes may be 1-, 2-, 3-, 4-, or 6-fold. A 
1-fold axis indicates no symmetry at all, while a 5-fold axis or one of higher 
degree than 6 is impossible, in the sense that unit cells having such sym- 
metry cannot be made to fill up space without leaving gaps. 

A body has an inversion center if corresponding points of the body are 
located at equal distances from the center on a line drawn through the 
center. A body having an inversion center will come into coincidence 
with itself if every point in the body is inverted, or "reflected," in the 
inversion center. A cube has such a center at the intersection of its body 
diagonals [Fig. 2-6(c)]. Finally, a body may have a rotation-inversion 
axis, either 1-, 2-, 3-, 4-, or 6-fold. If it has an n-fold rotation-inversion 
axis, it can be brought into coincidence with itself by a rotation of 360/n 
about the axis followed by inversion in a center lying on the axis. ; Figure 
2-6(d) illustrates the operation of a 4-fold rotation-inversion axis on a cube. 

^Now, the possession of a certain minimum set of symmetry elements 
is a fundamental property of each crystal system, and one system is dis- 
tinguished from another just as much by its symmetry elements as by the 
values of its axial lengths and angles'* In fact, these are interdependent 
The minimum number of symmetry elements possessed by each crystal 
system is listed in Table 2-2. { Some crystals may possess more than the 
minimum symmetry elements required by the system to which they belong, 
but none may have less.) 

Symmetry operations apply not only to the unit cells]shown in Fig. 2-3J 
considered merely as geometric shapes, but also to the point lattices asso- 
ciated with them. The latter condition rules out the possibility that the 
cubic system, for example, could include a base-centered point lattice, 
since such an array of points would not have the minimum set of sym- 
metry elements required by the cubic system, namely four 3-fold rotation 
axes. Such a lattice would be classified in the tetragonal system, which 
has no 3-fold axes and in which accidental equality of the a and c axes is 



TABLE 2-2 
SYMMETRY ELEMENTS 



System 



Minimum symmetry elements 



Cubic 

Tetragonal 

Orthorhombi c 

Rhombohedral 

Hexagonal 

Monoclinic 

Triclinic 



Four 3 - fold rotation axes 

One 4 -fold rotation (or rotation - inversion) axis 

Three perpendicular 2 -fold rotation (or rotation - inversion) axes 

One 3 -fold rotation (or rotation - inversion) axis 

One 6 -fold rotation (or rotation - inversion) axis 

One 2 -fold rotation (or rotation - Inversion) axis 

None 



36 



THE GEOMETRY OF CRYSTALS 



[CHAP. 2 



allowed; as mentioned before, however, this lattice is simple, not base- 
centered, tetragonal. 

Crystals in the rhombohedral (trigonal) system can be referred to either 
a rhombohedral or a hexagonal lattice.^ Appendix 2 gives the relation 
between these two lattices and the transformation equations which allow 
the Miller indices of a. plane (see Sec. 2-6) to be expressed in terms of 
either set of axes. 

2-5 Primitive and nonprimitive cells. In any point lattice a unit cell 
may be chosen in an infinite number of ways and may contain one or more 
lattice points per cell. It is important to note that unit cells do not "exist" 
as such in a lattice: they are a mental construct and can accordingly be 
chosen at our convenience. The conventional cells shown in Fig. 2-3 are 
chosen simply for convenience and to 
conform to the symmetry elements 
of the lattice. 

Any of the fourteen Bravais lattices 
may be referred to a primitive unit 
cell. For example, the face-centered 
cubic lattice shown in Fig. 2-7 may 
be referred to the primitive cell indi- 
cated by dashed lines. The latter cell 
is rhombohedral, its axial angle a is 
60, and each of its axes is l/\/2 
times the length of the axes of the 
cubic cell. Each cubic cell has four 
lattice points associated with it, each 
rhombohedral cell has one, and the 

former has, correspondingly, four times the volume of the latter. Never- 
theless, it is usually more convenient to use the cubic cell rather than the 
rhombohedral one because the former immediately suggests the cubic 
symmetry which the lattice actually possesses. Similarly, the other cen- 
tered nonprimitive cells listed in Table 2-1 are preferred to the primitive 
cells possible in their respective lattices. 

If nonprimitive lattice cells are used, the vector from the origin to any 
point in the lattice will now have components which are nonintegral mul- 
tiples of the unit-cell vectors a, b, c. The position of any lattice point in a 
cell may be given in terms of its coordinates] if the vector from the origin 
of the unit cell to the given point has components xa, yb, zc, where x, y, 
and z are fractions, then the coordinates of the point are x y z. Thus, 
point A in Fig. 2-7, taken as the origin, has coordinates 000 while points 
Bj C, and D, when referred to cubic axes, have coordinates Off, f f , 
and f f 0, respectively. Point E has coordinates f \ 1 and is equivalent 




FIG. 2-7. Face-centered cubic point 
lattice referred to cubic and rhombo- 
hedral cells. 



2-6] 



LATTICE DIRECTIONS AND PLANES 



37 



to point Z), being separated from it by the vector c. The coordinates of 
equivalent points in different unit cells can always be made identical by 
the addition or subtraction of a set of integral coordinates; in this case, 
subtraction of 1 from f ^ 1 (the coordinates of E) gives ^ f (the 
coordinates of D). 

Note that the coordinates of a body-centered point, for example, are 
always | ^ ^ no matter whether the unit cell is cubic, tetragonal, or ortho- 
rhombic, and whatever its size. The coordinates of a point position, such 
as ^ ^ \, may also be regarded as an operator which, when "applied" to a 
point at the origin, will move or translate it to the position \ \ \, the 
final position being obtained by simple addition of the operator \ \ \ 
and the original position 000. In this sense, the positions 000, \ \ \ 
are called the "body-centering translations," since they will produce the 
two point positions characteristic of a body-centered cell when applied to 
a point at the origin. Similarly, the four point positions characteristic of a 
face-centered cell, namely 0, \ ^, \ ^, and \ \ 0, are called the 
face-centering translations. The base-centering translations depend on 
which pair of opposite faces are centered; if centered on the C face, for 
example, they are 0, \ \ 0. 

2-6 Lattice directions and planes. The direction of any line in a lat- 
tice may be described by first drawing a line through the origin parallel 
to the given line and then giving the coordinates of any point on the line 
through the origin. Let the line pass through the origin of the unit cell 
and any point having coordinates u v w, where these numbers are not neces- 
sarily integral. (This line will also pass through the points 2u 2v 2w, 
3u 3v 3w, etc.) Then [uvw], written in square brackets, are the indices 
of the direction of the line. They are also the indices of any line parallel 
to the given line, since the lattice is infinite and the origin may be taken 
at any point. Whatever the values of i/, v, w, they are always converted 
to a set of smallest integers by multi- 
plication or division throughout: thus, 
[||l], [112], and [224] all represent 
the same direction, but [112] is the 
preferred form. Negative indices are 
written with a bar over the number, 
e.g., [uvw]. Direction indices are illus- 
trated in Fig. 2-8. 

Direction^ related by symmetry are 
called directions of a form, and a set 
of these are|Pepresented by the indices 
of one of them enclosed in angular 
bracHts; for example, the four body Fib/^-8. 



[100] 



[233] 



[001] 




[111] 



[210] 



HO 

[100] 
'[120] 

Indices of directions. 



38 THE GEOMETRY OF CRYSTALS [CHAP. 2 

diagonals of a cube, [111], [ill], [TTl], and [Til], may all be represented 
by the symbol (111). 

The orientation of planes in a lattice may also be represented sym- 
bolically, according to a system popularized by the English crystallographer 
Miller. In the general case, the given plane will be tilted with respect to 
the crystallographic axes, and, since these axes form a convenient frame 
of reference, we might describe the orientation of the plane by giving the 
actual distances, measured from the origin, at which it intercepts the 
three axes. Better still, by expressing these distances as fractions of the 
axial lengths, we can obtain numbers which are independent of the par- 
ticular axial lengths involved in the given lattice. But a difficulty then 
arises when the given plane is parallel to a certain crystallographic axis, 
because such a plane does not intercept that axis, i.e., its "intercept" can 
only be described as "infinity." To avoid the introduction of infinity into 
the description of plane orientation, we can use the reciprocal of the frac- 
tional intercept, this reciprocal being zero when the plane and axis are 
parallel. We thus arrive at a workable symbolism for the orientation of a 
plane in a lattice, the Miller indices, which are defined as the reciprocals of 
the fractional intercepts which the plane makes with the crystallographic axes. 
For example, if the Miller indices of a plane are (AW), written in paren- 
theses, then the plane makes fractional intercepts of I/A, I/A*, \/l with the 
axes, and, if the axial lengths are a, 6, c, the plane makes actual intercepts 
of a/A, b/k, c/l, as shown in Fig. 2-9(a). Parallel to any plane in any lat- 
tice, there is a whole set of parallel equidistant planes, one of which passes 
through the origin; the Miller indices (hkl) usually refer to that plane in 
the set which is nearest the origin, although they may be taken as referring 
to any other plane in the set or to the whole set taken together. 

We may determine the Miller indices of the plane shown in Fig. 2-9 (b) 
as follows : 





1A 2A 3A 4A 



(a) (b) 

FIG. 2-9. Plane designation by Miller indices. 



2-6] 



LATTICE DIRECTIONS AND PLANES 



39 



Axial lengths 
Intercept lengths 
Fractional intercepts 

Miller indices 



4A 
2A 

I 

I 2 
16 



8A 
6A 

3 
1 

4 



3A 

3A 

1 

1 

3 



Miller indices are always cleared of fractions, as shown above. As stated 
earlier, if a plane is parallel to a given axis, its fractional intercept on that 
axis is taken as infinity and the corresponding Miller index is zero. If a 
plane cuts a negative axis, the corresponding index is negative and is writ- 
ten with a bar over it. Planes whose indices are the negatives of one 
another are parallel and lie on opposite sides of the origin, e.g., (210) and 
(2lO). The planes (nh nk nl) are parallel to the planes (hkl) and have 1/n 
the spacing. The same plane may belong to two different sets, the Miller 
indices of one set being multiples of those of the other; thus the same plane 
belongs to the (210) set and the (420) set, and, in fact, the planes of the 
(210) set form every second plane in the (420) set. jjn the cubic system, 
it is convenient to remember that a direction [hkl] is always perpendicular 
to a plane (hkl) of the same indices, but this is not generally true in other 
systems. Further familiarity with Miller indices can be gained from a 
study of Fig. 2-10. 

A slightly different system of plane indexing is used in the hexagonal 
system. The unit cell of a hexagonal lattice is defined by two equal and 
coplanar vectors ai and a 2 , at 120 to one another, and a third axis c at 
right angles [Fig. 2-11 (a)]. The complete lattice is built up, as usual, by 




HfeocH 





(110) 




(110) (111) 

FIG. 2-10. Miller indices of lattice planes. 



(102) 



40 



THE GEOMETRY OF CRYSTALS 



[CHAP. 2 



[001] 




(0001) 



(1100)- 



[100] ' 




[Oil] 



(1210) 



[010] 



(1011) 



'[210] 



(a) (b) 

FIG. 2-11. (a) The hexagonal unit cell and (b) indices of planes and directions. 

repeated translations of the points at the unit cell corners by the vectors 
EI, a 2 , c. Some of the points so generated are shown in the figure, at the 
ends of dashed lines, in order to exhibit the hexagonal symmetry of the 
lattice, which has a 6-fold rotation axis parallel to c. The third axis a 3 , 
lying in the basal plane of the hexagonal prism, is so symmetrically related 
to EI and a 2 that it is often used in conjunction with the other two. Thus 
the indices of a plane in the hexagonal system, called Miller-Bra vais 
indices, refer to four axes and are written (hkil). The index i is the recipro- 
cal of the fractional iiltercept on the a 3 axis. Since the intercepts of a 
plane on ai and a 2 determine its intercept on a 3 , the value of i depends on 
the values of h and k. The relation is 



h + k = -i. 



(2-2) 



Since i is determined by h and A;, it is sometimes replaced by a dot and 
the plane symbol written (hk-l). However, this usage defeats the pur- 
pose for which Miller-Bra vais indices were devised, namely, to give similar 
indices to similar planes. For example, the side planes of the hexagonal 
prism in Fig. 2-1 l(b) are all similar and symmetrically located, and their 
relationship is clearly shown in their full Miller-Bra vais symbols: (10K)), 
(OlTO), (TlOO), (T010), (OTlO), (iTOO). On the other hand, the_abbreviated 
symbols of these planes, (10-0), (01-0), (11-0), (10-0), (01-0), (11-0) 
do not immediately suggest this relationship. 

Directions in a hexagonal lattice are best expressed in terms of the three 
basic vectors ai, a 2 , and c. Figure 2-1 l(b) shows several examples of 
both plane and direction indices. (Another system, involving four indices, 
is sometimes used to designate directions. The required direction is broken 
up into four component vectors, parallel to ai, a 2 , aa, and c and so chosen 
that the third index is the negative of the sum of the first two. Thus 



2-6] 



LATTICE DIRECTIONS AND PLANES 



41 



[100], for example, becomes [2110], [210] becomes [1010], [010] becomes 
[T210], etc.) 

In any crystal system there are sets of equivalent lattice planes related 
by symmetry. These are called planes of a form, and the indices of any 
one plane, enclosed in braces )M/}, stand for the whole set. In general, 
planes of a form have the same spacing but different Miller indices. For 
example, the faces of a cube, (100), (010), (TOO), (OTO), (001), and (001), 
are planes of the form {100}, since all of them may be generated from 
any one by operation of the 4-fold rotation axes perpendicular to the cube 
faces. In the tetragonal system, however, only the planes (100), (010), 
(TOO), and (OTO) belong to the form |100); the other two planes, (001) 
and (OOT), belong to the different form {001) ; the first four planes men- 
tioned are related by a 4-fold axis and the last two by a 2-fold axis.* 

Planes of a zone are planes which are all parallel to one line, called the 
zone axis, and the zone, i.e., the set of planes, is specified by giving the 



indices of the zone axis. Such planes 
may have quite different indices and 
spacings, the only requirement being 
their parallelism to a line. Figure 
2-12 shows some examples. If the 
axis of a zone has indices [uvw], then 
any plane belongs to that zone whose 
indices (hkl) satisfy the relation 

hu + kv + Iw = 0. (2-3) 

(A proof of this relation is given in 
Section 4 of Appendix 15.) Any two 
nonparallel planes are planes of a zone 
since they are both parallel to their 
line of intersection. If their indices 
are (/hfci/i) and (h^kj^j then the in- 
dices of their zone axis [uvw] are given 
by the relations 



[001] 



(210) 
UOO) \ 



(11) (210) 




,(100) 



FIG, 2-12, All shaded planes in the 
cubic lattice shown are planes of the 
zone [001]. 



(2-4) 



W = /&1/T2 h?jk\. 



* Certain important crystal planes are often referred to by name without any 
mention of their Miller indices. Thus, planes of the form ( 111 | in the cubic sys- 
tem are often called octahedral planes, since these are the bounding planes of an 
octahedron. In the hexagonal system, the (0001) plane is called the basal plane, 
planes of the form { 1010) are called prismatic planes, and planes of the form { 1011 ) 
are called pyramidal planes. 



42 



THE GEOMETRY OF CRYSTALS 



[CHAP. 2 




(13) 



FIG. 2-13. Two-dimensional lattice, showing that lines of lowest indices have 
the greatest spacing and the greatest density of lattice points. 

The various sets of planes in a lattice have various values of interplanar 
spacing. The planes of large spacing have low indices and pass through a 
high density of lattice points, whereas the reverse is true of planes of small 
spacing. Figure 2-13 illustrates this for a two-dimensional lattice, and 
it is equally true in three dimensions. The interplanar spacing rf^./, meas- 
ured at right angles to the planes, is a function both of the plane indices 
(hkl) and the lattice constants (a, />, r, a, 0, 7). The exact relation de- 
pends on the crystal system involved and for the cubic system takes on 
the relatively simple form 

(Cubic) d hk i = -^-JL===. (2-5) 

In the tetragonal system the spacing equation naturally involves both 
a and c since these are not generally equal : 



(Tetragonal) d h ki = 



(2-0) 



Interplanar spacing equations for all systems are given in Appendix 1 . 

2-7 Crystal structure. So far we have discussed topics from the field 
of mathematical (geometrical) crystallography and have said practically 
nothing about actual crystals and the atoms of which they are composed. 
In fact, all of the above was well known long before the discovery of x-ray 
diffraction, i.e., long before there was any certain knowledge of the interior 
arrangements of atoms in crystals. 

It is now time to describe the structure of some actual crystals and to 
relate this structure to the point lattices, crystal systems, and symmetry 



2-7] 



CRYSTAL STRUCTURE 



43 




BCC FCC 

FIG. 2-14. Structures of some com- 
mon metals. Body-centered cubic: a- 
Fe, Cr, Mo, V, etc.; face-centered 
cubic: 7-Fe, Cu, Pb, Ni, etc. 



elements discussed above. The cardi- 
nal principle of crystal structure is 
that the atoms of a crystal are set in 
space either on the points of a Bravais 
lattice or in some fixed relation to those 
points. It follows from this th the 
atoms of a crystal will be arranged 
periodically in three dimensions and 
that this arrangement of atoms will 
exhibit many of the properties of a 
Bravais lattice, in particular many of 
its symmetry elements. 

The simplest crystals one can imagine are those formed by placing atoms 
of the same kind on the points of a Bravais lattice. Not all such crystals 
exist but, fortunately for metallurgists, many metals crystallize in this 
simple fashion, and Fig. 2-14 shows two common structures based on the 
body-centered cubic (BCC) and face-centered cubic (FCC) lattices. The 
former has two atoms per unit cell and the latter four, as we can find by 
rewriting Eq. (2-1) in terms of the number of atoms, rather than lattice 
points, per cell and applying it to the unit cells shown. 

The next degree of complexity is encountered when two or more atoms 
of the same kind are "associated with" each point of a Bravais lattice, as 
exemplified by the hexagonal close-packed (HCP) structure common to 
many metals. This structure is simple hexagonal and is illustrated in 
Fig. 2-15. There are two atoms per unit cell, as shown in (a), one at 
and the other at \ | (or at \ f f , which is an equivalent position). 
Figure 2-15(b) shows the same structure with the origin of the unit cell 
shifted so that the point 1 in the new cell is midway between the atoms 
at 1 and \ | in (a), the nine atoms shown in (a) corresponding to the 
nine atoms marked with an X in (b). The ' 'association" of pairs of atoms 
with the points of a simple hexagonal Bravais lattice is suggested by the 
dashed lines in (b). Note, however, that the atoms of a close-packed 
hexagonal structure do not themselves form a point lattice, the surround- 
ings of an atom at being different from those of an atom at 3 ^. 
Figure 2-15(c) shows still another representation of the HCP structure: 
the three atoms in the interior of the hexagonal prism are directly above 
the centers of alternate triangles in the base and, if repeated through space 
by the vectors ai and a 2 , would alsd form a hexagonal array just like 
the atoms in the layers above and below. 

The HCP structure is so called because it is one of the two ways in 
which spheres can be packed together in space with the greatest possible 
density and still have a periodic arrangement. Such an arrangement of 
spheres in contact is shown in Fig. 2-15(d). If these spheres are regarded 



44 



THE GEOMETRY OF CRYSTALS 





(a) 





(c) 
FIG. 2-15. The hexagonal close-packed structure, shared by Zn, Mg, He, a-Ti, etc. 

as atoms, then the resulting picture of an HCP metal is much closer to 
physical reality than is the relatively open structure suggested by the 
drawing of Fig. 2-15(c), and this is true, generally, of all crystals. On the 
other hand, it may be shown that the ratio of c to a in an HCP structure 
formed of spheres in contact is 1 .633 whereas the c/a ratio of metals having 
this structure varies from about 1.58 (Be) to 1.89 (Cd). As there is no 
reason to suppose that the atoms in these crystals are not in contact, it 
'follows that they must be ellipsoidal in shape rather than spherical. 

The FCC structure is an equally close-packed arrangement. Its rela- 
tion to the HCP structure is not immediately obvious, but Fig. 2-16 shows 
that the atoms on the (111) planes of the FCC structure are arranged in a 
hexagonal pattern just like the atoms on the (0002) planes of the HCP 
structure. The only difference between the two structures is the way in 
which these hexagonal sheets of atoms are arranged above one another. 
In an HCP metal, the atoms in the second layer are above the hollows in 



2-7] 



CRYSTAL STRUCTURE 

i HID 



45 







[001] 











HEXAGONAL CLOSE-PACKED 



FIG. 2-16. Comparison of FCC and HCP structures. 



46 



THE GEOMETRY OF CRYSTALS 



[CHAP. 2 



j; 

HH 



FIG. 2-17. The structure of a-uranium. 
59, 2588, 1937.') 



(C. W. Jacob and B. E. Warren, J.A.C.S 



the first layer and the atoms in the third layer are above the atoms in the 
first layer, so that the layer stacking sequence can be summarized as 
A B A B A B . . . . The first two atom layers of an FCC metal are put down 
in the same way, but the atoms of the third layer are placed in the hollows 
of the second layer and not until the fourth layer does a position repeat. 
FCC stacking therefore has the sequence A B C ABC ... . These stack- 
ing schemes are indicated in the plan views shown in Fig. 2-1 (>. 

Another example of the "association" of more than one atom with each 
point of a Bravais lattice is given by uranium. The structure of the form 
stable at room temperature, a-uranium, is illustrated in Fig. 2-17 by plan 
and elevation drawings. In such drawings, the height of an atom (ex- 
pressed as a fraction of the axial length) above the plane of the drawing 
(which includes the origin of the unit cell and two of the cell axes) is given 
by the numbers marked on each atom. The Bravais lattice is base-centered 
orthorhombic, centered on the C face, and Fig. 2-17 shows how the atoms 
occur in pairs through the structure, each pair associated with a lattice 
point. There are four atoms per unit cell, located at Or/-}, y f , 
\ (\ + y} T> an d i (2 "~ y) T Here we have an example of a variable 
parameter y in the atomic coordinates. Crystals often contain such vari- 
able parameters, which may have any fractional value without destroying 
any of the symmetry elements of the structure. A quite different sub- 
stance might have exactly the same structure as uranium except for slightly 
different values of a, 6, c, and y. For uranium y is 0.105 0.005. 

Turning to the crystal structure of compounds of unlike atoms, we find 
that the structure is built up on the skeleton of a Bravais lattice but that 
certain other rules must be obeyed, precisely because there are unlike 
atoms present. Consider, for example, a crystal of A x E y which might be 
an ordinary chemical compound, an intermediate phase of relatively fixed 
composition in some alloy system, or an ordered solid solution. Then the 
arrangement of atoms in A x E y must satisfy the following conditions: 



2-7] 



CRYSTAL STRUCTURE 



47 



O CB+ 





[010] 



(a) CsCl 



(b) NaCl 



FIG. 2-18. The structures of (a) CsCl (common to CsBr, NiAl, ordered /3-brass, 
ordered CuPd, etc.) and (b) NaCl (common to KC1, CaSe, Pbf e, etc.). 

(1) Body-, face-, or base-centering translations, if present, must begin 
and end on atoms of the same kind. For example, if the structure is based 
on a body-centered Bravais lattice, then it must be possible to go from an 
A atom, say, to another A atom by the translation ^ ^ f . 

(2) The set of A atoms in the crystal and the set of B atoms must sep- 
arately possess the same symmetry elements as the crystal as a whole, 
since in fact they make up the crystal. In particular, the operation of 
any symmetry element present must bring a given atom, A for example, 
into coincidence with another atom of the same kind, namely A. 

Suppose we consider the structures of a few common crystals in light 
of the above requirements. Figure 2-18 illustrates the unit cells of two 
ionic compounds, CsCl and NaCl. These structures, both cubic, are com- 
mon to many other crystals and, wherever they occur, are referred to as 
the "CsCl structure" and the "NaCl structure. " In considering a crystal 
structure, one of the most important things to determine is its Bravais 
lattice, since that is the basic framework on which the crystal is built and 
because, as we shall see later, it has a profound effect on the x-ray diffrac- 
tion pattern of that crystal. 

What is the Bravais lattice of CsCl? Figure 2-1 8 (a) shows that the 
unit cell contains two atoms, ions really, since this compound is com- 
pletely ionized even in the solid state: a caesium ion at and a chlo- 
rine ion at ^ \ \ . The Bravais lattice is obviously not face-centered, but 
we note that the body-centering translation \ \ \ connects two atoms. 
However, these are unlike atoms and the lattice is therefore not body- 



48 THE GEOMETRY OF CRYSTALS [CHAP. 2 

centered. It is, by elimination, simple cubic. If one wishes, one may 
think of both ions, the caesium at and the chlorine at \ \ ^, as be- 
ing associated with the lattice point at 0. It is not possible, however, 
to associate any one caesium ion with any particular chlorine ion and re- 
fer to them as a CsCl molecule; the term "molecule" therefore has no real 
physical significance in such a crystal, and the same is true of most inor- 
ganic compounds and alloys. 

Close inspection of Fig. 2-18(b) will show that the unit cell of NaCl 
contains 8 ions, located as follows: 

4 Na + at 0, \ \ 0, \ |, and \ \ 
4 Cl~ at \\\, \, \ 0, and ^00. 

The sodium ions are clearly face-centered, and we note that the face-center- 
ing translations (0 0, \ \ 0, \ \, \ ^), when applied to the chlorine 
ion at \\\, will reproduce all the chlorine-ion positions. The Bravais 
lattice of NaCl is therefore face-centered cubic. The ion positions, inci- 
dentally, may be written in summary form as: 

4 Na 4 " at + face-centering translations 
4 Cl~ at \ \ \ + face-centering translations. 

Note also that in these, as in all other structures, the operation of any 
symmetry element possessed by the lattice must bring similar atoms or 
ions into coincidence. For example, in Fig. 2-18(b), 90 rotation about 
the 4-fold [010] rotation axis shown brings the chlorine ion at 1 \ into 
coincidence with the chlorine ion at ^11, the sodium ion at 1 1 with 
the sodium ion at 1 1 1, etc. 

Elements and compounds often have closely similar structures. Figure 
2-19 shows the unit cells of diamond and the zinc-blende form of ZnS. 
Both are face-centered cubic. Diamond has 8 atoms per unit cell, lo- 
cated at 

000 + face-centering translations 

1 i I + face-centering translations. 

The atom positions in zinc blende are identical with these, but the first 
set of positions is now occupied by one kind of atom (S) and the other by 
a different kind (Zn). 

Note that diamond and a metal like copper have quite dissimilar struc- 
tures, although both are based on a face-centered cubic Bravais lattice. 
To distinguish between these two, the terms "diamond cubic" and "face- 
centered cubic'' are usually used. 



2-7] 



CRYSTAL STRUCTURE 



51 




O Fe 

C position 



< 



(a) 



(b) 



FIG. 2-21. Structure of solid solutions: (a) Mo in Cr (substitutional) ; (b) C in 
a-Fe (interstitial). 

on the lattice of the solvent, while in the latter, solute atoms fit into the 
interstices of the solvent lattice. The interesting feature of these struc- 
tures is that the solute atoms are distributed more or less at random. For 
example, consider a 10 atomic percent solution of molybdenum in chro- 
mium, which has a BCC structure. The molybdenum atoms can occupy 
either the corner or body-centered positions of the cube in a random, ir- 
regular manner, and a small portion of the crystal might have the appear- 
ance of Fig. 2-21 (a). Five adjoining unit cells are shown there, contain- 
ing a total of 29 atoms, 3 of which are molybdenum. This section of the 
crystal therefore contains somewhat more than 10 atomic percent molyb- 
denum, but the next five cells would probably contain somewhat less. 
Such a structure does not obey the ordinary rules of crystallography: 
for example, the right-hand cell of the group shown does not have cubic 
symmetry, and one finds throughout the structure that the translation 
given by one of the unit cell vectors may begin on an atom of one kind 
and end on an atom of another kind. All that can be said of this structure 
is that it is BCC on the average, and experimentally we find that it displays 
the x-ray diffraction effects proper to a BCC lattice. This is not surpris- 
ing since the x-ray beam used to examine the crystal is so large compared 
to the size of a unit cell that it observes, so to speak, millions of unit cells 
at the same time and so obtains only an average "picture" of the structure. 
The above remarks apply equally well to interstitial solid solutions. 
These form whenever the solute atom is small enough to fit into the sol- 
vent lattice without causing too much distortion. Ferrite, the solid solu- 
tion of carbon in a-iron, is a good example. In the unit cell shown in 
Fig. 2-21 (b), there are two kinds of "holes" in the lattice: one at | 
(marked ) and equivalent positions in the centers of the cube faces and 
edges, and one at J ^ (marked x) and equivalent positions. All the 
evidence at hand points to the fact that the carbon atoms in ferrite are 
located in the holes at f f and equivalent positions. On the average, 
however, no more than about 1 of these positions in 500 unit cells is occu- 



2-8] ATOM SIZES AND COORDINATION 53 

the distance of closest approach in the three common metal structures: 



BCC = 



2 ' 

V2 
2 a > (2-7) 

HCP a (l)etwcen atoms in basal plane), 



a 2 c 2 (between atom in basal plane 
\ 3 4 and neighbors above or below). 

Values of the distance of closest approach, together with the crystal struc- 
tures and lattice parameters of the elements, are tabulated in Appendix 13. 
To a first approximation, the size of an atom is a constant. In other 
words, an iron atom has the same size whether it occurs in pure iron, an 
intermediate phase, or a solid solution This is a very useful fact to re- 
member when investigating unknown crystal structures, for it enables us 
to predict roughly how large a hole is necessary in a proposed structure to 
accommodate a given atom. More precisely, it, is known that the size of 
an atom has a slight dependence on its coordination number, which is the 
number of nearest neighbors of the given atom arid which depends on 
crystal structure. The coordination number of an atom in the FCC or 
HCP structures is 12, in BCC 8, and in diamond cubic 4. The smaller 
the coordination number, the smaller the volume occupied by a given 
atom, and the amount of contraction to be expected with decrease in co- 
ordination number is found to be: 

Change in coordination Size contraction, percent 

12 - 8 3 

12 -> 6 4 

12 -> 4 12 

This means, for example, that the diameter of an iron atom is greater if 
the iron is dissolved in FCC copper than if it exists in a crystal of BCC 
a-iron. If it were dissolved in copper, its diameter would be approximately 
2.48/0.97, or 2.56A. 

The size of an atom in a crystal also depends on whether its binding is 
ionic, covalent, metallic, or van der Waals, and on its state of ionization. 
The more electrons are removed from a neutral atom the smaller it be- 
comes, as shown strikingly for iron, whose atoms and ions Fe, 
Fe" 1 " 1 " 4 " have diameters of 2.48, 1.66, and L34A, respectively. 



54 THE GEOMETRY OF CRYSTALS [CHAP. 2 

2-9 Crystal shape. We have said nothing so far about the shape of 
crystals, preferring to concentrate instead on their interior structure. 
However, the shape of crystals is, to the layman, perhaps their most char- 
acteristic property, and nearly everyone is familiar with the beautifully 
developed flat faces exhibited by natural minerals or crystals artificially 
grown from a supersaturated salt solution. In fact, it was with a study 
of these faces and the angles between them that the science of crystallog- 
raphy began. 

Nevertheless, the shape of crystals is really a secondary characteristic, 
since it depends on, and is a consequence of, the interior arrangement of 
atoms. Sometimes the external shape of a crystal is rather obviously re- 
lated to its smallest building block, the unit cell, as in the little cubical 
grains of ordinary table salt (NaCl has a cubic lattice) or the six-sided 
prisms of natural quartz crystals (hexagonal lattice). In many other 
cases, however, the crystal and its unit cell have quite different shapes; 
gold, for example, has a cubic lattice, but natural gold crystals are octa- 
hedral in form, i.e., bounded by eight planes of the form {111}. 

An important fact about crystal faces was known long before there was 
any knowledge of crystal interiors. It is expressed as the law of rational 
indices, which states that the indices of naturally developed crystal faces 
are always composed of small whole numbers, rarely exceeding 3 or 4. 
Thus, faces of the form { 100 } , { 1 1 1 } , { iTOO ) , { 210 ) , etc., are observed but 
not such faces as (510}, {719}, etc. We know today that planes of low 
indices have the largest density of lattice points, and it is a law of crystal 
growth that such planes develop at the expense of planes with high indices 
and few lattice points. 

To a metallurgist, however, crystals with well-developed faces are in 
the category of things heard of but rarely seen. They occur occasionally 
on the free surface of castings, in some electrodeposits, or under other 
conditions of no external constraint. To a metallurgist, a crystal is most 
usually a "grain," seen through a microscope in the company of many 
other grains on a polished section. If he has an isolated single crystal, it 
will have been artificially grown either from the melt, and thus have the 
shape of the crucible in which it solidified, or by recrystallization, and 
thus have the shape of the starting material, whether sheet, rod, or wire. 

The shapes of the grains in a polycrystalline mass of metal are the re- 
sult of several kinds of forces, all of which are strong enough to counter- 
act the natural tendency of each grain to grow with well-developed flat 
faces. The result is a grain roughly polygonal in shape with no obvious 
aspect of crystallinity. Nevertheless, that grain is a crystal and just as 
"crystalline" as, for example, a well-developed prism of natural quartz, 
since the essence of crystallinity is a periodicity of inner atomic arrange- 
ment and not any regularity of outward form. 



2-10] TWINNED CRYSTALS 55 

2-10 Twinned crystals. Some crystals have two parts symmetrically 
related to one another. These, called twinned crystals, are fairly common 
both in minerals and in metals and alloys. 

The relationship between the two parts of a twinned crystal is described 
by the symmetry operation which will bring one part into coincidence 
with the other or with an extension of the other. Two main kinds of 
twinning are distinguished, depending on whether the symmetry opera- 
tion is (a) 180 rotation about an axis, called the twin axis, or (6) reflec- 
tion across a plane, called the twin plane. The plane on which the two 
parts of a twinned crystal are united is called the composition plane. In 
the case of a reflection twin, the composition plane may or may not coin- 
cide with the twin plane. 

Of most interest to metallurgists, who deal mainly with FCC, BCC, 
and HCP structures, are the following kinds of twins: 

(1) Annealing twins, such as occur in FCC metals and alloys (Cu, Ni, 
a-brass, Al, etc.), which have been cold-worked and then annealed to 
cause recrystallization. 

(2) Deformation twins, such as occur in deformed HCP metals (Zn, 
Mg, Be, etc.) and BCC metals (a-Fe, W, etc.). 

Annealing twins in FCC metals are rotation twins, in which the two 
parts are related by a 180 rotation about a twin axis of the form (111). 
Because of the high symmetry of the cubic lattice, this orientation rela- 
tionship is also given by a 60 rotation about the twin axis or by reflec- 
tion across the { 111 j plane normal to the twin axis. In other words, FCC 
annealing twins may also be classified as reflection twins. The twin plane 
is also the composition plane. 

Occasionally, annealing twins appear under the microscope as in Fig. 
2-22 (a), with one part of a grain (E) twinned with respect to the other 
part (A). The two parts are in contact on the composition plane (111) 
which makes a straight-line trace on the plane of polish. More common, 
however, is the kind shown in Fig. 2-22 (b). The grain shown consists of 
three parts: two parts (Ai and A 2 ) of identical orientation separated by a 
third part (B) which is twinned with respect to A\ and A 2 . B is known as 
a twin band. 




(a) 



FIG. 2-22. Twinned grains: (a) and (b) FCC annealing twins; (c) HCP defor- 
mation twin. 



56 



THE GEOMETRY OF CRYSTALS 



[CHAP. 2 



C A B C 




PLAN OF CRYSTAL PLAN OF TWIN 



FIG. 2-23. Twin band in FCC lattice. Plane of main drawing is (110). 



2-10] 



TWINNED CRYSTALS 



59 



twinning 
shear 




[211] 



(1012) 
twin plane 



PLAN OF CRYSTAL 



PLAN OF TWIN 



FIG. 2-24. Twin band in HCP lattice. Plane of main drawing is (1210). 



60 



THE GEOMETRY OF CRYSTALS 



[CHAP. 2 



are said to be first-order, second-order, etc., twins of the parent crystal A. 
Not all these orientations are new. In Fig. 2-22 (b), for example, B may 
be regarded as the first-order twin of AI, and A 2 as the first order twin 
of B. -4-2 is therefore the second-order twin of AI but has the same orien- 
tation as A i. 

2-11 The stereographic projection. Crystal drawings made in perspec- 
tive or in the form of plan and elevation, while they have their uses, are 
not suitable for displaying the angular relationship between lattice planes 
and directions. But frequently we are more interested in these angular 
relationships than in any other aspect of the crystal, and we then need a 
kind of drawing on which the angles between planes can be accurately 
measured and which will permit graphical solution of problems involving 
such angles. The stereographic projection fills this need. 

The orientation of any plane in a crystal can be just as well represented 
by the inclination of the normal to that plane relative to some reference 
plane as by the inclination of the plane itself. All the planes in a crystal 
can thus be represented by a set of plane normals radiating from some one 
point within the crystal. If a reference sphere is now described about 
this point, the plane normals will intersect the surface of the sphere in a 
set of points called poles. This procedure is illustrated in Fig. 2-25, which 
is restricted to the {100} planes of a cubic crystal. The pole of a plane 
represents, by its position on the sphere, the orientation of that plane. 

A plane may also be represented by the trace the extended plane makes 
in the surface of the sphere, as illustrated in Fig. 2-26, where the trace 
ABCDA represents the plane whose pole is PI. This trace is a great circle, 
i.e., a circle of maximum diameter, if the plane passes through the center 
of the sphere. A plane not passing through the center will intersect the 
sphere in a small circle. On a ruled globe, for example, the longitude lines 



100 




010 



FIG. 2-25. 
crystal. 



100 
{1001 poles of a cubic 




M 



FIG. 2-26. Angle between two planes. 



2-1 1J 



THE 8TEREOGRAPHIC PROJECTION 



61 



(meridians) are great circles, while the latitude lines, except the equator, 
are small circles. 

The angle a between two planes is evidently equal to the angle between 
their great circles or to the angle between their normals (Fig. 2-26). But 
this angle, in degrees, can also be measured on the surface of the sphere 
along the great circle KLMNK connecting the poles PI and P 2 of the two 
planes, if this circle has been divided into 360 equal parts. The measure- 
ment of an angle has thus been transferred from the planes themselves 
to the surface of the reference sphere. 

Preferring, however, to measure angles on a flat sheet of paper rather 
than on the surface of a sphere, we find ourselves in the position of the 



, projection plane 
- basic circle 



reference 
sphere 



\ 

point of 
projection 



4 

observer 



SECTION THROUGH 
AB AND PC 




FIG. 2-27. The stereographic projection. 



62 THE GEOMETRY OF CRYSTALS [CHAP. 2 

geographer who wants to transfer a map of the world from a globe to a 
page of an atlas. Of the many known kinds of projections, he usually 
chooses a more or less equal-area projection so that countries of equal area 
will be represented by equal areas on the map. In crystallography, how- 
ever, we prefer the equiangular stereographic projection since it preserves 
angular relationships faithfully although distorting areas. It is made by 
placing a plane of projection normal to the end of any chosen diameter 
of the sphere and using the other end of that diameter as the point of 
projection. In Fig. 2-27 the projection plane is normal to the diameter 
AB, and the projection is made from the point B. If a plane has its pole 
at P, then the stereographic projection of P is at P', obtained by draw- 
ing the line BP and producing it until it meets the projection plane. Al- 
ternately stated, the stereographic projection of the pole P is the shadow 
cast by P on the projection plane when a light source is placed at B. The 
observer, incidentally, views the projection from the side opposite the 
light source. 

The plane NESW is normal to AB and passes through the center C. 
It therefore cuts the sphere in half and its trace in the sphere is a great 
circle. This great circle projects to form the basic circk N'E'S'W on the 
projection, and all poles on the left-hand hemisphere will project within 
this basic circle. Poles on the right-hand hemisphere will project outside 
this basic circle, and those near B will have projections lying at very large 
distances from the center. If we wish to plot such poles, we move the 
point of projection to A and the projection plane to B and distinguish 
the new set of points so formed by minus signs, the previous set (projected 
from B) being marked with plus signs. Note that movement of the pro- 
jection plane along AB or its extension merely alters the magnification; 
we usually make it tangent to the sphere, as illustrated, but we can also 
make it pass through the center of the sphere, for example, in which case 
the basic circle becomes identical with the great circle NESW. 

A lattice plane in a crystal is several steps removed from its stereo- 
graphic projection, and it may be worth-while at this stage to summarize 
these steps: 

(1) The plane C is represented by its normal CP. 

(2) The normal CP is represented by its pole P, which is its intersec- 
tion with the reference sphere. 

(3) The pole P is represented by its stereographic projection P'. 
After gaining some familiarity with the stereographic projection, the 

student will be able mentally to omit these intermediate steps and he will 
then refer to the projected point P' as the pole of the plane C or, even 
more directly, as the plane C itself. 

Great circles on the reference sphere project as circular arcs on the pro- 
jection or, if they pass through the points A and B (Fig. 2-28), as straight 



2-11] 



THE STEREOGRAPHIC PROJECTION 



63 



lines through the center of the projection. Projected great circles always 
cut the basic circle in diametrically opposite points, since the locus of a 
great circle on the sphere is a set of diametrically opposite points. Thus 
the great circle ANBS in Fig. 2-28 projects as the straight line N'S' and 
AW BE as WE'\ the great circle NGSH, which is inclined to the plane of 
projection, projects as the circle arc N'G'S'. If the half great circle WAE 
is divided into 18 equal parts and these points of division projected on 
WAE' , we obtain a graduated scale, at 10 intervals, on the equator of 
the basic circle. 




FIG. 2-28. Stereographic projection of great and small circles. 



64 



THE GEOMETRY OP CRYSTALS 



[CHAP. 2 




FIG. 2-29. Wulff net drawn to 2 intervals. 

Small circles on the sphere also project as circles, but their projected 
center does not coincide with their center on the projection. For example, 
the circle AJEK whose center P lies on AW BE projects as AJ'E'K'. Its 
center on the projection is at C, located at equal distances from A and ', 
but its projected center is at P', located an equal number of degrees (45 
in this case) from A and E'. 

The device most useful in solving problems involving the stereographic 
projection is the Wulff net shown in Fig. 2-29. It is the projection of a 
sphere ruled with parallels of latitude and longitude on a plane parallel 
to the north-south axis of the sphere. The latitude lines on a Wulff net 
are small circles extending from side to side and the longitude lines (merid- 
ians) are great circles connecting the north and south poles of the net. 



2-11] 



THE STEREOGRAPHIC PROJECTION 



65 



PROJECTION 



Wulff net 




FIG. 2-30. Stereographie projection superimposed on Wulff net for measurement 
of angle between poles. 

These nets are available in various sizes, one of 18-cm diameter giving an 
accuracy of about one degree, which is satisfactory for most problems; 
to obtain greater precision, either a larger net or mathematical calculation 
must be used. Wulff nets are used by making the stereographic projec- 
tion on tracing paper and with the basic circle of the same diameter as 
that of the Wulff net; the projection is then superimposed on the Wulff 
net and pinned at the center so that it is free to rotate with respect to the 
net. 

To return to our problem of the measurement of the angle between 
two crystal planes, we saw in Fig. 2-26 that this angle could be measured 
on the surface of the sphere along the great circle connecting the poles of 
the two planes. This measurement can also be carried out on the stereo- 
graphic projection if, and only if, the projected poles lie on a great circle. 
In Fig. 2-30, for example, the angle between the planes* A and B or C 
and D can be measured directly, simply by counting the number of de- 
grees separating them along the great circle on which they lie. Note that 
the angle C-D equals the angle E-F, there being the same difference in 
latitude between C and D as between E and F. 

If the two poles do not lie on a great circle, then the projection is rotated 
relative to the Wulff net until they do lie on a great circle, where the de- 

* We are here using the abbreviated terminology referred to above. 



66 



PROJECTION 




(a) 




FIG. 2-31. (a) Stereo- 
graphic projection of poles 
Pi and P 2 of Fig. 2-26. (b) 
Rotation of projection to put 
poles on same great circle of Wulff 
net. Angle between poles = 30. 



(b) 



2-11] 



THE STEREOGRAPHIC PROJECTION 



67 



sired angle measurement can then be made. Figure 2-31 (a) is a projec- 
tion of the two poles PI and P 2 shown in perspective in Fig. 2-26, and the 
angle between them is found by the rotation illustrated in Fig. 2-3 l(b). 
This rotation of the projection is equivalent to rotation of the poles on 
latitude circles of a sphere whose north-south axis is perpendicular to the 
projection plane. 

As shown in Fig. 2-26, a plane may be represented by its trace in the 
reference sphere. This trace becomes a great circle in the stereographic 
projection. Since every point on this great circle is 90 from the pole of 
the plane, the great circle may be found by rotating the projection until 
the pole falls on the equator 'of the underlying Wulff net and tracing that 
meridian which cuts the equator 90 from the pole, as illustrated in Fig. 
2-32. If this is done for two poles, as in Fig. 2-33, the angle between the 
corresponding planes may also be found from the angle of intersection of 
the two great circles corresponding to these poles; it is in this sense that 
the stereographic projection is said to be angle-true. This method of an- 
gle measurement is not as accurate, however, as that shpwn in Fig. 2-3 l(b). 




FIG. 2-32. Method of finding the trace of a pole (the pole P 2 ' in Fig. 2-31). 



68 



THE GEOMETRY OF CRYSTALS 



[CHAP. 2 



PROJECTION 




FIG. 2-33. Measurement of an angle between two poles (Pi and P 2 of Fig. 2-26) 
by measurement of the angle of intersection of the corresponding traces. 



PROJECTION 




FIG. 2-34. Rotation of poles about NS axis of projection. 



2-11] THE STEREOGRAPHIC PROJECTION 69 

We often wish to rotate poles around various axes. We have already 
seen that rotation about an axis normal to the projection is accomplished 
simply by rotation of the projection around the center of the Wulff net. 
Rotation about an axis lying in the plane of the projection is performed 
by, first, rotating the axis about the center of the Wulff net until it coin- 
cides with the north-south axis if it does not already do so, and, second, 
moving the poles involved along their respective latitude circles the re- 
quired number of degrees. Suppose it is required to rotate the poles A\ 
and BI shown in Fig. 2-34 by 60 about the NS axis, the direction of mo- 
tion being from W to E on the projection. Then AI moves to A 2 along 
its latitude circle as shown. #1, however, can rotate only 40 before 
finding itself at the edge of the projection; we must then imagine it to move 
20 in from the edge to the point B[ on the other side of the projection, 
staying always on its own latitude circle. The final position of this pole 
on the positive side of the projection is at B 2 diametrically opposite B\. 

Rotation about an axis inclined to the plane of projection is accomplished 
by compounding rotations about axes lying in and perpendicular to the 
projection plane. In this case, the given axis must first be rotated into 
coincidence with one or the other of the two latter axes, the given rota- 
tion performed, and the axis then rotated back to its original position. 
Any movement of the given axis must be accompanied by a similar move- 
ment of all the poles on the projection. 

For example, we may be required to rotate AI about BI by 40 in a 
clockwise direction (Fig. 2-35). In (a) the pole to be rotated A } and the 
rotation axis BI are shown in their initial position. In (b) the projection 
has been rotated to bring BI to the equator of a Wulff net. A rotation of 
48 about the NS axis of the net brings BI to the point B 2 at the center 
of the net; at the same time AI must go to A 2 along a parallel of latitude. 
The rotation axis is now perpendicular to the projection plane, and the 
required rotation of 40 brings A 2 to A 3 along a circular path centered 
on B 2 . The operations which brought BI to B 2 must now be reversed in 
order to return B 2 to its original position. Accordingly, B 2 is brought to 
JBs and A% to A*, by a 48 reverse rotation about the NS axis of the net. 
In (c) the projection has been rotated back to its initial position, construc- 
tion lines have been omitted, and only the initial and final positions of the 
rotated pole are shown. During its rotation about B^ AI moves along 
the small circle shown. This circle is centered at C on the projection and 
not at its projected center BI. To find C we use the fact that all points 
on the circle must lie at equal angular distances from BI] in this case, 
measurement on a Wulff net shows that both AI and A are 76 from B\. 
Accordingly, we locate any other point, such as D, which is 76 from B\, 
and knowing three points on the required circle, we can locate its center C. 



70 



THE GEOMETRY OP CRYSTALS 



[CHAP. 2 



48 



40 



(b) 





(a) (c) 

FIG. 2-35. Rotation of a pole about an inclined axis. 



2-11] 



THE 8TEREOGRAPHIC PROJECTION 



71 



In dealing with problems of crystal orientation a standard projection is 
of very great value, since it shows at a glance the relative orientation of 
all the important planes in the crystal. Such a projection is made by se- 
lecting some important crystal plane of low indices as the plane of pro- 
jection [e.g., (100), (110), (111), or (0001)] and projecting the poles of 
various crystal planes onto the selected plane. The construction of a 
standard projection of a crystal requires a knowledge of the interplanar 
angles for all the principal planes of the crystal. A set of values applicable 
to all crystals in the cubic system is given in Table 2-3, but those for 
crystals of other systems depend on the particular axial ratios involved 
and must be calculated for each case by the equations given in Appendix 1. 
Much time can be saved in making standard projections by making use 
of the zonal relation: the normals to all planes belonging to one zone are 
coplanar and at right angles to the zone axis. Consequently, the poles 
of planes of a zone will all lie on the same great circle on the projection, 
and the axis of the zone will be at 90 from this great circle. Furthermore, 
important planes usually belong to more than one zone and their poles 
are therefore located at the intersection of zone circles. It is also helpful 
to remember that important directions, which in the cubic system are 
normal to planes of the same indices, are usually the axes of important 



zones. 



Figure 2-36 (a) shows the principal poles of a cubic crystal projected on 
the (001) plane of the crystal or, in other words, a standard (001) projec- 
tion. The location of the {100} cube poles follows immediately from Fig. 
2-25. To locate the {110} poles we first note from Table 2-3 that they 
must lie at 45 from {100} poles, which are themselves 90 apart. In 



100 



100 



no 



no 



110 




1)10 Oil 



no 



111 




FIG. 2-36. Standard projections of cubic crystals, (a) on (001) and (b) on (Oil). 



72 



THE GEOMETRY OF CRYSTALS 



[CHAP. 2 



TABLE 2-3 

INTERPLANAR ANGLES (IN DEGREES) IN CUBIC CRYSTALS BETWEEN 
PLANES OF THE FORM \hik\li\ AND 



1W2I 


IWi! 


100 


110 


in 


210 


211 


221 


310 


100 



















90 














110 


45 

















90 


60 
















90 












111 


54.7 


35.3 

















90 


70.5 
















109.5 










210 


26.6 


18.4 


39.2 













63.4 


50.8 


75.0 


36.9 










90 


71.6 




53.1 








211 


35.3 


30 


19.5 


24.1 











65.9 


54.7 


61.9 


43.1 


33.6 










73.2 


90 


56.8 


48.2 










90 












221 


48.2 


19.5 


15.8 


26.6 


17.7 









70.5 


45 


54.7 


41.8 


35.3 


27.3 








76.4 


78.9 


53.4 


47.1 


39.0 








90 












310 


18.4 


26.6 


43.1 


8.1 


25.4 


32.5 







71.6 


47.9 


68.6 


58.1 


49.8 


42.5 


25.9 




90 


63.4 




45 


58.9 


58.2 


36.9 






77.1 












311 


25.2 


31.5 


29.5 


19.3 


10.0 


25.2 


17.6 




72.5 


64.8 


58.5 


47.6 


42.4 


45.3 


40.3 






90 


80.0 


66.1 


60.5 


59.8 


55.1 


320 


33.7 


11.3 


61.3 


7.1 


25.2 


22.4 


15.3 




56.3 


54.0 


71.3 


29.8 


37.6 


42.3 


37.9 




90 


66.9 




41.9 


55.6 


49.7 


52.1 


321 


36.7 


19.1 


22.2 


17.0 


10.9 


11.5 


21.6 




57.7 


40.9 


51.9 


33.2 


29.2 


27.0 


32.3 




74.5 


55.5 


72.0 


53.3 


40.2 


36.7 


40.5 








90 










331 


46.5 


13.1 


22.0 










510 


11.4 














511 


15.6 














711 


11.3 















Largely from R. M. Bozorth, Phys. Rev. 26, 390 (1925); rounded 
off to the nearest 0.1. 



2-11] 



THE STEREOGRAPHIC PROJECTION 



73 



[112] 
zone 



mi] 



1110] 



[001] 
zone 



[100] // 
zone 




FIG. 2-37. Standard (001) projection of a cubic crystal. (From Structure of 
Metals, by C. S. Barrett, McGraw-Hill Book Company, Inc., 1952.) 

this way we locate (Oil), for example, on the great circle joining (001) 
and (010) and at 45 from each. After all the {110} poles are plotted, 
we can find the { 111 } poles at the intersection of zone circles. Inspection 
of a crystal model or drawing or use of the zone relation given by JEq. 
(2-3) will show that (111), for example, belongs to both the zone [101] 
and the zone [Oil]. The pole of (111) is thus located at the intersection 
of the zone circle through (OlO), (101), and (010) and the zone circle 
through (TOO), (Oil), and (100). This location may be checked by meas- 
urement of its angular distance from (010) or (100), which should be 
54.7. The (Oil) standard projection shown in Fig. 2-36(b) is plotted in 
the same manner. Alternately, it may be constructed by rotating all the 
poles in the (001) projection 45 to the left about the NS axis of the pro- 
jection, since this operation will bring the (Oil) pole to the center. In 
both of these projections symmetry symbols have been given each pole 
in conformity with Fig. 2-6(b), and it will be noted that the projection 
itself has the symmetry of the axis perpendicular to its plane, Figs. 2-36(a) 
and (b) having 4-fold and 2-fold symmetry, respectively. 



74 



THE GEOMETRY OF CRYSTALS 



[CHAP. 2 



Jl20 



T530, 




1321 



0113. 



' " 
14 l013 5,, 4 "<>' 




1014 .2203 




3 *' QI5 - %>* 




0114 TlO4 l 3 





*OII5 Tl05 


23?l 


0001 . 




12 F4 ?2l2 


T2H 


"05 .0115 




104 . *i 4 OM3 


9 



foil 



no. ioTs . 




53TO 



320 



FIG. 2-38. Standard (0001) projection for zinc (hexagonal, c/a = 1.86). (From 
Structure of Metals, by C. S. Barrett, McGraw-Hill Book Company, Inc., 1952.) 

Figure 2-37 is a standard (001) projection of a cubic crystal with con- 
siderably more detail and a few important zones indicated. A standard 
(0001) projection of a hexagonal crystal (zinc) is given in Fig. 2-38. 

It is sometimes necessary to determine the Miller indices of a given 
pole on a crystal projection, for example the pole A in Fig. 2-39(a), which 
applies to a cubic crystal. If a detailed standard projection is available, 
the projection with the unknown pole can be superimposed on it and its 
indices will be disclosed by its coincidence with one of the known poles 
on the standard. Alternatively, the method illustrated in Fig. 2-39 may 
be used. The pole A defines a direction in space, normal to the plane 
(hkl) whose indices are required, and this direction makes angles p, <r, r 
with the coordinate axes a, b, c. These angles are measured on the pro- 
jection as shown in (a). Let the perpendicular distance between the ori- 
gin and the (hkl) plane nearest the origin be d [Fig. 2-39(b)], and let the 
direction cosines of the line A be p, g, r. Therefore 



cosp 



d 
o/fc' 



cos a 



d 
bjk 



d 



cos r 



2-11] 



THE STEREOGRAPHIC PROJECTION 



75 





100 

(a) (b) 

FIG. 2-39. Determination of the Miller indices of a pole. 

h:k:l = pa:qb:rc. (2-8) 

For the cubic system we have the simple result that the Miller indices 
required are in the same ratio as the direction cosines. 

The lattice reorientation caused by twinning can be clearly shown on 
the stereographic projection. In Fig. 2-40 the open symbols are the { 100} 
poles of a cubic crystal projected on the (OOl)jplane. If this crystal is 
FCC, then one of its possible twin planes is (111), represented on the 
projection both by its pole and its trace. The cube poles of the twin 
formed by reflection in this plane are shown as solid symbols; these poles 
are located by rotating the projection on a Wulff net until the pole of the 
twin plane lies on the equator, after which the cube poles of the crystal 
can be moved along latitude circles of the net to their final position. 

The main principles of the stereographic projection have now been pre- 
sented, and we will have occasion to use them later in dealing with various 
practical problems in x-ray metal- 
lography. The student is reminded, 
however, that a mere reading of this 
section is not sufficient preparation 
for such problems. In order to gain 
real familiarity with the stereographic 
projection, he must practice, with 
Wulff net and tracing paper, the 
operations described above and solve 
problems of the kind given below. 
Only in this way will he be able to 
read and manipulate the stereo- 
graphic projection with facility and 
think in three dimensions of what is 
represented in two. 



100 



010 



010 




(111) 

twin plane 



100 



FIG. 2-40. Stereographic projection 
of an FCC crystal and its twin. 



76 THE GEOMETRY OP CRYSTALS [CHAP. 2 

PROBLEMS 

2-1. Draw the following planes and directions in a tetragonal unit cell: (001), 
(Oil), (113), [110], [201], [I01]._ 

2-2. Show by means of a (110) sectional drawing that [111] is perpendicular to 
(111) in the cubic system, but not, in general, in the tetragonal system. 

2-3. In a drawing of a hexagonal prism, indicate the following planes and di- 
rections: (1210), (1012), (T011), [110], [111), [021]. 

2-4. Derive Eq. (2-2) of the text. 

2-5. Show that the planes (110), (121), and (312) belong to the zone [111]^ 

2-6. Do the following planes all belong to the same zone: (110), (311), (132)? 
If so, what is the zone axis? Give the indices of any other plane belonging to this 
zone. 

2-7. Prepare a cross-sectional drawing of an HCP structure which will show that 
all atoms do not have identical surroundings and therefore do not lie on a point 
lattice. 

2-8. Show that c/a for hexagonal close packing of spheres is 1.633. 

2-9. Show that the HCP structure (with c/a = 1.633) and the FCC structure 
are equally close-packed, and that the BCC structure is less closely packed 
than either of the former. 

2-10. The unit cells of several orthorhombic crystals are described below. 
What is the Bravais lattice of each and how do you know? 

(a) Two atoms of the same kind per unit cell located at J 0, \. 

(6) Four atoms of the same kind per unit cell located at z, J z, f (^ + z), 

00(| + 2). 

(c) Four atoms of the same kind per unit cell located at x y z, x y z, ( J + x) 

(I - y) *, (I -*)(* + y) * 

(d) Two atoms of one kind A located at J 0, J J; and two atoms of another 
kind B located at \, \\ 0. 

2-11. Make a drawing, similar to Fig. 2-23, of a (112) twin in a BCC lattice 
and show the shear responsible for its formation. Obtain the magnitude of the 
shear strain graphically. 

2-12. Construct a Wulff net, 18 cm in diameter and graduated at 30 intervals, 
by the use of compass, dividers, and straightedge only. Show all construction lines. 

In some of the following problems, the coordinates of a point on a stereographic pro- 
jection are given in terms of its latitude and longitude, measured from the center of the 
projection. Thus, the N pole is 90N, 0E, the E pole is 0N, 90E, etc. 

2-13. Plane A is represented on a stereographic projection by a great circle 
passing through the N and S poles and the point 0N, 70W. The pole of plane B 
is located at 30N, 50W. 

(a) Find the angle between the two planes. 

(b) Draw the great circle of plane B and demonstrate that the stereographic 
projection is angle-true by measuring With a protractor the angle between 
the great circles of A and B. 



PROBLEMS 77 

2-14. Pole A, whose coordinates are 20N, 50E, is to be rotated about the 
axes described below. In each case, find the coordinates of the final position of 
pole A and show the path traced out during its rotation. 

(a) 100 rotation about the NS axis, counterclockwise looking from N to 8. 

(b) 60 rotation about an axis normal to the plane of projection, clockwise to 
the observer. 

(c) 60 rotation about an inclined axis B, whose coordinates are 10S, 30W, 
clockwise to the observer. 

2-16. Draw a standard (111) projection of a cubic crystal, showing all poles of 
the form { 100} , { 1 10 1 , (111) and the important zone circles between them. Com- 
pare with Figs. 2-36(a) and (b). 

2-16. Draw a standard (001) projection of white tin (tetragonal, c/a = 0.545), 
showing all poles of the form 1 001 1 , { 100 ) , { 1 10 ) , ( 01 1 1 , { 1 1 1 ) and the important 
zone circles between them. Compare with Fig. 2-36(a). 

2-17. Draw a standard (0001) projection of beryllium (hexagonal, c/a = 1.57), 
showing all poles of the form {2l70j, {lOTO}, {2TTl|, (10Tl| and the important 
zone circles between them. Compare with Fig. 2-38. 

2-18. On a standard (001) projection of a cubic crystal, in the orientation of 
Fig. 2~36(a), the pole of a certain plane has coordinates 53.3S, 26.6E. What 
are its Miller indices? Verify your answer by comparison of measured angles 
with those given in Table 2-3. 

2-19. Duplicate the operations shown in Fig. 2-40 and thus find the locations 
of the cube poles of a (TTl) reflection twin in a cubic crystal. What are their 
coordinates? 

2-20. Show that the twin orientation found in Prob. 2- 1 9 can also be obtained 

by 

(a) Reflection in a 1112) plane. Which one? 

(6) 180 rotation about a (ill) axis. Which one? 

(c) 60 rotation about a (ill) axis. Which one? 

In (c), show the paths traced out by the cube poles during their rotation. 



CHAPTER 3 
DIFFRACTION I: THE DIRECTIONS OF DIFFRACTED BEAMS 

3-1 Introduction. After our preliminary survey of the physics of x-rays 
and the geometry of crystals, we can now proceed to fit the two together 
and discuss the phenomenon of x-ray diffraction, which is an interaction 
of the two. Historically, this is exactly the way this field of science de- 
veloped. For many years, mineralogists and crystallographers had accumu- 
lated knowledge about crystals, chiefly by measurement of interfacial 
angles, chemical analysis, and determination of physical properties. There 
was little knowledge of interior structure, however, although some very 
shrewd guesses had been made, namely, that crystals were built up by 
periodic repetition of some unit, probably an atom or molecule, and that 
these units were situated some 1 or 2A apart. On the other hand, there 
were indications, but only indications, that x-rays might be electromag- 
netic waves about 1 or 2A in wavelength. In addition, the phenomenon 
of diffraction was well understood, and it was known that diffraction, as 
of visible light by a ruled grating, occurred whenever wave motion en- 
countered a set of regularly spaced scattering objects, provided that the 
wavelength of the wave motion was of the same order of magnitude as the 
repeat distance between the scattering centers. 

Such was the state of knowledge in 1912 when the German physicist 
von Laue took up the problem. He reasoned that, if crystals were com- 
posed of regularly spaced atoms which might act as scattering centers for 
x-rays, and if x-rays were electromagnetic waves of wavelength about 
equal to the interatomic distance in crystals, then it should be possible to 
diffract x-rays by means of crystals. Under his direction, experiments to 
test this hypothesis were carried out: a crystal of copper sulfate was set 
up in the path of a narrow beam of x-rays and a photographic plate was 
arranged to record the presence of diffracted beams, if any. The very 
first experiment was successful and showed without doubt that x-rays 
were diffracted by the crystal out of the primary beam to form a pattern 
of spots on the photographic plate. These experiments proved, at one 
and the same time, the wave nature of x-rays and the periodicity of the 
arrangement of atoms within a crystal. Hindsight is always easy and 
these ideas appear quite simple to us now, when viewed from the vantage 
point of more than forty years' development of the subject, but they were 
not at all obvious in 1912, and von Laue's hypothesis and its experimental 
verification must stand as a great intellectual achievement. 

78 



3-2] 



DIFFRACTION 



79 



The account of these experiments was read with great interest by two 
English physicists, W. H. Bragg and his son W. L. Bragg. The latter, 
although only a young student at the time it was still the year 1912 
successfully analyzed the Laue experiment and was able to express the 
necessary conditions for diffraction in a somewhat simpler mathematical 
form than that used by von Laue. He also attacked the problem of crystal 
structure with the new tool of x-ray diffraction and, in the following year, 
solved the structures of NaCl, KC1, KBr, and KI, all of which have the 
NaCl structure; these were the first complete crystal-structure determina- 
tions ever made. 

3-2 Diffraction. Diffraction is due essentially to the existence of cer- 
tain phase relations between two or more waves, and it is advisable, at 
the start, to get a clear notion of what is meant by phase relations. Con- 
sider a beam of x-rays, such as beam 1 in Fig. 3-1, proceeding from left to 
right. For convenience only, this beam is assumed to be plane-polarized 
in order that we may draw the electric field vector E always in one plane. 
We may imagine this beam to be composed of two equal parts, ray 2 and 
ray 3, each of half the amplitude of beam 1. These two rays, on the wave 
front AA', are said to be completely in phase or in step; i.e., their electric- 
field vectors have the same magnitude and direction at the same instant 
at any point x measured along the direction of propagation of the wave. 
A wave front is a surface perpendicular to this direction of propagation. 




FIG. 3-1. Effect of path difference on relative phase. 



80 DIFFRACTION II THE DIRECTIONS OF DIFFRACTED BEAMS [CHAP. 3 

Now consider an imaginary experiment, in which ray 3 is allowed to 
continue in a straight line but ray 2 is diverted by some means into a 
curved path before rejoining ray 3. What is the situation on the wave 
front BB' where both rays are proceeding in the original direction? On 
this front, the electric vector of ray 2 has its maximum value at the instant 
shown, but that of ray 3 is zero. The two rays are therefore out of phase. 
If we add these two imaginary components of the beam together, we find 
that beam 1 now has the form shown in the upper right of the drawing. 
If the amplitudes of rays 2 and 3 are each 1 unit, then the amplitude of 
beam 1 at the left is 2 units and that of beam 1 at the right is 1.4 units, if 
a sinusoidal variation of E with x is assumed. 

Two conclusions may be drawn from this illustration : 

(1) Differences in the length of the path traveled lead to differences in 
phase. 

(2) The introduction of phase differences produces a change in ampli- 
tude. 

The greater the path difference, the greater the difference in phase, since 
the path difference, measured in wavelengths, exactly equals the phase 
difference, also measured in wavelengths. If the diverted path of ray 2 in 
Fig. 3-1 were a quarter wavelength longer than shown, the phase differ- 
ence would be a half wavelength. The two rays would then be completely 
out of phase on the wave front BB' and beyond, and they would therefore 
annul each other, since at any point their electric vectors would be either 
both zero or of the same magnitude and opposite in direction. If the dif- 
ference in path length were made three quarters of a wavelength greater 
than shown, the two rays would be one complete wavelength out of phase, 
a condition indistinguishable from being completely in phase since ir + 
cases the two waves would combine to form a beam of amplitude 2 
just like the original beam. We may conclude that two rays are 
pletely in phase whenever their path lengths differ either by zero or > 
whole number of wavelengths. 

Differences in the path length of various rays arise quite naturally v 
we consider how a crystal diffracts x-rays. Figure 3-2 shows a section 
crystal, its atoms arranged on a set of parallel planes A, 5, C, D, 
normal to the plane of the drawing and spaced a distance d' apart. Ass 
that a beam of perfectly parallel, perfectly monochromatic x-rays of \v 
length X is incident on this crystal at an angle 0, called the Bragg a, 
where is measured between the incident beam and the particular cr; 
planes under consideration. 

We wish to know whether this incident beam of x-rays will be diffrd 
by the crystal and, if so, under what conditions. A diffracted beam me 
defined as a beam composed of a large number of scattered rays mutually 
forcing one another. Diffraction is, therefore, essentially a scattering- 



3-2| DIFFRACTION 83 

We have here regarded a diffracted beam as being built up of rays scat- 
tered by successive planes of atoms within the crystal. It would be a 
mistake to assume, however, that a single plane of atoms A would diffract 
x-rays just as the complete crystal does but less strongly. Actually, the 
single plane of atoms would produce, not only the beam in the direction 1' 
as the complete crystal does, but also additional beams in other directions, 
some of them not confined to the plane of the drawing. These additional 
beams do not exist in the diffraction from the complete crystal precisely 
because the atoms in the other planes scatter beams which destructively 
interfere with those scattered by the atoms in plane A, except in the direc- 
tion I 7 . 

At first glance, the. diffraction of x-rays by crystals and the reflection of 
visible light by mirrors appear very similar, since in both phenomena the 
angle of incidence is equal to the angle of reflection. It seems that we 
might regard the planes of atoms as little mirrors which "reflect" the 
x-rays. Diffraction and reflection, however, differ fundamentally in at 
least three aspects: 

(1) The diffracted beam from a crystal is built up of rays scattered by 
all the atoms of the crystal which lie in the path of the incident beam. 
The reflection of visible light takes place in a thin surface layer only. 

(2) The diffraction of monochromatic x-rays takes place only at those 
particular angles of incidence which satisfy the Bragg law. The reflection 
of visible light takes place at any angle of incidence. 

(3) The reflection of visible light by a good mirror is almost 100 percent 
efficient. The intensity of a diffracted x-ray beam is extremely small com- 
pared to that of the incident beam. 

Despite these differences, we often speak of "reflecting planes" and 
"reflected beams" when we really mean diffracting planes and diffracted 
beams. This is common usage and, from now on, we will frequently use 
these terms without quotation marks but with the tacit understanding that 
we really mean diffraction and not reflection. * 

To sum up, diffraction is essentially a scattering phenomenon in which 
a large number of atoms cooperate. Since the atoms are arranged period- 
ically on a lattice, the rays scattered by them have definite phase relations 
between them ; these phase relations are such that destructive interference 
occurs in most directions of scattering, but in a few directions constructive 
interference takes place and diffracted beams are formed. The two essen- 
tials are a wave motion capable of interference (x-rays) and a set of periodi- 
cally arranged scattering centers (the atoms of a crystal). 

* For the sake of completeness, it should be mentioned that x-rays can be totally 
reflected by a solid surface, just like visible light by a mirror, but only at very 
small angles of incidence (below about one degree). This phenomenon is of little 
practical importance in x-ray metallography and need not concern us further. 



84 DIFFRACTION i: THE DIRECTIONS OF DIFFRACTED BEAMS [CHAP. 3 

3-3 The Bragg law. Two geometrical facts are worth remembering: 

(1) The incident beam, the normal to the reflecting plane, and the dif- 
fracted beam are always coplanar. 

(2) The angle between the diffracted beam and the transmitted beam 
is always 26. This is known as the diffraction angle, and it is this angle, 
rather than 6, which is usually measured experimentally. 

As previously stated, diffraction in general occurs only when the wave- 
length of the wave motion is of the same order of magnitude as the repeat 
distance between scattering centers. This requirement follows from the 
Bragg law. Since sin cannot exceed unity, we may write 

n\ 

= sin0<l. (3-2) 

2rf' 

Therefore, n\ must be less than 2d'. For diffraction, the smallest value of 
n is 1. (n = corresponds to the beam diffracted in the same direction 
as the transmitted beam. It cannot be observed.) Therefore the condi- 
tion for diffraction at any observable angle 26 is 

X < 2d'. (3-3) 

For most sets of crystal planes d r is of the order of 3A or less, which means 
that X cannot exceed about 6A. A crystal could not possibly diffract ultra- 
violet radiation, for example, of wavelength about 500A. On the other 
hand, if X is very small, the diffraction angles are too small to be con- 
veniently measured. 
The Bragg law may be written in the form 

X = 2 - sin 6. (3-4) 

n 

Since the coefficient of X is now unity, we can consider a reflection of any 
order as a first-order reflection from planes, real or fictitious, spaced at a 
distance 1/n of the previous spacing. This turns out to be a real con- 
venience, so we set d = d'/n and write the Bragg law in the form 

(3-5) 

This form will be used throughout this book. 

This usage is illustrated by Fig. 3-3. Consider the second-order 100 re- 
flection* shown in (a). Since it is second-order, the path difference ABC 
between rays scattered by adjacent (100) planes must be Jwo whole wave- 

*This means the ^reflection from the (100) planes. Conventionally, the Miller 
indices of a reflecting plane hkl, written without parentheses, stand for the re- 
flected beam from the plane (hkl). 




3-4] 



X-RAY SPECTROSCOPY 



85 




(100) 




(200) 



FIG. 3-3. Equivalence of (a) a second-order 100 reflection and (b) a first-order 
200 reflection. 

lengths. If there is no real plane of atoms between the (100) planes, we 
can always imagine one as in Fig. 3-3 (b), where the dotted plane midway 
between the (100) planes forms part of the (200) set of planes. For the 
same reflection as in (a), the path difference DEF between rays scattered 
by adjacent (200) planes is now only one whole wavelength, so that this 
reflection can properly be called a first-order 200 reflection. Similarly, 
300, 400, etc., reflections are equivalent to reflections of the third, fourth, 
etc., orders from the (100) planes. In general, an nth-order reflection 
from (hkl) planes of spacing d f may be considered as a first-order reflection 
from the (nh nk nl) planes of spacing d = d' /n. Note that this convention 
is in accord with the definition of Miller indices since (nh nk nl) are the 
Miller indices of planes parallel to the (hkl) planes but with 1/n the spacing 
of the latter. 

3-4 X-ray spectroscopy. Experimentally, the Bragg law can be uti- 
lized in two ways. By using x-rays of known wavelength X and measuring 
6, we can determine the spacing d of various planes in a crystal: this is 
structure analysis and is the subject, 
in one way or another, of the greater 
part of this book. Alternatively, we 
can use a crystal with planes of known 
spacing d, measure 0, and thus deter- 
mine the wavelength X of the radia- 
tion used: this is x-ray spectroscopy. 

The essential features of an x-ray 
spectrometer are shown in Fig. 3-4. 
X-rays from the tube T are incident 
on a crystal C which may be set at 
any desired angle to the incident FIG. 3-4. The x-ray spectrometer. 




86 DIFFRACTION i: THE DIRECTIONS OF DIFFRACTED BEAMS [CHAP. 3 

beam by rotation about an axis through 0, the center of the spectrometer 
circle. D is an ionization chamber or some form of counter which measures 
the intensity of the diffracted x-rays; it can also be rotated about and 
set at any desired angular position. The crystal is usually cut or cleaved 
so that a particular set of reflecting planes of known spacing is parallel to 
its surface, as suggested by the drawing. In use, the crystal is positioned 
so that its reflecting planes make some particular angle 6 with the incident 
beam, and D is set at the corresponding angle 26. The intensity of the 
diffracted beam is then measured and its wavelength calculated from the 
Bragg law, this procedure being repeated for various angles 6. It is in this 
way that curves such as Fig. 1-5 and the characteristic wavelengths tabu- 
lated in Appendix 3 were obtained. W. H. Bragg designed and used the 
first x-ray spectrometer, and the Swedish physicist Siegbahn developed it 
into an instrument of very high precision. 

Except for one application, the subject of fluorescent analysis described 
in Chap. 15, we are here concerned with x-ray spectroscopy only in so 
far as it concerns certain units of wavelength. Wavelength measurements 
made in the way just described are obviously relative, and their accuracy 
is no greater than the accuracy with which the plane spacing of the crystal 
is known. For a cubic crystal this spacing can be obtained independently 
from a measurement of its density. For any crystal, 

weight of atoms in unit cell 

Density = - - > 

volume of unit cell 

ZA 
p = , (3-6) 

NV 

where p = density (gm/cm 3 ), SA = sum of the atomic weights of the 
atoms in the unit cell, N = Avogadro's number, and V = volume of unit 
cell (cm 3 ). NaCl, for example, contains four sodium atoms and four chlo- 
rine atoms per unit cell, so that 

SA = 4(at. wt Na) + 4 (at. wt Cl). 

If this value is inserted into Eq. (3-6), together with Avogadro's number 
and the measured value of the density, the volume of the unit cell V can 
be found. Since NaCl is cubic, the lattice parameter a is given simply by 
the cube root of V. From this value of a and the cubic plane-spacing 
equation (Eq. 2-5), the spacing of any set of planes can be found. 

In this way, Siegbahn obtained a value of 2.8 14 A for the spacing of the 
(200) planes of rock salt, which he could use as a basis for wavelength 
measurements. However, he was able to measure wavelengths in terms 
of this spacing much more accurately than the spacing itself was known, 
in the sense that he could make relative wavelength measurements accurate 



3-4] X-RAY 8PECTRO8COPY 87 

to six significant figures whereas the spacing in absolute units (angstroms) 
was known only to four. It was therefore decided to define arbitrarily 
the (200) spacing of rock salt as 2814.00 X units (XU), this new unit being 
chosen to be as nearly as possible equal to 0.001A. 

Once a particular wavelength was determined in terms of this spacing, 
the spacing of a given set of planes in any other crystal could be measured. 
Siegbahn thus measured the (200) spacing of calcite, which he found more 
suitable as a standard crystal, and thereafter based all his wavelength 
measurements on this spacing. Its value is 3029.45 XU. Later on, the 
kilo X unit (kX) was introduced, a thousand times as large as the X unit 
and nearly equal to an angstrom. The kX unit is therefore defined by the 
relation 

(200) plane spacing of calcite 

1 kX = (37) 

3.02945 V ; 

On this basis, Siegbahn and his associates made very accurate measure- 
ments of wavelength in relative (kX) units and these measurements form 
the basis of most published wavelength tables. 

It was found later that x-rays could be diffracted by a ruled grating 
such as is used in the spectroscopy of visible light, provided that the angle 
of incidence (the angle between the incident beam and the plane of the 
grating) is kept below the critical angle for total reflection. Gratings thus 
offer a means of making absolute wavelength measurements, independent 
of any knowledge of crystal structure. By a comparison of values so ob- 
tained with those found by Siegbahn from crystal diffraction, it was pos- 
sible to calculate the following relation between the relative and absolute 
units: 

(3-8) 



1 kX = 1.00202A 



This conversion factor was decided on in 1946 by international agreement, 
and it was recommended that, in the future, x-ray wavelengths and the 
lattice parameters of crystals be expressed in angstroms. If V in Eq. (3-6) 
for the density of a crystal is expressed in A 3 (not in kX 3 ) and the currently 
accepted value of Avogadro's number inserted, then the equation becomes 

1.66020S4 
P = (3-9) 

The distinction between kX and A is unimportant if no more than 
about three significant figures are involved. In precise work, on the other 
hand, units must be correctly stated, and on this point there has been con- 
siderable confusion in the past. Some wavelength values published prior 
to about 1946 are stated to be in angstrom units but are actually in kX 
units. Some crystallographers have used such a value as the basis for a 



'88; DIFFRACTION II THE DIRECTIONS OF DIFFRACTED BEAMS [CHAP. 3 

precise measurement of the lattice parameter of a crystal and the result 
has been stated, again incorrectly, in angstrom units. Many published 
parameters are therefore in error, and it is unfortunately not always easy 
to determine which ones are and which ones are not. The only safe rule 
to follow, in stating a precise parameter, is to give the wavelength of the 
radiation used in its determination. Similarly, any published table of 
wavelengths can be tested for the correctness of its units by noting the 
wavelength given for a particular characteristic line, Cu Ka\ for example. 
The wavelength of this line is 1.54051A or 1.53740 kX. 

3-5 Diffraction directions. What determines the possible directions, 
i.e., the possible angles 20, in which a given crystal can diffract a beam of 
monochromatic x-rays? Referring to Fig. 3-3, we see that various diffrac- 
tion angles 20i, 20 2 , 20 3 , ... can be obtained from the (100) planes by 
using a beam incident at the correct angle 0i, 2 , 0s, and producing 
first-, second-, third-, . . . order reflections. But diffraction can also be 
produced by the (110) planes, the (111) planes, the (213) planes, and so 
on. We obviously need a general relation which will predict the diffrac- 
tion angle for any set of planes. This relation is obtained by combining 
the Bragg law and the plane-spacing equation (Appendix 1) applicable to 
the particular crystal involved. 

For example, if the crystal is cubic, then 

X = 2d sin 
and 

1 (ft 2 + fc 2 + I 2 } 



Combining these equations, we have 

X 2 

sin 2 = - (h 2 + k 2 + l 2 ). (3-10) 

4a 2 

This equation predicts, for a particular incident wavelength X and a par- 
ticular cubic crystal of unit cell size a, all the possible Bragg angles at 
which diffraction can occur from the planes (hkl). For (110) planes, for 
example, Eq. (3-10) becomes 



If the crystal is tetragonal, with axes a and c, then the corresponding gen- 
eral equation is 



4 a 2 c 2 

and similar equations can readily be obtained for the other crystal systems. 



3-6] DIFFRACTION METHODS 89 

These examples show that the directions in which a beam of given wave- 
length is diffracted by a given set of lattice planes is determined by the 
crystal system to which the crystal belongs and its lattice parameters. In 
short, diffraction directions are determined solely by the shape and size of the 
unit cell. This is an important point and so is its converse: all we can pos- 
sibly determine about an unknown crystal by measurements of the direc- 
tions of diffracted beams are the shape and size of its unit cell. We will 
find, in the next chapter, that the intensities of diffracted beams are deter- 
mined by the positions of the atoms within the unit cell, and it follows that 
we must measure intensities if we are to obtain any information at all 
about atom positions. We will find, for many crystals, that there are 
particular atomic arrangements which reduce the intensities of some dif- 
fracted beams to zero. In such a case, there is simply no diffracted beam 
at the angle predicted by an equation of the type of Eqs. (3-10) and (3-11). 
It is in this sense that equations of this kind predict all possible diffracted 
beams. 

3-6 Diffraction methods. Diffraction can occur whenever the Bragg 
law, X = 2d sin 0, is satisfied. This equation puts very stringent condi- 
tions on X and 6 for any given crystal. With monochromatic radiation, 
an arbitrary setting of a single crystal in a beam of x-rays will not in gen- 
eral produce any diffracted beams. Some way of satisfying the Bragg law 
must be devised, and this can be done by continuously varying either X 
or 6 during the experiment. The ways in which these quantities are varied 
distinguish the three main diffraction methods: 



Laue method Variable Fixed 

Rotating-crystal method Fixed Variable (in part) 

Powder method Fixed Variable 

The Laue method was the first diffraction method ever used, and it re- 
produces von Laue's original experiment. A beam of white radiation, the 
continuous spectrum from an x-ray tube, is allowed to fall on a fixed single 
crystal. The Bragg angle 6 is therefore fixed for every set of planes in the 
crystal, and each set picks out and diffracts that particular wavelength 
which satisfies the Bragg law for the particular values of d and involved. 
Each diffracted beam thus has a different wavelength. 

There are two variations of the Laue method, depending on the relative 
positions of source, crystal, and film (Fig. 3-5). In each, the film is flat 
and placed perpendicular to the incident beam. The film in the trans- 
mission Laue method (the original Laue method) is placed behind the crys- 
tal so as to record the beams diffracted in the forward direction. This 



90 DIFFRACTION i: THE DIRECTIONS OF DIFFRACTED BEAMS [CHAP. 3 





(a) (b) 

FIG. 3-5. (a) Transmission and (b) back-reflection Laue methods. 

method is so called because the diffracted beams are partially transmitted 
through the crystal. In the back-reflection Laue method the film is placed 
between the crystal and the x-ray source, the incident beam passing through 
a hole in the film, and the beams diffracted in a backward direction are 
recorded. 

In either method, the diffracted beams form an array of spots on the 
film as shown in Fig. 3-6. This array of spots is commonly called a pat- 
tern, but the term is not used in any strict sense and does not imply any 
periodic arrangement of the spots. On the contrary, the spots are seen 
to lie on certain curves, as shown by the lines drawn on the photographs. 




(a) 



FIG. <H*. (a) Transmission and (b) back-reflection Laue patterns of an alumi- 
num crystal (cubic). Tungsten radiation, 30 kv, 19 ma. 



3-6] 



DIFFRACTION METHODS 



91 



Z.A. 




(b) 

FIG. 3-7. Location of Laue spots (a) on ellipses in transmission method and (b) 
on hyperbolas in back-reflection method. (C = crystal, F film, Z.A. = zone 
axis.) 

These curves are generally ellipses or hyperbolas for transmission patterns 
[Fig. 3-6(a)] and hyperbolas for back-reflection patterns [Fig. 3-6(b)]. 

The spots lying on any one curve are reflections from planes belonging 
to one zone. This is due to the fact that the Laue reflections from planes 
of a zone all lie on the surface of an imaginary cone whose axis is the zone 
axis. As shown in Fig. 3-7 (a), one side of the cone is tangent to the trans- 
mitted beam, and the angle of inclination <f> of the zone axis (Z.A.) to the 
transmitted beam is equal to the semi-apex angle of the cone. A film 
placed as shown intersects the cone in an imaginary ellipse passing through 
the center of the film, the diffraction spots from planes of a zone being 
arranged on this ellipse. When the angle <t> exceeds 45, a film placed 
between the crystal and the x-ray source to record the back-reflection pat- 
tern will intersect the cone in a hyperbola, as shown in Fig. 3-7 (b). 



92 



DIFFRACTION i: THE DIRECTIONS OF DIFFRACTED BEAMS [CHAP. 3 



Z.A. 





FIG. 3-8. Stereographic projection 
of transmission Laue method. 



FIG. 3-9. Rotating-crystal method. 



The fact that the Laue reflections from planes of a zone lie on the surface 
of a cone can be nicely demonstrated with the stereographic projection. 
In Fig. 3-8, the crystal is at the center of the reference sphere, the incident 
beam 7 enters at the left, and the transmitted beam T leaves at the right. 
The point representing the zone axis lies on the circumference of the basic 
circle and the poles of five planes belonging to this zone, PI to P 5 , lie on 
the great circle shown. The direction of the beam diffracted by any one 
of these planes, for example the plane P 2 , can be found as follows. 7, P 2 , D 2 
(the diffraction direction required), and T are all coplanar. Therefore 7> 2 
lies on the great circle through 7, P 2 , and T. The angle between 7 and P 2 
is (90 0), and 7) 2 must lie at an equal angular distance on the other 
side of P 2 , as shown. The diffracted beams so found, D\ to Z> 5 , are seen 
to lie on a small circle, the intersection with the reference sphere of a cone 
whose axis is the zone axis. 

The positions of the spots on the film, for both the transmission and the 
back-reflection method, depend on the orientation of the crystal relative 
to the incident beam, and the spots themselves become distorted and 
smeared out if the crystal has been bent or twisted in any way. These 
facts account for the two main uses of the Laue methods: the determina- 
tion of crystal orientation and the assessment of crystal perfection. 

In the rotating-crystal method a single crystal is mounted with one of 
its axes, or some important crystallographic direction, normal to a mono- 
chromatic x-ray beam. A cylindrical film is placed around it and the 
crystal is rotated about the chosen direction, the axis of the film coinciding 
with the axis of rotation of the crystal (Fig. 3-9). As the crystal rotates, 



3-6] DIFFRACTION METHODS 93 




^m^mm 
^'S'lililtt 



FIG. 3-10. Rotating-crystal pattern of a quartz crystal (hexagonal) rotated 
about its c axis. Filtered copper radiation. (The streaks are due to the white radi- 
ation not removed by the filter.) (Courtesy of B. E. Warren.) 

a particular set of lattice planes will, for an instant, make the correct 
Bragg angle for reflection of the monochromatic incident beam, and at 
that instant a reflected beam will be formed. The reflected beams are 
again located on imaginary cones but now the cone axes coincide with the 
rotation axis. The result is that the spots on the film, when the film is 
laid out flat, lie on imaginary horizontal lines, as shown in Fig. 3-10. 
Since the crystal is rotated about only one axis, the Bragg angle does not 
take on all possible values between and 90 for every set of planes. Not 
every set, therefore, is able to produce a diffracted beam ; sets perpendicular 
or almost perpendicular to the rotation axis are obvious examples. 

The chief use of the rotating-crystal method and its variations is in the 
determination of unknown crystal structures, and for this purpose it is 
the most powerful tool the x-ray crystallographer has at his disposal. How- 
ever, the complete determination of complex crystal structures is a subject 
beyond the scope of this book and outside the province of the average 
metallurgist who uses x-ray diffraction as a laboratory tool. For this 
reason the rotating-crystal method will not be described in any further 
detail, except for a brief discussion in Appendix 15. 

In the powder method, the crystal to be examined is reduced to a very 
fine powder and placed in a beam of monochromatic x-rays. Each particle 
of the powder is a tiny crystal oriented at random with respect to the inci- 
dent beam. Just by chance, some of the particles will be correctly oriented 
so that their (100) planes, for example, can reflect the incident beam. 
Other particles will be correctly oriented for (110) reflections, and so on. 
The result is that every set of lattice planes will be capable of reflection. 
The mass of powder is equivalent, in fact, to a single crystal rotated, not 
about one axis, but about all possible axes. 

Consider one particular hkl reflection. One or more particles of powder 
will, by chance, be so oriented that their (hkl) planes make the correct 



94 



DIFFRACTION 1 1 THE DIRECTIONS OF DIFFRACTED BEAMS [CHAP. 3 





(a) 



FIG. 3-11. Formation of a diffracted cone of radiation in the powder method. 

Bragg angle for reflection; Fig. 3-11 (a) shows one plane in this set and 
the diffracted beam formed. If this plane is now rotated about the incident 
beam as axis in such a way that 6 is kept constant, then the reflected beam 
will travel over the surface of a cone as shown in Fig. 3-1 l(b), the axis of 
the cone coinciding with the transmitted beam. This rotation does not 
actually occur in the powder method, but the presence of a large number 
of crystal particles having all possible orientations is equivalent to this 
rotation, since among these particles there will be a certain fraction whose 
(hkl) planes make the right Bragg angle with the incident beam and which 
at the same time lie in all possible rotational positions about the axis of 
the incident beam. The hkl reflection from a stationary mass of powder 
thus has the form of a cone of diffracted radiation, and a separate cone is 
formed for each set of differently spaced lattice planes. 

Figure 3-12 shows four such cones and also illustrates the most common 
powder-diffraction method. In this, the Debye-Scherrer method, a narrow 
strip of film is curved into a short cylinder with the specimen placed op 
its axis and the incident beam directed at right angles to this axis. The 
cones of diffracted radiation intersect the cylindrical strip of film in lines 
and, when the strip is unrolled and laid out flat, the resulting pattern has 
the appearance of the one illustrated in Fig. 3-12(b). Actual patterns, 
produced by various metal powders, are shown in Fig. 3-13. Each diffrac- 
tion line is made up of a large number of small spots, each from a separate 
crystal particle, the spots lying so close together that they appear as a 
continuous line. The lines are generally curved, unless they occur exactly 
at 26 == 90 when they will be straight. From the measured position of a 
given diffraction line on the film, 6 can be determined, and, knowing X, we 
can calculate the spacing d of the reflecting lattice planes which produced 
the line. > 

Conversely, if the shape and size of the unit cell of the crystal are known, 
we can predict the position of all possible diffraction lines on the film. The 
line of lowest 28 value is produced by reflection from planes of the greatest 



3-6] 



DIFFRACTION METHODS 



95 




point where 
incident beam 
enters (26 = 180) -/ 



(a) 



\ 



26 = 







1 t "] 

o 

I 


1 




b ) 





(b) 

FIG. 3-12. Debye-Scherrer powder method: (a) relation of film to specimen and 
incident beam; (b) appearance of film when laid out flat. 



26 = 180 



26 = 




ii 



(a) 





FIG. 3-13. Debye-Scherrer powder patterns of (a) copper (FCC), (b) tungsten 
(BCC), and (c) zinc (HCP). Filtered copper radiation, camera diameter * 5.73 



cm. 



96 DIFFRACTION i: THE DIRECTIONS OF DIFFRACTED BEAMS [CHAP. 3 

spacing. In the cubic system, for example, d is a maximum when 
(h 2 + k 2 + I 2 ) is a minimum, and the minimum v#lue of this term is 1, 
corresponding to (hkl) equal to (100). The 100 reflection is accordingly 
the one of lowest 20 value. The next reflection will have indices hkl corre- 
sponding to the next highest value of (h 2 + k 2 + / 2 ), namely 2, in which 
case (hkl) equals (110), and so on. 

The Debye-Scherrer and other variations of the powder method are very 
widely used, especially in metallurgy. The powder method is, of course, 
the only method that can be employed when a single crystal specimen is 
not available, and this is the case more often than not in metallurgical 
work. The method is especially suited for determining lattice parameters 
with high precision and for the identification of phases, whetrier they occur 
alone or in mixtures such as polyphase alloys, corrosion products, refrac- 
tories, and rocks. These and other uses of the powder method will be fully 
described in later chapters. 

Finally, the x-ray spectrometer can be used as a tool in diffraction anal- 
ysis. This instrument is known as a diffractometer when it is used with 
x-rays of known wavelength to determine the unknown spacing of crystal 
planes, and as a spectrometer in the reverse case, when crystal planes of 
known spacing are used to determine unknown wavelengths. The diffrac- 
tometer is always used with monochromatic radiation and measurements 
may be made on either single crystals or polycry stalline specimens ; in the 
latter case, it functions much like a Debye-Scherrer camera in that the 
counter intercepts and measures only a short arc of any one cone of dif- 
fracted rays. 

3-7 Diffraction under nonideal conditions. Before going any further, 
it is important to stop and consider with some care the derivation of the 
Bragg law given in Sec. 3-2 in order to understand precisely under what 
conditions it is strictly valid. In our derivation we assumed certain ideal 
conditions, namely a perfect crystal and an incident beam composed of 
perfectly parallel and strictly monochromatic radiation. These conditions 
never actually exist, so we must determine the effect on diffraction of vari- 
ous kinds of departure from the ideal. 

In particular, the way in which destructive interference is produced in 
all directions except those of the diffracted beams is worth considering in 
some detail, both because it is fundamental to the theory of diffraction 
and because it will lead us to a method for estimating the size of very small 
crystals. We will find that only the infinite crystal is really perfect and 
that small size alone, of an otherwise perfect crystal, can be considered a 
crystal imperfection. 

The condition for reinforcement used in Sec. 3-2 is that the waves in- 
volved must differ in path length, that is, in phase, by exactly an integral 



3-7J 



DIFFRACTION UNDER NONIDEAL CONDITIONS 



97 



number of wavelengths. But suppose that the angle 9 in Fig. 3-2 is such 
that the path difference for rays scattered by the first and second planes 
is only a quarter wavelength. These rays do not annul one another but, 
as we saw in Fig. 3-1, simply unite to form a beam of smaller amplitude 
than that formed by two rays which are completely in phase. How then 
does destructive interference take place? The answer lies in the contribu- 
tions from planes deeper in the crystal. Under the assumed conditions, 
the rays scattered by the second and third planes would also be a quarter 
wavelength out of phase. But this means that the rays scattered by the 
first and third planes are exactly half a wavelength out of phase and would 
completely cancel one another. Similarly, the rays from the second and 
fourth planes, third and fifth planes, etc., throughout the crystal, are com- 
pletely out of phase; the result is destructive interference and no diffracted 
beam. Destructive interference is therefore just as much a consequence 
of the periodicity of atom arrangement as is constructive interference. 

This is an extreme example. If the path difference between rays scat- 
tered by the first two planes differs only slightly from an integral number 
of wavelengths, then the plane scattering a ray exactly out of phase with 
the ray from the first plane will lie deep within the crystal. If the crystal 
is so small that this plane does not exist, then complete cancellation of all 
the scattered rays will not result. It follows that there is a connection 
between the amount of "out-of-phaseness" that can be tolerated and the 
size of the crystal. 

Suppose, for example, that the crystal has a thickness t measured in a 
direction perpendicular to a particular set of reflecting planes (Fig. 3-14). 
Let there be (m + 1) planes in this set. We will regard the Bragg angle 6 
as a variable and call OB the angle 
which exactly satisfies the Bragg law 
for the particular values of X and d 
involved, or 

X = 2d sin 6 B . 

In Fig. 3-14, rays A, D, . . . , M make 
exactly this angle OB with the re- 
flecting planes. Ray D', scattered by 
the first plane below the surface, is 
therefore one wavelength out of phase 
with A'; and ray M', scattered by the 
mth plane below the surface, is m 
wavelengths out of phase with A'. 
Therefore, at a diffraction angle 20#, 
rays A', D', . . . , M' are completely 
in phase and unite to form a diffracted 





FIG. 3-14. 
diffraction. 



Effect of crystal size on 



98 



DIFFRACTION i: THE DIRECTIONS OF DIFFRACTED BEAMS [CHAP. 3 



beam of maximum amplitude, i.e., a beam of maximum intensity, since the 
intensity is proportional to the square of the amplitude. 

When we consider incident rays that make Bragg angles only slightly 
different from 0#, we find that destructive interference is not complete. 
Ray B, for example, makes a slightly larger angle 0i, such that ray L' 
from the mth plane below the surface is (m + 1) wavelengths out of ph6.se 
with B', the ray from the surface plane. This means that midway in the 
crystal there is a plane scattering a ray which is one-half (actually, an 
integer plus one-half) wavelength out of phase with ray B' from the surface 
plane. These rays cancel one another, and so do the other rays from sim- 
ilar pairs of planes throughout the crystal, the net effect being that rays 
scattered by the top half of the crystal annul those scattered by the bottom 
half. The intensity of the beam diffracted at an angle 20i is therefore zero. 
It is also zero at an angle 20 2 where 2 is such that ray N' from the mth 
plane below the surface is (m 1) wavelengths out of phase with ray C' 
from the surface plane. It follows that the diffracted intensity at angles 
near 2fe, but not greater than 26 1 or less than 20 2 , is not zero but has a 
value intermediate between zero and the maximum intensity of the beam 
diffracted at an angle 20s- The curve of diffracted intensity vs. 28 will 
thus have the form of Fig. 3-15(a) in contrast to Fig. 3-15(b), which illus- 
trates the hypothetical case of diffraction occurring only at the exact Bragg 
angle. 

The width of the diffraction curve of Fig. 3-1 5 (a) increases as the thick- 
ness of the crystal decreases. The width B is usually measured, in radians, 
at an intensity equal to half the maximum intensity. As a rough measure 




20 2 



20i 



20 



20* 
20- 



(a) (b) 

FIG. 3-15. Effect of fine particle size on diffraction curves (schematic). 



3-7] DIFFRACTION UNDER NONIDEAL CONDITIONS 99 

of J5, we can take half the difference between the two extreme angles at 
which the intensity is zero, or 

B = f (20i - 20 2 ) = 0i - 2 . 
The path-difference equations for these two angles are 

2t sin 2 = (m - 1)X. 
By subtraction we find 

(sin 0i sin 2 ) = X, 

(/> i n \ //) /) \ 
CM ~"T~ f 2 \ i ^1 ^2 \ 
1 sin I ) = X. 
2 / \ 2 / 

But 0i and 2 are both very nearly equal to 0#, so that 

0i + 02 = 200 (approx.) 
and 

sin f ^J = f j (approx.). 

Therefore 

2t[ -) cos B = X, 



t = (3-12) 

JS cos SB 

A more exact treatment of the problem gives 

, . _*_. (3-13) 

B cos B R 

which is known as the Scherrer formula. It is used to estimate the particle 
size of very small crystals from the measured width of their diffraction 
curves. What is the order of magnitude of this effect? Suppose X = 1.5A, 
d = LOA, and = 49. Then for a crystal 1 mm in diameter the breadth 
J5, due to the small crystal effect alone, would be about 2 X 10~ 7 radian 
(0.04 sec), or too small to be observable. Such a crystal would contain 
some 10 7 parallel lattice planes of the spacing assumed above. However, 
if the crystal were only 500A thick, it would contain only 500 planes, and 
the diffraction curve would be relatively broad, namely about 4 X 10~~ 3 
radian (0.2). 

Nonparallel incident rays, such as B and C in Fig. 3-14, actually exist 
in any real diffraction experiment, since the "perfectly parallel beam" 



100 DIFFRACTION i: THE DIRECTIONS OF DIFFRACTED BEAMS [CHAP. 3 

assumed in Fig. 3-2 has never been produced in the laboratory. As will 
be shown in Sec. 5-4, any actual beam of x-rays contains divergent and 
convergent rays as well as parallel rays, so that the phenomenon of dif- 
fraction at angles not exactly satisfying the Bragg law actually takes 
place. 

Neither is any real beam ever strictly monochromatic. The usual 
"monochromatic" beam is simply one containing the strong Ka component 
superimposed on the continuous spectrum. But the Ka line itself has a 
width of about 0.001 A and this narrow range of wavelengths in the nom- 
inally monochromatic beam is a further cause of line broadening, i.e., of 
measurable diffraction at angles close, but not equal, to 20#, since for each 
value of A there is a corresponding value of 8. (Translated into terms of 
diffraction line width, a range of wavelengths extending over 0.001 A leads 
to an increase in line width, for X = 1.5A and 8 = 45, of about 0.08 
over the width one would expect if the Incident beam were strictly mono- 
chromatic.) Line broadening due to this natural "spectral width" is 
proportional to tan 8 and becomes quite noticeable as 8 approaches 90. 

Finally, there is a kind of crystal 
imperfection known as mosaic struc- 
ture which is possessed by all real 
crystals to a greater or lesser degree 
and which has a decided effect on 
diffraction phenomena. It is a kind 
of substructure into which a "single" 
crystal is broken up and is illustrated 
in Fig. 3-16 in an enormously ex- 
aggerated fashion. A crystal with 
mosaic structure does not have its 
atoms arranged on a perfectly regular 
lattice extending from one side of the 
crystal to the other; instead, the lattice is broken up into a number of tiny 
blocks, each slightly disoriented one from another. The size of these blocks 
is of the order of 1000A, while the maximum angle of disorientation be- 
tween them may vary from a very small value to as much as one degree, 
depending on the crystal. If this angle is , then diffraction of ^a parallel 
monochromatic beam from a "single" crystal will occur not only at an 
angle of incidence 0# but at all angles between 8s and OR + c. Another 
effect of mosaic structure is to increase the intensity of the reflected beam 
relative to that theoretically calculated for an ideally perfect crystal. 

These, then, are some examples of diffraction under nonideal conditions, 
that is, of diffraction as it actually occurs. We should not regard these as 
"deviations" from the Bragg law, and we will not as long as we remember 
that this law is derived for certain ideal conditions and that diffraction is 




FIG. 3-K). The mosaic structure of 
a real crystal. 



3-7] 



DIFFRACTION UNDER NONIDEAL CONDITIONS 



101 




(a) 






(1)) 

FIG. 3-17. (a) Scattering by 
atom, (b) Diffraction by a crystal. 



crystal 





liquid or amorphous solid 




90 180 

DIFFRAC TION (SCATTERING) 
ANGLE 28 (degrees) 

FIG. 3-18. Comparative x-ray scat- 
tering by crystalline solids, amorphous 
solids, liquids, and monatomic gases 
(schematic). 



only a special kind of scattering. This latter point cannot be too strongly 
emphasized. A single atom scatters an incident beam of x-rays in all 
directions in space, but a large number of atoms arranged in a perfectly 
periodic array in three dimensions to form a crystal scatters (diffracts) 
x-rays in relatively few directions, as illustrated schematically in Fig. 3-17. 
It does so precisely because the periodic arrangement of atoms causes 
destructive interference of the scattered rays in all directions except those 
predicted by the Bragg law, and in these directions constructive inter- 
ference (reinforcement) occurs. It is not surprising, therefore, that meas- 
urable diffraction (scattering) occurs at non-Bragg angles whenever any 
crystal imperfection results in the partial absence of one or more of the 
necessary conditions for perfect destructive interference at these angles. 



102 DIFFRACTION i: THE DIRECTIONS OF DIFFRACTED BEAMS [CHAP. 3 

These imperfections are generally slight compared to the over-all regularity 
of the lattice, with the result that diffracted beams are confined to very 
narrow angular ranges centered on the angles predicted by the Bragg law 
for ideal conditions. 

This relation between destructive interference and structural periodicity 
can be further illustrated by a comparison of x-ray scattering by solids, 
liquids, and gases (Fig. 3-18). The curve of scattered intensity vs. 26 for a 
crystalline solid is almost zero everywhere except at certain angles where 
high sharp maxima occur: these are the diffracted beams. Both amorphous 
solids and liquids have structures characterized by an almost complete 
lack of periodicity and a tendency to "order" only in the sense that the 
atoms are fairly tightly packed together and show a statistical preference 
for a particular interatomic distance; the result is an x-ray scattering curve 
showing nothing more than one or two broad maxima. Finally, there are 
the monatomic gases, which have no structural periodicity whatever; in 
such gases, the atoms are arranged perfectly at random and their relative 
positions change constantly with time. The corresponding scattering 
curve shows no maxima, merely a regular decrease of intensity with in- 
crease in scattering angle. 

PROBLEMS 

3-1. Calculate the "x-ray density" [the density given by Eq. (3-9)] of copper 
to four significant figures. 

3-2. A transmission Laue pattern is made of a cubic crystal having a lattice 
parameter of 4.00A. The x-ray beam is horizontal. _ The [OlO] axis of the crystal 
points along the beam towards the x-ray tube, the [100] axis points vertically up- 
ward, and the [001] axis is horizontal and parallel to the photographic film. The 
film is 5.00 cm from the crystal. 

(a) What is the wavelength of the radiation diffracted from the (3TO) planes? 
(6) Where will the 310 reflection strike the film? 

3-3. A back-reflection Laue pattern is made of a cubic crystal in the orientation 
of Prob. 3-2. By means of a stereographic projection similar to Fig. 3-8, show that 
the beams diffracted by the planes (120), (T23), and (121), all of which belong to 
the zone [210], lie on the surface of a cone whose axis is the zone axis. What is 
the angle <f> between the zone axis and the transmitted beam? 

3-4. Determine the values of 20 and (hkl) for the first three lines (those of low- 
est 26 values) on the powder patterns of substances with the following structures, 
the incident radiation being Cu Ka: 

(a) Simple cubic (a = 3.00A) 

(6) Simple tetragonal (a = 2.00A, c = 3.00A) 

(c) Simple tetragonal (a == 3.00A, c = 2.00A) 

(d) Simple rhombohedral (a = 3.00A, a = 80) 



PROBLEMS 103 

3-6. Calculate the breadth B (in degrees of 26), due to the small crystal effect 
alone, of the powder pattern lines of particles of diameter 1000, 750, 500, and 250A. 
Assume 6 = 45 and X = 1.5A. For particles 250A in diameter, calculate the 
breadth B for = 10, 45, and 80. 

3-6. Check the value given in Sec. 3-7 for the increase in breadth of a diffrac- 
tion line due to the natural width of the Ka emission line. (Hint: Differentiate 
the Bragg law and find an expression for the rate of change of 26 with X.) 



CHAPTER 4 
DIFFRACTION II: THE INTENSITIES OF DIFFRACTED BEAMS 

4-1 Introduction. As stated earlier, ^.he positions of the atoms in the 
unit cell affect the intensities but not the directions of the diffracted beams. 
That this must be so may be seen by considering the two structures shown 
in Fig. 4-1. Both are orthorhombic with two atoms of the same kind per 
unit cell, but the one on the left is base-centered and the one on the right 
body-centered. Either is derivable from the other by a simple shift of 
ope atom by the vector ^c. 

/ Consider reflections from the (001) planes which are shown in profile in 
Ftg. 4-2. For the base-centered lattice shown in (a), suppose that the 
Bragg law is satisfied for the particular values of X and 6 employed. This 
means that the path difference ABC between rays 1' and 2' is one wave- 
length, so that rays 1' and 2' are in phase and diffraction occurs in the 
direction shown. Similarly, in the body-centered lattice shown in (b), 
rays 1' and 2' are in phase, since their path difference ABC is one wave- 
length. However, in this case, there is another plane of atoms midway 
between the (001) planes, and the path difference DEF between rays 1' 
and 3' is exactly half of ABC, or one half wavelength. Thus rays 1' and 
3' are completely out of phase and annul each other. Similarly, ray 4' 
from the next plane down (not shown) annuls ray 2', and so on throughout 
the crystal. There is no 001 reflection from the body-centered latticeTJ 

This example shows how a simple rearrangement of atoms within the 
unit cell can eliminate a reflection completely. More generally, the in- 
tensity of a diffracted beam is changed, not necessarily to zero, by any 
change in atomic positions, and, conversely, we can only determine atomic 
positions by observations of diffracted intensities. To establish an exact 
relation between atom position and intensity is the main purpose of this 
chapter. The problem is complex because of the many variables involved, 
and we will have to proceed step by step : we will consider how x-rays are 
scattered first by a single electron, then by an atom, and finally by all the 





,$ (a) (b) 

FIG. 4-1. (a) Base-centered and (b) body-centered orthorhombic unit cells. 

104 



4-2] 




SCATTERING BY AN ELECTRON 

r i 

3 



105 




(a) 



(b) 



FIG. 4-2. Diffraction from the (001) planes of (a) base-centered and (b) body- 
centered orthorhombir lattices. 

atoms in the unit cell. We will apply these results to the powder method 
of x-ray diffraction only, and, to obtain an expression for the intensity of a 
powder pattern line, we will have to consider a number of other factors 
which affect the way in which a crystalline powder diffracts x-rays. 

4-2 Scattering by an electron. We have seen in Chap. 1 that aq| x-ray 
beam is an electromagnetic wave characterized by an electric field whose 
strength varies sinusoidally with time at any one point in the beam., Sipce 
anVlectric field exerts a force on a Charged particle such as an electron^lhe 
oscillating electric field of an x-ray beam will set any electron it encounters 
into oscillatory motion about its mean position.} 

Wow an accelerating or decelerating electron emits an electromagnetic 
wave. We have already seen an example of this phenoinejionjn the x-ray 
tube, where x-rays are emitted because of the rapid deceleration of the 
electrons striking the target. Similarly, an electron which has been set 
into oscillation by an x-ray beam is continuously accelerating and de- 
celerating during its motion and therefore emits an electromagnetic, .wjave. 
In this sense, an electron is said to scatter x-rays, the scattered beam being 
simply ITie beam radiated by the electron under the action of the incident 
beam. The scattered beam has the same wavelength and frequency as 
the incident beam and is said to be coherent with it, since there is a definite 
relationship T>etwee7fT1ie "phase of lite scattereHbeam anJTEat of the inci- 
denFfieam which produced it. \ """' 

Although x-rays are scattered in all directions by an electron, the in- 
tensity of the scattered beam depends on the angle of scattering, in a way 
which was first worked out by J. J. Thomson. He found that the intensity 
/ of the beam scattered by a single electron of charge e and mass m, at a 
^stance r from the electron, is given by 



sin 2 a, 



(4-1) 



106 DIFFRACTION II : THE INTENSITIES OF DIFFRACTED BEAMS [CHAP. 4 

where /o = intensity of the incident beam, c = velocity of light, and 
a = angle between the scattering direction and the direction of accelera- 
tion of the electron. Suppose the incident beam is traveling in the direc- 
tion Ox (Fig. 4-3) and encounters an electron at 0. We wish to know the 
scattered intensity at P in the xz plane where OP is inclined at a scattering 
angle of 26 to the incident beam. An unpolarized incident beam, such as 
that issuing from an x-ray tube, has its electric vector E in a random 
direction in the yz plane. This beam may be resolved into two plane- 
polarized components, having electric vectors E y and E 2 where 



On the average, E y will be equal to E, since the direction of E is perfectly 
random. Therefore 

E, 2 = E z 2 = E 2 . 

The intensity of these two components of the incident beam is proportional 
to the square of their electric vectors, since E measures the amplitude of 
the wave and the intensity of a wave is proportional to the square of its 
amplitude. Therefore 

IQ V = IQ Z = 2^0- 

The y component of the incident beam accelerates the electron in the 
direction Oy. It therefore gives rise to a scattered beam whose intensity 
at P is found from Eq. (4-1) to be 



r 2 ra 2 c 4 



since a = ^yOP = w/2. Similarly, the intensity of the scattered z com- 
ponent is given by 



since a = r/2 20. The total scattered intensity at P is obtained by 
summing the intensities of these two scattered components: 

IP = Ip v + Ip z 

e 4 
= -r-r-r (7o + hz cos 2 20) 



r'm'c' 



e 4 //o /o 2o \ 
= ( ~ -^ cos 2 2^ ) 

r 2 m 2 c 4 \2 2 / 

^V 



+ cos 2 



4-2] 



SCATTERING BY AN ELECTRON 



107 




\ 



before impact 




FIG. 4-3. Coherent scattering of x- 
rays by a single electron. 



after impart 

FIG. 4-4. Elastic collision of photon 
and electron (Compton effect). 



This is the Thomson equation for the scattering of an x-ray beam by a 
single electron. If the values of the constants e, r, m, and c are inserted 
into this equation, it will be found that the intensity of the scattered beam 
is only a minute fraction of the intensity of the incident beam. The equa- 
tion also shows that the scattered intensity decreases as the inverse square 
of the distance from the scattering atom, as one \vould expect, and that 
the scattered beam is stronger in forward or backward directions than in a 
direction at right angles to the incident beam. 

The Thomson equation gives the absolute intensity (in ergs/sq cm/sec) 
of the scattered beam in terms of the absolute intensity of the incident 
beam. These absolute intensities are both difficult to measure and difficult 
to calculate, so it is fortunate that relative values are sufficient for our 
purposes in practically all diffraction problems. In most cases, all factors 
in Eq. (4-2) except the last are constant during the experiment and can 
be omitted.* This last factor, ^(1 + cos 2 26), is called the polamation 
factor; this is a rather unfortunate term because, as we have just seen, this 
factor enters the equation simply because the incident beam is unpolarized. 
The polarization factor is common to all intensity calculations, and we 
will use it later in our equation for the intensity of a beam diffracted by a 
crystalline powder. 

There is another and quite different way in which an electron can scatter 

x-rays, and that is manifested in the Compton effect. This effect, discovered 

by A. H. Compton in 1923, occurs whenever x-rays encounter loosely 

bound or free electrons and can be best understood by considering the 

incident beam, not as a wave motion, but as a stream of x-ray quanta or 

photons, each of energy hvi. When such a photon strikes a loosely bound 

electron, the collision is an elastic one like that of two billiard balls (Fig. 

\ The electron is knocked aside and the photon is deviated through 

Jigle 26. Since some of the energy of the incident photon is used in 

/iding kinetic energy for the electron, the energy hv 2 of the photon 



108 DIFFRACTION II! THE INTENSITIES OF DIFFRACTED BEAMS [CHAP. 4 

after impact is less than its energy hv\ before impact. The wavelength 
X 2 of the scattered radiation is thus slightly greater than the wavelength 
Xi of the incident beam, the magnitude of the change being given by the 
equation 



The increase in wavelength depends only on the scattering angle, and it 
varies from zero in the forward direction (26 = 0) to 0.05A in the extreme 
backward direction (20 = 180). 

Radiation so scattered is called Compton modified radiation, and, be- 
sides having its wavelength increased, it has the important characteristic 
that its phase has no fixed relation to the phase of the incident beam. For 
this reason it is also known as incoherent radiation. It cannot take part 
in diffraction because its phase is only randomly related to that of the inci- 
dent beam and cannot therefore produce any interference effects. Comp- 
ton modified scattering cannot be prevented, however, and it has the 
undesirable effect of darkening the background of diffraction patterns. 

[It should be noted that the quantum theory can account for both the 
coherent and the incoherent scattering, whereas the wave theory is only 
applicable to the former. In terms of the quantum theory, coherent scat- 
tering occurs when an incident photon bounces off an electron which is so 
tightly bound that it receives no momentum from the impact, The scat- 
tered photon therefore has the same energy, and hence wavelength, as it 
had before 

4-3 Scattering by an atom. 1 When an x-ray beam encounters an atom, 
each electron in it scatters part of the radiation coherently in accordance 
with the Thomson equation. One might also expect the nucleus to take 
part in the coherent scattering, since it also bears a charge and should be 
capable of oscillating under the influence of the incident beam,} However, 
the nucleus has an extremely large mass relative to that of tne electron 
and cannot be made to oscillate to any appreciable extent; in fact, the 
Thomson equation shows that the intensity of coherent scattering is in- 
versely proportional to the square of the mass of the scattering particle. 
The net effect is that coherent scattering by an atom is due only to the 
electrons contained in that atom. 

The following question then arises: is the wave scattered by an atom 
simply the sum of the waves scattered by its component electrons? More 
precisely, does an atom of atomic number Z, i.e., an atom containing Z 
electrons, scatter a wave whose amplitude is Z times the amplitude of 
the wave scattered by a single electron? The answer is yes, if the scatter- 
ing is in the forward direction (20 = 0), because the waves scattered 1 " by 
all the electrons of the atom are then in phase and the amplitudes o f all 
the scattered waves can be added directly. 



4-3] SCATTERING BY AN ATOM It9 

This is not true for other directions of scattering. iThe fact that the 
electrons of an atom are situated at different points in space introduces 
differences in phase between the waves scattered by different electrons:^ 
Consider Fig. 4-5, in which, for simplicity, the electrons are shown as 
points arranged around the central nucleus. The waves scattered in the 
forward direction by electrons A and_J^are exactly* in phase on_a_3Kave 
front such as XX', because each wave has traveled the same distance 
before and after scattering. The other scattered waves shown in' the 'fig- 
ure, however, have a path difference equal to (CB AD) and are thus 
somewhat out of phase along a wave front such as YY', the path differ- 
ence being less than one wavelength. Partial interference occurs between 
the waves scattered by A and 5, with the result that the net amplitude of 
the wave scattered in this direction is less than that of the wave scattered 
by the same electrons in the forward direction. 

I A quantity /, the atomic scattering factor, is used to describe the "effi- 
ciency" of scattering of a given atom in a given direction. It is defined 
as a ratio of amplitudes : 



/ = 



amplitude of the wave scattered by an atom 
amplitude of the wave scattered by one electron f 



From what has been* said already, lit is clear that / = Z f or any atom 
scattering in the forward direction^ As increases, however, the waves 
scattered by individual electrons become more and more out of phase and 
/ decreases. The atomic scattering factor also depends on the wavelength 
of the incident beam : at a fixed value of 0, f will be smaller the shorter the 




X' 



FIG, 4-5. X-ray scattering by an atom. 




FIG. 4-6. The atomic scattering fac- 
tor of copper. 



110 DIFFRACTION II : THE INTENSITIES OF DIFFRACTED BEAMS [CHAP. 4 

wavelength, since the path differ- 
ences will be larger relative to the 
wavelength, leading to greater in- 
terference between the scattered 
beams. The actual calculation of / 
involves sin 6 rather than 6, so that 
the net effect is that / decreases as 
the quantity (sin 0)/X increases! 

Calculated values of / for various 
atoms and various values of (sin 0)/X 
are tabulated in Appendix 8, and a 
curve showing the typical variation 
of/, in this case for copper, is given 
in Fig. 4-6. Note again that the 
curve begins at the atomic number 
of copper, 29, and decreases to very 
low values for scattering in the back- 
ward direction (0 near 90) or for 
very short wavelengths. Since the intensity of a wave is proportional to 
the square of its amplitude, a curve of scattered intensity fit)m an atom 
can be obtained simply by squaring the ordinates of a curve such a& Fig. 
4-6. (The resulting curve closely approximates the observed scattered in- 
tensity per atom of a monatomic gas, as shown in Fig. 3-18.) 

The scattering just discussed, whose amplitude is expressed in terms of 
the atomic scattering factor, is coherent, or unmodified, scattering, which 
is the only kind capable of being diffracted. On the other hand, incoherent, 
or Compton modified, scattering is occurring at the same time. Since the 
latter is due to collisions of quanta with loosely bound electrons, its in- 
tensity relative to that of the unmodified radiation increases as the pro- 
portion of loosely bound electrons increases. The intensity of Compton 
modified radiation thus increases as the atomic number Z decreases. It 
is for this reason that it is difficult to obtain good diffraction photographs 
of organic materials, which contain light elements such as carbon, oxygen, 
and hydrogen, since the strong Compton modified scattering from these 
substances darkens the background of the photograph and makes it diffi- 
cult to see the diffraction lines formed by the unmodified radiation. It is 
also found that the intensity of the modified radiation increases as the 
quantity (sin 0)/X increases. The intensities of modified scattering and of 
unmodified scattering therefore vary in opposite ways with Z and with 
(sin0)/X. i 

To summarize,|when a monochromatic beam of x-rays strikes an atom, 
two scattering processes occur 4 Tightly bound electrons are jet, into pscTP" 
lation and radiate x-rays of the saiffi wavelength as that of the incident 



SCATTERING BY A UNIT CELL 

incident beam 



absorbing substance 




fluorescent x-rays 



unmodified 
(coherent) 



Compton modified 
(incoherent) 



Compton recoil 
electrons 



photoelectrons 



FIG. 4-7. Effects produced by the passage of x-rays through matter. (After 
N. F. M. Henry, H. Lipson, and W. A. Wooster, The Interpretation of X-Ray Dif- 
fraction Photographs, Macmillan, London, 1951.) 

beam. More loosely bound electrons scatter part of the incident beam 
and slightly increase its wavelength in the process, the exact amount of 
increase depending on the scattering angle. The former is called coherent 
or unmodified scattering and the latter incoherent or modified ; both kinds 
occur simultaneously and in all directions. If the atom is a part of a large 
group of atoms arranged in spaceTh a Tegular periodic fashion as in a crys- 
tal, then another phenomenon occurs. The coherently scattered radiation 
from all the atoms undergoes reinforcement in certain directions and can- 
cellation in other directions, thus producing diffracted beams. Djttjw^p^ 
is, essentially, reinforced coherent scattering. I ^1 

We are now in a position to summarize, from the preceding sections and 
from Chap. 1, the chief effects associated with the passage of x-rays through 
matter. This is done schematically in Fig. 4-7. The incident x-rays are 
assumed to be of high enough energy, i.e., of short enough wavelength, 
to cause the emission of photoelectrons and characteristic fluorescent radia- 
tion. The Compton recoil electrons shown in the diagram are the loosely 
bound electrons knocked out of the atom by x-ray quanta, the interaction 
giving rise to Compton modified radiation. 

Scattering by a unit cell. To arrive at an expression for the in- 
tensity of a diffracted beam, we must now restrict ourselves to a considera- 
tion of the coherent scattering, not from an isolated atom, but from all 
the atoms making up the crystal. The mere fact that the atoms are Ar- 
ranged in a periodic fashion in space mftans that the scattered radiation is 
nowjeverely limited~to certain definite directions and is now referred to 
as a set of diffracted beams. 'The directions of these beams are fixed by 



112 DIFFRACTION II : THE INTENSITIES OF DIFFRA' 

2' 

-(MO) 



p. 4 



o 




FIG. 4-8. The effect of atom position on the phase difference between diffracted 
rays. 

the Bragg law, Avhich is, in a sense, a negative law. If the Bragg law is not 
satisfied, no. diffracted beam can occur; however, the Bragg law may be 
satisfied for a certain set of atomic planes and yet no diffraction may occur, 
as in the example given at the beginning of this chapter, because of a 
particular arrangement of atoms within the unit cell [Fig. 4-2(b)]. 

Vssuming that the Bragg law is satisfied, we wish to find the intensity 
oMhhe frftftm diffracted by " . fgrgjgjjis fl fijnrtinn nf fl.tnrjijvisit.inn Since 

the crystal is merely a repetition of the fundamental unit cell, it is enough 
to consider the way in which the arrangement of atoms within a single 
unit cell affects the diffracted intensity.\ 

Qualitatively, the effect is similar to*the scattering from ar^ atom, dis- 
cussed in the previous section. [There we found that phase differences 
occur in the waves scattered by thejndividual plentrnns j for any direction 
of scattering except the.extreme forward direction. Similarly, the waves 
scattered by the individual atoms of a unit cell are not necessarily in phase 
except in the forward direction,! and we must now determine how the 
phase difference depends on the arrangement of the atoms. 

|This problem is most simply approached by finding the phase difference 
between waves scattered by an atom at the origin and another atom whose 
position is variable in the x direction only. \ For convenience. consklex*an 
orjJvjgoriaJunit cell, a section of which is shown in Fig. 4-8. Taice.aiDm 
^as the origm^and let diffraction occur from the (AOO) planes shown as 
heavy hnftsJiTthe drawings This means that the Bragg law is satisfied for 
this reflection and that 5 2 'iV$he path difference between ray 2' and ray 

^ t I . _ ^^ ' ... . . .. f. ( ) _*. I IM.I ........ |/ 

5 2 'i' = MCN = 2rf/, 00 sin = X. 



4-4] SCATTERING BY A UNIT CELL 113 

From the definition of Miller indices, 

a 
= AC = - 

n 



How is this reflection affected by x-rays scattered in the same direction 
by atom B, located at a distance x from Al Note that only this direction 
need be considered since only in this direction is the Bragg law satisfied 
for the AGO reflection. Clearly, the path difference between ra%._ 3' and. 
ray 1', 6 3 'i>, will be less than X; by simple proportion it is found to be 



(X) = (X). 
AC ... _ a/ft 



Phase differences may be expressed in angular measure as well as in 
wavelength: two rays, differing in path length by one whole wavelength, 
are said to differ in phase by 360, or 2?r radians. If the path difference is 
6, then the 'phase difference jjn_ 



= - (27T). . 

The use of angular measure is convenient because it makes the expression 
of phase differences independent of wavelength, whereas the use of a path 
difference to describe a phase difference is meaningless unless the wave- 
length is specified. 

The phase difference, then, between the wave scattered by atom B and 
that scattered by atom A at the origin is given by 

5vi' 2irhx ^ 



If the position of atom B is specified by its fractional coordinate u = - , 
then the phase difference becomes 



This reasoning may be extended to three dimensions, as in Fig. 4-9, in 

xyz 

which atom B has actual coordinates x y z or fractional coordinates - - - 

a o c 

equal to u v w, respectively. We then arrive at the following important 
relation for the phase difference between the wave scattered by atom B 
and that scattered by atom A at the origin, for the hkl reflection: 

faL^bJm). (4-4) 



This relation is general and applicable to a unit cell of any shape. 



114 DIFFRACTION II : THE INTENSITIES OF DIFFRACTED BEAMS [CHAP. 4 




FIG. 4-9. The three-dimensional analogue of Fig. 4-8. 

These two waves may differ, not only in phase, jbut^also in amplitude if 
atom B and the atonTstr-trre ongih"^l^^d^fferent kinds. In that case, 
v .ie amplitudes of these waves are given, relative to the amplitude of the 
wave scattered by a single electron, by the appropriate values of /, the 
atomic scattering factor. 

We now see that the problem of scattering from a unit cell resolves itself 
into one of adding waves of different phase and amplitude in order to find 
the resultant wave. Waves scattered by all the atoms of the unit cell, 
including the one at the origin, must be added. The most convenient way 
of carrying out this summation is by expressing each wave as & complex 
exponential function. 



+E 




FIG. 4-10. The addition of sine waves of different phase and amplitude. 



4-4] 



SCATTERING BY A UNIT CELL 



117 





~-2 



FIG. 4-11. Vector addition of waves. 



FIG. 4-12. A 
complex plane. 



wave vector in the 



The two waves shown as full lines in Fig. 4-10 represent the variations 
in electric field intensity E with time t of two rays on any given wave front 
in a diffracted x-ray beam. Their equations may be written 

EI = A\ sin (2irvt ^i), (4-5) 

E 2 = A 2 sin (2wt - $2). ( 4 ~^) 



These waves are of the same frequency v and therefore of the same wave- 
length A, but differ in amplitude A and in phase </>. The dotted curve 
shows their sum E 3 , which is also a sine wave, but of different amplitude 

and phase. 

Waves differing in amplitude and phase may also be added by represent- 
ing them as vectors. In Fig. 4-11, each component wave is represented 
by a vector whose length is equal to the amplitude of the wave and which 
is inclined to the :r-axis at an angle equal to the phase angle. The ampli- 
tude and phase of the resultant wave is then found simply by adding the 
vectors by the parallelogram law. 

This geometrical construction may be avoided by use of the following 
analytical treatment, in which complex numbers are used to represent the 
vectors. A complex number is the sum of a real and anjmaginary num- 
ber, such as (a + 6z), where a and 6 are real andjt = V-il is imaginary. 
Such numbers may be plotted in the "complex plane," in which real num- 
bers are plotted as abscissae and imaginary numbers as ordinates. Any 
point in this plane or the vector drawn from the origin to this point then 
represents a particular complex number (a + bi). 

To find an analytical expression for a vector representing a wave, we 
draw the wave vector in the complex plane as in Fig. 4-12. Here again 
the amplitude and phase of the wave is given by A, the length of the vector, 
and 0, the angle between the vector and the axis of real numbers. The 
analytical expression for the wave is now the complex number (A cos <t> + 
iA sin </>), since these two terms are the horizontal and vertical components 



DIFFRACTION II : THE INTENSITIES OF DIFFRACTED BEAMS [CHAP. 4 

md ON of the vector. Note that multiplication of a vector by i 
jtates it counterclockwise by 90; thus multiplication by i converts the 
horizontal vector 2 into the vertical vector 2i. Multiplication twice by i, 
that is, by i 2 = 1, rotates a vector through 180 or reverses its sense; 
thus multiplication twice by i converts the horizontal vector 2 into the 
horizontal vector 2 pointing in the opposite direction. 

If we write down the power-series expansions of e ix , cos x y and sin x, we 
find that 

e ix = cos x + i sin x (4-7) 

or 

Ae* = A cos <t> + Ai sin 4. (4-8) 

Thus the wave vector may be expressed analytically by either side of 
Eq. (4-8). The expression on the left is called a complex exponential 
function. 

Since the intensity of a wave is proportional to the square of its ampli- 
tude, we now need an expression for A 2 , the square of the absolute value 
of the wave vector. When a wave is expressed in complex form, this quan- 
tity is obtained by multiplying the complex expression for the wave by 
its complex conjugate, which is obtained simply by replacing i by i. 
Thus, the complex conjugate of Ae l * is Ae~ l *. We have 



\Ae l *\ 2 = Ae l +Ae-* = A 2 , (4-9) 

which is the quantity desired. Or, using the other form given by Eq. (4-8), 
we have 

A (cos + i sin 4)A(cos < i sin <) = A 2 (cos 2 <t> + sin 2 </>) == A 2 . 

We return now to the problem of adding the scattered waves from each 
of the atoms in the unit cell. The amplitude of each wave is given by the 
appropriate value of / for the scattering atom considered and the value 
of (sin 0)/X involved in the reflection. The phase of each wave is given by 
Eq. (4-4) in terms of the hkl reflection considered and the uvw coordinates 
of the atom. Using our previous relations, we can then express any scat- 
tered wave in the complex exponential form 



(4-10) 

The resultant wave scattered by alljbhe atoms of the unit cell is called the 
structure factor and is designated by the symBol F. It" is" obtained by simply 
adding together all the waves scattered by the individual atoms> If a 
unit cell contains atoms 1, 2, 3, . . . , N, with fractional coordinates 
Ui vi !!, u 2 v 2 tt? 2 , MS *>3 MS, ... and atomic scattering factors /i, / 2 , /a, . . . , 
then the structure factor for the hkl reflection is given by 

^ y e 2*i(hu2+kvi+lwti i / g 2iri(Au3-H;i>s-f Iwi) i . . . 



4-4] SCATTERING BY A UNIT CELL 117 

This equation may be written more compactly as 



N 



1 hkl Z^Jn 
1 



\~* f 



14-11) 



the summation extending over all the atoms of the unit cell. 

F is, in general, a complex number, and it expresses both the amplitude 
and phase of the resultant wave. {Its absolute value |F| gives the ampli- 
tude of the resultant wave in termsofr tne amplitude of the wa/ve^scaTEered 
ay a single elect ron.~Like the atomic scattering factoFJT |^'| is~definect as 
i ratio of amplitudes :\ **" 

amplitude of the wave scattered by all the atoms of a unit cell 

|/P| = - 

amplitude of the wave scattered by one electron 
.4 
The intensity of the beanL diffracted by all the atoms of the unit cell in a 

direction predicted by the Bragg law is proportional simply to |f| 2 , the 
square of the amplitude oQiiejresul^^^ |F| 2 is ^obtained ITy 

multiplying the expression given for F in Eq. (4-1 1) by its complex con- 
jugate* Equation (4-11) is therefore a very important relation in x-ray 
crystallography, since it permits a calculation of the intensity of any hkl 
reflection from a knowledge of the atomic positions. 

We have found the resultant scattered wave by adding together waves, 
differing in phase, scattered by individual atoms in the unit cell. Note 
that the phase difference between rays scattered by any two atoms, such 
as A and B in Fig. 4-8, is constant for every unit cell. There is no question 
here of these rays becoming increasingly out of phase as we go deeper in 
the crystal as there was when we considered diffraction at angles not 
exactly equal to the Bragg angle OB- In the direction predicted by the 
Bragg law, the rays scattered by all the atoms A in the crystal are exactly 
in phase and so are the rays scattered by all the atoms B, but between 
these two sets of rays there is a definite phase difference which depends on 
the relative positions of atoms A and B in the unit cell and which is given 
by Eq. (4-4). 



Although it is more unwieldy, the following trigonometric equation may be 
used instead of Eq. (4-11): 

N 

F = Z/n[cOS 2ir(7Wn + kVn + lw n ) + I SU1 2v(hu n + kVn + lWn)]. 
1 

One such term must be written down for each atom in the unit cell. In general, 
the summation will be a complex number of the form 

F = a + ib, 



118 DIFFRACTION III THE INTENSITIES OF DIFFRACTED BEAMS [CHAP. 4 
where 

N 

<* a = /n cos 2ir(hu n + kv n + Jw n ), 



JV 

b = /n sin 27r(/m n + ^ n + lw n ), 

1 

\F\ 2 - (a + ib)(a - ib) = a 2 + & 2 . 

Substitution for a and fe gives the final form of the equation: 
\F\* = [/i cos 2r(hui + fan + Ztin) + / 2 cos 2r(Atii + fa* + ^2) + ] 2 

+ [/i sin 2ir(hui + kvi + Iwi) + /, sin 2ir(fctt2 + kv 2 + Iw 2 ) + -] 2 - 



Equation (4-11) is much easier to manipulate, compared to this trigonometric 
form, particularly if the structure is at all complicated, since the exponential 
form is more compact. 

4-5 Some useful relations. In calculating structure factors by com- 
plex exponential functions, many particular relations occur often enough 
to be worthwhile stating here. They may be verified by means of Eq. 
(4-7). 

(a) e vi = e* Ti == e 5iri = -1, 

(fc) c*' **< = 6 " +1, 

(c) In general, e nTl = ( l) n , where n is any integer, 

(d) e nvi = e~~ nTl , where n is any integer, 

(e) e lx + e~ lx = 2 cos z. 

4r-6 Structure-factor calculations. Facility in the use of Eq. (4-11) can 
be gained only by working out some actual examples, and we shall con- 
sider a few such problems here and again in Chap. 10. 

(a) The simplest case is that of a unit cell containing only one atom at 
the origin, i.e., having fractional coordinates 000. Its structure factor is 

F = /e 2Tl(0) = / 
and 

F 2 =/ 2 . 

F 2 is thus independent of A, fc, and I and is the same for all reflections. 

(6) Consider now the base-centered cell discussed at the beginning of 
this chapter and shown in Fig. 4-1 (a). It has two atoms of the same kind 
per unit cell located at 0,and J J 0. 



/[I 



4-6] STRUCTURE-FACTOR CALCULATIONS 1U 

This expression may be evaluated without multiplication by the complex 
conjugate, since (h + fc) is always integral, and the expression for F ig 
thus real and not complex. If h and fc are both even or both odd, :Te., 
"unmixed," then their sum is always even and e* l(h+k} has the value 1. 

Therefore 

F = 2/ for h and k unmixed; 

F 2 = 4/ 2 . 

On the other hand, if h and k are one even and one odd, i.e., "mixed," 
then their sum is odd and e 7rl ^ +/r) has the value 1. Therefore 

F = for h and k mixed; 
F 2 = 0. 

Note that, in either case, the value of the I index has no effect on the struc- 
ture factor. For example, the reflections 111, 112, 113, and 021, 022, 023 
all have the same value of F, namely 2/. Similarly, the reflections Oil, 
012, 013, and 101, 102, 103 all have a zero structure factor. 

(c) The structure factor of the hody-ppntfifpH r,el] ahnwn In Fig. 4-1 (b^ 
may also be calculated. This cell has two atoms of the same kind located 
at and f | |. 

F = f e 27n(0) + S e 2iri(h/2+k/2+l/2) 



F = 2f when (h + k + I) is even; 
F 2 = 4/ 2 . 
F = when (h + k + I) is odd; 



We had previously concluded from geometrical considerations that the 
base-centered cell would produce a 001 reflection but that the body-centered 
cell would not. This result is in agreement with the structure-factor equa- 
tions for these two cells. A detailed examination of the geometry of all 
possible reflections, however, would be a very laborious process compared 
to the straightforward calculation of the structure factor, a calculation 
that yields a set of rules governing the value of F 2 for all possible values of 
otene indices. 

(d) A face-centered cubic cell, such as that shown in Fig. 2-14, may 
now be considered. Assume it to contain four atoms of the same kind, 
located at 0, | f 0, \ |, and \ \. 



t20 DIFFRACTION II : THE INTENSITIES OF DIFFRACTED BEAMS [CHAP. 4 

If A, fc, and I are unmixed, then all three sums (h + ft), (h + Z), and (fc + 
are even integers, and each term in the above equation has the value 1. 

F = 4f for unmixed indices; 
F 2 = 16/ 2 . ^ *> V; ^ < 

If ft, /c, and Z are mixed, then the sum of the three exponentials is 1, 
whether two of the indices are odd and one even, or two even and one odd. 
Suppose for example, that h and I are even and k is odd, e.g., 012. Then 
F = /(I 1 -f 1 1) = 0, and no reflection occurs. 

F = for mixed indices; 
F 2 = 

Thus, reflections will occur for such planes as (111), (200), and (220) but 
not for the planes (100), (210), (112), etc. 

The reader may have noticed in the previous examples that some of the 
information given was not used in the calculations. In (a), for example, 
the cell was said to contain only one atom, but the shape of the cell was 
not specified; in (6) and (c), the cells were described as orthorhombic and 
in (d) as cubic, but this information did not enter into the structure-factor 
calculations. This illustrates the important point that the structure factor 
is independent of the shape and size of the unit cell. For example, any body- 
centered cell will have missing reflections for those planes which have 
(h + fc + I) equal to an odd number, whether the cell is cubic, tetragonal, 
or orthorhombic. The rules we have derived in the above examples are 
therefore of wider applicability than would at first appear and demonstrate 
the close connection between the Bravais lattice of a substance and its 
diffraction pattern. They are summarized in Table 4-1. These rules are 
subject to some qualification, since some cells may contain more atoms 
than the ones given in examples (a) through (d), and these atoms may be 
in such positions that reflections normally present are now missing. For 
example, diamond has a face-centered cubic lattice, but it contains eight 

TABLE 4-1 



Bravais lattice 


Reflections present 


Reflections absent 


Simple 
Base -centered 
Body-centered 
Face-centered 


all 
h and k unmixed 
(h + k + I) even 
h t k, and / unmixed 


none 
h and k mixed 
(h + k + l) odd 
h, k, and / mixed 



* These relations apply to a cell centered on the C face. If reflections are present 
only when h and I are unmixed, or when k and I are unmixed, then the cell is cen- 
tered on the B or A face, respectively. 



4-6] STRUCTURE-FACTOR CALCULATIONS 121 

carbon atoms per unit cell. All the reflections present have unmixed 
indices, but reflections such as 200, 222, 420, etc., are missing. The fact 
that the only reflections present have unmixed indices proves that the lat- 
tice is face-centered, while the extra missing reflections are a clue to the 
actual atom arrangement in this crystal. 

(e) This point may be further illustrated by the structure of NaCl 
(Fig. 2-18). This crystal has a cubic lattice with 4 Na and 4 Cl atoms 
per unit cell, located as follows: 

Na 000 f | | Off 

Cl HI 00 i OfO fOO 



In this case, the proper atomic scattering factors for each atom must be 
inserted in the structure-factor equation : 



F = /Na[l + e 

+ e'* 7 + e* lk + 



As discussed in Sec?. 2-7, the sodium-atom positions are related by the 
face-centering translations and so are the chlorine-atom positions. When- 
ever a lattice contains common translations, the corresponding terms in 
the structure-factor equation can always be factored out, leading to con- 
siderable simplification. In this case we proceed as follows : 

F = /Natl + 



The signs of the exponents in the second bracket may be changed, by rela- 
tion (d) of Sec. 4-5. Therefore 



Here the terms corresponding to the face-centering translations appear in 
the first factor. These terms have already appeared in example (d), and 
they were found to have a total value of zero for mixed indices and 4 for 
unmixed indices. This shows at once that NaCl has a face-centered lattice 
and that 

F = for mixed indices; 



122 DIFFRACTION II : THE INTENSITIES OF DIFFRACTED BEAMS [CHAP. 4 

For unmixed indices, 



F - 4(/ N + /ci) if (h + k + is even; 

f 2 = 16(/ Na +/Cl) 2 . 

F - 4(/ Na - /ci) if (ft + fc + 9 is odd; 

F 2 - 16(/ Na ~ 



In this case, there are more than four atoms per unit cell, but the lattice 
is still face-centered. The introduction of additional atoms has not elim- 
inated any reflections present in the case of the four-atom cell, but it has 
decreased some in intensity. For example, the 111 reflection now involves 
the difference, rather than the sum, of the scattering powers of the two 
atoms. 

(/) One other example of structure factor calculation will be given here. 
The close-packed hexagonal cell shown in Fig. 2-15 has two atoms of the 
same kind located at and J. 



F = fe 2iri(0) 

- fM _|_ e 

For convenience, put [(h + 2/c)/3 + 1/2] = g. 

F = /(I + e 2 '). 

Since g may have fractional values, such as ^-, $, ^, etc., this expression 
is still complex. Multiplication by the complex conjugate, however, will 
give the square of the absolute value of the resultant wave amplitude F. 

|F| a =/ 2 (l + e 2 "')(l + c-" 2 ' t ') 

= / 2 (2 + e 2vi * + <T 2Tl *). 
By relation (e) of Sec. 4-5, this becomes 
|F| 2 = / 2 (2 + 2 cos 2*0) 

= / 2 [2 + 2(2 cos 2 *g - 1)] 



when (h + 2fc) is a multiple of 3 and I is odd. 



4-7] APPLICATION TO POWDER METHOD 123 

It is by these missing reflections, such as 11-1, 11-3, 22-1, 22-3, that a 
hexagonal structure is recognized as being close-packed. Not all the re- 
flections present have the same structure factor. For example, if (h + 2k) 
is a multiple of 3 and I is even, then 

/h + 2k l\ 

I -- h - ) = n, where n is an integer; 

\ o 2t/ 

cos irn = 1 , 
cos 2 trn = 1 , 
|F| 2 = 4/ 2 . 

When all possible values of h, k, and 7 are considered, the results may be 
summarized as follows: 



3n odd 

3w even 4/ 2 

3n 1 odd 3/ 2 

3 A? 1 even / 2 

4-7 Application to powder method. Any calculation of the intensity of 
a diffracted beam must always begin with the structure factor. The re- 
mainder of the calculation, however, varies with the particular diffraction 
method involved. For the Laue method, intensity calculations are so 
difficult that they are rarely made, since each diffracted beam has a differ- 
ent wavelength and blackens the film by a variable amount, depending 
on both the intensity and the film sensitivity for that particular wave- 
length. The factors governing diffracted intensity in the rotating-crystal 
and powder methods are somewhat similar, in that monochromatic radia- 
tion is used in each, but they differ in detail. The remainder of this chapter 
will be devoted to the powder method, since it is of most general utility 
in metallurgical work. 

There ^re_six_factorsaffecting the relative intensity of the diffraction 
lines on a powder pattern: 

(1) polarization factor, 

(2) structure factor, 

(3) multiplicity factor, 

(4) Lorentz factor, 

(5) absorption factor, 

(6) temperature factor^ 

The first two of these have already been described, and the others will be 
discussed in the following sections. 



124 DIFFRACTION II : THE INTENSITIES OF DIFFRACTED BEAMS [CHAP. 4 

4-8 Multiplicity factor. Consider the 100 reflection from a cubic lat- 
tice. In the powder specimen, some of the crystals will be so oriented that 
reflection can occur from their (100) planes. Other crystals of different 
orientation may be in such a position that reflection can occur from their 
(010) or (001) planes. Since all these planes have the same spacing, the 
beams diffracted by them all form part of the same cone. Now consider 
the 111 reflection. There are four sets of planes of the form {111) which 
have the same spacing but different orientation, namely, (111), (111), 
(111), and (ill), whereas there are only three sets of the form (100). 
Therefore, the probability that {111 } planes will be correctly oriented for 
reflection is f the probability that {100} planes will be correctly oriented. 
It follows that the intensity of the 1 11 reflection will be f that of the 100 
reflection, other things being equal. 

This relative proportion of planes contributing to the same reflection 
enters the intensity equation as the quantity p, the multiplicity factor, 
which may be defined as the number of different planes in a form having 
the same spacing. Parallel planes with different Miller indices, such as 
(100) and (TOO), are counted separately as different planes, yielding num- 
bers which are double those given in the preceding paragraph. Thus the 
multiplicity factor for the {100} planes of a cubic crystal is 6 and for the 
{111} planes 8. 

The value of p depends on the crystal system: in a tegragonal crystal, 
the (100) and (001) planes do not have the same spacing, so that the value 
of p for {100} planes is reduced to 4 and the value for {001} planes to 2. 
Values of the multiplicity factor as a function of hkl and crystal system 
are given in Appendix 9. 

4-9 Lorentz factor. We must now consider certain trigonometrical fac- 
tors which influence the intensity of the reflected beam. Suppose there is 
incident on a crystal [Fig. 4-13 (a)] a narrow beam of parallel monochro- 
matic rays, and let the" crystal be rotated at a uniform angular velocity 
about an axis through and normal to the drawing, so that a particular 
set of reflecting planes, assumed for convenience to be parallel to the crys- 
tal surface, passes through the angle fe, at which the Bragg law is exactly 
satisfied. As mentioned in Sec. 3-7, the intensity of reflection is greatest 
at the exact Bragg angle but still appreciable at angles deviating slightly 
from the Bragg angle, so that a curve of intensity vs. 20 is of the form 
shown in Fig. 4-13 (b). If all the diffracted beams sent out by the crystal 
as it rotates through the Bragg angle are received on a photographic film 
or in a counter, the total energy of the diffracted beam can be measured. 
This energy is called the integrated intensity of the reflection and is given 
by the area under the curve of Fig. 4-13 (b). The integrated intensity is 
of much more interest than the maximum intensity, since the former is 



4-9] 



LORENTZ FACTOR 



125 





(a) 



DIFFRACTION ANGLE 26 

(b) 



FIG. 4-13. Diffraction by a crystal rotated through the Bragg angle. 

characteristic of the specimen while the latter is influenced by slight adjust- 
ments of the experimental apparatus. Moreover, in the visual comparison 
of the intensities of diffraction lines, it is the integrated intensity of the 
line rather than the maximum intensity which the eye evaluates. 

The integrated intensity of a reflection depends on the particular value 
of BB involved, even though all other variables are held constant. We can 
find this dependence by considering, separately, two aspects of the diffrac- 
tion curve: the maximum intensity and the breadth. When the reflecting 
planes make an angle BB with the incident beam, the Bragg law is exactly 
satisfied and the intensity diffracted in the direction 26s is a maximum. 
But some energy is still diffracted in this direction when the angle of inci- 
dence differs slightly from fe, (and the total energy diffracted in the direc- 
tion 20# as the crystal is rotated through the Bragg angle is given by the 
value of / m ax of the curve of Fig. 4-13(b). ^The value of / ma x therefore 
depends on the angular range of crystal rotation over which the energy 
diffracted in the direction 20 is appreciable.) In Fig. 4-14(a), the dashed 
lines show the position of the crystal after rotation through a small angle 

2, 



2' 





(a) (b) 

FIG. 4-14. Scattering in a fixed direction during crystal rotation. 



126 DIFFRACTION II! THE INTENSITIES OF DIFFRACTED BEAMS [CHAP. 4 

A0 from the Bragg position. The incident beam and the diffracted beam 
under consideration now make unequal angles with the reflecting planes, 
the former making an angle 0i = OB + A0 and the latter an angle 2 
OB A0. The situation on an atomic scale is shown in Fig. 4-14(b). Here 
we need only consider a single plane of atoms, since the rays scattered by 
all other planes are in phase with the corresponding rays scattered by the 
first plane. Let a equal the atom spacing in the plane and Na the total 
length* of the plane. The difference in path length for rays 1' and 2' 
scattered by adjacent atoms is given by 

5 r2 ' = AD - CB 

= a cos 6 2 a cos B\ 

= a[cos (Bs A0) - cos (SB + A0)]. 

By expanding the cosine terms and setting sin A0 equal to A0, since the 
latter is small, we find: 

$i> 2 ' = 2aA0 sin 0#, 

and the path difference between the rays scattered by atoms at either end 
of the plane is simply N times this quantity. When the rays scattered by 
the two end atoms are (N + 1) wavelengths out of phase, the diffracted 
intensity will be zero. (The argument here is exactly analogous to that 
used in Sec. 3-7.) The condition for zero diffracted intensity is therefore 

2JVaA0 sin B = (N + 1)X, 
or 

(AT + 1)X 



A0 



2Na sin 6 B 



This equation gives the maximum angular range of crystal rotation over 
which appreciable energy will be diffracted in the direction 20#. Since 
/max depends on this range, we can conclude that / max is proportional to 
I/sin 0fl. Other things being equal, / max is therefore large at low scatter- 
ing angles and small in the back-reflection region. 

The breadth of the diffraction curve varies in the opposite way, being 
larger at large values of 20#, as was shown in Sec. 3-7, where the half- 
maximum breadth B was found to be proportional to I/cos BB. The inte- 
grated intensity of the reflection is given by the area under the diffraction 
curve and is therefore proportional to the product / ma x-B, which is in turn 
proportional to (l/sin0#)(l/cos0B) or to I/sin 26 B . (Thus, as a crystal 
is rotated through the Bragg angle, the integrated intensity of a reflection, 
which is the quantity of most experimental interest, turns out to be greater 

* If the crystal is larger than the incident beam, then Na is the irradiated length 
of the plane; if it is smaller, Na is the actual length of the plane. 



4-9] 



LORENTZ FACTOR 



127 



for large and small values of 200 than for intermediate values, other things 
being equal. 

The preceding remarks apply just as well to the powder method as they 
do to the case of a rotating crystal, since the range of orientations available 
among the powder particles, some satisfying the Bragg law exactly, some 
not so exactly, are the equivalent of single-crystal rotation. 

However, in the powder method, a second geometrical factor arises when 
we consider thatfyhe integrated intensity of a reflection at any particular 
Bragg angle depends on the number of particles oriented at or near that 
angled This number is not constant even though the particles are oriented 
completely at random. In Fig. 4-15 
a reference sphere of radius r is drawn 
around the powder specimen located 
at 0. For the particular hkl reflec- 
tion shown, ON is the normal to this 
set of planes in one particle of the 
powder. Suppose that the range of 
angles near the Bragg angle over 
which reflection is appreciable is A0. 
Then, for this particular reflection, 
only those particles will be in a re- 
flecting position which have the ends 
of their plane normals lying in a band 
of width rA0 on the surface of the 
sphere. Since the particles are as- 
sumed to be oriented at random, the 
ends of their plane normals will be uniformly distributed over the surface 
of the sphere; the fraction favorably oriented for a reflection will be given 
by the ratio of the area of the strip to that of the whole sphere. If AAT is 
the number of such particles and N the total number, then 

AAT rA0 2nr sin (90 - B ) A0 cos 6 B 




FIG. 4-15. The distribution of plane 
normals for a particular cone of re- 
flected rays. 



The number of particles favorably oriented for reflection is thus propor- 
tional to cos B and is quite small for reflections in the backward direction. 
In assessing relative intensities, we do not compare the total diffracted 
energy in one cone of rays with that in another but rather the integrated 
intensity per unit length of one diffraction line with that of another. For 
example, in the most common arrangement of specimen and film, the 
Debye-Scherrer method, shown in Fig. 4-16, the film obviously receives a 
greater proportion of a diffraction cone when the reflection is in the forward 
or backward direction than it does near 20 = 90. Inclusion of this effect 



128 DIFFRACTION II! THE INTENSITIES OF DIFFRACTED BEAMS [CHAP. 4 




R sin 20/i 



FIG. 4-16. Intersection of cones of diffracted rays with Debye-Scherrer film. 

thus leads to a third geometrical factor affecting the intensity of a reflec- 
tion. The length of any diffraction line being 2vR sin 20s, where R is the 
radius of the camera, the relative intensity per unit length of line is pro- 
portional to I/sin 20B. 

In intensity calculations, the three factors just discussed are combined 
into one and called the Lorentz factor. Dropping the subscript on the 
Bragg angle, we have: 



Lorentz factor == ( ) [ cos 6 } [ I 

Vsin 207 \ / Vsin 207 



1 



CO80 



sin 2 28 4 sin 2 6 cos 



This in turn is combined with the polarization factor 
Sec. 4-2 to give the combined Lorentz- 
polarization factor which, with a con- 
stant factor of -^ omitted, is given by 

Lorentz-polarization factor = ^ 

o 

1 + cos 2 26 5| 

CSJ 

sin' 2 6 cos 6 3 



+ cos 2 26) of 



Values of this factor are given in 
Appendix 10 and plotted in Fig. 4-17 
as a function of 6. (jhe over-all effect 
of these geometrical factors is to de- 
crease the intensity of reflections at 
intermediate angles compared to those 
in forward or backward directions. 



10 








45 



90 



BRAGG ANGLE 6 (degrees) 
FIG. 4-17. Lorentz-polarization factor. 



4-10] 



ABSORPTION FACTOR 



129 



\ 




(a) 



(h) 



FIG. 4-18. Absorption in Debye-Scherrer specimens: (a) general case, (b) highly 
absorbing specimen. 

4-10 Absorption factor. Still another factor affecting the intensities of 
the diffracted rays must be considered, and that is the absorption which 
takes place in the specimen itself. The specimen in the Debye-Scherrer 
method has the form of a very thin cylinder of powder placed on the camera 
axis, and Fig. 4-1 8 (a) shows the cross section of such a specimen. For 
the low-angle reflection shown, absorption of a particular ray in the inci- 
dent beam occurs along a path such as AB] at 5 a small fraction of the 
incident energy is diffracted by a powder particle, and absorption of this 
diffracted beam occurs along the path BC. Similarly, for a high-angle 
reflection, absorption of both the incident and diffracted beams occurs 
along a path such as (DE + EF). The net result is that the diffracted 
beam is of lower intensity than one would expect for a specimen of no 
absorption. 

A calculation of this effect shows that the relative absorption increases 
as 6 decreases, for any given cylindrical specimen. That this must be so 
can be seen from Fig. 4-1 8 (b) which applies to a specimen (for example, 
tungsten) of very high absorption. The incident beam is very rapidly 
absorbed, and most of the diffracted beams originate in the thin surface 
layer on the left side of the specimen ,-f backward-reflected beams then 
undergo very little absorption, but forward-reflected beams have to pass 
through the whole specimen and are greatly absorbed.^ Actually, the 
forward-reflected beams in this case come almost entirely from the top and 
bottom edges of the specimen.* This difference in absorption between 

* The powder patterns reproduced in Fig. 3-13 show this effect. The lowest- 
angle line in each pattern is split in two, because the beam diffracted through the 
center of the specimen is so highly absorbed. It is important to keep the possi- 
bility of this phenomenon in mind when examining Debye-Scherrer photographs, 
or split low-angle lines may be incorrectly interpreted as separate diffraction lines 
from two different sets of planes. 



130 DIFFRACTION III THE INTENSITIES OF DIFFRACTED BEAMS [CHAP. 4 

high-0 and low-0 reflections decreases as the linear absorption coefficient 
of the specimen decreases, but the absorption is always greater for the 
low-0 reflections. (These remarks apply only to the cylindrical specimen 
used in the Debye-Scherrer method. The absorption factor has an entirely 
different form for the flat-plate specimen used in a diffractometer, as will 
be shown in Sec. 7-4.) 

Exact calculation of the absorption factor for a cylindrical specimen is 
often difficult, so it is fortunate that this effect can usually be neglected in 
the calculation of diffracted intensities, when the Debye-Scherrer method 
is used. Justification of this omission will be found in the next section. 

4-11 Temperature factor. So far we have considered a crystal as a 
collection of atoms located at fixed points in the lattice. Actually, the 
atoms undergo thermal vibration about their mean positions even at the 
absolute zero of temperature, and the amplitude of this vibration increases 
as the temperature increases. In aluminum at room temperature, the 
average displacement of an atom from its mean position is about 0.1 7 A, 
which is by no means negligible, being about 6 percent of the distance of 
closest approach of the mean atom positions in this crystal. 

^Thermal agitation decreases the intensity of a diffracted beam because 
it has the effect of smearing out the lattice planes;* atoms can be regarded 
as lying no longer on mathematical planes but rather in platelike regions 
of ill-defined thickness. Thus the reinforcement of waves scattered at the 
Bragg angle by various parallel planes, the reinforcement which is called a 
diffracted beam, is not as perfect as it is for a crystal with fixed atoms. 
This reinforcement requires that the path difference, which is a function 
of the plane spacing d, between waves scattered by adjacent planes be an 
integral number of wavelengths. Now the thickness of the platelike 
"planes' ' in which the vibrating atoms lie is, on the average, 2?/, where 
u is the average displacement of an atom from its mean position. Under 
these conditions reinforcement is no longer perfect, and it becomes more 
imperfect as the ratio u/d increases, i.e., as the temperature increases, 
since that increases u, or as increases, since high-0 reflections involve 
planes of low d value. TThus the intensity of a diffracted beam decreases 
as the temperature is raised, and, for a constant temperature, thermal 
vibration causes a greater decrease in the reflected intensity at high angles 
than at low angles. / 

The temperature effect and the previously discussed absorption effect 
in cylindrical specimens therefore depend on angle in opposite ways and, 
to a first approximation, cancel each other. In back reflection, for exam- 
ple, the intensity of a diffracted beam is decreased very little by absorption 
but very greatly by thermal agitation, while in the forward direction the 
reverse is true. The two effects do not exactly cancel one other at all 



4-11] TEMPERATURE FACTOR 131 

angles; however, if the comparison of line intensities is restricted to lines 
not differing too greatly in 6 values, the absorption and temperature effects 
can be safely ignored. This is. a fortunate circumstance, since both of 
these effects are rather difficult to calculate exactly. 

It should be noted here that thermal vibration of the atoms of a crystal 
does not cause any broadening of the diffraction lines; they remain sharp 
right up to the melting point, but their maximum intensity gradually de- 
creases. It is also worth noting that the mean amplitude of atomic vibra- 
tion is not a function of the temperature alone but depends also on the 
elastic constants of the crystal. At any given temperature, the less "stiff" 
the crystal, the greater the vibration amplitude u. This means that u 
is much greater at any one temperature for a soft, low-melting-point metal 
like lead than it is for, say, tungsten. Substances with low melting points 
have quite large values of u even at room temperature and therefore yield 
rather poor back-reflection photographs. 

The thermal vibration of atoms has another effect on diffraction pat- 
terns. Besides decreasing the intensity of diffraction lines, it causes some 
general coherent scattering in all directions. This is called temperature- 
diffuse scattering; it contributes only to the general background of the 
pattern and its intensity gradually increases with 26. Contrast between 
lines and background naturally suffers, so this effect is a very undesirable 
one, leading in extreme cases to diffraction lines in the back-reflection 
region scarcely distinguishable from the background. 

In the phenomenon of temperature-diffuse scattering we have another 
example, beyond those alluded to in Sec. 3-7, of scattering at non-Bragg 
angles. Here again it is not surprising that such scattering should occur, 
since the displacement of atoms from their mean positions constitutes a 
kind of crystal imperfection and leads to a partial breakdown of the con- 
ditions necessary for perfect destructive interference between rays scat- 
tered at non-Bragg angles. 

The effect of thermal vibration also illustrates what has been called 
"the approximate law of conservation of diffracted energy. " This law 
states that the total energy diffracted by a particular specimen under par- 
ticular experimental conditions is roughly constant. Therefore, anything 
done to alter the physical condition of the specimen does not alter the total 
amount of diffracted energy but only its distribution in space. This "law" 
is not at all rigorous, but it does prove helpful in considering many diffrac- 
tion phenomena. For example, at low temperatures there is very little 
background scattering due to thermal agitation and the diffraction lines 
are relatively intense; if the specimen is now heated to a high temperature, 
the lines will become quite weak and the energy which is lost from 
the lines will appear in a spread-out form as temperature-diffuse scat- 
tering. 



132 DIFFRACTION II : THE INTENSITIES OF DIFFRACTED BEAMS [CHAP. 4 

4-12 Intensities of powder pattern lines. We are now in a position to 
gather together the factors discussed in preceding sections into an equation 
for the relative intensity of powder pattern lines: 



Y 1 + C0s22g> ) , (4-12) 

\ sin 2 6 cos 6 / 



where I = relative integrated intensity (arbitrary units), F = structure 
factor, p = multiplicity factor, and 6 = Bragg angle. In arriving at this 
equation, we have omitted factors which are constant for all lines of the 
pattern. For example, all that is retained of the Thomson equation (Eq. 
4-2) is the polarization factor (1 + cos 2 26), with constant factors, such 
as the intensity of the incident beam and the charge and mass of the elec- 
tron, omitted. The intensity of a diffraction line is also directly propor- 
tional to the irradiated volume of the specimen and inversely proportional 
to the camera radius, but these factors are again constant for all diffraction 
lines and may be neglected. Omission of the temperature and absorption 
factors means that Eq. (4-12) is valid only for the Debye-Scherrer method 
and then only for lines fairly close together on the pattern; this latter 
restriction is not as serious as it may sound. Equation (4-12) is also re- 
stricted to the Debye-Scherrer method because of the particular way in 
which the Lorentz factor was determined; other methods, such as those 
involving focusing cameras, will require a modification of the Lorentz 
factor given here. In addition, the individual crystals making up the 
powder specimen must have completely random orientations if Eq. (4-12) 
is to apply. Finally, it should be remembered that this equation gives the 
relative integrated intensity, i.e., the relative area under the curve of in- 
tensity vs. 20. 

It should be noted that "integrated intensity" is not really intensity, 
since intensity is -expressed in terms of energy crossing unit area per unit 
of time. A beam diffracted by a powder specimen carries a certain amount 
of energy per unit time and one could quite properly refer to the total 
power of the diffracted beam. If this beam is then incident on a measuring 
device, such as photographic film, for a certain length of time and if a 
curve of diffracted intensity vs. 26 is constructed from the measurements, 
then the area under this curve gives the total energy in the diffracted beam. 
This is the quantity commonly referred to as integrated intensity. A 
more descriptive term would be "total diffracted energy," but the term 
"integrated intensity" has been too long entrenched in the vocabulary of 
x-ray diffraction to be changed now. 

4-13 Examples of intensity calculations. The use of Eq. (4-12) will 
be illustrated by the calculation of the position and relative intensities of 



EXAMPLES OF INTENSITY CALCULATIONS 



133 



4-13] 

the diffraction lines on a powder pattern of copper, made with Cu Ka. 
radiation. The calculations are most readily carried out in tabular form, 
as in Table 4-2. 

TABLE 4-2 



1 


2 


3 


4 


5 


6 


7 


8 


Line 


hkl 


h' 2 + A- 2 + l <2 


sin 2 


sinO 


e 


^ 'A"') 


A'u 


1 


111 


3 


0.1365 


0.369 


21.7 


0.240 


20.0 


2 


200 


4 


0.1820 


0.426 


25.2 


0.277 


18.7 


3 


220 


8 


0.364 


0.602 


37.0 


0.391 


15.6 


4 


311 


11 


0.500 


0.706 


44.9 


0.459 


14.0 


5 


222 


12 


0.546 


0.738 


47.6 


0.479 


13.7 


6 


400 


16 


0.728 


0.851 


58.3 


0.553 


12.4 


7 


331 


19 


0.865 


0.929 


68.3 


0.604 


11.7 


8 


420 


20 


0.910 


0.951 


72.0 


0.618 


11.4 



1 


9 


10 


11 


12 


13 


14 




rjo 




1 + cos 2 20 


Relative integrated intensity 


Line 


b z 


P 


2 

sin cos0 


Calc. 


Calc. 


Obs. 


1 


6400 


8 


12.10 


6.20 X 10 5 


10.0 


s 


2 


5600 


6 


8.50 


2.86 


4.6 


m 


3 


3890 


12 


3.75 


1.75 


2.8 


m 


4 


3140 


24 


2.87 


2.16 


3.5 


s 


5 


3000 


8 


2.75 


0.66 


1.1 


w 


6 


2460 


6 


3.18 


0.47 


0.8 


w 


7 


2190 


24 


4.75 


2.50 


4.0 


vs 


8 


2060 


24 


5.92 


2.96 


4.8 


vs 



Remarks: 

Column 2: Since copper is face-centered cubic, F is equal to 4/ Cu for lines of un- 
mixed indices and zero for lines of mixed indices. The reflecting plane indices, all 
unmixed, are written down in this column in order of increasing values of (h 2 -f- 
fc 2 + Z 2 ), from Appendix 6. 

Column 4: For a cubic crystal, values of sin 2 6 are given by Eq. (3-10) : 



sm"0 = j-gC/r -h /r -h r;. 

In this case, X = 1.542A (Cu Ka) and a = 3.615A (lattice parameter of copper). 
Therefore, multiplication of the integers in column 3 by X 2 /4a 2 = 0.0455 gives the 
values of sin 2 listed in column 4. In this and similar calculations, slide-rule 
accuracy is ample. 

Column 6: Needed to determine the Lorentz-polarization factor and (sin 0)/X. 

Column 7: Obtained from Appendix 7. Needed to determine / Cu - 

Column 8: Read from the curve of Fig. 4-6. 

Column 9: Obtained from the relation F 2 = 16/ Cu 2 - 

Column 10: Obtained from Appendix 9. 



134 DIFFRACTION II : THE INTENSITIES OF DIFFRACTED BEAMS [CHAP. 4 

Column 11: Obtained from Appendix 10. 

Column 12: These values are the product of the values in columns 9, 10, and 11. 

Column 13: Values from column 12 recalculated to give the first line an arbitrary 
intensity of 10. 

Column 14: These entries give the observed intensities, visually estimated ac- 
cording to the following simple scale, from the pattern shown in Fig. 3-1 3(a) 
(vs = very strong, s = strong, m = medium, w = weak). 

The agreement obtained here between observed and calculated intensities 
is satisfactory. For example, lines 1 and 2 are observed to be of strong 
and medium intensity, their respective calculated intensities being 10 and 
4.0. Similar agreement can be found by comparing the intensities of any 
pair of neighboring lines in the pattern. Note, however, that the com- 
parison must be made between lines which are not too far apart: for exam- 
ple, the calculated intensity of line 2 is greater than that of line 4, whereas 
line 4 is observed to be stronger than line 2. Similarly, the strongest lines 
on the pattern are lines 7 and 8, while calculations show line 1 to 
be strongest. Errors of this kind arise from the omission of the absorption 
and temperature factors from the calculation. 

A more complicated structure may now be considered, namely that of 
the zinc-blende form of ZnS, shown in Fig. 2-19(b). This form of ZnS is 
cubic and has a lattice parameter of 5.41A. We will calculate the relative 
intensities of the first six lines on a pattern made with Cu Ka radiation. 

As always, the first step is to work out the structure factor. ZnS has 
four zinc and four sulfur atoms per unit cell, located in the following posi- 
tions: 

'Zn: \ \ \ + face-centering translations, 

S: + face-centering translations. 

Since the structure is face-centered, we know that the structure factor 
will be zero for planes of mixed indices. We also know, from example (e) 
of Sec. 4-6, that the terms in the structure-factor equation corresponding 
to the face-centering translations can be factored out and the equation for 
unmixed indices written do\vn at once: 



|F| 2 is obtained by multiplication of the above by its complex conjugate: 



This equation reduces to the following form: 

|F| 2 = 16 I/!, 2 + / Zn 2 + 2/s/ Zn cos *- (h + k + J 



4-13] EXAMPLES OF INTENSITY CALCULATIONS 135 
Further simplification is possible for various special cases: 

\F\ 2 = 16(/ s 2 + / Zn 2 ) when (h + k + I) is odd; (4-13) 

\F\ 2 = 16(/ s - / Z n) 2 when (h + k + 1} is an odd multiple of 2; (4-14) 

|^| 2 = 16(/ s + /zn) 2 when (h + k + I) is an even multiple of 2. (4-15) 

The intensity calculations are carried out in Table 4-3, with some columns 
omitted for the sake of brevity. 

TABLE 4-3 



1 


2 


3 


4 


5 


6 


Line 


hU 


e 


^ (A' 1 ) 


/s 


J'/M 


1 


111 


14.3 


0.161 


11.5 


24.2 


2 


200 


16.6 


0.185 


11.0 


23.2 


3 


220 


23.8 


0.262 


9.5 


20.0 


4 


311 


28.2 


0.307 


8.9 


18.5 


5 


222 


29.6 


0.321 


8.6 


18.0 


6 


400 


34.8 


0.370 


8.1 


16.7 




















7 


8 


9 


10 


11 


1 !na 


//,J 




1 +cos 2 29 


Relative intensity 






P 


sin 2 9 cos 9 


Calc. 


Obs. 


1 


11490 


8 


30.1 


10.0 


vs 


2 


2380 


6 


21.9 


1.1 


w 


3 


13940 


12 


9.72 


5.9 


vs 


4 


6750 


24 


6.65 


3.9 


vs 


5 


1410 


8 


6.00 


0.2 


vw 


6 


9850 


6 


4.24 


0.9 


w 



Remarks: 

Columns 5 and 6: These values are read from scattering-factor curves plotted 
from the data of Appendix 8. 

Column 7: \F\~ is obtained by the use of Eq. (4-13), (4-14), or (4-15), depending 
on the particular values of hkl involved. Thus, Eq. (4-13) is used for the 111 re- 
flection and Eq. (4-15) for the 220 reflection. 

Columns 10 and 11: The agreement obtained here between calculated and ob- 
served intensities is again satisfactory. In this case, the agreement is good when 
any pair of lines is compared, because of the limited range of 6 values involved. 



One further remark on intensity calculations is necessary. In the powder 
method, two sets of planes with different Miller indices can reflect to the 
same point on the film: for example, the planes (411) and (330) in the 
cubic system, since they have the same value of (h 2 + k 2 + I 2 ) and hence 
the same spacing, or the planes (501) and (431) of the tetragonal system, 



JLJO DIFFRACTION II : THE INTENSITIES OF DIFFRACTED BEAMS [CHAP. 4 

since they have the same values of (h? + fc 2 ) and I 2 . In such a case, the 
intensity of each reflection must be calculated separately, since in general 
the two will have different multiplicity and structure factors, and then 
added to find the total intensity of the line. 

4-14 Measurement of x-ray intensity. In the examples just given, the 
observed intensity was estimated simply by visual comparison of one line 
with another. Although this simple procedure is satisfactory in a sur- 
prisingly large number of cases, there are problems in which a more precise 
measurement of diffracted intensity is necessary. Two methods are in 
general use today for making such measurements, one dependent on the 
photographic effect of x-rays and the other on the ability of x-rays to ionize 
gases and cause fluorescence of light in crystals. These methods have 
already been mentioned briefly in Sec. 1-8 and will be described more fully 
in Chaps. 6 and 7, respectively. 

PROBLEMS 

4-1. By adding Eqs. (4-5) and (4-6) and simplifying the sum, show that E 3 , 
the resultant of these two sine waves, is also a sine wave, of amplitude 

A 3 = [Ai 2 + A 2 * + 2A,A 2 cos fa - <*> 2 )] 

and of phase 

. AI sin fa + Az sin 92 
</> 3 = tan" 1 -; ^ , , 

AI COS fa + A 2 COS 02 

4-2. Obtain the same result by solving the vector diagram of Fig. 4-11 for the 
right-angle triangle of which A 3 is the hypotenuse. 

4^3. Derive simplified expressions for F 2 for diamond, including the rules gov- 
erning observed reflections. This crystal is cubic and contains 8 carbon atoms per 
unit cell, located in the following positions: 

000 HO $0i OH 

Hi Hi Hi Hi 

4-4. A certain tetragonal crystal has four atoms of the same kind per unit cell, 
located at H. i i, \ f, H- 

(a) Derive simplified expressions for F 2 . 

(b) What is the Bravais lattice of this crystal? 

(c) What are the values of F 2 for the 100, 002, 111, and Oil reflections? 

4-6. Derive simplified expressions for F 2 for the wurtzite form of ZnS, includ- 
ing the rules governing observed reflections. This crystal is hexagonal and con- 
tains 2 ZnS per unit cell, located in the following positions: 

Zn:000, Hi 
S:OOf,Hi 



PROBLEMS 137 

Note that these positions involve a common translation, which may be factored 
out of the structure-factor equation. 

4-6. In Sec. 4-9, in the part devoted to scattering when the incident and scat- 
tered beams make unequal angles witli the reflecting planes, it is stated that 
"rays scattered by all other planes are in phase with the corresponding rays scat- 
tered by the first plane." Prove this. 

4-7. Calculate the position (in terms of 6) and the integrated intensity (in rela- 
tive units) of the first five lines on the Debye pattern of silver made with Cu Ka 
radiation. Ignore the temperature and absorption factors. 

4-^8. A Debye-Scherrer pattern of tungsten (BCC) is made with Cu Ka radia- 
tion. The first four lines on this pattern were observed to have the following 8 
values: 

Line 6 

1 20.3 

2 29.2 

3 36.7 

4 43.6 

Index these lines (i.e., determine the Miller indices of each reflection by the use 
of Eq. (3-10) and Appendix 6) and calculate their relative integrated intensities. 

4-9. A Debye-Scherrer pattern is made of gray tin, which has the same struc- 
ture as diamond, with Cu Ka radiation. What are the indices of the first two lines 
on the pattern, and what is the ratio of the integrated intensity of the first to that 
of the second? 

4-10. A Debye-Scherrer pattern is made of the intermediate phase InSb with 
Cu Ka radiation. This phase has the zinc-blende structure and a lattice parameter 
of 6.46A. What are the indices of the first two lines on the pattern, and what is 
the ratio of the integrated intensity of the first to the second? 

4-11. Calculate the relative integrated intensities of the first six lines of the 
Debye-Scherrer pattern of zinc, made with Cu Ka radiation. The indices and ob- 
served 6 values of these lines are: 

Line hkl 6 



1 


00-2 


18.8 


2 


10-0 


20.2 


3 


10-1 


22.3 


4 


10-2 


27.9 


5 


11-0, 10-3 


36.0 


6 


00-4 


39.4 



(Line 5 is made up of two unresolved lines from planes of very nearly the same 
spacing.) Compare your results with the intensities observed in the pattern 
shown in Fig. 3-13(b). 



CHAPTER 5 
LAUE PHOTOGRAPHS 

6-1 Introduction. The experimental methods used in obtaining diffrac- 
tion patterns will be described in this chapter and the two following ones. 
Here we are concerned with the Laue method only from the experimental 
viewpoint; its main applications will Be dealt with in Chap. 8. 

Laue photographs are the easiest kind of diffraction pattern to make and 
require only the simplest kind of apparatus. White radiation is necessary, 
and the best source is a tube with a heavy-meta! target, such as tungsten, 
since the intensity of the continuous spectrum is proportional to the atomic 
number of the target metal. Good patterns can also be obtained with 
radiation from other metals, such as molybdenum or copper. Ordinarily, 
the presence of strong characteristic components, such as W Lai, Cu Ka, 
Mo Ka, etc., in the radiation used, does not complicate the diffraction 
pattern in any way or introduce difficulties in its interpretation. Such a 
component will only be reflected if a set of planes in the crystal happens to 
be oriented in just such a way that the Bragg law is satisfied for that com- 
ponent, and then the only effect will be the formation of a Laue spot of 
exceptionally high intensity. 

The specimen used in the Laue method is a single crystal. This may 
mean an isolated single crystal or one particular crystal grain, not too 
small, in a polycrystalline aggregate. The only restriction on the size of a 
crystal in a polycrystalline mass is that it must be no smaller than the 
incident x-ray beam, if the pattern obtained is to correspond to that crystal 
alone. 

Laue spots are often formed by overlapping reflections of different 
orders. For example, the 100, 200, 300, . . . reflections are all superimposed 
since the corresponding planes, (100), (200), (300), ... are all parallel. 
The first-order reflection is made up of radiation of wavelength X, the 
second-order of X/2, the third-order of X/3, etc., down to XSWL, the short- 
wavelength limit of the continuous spectrum. 

The position of any Laue spot is unaltered by a change in plane spacing, 
since the only effect of such a change is to alter the wavelength of the 
diffracted beam. It follows that two crystals of the same orientation and 
crystal structure, but of different lattice parameter, will produce identical 
Laue patterns. 

5-2 Cameras. Laue cameras are so simple to construct that home- 
made models are found in a great many laboratories. Figure 5-1 shows 
a typical transmission camera, in this case a commercial unit, and Fig. 

138 



5-2] 



CAMERAS 



139 




FIG. 5-1. Transmission Laue camera. Specimen holder not shown. (Courtesy 
of General Electric Co., X-Ray Department.) 

5-2 illustrates its essential parts. A is the collimator, a device used to 
produce a narrow incident beam made up of rays as nearly parallel as pos- 
sible; it usually consists of two pinholes in line, one in each of two lead 
disks set into the ends of the collimator tube. (7 is the single-crystal 



specimen supported on the holder B. 
cassette, made of a frame, a removable 
metal back, and a sheet of opaque 
paper; the film, usually 4 by 5 in. in 
size, is sandwiched between the metal 
back and the paper. S is the beam 
stop, designed to prevent the trans- 
mitted beam from striking the film 
and causing excessive blackening. A 



F is the light-tight film holder, or 




FIG. 5-2. Transmission Laue camera. 



140 LAUE PHOTOGRAPHS [CHAP. 5 

small copper disk, about 0.5 mm thick, cemented on the paper film cover 
serves very well for this purpose: it stops all but a small fraction of the 
beam transmitted through the crystal, while this small fraction serves to 
record the position of this beam on the film. The shadow of a beam stop 
of this kind can be seen in Fig. 3-6(a). 

The Bragg angle corresponding to any transmission Laue spot is found 
very simply from the relation 

tan 20 = -> (5-1) 

D 

where r\ = distance of spot from center of film (point of incidence of trans- 
mitted beam) and D = specimen-to-film distance (usually 5 cm). Adjust- 
ment of the specimen-to-film distance is best made by using a feeler gauge 
of the correct length. 

The voltage applied to the x-ray tube has a decided effect on the appear- 
ance of a transmission Laue pattern. It is of course true that the higher 
the tube voltage, the more intense the spots, other variables, such as tube 
current and exposure time, being held constant. But there is still another 
effect due to the fact that the continuous spectrum is cut off sharply on 
the short-wavelength side at a value of the wavelength which varies in- 
versely as the tube voltage [Eq. (1-4)]. Laue spots near the center of a 
transmission pattern are caused by first-order reflections from planes in- 
clined at very small Bragg angles to the incident beam. Only short-wave- 
length radiation can satisfy the Bragg law for such planes, but if the tube 
voltage is too low to produce the wavelength required, the corresponding 
Laue spot will not appear on the pattern. It therefore follows that there 
is a region near the center of the pattern which is devoid of Laue spots and 
that the size of this region increases as the tube voltage decreases. The 
tube voltage therefore affects not only the intensity of each spot, but also 
the number of spots. This is true also of spots far removed from the center 
of the pattern; some of these are due to planes so oriented and of such a 
spacing that they reflect radiation of wavelength close to the short-wave- 
length limit, and such spots will be eliminated by a decrease in tube voltage 
no matter how long the exposure. 

A back-reflection camera is illustrated in Figs. 5-3 and 5-4.. Here the 
cassette supports both the film and the collimator. The latter has a re- 
duced section at one end which screws into the back plate of the cassette 
and projects a short distance in front of the cassette through holes punched 
in the film and its paper cover. 

The Bragg angle for any spot on a back-reflection pattern may be 
found from the relation 

tan (180 - 20) = -> (5-2) 



6-2] 



CAMERAS 



141 




t Back - re f ec L tion L aue camera. The specimen holder shown permits 

the h tm 1 th6 Spedme \ " We " as rotation about an ** Pail to 

the incident beam. The specimen shown is a coarse-grained polycrystaJline one 
poBitioned so that only a single, selected grain will be struck by the incident beam! 
FIG. 5-4. Back-reflection Laue camera (schematic). 

where r 2 = distance of spot from center of film and D = specimen-to-film 
distance (usually 3 cm). In contrast to transmission patterns, back-reflec- 
tion patterns may have spots as close to the center of the film as the size 
of the colhmator permits. Such spots are caused by high-order over- 
lapping reflections from planes almost perpendicular to the incident beam 
bmce each diffracted beam is formed of a number of wavelengths the only 
effect of a decrease in tube voltage is to remove one or more short-wave- 
ength components from some of the diffracted beams. The longer wave- 
lengths will still be diffracted, and the decrease in voltage will not in 
general, remove any spots from the pattern. ' 

Transmission patterns can usually be obtained with much shorter ex- 
posures than back-reflection patterns. For example, with a tungsten- 
target tube operating at 30 kv and 20 ma and an aluminum crystal about 
1 mm thick, the required exposure is about 5 min in transmission and 
30 mm in back reflection. This difference is due to the fact that the atomic 
scattering factor / decreases as the quantity (sin0)/A increases, and this 



142 



LAUE PHOTOGRAPHS 



[CHAP. 5 



quantity is much larger in back reflection than in transmission. Trans- 
mission patterns are also clearer, in the sense of having greater contrast 
between the diffraction spots and the background, since the coherent 
scattering, which forms the spots, and the incoherent (Compton modified) 
scattering, which contributes to the background, vary in opposite ways 
with (sin 0)/X. The incoherent scattering reaches its maximum value in 
the back-reflection region, as shown clearly in Fig. 3-6(a) and (b); it is 
in this region also that the temperature-diffuse scattering is most intense. 
In both Laue methods, the short-wavelength radiation in the incident 
beam will cause most specimens to emit K fluorescent radiation. If this 
becomes troublesome in back reflection, it may be minimized by placing a 
filter of aluminum sheet 0.01 in. thick in front of the film. 

If necessary, the intensity of a Laue spot may be increased by means 
of an intensifying screen, as used in radiography. This resembles a fluores- 
cent screen in having an active material coated on an inert backing such 
as cardboard, the active material having the ability to fluoresce in the 
visible region under the action of x-rays. When such a screen is placed 
with its active face in contact with the film (Fig. 5-5), the film is blackened 
not only by the incident x-ray beam but also by the visible light which 
the screen emits under the action of the beam. Whereas fluorescent screens 
emit yellow light, intensifying screens are designed to emit blue light, 
which is more effective than yellow in blackening the film. Two kinds of 
intensifying screens are in use today, one containing calcium tungstate 
and the other zinc sulfide with a trace of silver; the former is most effective 
at short x-ray wavelengths (about 0.5A or less), while the latter can be 
used at longer wavelengths. 

An intensifying screen should not be used if it is important to record 
fine detail in the Laue spots, as in some studies of crystal distortion, since 
the presence of the screen will cause the spots to become more diffuse than 



paper screen 

film / back plate 




r 




diffracted 
beam 




emulsion 
film base 



D 



active side 
of screen 

FIG. 5-5. Arrangement of film and 
intensifying screen (exploded view). 



(a) (b) 

FIG. 5-6. Effect of double-coated film 
on appearance of Laue spot: (a) section 
through diffracted beam and film; (b) 
front view of doubled spot on film. 



5-3] 



SPECIMEN HOLDERS 



143 



they would ordinarily bo. Each particle of the screen which is struck by 
x-rays emits light in all directions and therefore blackens the film outside 
the region blackened by the diffracted beam itself, as suggested in Fig. 5-5. 
This effect is aggravated by the fact that most x-ray film is double-coated, 
the two layers of emulsion being separated by an appreciable thickness of 
film base. Even when an intensifying screen is not used, double-coated 
film causes the size of a diffraction spot formed by an obliquely incident 
beam to be larger than the cross section of the beam itself; in extreme 
cases, an apparent doubling of the diffraction spot results, as shown in 
Fig. 5-0. 

5-3 Specimen holders. Before going into the question of specimen 
holders, we might consider the specimen itself Obviously, a specimen for 
the transmission method must have low enough absorption to transmit the 
diffracted beams; in practice, this means that relatively thick specimens 
of a light element like aluminum may be used but that the thickness of a 
fairly heavy element like copper must be reduced, by etching, for example, 
to a few thousandths of an inch On the other hand, the specimen must 
not be too thin or the diffracted intensity will be too low, since the intensity 
of a diffracted beam is proportional to the volume of diffracting material. 
In the back-reflection method, there is no restriction on the specimen 
thickness and quite massive specimens may be examined, since the dif- 
fracted beams originate in only a thin surface layer of the specimen. This 
difference between the two methods may be stated in another way and 
one which is well worth remembering: any information about a thick 
specimen obtained by the back-reflection method applies only to a 
thin surface layer of that specimen, 
whereas information recorded on a 
transmission pattern is represent at ive 
of the complete thickness of the speci- 
men, simply because the transmission 
specimen must necessarily be thin 
enough to transmit diffracted beams 
from all parts of its cross section.* 

There is a large variety of specimen 
holders in use, each suited to some 
particular purpose. The simplest 
consists of a fixed post to which the 
specimen is attached with wax or 
plasticine. A more elaborate holder is 
required when it is necessary to set a 
crystal in some particular orientation 




FIG* 5-7. Goniometer with 
rotation axes, (Courtesy of 
Supper Co,) 



' See Sec. 9-5 for further discussion of this point. 



144 LAUE PHOTOGRAPHS [CHAP. 5 

relative to the x-ray beam. In this case, a three-circle goniometer is used 
(Fig. 5-7) ; it has three mutually perpendicular axes of rotation, two hori- 
zontal and one vertical, and is so constructed that the crystal, cemented 
to the tip of the short metal rod at the top, is not displaced in space by 
any of the three possible rotations. 

In the examination of sheet specimens, it is frequently necessary to 
obtain diffraction patterns from various points on the surface, and this 
requires movement of the specimen, between exposures, in two directions 
at right angles in the plane of the specimen surface, this surface being per- 
pendicular to the incident x-ray beam. The mechanical stage from a 
microscope can be easily converted to this purpose. 

It is often necessary to know exactly where the incident x-ray beam 
strikes the specimen, as, for example, when one wants to obtain a pattern 
from a particular grain, or a particular part of a grain, in a polycrystalline 
mass. This is sometimes a rather difficult matter in a back-reflection 
camera because of the short distance between the film and the specimen. 
One method is to project a light beam through the collimator and observe 
its point of incidence on the specimen with a mirror or prism held near the 
collimator. An even simpler method is to push a stiff straight wire through 
the collimator and observe where it touches the specimen with a small 
mirror, of the kind used by dentists, fixed at an angle to the end of a rod. 

6-4 Collimators. Collimators of one kind or another are used in all 
varieties of x-ray cameras, and it is therefore important to understand their 
function and to know what they can and cannot do. To "collimate" 
means, literally, to "render parallel," and the perfect collimator would 
produce a beam composed of perfectly parallel rays. Such a collimator 
does not exist, and the reason, essentially, lies in the source of the radia- 
tion, since every source emits radiation in all possible directions. 

Consider the simplest kind of collimator (Fig. 5-8), consisting of two 
circular apertures of diameter d separated by a distance u, where u is 
large compared to d. If there is a point source of radiation at S, then all 
the rays in the beam from the collimator are nonparallel, and the beam is 
conical in shape with a maximum angle of divergence f$\ given by the 




FIG. 5-8. Pinhole collimator and small source. 



5-4] 
equation 



COLLIMATORS 

t Hi d/2 

tan = 

2 v 



145 



where v is the distance of the exit pinhole from the source. Since 1 is 
always very small, this relation can be closely approximated by the equa- 
tion 

d 

ft i = - radian. (5-3) 

v 

Whatever we do to decrease 0\ and therefore render the beam more 
nearly parallel will at the same time decrease the energy of the beam. We 
note also that the entrance pinhole serves no function when the source is 
very small, and may be omitted. 

No actual source is a mathematical point, and, in practice, we usually 
have to deal with x-ray tubes which have focal spots of finite size, usually 
rectangular in shape. The projected shape of such a spot, at a small target- 
to-beam angle, is either a small square or a very narrow line (Fig. 1-16), 
depending on the direction of projection. Such sources produce beams 
having parallel, divergent, and convergent rays. 

Figure 5-9 illustrates the case when the projected source shape is square 
and of such a height h that convergent rays from the edges of the source 
cross at the center of the collimator and then diverge. The maximum 
divergence angle is now given by 



,. 

($2 = radian, 
u 



(5-4) 



and the center of the collimator may be considered as the virtual source of 
these divergent rays. The beam issuing from the collimator contains not 
only parallel and divergent rays but also convergent ones, the maximum 
angle of convergence being given by 



u + w 



radian, 



(5-5) 




FIG. 5-9. Pinhole collimator and large source. S = source, (7 = crystal. 



146 LAUE PHOTOGRAPHS [CHAP. 5 

where w is the distance of the crystal from the exit pinhole. The size of 
the source shown in Fig. 5-9 is given by 



/2u \ 
-d(--l). 
\u / 



(5-6) 



In practice, v is very often about twice as large as u, which means that the 
conditions illustrated in Fig. 5-9 are achieved when the pinholes are about 
one-third the size of the projected source. If the value of h is smaller than 
that given by Eq. (5-6), then conditions will be intermediate between 
those shown in Figs. 5-8 and 5-9; as h approaches zero, the maximum 
divergence angle decreases from the value given by Eq. (5-4) to that given 
by Eq. (5-3) and the proportion of parallel rays in the beam and the max- 
imum convergence angle both approach zero. When h exceeds the value 
given by Eq. (5-6), none of the conditions depicted in Fig. 5-9 are changed, 
and the increase in the size of the source merely represents wasted energy. 

When the shape of the projected source is a fine line, the geometry of 
the beam varies between two extremes in two mutually perpendicular 
planes. In a plane at right angles to the line source, the shape is given by 
Fig. 5-8, and in a plane parallel to the source by Fig. 5-9. Aside from the 
component which diverges in the plane of the source, the resulting beam 
is shaped somewhat like a wedge. Since the length of the line source 
greatly exceeds the value given by Eq. (5-6), a large fraction of the x-ray 
energy is wasted with this arrangement of source and collimator. 

The extent of the nonparallelism of actual x-ray beams may be illus- 
trated by taking, as typical values, d = 0.5 mm, u = 5 cm, and w = 3 cm. 
Then Eq. (5-4) gives 2 = 1.15 and Eq. (5-5) gives a = 0.36. These 
values may of course be reduced by decreasing the size of the pinholes, for 
example, but this reduction will be obtained at the expense of decreased 
energy in the beam and increased exposure time. 

6-5 The shapes of Laue spots. We will see later that Laue spots be- 
come smeared out if the reflecting crystal is distorted. Here, however, 
we are concerned with the shapes of spots obained__from perfect, undis- 
torted crystals. These shapes are greatly influenced by the nature of the 
incident beam, i.e., by its convergence or divergence, and it is important 
to realize this fact, or Laue spots of "unusual" shape may be erroneously 
taken as evidence of crystal distortion. 

Consider the transmission case first, and assume that the crystal is thin 
and larger than the cross section of the primary beam at the point of inci- 
dence. If this beam is mainly divergent, which is the usual case in practice 
(Fig. 5-8 or 5-9), then a focusing action takes place on diffraction. Figure 
5-10 is a section through the incident beam and any diffracted beam; the 
incident beam, whose cross section at any point is circular, is shown issuing 



5-5] 



THE SHAPES OF LAUE SPOTS 



147 



H 



FIG. 5-10. Focusing of diffracted beam in the transmission Laue method. S T = 
source, C = crystal, F = focal point. 

from a small source, real or virtual. Each ray of the incident beam which 
lies in the plane of the drawing strikes the reflecting lattice planes of the 
crystal at a slightly different Bragg angle, this angle being a maximum i ' 
A and decreasing progressively toward B. The lowermost rays are there- 
fore deviated through a greater angle 28 than the upper ones, with the 
result that the diffracted beam converges to a focus at F. This is true 
only of the rays in the plane of the drawing; those in a plane at right angles 
continue to diverge after diffraction, with the result that the diffracted 
beam is elliptical in cross section. The film intersects different diffracted 
beams at diJerent distances from the crystal, so elliptical spots of various 
sizes are observed, as shown in Fig. 5-11. This is not a sketch of a Laue 
pattern but an illustration of spot size and shape as a function of spot 
position in one quadrant of the film. Note that the spots are all elliptical 
with their minor axes aligned in a radial direction and that spots near the 
center and edge of the pattern are thicker than those in intermediate posi- 
tions, the latter being formed by beams near their focal point. Spots 
having the shapes illustrated are fairly common, and Fig. 3-6(a) is an 
example. 

In back reflection, no focusing oc- 
curs and a divergent incident beam 
intinues to diverge in all directions 
ter diffraction. Back-reflection 
le spots are therefore more or less 
* near the center of the pat- 
1 they become increasingly 
ward the edge, due to the 
>nce of the rays on the 
)r axes of the ellipses 
lately radial. Figure FlG . ^_ 1L shape of transmission 
.al. Laue spots as a function of position. 



148 LAUE PHOTOGRAPHS [CHAP. 5 

PROBLEMS 

5-1. A transmission Laue pattern is made of an aluminum crystal with 40-kv 
tungsten radiation. The film is 5 cm from the crystal. How close to the center 
of the pattern can Laue spots be formed by reflecting planes of maximum spacing, 
namely (111), and those of next largest spacing, namely (200)? 

6-2. A transmission Laue pattern is made of an aluminum crystal with a speci- 
men-to-film distance of 5 cm. The (111) planes of the crystal make an angle of 
3 with the incident beam. What minimum tube voltage is required to produce a 
111 reflection? 

6-3. (a) A back-reflection Laue pattern is made of an aluminum crystal at 50 
kv. The (111) planes make an angle of 88 with the incident beam. What orders 
of reflection are present in the beam diffracted by these planes? (Assume that 
wavelengths larger than ? A are too weak and too easily absorbed by air to regis- 
ter on the film.) 

(6) What orders of the 111 reflection are present if the tube voltage is reduced 
' ) 40 kv? 



CHAPTER 6 
POWDER PHOTOGRAPHS 

6-1 Introduction. The powder method of x-ray diffraction was de- 
vised independently in 1916 by Debye and Scherrer in Germany and in 
1917 by Hull in the United States. It is the most generally useful of all 
diffraction methods and, when properly employed, can yield a great deal 
of structural information about the material under investigation. Basi- 
cally, this method involves the diffraction of monochromatic x-rays by a 
powder specimen. In this connection, "monochromatic" usually means 
the strong K characteristic component of the general radiation from an 
x-ray tube operated above the K excitation potential of the target mate-] 
rial. "Powder" can mean either an actual, physical powder held together 
with a suitable binder or any specimen in polycrystalline form. The 
method is thus eminently suited for metallurgical work, since single crys- 
tals are not always available to the metallurgist and such materials as 
polycrystalline wire, sheet, rod, etc., may be examined nondestructively 
without any special preparation. 

There are three main powder methods in use, differentiated by the rela- 
tive position of the specimen and film: 

(1) Debye-Scherrer method. The film is placed on the surface of a cylin- 
der and the specimen on the axis of the cylinder. 

(2) Focusing method. The film, specimen, and x-ray source are all placed 
on the surface of a cylinder. 

(3) Pinhole method. The film is flat, perpendicular to the incident x-ray 
beam, and located at any convenient distance from the specimen. 

In all these methods, the diffracted beams lie on the surfaces of cones 
whose axes lie along the incident beam or its extension; each cone of rays 
is diffracted from a particular set of lattice planes. In the Debye-Scherrer 
and focusing methods, only a narrow strip of film is used and the recorded 
diffraction pattern consists of short lines formed by the intersections of the 
cones of radiation with the film. In the pinhole method, the whole cone 
intersects the film to form a circular diffraction ring. 

6-2 Debye-Scherrer method. A typical Debye camera is shown in 
Fig. 6-1. It consists essentially of a cylindrical chamber with a light-tight 
cover, a collimator to admit and define the incident beam, a beam stop to 
confine and stop the transmitted beam, a means for holding the film 
tightly against the inside circumference of the camera, and a specimen 
holder that can be rotated. 

149 



150 



POWDER PHOTOGRAPHS 



[CHAP. 6 



\ 




FIG. 6-1. Debye-Scherrer camera, with cover plate removed. (Courtesy of 
North American Philips Company, Inc.) 

Camera diameters vary from about 5 to about 20 cm. The greater the 
diameter, the greater the resolution or separation of a particular pair of 
lines on the film. In spectroscopy, resolving power is the power of dis- 
tinguishing between two components of radiation which have wavelengths 
very close together and is given by X/AX, where AX is the difference be- 
tween the two wavelengths and X is their mean value; in crystal-structure 
analysis, we may take resolving power as the ability to separate diffraction 
lines from sets of planes of very nearly the same spacing, or as the value 
of d/M. * Thus, if S is the distance measured on the film from a particular 
diffraction line to the point where the transmitted beam would strike the 
film (Fig. 6-2), then 

S = 2dR 

* Resolving power is often defined by the quantity AX/X, which is the reciprocal 
of that given above. However, the power of resolving two wavelengths which are 
nearly alike is a quantity which should logically increase as AX, the difference be- 
tween the two wavelengths to be separated, decreases. This is the reason for the 
definition given in the text. The same argument applies to interplanar spacings d. 




6-2] DEBYE-SCHERRER METHOD 151 

and AS = #A20, (6-1) 

where R is the radius of the camera. Two sets of planes of very nearly 
the same spacing will give rise to two diffracted beams separated by a 
small angle A20; for a given value of A20, Eq. (6-1) shows that AS, the 
separation of the lines on the film, increases with R. The resolving power 
may be obtained by differentiating the Bragg law:* 

X = 2d sin 

d0 -1 

= tan 0. , (6-2) 

dd d 
But 

_ dS 
6 ~ 2R 
Therefore 

dS 2R ^ 1G ' ^"^' ^ eome ^ r y 

= '. an 0, Scherrer method. Section through 

dd d film and one diffraction cone. ^ 

d -2R 

Resolving power = = tan 0, (6-3, 

Arf AS 

where d is the mean spacing of the two sets of planes, Ad the difference in 
their spacings, and AS the separation of two diffraction lines which appear 
just resolved on the film. Equation (6-3) shows that the rcsolyjng power 
increases with the size of the camera; this increased resolution is obtained, 
however, at the cost of increased exposure time, and the smaller cameras 
are usually preferred for all but the most complicated patterns. A camera 
diameter of 5.73 cm is often used and will be found suitable for most work. 
This particular diameter, .equal to 1/10 the number of degrees in a radian, 
facilitates calculation, since 0, (in degrees) is obtained simply by multipli- 
cation of S (in cm) by 10, except for certain corrections necessary in pre- 
cise work. Equation (6-3) also shows that the resolving power of a given 
camera increases with 0, being directly proportional to tan 0. 

The increased exposure time required by an increase in camera diameter 
is due not only to the decrease in intensity <rf the diffracted beam with 
increased distance from the specimen, but also to the partial absorption 
of both the incident and diffracted beams by the air in the camera. For 
example, Prob. 1-7 and the curves of Fig. 6-3 show that, in a camera of 
19 cm diameter (about the largest in common use), the decrease in in- 
tensity due to air absorption is about 20 percent for Cu Ka radiation and 
about 52 percent for Cr Ka radiation. This decrease in intensity may be 

* A lower-case roman d is used throughout this book for differentials in order to 
avoid confusion with the symbol d for distance between atomic planes. 



152 



POWDER PHOTOGRAPHS 



[CHAP. 6 



avoided by evacuating the camera or 
by filling it with a light gas such as 
hydrogen or helium during the ex- 
posure. 

Correct design of the pinhole system 
which collimates the incident beam is 
important, especially when weak dif- 
fracted beams must be recorded. The 
exit pinhole scatters x-rays in all di- 
rections, and these scattered rays, if 
not prevented from striking the film, 
can seriously increase the intensity 
of the background. A "guarded- 
pinhole" assembly which practically 




5 10 15 20 

PATH LENGTH (cm) 

FIG. 6-3. Absorption of Cu Ka and 
Cr Ka radiation by air. 



eliminates this effect is shown in Fig. 6-4, where the divergent and con- 
vergent rays in the incident beam are ignored and only the parallel com- 
ponent is shown. The collimator tube is extended a considerable distance 
beyond the exit pinhole and constricted so that the end A is close enough 
to the main beam to confine the radiation scattered by the exit pinhole 
to a very narrow angular range and yet not close enough to touch the 
main beam and be itself a cause of further scattering. The beam stop is 
usually a thick piece of lead glass placed behind a fluorescent screen, the 
combination allowing the transmitted beam to be viewed with safety when 
adjusting the camera in front of the x-ray tube. Back scatter from the 
stop is minimized by extending the beam-stop tube backward and con- 
stricting its end B. Another reason for extending the collimator and 
beam-stop tubes as close to the specimen as possible is to minimize the 
extent to which the primary beam is scattered by air, as it passes through 
the camera. Both tubes are tapered to interfere as little as possible with 
low-angle and high-angle diffracted beams. 

Some cameras employ rectangular slits rather than pinholes to define 
the beam, the long edges of the slits being parallel to the axis of the speci- 




fluorescent 
screen 



FIG. 6-4. Design of collimator and beam stop (schematic). 



6-3] SPECIMEN PREPARATION 153 

men. The use of slits instead of pinholes decreases exposure time by in- 
creasing the irradiated volume of the specimen, but requires more accurate 
positioning of the camera relative to the source and produces diffraction 
lines which are sharp only along the median line of the film. 

6-8 Specimen preparation. Metals and alloys may be converted to 
powder by filing or, if they are sufficiently brittle, by grinding in a small 
agate mortar. In either case, the powder should be filed or ground as 
fine as possible, preferably to pass a 325-mesh screen, in order to produce 
smooth, continuous diffraction lines. The screened powder is usually an- 
nealed in evacuated glass or quartz capsules in order to relieve the strains 
due to filing or grinding. 

Special precautions are necessary in screening two-phase alloys. If a 
small, representative sample is selected from an ingot for x-ray analysis, 
then that entire sample must be ground or filed to pass through the screen. 
The common method of grinding until an amount sufficient for the x-ray 
specimen has passed the screen, the oversize being rejected, may lead to 
very erroneous results. One phase of the alloy is usually more brittle than 
the other, and that phase will more easily be ground into fine particles; if 
the grinding and screening are interrupted at any point, then the material 
remaining on the screen will contain less of the more brittle phase than the 
original sample while the undersize will contain more, and neither will be 
representative. 

The final specimen for the Debye camera should be in the form of a thin 
rod, 0.5 mm or less in diameter and about 1 cm long. There are various 
ways of preparing such a specimen, one of the simplest being to coat the 
powder on the surface of a fine glass fiber with a small amount of glue or 
petroleum jelly. Other methods consist in packing the powder into a thin- 
walled tube made of a weakly absorbing substance such as cellophane or 
lithium borate glass, or in extruding a mixture of powder and binder 
through a small hole. Polycrystalline wires may be used directly, but 
since they usually exhibit some preferred orientation, the resulting diffrac- 
tion pattern must be interpreted with that fact in mind (Chap. 9). Strongly 
absorbing substances may produce split low-angle lines (see Sec. 4-10); 
if this effect becomes troublesome, it may be eliminated by diluting the 
substance involved with some weakly absorbing substance, so that the 
absorption coefficient of the composite specimen is low. Both flour and 
cornstarch have been used for this purpose. The diluent chosen should 
not produce any strong diffraction lines of its own and too much of it 
should not be used, or the lines from the substance being examined will 
become spotty. 

After the specimen rod is prepared, it is mounted in its holder so that it 
will lie accurately along the axis of the camera when the specimen holder 



154 POWDER PHOTOGRAPHS [CHAP. 6 

is rotated. (Rotation of the specimen during the exposure is common prac- 
tice but not an intrinsic part of the powder method; its only purpose is to 
produce continuous, rather than spotty, diffraction lines by increasing the 
number of powder particles in reflecting positions. | 

6-4 Film loading. Figure 6-5 illustrates three methods of arranging 
the film strip in the Debye method. The small sketches on the right show 
the loaded film in relation to the incident beam, while the films laid out 
flat are indicated on the left. In (a), a hole is punched in the center of the 
film so that the film may be slipped over the beam stop; the transmitted 
beam thus leaves through the hole in the film. The pattern is symmetrical 
on either side, and the 6 value of a particular reflection is obtained by 
measuring U, the distance apart of two diffraction lines formed by the 
same cone of radiation, and using the relation 

4BR = U. 

Photographic film always shrinks slightly during processing and drying, 
and this shrinkage effectively changes the camera radius. The film-shrink- 
age error may be allowed for by slipping the ends of the film under metal 
knife-edges which cast a sharp shadow near each end of the film. In this 
way, a standard distance is impressed on the film which will shrink in the 
same proportion as the distance between a given pair of diffraction lines. 
If the angular separation 40* of the knife-edges in the camera is known, 
either by direct measurement or by calibration with a substance of known 
lattice parameter, then the value of for a particular reflection may be 
obtained by simple proportion: 

6 U 



where UK is the distance apart of the knife-edge shadows on the film. 

Figure 6-5(b) illustrates a method of loading the film which is just the 
reverse of the previous one. Here the incident beam enters through the 
hole in the film, and is obtained from the relation 

(27T - 4S)R ^ V. 

Knife-edges may also be used in this case as a basis for film-shrinkage cor- 
rections. 

The unsymmetrical, or Straumanis, method of film loading is shown in 
Fig. 6-5 (c). Two holes are punched in the film so that it may be slipped 
over both the entrance collimator and the beam stop. Since it is possible 
to determine from measurements on the film where the incident beam en- 
tered the film circle and where the transmitted beam left it, no knife-edges 
are required to make the film-shrinkage correction. The point X (20 = 



6-4] 



FILM LOADING 



155 



5 4 



knife-edge 
shadow 



2 1 



12 



45 





26 



12 3 4 5 5 4 




(c) 



4 3 o 4 3 21 12 


(( 1 


r)) 




M 


r)) 




ir 


.s 






FIG. 6-5. Methods of film loading in Debye cameras, 
have the same numbers in all films. 



Corresponding lines 



180), where the incident beam entered, is halfway between the measured 
positions of lines 5,5; similarly, the point Y (26 = 0), where the trans- 
mitted beam left, is halfway between lines 1,1. The difference between 
the positions of X and Y gives W, and 6 is found by proportion : 

29 _ 8 
7 ~ W 

Unsymmetrical loading thus provides for the film-shrinkage correction 
without calibration of the camera or knowledge of any camera dimension. 
The shapes of the diffraction lines in Fig. 6-5 should be noted. The low- 
angle lines are strongly curved because they are formed by cones of radia- 
tion which have a small apex angle 48. The same is true of the high-angle 
lines, although naturally they are curved in the opposite direction. Lines 
for which 40 is nearly equal to 180 are practically straight. This change 
of line shape with change in 6 may also be seen in the powder photographs 
shown in Fig. 3-13. 



156 POWDER PHOTOGRAPHS [CHAP. 6 

6-6 Cameras for high and low temperatures. Metallurgical investiga- 
tions frequently require that the crystal structure of a phase stable only 
at high temperature be determined. In many cases, this can be accom- 
plished by quenching the specimen at a high enough rate to suppress the 
decomposition of the high-temperature phase and then examining the 
specimen in an ordinary camera at room temperature. In other cases, the 
transformation into the phases stable at room temperature cannot be sup- 
pressed, and a high-temperature camera is necessary in order that the 
specimen may be examined at the temperature at which the phase in ques- 
tion is stable. 

The design of high-temperature Debye cameras varies almost from 
laboratory to laboratory. They all involve a small furnace, usually of the 
electric-resistance type, to heat the specimen and a thermocouple to meas- 
ure its temperature. The main design problem is to keep the film cool 
without too great an increase in the camera diameter; this requires water- 
cooling of the body of the camera and/or the careful placing of radiation 
shields between the furnace and the film, shields so designed that they will 
not interfere with the diffracted x-ray beams. The furnace which sur- 
rounds the specimen must also be provided with a slot of some kind to 
permit the passage of the incident and diffracted beams. If the specimen 
is susceptible to oxidation at high temperatures, means of evacuating the 
camera or of filling it with an inert gas must be provided; alternately, the 
powder specimen may be sealed in a thin-walled silica tube. Because of 
the small size of the furnace in a high-temperature camera, the tempera- 
ture gradients in it are usually quite steep, and special care must be taken 
to ensure that the temperature recorded by the thermocouple is actually 
that of the specimen itself. Since the intensity of any reflection is de- 
creased by an increase in temperature, the exposure time required for a 
high-temperature diffraction pattern is normally rather long. 

Debye cameras are also occasionally required for work at temperatures 
below room temperature. Specimen cooling is usually accomplished by 
running a thin stream of coolant, such as liquid air, over the specimen 
throughout the x-ray exposure. The diffraction pattern of the coolant will 
also be recorded but this is easily distinguished from that of a crystalline 
solid, because the typical pattern of a liquid contains only one or two very 
diffuse maxima in contrast to the sharp diffraction lines from a solid. Scat- 
tering from the liquid will, however, increase the background blackening 
of the photograph. 

6-6 Focusing cameras. Cameras in which diffracted rays originating 
from an extended region of the specimen all converge to one point on the 
film are called focusing cameras. The design of all such cameras is based 
on the following geometrical theorem (Fig. 6-6) : all angles inscribed in a 



6-7] 



8EEMANN-BOHLIN CAMERA 



157 




FIG. 6-6. Geometry of focusing cameras. 

circle and based on the same arc SF are equal to one another and equal to 
half the angle subtended at the center by the same arc. Suppose that 
x-rays proceeding in the directions SA and SB encounter a powder speci- 
men located on the arc AB. Then the rays diffracted by the (hkl) planes 
at points A and B will be deviated through the same angle 26. But these 
deviation angles 26 are each equal to (180 a), which means that the 
diffracted rays must proceed along AF and BF, and come to a focus at F 
on a film placed along the circumference of the circle. 

6-7 Seemann-Bohlin camera. This focusing principle is utilized in the 
Seemann-Bohlin camera shown in Fig. 6-7. The slit S acts as a virtual 
line source of x-rays, the actual source being the extended focal spot on 
the target T of the x-ray tube. Only converging rays from the target can 
enter this slit and, after passing it, they diverge to the specimen AB. 
(Alternatively, if a tube with a fine-line focal spot is available, the slit 
may be eliminated and exposure time shortened by designing the camera 
to use the focal spot itself as a source of divergent radiation.) For a par- 
ticular hkl reflection, each ray is then diffracted through the same angle 
26, with the result that all diffracted rays from various parts of the spec- 
imen converge to a focus at F. As in any powder method, the diffracted 
beams lie on the surfaces of cones whose axes are coincident with the inci- 
dent beam; in this case, a number of incident beams contribute to each 
reflection and a diffraction line is formed by the intersection of a number 
of cones with the film. As in the Debye-Scherrer method, a diffraction 
line is in general curved, the amount of curvature depending on the par- 



158 



POWDER PHOTOGEAPHS 

A 



[CHAP. 6 



B 



N 




film 



FIG. 6-7. Seemann-Bohlin focusing camera. Only one hkl reflection is shown. 

ticular value of 6 involved. Figure 0-8 shows a typical powder pattern 
made with this camera. 

The ends of the film strip are covered by knife-edges M and N, which 
cast reference shadows on the film. The value of 6 for any diffraction line 
may be found from the distance U, measured on the film, from the line to 
the shadow of the low-angle knife-edge N, by use of the relation 



46R 



*rcSABN. 



(6-4) 



In practice, is found by calibrating the camera with a standard sub- 
stance of known lattice parameter, such as NaCl, rather than by the use 
of Eq. (6-4). Several patterns are prepared of the same standard with 
radiations of different wavelength, in order to obtain diffraction lines at a 
large number of 26 positions. Line positions are measured on each film, 
as well as the total length of the film between the knife-edge shadows M 
and N. Because of variable film shrinkage, these films will generally have 
unequal lengths. The length of one is taken as a standard, and a multiply- 
ing factor is found for each of the other films which will make its length 
equal to the standard length. This factor is then applied to the U value 
of each diffraction line. The corrected values of U are then plotted against 
calculated values of 6 to obtain a calibration curve for the camera. 



FIG. 6-8. Powder pattern of tungsten, made in a Seemann-Bohlin camera, 8.4 
cm in diameter. This camera covers a 28 range of 92 to 166. High-angle end of 
film at left. Filtered copper radiation. (Courtesy of John T. Norton.) 



6-7] SEEMANN-BOHLIN CAMERA 159 

A similar procedure is then followed when an "unknown" specimen is 
being examined. A correction factor is found which will convert the meas- 
ured film length of the unknown to the standard length. This factor is 
then applied to each measured U value before finding the corresponding 
value from the calibration curve. 

If more accuracy is desired than this graphical method can give, the 
calibration data can be handled analytically. Equation (6-4) is written 
in the form 

= K 1 U + K 2 , 

where KI and K 2 are constants. The values of these constants are then 
determined by the method of least squares (see Sec. 11-6). Once the 
constants are known, this equation can be used to calculate 0, or a table 
of corresponding and U values can be constructed. 
By differentiating Eq. (0-4), we obtain 

dU 

dd = 
4R 

This relation may be combined with Eq. ((5-2) to give 

dU 4R 

= tan 6. 

dd d 

d 4R 

Resolving power = = tan 6. (6-5) 

M AU 

The resolving power, or ability to separate diffraction lines from planes 
of almost the same spacing, is therefore twice that of a Debye-Scherrer 
camera of the same radius. In addition, the exposure time is much shorter, 
because of the fact that a much larger specimen is used (the arc AB of 
Fig. 6-7 is of the order of 1 cm) and diffracted rays from a considerable 
volume of material are all brought to one focus. The Seemann-Bohlin 
camera is, therefore, very useful in studying complex diffraction patterns, 
whether they are due to a single phase or to a mixture of phases such as 
occur in alloy systems. 

For metallurgical work, this camera has the further advantage that a 
massive polycrystalline specimen may be used as well as a powder. For 
example, a metallographic specimen, mounted in the usual 1-in. diameter 
bakelite mount for microscopic examination, can be fastened to the cir- 
cumference of the camera and used directly. When a flat specimen placed 
tangentially to the camera circle is substituted for a curved specimen, the 
focusing action of the camera is slightly decreased but not objectionably 
so, while the advantage of being able to examine the same area of the 
specimen both with the microscope and with x-rays is obvious. It is 



160 POWDER PHOTOGRAPHS [CHAP. 6 

worth noting also that both methods of examination, the optical and the 
x-ray, provide information only about the surface layer of the specimen, 
since the x-ray method here involved is of the reflection, and not the trans- 
mission, type. 

A powder specimen may also be used in this camera by fixing a thin 
layer of the powder to a piece of paper with glue or petroleum jelly. The 
paper is then curved and held against the camera circumference by an 
attachment provided with the camera. Whether the specimen is in the 
massive or powder form, smoother diffraction lines can be obtained by 
oscillating the specimen about the camera axis. 

On the debit side, the Seemann-Bohlin camera has the disadvantage that 
the reflections registered on the film cover only a limited range of 26 values, 
particularly on the low-angle side; for this reason, it is better to make a 
preliminary survey of the whole pattern with a Debye camera, reserving 
the focusing camera for a closer study of certain portions. Some investiga- 
tors use a set of three Seemann-Bohlin cameras, designed to cover practically 
the whole range of 26 values in overlapping angular ranges. 

Diffraction lines formed in a Seemann-Bohlin camera are normally 
broader than those in a Debye-Scherrer pattern. The focused line is, in a 
sense, an image of the slit, and decreasing the slit opening will decrease 
the line breadth but increase the exposure time. The line breadth in- 
creases as 26 Becomes smaller, since at low 26 values the diffracted rays 
strike the film at a very low angle. This effect is aggravated by the double- 
emulsion film normally used for x-ray diffraction. In special cases, it may 
pay to use single-emulsion film at the cost of increased exposure time. 

6-8 Back-reflection focusing cameras. The most precise measurement 
of lattice parameter is made in the back-reflection region, as discussed in 
greater detail in Chap. 11. The most suitable camera for such measure- 
ments is the symmetrical back-reflection focusing camera illustrated in 
Fig. 6-9. 

It employs the same focusing principle as the Seemann-Bohlin camera, 
but the film straddles the slit and the specimen is placed diametrically 
opposite the slit. Means are usually provided for slowly oscillating the 
specimen through a few degrees about the camera axis in order to produce 
smooth diffraction lines. A typical film, punched in the center to allow 
the passage of the incident beam, is shown in Fig. 6-10. The value of 6 
for any diffraction line may be calculated from the relation 

(4T - 86)R = V, (6-6) 

where V is the distance on the film between corresponding diffraction lines 
on either side of the entrance slit. 



6-8] 



BACK-REFLECTION FOCUSING CAMERAS 



161 




film 



FIG. 6-9. Symmetrical back-reflection focusing camera. Only one hkl reflec- 
tion is shown. 



Differentiation of Eq. (6-6) gives 

--'ft- 

4R \2/ 



A0 



(6-7) 



where A(F/2) is the separation on the film of two reflections differing in 
Bragg angle by A0. Combination of this equation with Eq. (6-2) shows 
that 

d 4R 
Resolving power = = tan 6. 

M A(F/2) 

The resolving power of this camera is therefore the same as that of a 
Seemann-Bohlin camera of the same diameter. 

In the pattern shown in Fig. 6-10, two pairs of closely spaced lines can 
be seen, lines 1 and 2 and lines 4 and 5. Each pair is a doublet formed by 

321 i 




6 5 4 



3 2 1 



FIG. 6-10. Powder photograph of tungsten made in a symmetrical back-reflec- 
tion focusing camera, 4.00 in. in diameter. Unfiltered copper radiation. 



162 POWDER PHOTOGRAPHS [CHAP. 6 

reflection from one set of planes of the two components, Ka\ and Ka^ 
which make up Ka radiation. These component lines are commonly found 
to be resolved, or separated, in the back-reflection region. (The ft lines in 
this photograph are not resolved since K/3 radiation consists only of a 
single wavelength.) To determine the conditions under which a given 
camera can separate two components of radiation which have almost the 
same wavelength, we must use the spectroscopic definition of resolving 
power, namely X/AX, where AX is the difference between the two wave- 
lengths and X is their mean value. For Cu Ka radiation, these wave- 
lengths are : 

\(CuKa 2 ) = 1.54433A 

X(Cu#a!) = 1.54051 A 

AX = 0.00382A 
Therefore 

X 1.542 

= = 404. 

AX 0.00382 

The resolving power of the camera must exceed this value, for the partic- 
ular reflection considered, if the component lines are to be separated on 
the film. 

By differentiating the Bragg law, we obtain 

X = 2d sin 0, 
d<9 1 tan tan 



dX 2d cos S 2d sin 
X tan S 



AX A0 
Substitution of Eq. (0-7) gives 



(6-8) 



X 4J?tan0 

Resolving power = = (6-9) 

AX A(7/2) 

The negative sign here can be disregarded; it merely means that an in- 
crease in X causes a decrease in F/2, since the latter is measured from the 
center of the film. Equation (6-9) demonstrates that the resolving power 
increases with the camera radius and with 6, becoming very large near 90. 
This latter point is clearly evident in Fig. 6-10, which shows a greater 
separation of the higher-angle 400 reflections as compared to the 321 re- 
flections. 

By use of Eq. (6-9), we can calculate the resolving power, for the 321 
reflections, of the camera used to obtain Fig. 6-10. The camera radius is 



6-9] 



PINHOLE PHOTOGRAPHS 



163 



2.00 in., and the mean 6 value for these reflections is about 65.7. The 
line breadth at half maximum intensity is about 0.04 cm. The two com- 
ponent lines of the doublet will be clearly resolved on the film if their 
separation is twice their breadth. Therefore 



0- 



2(0.04) = 0.08 cm, 



X 
AX 



(4) (2.00) (2.54) (tan 05.7) 
(0.08) 



= 5(8. 



Since this value exceeds the resolving power of 404, found above to be 
necessary for resolution of the Cu Ka doublet, we would expect this doublet 
to be resolved for the 321 reflection, and such is seen to be the case in 
Fig. (>-10. At some lower angle, this would not be true and the two com- 
ponents would merge into a single, unresolved line. The fact that resolu- 
tion of the Ka doublet normally occurs only in the back-reflection region 
can be seen from the Debye photographs reproduced in Fig. 3-13. 

6-9 Pinhole photographs. When monochromatic radiation is used to 
examine a poly crystalline specimen in a Latie camera, the result is called, 
for no particularly good reason, a pinhole photograph. Either a trans- 
mission or a back-reflection camera may be used. A typical transmission 
photograph, made of fine-grained aluminum sheet, is shown in Fig. 6-11. 

The pinhole method has the ad- 
vantage that an entire Debye ring, 
and not just a part of it, is recorded 
on the film. On the other hand, the 
range of 6 values which are recorded 
is rather limited : either low-angle or 
high-angle reflections may be ob- 
tained, but not those in the median 




FIG. 6-11. Transmission pinhole 
photograph of an aluminum sheet 
specimen. Filtered copper radiation. 
(The diffuse circular band near the 
center is caused by white radiation. 
The nonuniform blackening of the 
Debye rings is due to preferred orien- 
tation in the specimen; see Chap. 9.) 




FIG. 6-12. Angular relationships in 
the pinhole method. 



164 POWDER PHOTOGRAPHS [CHAP. 6 

range of 6 (see Fig. 6-12). In the transmission method, the value of for 
a particular reflection is found from the relation 

U 

tan 21? = . (6-10) 

2D 

where U = diameter of the" Debye ring and D = specimen-to-film dis- 
tance. The corresponding relation for the back-reflection method is 

tan (* - 28) = > (6-11) 

where V = diameter of the Debye ring. The distance D is usually of the 
order of 3 to 5 cm. 

Powder specimens may be prepared simply by spreading a bit of the 
powder mixed with a binder on a glass slide or a small piece of paper. 
However, the greatest utility of the pinhole method in metallurgical work 
lies in the fact that massive, poly crystalline specimens may be used. In 
back reflection, mounted metallographic specimens may be examined di- 
rectly, while the transmission method is of course restricted to wire and 
sheet specimens which are not too highly absorbing. 

There is an optimum specimen thickness for the transmission method, 
because the diffracted beams will be very weak or entirely absent if the 
specimen is either too thin (insufficient volume of diffracting material) or 
too thick (excessive absorption). As will be shown in Sec. 9-9, the speci- 
men thickness which produces the maximum diffracted intensity is given 
by I/M, where M is the linear absorption coefficient of the specimen. In- 
spection of Eq. (1-10) shows that this condition can also be stated as 
follows: a transmission specimen is of optimum thickness when the inten- 
sity of the beam transmitted through the specimen is 1/c, or about , of 
the intensity of the incident beam. Normally this optimum thickness is 
of the order of a few thousandths of an inch. There is one way, however, 
in which a partial transmission pattern can be obtained from a thick 
specimen and that is by diffraction from an edge (Fig. 6-13). Only the 
upper half of the pattern is recorded on the film, but that is all that is 
necessary in many applications. The same technique has also been used 
in some Debye-Scherrer cameras. 

The pinhole method is used in studies of preferred orientation, grain 
size, and crystal perfection. With a back-reflection camera, fairly precise 
parameter measurements can be made by this method. Precise knowledge 
of the specimen-to-film distance D is not necessary, provided the proper 
extrapolation equation is used (Chap. 1 1) or the camera is calibrated. The 
calibration is usually performed for each exposure, simply by smearing a 
thin layer of the calibrating powder over the surface of the specimen; in 
this way, reference lines of known 8 value are formed on each film. 



6-10] 



CHOICE OF RADIATION 



165 




specimen 



(a) 




film 



(b) 



FIG. 6-13. Transmission pinhole method for thick specimens: (a) section through 
incident beam; (b) partial pattern obtained. 

When the pinhole method is used for parameter measurements, the film 
or specimen, or both, is moved during the exposure to produce smooth, 
continuous diffraction lines. By rotating or oscillating the film about the 
axis of the incident beam, the reflections from each reflecting particle or 
grain are smeared out along the Debye ring. The specimen itself may be 
rotated about the incident beam axis or about any axis parallel to the 
incident beam, or translated back and forth in any direction in a plane 
parallel to the specimen surface. Such movements increase the number 
of grains in reflecting positions and allow a greater proportion of the total 
specimen surface to take part in diffraction, thus ensuring that the informa- 
tion recorded on the film is representative of the surface as a whole. Any 
camera in which the specimen can be so moved during the exposure that 
the incident beam traverses a large part of its surface is called an integrating 
camera. 



6-10 Choice of radiation. With any of the powder methods described 
above, the investigator must choose the radiation best suited to the prob- 
lem at hand. In making this choice, the two most important considera- 
tions are : 

(1) The characteristic wavelength used should not be shorter than the 
K absorption edge of the specimen, or the fluorescent radiation produced 
will badly fog the film. In the case of alloys or compounds, it may be 
difficult or impossible to satisfy this condition for every element in the 
specimen. 

(2) The Bragg law shows that the shorter the wavelength, the smaller 
the Bragg angle for planes of a given spacing. Decreasing the wavelength 
will therefore shift every diffraction line to lower Bragg angles and increase 
the total number of lines on the film, while increasing the wavelength will 
have the opposite effect. The choice of a short or a long wavelength de- 
pends on the particular problem involved. 



166 POWDER PHOTOGRAPHS [CHAP. 6 

The characteristic radiations usually employed in x-ray diffraction are 

the following: 

MoKa: 0.711A 

CuKa: 1.542 

CoKa: 1.790 

YeKa: 1.937 

CrKa: 2.291 

In each case, the appropriate filter is used to suppress the K/3 component 
of the radiation. All in all, Cu K a radiation is generally the most useful. 
It cannot be employed with ferrous materials, however, since it will cause 
fluorescent radiation from the iron in the specimen; instead, Co Ka, Fe Ka 
or Cr Ka radiation should be used. 

Precise lattice-parameter measurements require that there be a num- 
ber of lines in the back-reflection region, while some specimens may yield 
only one or two. This difficulty may be avoided by using unfiltered radia- 
tion, in order to have Kfi as well as Ka lines present, and by using an alloy 
target. For example, if a 50 atomic percent Fe-Co alloy is used as a tar- 
get, and no filter is used in the x-ray beam, the radiation will contain the 
Fe Ka, Fe K0, Co Ka, and Co K/3 wavelengths, since each element will 
emit its characteristic radiation independently. Of course, special targets 
can be used only with demountable x-ray tubes. 



Background radiation. A good powder photograph has sharp in- 
tense lines superimposed on a background of minimum intensity. How- 
ever, the diffraction lines themselves vary in intensity, because of the struc- 
ture of the crystal itself, and an appreciable background intensity may 
exist, due to a number of causes. The two effects together may cause the 
weakest diffraction line to be almost invisible in relation to the background. 
This background intensity is due to the following causes: 
(1) Fluorescent radiation emitted by the specimen. It cannot be too 
strongly emphasized that the characteristic wavelength used should be 
longer than the K absorption edge of the specimen, in order to prevent 
the emission of fluorescent radiation. Incident radiation so chosen, how- 
ever, will not completely eliminate fluorescence, since the short-wavelength 
components of the continuous spectrum will also excite K radiation in the 
specimen. For example, suppose a copper specimen is being examined 
with CuKa radiation of wavelength 1.542A from a tube operated at 
30 kv. Under these conditions the short-wavelength limit is 0.413A. The 
K absorption edge of copper is at 1.380A. The Ka component of the 
incident radiation will not cause fluorescence, but all wavelengths between 
0.413 and 1.380A will. If a nickel filter is used to suppress the K/3 com- 
ponent of the incident beam, it will also have the desirable effect of reducing 



6-11] BACKGROUND RADIATION 167 

the intensity of some of the short wavelengths which cause fluorescence, 
but it will not, of course, eliminate them completely, particularly in the 
wavelength region near 0.6A, where the intensity of the continuous spec- 
trum is high and the absorption coefficient of nickel rather low. 

It is sometimes possible to filter part of the fluorescent radiation from 
the specimen by placing the proper filter over the film. For example, if 
a steel specimen is examined with copper radiation, which is not generally 
advisable, the situation may be improved by covering the film with alu- 
minum foil, because aluminum has a greater absorption for the fluorescent 
Fe KOL radiation contributing to the background than for the Cu Ka radia- 
tion forming the diffraction lines. In fact, the following is a good general 
rule to follow: if it is impossible to use a wavelength longer than the K 
absorption edge of the specimen, choose one which is considerably shorter 
and cover the film with a filter. Sometimes the air itself will provide 
sufficient filtration. Thus excellent patterns of aluminum can be obtained 
with CuKa radiation, even though this wavelength (1.54A) is much 
shorter than the K absorption edge of aluminum (6.74A), simply because 
the Al Ka radiation excited has such a long wavelength (8.34A) that it is 
almost completely absorbed in a few centimeters of air. 

(2) Diffraction of the continuous spectrum. Each crystal in a powder 
specimen forms a weak Laue pattern, because of the continuous radiation 
component of the incident beam. This is of course true whether or not 
that particular crystal is in the correct position to reflect the characteristic 
component into the Debye ring. Many crystals in the specimen are there- 
fore contributing only to the background of the photograph and not to 
the diffraction ring, and the totality of the Laue patterns from all the 
crystals is a continuous distribution of background radiation. If the inci- 
dent radiation has been so chosen that very little fluorescent radiation is 
emitted, then diffraction of the continuous spectrum is the largest single 
cause of high background intensity in powder photographs. 

(3) Diffuse scattering from the specimen itself. 

(a) Incoherent (Compton modified) scattering. This kind of scat- 
tering becomes more intense as the atomic number of the specimen 
decreases. 
(6) Coherent scattering. 

(i) Temperature-diffuse scattering. This form is more intense 

with soft materials of low melting point. 

(ii) Diffuse scattering due to various kinds of imperfection in 

the crystals. Any kind of randomness or strain will cause such 

scattering. 

(4) Diffraction and scattering from other than the specimen material. 

(a) Collimator and beam stop. This kind of scattering can be mini- 
mized by correct camera design, as discussed in Sec. 6-2. 



168 POWDER PHOTOGRAPHS [CHAP. 6 

(b) Specimen binder, support, or enclosure. The glue or other 
adhesive used to compact the powder specimen, the glass fiber to 
which the powder is attached, or the glass or fused-quartz tube in 
which it is enclosed all contribute to the background of the photo- 
graph, since these are all amorphous substances. The amount of 
these materials should be kept to the absolute minimum. 

(c) Air. Diffuse scattering from the air may be avoided by evacu- 
ating the camera or filling it with a light gas such as hydrogen or 
helium. 

6-12 Crystal monochromators. The purest kind of radiation to use in 
a diffraction experiment is radiation which has itself been diffracted, since 
it is entirely monochromatic.* If a single crystal is set to reflect the strong 
Ka component of the general radiation from an x-ray tube and this reflected 
beam is used as the incident beam in a diffraction camera, then the causes of 
background radiation listed under (1) and (2) above can be completely elimi- 
nated. Since the other causes of background scattering are less serious, the 
use of crystal-monochromated radiation produces diffraction photographs of 
remarkable clarity. There are two kinds of monochromators in use, depend- 
ing on whether the reflecting crystal is unbent or bent and cut. 

An unbent crystal is not a very efficient reflector, as can be seen from 
Fig. 6-14. The beam from an x-ray tube is never composed only of parallel 
rays, even when defined by a slit or collimator, but contains a large pro- 
portion of convergent and divergent radiation. When the crystal is set 
at the correct Bragg angle for the parallel component of the incident beam, 
it can reflect only that component and none of the other rays, with the 



* This statement requires some qualification. When a crystal monochromator 
is set to diffract radiation of wavelength X from a particular set of planes, then 
these same planes will also diffract radiation of wavelength A/2 and A/3 in the 
second and third order, respectively, and at exactly the same angle 26. These 
components of submultiple wavelength are of relatively low intensity when the 
main component is Ka characteristic radiation but, even so, their presence is un- 
desirable whenever precise calculations of the intensity diffracted by the specimen 
must be made. The submultiple components may be eliminated from the beam 
from the monochromator by reducing the tube voltage to the point where these 
wavelengths are not produced. If the main component is Cu Ka radiation, this 
procedure is usually impractical because of the decrease in intensity attendant on 
a reduction in tube voltage to 16 kv (necessary to eliminate the A/2 and A/3 com- 
ponents). Usually, a compromise is made by operating at a voltage just insuffi- 
cient to generate the A/3 component (24 kv for copper radiation) and by using a 
crystal which has, for a certain set of planes, a negligible reflecting power for the 
A/2 component. Fluorite (CaF 2 ) is such a crystal, the structure factor for the 222 
reflection being much less than for the 111. The diamond cubic crystals, silicon 
and germanium, are even better, since their structure factors for the 222 reflec- 
tion are actually zero. 



6-12] 



CRYSTAL MONOCHROMATORS 



169 




FIG. 6-14. Monochromatic reflec- 
tion when the incident beam is non- 
parallel. 



result that the reflected beam is of 
very low intensity although it is itself 
perfectly parallel, at least in the plane 
of the drawing. In a plane at right 
angles, the reflected beam may con- 
tain both convergent and divergent 
radiation. 

A large gain in intensity may be 
obtained by using a bent and cut crys- 
tal, which operates on the focusing 
principle illustrated in Fig. 6-15. A 

line source of x-rays, the focal line on the tube target, is located at S per- 
pendicular to the plane of the drawing. The crystal AB is in the form of a 
rectangular plate and has a set of reflecting planes parallel to its surface. 
It is elastically bent into a circular form so that the radius of curvature of 
the plane through C is 2R = CM; in this way, all the plane normals are 
made to pass through M, which is located on the same circle, of radius J?, 
as the source S. If the face of the crystal is then cut away behind the 
dotted line to a radius of 72, then all rays diverging from the source S will 
encounter the lattice planes at the same Bragg angle, since the angles 
SDM, SCM, and SEM are all equal to one another, being inscribed on the 
same arc SM, and have the value (ir/2 8). 

When the Bragg angle is adjusted to that required for reflection of the 
Ka component of the incident beam, then a strong monochromatic beam 




focusing 
circle 



FIG. 6-15. Focusing monochromator (reflection type). 



170 POWDER PHOTOGRAPHS [CHAP. 6 

will be reflected by the crystal. Moreover, since the diffracted rays all 
originate on a circle passing through the source S, they will converge to a 
focus at F, located on the same circle as S and at the same distance from 
C, in much the same way as in the focusing cameras previously discussed. 
In practice the crystal is not bent and then cut as described above, but 
the unbent crystal, usually of quartz, is first cut to a radius of 2R and then 
bent against a circular form of radius R. This procedure will produce the 
same net result. The value of 6 required for the diffraction of a particular 
wavelength X from planes of spacing d is given by the Bragg law: 

X = 2rfsin0. (0-12) 

The source-to-crystal distance 8C, which equals the crystal-to-focus dis- 
tance CF, is given by 

SC = 2fl cos (- - 0V (0-13) 

By combining Eqs. (6-12) and (0-13), we obtain 

SC = R-- (0-M) 

d 

For reflection of Cu Ka radiation from the (10-1) planes of quartz, the 
distance SC is 14.2 cm for a value of K of 30 em. 

The chief value of the focusing monochromator lies in the fact that all 
the monochromatic rays in the incident beam are utilized and the diffracted 
rays from a considerable area of the crystal surface are all brought to a 
focus. This leads to a large concentration of energy and a considerable 
reduction in exposure time compared to the unbent-crystal monochromator 
first described. However, the latter does produce a semiparallel beam of 
radiation, and, even though it is of very low intensity, such a beam is re- 
quired in some experiments. 

If the monochromating crystal is bent but not cut, some concentration 
of energy will be achieved inasmuch p,s the reflected beam will be con- 
vergent, but it will not converge to a perfect focus. 

The focusing monochromator is best used with powder cameras especially 
made to take advantage of the particular property of the reflected beam, 
namely its focusing action. Figure 0-10(a) shows the best arrangement. 
A cylindrical camera is used with the specimen and film arranged on the 
surface of the cylinder. Low-angle reflections are registered with the cam- 
era placed in position C, in which case the specimen D must be thin enough 
to be examined in transmission. High-angle reflections are obtained by 
back reflection with the camera in position C", shown dotted, and the 
specimen at D 1 . In the latter case, the geometry of the camera is exactly 
similar to that of the Seemann-Bohlin camera, the focal point F of the 



6-12] 



CRYSTAL MONOCHROMATORS 



171 





(h) '/" 

FIG. 6-16. Cameras used with focusing monochromators (a) focusing cameras; 
(b) Debye-Scherrer and flat-film cameras Only one diffracted beam is shown in 
each case. (After A. (Juinier, X-ray Crystallographic Technology, Hilger and Watts, 
Ltd., London, 1952) 

monochromatic beam acting as a virtual source of divergent radiation. 
In either case, the diffracted rays from the specimen are focused on the 
film for all hkl reflections; the only requirement is that the film be located 
on a circle passing through the specimen and the point F. 

A Debye-Scherrer or flat-film camera may also be used with a focusing 
monochromator, if the incident-beam collimator is removed. Figure 
6-1 6(b) shows such an arrangement, where D is the specimen, E is a 
Debye camera, and PP' is the position where a flat film may be placed. 
In neither case, however, is the above-mentioned focusing requirement 
satisfied, with the result that no more than one diffracted beam, corre- 
sponding to one particular hkl reflection, can be focused on the film at the 
same time. 

A bent crystal may also be used in transmission as a focusing mono- 
chromator. It must be thin enough to transmit a large fraction of the 
incident radiation arid have a set of reflecting planes at right angles to its 
surface; mica is often used. In Fig. 6-17, the line ACB represents the 
crystal, bent to a radius 2/2, its center of curvature located at M. Three 
of its transverse reflecting planes are shown. If radiation converging to 
A' were incident on these planes and reflected at the points //, C, and (/, 
the reflected radiation would converge to a perfect focus at F, all the 
points mentioned being on a focusing circle of radius R centered at 0. 
But the reflecting planes do not actually extend out of the crystal surface 
in the way shown in the drawing and reflection must occur at the points 



172 



POWDER PHOTOGRAPHS 



[CHAP. 6 




focusing 
circle 



FIG. 6-17. Focusing monoohromator (transmission type). 

D, C, and K. Under these conditions the reflected rays from all parts of 
the crystal do not converge to a perfect focus at F. Nevertheless there is 
sufficient concentration of diffracted energy in a very narrow region near 
F to make this device a quite efficient and usable monochromator. The 
crystal-to-focus distance CF is given by 



CF = 2R cos 0. 



(0-15) 



Combination of this equation with the Bragg law will give the bending 
radius required for specific applications. 

The use of a monochromator produces a change in the relative intensities 
of the beams diffracted by the specimen. Equation (4-12), for example, 
was derived for the completely unpolarized incident beam obtained from 
the x-ray tube. Any beam diffracted by a crystal, however, becomes par- 
tially polarized by the diffraction process itself, which means that the 
beam from a crystal monochromator is partially polarized before it reaches 
the specimen. Under these circumstances, the usual polarization factor 
(1 + cos 2 20)/2, which is included in Eq. (4-12), must be replaced by the 
factor (I + cos 2 2 cos 2 20)/(l + cos 2 2a), where 2a is the diffraction 
angle in the monochromator [Fig. 6-16(b)]. Since the denominator in 
this expression is independent of 0, it may be omitted; the combined 
Lorentz-polarization factor for a Debye-Scherrer camera and crystal- 
monochromated radiation is therefore (1 + cos 2 2a cos 2 20) /sin 2 cos 6. 



6-14] 



MEASUREMENT OP LINE INTENSITY 



173 




FIG. 6-18. Film-measuring device. 
Department.) 



(Courtesy of General Electric Co., X-Ray 



6-13 Measurement of line position. The solution of any powder pho- 
tograph begins with the measurement of the positions of the diffraction 
lines on the film. A device of the kind shown in Fig. 6-18 is commonly 
used for this purpose. It is essentially a box with an opal-glass plate on 
top, illuminated from below, on which the film to be measured is placed. 
On top of the glass plate is a graduated scale carrying a slider equipped 
with a vernier and cross-hair; the cross-hair is moved over the illuminated 
film from one diffraction line to another and their positions noted. The 
film is usually measured without magnification. A low-power hand lens 
may be of occasional use, but magnification greater than 2 or 3 diameters 
usually causes the line to merge into the background and become invisible, 
because of the extreme graininess of x-ray film. 

6-14 Measurement of line intensity. Many diffraction problems re- 
quire an accurate measurement of the integrated intensity, or the breadth 
at half maximum intensity, of a diffraction line on a powder photograph. 
For this purpose it is necessary to obtain a curve of intensity vs. 26 for 
the line in question. 

The intensity of an x-ray beam may be measured by the amount of 
blackening it causes on a photographic film. The photographic density D, 
or blackening, of a film is in turn measured by the amount of visible light 



174 POWDER PHOTOGRAPHS [CHAP. 6 

it will transmit and is defined by the relation 

, /0 
D = Iog 10 y 

where / = intensity of a beam of light incident on the film and / = inten- 
sity of the transmitted beam. For most x-ray films, the density is directly 
proportional to the exposure up to a density of about 1.0 (which corre- 
sponds to 10 percent transmission of the incident light). Here, "exposure" 
is defined by the relation 

Exposure = (intensity of x-ray beam) (time). 

Since the time is constant for all the diffraction lines on one film, this means 
that the photographic density is directly proportional to the x-ray in- 
tensity. 

Density is measured by means of a microphotometer. There are sev- 
eral forms of such instruments, the simplest consisting of a light source 
and an arrangement of lenses and slits which allows a narrow beam of 
light to pass through the x-ray film and strike a photocell or thermopile 
connected to a recording galvanometer. Since the current through the 
galvanometer is proportional to the intensity of the. light striking the 
photocell, the galvanometer deflection 8 is proportional to the transmitted 
light intensity /. 

The light beam is rectangular in cross section, normally about 3 mm 
high and 0.1 mm wide. With movement of the film, this beam is made to 
traverse the film laterally, crossing one diffraction line after another [Fig. 
6-19(a)]. The resulting galvanometer record [Fig. 6-1 9(b)] shows gal- 
vanometer deflection as ordinate and distance along the film as abscissa, 
the latter being increased by a factor of about 5 in order to spread the 
lines out. The line A at the top of the record marks zero deflection of the 
galvanometer; the line B at the bottom marks the maximum galvanometer 
deflection S when the light beam passes through an unexposed portion of 
the film, a portion which has been shielded from all scattered x-rays. S Q is 
therefore constant and proportional to the incident light intensity 7 . In 
this way the readings are corrected for the normal background fog of 
unexposed film. The density of any exposed part of the film is then ob- 
tained from the relation 

r, i /0 i SQ 
D = Iog 10 -- = logio 

1 o 

Finally, a curve is constructed of x-ray intensity as a function of 26 [Fig. 
6-19(c)]. Such a plot is seen to consist of a number of diffraction peaks 
superimposed on a curve of slowly varying background intensity, due to 
fluorescent radiation, diffraction of the continuous spectrum, Compton 



6-14] 



MEASUREMENT OF LINE INTENSITY 



175 






DISTANCE ALONCJ FILM (Yj>) 



20 

1C) 



FIG. 6-19. Measurement of line intensity with a microphotomctci (schematic) 
(a) film; (b) galvanometei recoul, (c) x-iay intensity curve 

scattering, etc., as previously discussed. A continuous background line 
is drawn in below each peak, after which measurements of the integrated 
intensity and the breadth K at half maximum intensity can be made. 
Note that the integrated intensity is given by the shaded area, measured 
above the background. A microphotorneter record of an actual pattern is 
shown in Fig (>-20. 

In very precise work, or when the line density exceeds a value of 1.0, it 
is no longer safe to assume that the density is proportional to the x-ray 
exposure Instead, each film should be calibrated by exposing a strip near 
its edge to a constant-intensity x-ray beam for increasing amounts of time 
so that a series of stepwise increasing exposures is obtained. The exact 
relation between density and x-ray exposure can then be determined ex- 
perimentally. 




1*10. 6-20. Powder pattern of quartz (above) and corresponding mirrophotom- 
eter trace (below). (J. W. Ballard, H. I. Oshry, and II. II Schrcrik, T. S. Bur 
Mines R. I. 520. Courtesy of U. S. Bureau of Mines.) 



176 POWDER PHOTOGRAPHS [CHAP. 6 

PROBLEMS 

6-1. Plot a curve similar to that of Fig. 6-4 showing the absorption of Fe Ka 
radiation by air. Take the composition of air as 80 percent nitrogen and 20 per- 
cent oxygen, by weight. If a 1-hr exposure in air is required to produce a certain 
diffraction line intensity in a 19-cm-diameter camera with Fe Ka radiation, what 
exposure is required to obtain the same line intensity with the camera evacuated, 
other conditions being equal? 

6-2. Derive an equation for the resolving power of a Debye-Scherrer camera 
for two wavelengths of nearly the same value, in terms of AS, where S is defined 
by Fig. 6-2. 

6-3. For a Debye pattern made in a 5.73-crn-diameter camera with Cu Ka radi- 
ation, calculate the separation of the components of the Ka doublet in degrees 
and in centimeters for = 10, 35, 60, and 85. 

6-4. What is the smallest value of 6 at which the Cr Ka doublet will be resolved 
in a 5.73-cm-diameter Debye camera? Assume that the line width is 0.03 cm and 
that the separation must be twice the width for resolution. 

6-5. A powder pattern of zinc is made in a Debye-Scherrer camera 5.73 cm in 
diameter with Cu Ka radiation. 

(a) Calculate the resolving power necessary to separate the 11.0 and 10.3 diffrac- 
tion lines. Assume that the line width is 0.03 cm. 

(b) Calculate the resolving power of the camera used, for these lines. 

(c) What minimum camera diameter is required to produce resolution of these 
lines? 

(See Fig. 3-13(c), which shows these lines unresolved from one another. They 
form the fifth line from the low-angle end.) 

6-6. A transmission pinhole photograph is made of copper with Cu Ka radia- 
tion. The film measures 4 by 5 in. What is the maximum specimen-to-film dis- 
tance which can be used and still have the first two Debye rings completely re- 
corded on the film? 

6-7. A powder pattern of iron is made with Cu Ka radiation. Assume that 
the background is due entirely to fluorescent radiation from the specimen. The 
maximum intensity (measured above the background) of the weakest line on the 
pattern is found to be equal to the background intensity itself at that angle. If 
the film is covered with aluminum foil 0.0015 in. thick, what will be the ratio of 
/max for this line to the background intensity? 

6-8. A microphotometer record of a diffraction line shows the following gal- 
vanometer deflections: 

Position of Light Beam Deflection 

On unexposed film 5 . cm 

On background, just to left of line 3.0 

On background, just to right of line 3.2 

On center of diffraction line 1 . 2 

Assume that x-ray intensity is proportional to photographic density. Calculate 
the ratio of 7 ma x for the diffraction line (measured above the background) to the 
intensity of the background at the same Bragg angle. 



CHAPTER 7 
DIFFRACTOMETER MEASUREMENTS 

7-1 Introduction. The x-ray spectrometer, briefly mentioned in Sec. 
3-4, has had a long and uneven history in the field of x-ray diffraction. It 
was first used by W. H. and W. L. Bragg in their early work on x-ray 
spectra and crystal structure, but it then passed into a long period of rela- 
tive disuse during which photographic recording in cameras was the most 
popular method of observing diffraction effects. The few spectrometers in 
use were all home made and confined largely to the laboratories of research 
physicists. In recent years, however, commercially made instruments 
(based mainly on a design developed by Friedman about 1943) have be- 
come available, and their use is growing rapidly because of certain par- 
ticular advantages which they offer over film techniques. Initially a 
research tool, the x-ray spectrometer has now become an instrument for 
control and analysis in a wide variety of industrial laboratories. 

Depending solely on the way it is used, the x-ray spectrometer is really 
two instruments: 

s (1) An instrument for measuring x-ray spectra by means of a crystal of 
known structure. 

(2) An instrument for studying crystalline (and noncrystalline) mate- 
rials by measurements of the way in which they diffract x-rays of known 
wavelength. 

The term spectrometer has been, and still is, used to describe both instru- 
ments, but, properly, it should be applied only to the first instrument. 
The second instrument has been aptly called a diffractometer: this is a term 
of quite recent coinage but one which serves well to emphasize the par- 
ticular use to which the instrument is being put, namely, diffraction anal- 
ysis rather than spectrometry. In this chapter, the design and operation 
of diffractometers will be described with particular reference to the com- 
mercial models available. 

7-2 General features. In a diffraction camera, the intensity of a dif- 
fracted beam is measured through the amount of blackening it produces 
on a photographic film, a microphotometer measurement of the film being 
required to convert "amount of blackening" into x-ray intensity. In the 
diffractometer, the intensity of a diffracted beam is measured directly, 
.either by means of the ionization it produces in a gas or the fluorescence 

177 



178 DIFFRACTOMETER MEASUREMENTS [CHAP. 7 

it produces in a solid. As we saw in Sec. 1-5, incident x-ray quanta can 
eject electrons from atoms and thus convert them into positive ions. If 
an x-ray beam is passed into a chamber containing a gas and two elec- 
trodes, one charged positively and the other negatively, then the ejected 
electrons will be drawn to the positive electrode (the anode) and the posi- 
tive ions to the negative electrode (the cathode). A current therefore 
exists in the external circuit connecting anode to cathode. Under special 
conditions, which are described later in detail, this current can be caused 
to surge or pulse rather than be continuous; each pulse results from the 
ionization caused by a single entering x-ray quantum. By use of the 
proper external circuit, the number of current pulses produced per unit 
of time can be counted, and this number is directly proportional to the 
intensity of the x-ray beam entering the gas chamber. Appropriately, 
this device is called a counter, and two varieties are in common use, the 
proportional counter and the Geiger counter. In another type, the scintil- 
lation counter, incident x-ray quanta produce flashes or scintillations of 
fluorescent blue light in a crystal and these light flashes are converted into 
current pulses in a phototube. 

Basically, a diffractometer is designed somewhat like a Debye-Scherrer 
camera, except that a movable counter replaces the strip of film. In both 
instruments, essentially monochromatic radiation is used and the x-ray 
detector (film or counter) is placed on the circumference of a circle cen- 
tered on the powder specimen. The essential features of a diffractometer 
are shown in Fig. 7-1. A powder specimen C, in the form of a flat plate, 
is supported on a table H, which can be rotated about an axis*0 perpen- 
dicular to the plane of the drawing. The x-ray source is S, the line focal 
spot on the target T of the x-ray tube; S is also normal to the plane of the 
drawing and therefore parallel to the diffractometer axis 0. X-rays di- 
verge from this source and are diffracted by the specimen to form a con- 
vergent diffracted beam which comes to a focus at the slit F and then 
enters the counter G. A and B are special slits which define and collimate 
the incident and diffracted beams. 

The receiving slits and counter are supported on the carriage E y which 
may be rotated about the axis and whose angular position 26 may be 
read on the graduated scale K. The supports E and H are mechanically 
coupled so that a rotation of the counter through 2x degrees is automatically 
accompanied by rotation of the specimen through x degrees. This cou- 
pling ensures that the angles of incidence on, and reflection from, the flat 
specimen will always be equal to one another and equal to half the total 
angle of diffraction, an arrangement necessary to preserve focusing con- 
ditions. The counter may be power-driven at a constant angular velocity 
about the diffractometer axis or moved by hand to any desired angular 
position. 



7-2] 



GENERAL FEATURES 



179 




FKJ 7 1. X-ray difTrartoinetei (schematic) 

Figures 7-2 and 7-3 illustrate two commercial instruments. Basically, 
both adhere to the design principles described above, but they differ in 
detail and in positioning: in the (General Electric unit, the diffract ometer 
axis is vertical and the counter moves in a horizontal plane, whereas the 
axis of the Xorelco unit is horizontal and the counter moves in a vertical 
plane. 

The way in which a diffract ometer is used to measure a diffraction pat- 
tern depends on the kind of circuit used to measure the rate of production 
of pulses in the counter. The pulse rate may be measured in t\\o different 
uuys: 

(1) The succession of current pulses is converted into a steady current, 
which is measured on a meter called a counting-rate meter, calibrated in 
such units as counts (pulses) per second. Such a circuit gives a continuous 
indication of x-ray intensity. 

(2) The pulses of current are counted electronically in a circuit called a 
sealer, and the average counting rate is obtained simply by dividing the 
number of pulses counted by the time spent in counting. This operation 
is essentially discontinuous because of the time spent in counting, and a 
scaling circuit cannot be used to follow continuous changes in x-ray in- 
tensity. 

Corresponding to these two kinds of measuring circuits, there are two 
ways in which the diffraction pattern of an unknown substance may be 
obtained with a diffract ometer: 



180 



DIFFRACTOMETER MEASUREMENTS 



[CHAP. 7 














FIG. 7-2. General Electric diffractometer. (Courtesy of General Electric Co., 
X-Ray Department.) 



7-2] 



GENERAL FEATURES 



181 













' ' ' 



FIG. 7-3. Norelco diffractometer. In this particular photograph, the specimen 
holder for a thin rod specimen is shown instead of the usual holder for a flat plate 
specimen. X-ray tube not shown. (Courtesy of North American Philips Co., Inc.) 



Ig2 DIFFRACTOMETER MEASUREMENTS [CHAP. 7 

(1) Continuous. The counter is set near 26 = and connected to a 
counting-rate meter. The output of this circuit is fed into a fast-acting 
automatic recorder of the kind used to record temperature changes as 
measured by a thermocouple. The counter is then driven at a constant 
angular velocity through increasing values of 20 until the whole angular 
range is "scanned." At the same time, the paper chart on the recorder 
moves at a constant speed, so that distances along the length of the chart 
are proportional to 26. The result is a chart, such as Fig. 7-4, which gives 
a record of counts per second (proportional to diffracted intensity) vs. dif- 
fraction angle 26. 

(2) Intermittent. The counter is connected to a sealer and set at a fixed 
value of 26 for a time sufficient to make an accurate count of the pulses 
obtained from the counter. The counter is then moved to a new angular 
position and the operation repeated. The whole range of 26 is covered in 
this fashion, and the curve of intensity vs. 26 is finally plotted by hand. 
When the continuous background between diffraction lines is being meas- 
ured, the counter may be moved in steps of several degrees, but determina- 
tions of line profile may require measurements of intensity at angular 
intervals as small as 0.01 . This method of obtaining a diffraction pattern 
is much slower than that involving a rate meter and automatic recorder 
but it yields more precise measurements of intensity. 

There is a fundamental difference between the operation of a powder 
camera and a diffractometer. In a camera, all diffraction lines are recorded 
simultaneously, and variations in the intensity of the incident x-ray beam 
during the exposure can have no effect on relative line intensities. On 
the other hand, with a diffractometer, diffraction lines are recorded one 
after the other, and it is therefore imperative to keep the incident-beam 
intensity constant when relative line intensities must be measured accu- 
rately. Since the usual variations in line voltage are quite appreciable, 
the x-ray tube circuit of a diffractometer must include a voltage stabilizer 
and a tube-current stabilizer, unless a monitoring system is used (see 
Sec. 7-8). 

The kind of specimen used depends on the form and amount of material 
available. Flat metal sheet or plate may be examined directly; however, 
such materials almost always exhibit preferred orientation and this fact 
must be kept in mind in assessing relative intensities. This is also true of 
wires, which are best examined by cementing a number of lengths side by 
side to a glass plate. This plate is then inserted in the specimen holder 
so that the wire axes are at right angles to the diffractometer axis. 
Powder specimens are best prepared by placing the powder in a recess in 
a glass or plastic plate, compacting it under just sufficient pressure to 
cause cohesion without use of a binder, and smoothing off the surface. 
Too much pressure causes preferred orientation of the powder particles. 
Alternately, the powder may be mixed with a binder and smeared on the 



7-2] 



GEKKRAL FEATURES 



183 




(wb) 3TVDS A1ISN31NI 



184 DIFFRACTOMETER MEASUREMENTS [CHAP. 7 

surface of a glass slide. The powder should be ground extremely fine, to a 
size of 10 microns or less, if relative line intensities are to be accurately 
reproducible; since the flat specimen is not rotated as a Debye-Scherrer 
specimen is, the only way of obtaining an adequate number of particles 
having the correct orientation for reflection is to reduce their average size. 
Surface roughness also has a marked effect on relative line intensities. If 
the surface is rough, as in the case of a coarse powder compact, and the 
linear absorption coefficient high, the intensities of low-angle reflections 
will be abnormally low, because of the absorption of the diffracted rays in 
each projecting portion of the surface. The only way to avoid this effect 
is to use a flat-surfaced compact of very fine powders or a specimen with 
a polished surface. 

If not enough powder is available for a flat specimen, a thin-rod speci- 
men of the kind used in Debye-Scherrer cameras may be used ; it is mounted 
on the diffractometer axis and continuously rotated by a small motor 
(see Fig. 7-3). However, the use of such a small specimen should be 
avoided if possible, since it leads to intensities very much lower than those 
obtainable with a flat, specimen. 

Single-crystal specimens may also be examined in a diffractometer by 
mounting the crystal on a three-circle goniometer, such as that shown in 
Fig. 5-7, which will allow independent rotation of the specimen and coun- 
ter about the diffractometer axis. 

A diffractometer may be used for measurements at high or low tempera- 
tures by surrounding the specimen with the appropriate heating or cooling 
unit. Such an adaptation of the instrument is much easier with the dif- 
fractometer than with a camera because of the generally larger amount of 
free working space around the specimen in the former. 

In the succeeding sections, the various parts of the diffractometer will 
be described in greater detail. This summary of the general features of 
the instrument is enough to show its principal advantage over the powder 
camera: the quantitative measurement of line position and intensity is 
made in one operation with a diffractometer, whereas the same measure- 
ment with film technique requires three steps (recording the pattern on 
film, making a microphotometer record of the film, and conversion of 
galvanometer deflections to intensities) and leads to an over-all result 
which is generally of lower accuracy. This superiority of the diffractometer 
is reflected in the much higher cost of the instrument, a cost due not only 
to the precision machining necessary in its mechanical parts but also to 
the expensive circuits needed to stabilize the power supply and measure 
the intensity of diffracted beams. 

7-3 X-ray optics. The chief reason for using a flat specimen is to take 
advantage of the focusing action described in Sec. 6-6 and so increase the 



7-3] 




X-RAY OPTICS 

iffract omc'tor circle 



185 




(a) 



FIG. 7-5. Focusing geometry for flat specimens in (a) forward reflection and 
(h) hack reflection. 

intensity of weak diffracted beams to a point where they can he accurately 
measured. Figure 7-5 shows how this is done. For any position of the 
counter, the receiving slit F and the x-ray source S are always located on 
the difTractometer circle, which means that the face of the specimen, be- 
cause of its mechanical coupling with the counter, is always tangent to a 
focusing circle centered on the normal to the specimen and passing through 
F and $. The focusing circle is not of constant size but increases in radius 
as the angle 26 decreases, as indicated in Fig. 7-5. Perfect focusing at F 
requires that the specimen be curved to fit the focusing circle, but that is 
not practical because of the changing radius of curvature of the circle. 
This inevitably causes some broadening of the diffracted beam at F but 
not to any objectionable degree, so long as the divergence of the incident 
beam is not too large. 

The line source $ extends considerably above and below the plane of 
the drawing of Fig. 7-5 and emits radiation in all directions, but the focus- 
ing described above requires that all rays in the incident beam be parallel 
to the plane of the drawing. This condition is realized as closely as pos- 
sible experimentally by passing the incident beam through a Soller slit 
(Fig. 7-0), slit A in Fig. 7-1, which contains a set of closely spaced, thin 
metal plates parallel to the plane of the diffractometer circle. These plates 
remove a large proportion of rays inclined to the plane of the diffractometer 
circle and still allow the use of a line source of considerable length. Typical 
dimensions of a Soller slit are: length of plates 32 mm, thickness of 
plates 0.05 mm, clear distance between plates 0.43 mm. At either end of 
the slit assembly are rectangular slits a and 6, the entrance slit a next to 
the source being narrower than the exit slit b. The combination of slits 
and plates breaks up the incident beam into a set of triangular wedges of 
radiation, as indicated in Fig. 7-6. There are, of course, some rays, not 
shown in the drawing, which diverge in planes perpendicular to the plane 



186 



DIFFRACTOMETER MEASUREMENTS 



[CHAP. 7 




o 

TJ 



o 

03 
G 



c3 
G 

6 



r 



7-3] 



X-RAY OPTICS 187 

incident-beam slits 







specimen 



S 




*-- iecei\ ing slit 



.^ ^_ to counter 



FIG. 7-7. Arrangement of slits in diffractometer. 



of the plates, and these rays cause the wedges of radiation to merge into 
one another a short distance away from the exit slit. However, the long, 
closely spaced plates do restrict this unwanted divergence to an angle of 
about 1.5. Slits a and b define the divergence of the incident beam in the 
plane of the diffractometer circle. The slits commonly available have 
divergence angles ranging from very small values up to about 4. In the 
forward-reflection region, a divergence angle of 1 is sufficient because of 
the low inclination of the specimen surface to the incident beam, but in 
back reflection an increase in divergence angle to 3 or 4 will increase the 
diffracted intensity. But if line intensities are to be compared over the 
whole range of 26, the same divergence must be used throughout and the 
specimen must be wider than the beam at all angles. 

The beam diffracted by the specimen passes through another Soller-slit 
assembly and the receiving slit F before entering the counter. Since the 
receiving slit defines the width of the beam admitted to the counter, an 
increase in its width will increase the maximum intensity of any diffraction 
line being measured but at the expense of some loss of resolution. On the 
other hand, the relative integrated intensity of a diffraction line is inde- 
pendent of slit width, which is one reason for its greater fundamental im- 
portance. * Figure 7-7 illustrates the relative arrangement of the various 

*A number of things besides slit width (e.g., x-ray tube current) will change 
the integrated intensity of a single diffraction line. The important thing to note, 
however, is that a change in any one of the operating variables changes the inte- 
grated intensities of all diffraction lines in the same ratio but can produce very 
unequal effects on maximum intensities. Thus, 'if /i//2 is the ratio of the inte- 
grated intensities of two lines measured with a certain slit width and Mi/M 2 the 
ratio of their maximum intensities, then another measurement with a different 
slit width will result in the same ratio I\/h for the integrated intensities, but the 
ratio of the maximum intensities will now, in general, differ from Af i/Af 2. 




188 DIFFRACTOMETER MEASUREMENTS [CHAP. 7 

slits in a typical diffractometer and shows the passage of a few selected 
rays from source to counter. 

Because of the focusing of the diffracted rays and the relatively large 
radius of the diffractometer circle, about 15 cm in commercial instruments, 
a diffractometer can resolve very closely spaced diffraction lines. Indica- 
tive of this is the fact that resolution of the Cu Ka doublet can be obtained 
at 20 angles as low as about 40. Such resolution can only be achieved 
with a correctly adjusted instrument, and it is necessary to so align the 
component parts that the following conditions are satisfied for all diffrac- 
tion angles : 

(1) line source, specimen surface, and receiving-slit axis are all parallel, 

(2) the specimen surface coincides with the diffractometer axis, and 

(3) the line source and receiving slit both lie on the diffractometer circle. 

7-4 Intensity calculations. The calculation of the relative integrated 
intensities of beams diffracted by a powder specimen in a diffractometer 
follows the general principles de- 
scribed in Chap. 4, but the details of 
the calculation depend on the form 
of the specimen. 

The use of a flat-plate specimen, 
making equal angles with the incident 
and diffracted beams, not only pro- 
duces focusing as described above but 
makes the absorption factor inde- FIG. 7-*- Diffraction from a flat 
pendent of the angle 0. We can prove f late: ^ id f nt an f d diffmcted beams 
, i i i rr t i have a thickness of 1 cm in a direction 

this by calculating the effect of absorp- normfll t() the plane ()f the drawing . 

tion in the specimen on the intensity 

of the diffracted beam, and, since this effect will come up again in later 
parts of this book, we will make our calculation quite general. In Fig. 7-8, 
the incident beam has intensity 7 (ergs/cm 2 /ec), is 1 cm square in cross 
section, and is incident on the powder plate at an angle a. We consider 
the energy diffracted from this beam by a layer of the powder of length / 
and thickness dr, located at a depth x below the surface. Since the inci- 
dent beam undergoes absorption by the specimen over the path length 
AB, the energy incident per second on the layer considered is I e~^ (AB} 
(ergs/sec), where M is the linear absorption coefficient of the powder com- 
pact. Let a be the volume fraction of the specimen containing particles 
having the correct orientation for reflection of the incident beam, and b 
the fraction of the incident energy which is diffracted by unit volume. 
Then the energy diffracted by the layer considered, which has a volume 
Idx, is given by aW/ e~" u *' } dx. But this diffracted energy is also de- 
creased by absorption, by a factor of e~~ (BC \ since the diffracted rays 




7-4] INTENSITY CALCULATIONS 189 

have a path length of BC in the specimen. The energy flux per second in 
the diffracted beam outside the specimen, i.e., the integrated intensity, is 
therefore given by 

dI D = ablI c- (AB+BC} dx (ergs/sec). (7-1) 

But 

1 x x 



sin a sin a sin ft 

Therefore, 

in /J) dj (7_ 2 ) 



sn a. 



For the particular specimen arrangement used in the diffractometer, 
a = ^ 0, and the above equation becomes 



m 9 dx (7 _ 3) 

sin 6 

The total diffracted intensity is obtained by integrating over an infinitely 
thick specimen : x 

ID = 



Here 7 , 6, and M are constant for all reflections (independent of 8) and we 
may also regard a as constant. Actually, a varies with 0, but this variation 
is already taken care of by the cos0 portion of the Lorentz factor (see 
Sec. 4-9) and need not concern us here. We conclude that the absorption 
factor, l/2/i, is independent of for a flat specimen making equal angles 
with the incident and diffracted beams, provided the specimen fills the 
incident beam at all angles and is effectively of infinite thickness. * This 



* ' 



1 The criterion adopted for "infinite thickness" depends on the sensitivity of pur 
intensity measurements or on what we regard as negligible diffracted intensity. 
For example, we might arbitrarily but quite reasonably define infinite thickness as 
that thickness t which a specimen must have in order that the intensity diffracted 
by a thin layer on the back side be T ^Vo f tne intensity diffracted by a thin layer 
on the front side. Then, from Eq. (7-3) we have 

dip (at x = 0) = ^ t/Bm e = ]0()0 

d! D (at x -= 
from which . 

_ 3. 45 sin 8 

M 

This expression shows that "infinite thickness," for a metal specimen, is very 
small indeed. For example, suppose a specimen of nickel powder is being ex- 
amined with Cu KOL radiation at 8 values approaching 90. The density of the 
powder compact may be taken as about 0.6 the density of bulk nickel, which is 
8.9 gm/cm 3 , leading to a value of M for the compact of 263 cm" 1 . The value of t 
is therefore 1.31 X 10 ~ 2 cm, or about five thousandths of an inch. 



190 DIFFRACTOMETER MEASUREMENTS [CHAP. 7 

independence of 6 is due to the exact balancing of two opposing effects. 
When 6 is small, the specimen area irradiated by an incident beam of fixed 
cross section is large, but the effective depth of x-ray penetration is small ; 
when is large, the irradiated area is small, but the penetration depth is 
relatively large. The net effect is that the effective irradiated volume is 
constant and independent of 6. Absorption occurs in any case, however, 
and the larger the absorption coefficient of the specimen, the lower the in- 
tensity of the diffracted beams, other things being equal. The important 
fact to note is that absorption decreases the intensities of all diffracted 
beams by the same factor and therefore does not enter into the calculation 
of relative intensities. This means that Eq. (4-1 2) for the relative integrated 
intensity of a diffraction line from a powder specimen, namely, 

+ cos 2 20 



sm 2 6 cos 8 

needs only the insertion of a temperature factor to make it precise, for the 
case of a flat specimen examined in a diffractometer. As it stands, it may 
still be used to calculate the approximate relative intensities of two adja- 
cent lines on the pattern, but the calculated intensity of the higher-angle 
line, relative to that of the lower-angle one, will always be somewhat too 
large because of the omission of the temperature factor. 

When the specimen used in the diffractometer has the form of a thin 
rod, no focusing occurs and the incident-beam slits are chosen to produce 
a thin, essentially parallel beam. The x-ray geometry is then entirely 
equivalent to that of a Debye-Scherrer camera equipped with slits, and 
Eq. (4-12) applies, with exactly the same limitations as mentioned in 
Sec. 4-12. 

7-5 Proportional counters. Proportional, Geiger, and scintillation 
counters may be used to detect, not only x- and 7-radiation, but also 
charged particles such as electrons or a-particles, and the design of the 
counter and associated circuits depends to some extent on what is to be 
detected. Here we are concerned only with counters for the detection of 
x-rays of the wavelengths commonly employed in diffraction. 

Consider the device shown in Fig. 7-9, consisting of a cylindrical metal 
shell (the cathode) filled with a gas and containing a fine metal wire (the 
anode) running along its axis. Suppose there is a constant potential dif- 
ference of about 200 volts between anode and cathode. One end of the 
cylinder is covered with a window material, such as mica or beryllium, of 
high transparency to x-rays. Of the x-rays which enter the cylinder, a 
small fraction passes right through, but the larger part is absorbed by the 
gas, and this absorption is accompanied by the ejection of photoelectrons 



7-5] 



PROPORTIONAL COUNTERS 



191 



insulator 




to 

detector 
circuit 



and Compton recoil electrons from 
the atoms of the gas. The net result 
is ionization of the gas, producing 
electrons, which move under the in- 
fluence of the electric field toward 
the wire anode, and positive gas ions, 
which move toward the cathode shell. 
At a potential difference of about 
200 volts, all these electrons and ions 
will be collected on the electrodes, 
and, if the x-ray intensity is constant, FlG ? _ g Gas counter ( pro p rtional 
there will be a small constant current orGeiger) and basic circuit connections, 
of the order of 10~ 12 amp or less 

through the resistance R\. This current is a measure of the x-ray in- 
tensity. When operated in this manner, this device is called an ionization 
chamber. It was used in the original Bragg spectrometer but is now 
obsolete for the measurement of x-ray intensities because of its low sensi- 
tivity. 

The same instrument, however, can be made to act as a proportional 
counter if the voltage is raised to the neighborhood of 600 to 900 volts. 
A new phenomenon now occurs, namely, multiple ionization or "gas ampli- 
fication." The electric-field intensity is now so high that the electrons 
produced by the primary ionization are rapidly accelerated toward the 
wire anode and at an ever increasing rate of acceleration, since the field 
intensity increases as the wire is approached. The electrons thus acquire 
enough energy to knock electrons out of other gas atoms, and these in turn 
cause further ionization and so on, until the number of atoms ionized by 
the absorption of a single x-ray quantum is some 10 3 to 10 5 times as large 
as the number ionized in an ionization chamber. As a result of this ampli- 
fication a veritable avalanche of electrons hits the wire and causes an easily 
detectible pulse of current in the external circuit. This pulse leaks away 
through the large resistance RI but not before the charge momentarily 
added to the capacitor Ci has been detected by the ratemeter or scaling 
circuit connected to Ci. At the same time the positive gas ions move to 
the cathode but at a much lower rate because of their larger mass. This 
whole process, which is extremely fast, is triggered by the absorption of 
one x-ray quantum. 

We can define a gas amplification factor A as follows : if n is the number 
of atoms ionized by one x-ray quantum, then An is the total number 
ionized by the cumulative process described above. Figure 7-10 shows 
schematically how the gas amplification factor varies with the applied 
voltage. At the voltages used in ionization chambers, A = 1; i.e., there 
is no gas amplification, since the electrons produced by the primary ioniza- 



192 



DIFFRACTOMETER MEASUREMENTS 



[CHAP. 7 



l() n 

O 10"' 

E-I 

< 

^ 10 s 

^ 1() 7 

O 



r< !(). 

E KM 

i io * 

^ H) 2 



10 

1 




glow 
discharge 



corona 
discharge 



avalanche 
region 



VOLTAC5E 

FIG. 7-10. Effect of voltage on the gas amplification factor. (H. Friedman, 
Proc. /.#.#. 37,791, 1949.) 

tion do not acquire enough energy to ionize other atoms. But when the 
voltage is raised into the proportional counter region, A becomes of the 
order of 10 3 to 10 5 . 

The current pulse in the anode wire is normally expressed in terms of the 
momentary change of voltage in the wire, and this change is of the order 
of a few millivolts. The proportional counter receives its name from the 
fact that the size of this pulse, for a given applied voltage, is directly pro- 
portional to n, the number of ions formed by the primary ionization process, 
and this number is in turn proportional to the energy of the x-ray quantum 
absorbed. Thus, if absorption of a Cu Ka quantum (hv = 9,000 ev) pro- 
duces a voltage pulse of 1 .0 mv, then absorption of a Mo Ka quantum 
(hv = 20,000 ev) will produce a pulse of (20,000/9 ,000) (1.0) = 2.2 mv. 

The proportional counter is essentially a very fast counter; i.e., it can 
resolve separate pulses arriving at a rate as high as 10 per second. It can 
do this because each avalanche is confined to an extremely narrow region 
of the wire, 0.1 mm or less, and does not spread longitudinally along the 
counter tube. This is an important feature of the process and one to w r hich 
we will return in the next section. 



By inserting special circuits between a proportional counter and the measuring 
instrument (sealer or ratemeter), it is possible to take advantage of the fact that 
the sizes of the pulses produced are inversely proportional to the wavelengths of 
the x-rays producing them. For example, one such circuit allows only pulses 
larger than a certain selected size to pass and discriminates against smaller ones; 
it is called a pulse-height discriminator. If two such circuits are used together, one 



7-6] GEIGER COUNTERS 193 

set to pass only those pulses larger than Vi volts and the other only those larger 
than ^2 volts, then the difference between their two outputs is due only to pulses 
having sizes in the V\- to VVvolt range. This subtraction may be done electroni- 
cally, in which case the composite circuit is called a single-channel pulse-height 
analyzer. 

Such a device allows a proportional counter to be operated under essentially 
monochromatic conditions. For example, if a diffraction pattern is being obtained 
with copper radiation, the analyzer can be set to pass only pulses due to Cu Ka 
radiation and reject those due to other wavelengths, such as Cu Kft, fluorescent 
radiation from the specimen, white radiation, etc. 

7-6 Geiger counters. If the voltage on a proportional counter is in- 
creased some hundreds of volts, it will act as a Geiger counter. The exact 
operating voltage is determined in the following way. The counter is 
exposed to a beam of x-rays of constant intensity and connected to an 
appropriate circuit which will measure its counting rate, i.e., the rate of 
production of current pulses in the external circuit. The applied voltage 
is then gradually increased from zero, and the counting rate is found to 
vary with voltage in the manner shown in Pig. 7-11. No counts are ob- 
tained below a certain minimum voltage called the starting voltage,* but 
above this value the counting rate increases rapidly with voltage until 
the threshold of the Geiger region is reached. In this region, called the 

plateau, the counting rate is almost ^ plateau 

independent of voltage. At voltages 

beyond the plateau, the counter goes & i^ ( -l!L g( l! 

into a state of continuous discharge. p \\ \ continuous 

A Geiger counter is operated on the ~ 4 4 1^^ ' 

plateau, normally at an overvoltage 

of about 100 volts, i.e., at 100 volts 

higher than threshold. The plateau APPLIMI) \()LTA(JK 

has a finite slope, about 0.05 per- 

. . . , , PIG. 7-11. Effect of voltage on 

cent/volt, which means that the oper- ( . ountmg rato for (>onstant x . ray in _ 

ating voltage must be stabilized if the tensity, 
counting rate is to be accurately pro- 
portional to x-ray intensity. (The same is true of proportional counters.) 
No exact figures can be given for the starting voltage, threshold voltage, 
and length of plateau of Geiger counters, as these depend on such variables 
as counter dimensions and nature of the gas mixture, but the operating 

* Pulses are produced below /his voltage, but they are too small to be counted 
by the measuring circuit (sealer or ratemeter). Below the starting voltage, the 
counter is acting as a proportional counter and the pulses are much smaller than 
those produced in the (Jeiger region. Since the measuring circuit used with a 
Geiger counter is designed to operate only on pulses larger than a certain size, 
usually 0.25 volt, no pulses are counted at voltages less than the starting voltage 




194 DIFFRACTOMETER MEASUREMENTS [CHAP. 7 

voltage of most counters is commonly found to lie in the range of 1000 to 
1500 volts. It should be noted that some counters can be permanently 
damaged if subjected, even for brief periods, to voltages high enough to 
cause a continuous discharge. 

There are several important differences between the action of a Geiger 
counter and that of a proportional counter: 

(1) The absorption of an x-ray quantum anywhere within the volume 
of a Geiger counter triggers an avalanche that extends over the whole 
length of the counter. 

(2) The gas amplification factor of a Geiger counter is therefore much 
larger, about 10 8 to 10 9 (see Fig. 7-10), and so is the voltage pulse in the 
wire, now about 1 to 10 volts. This means that less amplification is needed 
in the external circuit. (Pulses from either kind of counter are always 
amplified before being fed to a sealer or ratemeter.) 

(3) At a constant applied voltage, all Geiger pulses are of the same size, 
independent of the energy of the x-ray quantum that caused the primary 
ionization. x . ra> (iuantimi 

These differences are illustrated absorbed hm- 

schematically in Fig. 7-12. The ab- 
sorption of an x-ray quantum in a 

proportional counter produces a very pHU])()lrn()NAL ( , n NT ,, K 
localized radial column ot ions ana 
electrons. In a Geiger counter, on the 
other hand, the applied voltage is so 
high that not only are some atoms 
ionized but others are raised to ex- 
cited states and caused to emit ultra- r.KKiKK < 'orvrat 
violet radiation. These ultraviolet FIG. 7-12. Differences in the extent 
photons then travel throughout the of ionization between proportional and 
counter at the speed of light, knock- Geiger counters. Each plus (or minus) 
. ,, . symbol represents a large number ol 
mg electrons out of other gas atoms positiye kms (or ele( , trons) . 

and out of the cathode shell. All the 

electrons so produced trigger other avalanches, and the net result is that 
one tremendous avalanche of electrons hits the whole length of the anode 
wire whenever an x-ray quantum is absorbed anywhere in the tube. 

All these electrons hit the wire in less than a microsecond, but the slowly 
moving positive ions require about 200 microseconds to reach the cathode. 
This means that the electron avalanche in a Geiger counter leaves behind 
it a cylindrical sheath of positive ions around the anode wire. The presence 
of this ion sheath reduces the electric field between it and the wire below 
the threshold value necessary to produce a Geiger pulse. Until this ion 
sheath has moved far enough away from the wire, the counter is insensitive 
to entering x-ray quanta. If these quanta are arriving at a very rapid 



7-6] 



(a) 



6EIGER COUNTERS 



195 



VOLTAGE 


A 


A 


A 


A 


A 


A 


A 



TIME 



O 



(b) 





TIME 



TIME 



input sensitivity 
of detector circuit 




-dead time /,/ *-| 

resolving time / s 
recovery time /, 



TIME 



FIG. 7-13. Dependence of pulse amplitude on pulse spacing. 

rate, it follows that not every one will cause a separate pulse and the coun- 
ter will become "choked." This places an upper limit on the rate at 
which entering quanta can be accurately counted without losses. This 
limit is much lower than that of a proportional counter, since the positive 
ions produced by a discharge are very localized in the proportional counter 
and do not render the rest of the counter volume insensitive. 

The way in which pulses occur in a Geiger counter is worth examining 
in some detail. It must be remembered that the arrival of x-ray quanta 
in the counter is random in time. Therefore pulse production in the coun- 
ter is also random in time, and a curve showing the change in voltage of 
the anode wire with time would have the appearance of Fig. 7-13 (a). 
During each pulse, the voltage rises very rapidly to a maximum and then 



196 DIFFRACTOMETER MEASUREMENTS [CHAP. 7 

decreases more slowly to its normal value. All pulses have the same ampli- 
tude and are spaced at random time intervals. 

But if the rate of pulse production is so high that two successive pulses 
occur too closely together, it is found that the second one has less than 
normal amplitude, as indicated in Fig. 7-13(b) on enlarged voltage-time 
scales. If the interval between pulses becomes smaller than that shown in 
(b), then the amplitude of the second pulse becomes still smaller, as shown 
in (c). Figure 7-13(d) sums up a number of curves of this kind; i.e., it is 
a superposition of a number of curves like (b) and (c), and it shows the 
amplitude which any given pulse will have when it follows the initial pulse 
at the time interval indicated by its position on the time axis. This de- 
crease in pulse height with decrease in pulse spacing has been correlated 
with the phenomena occurring in the counter as follows. When the ava- 
lanche of electrons hits the anode wire to form the initial pulse, the voltage 
rapidly builds up to its maximum value and then decays more slowly to 
zero as the charge on the wire leaks away. But, as stated above, the posi- 
tive ion sheath left behind reduces the field strength between it and the 
wire. The field strength increases as the ions move away from the wire, 
and the time at which the field reaches the threshold value marks theVhid 
of the dead time ,/, during which the counter is absolutely insensitive to 
entering quanta. The arrival of the ion sheath at the cathode restores the 
field to its normal strength and marks the end of the recovery time t r . Be- 
tween id and t r the field is above threshold but not yet back to normal; 
during this interval entering quanta can cause pulses, but they will not 
have the full amplitude characteristic of the applied voltage. The recov- 
ery time, at which the pulses regain their full amplitude, is fixed by the 
counter design and generally is of the order of 2 X 10~ 4 sec. However, 
the detecting circuit can usually detect pulses smaller than maximum 
amplitude, and we can therefore speak of the resolving time t s of the counter- 
circuit combination, defined ,by the time after the initial pulse at which a 
following pulse can first be detected. 

If the arrival, and absorption, of entering quanta were absolutely periodic 
in time, the maximum counting rate without losses would be given simply 
by \/t 8 . But even if their average rate of arrival is no greater than l/t 8 , 
some successive quanta may be spaced less than t 8 apart because of their 
randomness in time. It follows that counting losses will occur at rates 
less than \/t 8 and that the losses will increase as the rate increases, as 
shown in Fig. 7-14. Here "quanta absorbed per second" are directly 
proportional to the x-ray intensity, so that this curve has an important 
bearing on diffractometer measurements, since it shows the point at which 
the observed counting rate is no longer proportional to the x-ray intensity. 
The straight line shows the ideal response which can be obtained with a 
proportional counter at the rates shown. 



7-6] 



GEIGER COUNTERS 



197 



5000 r- 



single-chain her 
CJcieer counter 




1000 2000 3000 4000 5000 

QUANTA ABSOHBKI) PKK SECOND 
FIG. 7-14. The effect of counting rate on counting losses (schematic). 

Since the resolving time of the ordinary Geiger counter is of the order 
of 10~ 4 sec, counting-rate curves should be linear up to about 10,000 cps 
(counts per second) if the arrival of quanta were periodic in time. How- 
ever, counting losses are observed to begin at much lower rates, namely, 
at a few hundred counts per second, as shown in Fig. 7-14. In the multi- 
chamber counter the counting rate is linear up to more than 1000 cps; 
such a counter has a number of chambers side by side, each with its own 
anode wire, and one chamber can therefore register a count while another 
one is in its insensitive period. (The proportional counter, much faster 
than either of these, has a linear counting curve up to about 10,000 cps. 
Its resolving time is less than a microsecond; this is the time required for 
an electron avalanche to hit the wire, immediately after which the pro- 
portional counter is ready to register another pulse, since the positive ions 
formed produce no interference.) 

The particular counting rate where losses begin with a particular Geiger- 
counter-scaler combination must be determined experimentally, and this 
can be done as follows. Position the counter to receive a strong diffracted 
beam, and insert in this beam a sufficient number of metal foils of uniform 
thickness to reduce the counting rate almost to the cosmic background. 
(Cosmic rays, because of their high penetrating power, pass right through 
the walls of the counter and continually produce a few counts per second.) 
Measure the counting rate, remove one foil, measure the counting rate, and 
continue in this manner until all the foils have been removed. Since each 



198 



DIFPRACTOMETER MEASUREMENTS 



[CHAP. 7 



10,000 



w 

H 
<S 
tf 



CQ 
O 




1000 



2 4 S 10 

Nt'MHKK OK FOILS REMOVED 

FIG. 7-15. Calibration curves of a multichamber Geiger counter for two values 
of the x-ray tube peak voltage. Cu A'a radiation. Nickel foils, each 0.01 mm 
thick, used as absorbers. 

foil produces the same fractional absorption of the energy incident on it, 
a plot of observed counting rate (on a logarithmic scale) vs. number of 
foils removed from the beam (on a linear scale) will be linear up to the 
point where losses begin and will in fact resemble Fig. 7-14. A curve of 
this kind is shown in Fig. 7-15. Once the length of the linear portion of 
the calibration curve has been determined, it is best to make all further 
measurements in this region. Of course, the losses attendant on very high 
counting rates can be determined from the calibration curve and used to 
correct the observed rate, but it is usually safer to reduce the intensity of a 
very strong beam, by means of foils of known absorption, to a point where 
the observed counting rate is on the linear portion of the curve. 

Figure 7-15 also shows that the range of linearity of a counting rate 
curve is dependent on the x-ray tube voltage and is shorter for lower voltages. 
The reason for this dependence is the fact that the x-ray tube emits charac- 
teristic x-rays not continuously but only in bursts during those times when 



7-6] 



GEIGER COUNTERS 



199 




-\ cycle - 



-I 



critical excitation 

voltage 
TIME 

FIG. 7-16. Variation of tube volt- 
age with time for a full-wave rectified 
x-ray tube (schematic). 



the tube voltage exceeds the critical 
excitation voltage of the target mate- 
rial. Suppose, for example, that a 
copper target (excitation voltage = 
9 kv) is operated at a peak voltage 
of 50 kv. Then, if the wave form is 
like that shown in Fig. 7-16, Cu Ka 
radiation will be emitted during the 
time intervals ^2 and t^ but not 
during < 2 fe- But if the peak voltage 
is decreased to 25 kv, Cu Ka emission 
is limited to the shorter time intervals 
t 5 t G and / 7 < 8 . If the x-ray intensity is 
made the same at both voltages by 
adjusting the tube current, then it fol- 
lows that the same number of Cu Ka 
quanta are bunched into shorter times at the lower tube voltage than at 
the higher. Lowering the tube voltage therefore decreases the average 
time interval between quanta entering the Geiger counter during each 
half-cycle and may cause counting losses to occur at rates at which no 
losses are produced at higher tube voltages. It follows that a counter cali- 
bration curve applies only to measurements made at voltages not less than 
the voltage at which the calibration was performed. 

One other aspect of Geiger-counter operation deserves mention, and 
that is the method used to prevent the discharge actuated by the absorp- 
tion of one quantum from continuing indefinitely. If the counter is filled 
with a single gas such as argon, the positive argon ions on reaching the 
cathode are able to eject electrons from the cathode material. These 
electrons are accelerated to the anode and initiate another chain of ioniza- 
tion, with the result that a continuous discharge is set up in the counter, 
rendering it incapable of counting any entering quanta after the first one. 
This discharge may be prevented or "quenched" if an external circuit is 
used which abruptly lowers the voltage on the counter after each pulse 
to a value below that necessary to maintain a discharge but high enough 
to clear all ions from the gas. As soon as the ions are neutralized at the 
cathode, the high voltage is reapplied and the counter is again sensitive. 
To avoid the necessity for a quenching circuit, counters have been designed 
which are self-quenching by virtue of the gas mixture they contain. To 
the main gas in the counter, usually argon or krypton, is added a small 
proportion of "quench gas," which is either a polyatomic organic vapor, 
such as alcohol, or a halogen, such as chlorine or bromine. As its name 
implies, the quench gas plays the role of the quenching circuit used with 
single-gas counters and prevents the initial avalanche of ionization from 



200 



DIFFRACTOMETER MEASUREMENTS 



[CHAP. 7 



becoming a continuous discharge. In an argon-chlorine counter, for exam- 
ple, ionized argon atoms acquire electrons from chlorine molecules by 
collision, forming neutral argon atoms and ionized chlorine molecules. 
The latter are merely neutralized on reaching the cathode and do not re- 
lease electrons as argon ions do. Most counters used today are of the 
self-quenching variety. 

The efficiency of a Geiger or proportional counter and its associated 
circuits is given by the product of two efficiencies, that of quantum ab- 
sorption and that of quantum detection. The absorption efficiency de- 
pends on the absorption coefficient and thickness of the counter window, 
both of which should be as small as possible, and on the absorption coeffi- 
cient of the counter gas and the length of the counter, both of which should 
be as large as possible. The detection efficiency of a Geiger counter, as we 
have seen, depends on the counting rate and is effectively 100 percent at 
low rates; with a proportional counter this efficiency is near 100 percent 
at any rate likely to be encountered in diffraction experiments. The over- 
all efficiency of either counter at low rates is therefore determined by the 
absorption efficiency, which is commonly about 60 to 80 percent. 

The absorption efficiency, however, is very much dependent on the 
x-ray wavelength, the kind of gas used, and its pressure, since these factors 
determine the amount of radiation absorbed in a counter of given length. 
Figure 7-17 shows how the amount absorbed depends on wavelength for 
the two gases most often used in x-ray counters. Note that a krypton- 
filled counter has high sensitivity for 
all the characteristic radiations nor- 
mally used in diffraction but that an 
argon-filled counter is sensitive only 
to the longer wavelengths. This latter 
characteristic may be advantageous 
in some circumstances. For example, 
if a diffraction pattern is made with 
filtered radiation from a copper tar- 
get, use of an argon-filled counter w r ill 
produce semimonochromatic condi- 
tions, in that the counter will be 
highly sensitive to Cu Ka radiation 
and relatively insensitive to the short 
wavelength radiation which forms the 
most intense part of the continuous 
spectrum. The diffraction background 
will therefore be lower than if a 
krypton-filled counter had been 
used. 



Mo A 



c KM) 




05 10 1.5 
WAVELENGTH 



2.0 

(A) 



FIG. 7-17. Absorption of x-rays in 
a 10-cm path length of krypton and 
argon, each at a pressure of 65 cm 
Hg. 



7-7] 



SCINTILLATION COUNTERS 



201 



7-7 Scintillation counters. This type of counter utilizes the ability of 
x-rays to cause certain substances to fluoresce visible light. The amount 
of light emitted is proportional to the x-ray intensity and can be measured 
by means of a phototube. Since the amount of light emitted is small, a 
special kind of phototube called a photomultiplier has to be employed in 
order to obtain a measurable current output. 

The substance generally used to detect x-rays is a sodium iodide crystal 
activated with a small amount of thallium. It emits blue light under 
x-ray bombardment. The crystal is cemented to the face of a photo- 
.multiplier tube, as indicated in Fig. 7-18, and shielded from external light 
by means of aluminum foil. A flash of light is produced in the crystal for 
every x-ray quantum absorbed, and this light passes into the photomulti- 
plier tube and ejects a number of electrons from the photocathode, vhich 
is a photosensitive material generally made of a caesium-antimony inter- 
metallic compound. (For simplicity, only one of these electrons is shown 
in Fig. 7-18.) The emitted electrons are then drawn to the first of several 
metal dynodes, each maintained at a potential about 100 volts more posi- 
tive than the preceding one, the last one being connected to the measuring 
circuit. On reaching the first dynode, each electron from the photocathode 
knocks two electrons, say, out of the metal surface, as indicated in the 
drawing. These are drawn to the second dynode where each knocks out 
two more electrons and so on. Actually, the gain at each dynode may 
be 4 or 5 and there are usually at least 10 dynodes. If the gain per dynode 
is 5 and there are 10 dynodes, then the multiplication factor is 5 10 = 10 7 . 
Thus the absorption of one x-ray quantum in the crystal results in the 
collection of a very large number of electrons at the final dynode, producing 
a pulse about as large as a Geiger pulse, i.e., of the order of volts. Further- 
more, the whole process requires less than a microsecond, so that a scintil- 
lation counter can operate at rates as high as 10 5 counts per second without 



As in the proportional counter, the pulses produced in a scintillation 
counter have sizes proportional to the energy of the quanta absorbed. 



photocathode dynodes 



vacuum 

\ 




crystal photoniultiplicr tube 

FIG. 7-18. Scintillation counter (schematic). Electrical connections not shown. 



202 DIFFRACTOMETER MEASUREMENTS [CHAP. 7 

But the pulse size corresponding to a certain quantum energy is much 
less sharply defined than in a proportional counter; i.e., scintillation- 
counter pulses produced by x-ray quanta of a given energy have a mean 
size characteristic of that energy, but there is also a fairly wide distribu- 
tion of pulse size about this mean. As a result, it is difficult to discriminate 
between x-ray quanta of different energies on the basis of pulse size. 

The efficiency of a scintillation counter approaches 100 percent over the 
whole range of x-ray wavelengths, short and long, because all incident 
x-ray quanta are absorbed in the crystal. Its chief disadvantage is its 
rather high background count; a so-called "dark current" of pulses is pro- 
duced even when no x-ray quanta are incident on the counter. The main 
source of this dark current is thermionic emission of electrons from the 
photocathode. 

7-8 Sealers. A sealer is an electronic device which counts each pulse 
produced by the counter. Once the number of pulses over a measured 
period of time is known, the average counting rate is obtained by simple 
division. If the rate of pulse production were always low, say a few counts 
per second, the pulses could be counted satisfactorily by a fast mechanical 
counter, but such devices cannot handle high counting rates. It is there- 
fore necessary to divide, or scale down, the pulses by a known factor before 
feeding them to the mechanical counter. As its name implies, the sealer 
fulfills this latter function. There are two main kinds, the binary sealer, 
in which the scaling factor is some power of 2, and the decade sealer, in 
which it is a power of 10. 

We will consider sealer operation only in terms of binary sealers but 
the principles involved are applicable to either type. A typical binary 
sealer has several scaling factors available at the turn of a switch, ranging 
from 2 (= 1) to about 2 14 (= 16384). The scaling circuit is made up of a 
number of identical "stages" connected in series, the number of stages 
being equal to n where 2 n is the desired scaling factor. Each stage is com- 
posed of a number of vacuum tubes, capacitors, and resistors so connected 
that only one pulse of current is transmitted for every two pulses received. 
Since the output of one stage is connected to the input of another, this 
division by two is repeated as many times as there are stages. The output 
of the last stage may be connected to a mechanical counter which will 
register one count for every pulse transmitted to it by the last stage. Thus, 
if N pulses from a counter are passed through a circuit of n stages, only 
N/2 n will register on the mechanical counter. 

There are two ways of using a sealer to obtain an average counting rate : 
counting for a fixed time and counting a fixed number of pulses. In the 
first method, the sealer is turned on for a time t and then shut off. If the 
mechanical counter then shows N Q counts, the number of input pulses 



7-8] 



8CALER8 



203 



must have been 

AT = N (2 n ) + a, (7-5) 

where a is an integer ranging from up to (2 n 1). The integer a gives 
the number of pulses still "in the circuit" when the input pulses were shut 
off, and its value is found by noting which of several neon interpolation 
lamps connected to the several stages are still on. As indicated in Fig. 
7-19 for a scale-of-16 circuit, there is a neon lamp connected to each stage 
and the number opposite each lamp is 2 n ~ 1 where n is the number of the 
stage. The initial pulse entering a stage turns the lamp on and the second 
pulse turns it off. Since the second entering pulse causes a pulse to be 
transmitted to the next stage, the lamp on that stage goes on at the same 
time that the lamp on the preceding stage goes out. The integer a is 
therefore given by the sum of the numbers opposite lighted neon lamps. 
The total count shown in Fig. 7-19, for example, is N = 18(16) + (2 + 4) 
= 294. Once the total number of counts is known, the average counting 
rate is given simply by N/t. 

In the second method of scaling (counting a fixed number of pulses), 
the mechanical counter is replaced by an electric timer. The timer is con- 
nected to the circuit in such a way that it starts when the sealer is started 
and stops at the instant a pulse is transmitted from the last stage. For 
example, if the timer is connected to a 10-stage sealer, it will stop when 
exactly 1024 (= 2 10 ) pulses have entered the first stage, because at that 
instant the tenth stage will transmit its first pulse; the average counting 
rate is then given by the quotient of 1024 and the time shown on the timer. 
Such a circuit requires no interpolation since no counts remain in the circuit 
at the instant the final stage transmits its pulse to the timer; i.e., all the 
neon lights are off. The total number of counts, which must be a power 
of 2 in a binary sealer, is selected by a switch which connects the timer to 
any desired stage, thus making that stage the final stage and short-circuit- 
ing the remainder. 

Because the arrival of x-ray quanta in the counter is random in time, 
the accuracy of a counting rate measurement is governed by the laws of 
probability. Two counts of the same x-ray beam for identical periods of 
time will not be precisely the same because of the random spacing between 



interpolation x v 
numbers ~~^ \i) 














mechanical 
counter 









stage 2 




stage 3 




stage 4 


input 
pulses 


















FIG. 7-19. Determination of sealer counts. 



204 DIFFRACTOMETER MEASUREMENTS [CHAP. 7 

pulses, even though the counter and sealer are functioning perfectly. 
Clearly, the accuracy of a rate measurement of this kind improves as the 
time of counting is prolonged, and it is therefore important to know how 
long to count in order to attain a specified degree of accuracy. The prob- 
able error* in a single count of N pulses, relative to an average value 
obtained by a great many repetitions of the same counting operation, is 

given by 

67 
E N = = percent, (7-6) 



so long as N is fairly large. For some of the total counts obtainable from 
a binary sealer, this expression gives the following errors: 



Total number of 
pulses counted 


Percent 
probable error 


256 ( = 2 8 ) 
512 ( = 2 9 ) 
1024 (= 2 10 ) 
2048 ( = 2 11 ) 
4096 (= 2 12 ) 
8192(= 2 13 ) 
16384( = 2 14 ) 


4.2 
3.0 
2.1 
1.5 
1.0 
0.7 
0.5 



Note that the error depends only on the number of pulses counted and not 
on their rate, which means that high rates and low rates can be measured 
with the same accuracy, if the counting times are chosen to produce the 
same total number of counts in each measurement. It also follows that 
the second scaling method outlined above, in which the time is measured 
for a fixed number of counts, is generally preferable to the first, since it 
permits intensity measurements of the same precision of both high- and 
low-intensity beams. 

Equation (7-6) is valid only when the counting rate due to the radiation 
being measured is large relative to the background. (Here "background" 
means the unavoidable background counting rate measured with the x-ray 
tube shut off, and not the "diffraction background" at non-Bragg angles 
due to any of the several causes listed in Sec. 6-11 and of which fluorescent 
radiation is usually the most important. The unavoidable background is 
due to cosmic rays and may be augmented, in some laboratories, by stray 

* The probable error is that which is just as likely to be exceeded as not. Three 
times the probable error is a somewhat more useful figure, as the probability that 
this will be exceeded is only 0.04. Thus, if a single measurement gives 1000 counts, 
then the probable error is 67/^/1000 = 2.1 percent or 21 counts. Then the prob- 
ability is 0.5 that this count lies in the range Nt 21, where N t is the true number 
of counts, while the probability is 0.96 that the measured value lies in the range 
N t 63. 



7-8] SCALER8 205 

radiation from nearby radioactive material; it may be rather high, if a 
scintillation counter is used, because of the dark current of this counter.) 
Suppose a measurement is required of the diffraction background, always 
rather low, in the presence of a fairly large unavoidable background. In 
these circumstances, Eq. (7-6) does not apply. Let TV be the number of 
pulses counted in a given time with the x-ray tube on, and Nb the number 
counted in the same time with the tube off. Then Nb counts are due to 
the unavoidable background and (N Nb) to the diffraction background 
being measured, and the relative probable error in (N Nb) is 



07V 'N + N b 

E *-x* = -~7^ ^7T~ P ercent - (7-7) 

(N - N b ) 

Comparison of Eqs. (7-0) and (7-7) shows that longer counts must be 
made when the unavoidable background is of comparable intensity to 
the radiation being measured than when the unavoidable background is 
completely negligible by comparison, if the same accuracy is to be obtained 
in both measurements. 

As indicated in Sec. 7-2, the integrated intensity of a diffraction line 
may be measured with a sealer by determining the average counting rate 
at several angular positions of the counter. The line profile, the curve of 
intensity vs. 26, is then plotted on graph paper, and the area under the 
curve, and above the continuous background, is measured with a planimeter. 
To obtain the same relative accuracy of both the line profile and the adja- 
cent background, all measurements should be made by counting a fixed 
number of pulses. Three other methods of measuring integrated intensities 
have been used, all of which utilize the integrating properties of the scaling 
circuit to replace the curve plotting and planimeter measurement: 

(1) The line is scanned from one side to the other at a constant angular 
rate, the sealer being started at the beginning of the scan and stopped at 
its end. The total number of counts registered by the sealer, minus the 
number of counts due to the background, is then proportional to the in- 
tegrated intensity of the line. All lines on the pattern must be measured 
with the same receiving slit and the same scanning rate. The background 
adjacent to, and on either side of, the line may be measured by the same 
procedure, i.e., by scanning at the same rate over the same angular range, 
or by counting at a fixed position for the same time required to scan the 
line. 

(2) The counter is moved stepwise across the line and maintained in 
each position for the same length of time, the sealer being operated con- 
tinuously except when changing counter positions. The total count accu- 
mulated by the sealer, minus the background correction, is again propor- 
tional to the integrated intensity. A wide receiving slit is used, and the 



206 DIFFRACTOMETER MEASUREMENTS [CHAP. 7 

angular interval between counter positions is so chosen that the overlap 
between adjacent settings of the slit is negligibly small and constant and 
never coincides with the maximum intensity of the line being measured. 

(3) A receiving slit is used which is wider than the line being measured. 
The slit is centered on the line and a count made for a given time. The 
background is measured by counting at a position adjacent to the line 
with the same slit for the same length of time. 

Because all these methods involve counting for a fixed time, the back- 
ground and low-intensity portions of the diffraction line are measured 
with less accuracy than the high-intensity portions. The counting time 
should be chosen so that the low intensities are measured to the accuracy 
required by the particular problem involved; it will then follow that the 
high intensities are measured with unnecessarily high accuracy, but that 
is unavoidable in fixed-time methods such as these. 

The integrating ability of a sealer is also put to use in x-ray tube moni- 
tors. In Sec. 7-2 it was mentioned that the incident-beam intensity had 
to be maintained absolutely constant in a diffractometer and that this 
constancy required tube current and voltage stabilizers. These stabilizing 
circuits are not needed if an extra counter and sealer are available to 
"watch," or monitor, the tube output. The monitor counter may be posi- 
tioned to receive the direct beam, suitably filtered to reduce its intensity, 
from another window of the x-ray tube, or an auxiliary crystal may be set 
to diffract a portion of the beam used in the diffractometer into the monitor 
counter. In either case, every intensity measurement with the diffrac- 
tometer is made by starting the diffractometer sealer and monitor sealer 
simultaneously and stopping both when the monitor sealer has registered 
a constant number of counts N. In this way, every intensity measurement 
is made in terms of the same amount of energy incident on the specimen, 
and variations in tube output have no effect. 

7-9 Ratemeters. The counting-rate meter, as its name implies, is a 
device which indicates the average counting rate directly without requir- 
ing, as in the sealer-timer combination, separate measurements of the 
number of counts and the time. It does this by a circuit which, in effect, 
smooths out the succession of randomly spaced pulses from the counter 
into a steady current, whose magnitude is proportional to the average 
rate of pulse production in the counter. 

The heart of a ratemeter circuit is a series arrangement of a capacitor 
and resistor. To understand the action of a ratemeter, we must review 
some of the properties of such a circuit, notably the way in which the 
current and voltage vary with time. Consider the circuit shown in Fig. 
7-20(a), in which the switch S can be used either to connect a to c and thus 
apply a voltage to the capacitor, or to connect b to c and thus short-circuit 



7-9] 



RATEMETERS 



207 





FIG. 7-20. The capacitor-resistor circuit. 

the capacitor and resistor. When a is suddenly connected to c, the voltage 
across the capacitor reaches its final value V not instantaneously but only 
over a period of time, and at a rate which depends on the resistance R and 
the capacitance C, as shown in Fig. 7-20(b). The product of R and C has 
the dimensions of time (seconds, in fact, if R is in megohms and C in micro- 
farads), and it may be shown that the voltage across the capacitor reaches 
63 percent of its final value in a time given by RC, known as the time 
constant of the circuit. The time required to reach 99 percent of its final 
value is 4.6RC. Conversely, if the fully charged capacitor, bearing a 
charge Q = CV, is suddenly shorted through the resistor by connecting 
b to c, the charge does not immediately disappear but leaks away at a rate 
dependent on the time constant. The charge drops to 37 percent of its 
initial value in a time equal to RC and to 1 percent in a time equal to 



A complete ratemeter circuit consists of two parts. The first is a pulse- 
amplifying and pulse-shaping portion which electronically converts the 
counter pulses, which vary in amplitude and shape from counter to counter, 
into rectangular pulses of fixed dimensions in voltage and time. These 
pulses are then fed into the second portion, which is the measuring circuit 
shown in Fig. 7-21, a circuit basically pulse input 
similar to that of Fig. 7-20 (a) and 
having a time constant #2^2- $, 
shown as a simple switch, is actually 
an electronic circuit which connects a 
to c each time a pulse arrives and then 
connects b to c immediately after- 
wards. A constant charge is thus 
added to the capacitor for each pulse p IG . 7.21. Measuring portion of 
received and this charge leaks away ratemeter circuit. 




208 DIFFRACTOMETER MEASUREMENTS [CHAP. 7 

through the resistor until, at equilibrium, the rate of addition of charge is 
just balanced by the rate of leakage. The rate of charge leakage is simply 
the current through the microammeter M, which therefore indicates the 
rate of pulse production in the counter and, in turn, the x-ray intensity. 
The circuit usually contains, in addition to the indicating meter, a chart 
recorder which produces a continuous record of the intensity. 

Even when the x-ray intensity is constant (constant average counting 
rate), the spacing of the counter pulses is random in time, which means 
that the counting rate actually varies with time over short periods. The 
ratemeter responds to these statistical fluctuations in the counting rate, 
and its response speed is greater the smaller the time constant. This fol- 
lows from the discussion of the capacitor-resistor circuit: any change in 
the pulse rate causes a change in the current through the circuit, but the 
latter change always lags behind the former; the amount of lag is less for a 
small time constant than for a large one. Random fluctuations in the 
counting rate are therefore more evident with a small time constant, be- 
cause the current in the circuit then follows the changes in counting rate 
more closely. This feature is illustrated in Fig. 7-22, which shows the 
automatically recorded output of a ratemeter when the counter is receiving 
a constant-intensity x-ray beam. The large fluctuations at the left have 
been reduced in magnitude by successive increases in the time constant, 
effected by changing the value of C 2 . Evidently, a single reading of the 
position of the indicating meter needle or the recorder pen of a ratemeter 
may be seriously in error, and more so at low time constants than at high. 
In Sec. 7-8 we saw that the error in a counting-rate measurement de- 
creased as the number of counts increased. Now it may be shown that a 
ratemeter acts as if it counted for a time 2R 2 C 2 , in the sense that the 
accuracy of any single reading is equivalent to a count made with a sealer 
for a time 2R 2 C 2 . Therefore, the relative probable error in any single 
ratemeter reading is given by the counterpart of Eq. (7-G), namely by 

C7 

E = ; percent, (7-8) 

\/2nR 2 C 2 

where n is the average counting rate. This equation also shows that the 
probable error is less for high counting rates than for low, when the time 
constant remains the same. This is illustrated graphically in Fig. 7-23, 
which shows how the recorded fluctuations in the counting rate decrease 
as the rate itself is increased. 

The most useful feature of a ratemeter is its ability to follow changes in 
the average counting rate, a function which the sealer is totally unable to 
perform, since a change in the average counting rate occurring during the 
time a count is being made with a sealer will go entirely undetected. It is 
this feature of a ratemeter which is so useful in diffractometry. A diffrac- 



7-9] 



RATEMETERS 



209 




FIG. 7-22. Effect of time constant (T.C.) on recorded fluctuations in counting 
rate at constant x-ray intensity (schematic). Time constants changed abruptly at 
times shown. (T.C.)i < (T.C.) 2 < (T.C.) 3 . 

tion pattern can be scanned from one end to the other, and the moving 
counter automatically transmits, through the ratemeter, a continuous 
record of the intensity it observes as the diffraction angle is changed. On 
the other hand, the ratemeter is less accurate than the sealer, both because 
of the unavoidable statistical fluctuations in its output and because of the 
errors inherent in its indicating or recording instruments. 

As mentioned earlier, a large time constant smooths out fluctuations in 
the average counting rate by increasing the response time to changes in 
rate. But when a sharp diffraction line is being scanned, the average 
counting rate is changing rapidly and we would like the ratemeter to indi- 
cate this change as accurately as possible. From this point of view a short 
response time, produced by a small time constant, is required. A rate- 
meter must therefore be designed with these two conflicting factors in 





TIME 



FIG. 7-23. Effect of average counting rate on recorded fluctuations in counting 
rate, for a fixed time constant (schematic). X-ray intensity changed abruptly at 
times shown. 



210 DIFFRACTOMETBR MEASUREMENTS [CHAP. 7 

mind, and the time constant should be chosen large enough to smooth out 
most of the statistical fluctuations and yet small enough to give a reason- 
ably short response time. 

Most commercial ratemeters have several scales available to cover var- 
ious ranges of x-ray intensity (100, 1000, and 10,000 cps for full-scale 
deflection of the recorder pen, for example). Smaller time constants are 
used with the higher scales, just as short counting times are used with a 
sealer when the counting rate is high. In some instruments, the time con- 
stant appropriate to each scale is fixed by the manufacturer, and in others 
the operator can select any one of several time constants, ranging from 
about 0.5 to 15 sec, by switches which insert the proper capacitance in the 
circuit. The proper time constant to use is, of course, not unrelated to the 
scanning speed, for a fast scan demands a fast response from the ratemeter 
and therefore a short time constant. A time constant which is too large 
for the scanning speed used will slightly shift the peaks of diffraction lines 
in the direction of the scan and lower their maximum intensity and, be- 
cause of its excessive smoothing action, may actually obliterate weak dif- 
fraction lines and cause them to go unnoticed. In choosing a time constant, 
it is therefore better to err on the short side. A good rule to follow is to 
make the time constant less than half the time width of the receiving slit, 
where the time width is defined as the time required for the slit to travel 
its own width. For example, if a 0.2 slit is used at a scanning speed of 
2/min, then the time width of the slit is (0.2/2) (60) = 6 sec, and the 
time constant should therefore be less than 3 sec. The same rule can be 
used to find the proper slit width for a given scanning speed when the time 
constant is fixed. 

The relation between the x-ray intensity, i.e., the average counting rate, 
and the deflection of the indicating meter needle or recorder pen is linear 
for some ratemeters and logarithmic for others. The exact relation may 
be found by a calibration procedure similar to that used for the Geiger 
counter and sealer, as outlined in Sec. 7-8. A number of identical metal 
foils are placed in a strong diffracted beam entering the counter and these 
are withdrawn one by one, with the counter in a fixed position. After 
each withdrawal, the counting rate is measured accurately with a sealer, 
and the ratemeter operated for a time at least equal to the scaling time, 
the recording chart speed being selected to give a trace of reasonable 
length. An average straight line is then drawn through each trace, in such 
a way as to make the positive and negative fluctuations as nearly equal as 
possible. (Figure 7-23 shows a portion of a calibration run made in this 
way.) Finally, the distances of these straight lines from the chart zero are 
plotted against the corresponding average counting rates as determined by 
the sealer, and the calibration curve so obtained is used as a basis for* future 
intensity measurements with the ratemeter-recorder combination. 



7-10] USE OF MONOCHROMATORS 211 

7-10 Use of monochromators. Some research problems, notably the 
measurement of diffuse scattering at non-Bragg angles, require a strictly 
monochromatic incident beam if the effects to be measured are not to be 
blotted out by the continuous spectrum. In such a case, the focusing 
crystal monochromator described in Sec. 6-12 may be used in conjunction 
with a diffractometer in the manner shown in Fig. 7-24. Rays from the 
physical line source S on the x-ray tube target T are diffracted by the bent 
and cut crystal M to a line focus at S', located on the diffractometer circle, 
and then diverge to the specimen C. After diffraction from the specimen, 
they are again focused at F, the counter receiving slit. The diffractometer 
geometry is therefore identical with that shown in Fig. 7-1 but with the 
important difference that the rays incident on the specimen are mono- 
chromatic and issue from the virtual source S', the focal line of the mono- 
chromating crystal. 

There is another method of operating under essentially monochromatic 
conditions, a method peculiar to the diffractometer, and that is by the 
use of Ross filters, also called balanced filters. This method depends on 
the fact that the absorption coefficients of all substances vary in the same 
way with wavelength; i.e., they are proportional to X 3 , as shown by Eq. 
(1-13). If filters are made of two substances differing in atomic number 
by one, and their thicknesses adjusted so that they produce the same ab- 
sorption for a particular wavelength, then they will have the same absorp- 
tion for all wavelengths except those lying in the narrow wavelength region 
between the K absorption edges of the two substances. This region is 
called the pass band of the filter combination. If these filters are placed 
alternately in a heterochromatic x-ray beam, i.e., a beam containing rays 
of different wavelengths, then the difference between the intensities trans- 
mitted in each case is due only to wavelengths lying in the pass band. 




FIG. 7-24. Use of crystal monochromator with diffractometer. 



212 DIFFRACTOMETER MEASUREMENTS [CHAP. 7 

When the pass band is chosen to include a strong characteristic component 
of the spectrum, then the net effect is that of a strong monochromatic 
beam. 

The isolation of Cu Ka radiation may be taken as an example. Its 
wavelength is 1.542A, which means that cobalt and nickel can be used as 
filter materials since their K absorption edges (1.608 and 1.488A, respec- 
tively) effectively bracket the Cu Ka line. Their linear absorption coeffi- 
cients M are plotted in Fig. 7-25 (a), which shows that balancing can be 
obtained by making the nickel filter somewhat thinner than the cobalt one. 
When their thicknesses x are adjusted to the correct ratio, then MN^NI = 
MCO^CO except in the pass band, and a plot of px vs. X has the appearance 
of Fig. 7-25(b). Since /LT = In / x // , the transmission factors /V/o 
(ratio of transmitted to incident intensity) of the two filters are now equal 
for all wavelengths except those in the pass band, which is only 0.1 2A 
wide. At each angle 20 at which the intensity is to be measured with the 
diffractometer, first one filter and then the other is placed in the diffracted 
beam before it enters the counter. The intensity of the diffracted beam 
passing through each filter is then measured, and the difference in the 
measurements gives the diffracted intensity of only the Cu A'a line and 
the relatively weak wavelengths immediately adjacent to it in the pass 
band. 

It should be emphasized that the beam entering the counter is never 
physically monochromatic, as it is when a crystal monochromator is used. 
Radiation with a great many wavelengths enters the counter when either 
filter is in place, but every wavelength transmitted by one filter has the 
same intensity as that transmitted by the other filter, except those wave- 
lengths lying in the pass band, and these are transmitted quite unequally 
by the two filters. Therefore, when the intensity measured with one filter 
is subtracted from that measured with the other filter, the difference is 
zero for every wavelength except those in the pass band. 

In practice, balancing of the filters is carried out by inserting two foils 
of approximately the same thickness into suitable holders which can be 
slipped into place in the beam entering the counter. One foil is always 
perpendicular to the x-ray beam, while the other may be rotated about 
an axis at right angles to the beam; in this way the second foil may be in- 
clined to the beam at such an angle that its effective thickness x equals 
the thickness required for balancing. Perfect balancing at all wavelengths 
outside the pass band is not possible, although it may be approached quite 
closely, because n does not vary exactly as X 3 and because the magnitude 
of the K absorption jump (ratio of absorption coefficients for wavelengths 
just shorter and just longer than the K edge) is not exactly the same for 
all elements. 



7-10] 



USE OF MONOCHROMATORS 



213 






c3 



O 
-3 

c 
<u 

1 







i 

,0 




o 



- 
g 

8 

^ 

5 



II 



MDLl^KX) XOLIxIHOSHV HVMXTI 



214 DIFFRACTOMETER MEASUREMENTS [CHAP. 7 

PROBLEMS 

7-1. A powder specimen in the form of a rectangular plate has a width of 0.5 
in., measured in the plane of the diffractometer circle, which has a radius of 5.73 
in. If it is required that the specimen entirely fill the incident beam at all angles 
and that measurements must be made to angles as low as 26 = 10, what is the 
maximum divergence angle (measured in the plane of the diffractometer circle) 
that the incident beam may have? 

7-2. Prove the statement made in Sec. 7-4 that the effective irradiated volume 
of a flat plate specimen in a diffractometer is constant and independent of 6. 

7-3. In measuring the maximum intensity of a certain diffraction line with a 
sealer, 2048 pulses were counted in 1.9 sec. When the "diffraction background" 
a few degrees away from the line was measured, 2048 pulses were counted in 182 
seconds. The average counting rate determined over a long period of time with 
the x-ray tube shut off was 2.2 cps. 

(a) What is the ratio of the maximum intensity of the line to that of the "dif- 
fraction background"? 

(6) What is the probable error in each of these intensities? 

(c) How long must the "diffraction background" be counted in order to obtain 
its intensity with the same accuracy as that of the diffraction line? 

7-4. (a) Calculate the ratio of the effective thicknesses of cobalt and nickel 
filters when they are balanced for all wavelengths except Cu Ka. (Obtain an av- 
erage value applicable to a wavelength range extending from about 0.5A to about 
2A.) 

(6) When the filters are balanced, calculate the ratio of the intensity of Cu Ka 
radiation transmitted by the nickel filter to that transmitted by the cobalt filter, 
assuming the same incident intensity in each case. The effective thickness of the 
nickel filter is 0.00035 in. 



CHAPTER 8 
ORIENTATION OF SINGLE CRYSTALS 

8-1 Introduction. Much of our understanding of the properties of poly- 
crystalline materials has been gained by studies of isolated single crystals, 
since such studies permit measurement of the properties of the individual 
building blocks in the composite mass. Because single crystals are usually 
anisotropic, research of this kind always requires accurate knowledge of 
the orientation of the single crystal test specimen in order that measure- 
ments may be made along known crystallographic directions or planes. 
By varying the crystal orientation, we can obtain data on the property 
measured (e.g., yield strength, electrical resistivity, corrosion rate) as a 
function of crystal orientation. 

In this chapter the three main x-ray methods of determining crystal 
orientation will be described: the back-reflection Laue method, the trans- 
mission Laue method, and the diffractometer method. It is also con- 
venient to treat here the question of crystal deformation and the measure- 
ment of this deformation by x-ray methods. Finally, the subject of rela- 
tive crystal orientation is discussed, and methods are given for determining 
the relative orientation of two naturally associated crystals, such as the 
two parts of a twin or a precipitated crystal and its parent phase. 

8-2 The back-reflection Laue method. As mentioned in Sec. 3-6, the 
Laue pattern of a single crystal consists of a set of diffraction spots on the 
film and the positions of these spots depend on the orientation of the crys- 
tal. This is true of either Laue method, transmission or back-reflection, 
so either can be used to determine crystal orientation. However, the back- 
reflection method is the more widely used of the two because it requires no 
special preparation of the specimen, which may be of any thickness, 
whereas the transmission method requires relatively thin specimens of low 
absorption. 

In either case, since the orientation of the specimen is to be determined 
from the location of the Laue spots on the film, it is necessary to orient 
the specimen relative to the film in some known manner. The single 
crystal specimens encountered in metallurgical work are usually in the 
form of wire, rod, sheet, or plate, but crystals of irregular shape must occa- 
sionally be dealt with. Wire or rod specimens are best mounted with 
their axis parallel to one edge of the square or rectangular film; a fiducial 
mark on the specimen surface, for example on the side nearest the film, 
then fixes the orientation of the specimen completely. It is convenient to 

215 



216 



ORIENTATION OF SINGLE CRYSTALS 
ZA 



[CHAP. 8 




FIG. 8-1. Intersection of a conical array of diffracted beams with a film placed 
in the back-reflection position. C = crystal, F = film, Z.A = zone axis. 

mount sheet or plate specimens with their plane parallel to the plane of 
the film and one edge of the sheet or plate parallel to an edge of the film. 
Irregularly shaped crystals must have fiducial marks on their surface which 
will definitely fix their orientation relative to that of the film. 

The problem now is to determine the orientation of the crystal from the 
position of the back-reflection Laue spots on the film. If wo wished, we 
could determine the Bragg angle corresponding to each Laue spot from 
Eq. (5-2), but that would be no help in identifying the planes producing 
that spot, since the wavelength of the diffracted beam is unknown. We 
can, however, determine the orientation of the normal to the pianos caus- 
ing each spot, because the plane normal always bisects the angle between 
incident and diffracted beams. The directions of tho piano normals can 
then bo plotted on a steroographic projection, the angles between thorn 
measured, and the planes identified by comparison with a list of known 
interplanar angles for the crystal involved. 

Our first problem, therefore, is to derive, from the measured position of 
each diffraction spot on the film, the position on a stereographic projection 
of the pole of the plane causing that spot. In doing this it is helpful to 
recall that all of the planes of one zone reflect beams which lie on tho sur- 
face of a cone whoso axis is tho zono axis and whoso somi-apox angle is 
equal to the angle <t> at which tho zono axis is inclined to the transmitted 
beam (Fig. 8-1). If <t> doos not exceed 45, tho cone will not intersect a 
film placed in tho back-reflection region; if < lies between 45 and 90, the 
cone intersects tho film in a hyperbola; and, if </> oquals 90, the intersection 
is a straight line passing through tho incident beam. (If </> exceeds 90, 
the cone shifts to a position below tho transmitted beam and intersects 
the lower half of the film, as may be soon by viewing Fig. 8-1 upside down.) 
Diffraction spots on a back-reflection Laue film therefore lie on hyper- 



8-2] 



THE BACK-REFLECTION LAUE METHOD 



217 



film 




[ornri cut tor 
identification 



Fid S 2 Locution oi buck-ieflertion hnuo spot. Note that 7 =- 1)0 0. 

bolas or straight lines, and the distance of any hyperbola from the center 
of the film is a measure of the inclination of the zone axis. 

In Fig. 8 2 the film is \ie\\ed from the crystal. Coordinate axes are 
set up such that the incident beam proceed* along the z-axis in the direc- 
tion Oz and the .r- and //-axes he in the plane of the film. The beam re- 
flected by the plane shown strikes the film at S. The normal to this reflect- 
ing plane is (\\ and the plane itself is assumed to belong to a zone \\hose 
axis lies in the //; -plane. If we imagine this plane to rotate about the zone 
axis, it will pass through all the positions at which planes of this zone in an 
actual crystal might lie. During this rotation, the plane normal would cut 
the film in the straight line AB and the reflected beam in the hyperbola HK. 
AB is therefore the locus of plane normal intersections with the film and 
HK the locus of diffracted beam intersections. The plane \\hich reflects a 
beam to N, for example, has a normal which intersects the film at N, since 
the incident beam, plane normal, and diffracted beam are coplanar. Since 
the orientation of the plane normal in space can be described by its angular 
coordinates 7 and 6, the problem is to determine 7 and 6 from the measured 
coordinates x and // of the diffraction spot S on the film. 

A graphical method of doing this was devised by (Ireninger who devel- 
oped a chart which, when placed on the film, gives directly the 7 and 5 
coordinates corresponding to any diffraction spot. To plot such a chart, 
we note from Fig. 8-2 that 

jc = OK sin /x, U = OS cos M, and OS = OC tan 2a, 



218 ORIENTATION OF SINGLE CRYSTALS [CHAP. 8 

where OC = D = specimen-film distance. The angles ju and <r are ob- 
tained from 7 and d as follows: 

FN CF tan 8 tan 6 

tan M = = = 

FO CF sin 7 sin 7 



OC \sin M/ \CF cos y/ \ sin M / \CF cos 7 
tan 5 



sin /i cos 7 

With these equations, the position (in terms of x and y) of any diffraction 
spot can be plotted for given values of 7 and d and any desired specimen- 
film distance D. The result is the Greninger chart, graduated at 2 inter- 
vals shown in Fig. 8-3. The hyperbolas running from left to right are 
curves of constant 7, and any one of these curves is the locus of diffraction 
spots from planes of a zone whose axis is tilted away from the plane of the 
film by the indicated angle 7. If points having the same value of d are 
joined together, another set of hyperbolas running from top to bottom is 
obtained. The lower half of the chart contains a protractor whose use 
will be referred to later. Greninger charts should have dark lines on a 
transparent background and are best prepared as positive prints on photo- 
graphic film. 

In use, the chart is placed over the film with its center coinciding with 
the film center and with the edges of chart and film parallel. The 7 and 
S coordinates corresponding to any diffraction spot are then read directly. 
Note that use of the chart avoids any measurement of the actual coordinate 
distances x and y of the spot. The chart gives directly, not the x and y 
coordinates of the spot, but the angular coordinates y and d of the normal to 
the plane causing the spot. 

Knowing the 7 and 8 coordinates of any plane normal, for example CN 
in Fig. 8-2, we can plot the pole of the plane on a stereographic projection. 
Imagine a reference sphere centered on the crystal in Fig. 8-2 and tangent 
to the film, and let the projection plane coincide with the film. The point 
of projection is taken as the intersection of the transmitted beam and the 
reference sphere. Since the plane normal CN intersects the side of the 
sphere nearest the x-ray source, the projection must be viewed from that 
side and the film "read" from that side. In order to know, after processing, 
the orientation the film had during the x-ray exposure, the upper right- 
hand corner of the film (viewed from the crystal) is cut away before it is 
placed in the cassette, as shown in Fig. 8-2. When the film is read, this 



8-2] 



THE BACK-REFLECTION LAUE METHOD 



219 



6 20 




7 = 20 



7 = 10 



7=0 



21) 



FIG. (S-3. (jreniiifter chait for the solution of back-reflection Laue patterns, 
reproduced in the correct size for a specimen-to-film distance D of 3 cm. 

cut corner must therefore be at the upper left, as shown in Fig. 8-4(a). 
The angles 7 and 6, read from the chart, are then laid out on the projection 
as indicated in Fig. 8-4 (b). Note that the underlying Wulff net must be 
oriented so that its meridians run from side to side, not top to bottom. 
The reason for this is the fact that diffraction spots which lie on curves of 
constant y come from planes of a zone, and the poles of these planes must 



220 



ORIENTATION OF SINGLE CRYSTALS 

cut corner y 



[CHAP. 8 




RO.JI-XTION 




(b) 

P'IG. <S-4. Use of the Greninger chart to plot the pole of a reflecting plane on a 
stereographic projection. Pole 1' m (b) is the pole of the plane causing diffraction 
spot 1 in (a). 



THE BACK-REFLECTION LAUE METHOD 



8-2] 

therefore lie on a great circle on the 
projection. The 7,6 coordinates cor- 
responding to diffraction spots on the 
lower half of the film are obtained 
simply by reversing the Greninger 
chart end for end. 

This procedure may be illustrated 
by determining the orientation of the 
aluminum crystal whose back-reflec- 
tion Laue pattern is shown in Fig. 
3-(>(b)- Fig. 8-5 is a tracing of this 
photograph, showing the more im- 
portant spots numbered for reference. 
The poles of the planes causing these 
numbered spots are plotted stereo- 
graphically in Fig. 8-0 by the method 
of Fig. 8-4 and are shown as solid 
circles. 



221 



FIG. X-5. Selected diffraction spots 
of back-reflection Laue pattern of an 
aluminum crystal, traced from Fig. 
3-6(b). 




FIG. 8-6. Stereographic projection corresponding to back-reflection pattern of 
Fig. 8-5. 



222 ORIENTATION OF SINGLE CRYSTALS [CHAP. 8 

The problem now is to "index" these planes, i.e., to find their Miller 
indices, and so disclose the orientation of the crystal. With the aid of a 
Wulff net, great circles are drawn through the various sets of poles corre- 
sponding to the various hyperbolas of spots on the film. These great 
circles connect planes of a zone, and planes lying at their intersections are 
generally of low indices, such as j 100 } , {110}, { 11 1 } , and {112}. The axes 
of the zones themselves are also of low indices, so it is helpful to locate 
these axes on the projection. They are shown as open circles in Fig. 8-6, 
PA being the axis of zone .4, PB the axis of zone B, etc. We then measure 
the angles between important poles (zone intersections and zone axes) 
and try to identify the poles by comparison of these measured angles with 
those calculated for cubic crystals (Table 2-3). The method is essentially 
one of trial and error. We note, for example, that the angles P A PB, 
P A _ 5' ? an d p B 5' ar e all 90. This suggests that one or more of these 
poles might be |100| or {110}, since the angle between two {100} poles 
or between two jllOj poles is 90. Suppose we tentatively assume that 
PA, PB, and 5' are all J100} poles.* Then P E < which lies on the great 
circle between P A and P B and at an angular distance of 45 from each, 
must be a j 1 10} pole. We then turn our attention to zone C and find that 
the distance between pole 6' and either pole 5' or PR is also 45. But 
reference to a standard projection, such as Fig. 2-37, shows that there is 
no important pole located midway on the great circle between {100}, 
which we have identified with 5', and {110}, which we have identified 
with PR. Our original assumption is therefore wrong. We therefore make 
a second assumption, which is consistent with the angles measured so far, 
namely that 5' is a {100! pole, as before, but that P A and P B are {110} 
poles. PE must then be a {100} pole and & a {110} pole. We can check 
this assumption by measuring the angles in the triangle a b 5'. Both 
a and b are found to be 55 from 5', and 71 from each other, which con- 
clusively identifies a and b as {111} poles. We note also, from a standard 
projection, that a {111} pole must lie on a great circle between { 100 j and 
{110}, which agrees with the fact that a, for example, lies on the great 
circle between 5', assumed to be {100}, and PB, assumed to be {110}- 
Our second assumption is therefore shown to be correct. 



* i 



1 The reader may detect an apparent error in nomenclature here. Pole 5' for 
example, is assumed to be a {100} pole and spot 5 on the diffraction pattern is 
assumed, tacitly, to be due to a 100 reflection. But aluminum is face-centered 
cubic and we know that there is no 100 reflection from such a lattice, since hkl 
must be unmixed for diffraction to occur. Actually, spot 5, if our assumption is 
correct, is due to overlapping reflections from the (200), {400}, (600), etc., planes. 
But these planes are all parallel and are represented on the stereographic projec- 
tion by one pole, which is conventionally referred to as { 100} . The corresponding 
diffraction spot is also called, conventionally but loosely, the 100 spot. 



8-2] 



THE BACK-REFLECTION LAUE METHOD 



223 




FIG. 8-7. Stereographic projection of Fig. 8-6 with poles identified. 

Figure 8-7 shows the stereographic projection in a more complete form, 
with all poles of the type {100}, {110}, and {111} located and identified. 
Note that it was not necessary to index all the observed diffraction spots 
in order to determine the crystal orientation, which is specified completely, 
in fact, by the locations of any two { 100 } poles on the projection. The 
information given in Fig. 8-7 is therefore all that is commonly required. 
Occasionally, however, we may wish to know the Miller indices of a par- 
ticular diffraction spot on the film, spot 11 for example. To find these 
indices, we note that pole IT is located 35 from (001) on the great circle 
passing through (001) and (111). Reference to a standard projection and 
a table of interplanar angles shows that its indices are (112). 

As mentioned above, the stereographic projection of Fig. 8-7 is a com- 
plete description of the orientation of the crystal. Other methods of 
description are also possible. The crystal to which Fig. 8-7 refers had the 
form of a square plate and was mounted with its plane parallel to the plane 
of the film (and the projection) and its edges parallel to the film edges, 
which are in turn parallel to the NS and EW axes of the projection. Since 
the (001) pole is near the center of the projection, which corresponds to 



224 



ORIENTATION OF SINGLE CRYSTALS 



[CHAP. 8 



the specimen normal, and the (010) pole near the edge of the projection 
and approximately midway between the K and *S Y poles, we may very 
roughly describe the crystal orientation as follows: one set of cube planes 
is approximately parallel to the surface of the plate while another set 
passes diagonally through the plate and approximately at right angles to 
its surface. 

Another method of description may be used when only one direction in 
the crystal is of physical significance, such as the plate normal in the pres- 
ent case. For example, we may wish to make a compression test of this 
crystal, with the axis of compression normal to the plate surface. We are 
then interested in the orientation of the crystal relative to the compression 
axis (plate normal) or, stated inversely, in the orientation of the compres- 
sion axis relative to certain directions of low indices in the crystal. Now 
inspection of a standard projection such as Fig. 2-36(a) shows that each 
half of the reference sphere is covered by 24 similar and equivalent spherical 
triangles, each having f 100}, 1 110), and j 1 1 1 1 as its vertices The plate 
normal will fall in one of these triangles and it is necessary to draw only 
one of them in order to describe the precise location of the normal. In 
Fig. 8-7, the plate normal lies in the (001)-(101)-(1 Jl) triangle which is 
redrawn in Fig. 8-8 in the conventional orientation, as though it formed 
part of a (001) standard projection. To locate the plate normal on this 
new drawing, we measure the angles between the center of the projection 
in Fig. 8-7 and the three adjacent poles. Let these angles be pooi, Pioi, 
and pin- These angles are then used to determine the three arcs shown 
in Fig. 8-8. These are circle arcs, 
but they are not centered on the cor- 
responding poles; rather, each one is 
the locus of points located at an equal 
angular distance from the pole in- 
volved and their intersection there- 
fore locates the desired point. An 
alternate method of arriving at Fig. 
8-8 from Fig. 8-7 consists simply in 
rotating the whole projection, poles 
and plate normal together, from the 
orientation shown in Fig. 8-7 to that 
of a standard (001) projection. 

Similarly, the orientation of a 

single-crystal wire or rod may be de- FIG. 8-8. Use of the unit stereo- 
scribed in terms of the location of its ff?P hic . tHan ^ e t() . A &* ^ l 

. . xl . A , , . , . , orientation. The point inside the tn- 

axis in the unit stenographic triangle. angle ig ^ normal to the gingle cryg . 

Note that this method does not tal plate whose orientation is shown 
completely describe the orientation in Fig. 8-7. 




g-2] THE BACK-REFLECTION LAUE METHOD 225 

of the crystal, since it allows one rotational degree of freedom about the 
specimen axis. This is of no consequence, however, when we are only 
interested in the value of some measured physical or mechanical property 
along a particular direction in the crystal. 

There arc alternate ways of manipulating both the Gremnger chart and the 
stereographic projection, and the particular method used is purely a matter of 
personal preference For example, we may ignore the individual spots on the film 
and focus our attention instead on the various hyperbolas on which they lie. The 
spots on one hyperbola are due to reflections from planes of one zone and, by means 
of the Greninger chart, we can plot directly the axis of this zone without plotting 
the poles of any of the planes belonging to it. The procedure is illustrated in Fig. 
S-9. Keeping the centers of film and chart coincident, we rotate the film about 
this center until a particular hyperbola of spots coincides with a curve of constant 
7 on the chart, as in (a). The amount of rotation required is read from the inter- 
section of a vertical pencil line, previously ruled through the center of the film and 
parallel to one edge, with the protractor of the Greninger chart. Suppose this 
angle is e. Then the projection is mtftted by the same angle c with respect to the 
underlying Wulff net and the zojueaxis is j plotted on the vertical axis of the pro- 
jection at an angle 7 {wm&e 'circumference, as in (b). (Note that zone A itself 
is represented by a greatyrcle located at an angle 7 above the center of the pro- 
jection. However, tlprpK)tting of the zone circle is not ordinarily necessary since 
the zone axis adequately represents the whole zone.)* Proceeding in this way, 
we plot the poles of all the important zones and, by the method of Fig. 8-4, the 
pole of the plane causing the most important spot or spots on the pattern. (The 
latter are, like spot 5 of Fig. S-5, of high intensity, at the intersection of a number 
of hyperbolas, and well separated from their neighbors.) The points so obtained 
are always of low indices and can usually be indexed without difficulty. 

An alternate method of indexing plotted poles depends on having available a 
set of detailed standard projections in a number of orientations, such as {100(, 
( 1 10|, and ( 11 1 1 for cubic crystals. It is also a trial and error method and may 
be illustrated with reference to Fig S-6. First, a prominent zone is selected and 
an assumption is made as to its indices, for example, we might assume that zone 
B is a (100) zone. This assumption is then tested by (a) rotating the projection 
about its center until PH lies on the equator of the Wulff net and the ends of the 
zone circle coincide uith the N and A> poles of the net, and (b) rotating all the im- 
portant points on the projection about the MS-axis of the net until PB lies at the 
center and the zone circle at the circumference. The new projection is then super- 
imposed on a (100) standard projection and rotated about the center until all 
points on the projection coincide with those on the standard. If no such coinci- 
dence is obtained, another standard projection is tried. For the particular case 

* Note that, when a hyperbola of spots is lined up with a horizontal hyperbola 
on the chart as in Fig. 8-9(a), the vertical hyperbolas can be used to measure the 
difference in angle 5 for any two spots and that this angle is equal to the angle be- 
tween the planes causing those spots, just as the angle between two poles lying 
on a meridian of a Wulff net is given by their difference in latitude. 



226 



ORIENTATION OF SINGLE CRYSTALS 



[CHAP. 8 



cut corner 



row of .spots 

from planes of 

zone A 




(b) 



FIG. 8-9. Use of the Greninger chart to plot the axis of a zone of planes on the 
stereographic projection. PA is the axis of zone A. 



8-2] 



THE BACK-REFLECTION LAUE METHOD 



227 




plane 



FIG. 8-10. Relation between diffraction spot 8 and stereographic projection P 
of the plane causing the spot, for back reflection. 

of Fig. 8-6, a coincidence would be obtained only on a { 1 1 1 standard, since PB 
is actually a ( 110) pole. Once a match has been found, the indices of the unknown 
poles are given simply by the indices of the poles on the standard with which they 
coincide. 

In the absence of a Greninger chart, the pole corresponding to any observed 
Laue spot may be plotted by means of an easily constructed "stereographic ruler." 
The construction of the ruler is based on the relations shown in Fig. 8-10. This 
drawing is a section through the incident beam OC and any diffracted beam CS. 
Here it is convenient to use the plane normal ON' rather than ON and to make the 
projection from T, the intersection of the reference sphere with the incident beam. 
The projection of the pole N' is therefore at P. From the measured distance OS 
of the diffraction spot from the center of the film, we can find the distance PQ of 
the projected pole from the center of the projection, since 



and 



OS = OC tan (180 - 20) = D tan (180 - 26) 



PQ = TQ tan 



~ f ) = 2r 



tan 



(8-1) 
(8-2) 



where D is the specimen-film distance and r the radius of the reference sphere. 
The value of r is fixed by the radius R of the Wuiff net used, since the latter equals 
the radius of the basic circle of the projection. We note that, if the pole of the 



228 



ORIENTATION OP SINGLE CRYSTALS 



[CHAP. 8 



PROJECTION 




FIG. 8-11. Use of a stereographic ruler to plot the pole of a reflecting plane on a 
stereographic projection in the back- reflection Laue method. Pole 1' is the pole 
of the plane causing diffraction spot 1 . 

plane were in its extreme position at M, then its projection would he at U. The 
point U therefore lies on the basic circle of the projection, and UQ is the radius R 
of the basic circle. Because the triangles TUQ and TMC are similar, ft = 2r and 



PQ = R tan ^45 - -) 



(8-3) 



The ruler is constructed by marking off, from a central point, a scale of centi- 
meters by which the distance ON may be measured. The distance PQ correspond- 
ing to each distance OS is then calculated from Eqs. (S-l) and (S 3), and marked 
off from the center of the ruler in the opposite direction. Corresponding gradua- 
tions are given the same number and the result is the rulei shown in Fig. 8-11, 
which also illustrates the method of using it. [Calculation of the various distances 
PQ can be avoided by use of the Wulff net itself. Fig. 8-10 shows that the pole 
of the reflecting plane is located at an angle 6 from the edge of the projection, and 
6 is given for each distance OS by Eq. (8-1). The ruler is laid along the equator 
of the Wulff net, its center coinciding with the net center, and the distance PQ 
corresponding to each angle 6 is marked off with the help of the angular scale on 
the equator.] 

From the choice of plane normal made in Fig. 8-10, it is apparent that the pro- 
jection must be viewed from the side opposite the x-ray source. This requires 
that the film be read from that side also, i.e., with its cut corner in the upper right- 
hand position. The projection is then placed over the film, illuminated from be- 
low, as shown in Fig. 8-11. With the center of the ruler coinciding with the cen- 
ter of the projection, the ruler is rotated until its edge passes through a particular 



8-3] 



TRANSMISSION LAUE METHOD 



229 



diffraction spot. The distance 08 is noted and the corresponding pole plotted as 
shown, on the other side of center and at the corresponding distance PQ. This 
procedure is repeated for each important diffraction spot, after which the projec- 
tion is transferred to a Wulff net and the poles indexed by either of the methods 
previously described. Note that this procedure gives a projection of the crystal 
from the side opposite the x-ray source, whereas the Oreninger chart gives a pro- 
jection of the crystal as seen from the x-ray source. A crystal orientation can, 
of course, be described just as well from one side as the other, and either projec- 
tion can be made to coincide with the other by a 180 rotation of the projection 
about its EW-axis. Although simple to use and construct, the stereographic ruler 
is not as accurate as the Greninger chart in the solution of back-reflection patterns. 

The methods of determining and describing crystal orientation have 
been presented here exclusively in terms of cubic crystals, because these 
are the simplest kind to consider and the most frequently encountered. 
These methods are quite general, however, and can be applied to a crystal 
of any system as long as its interplariar angles are known. 

8-3 Transmission Laue method. Given a specimen of sufficiently low 
absorption, a transmission Laue pattern can be obtained and used, in much 
the same way as a back-reflection Laue pattern, to reveal the orientation 
of the crystal. 

In either Laue method, the diffraction spots on the film, due to the 
planes of a single zone in the crystal, always lie on a curve which is some 
kind of conic section. When the film is in the transmission position, this 
curve is a complete ellipse for sufficiently small values of </>, the angle be- 
tween the zone axis and the transmitted beam (Fig. 8-12). For somewhat 
larger values of </>, the ellipse is incomplete because of the finite size of the 
film. When = 45, the curve becomes a parabola, when </> exceeds 45, a 




FIG. 8-12. Intersection of a conical array of diffracted beams with a film placed 
in the transmission position. C = crystal, F = film, Z.A. = zone axis. 



230 



ORIENTATION OF SINGLE CRYSTALS 



[CHAP. 8 




Z A 



FIG. 8-13, Relation between plane normal orientation and diffraction spot posi- 
tion in the transmission Laue method. 

hyperbola, and when </> = 90, a straight line. In all cases, the curve 
passes through the central spot formed by the transmitted beam. 

The angular relationships involved in the transmission Laue method 
are illustrated in Fig. 8-13. Here a reference sphere is described about 
the crystal at C, the incident beam entering the sphere at / and the trans- 
mitted beam leaving at 0. The film is placed tangent to the sphere at 0, 
and its upper right-hand corner, viewed from the crystal, is cut off for 
identification of its position during the x-ray exposure. The beam reflected 
by the lattice plane shown strikes the film at R, and the normal to this 
plane intersects the sphere at P. 

Suppose we consider diffraction from a zone of planes whose axis lies in 
the jyz-plane at an angle <t> to the transmitted (or incident) beam. If a 
single plane of this zone is rotated so that its pole, initially at A, travels 
along the great circle APEBWA, then it will pass through all the orienta- 
tions in which planes of this zone might occur in an actual crystal. During 
this rotation, the diffraction spot on the film, initially at D, would travel 
along the elliptical path DROD shown by the dashed line. 

Any particular orientation of the plane, such as the one shown in the 
drawing, is characterized by particular values of <t> and 5, the angular co- 



8-3] 



TRANSMISSION LAUE METHOD 
10 20 



231 




60 



FIG. (S-14. Leonhardt chart for the solution of transmission Laue patterns, re- 
produced in the correct size for a specimen-to-film distance of 3 cm. The dashed 
lines are lines of constant </>, and the solid lines are lines of constant 5. (Courtesy 
of C. G. Dunn.) 

ordinates of its pole. These coordinates in turn, for a given crystal-film 
distance D (= TO), determine the x,y coordinates of the diffraction spot 
R on the film. From the spot position we can therefore determine the 
plane orientation, and one way of doing this is by means of the Leonhardt 
chart shown in Fig. 8-14. 

This chart is exactly analogous to the Greninger chart for solving back- 
reflection patterns and is used in precisely the same way. It consists of a 
grid composed of two sets of lines: the lines of one set are lines of constant <t> 
and correspond to the meridians on a Wulff net, and the lines of the other 
are lines of constant 5 and correspond to latitude lines. By means of this 
chart, the pole of a plane causing any particular diffraction spot may be 
plotted stereographically. The projection plane is tangent to the sphere 
at the point / of Fig. 8-13 and the projection is made from the point 0. 
This requires that the film be read from the side facing the crystal, i.e., 



232 



ORIENTATION OF SINGLE CRYSTALS 

r 



[CHAP. 8 



FILM 



10 20 30 




PROJECTION 



underlying 
Wulff net 




FIG. 8-15. Use of the Leonhardt chart to plot the pole of a plane on a stereo- 
graphic projection. Pole 1' in (b) is the pole of the plane causing diffraction spot 
1 in (a). 



8-3] 



TRANSMISSION LAUE METHOD 

FILM 



233 



(a) 



Ellipse of spots from 
plant' of zone A 

10 20 / JO 




cut cornel 



(b) 



FIG. S-16. Use of the Leonhardt chart to plot the axis of a zone of planes on the 
projection. PA is the axis of zone A. 

with the cut corner at the upper right. Figure 8-15 shows how the pole 
corresponding to a particular spot is plotted when the film and chart are 
in the parallel position. An alternate way of using the chart is to rotate 
it about its center until a line of constant <t> coincides with a row of spots 
from planes of a single zone, as shown in Fig. 8-16; knowing and the 
rotation angle 6, we can then plot the axis of the zone directly. 



234 



ORIENTATION OF SINGLE CRYSTALS 



[CHAP. 8 



reference 
sphere 




film /? 

FIG. 8-17. Relation between diffraction spot S and stereographic projection P 
of the plane causing the spot, in transmission. 




FIG. 8-18. Use of a stereographic ruler to plot the pole of a reflecting plane on 
a stereographic projection in the transmission Laue method. Pole 1' is the pole of 
the plane causing diffraction spot 1 . 



8-3] TRANSMISSION LAUE METHOD 235 

A stereographic ruler may be constructed for the transmission method 
and it will give greater accuracy of plotting than the Leonhardt chart, 
particularly when the angle <t> approaches 90. Figure 8-17, which is a 
section through the incident beam and any diffracted beam, shows that 
the distance of the diffraction spot from the center of the film is given by 

OS = D tan 20. 

The distance of the pole of the reflecting plane from the center of the pro- 
jection is given by 

PQ = R tan ( 45 - - 
V 2 

Figure 8-18 illustrates the use of a ruler constructed according to these equa- 
tions. In this case, the projection is made on a plane located on the same 
side of the crystal as the film and, accordingly, the film must be read with 
its cut corner in the upper left-hand position. 

Whether the chart or the ruler is employed to plot the poles of reflecting 
planes, they are indexed in the same way as back-reflection patterns. For 
example, the transmission Laue pattern shown in Fig. 8-19 in the form 
of a tracing yields the stereographic projection shown in Fig. 8-20. The 
solid symbols in the latter are the poles of planes responsible for spots on 
the film and are numbered accordingly; the open symbols are poles derived 
by construction. (The reader will note that the poles of planes responsible 
for observed spots on a transmission film are all located near the edge of 
the projection, since such planes must necessarily be inclined at small 
angles to the incident beam. The reverse is true of back-reflection pat- 
terns, as inspection of Fig. 8-6 will show.) The solution of Fig. 8-20 
hinged on the identification of the zone axes PA, PB, and PC. Measure- 
ment showed that the stereographic triangle formed by these axes had 
sides equal to 35 (P A - P B ), 45 (P B - PC), and 30 (P c - PA), which 
identified P A , PB, and PC as {211}, {100}, and {110} poles, respectively. 
Now the transmission pattern shown in Fig. 8-19 and the back-reflection 
pattern shown in Fig. 8-5 were both obtained from the same crystal in the 
same orientation relative to the incident beam. The corresponding pro- 
jections, Figs. 8-20 and 8-7, therefore refer to a crystal of the same orien- 
tation. But these were made from opposite sides of the crystal and so 
appear completely dissimilar. However, a rotation of either projection 
by 180 about its EW-&xis will make it coincide with the other, although 
no attempt has been made to make the indexing of one projection con- 
sistent with that of the other. 



236 



ORIENTATION OF SINGLE CRYSTALS 



[CHAP. 8 



----^ 3 



FIG. 8-19. Transmission Laue pattern of an aluminum crystal, traced from Fig. 
3-6 (a). Only selected diffraction spots are shown. 




FIG. 8-20. Stereographic projection corresponding to transmission pattern of 
Fig. 8-19. 



8-4] DIFFRACTOMETER METHOD 237 

8-4 Diffractometer method. Still another method of determining crys- 
tal orientation involves the use of the diffractometer and a procedure radi- 
cally different from that of either Laue method. With the essentially 
monochromatic radiation used in the diffractometer, a single crystal will 
produce a reflection only when its orientation is such that a certain set of 
reflecting planes is inclined to the incident beam at an angle 6 which satis- 
fies the Bragg law for that set of planes and the characteristic radiation 
employed. But when the counter, fixed in position at the corresponding 
angle 20, discloses that a reflection is produced, then the inclination of the 
reflecting planes to any chosen line or plane on the crystal surface is known 
from the position of the crystal. Two kinds of operation are required: 

(1) rotation of the crystal about various axes until a position is found 
for which reflection occurs, 

(2) location of the pole of the reflecting plane on a stereographic projec- 
tion from the known angles of rotation. 

The diffractometer method has many variations, depending on the par- 
ticular kind of goniometer used to hold and rotate the specimen. Only one 
of these variations will be described here, that involving the goniometer 
used in the reflection method of determining preferred orientation, since 
that is the kind most generally available in metallurgical laboratories. 
This specimen holder, to be described in detail in Sec. 9-9, needs very 
little modification for use with single crystals, the chief one being an in- 
crease in the width of the primary beam slits in a direction parallel to the 
diffractometer axis in order to increase the diffracted intensity. This type 
of holder provides the three possible rotation axes shown in Fig. 8-21 : one 
coincides with the diffractometer axis, the second (A A') lies in the plane 
of the incident beam / and diffracted beam D and tangent to the specimen 
surface, shown here as a flat plate, while the third (BB r ) is normal to the 
specimen surface. 

Suppose the orientation of a cubic crystal is to be determined. For such 
crystals it is convenient to use the {111) planes as reflectors; there are 
four sets of these and their reflecting power is usually high. First, the 26 
value for the 111 reflection (or, if desired, the 222 reflection) is computed 
from the known spacing of the {111} planes and the known wavelength of 
the radiation used. The counter is then fixed in this 28 position. The 
specimen holder is now rotated about the diffractometer axis until its sur- 
face, and the rotation axis A A', is equally inclined to the incident beam 
and the diffracted beam, or rather, to the line from crystal to counter with 
which the diffracted beam, when formed, will coincide. The specimen 
holder is then fixed in this position, no further rotation about the diffrac- 
tometer axis being required. Then, by rotation about the axis BB f , one 
edge of the specimen or a line drawn on it is made parallel to the diffrac- 
tometer axis. This is the initial position illustrated in Fig. 8-21. 



238 



ORIENTATION OF SINGLE CRYSTALS 



[CHAP. 8 



The crystal is then slowly rotated 
about the axes A A ' and BE' until an 
indication of a reflection is observed 
on the counting-rate meter. Once a 
reflecting position of the crystal has 
been found, we know that the normal 
to one set of (111! planes coincides 
with the line CN 9 that is, lies in the 
plane of the diffractometer circle and 
bisects the angle between incident 
and diffracted beams. The pole of 
these diffracting planes may now be 
plotted stereographically, as shown in 
Fig. 8-22. The projection is made 
on a plane parallel to the specimen 
surface, and with the MS-axis of the 
projection parallel to the reference 
edge or line mentioned above. When 

the crystal is rotated degrees about BB' from its initial position, the 
projection is also rotated degrees about its center. The direction CAT, 
which might be called the normal to "potential" reflecting planes, is repre- 




FIG. 8-21. Crystal rotation axes 
for the diffractometer method of de- 
termining orientation. 



PROJECTION 




FIG. 8-22. Plotting method used when determining crystal orientation with the 
diffractometer. (The directions of the rotations shown here correspond to the 
directions of the arrows in Fig. 8-21.) 



8-4] DIFFEACTOMETER METHOD 239 

sented by the pole N f , which is initially at the center of the projection but 
which moves y degrees along a radius when the crystal is rotated y degrees 
about A A'. 

What we are trying to do, essentially, is to make N f coincide with a 
{ 111 J pole and so disclose the location of the latter on the projection. The 
search may be made by varying y continuously for fixed values of 4 or 5 
apart; the projection is then covered point by point along a series of radii. 
It is enough to examine one quadrant in this way since there will always 
be at least one {111} pole in any one quadrant. Once one pole has been 
located, the search for the second is aided by the knowledge that it must 
be 70.5 from the first. Although two {111) poles are enough to fix the 
orientation of the crystal, a third should be located as a check. 

Parenthetically, it should be noted that the positioning of the crystal 
surface and the axis A A' at equal angles to the incident and diffracted 
beams is done only for convenience in plotting the stereographic projec- 
tion. There is no question of focusing when monochromatic radiation is 
reflected from an undeformed single crystal, and the ideal incident beam 
for the determination of crystal orientation is a parallel beam, not a di- 
vergent one. 

In the hands of an experienced operator, the diffractometer method is 
faster than either Laue method. Furthermore, it can yield results of 
greater accuracy if narrow slits are used to reduce the divergence of the 
incident beam, although the use of extremely narrow slits will make it 
more difficult to locate the reflecting positions of the crystal. On the other 
hand, the diffractometer method furnishes no permanent record of the 
orientation determination, whereas Laue patterns may be filed away for 
future reference. But what is more important, the diffractometer method 
does not readily disclose the state of perfection of the crystal, whereas a 
Laue pattern yields this kind of information at a glance, as we will see in 
Sec. 8-6, and in many investigations the metallurgist is just as much inter- 
ested in the relative perfection of a single crystal as he is in its orientation. 

All things considered, the Laue methods are preferable when only occa- 
sional orientation determinations are required, or when there is any doubt 
as to the perfection of the crystal. When the orientations of large num- 
bers of crystals have to be determined in a routine manner, the diffrac- 
tometer method is superior. In fact, this method was developed largely 
for just such an application during World War II, when the orientation of 
large numbers of quartz crystals had to be determined. These crystals 
were used in radio transmitters to control, through their natural frequency 
of vibration, the frequency of the transmitted signal. For this purpose 
quartz wafers had to be cut with faces accurately parallel to certain crys- 
tallographic planes, and the diffractometer was used to determine the 
orientations of these planes in the crystal. 



240 



ORIENTATION OF SINGLE CRYSTALS 



[CHAP. 8 



8-5 Setting a crystal in a required orientation. Some x-ray investiga- 
tions require that a diffraction pattern be obtained of a single crystal 
having a specified orientation relative to the incident beam. To obtain 
this orientation, the crystal is mounted in a three-circle goniometer like 
that shown in Fig. 5-7, whose arcs have been set at zero, and its orienta- 
tion is determined by, for example, the back-reflection Laue method. A 
projection of the crystal is then made, and from this projection the goni- 
ometer rotations which will bring the crystal into the required orientation 
are determined. 

For example, suppose it is required to rotate the crystal whose orienta- 
tion is given by Fig. 8-7 into a position where [Oil] points along the inci- 
dent beam and [100] points horizontally to the left, i.e., into the standard 
(Oil) orientation shown by Fig. 2-36 (b) if the latter were rotated 90 
about the center. The initial orientation (Position 1) is shown in Fig. 8-23 
by the open symbols, referred to NSEW-&xes. Since (01 1) is to be brought 
to the center of the projection and (100) to the left side, (010) will lie on 
the vertical axis of the projection when the crystal is in its final position. 
The first step therefore is to locate a point 90 away from (Oil) on the 
great circle joining (010) to (Oil), because this point must coincide with 
the north pole of the final projection. This is simply a construction point; 




FIG. 8-23. Crystal rotation to produce specified orientation. Positions 1 and 2 
are indicated by open symbols, position 3 by shaded symbols, and position 4 by 
solid symbols. 



8-5] 



SETTING A CRYSTAL IN A REQUIRED ORIENTATION 



241 



in the present case it happens to coincide with the (Oil) pole, but gen- 
erally it is of no crystallographic significance. The projection is then 
rotated 22 clockwise about the incident-beam axis to bring this point 
onto the vertical axis of the underlying Wulff net. (In Fig. 8-23, the 
latitude and longitude lines of this net have been omitted for clarity.) The 
crystal is now in Position 2, shown by open symbols referred to N'S'E'W- 
axes. The next rotation is performed about the .EW-axis, which requires 
that the underlying Wulff net be arranged with its equator vertical so that 
the latitude lines will run from top to bottom. This rotation, of 38, moves 
all poles along latitude lines, shown as dashed small circles, and brings 
(Oil) to the N'-pole, and (100) and (Oil) to the E'W'-'dxis of the projec- 
tion, as indicated by the shaded symbols (Position 3). The final orienta- 
tion is obtained by a 28 rotation about the JV'S'-axis, with the equator 
of the underlying Wulff net now horizontal ; the poles move to the positions 
shown by solid symbols (Position 4). 

The necessity for selecting a construction point 90 from (Oil) should 
now be evident. If this point, which here happens to be (Oil), is brought 
to the Af'-pole, then (Oil) and (100) must of necessity lie on the SW-axis; 
the final rotation about N'S' will then move the latter to their required 
positions without disturbing the position of the (Oil) pole, since [Oil] 
coincides with the N'$'-axis. 

The order of these three rotations is not arbitrary. The stereographic 
rotations correspond to physical rotations on the goniometer and must be 
made in such a way that one rotation does not physically alter the position 
of any axis about which a subsequent rotation is to be made. The goni- 
ometer used here was initially set with the axis of its uppermost arc hori- 
zontal and coincident with the primary beam, and with the axis of the 
next arc horizontal and at right angles to the incident beam. The first 
rotation about the beam axis there- 
fore did not disturb the position of 
the second axis (the UW-axis), and 
neither of the first two rotations dis- 



FIG. 8-24. Back-reflection Laue pat- 
tern of an aluminum crystal. The in- 
cident beam is parallel to [Oil], [Oil] 
points vertically upward, and [100] points 
vertically to the left. Tungsten radia- 
tion, 30 kv, 19 ma, 40 min exposure, 5 cm 
specimen-to-film distance. (The shadow 
at the bottom is that of the goni- 
ometer which holds the specimen.) 




242 ORIENTATION OF SINGLE CRYSTALS [CHAP. 8 

turbed the position of the third axis (the vertical WS'-axis). Whether or 
not the stereographic orientations are performed in the correct order makes 
a great difference in the rotation angles found, but once the right angles 
are determined by the correct stereographic procedure, the actual physical 
rotations on the goniometer may be performed in any sequence. 

The back-reflection Laue pattern of an aluminum crystal rotated into 
the orientation described above is shown in Fig. 8-24. Note that the 
arrangement of spots has 2-fold rotational symmetry about the primary 
beam, corresponding to the 2-fold rotational symmetry of cubic crystals 
about their (110) axes. (Conversely, the observed symmetry of the Laue 
pattern of a crystal of unknown structure is an indication of the kind of 
symmetry possessed by that crystal. Thus the Laue method can be used 
as an aid in the determination of crystal structure.) 

There is another method of setting a crystal in a standard orientation, 
which does not require either photographic registration of the diffraction 
pattern or stereographic manipulation of the data. It depends on the fact 
that the diffracted beams formed in the transmission Laue method are so 
intense, for a crystal of the proper thickness, that the spots they form on a 
fluorescent screen are visible in a dark room. The observer merely rotates 
the crystal about the various arcs of the goniometer until the pattern cor- 
responding to the required orientation appears on the screen. Obviously, 
he must be able to recognize this pattern when it appears, but a little 
study of a few Laue photographs made of crystals in standard orientations 
will enable him to do this. The necessity for working in a darkened room 
may be avoided by use of a light-tight viewing box, if the job of crystal 
setting occurs sufficiently often to justify its construction. This box en- 
closes the fluorescent screen which the observer views through a binocular 
eyepiece set in the wall of the box, either directly along the direction of 
the transmitted beam, or indirectly in a direction at right angles by means 
of a mirror or a right-angle prism. For x-ray protection, the optical system 
should include lead glass, and the observer's hands should be shielded 
during manipulation of the crystal. 

8-6 The effect of plastic deformation. Nowhere have x-ray methods 
been more fruitful than in the study of plastic deformation. The way in 
which a single crystal deforms plastically is markedly anisotropic, and 
almost all of our knowledge of this phenomenon has been gained by x-ray 
diffraction examination of crystals at various stages during plastic defor- 
mation. At the outset we can distinguish between two kinds of deforma- 
tion, that of the crystal lattice itself and that of the crystal as a whole. 
This distinction is worth while because crystal deformation, defined as a 
change in the shape of the crystal due to lattice rotation, can occur with 
or without lattice deformation, defined as the bending and/or twisting of 



8-6] 



THE EFFECT OF PLASTIC DEFORMATION 



243 



originally flat lattice planes. On the other hand, lattice deformation 
cannot occur without some deformation of the crystal as a whole. 

A crystal lattice can therefore behave in two quite different ways during 
plastic deformation : it can simply rotate without undergoing deformation 
itself, or it can become bent and/or twisted. Laue photographs can easily 
decide between these two possibilities. In the Laue method, any change 
in the orientation of the reflecting planes is accompanied by a correspond- 
ing change in the direction (and wavelength) of the reflected beam. In 
fact, Laue reflection of x-rays is often compared to the reflection of visible 
light by a mirror. If the lattice simply rotates during deformation, then 
Laue patterns made before and after will merely show a change in the 
position of the diffraction spots, corresponding to the change in orientation 
of the lattice, but the spots themselves will remain sharp. On the other 
hand, if the lattice is bent or twisted, the Laue spots will become smeared 
out into streaks because of the continuous change in orientation of the 
reflecting planes, just as a spot of light reflected by a flat mirror becomes 
elongated when the mirror is curved. 

A classic example of simple lattice rotation during crystal deformation 
is afforded by the tensile elongation of long cylindrical single metal crystals. 
When such a crystal is extended plastically, Laue photographs of the 
center section made before and after the extension show that the lattice 
has been rotated but not deformed. Yet the crystal itself has undergone 
considerable deformation as evidence by its change in shape it has be- 
come longer and thinner. How this occurs is suggested by Fig. 8-25. 
The initial form of the crystal is shown in (a), with the potential slip 
planes seen in profile. The applied tensile forces can be resolved into 



^ . bonding 







bonding 



t ? 

(b) Co 

P'iG. 8-25. Slip in tension (schematic). 



244 



ORIENTATION OF SINGLE CRYSTALS 



[CHAP. 8 



shearing forces parallel to these slip planes and tensile forces normal to 
them. The normal forces have no effect, but the shearing forces cause 
slip to occur, and the crystal would, as a result, assume the shape shown 
in (b) if the ends were not constrained laterally. However, the grips of 
the tensile machine keep the ends of the crystal aligned, causing bending 
of the crystal lattice near each grip, as indicated in (c), which illustrates 
the appearance of the crystal after considerable extension. Note that the 
lattice of the central portion has undergone reorientation but not distor- 
tion. This reorientation clearly consists in a rotation which makes the 
active slip plane more nearly parallel to the tension axis. 

Analysis of the Laue patterns yields further information about the 
deformation process. The changes in orientation which occur in the cen- 
tral section can be followed stereographically, either by plotting the before 
and after orientations of the crystal on a fixed projection plane, or by 
plotting the before and after orientations of the specimen axis in the unit 
stereographic triangle. The latter method is the more common one and 
is illustrated by Fig. 8-26, which applies to a face-centered cubic crystal. 
The initial position of the tension axis is represented by point 1. After 
successive extensions, the position of this axis is found to be at points 
2, 3, 4, . . . ; i.e., the axis moves 
along a great circle passing through 
the initial position and the direction 
[T01], which is the direction of slip. 
During this extension the active slip 
plane is (111). We can conclude that 
the lattice reorientation occurs in such 
a way that both the slip plane and 
the slip direction in that plane rotate 
toward the axis of tension. This 
process becomes more complicated at 
later stages of the deformation, and 
the interested reader is referred to 
books on crystal plasticity for further 
details. Enough has been said here 
to indicate the way in which x-ray 
diffraction may be applied to this 
particular problem. 

One other example of lattice reorientation during slip may be given in 
order to illustrate the alternate method of plotting the data. In Fig. 8-27, 
the successive orientations which a cylindrical magnesium crystal assumes 
during plastic torsion are plotted on a fixed projection plane parallel to the 
specimen axis (the axis of torsion). Since the poles of reflecting planes 
are found to move along latitude circles on the projection, it follows that 




slip plane 



-NAiii 



FIG. 8-26. Lattice rotation during 
slip in elongation of FCC metal crys- 
tal. 



8-6] 



THE EFFECT OF PLASTIC DEFORMATION 



245 



-SPECIMEN AXIS 




AFTER TWIST OF 2 
AFTER TWIST OF 11 

FIG. 8-27. Change in lattice orientation during plastic torsion of a magnesium 
crystal. The active slip plane is (0001), the basal plane of the hexagonal lattice. 
(S. S. Hsu and B. D. Cullity, Trans. A.I.M.E. 200, 305, 1954.) 

the lattice reorientation is mainly one of rotation about the specimen axis. 
Some lattice distortion also occurs, since special x-ray methods reveal 
that twisting of the lattice planes takes place, but the main feature of the 
deformation is the lattice rotation described above. Similarly, in the 
plastic elongation of single crystals, it should not be supposed that abso- 
lutely no lattice deformation occurs. Here again the main feature is lattice 
rotation, but sensitive x-ray methods will always show some bending or 
twisting of lattice planes, and in some cases this lattice distortion may be 
so severe that ordinary Laue patterns will reveal it. 

A good example of severe lattice distortion is afforded by those parts of 
a single-crystal tension specimen immediately adjacent to the grips. As 
mentioned earlier, these portions of the crystal lattice are forced to bend 
during elongation of the specimen, and Laue photographs made of these 
sections will accordingly show elongated spots. If the bending is about a 
single axis, the Miller indices of the bending axis can usually be determined 



246 



ORIENTATION OP SINGLE CRYSTALS 



[CHAP. 8 





(a) Transmission 



(b) Back reflection 



FIG. 8-28. Laue photographs of a deformed aluminum crystal. Specimen-to- 
film distance 3 cm, tungsten radiation, 30 kv. 

stereographically; each Laue streak is plotted as an arc representing the 
range of orientation of the corresponding lattice plane, and a rotation axis 
which will account for the directions of these arcs on the projection is 
found. The angular lengths of the arcs are a measure of the amount of 
bending which has occurred. In measuring the amount of bending by 
this method, it must be remembered that the wavelengths present in the 
incident beam do not cover an infinite range. There is no radiation of 
wavelength shorter than the short-wavelength limit, and on the long- 
wavelength side the intensity decreases continuously as the wavelength 
increases.- This means that, for a given degree of lattice bending, some 
Laue streaks may not be as long as they might be if a full range of wave- 
lengths were available. The amount of bending estimated from the lengths 
of these streaks would therefore be smaller than that actually present. 

Transmission and back-reflection Laue patterns made from the same 
deformed region usually differ markedly in appearance. The photographs 
in Fig. 8-28 were made, under identical conditions, of the same region of a 
deformed aluminum crystal having the same orientation relative to the 
incident beam for each photograph. Both show elongated spots, which 
are evidence of lattice bending, but the spots are elongated primarily in a 
radial direction on the transmission pattern while on the back-reflection 
pattern they tend to follow zone lines. The term asterism (from the Greek 
aster = star) was used initially to describe the starlike appearance of a 
transmission pattern such as Fig. 8-28 (a), but it is now used to describe any 
form of streaking, radial or nonradial, on either kind of Laue photograph. 

The striking difference between these two photographs is best under- 
stood by considering a very general case. Suppose a crystal is so deformed 



THE EFFECT OF PLASTIC DEFORMATION 



247 



film 




FIG. 8-29. Effect of lattice distortion 
on the shape of a transmission Laue 
spot. CN is the normal to the reflect- 
ing plane. 



FIG. 8-30. Effect of lattice distortion 
on the shape of a back-reflection Laue 
spot. CN is the normal to the reflecting 
plane. 



that the normal to a particular set of reflecting lattice planes describes a 
small cone of apex angle 2e; i.e., in various parts of the crystal the normal 
deviates by an angle c in all directions from its mean position. This is 
equivalent to rocking a flat mirror through the same angular range and, 
as Fig. 8-29 shows, the reflected spot S is roughly elliptical on a film placed 
in the transmission position. When the plane normal rocks through the 
angle 2c in the plane ACN, the reflected beam moves through an angle 4c, 
and the major axis of the ellipse is given approximately by t(AC) when 
26 is small. On the other hand, when the plane normal rocks through the 
angle 2e in a direction normal to the plane of reflection ACN, the only 
effect is to rock the plane of reflection through the same angle 2c about 
the incident beam. The minor axis of the elliptical spot is therefore given 
by 2e(AS) 2e(AC) tan 26 2e(AC)26. The shape of the spot is charac- 

terized by the ratio 

Major axis 



Minor axis 2e(AC)26 6 
For 26 = 10, the major axis is some 12 times the length of the minor axis. 




248 ORIENTATION OF SINGLE CRYSTALS [CHAP. 8 

In the back-reflection region, the situation is entirely different and the 
spot S is roughly circular, as shown in Fig. 8-30. Both axes of the spot 
subtend an angle of approximately 4c at the crystal. We may therefore 
conclude that the shape of a back-reflection spot is more directly related 
to the nature of the lattice distortion than is the shape of a transmission 
spot since, in the general case, circular motion of the end of the reflecting 
plane normal causes circular motion of the backward-reflected beam but 
elliptical motion of the forward-reflected beam. For this reason, the back- 
reflection method is generally preferable for studies of lattice distortion. 
It must not be supposed, however, that only radial streaking is possible on 
transmission patterns. The direction of streaking depends on the orienta- 
tion of the axis about which the reflecting planes are bent and if, for exam- 
ple, they are bent only about an axis lying in the plane ACN of Fig. 8-29, 
then the spot will be elongated in a direction at right angles to the radius 
AS. 

^ x enlaiged 

\ Laue spot 

Laue spot - ^^ \ 

\ 

1 Deb\ e a ic - 

potential -*- ; ' 

Debye ring / 

(a) Undeformed crystal (l>) Deformed cnstal 

FIG. 8-31. Formation of Debye arcs on Laue patterns of deformed crystals. 

One feature of the back-reflection pattern of Fig. 8-28 deserves some 
comment, namely, the short arcs, concentric with the film center, which 
pass through many of the elongated Laue spots. These are portions of 
Debye rings, such as one might expect on a pinhole photograph made of a 
polycrystalline specimen with characteristic radiation (Sec. 6-9). With 
a polycrystalline specimen of random orientation a complete Debye ring 
is formed, because the normals to any particular set of planes (hkl) have 
all possible orientations in space; in a deformed single crystal, the same 
normals are restricted to a finite range of orientations with the result that 
only fragments of Debye rings appear. We may imagine a circle on the 
film along which a Debye ring would form if a polycrystalline specimen 
were used, as indicated in Fig. 8-31. If a Laue spot then becomes enlarged 
as a result of lattice deformation and spreads over the potential Debye 
ring, then a short portion of a Debye ring will form. It will be much 
darker than the Laue spot, since the characteristic radiation* which 

* In Fig. 8-28(b), the characteristic radiation involved is tungsten L radiation. 
The voltage used (30 kv) is too low to excite the K lines of tungsten (excitation 
voltage = 70 kv) but high enough to excite the L lines (excitation voltage =12 
kv). 



8-6] 



THE EFFECT OF PLASTIC DEFORMATION 



249 




HKXT 



POLYGOMZKD 



FlG. 8-32. 
(schematic). 



Reflection of white radiation by bent and polygonized lattices 



forms it is much more intense than the wavelengths immediately adjacent 
to it in the continuous spectrum. In fact, if the x-ray exposure is not 
sufficiently long, only the Debye arcs may be visible on the film, and the 
observer may be led to erroneous conclusions regarding the nature and 
extent of the lattice deformation. 

With these facts in mind, re-examination of the patterns shown in Fig. 
8-28 leads to the following conclusions: 

(1) Since the asterism on the transmission pattern is predominantly 
radial, lattice planes inclined at small angles to the incident beam are bent 
about a number of axes, in such a manner that their plane normals are 
confined to a small cone in space. 

(2) Since the asterism on the back-reflection pattern chiefly follows zone 
lines, the major portion of planes inclined at large angles to the incident 
beam are bent about a single axis. However, the existence of Debye arcs 
shows that there are latent Laue spots of considerable area superimposed 
on the visible elongated spots, and that a small portion of the planes 
referred to are therefore bent about a number of axes. 

On annealing a deformed crystal at a sufficiently high temperature, one 
of the following effects is usually produced: 

(1) Polygonization. If the deformation is not too severe, plastically 
bent portions of the crystal break up into smaller blocks, which are strain- 
free and disoriented by approximately the same total amount (never more 
than a few degrees) as the bent fragment from which they originate, as 
suggested by Fig. 8-32. (The term "polygonization" describes the fact 
that a certain crystallographic direction [uvw] forms part of an arc before 
annealing and part of a polygon afterwards.) Moreover, the mean orienta- 
tion of the blocks is the same as that of the parent fragment. The effect 
of polygonization on a Laue pattern is therefore to replace an elongated 
Laue streak (from the bent lattice) with a row of small sharp spots (from 
the individual blocks) occupying the same position on the film, provided 
each block is sufficiently disoriented from its neighbor so that the beams 




250 ORIENTATION OF SINGLE CRYSTALS [CHAP 8 



FIG. 8-33. Enlarged transmission Laue 
spots from a thin crystal of silicon fer- 
rite (a-iron containing 3.3 percent silicon 
in solid solution) : (a) as bent to a radius 
of f in., (b) after annealing 10 min at 
950C, (c) after annealing 4 hr at 1 300C. 
(C. G. Dunn and F. W. Daniels, Trans. 
(c) A.I.M.E. 191, 147, 1951 ) 



reflected by adjoining blocks are resolved one from another. Figure 8-33 
shows an example of polygonization in a crystal of silicon ferrite. 

(2) Recrystalhzalion. If the deformation is severe enough, the crystal 
may recrystallize into a new set of strain-free grains differing completely 
in orientation from the original crystal. The appearance of the diffraction 
pattern then depends on the size of the new grains relative to the cross- 
sectional area of the incident x-ray beam. The appearance of such pat- 
terns is discussed and illustrated in Sec. 9-2. 

8-7 Relative orientation of twinned crystals. In this and the next sec- 
tion we shall consider, not single crystals, but pairs of crystals which are 
naturally associated one with another in certain particular ways. Twinned 
crystals are obvious examples of such pairs: the two parts of the twin have 
different orientations, but there is a definite orientation relationship be- 
tween the two. Furthermore, the two parts are united on a plane, the 
composition plane, which is also fixed and invariable, not merely a random 
surface of contact such as that between two adjacent grains in a poly- 
crystalline mass. Twinned crystals therefore present a twofold problem, 
that of determining the orientation relationship and that of determining 
the indices of the composition plane. 

The orientation relationship is established by finding the orientation of 
each part of the twin and plotting the two together on the same stereo- 
graphic projection. Determination of the composition-plane indices re- 
quires a knowledge of how to plot the trace, or line of intersection, of one 
plane in another, and we must digress at this point to consider that problem. 
Suppose that, on the polished surface of a twinned grain, the trace of the 
composition plane makes an angle a with some reference line NS t as shown 
in Fig. 8-34(a). Then, if we make the projection plane parallel to the 
plane of polish, the latter will be represented by the basic circle of the pro- 
jection and any directions in the plane of polish by diametrically opposite 
points on the basic circle. Thus, in Fig. 8-34 (b), the AT- and $-poles repre- 
sent the reference line NS and the points A and B, located at an angle a 



8-7] 



RELATIVE ORIENTATION OF TWINNED CRYSTALS 



251 





trace of 

composition 

plane 



(a) 
FIG. 8-34. Projection of the trace of a plane in a surface. 

from N and S, represent the trace. Note that the diameter ACB does not 
represent the trace; ACB represents a plane perpendicular to the plane of 
polish which could have caused the observed trace, but so could the in- 
clined planes ADB, AFB, and AGB. Evidently any number of planes 
could have caused the observed trace, and all we can say with certainty is 
that the pole of the composition plane lies somewhere on the diameter HK, 
where H and K are 90 from the trace direction A ,B. HK is called a trace 
normal. 

To fix the orientation of the composition plane requires additional infor- 
mation which can be obtained by sectioning the twinned grain by another 

AT 

^^- 

direction A, H 




(a) (b) 

FIG. 8-35. Projection of the trace of a plane in two surfaces. 



252 



ORIENTATION OF SINGLE CRYSTALS 



[CHAP. 8 



plane and determining the trace direction in this new plane. Suppose the 
section is made through a line WE, chosen for convenience to be at right 
angles to NS, and that the new plane of polish (Plane 2) makes an angle 4 
with the original one (Plane 1), as shown in Fig. 8-35(a). It is now con- 
venient to use the edge WE as a reference direction. Let the traces of the 
composition plane in surfaces 1 and 2 make angles of ft (equal to 90 - a) 
and 7 with the edge WE. Then, if the stenographic projection plane is 
again made parallel to surface 1, surface 2 is represented by a great circle 
through W and E and at an angle #> from the circumference [Fig. 8-35(b)J. 
The trace of the composition plane in surface 1 is then represented by A,B 
as before and the same trace in surface 2 by the direction C, both angles ft 




(c) A and B 

FIG 8-36 Back-reflection Laue photographs of two parts, A and B, of a twinned 
crystal of copper. Tungsten radiation, 30 kv, 20 ma. Film covered with 0.01-m.- 
thick aluminum to reduce the intensity of K fluorescent radiation from specimen. 



8-7J 



RELATIVE ORIENTATION OF TWINNED CRYSTALS 



253 



and y being measured from the edge \V,K. Two nonparallel lines in the 
unknown composition plane -Y are now known, namely the direction A,B 
and the direction C. A great circle drawn through B, (\ and A therefore 
describes the orientation of plane -Y, and PA' is its pole. 

An application of this method is afforded by annealing twins in copper. 
The back-reflection Lane photographs of Fig. 8-30 were obtained from a 
large grain containing a twin band; by shifting the specimen in its own 
plane between exposures, the incident beam was made to fall first on one 
part of the twin [pattern (a)], then on the other part [pattern (b)], and 
finally on each side of the trace of the composition plane [pattern (c)]. 
The latter photograph is therefore a double pattern of both parts of the 
twin together. 

The orientations derived from patterns (a) and (b) are shown in Fig. 
8-37, and certain poles of each part of the twin are seen to coincide, par- 
ticularly the (111) pole in the lower right quadrant. These coincidences 
are also evident in Fig. 8-3()(<0 in the form of coincident Laue spots. By 
measuring the directions ot the trace of the composition plane X in two 
surfaces, the orientation of X was determined, as shown in the projection. 
l>x is found to coincide with the (111) pole common to each part of the 




FIG. 8-37. Projection of part A (open symbols) and part B (solid symbols) of 
a twin in copper, made from Figs. 8-36(a) and (b). 



254 ORIENTATION OF SINGLE CRYSTALS [CHAP. 8 

twin, thus disclosing the indices of the composition plane. By the methods 
described in Sec. 2-11, it may also be shown that the two parts of the twin 
are related by reflection in this same (111) plane. The twinning plane 
(the plane of reflection) in copper is therefore shown to be identical with 
the composition plane. 

Similar problems arise in studies of plastic deformation. For example, 
we may wish to find the indices of slip planes responsible for the observable 
slip lines on a polished surface. Or we may wish to identify the composi- 
tion plane of a deformation twin. The simplest procedure, if it can be 
used, is to convert the test specimen into grains large enough so that the 
orientation of any selected grain can be directly determined by one of the 
Laue methods. The polished specimen is then strained plastically to pro- 
duce visible slip lines or deformation twins. The orientation of a grain 
showing such traces is determined and the directions of these traces are 
measured. If traces are measured on two surfaces, the method of solution 
is identical with that described above for twinned copper. If traces are 
measured only on one plane, then the trace normals are plotted on a stereo- 
graphic projection of the grain; the crystal orientation and the trace nor- 
mals are rotated into some standard orientation and superimposed on a 
detailed standard projection. Intersection of the normals with certain 
poles of the standard will then disclose the indices of the planes causing the 
observed traces. 

But it may happen that the grain size is too small to permit a deter- 
mination of grain orientation. The problem is now much more difficult, 
even when trace directions are measured on two surfaces. The first step 
is to plot the trace normals corresponding to the traces on both surfaces; 
these normals will be straight lines for the traces on the surface on which 
the projection is being made and great circles for the traces on the other 
surface. A standard (/hWi) projection is then superimposed on the pro- 
jection of the trace normals, and a rotation is sought which will bring 
[h\kili } poles into coincidence with the intersections of straight and curved 
trace normals. If such coincidence cannot be found, an (h^h) standard 
projection is tried, and so on. If the traces in either plane have more than 
one direction, it will be helpful to note how many different directions are 
involved. For example, if there are more than three different directions in 
one grain of a cubic metal, the traces cannot be caused by {100} planes; 
if more than four directions are observed, both {100} and {111} planes 
are ruled out; and so on. 

Up to this point we have been concerned with the problem of finding 
the indices of planes causing certain observed traces, generally in a grain 
of known orientation. The same problem may be solved in reverse: given 
traces in two surfaces of a plane of known indices (hkl\ the orientation of 
the crystal may be found without using x-rays. The trace normals are 



8-7] 



RELATIVE ORIENTATION OF TWINNED CRYSTALS 

N 



255 




FIG. 8-38. Determination of crystal orientation of copper from traces of two 
known twin planes in one surface. 

plotted on one sheet of paper and on this is superposed a standard projec- 
tion showing only {hkl\ planes. By trial and error, a rotation is found 
which will make the {hkl\ poles fall on the observed trace normals. 

By the same method, crystal orientation can also be determined from 
two nonparallel traces of planes of known indices in one surface. In this 
way, it is sometimes possible to determine the orientation of a single grain 
in a polycrystalline mass when the grain size is too small to permit direct 
x-ray determination. For example, we may use the fact that annealing 



256 



ORIENTATION OF SINGLE CRYSTALS 



[CHAP. 8 



twins in copper have {111} composition planes to determine the orienta- 
tion of the grain shown in Fig. 8-38 (a), where twin bands have formed on 
two different {111} planes of the parent grain. The trace normals are 
plotted in Fig. 8-38(b), and on this is placed a standard (001) projection 
containing only (111) poles. If the standard is rotated about its center 
to the position shown, then it is possible by a further rotation about the 
axis AB to bring the {111} poles of the standard, shown by open symbols, 
to positions lying on trace normals, shown by solid symbols. The solid 
symbols therefore show an orientation of the crystal which will account 
for the observed traces. Unfortunately, it is not the only one : the orienta- 
tion found by reflecting the one shown in the plane of projection is also a 
possible solution. A choice between these two possibilities can be made 
only by sectioning the crystal so as to expose trace directions in a second 
surface. 

8-8 Relative orientation of precipitate and matrix. When a supersatu- 
rated solid solution precipitates a second phase, the latter frequently 
takes the form of thin plates which lie parallel to certain planes of low 
indices in the matrix. The matrix plane on which the precipitate plate 
lies is called the habit plane and its indices always refer to the lattice of the 
matrix. There is also a definite orientation relationship between the lattice 
of the precipitate and that of the matrix. Both of these effects result from 
a tendency of the atomic arrangement in the precipitate to conform as 
closely as possible to the atomic arrangement in the matrix at the interface 
between the two. For example, precipitation of an HCP phase from an 
FCC solid solution often occurs in such a way that the basal (0001) plane 
of the precipitate is parallel to a (111) plane of the matrix, since on both 
of these planes the atoms have a 
hexagonal arrangement. 

Relations of this kind are illustrated 
on an atomic scale in Fig. 8-39. In 
this hypothetical case the habit plane 
is (HO) and the lattice relationship is 
such that the plane (010) of the pre- 
cipitate is parallel to the plane (110) 
of the matrix; the direction [100] in 
the former plane is parallel to the 
direction [110] in the latter, or, in the 
usual shorthand notation, 

MATRIX PRECIPITATE 
UNIT CELL UNIT CELL 
FIG. 8-39. Matrix-precipitate rela- 
tionship. 




where the subscripts p and m refer to 
precipitate and matrix, respectively. 



8-8] RELATIVE ORIENTATION OF PRECIPITATE AND MATRIX 257 




FIG. X-40. Widmanstatten structure (schematic). Cubic matrix has (100) 
habit. Top grain is intersected parallel to { 100). 

If a certain solid solution has an \hkl\ habit plane, then precipitation 
can of course take place on all planes of the form \hkl\. Thus one grain 
may contain sets of precipitate plates having quite different orientations. 
When such a grain is sectioned, the thin precipitate plates appear as 
needles on the plane of polish resulting in a structure such as that shown 
in Fig. 8-40 in a highly idealized form. This is called a Widmanstatten 
structure. It is very often the product of nucleation and growth reactions, 
such as precipitation and eutectoid decomposition. Somewhat similar 
structures are also observed as the result of the martensitic reaction and 
other diffusionless transformations. (There are some secondary differences, 
however: martensite often takes the form of needles as well as plates and 
the indices of its habit plane are often irrational, e.g., (259), and may even, 
as in the case of Fe-C martensite, change with composition.) 

The crystallographic problems presented by such structures are very 
much the same as those described in Sec. 8-7, except that the plates of the 
second phase almost always differ in crystal structure from the matrix, 
unlike the two parts of a twin or the material on either side of a slip plane. 
The habit plane is identified by the methods previously described for the 
identification of slip or twinning planes. The orientation relationship is 
easily determined if a single precipitate plate can be found which is large 
enough to permit determination of its orientation by one of the Laue 
methods. Ordinarily, however, the precipitate is so fine that this method 
cannot be applied and some variant of the rotating-crystal method must 
be used. 



258 ORIENTATION OF SINGLE CRYSTALS [CHAP. 8 

PROBLEMS 

8-1. A back-reflection Laue photograph is made of an aluminum crystal with 
a crystal-to-film distance of 3 cm. When viewed from the x-ray source, the Laue 
spots have the following ^-coordinates, measured (in inches) from the center of 
the film : 

x y x y 

+0.26 +0.09 -0.44 +1.24 

+0.45 +0.70 -1.10 +1.80 

+ 1.25 +1.80 -1.21 +0.40 

+ 1.32 +0.40 -1.70 +1.19 

+0.13 -1.61 -0.76 -1.41 

+0.28 -1.21 -0.79 -0.95 

+0.51 -0.69 -0.92 -0.26 

+0.74 -0.31 

Plot these spots on a sheet of graph paper graduated in inches. By means of a 
Greninger chart, determine the orientation of the crystal, plot all poles of the form 
(100), (110), and (111), and give the coordinates of the {100J poles in terms of 
latitude and longitude measured from the center of the projection. 

8-2. A transmission Laue photograph is made of an aluminum crystal with a 
crystal-to-film distance of 5 cm. To an observer looking through the film toward 
the x-ray source, the spots have the following ^-coordinates (in inches) : 
x y x y 

+0.66 +0.88 -0.10 +0.79 

+0.94 +2.44 -0.45 +2.35 

+ 1.24 +0.64 -0.77 +1.89 

+ 1.36 +0.05 -0.90 +1.00 

+ 1.39 +1.10 -1.27 +0.50 

+0.89 -1.62 -1.75 +1.55 

+ 1.02 -0.95 -1.95 +0.80 

+ 1.66 -1.10 -0.21 -0.58 

-0.59 -0.28 

-0.85 -1.31 

-1.40 -1.03 

-1.55 -0.36 

Proceed as in Prob. 8-1, but use a stereographic ruler to plot the poles of reflecting 
planes. 

8-3. Determine the necessary angular rotations about (a) the incident beam 
axis, (6) the east-west axis, and (c) the north-south axis to bring the crystal of 
Prob. 8-2 into the "cube orientation/' i.e., that shown by Fig. 2-36(a). 

8-4. With reference to Fig. 8-35(a), if ft = 120, y = 135, and <t> = 100, 
what are the coordinates (in terms of latitude and longitude) of the pole of the 
composition plane? 

8-6. Precipitate plates in a cubic matrix form a Widmanstatten structure. The 
traces of the plates in the plane of polish lie in three directions in one particular 
grain, making azimuthal angles of 15, 64, and 113, measured clockwise from a 
"vertical" NS reference line. Determine the indices of the habit plane and the 
orientation of the matrix grain (in terms of the coordinates of its {100} poles). 



CHAPTER 9 
THE STRUCTURE OF POLYCRYSTALLINE AGGREGATES 

9-1 Introduction. In the previous chapter we were concerned with the 
orientation and relative perfection of single crystals. But the single metal 
crystal is, after all, somewhat of a laboratory curiosity; the normal way in 
which metals and alloys are used is in the form of polycrystalline aggregates, 
composed of a great many individual crystals usually of microscopic size. 
Since the properties of such aggregates are of great technological impor- 
tance, they have been intensively studied in many ways. In such studies 
the two most useful techniques are microscopic examination and x-ray 
diffraction, and the wise investigator will use them both; one complements 
the other, and both together can provide a great deal of information about 
the structure of an aggregate. 

The properties (mechanical, electrical, chemical, etc.) of a single-phase 
aggregate are determined by two factors: 

(1) the properties of a single crystal of the material, and 

(2) the way in which the single crystals are put together to form the 
composite mass. 

In this chapter we will be concerned with the second factor, namely, the 
structure of the aggregate, using this term in its broadest sense to mean 
the relative size, perfection, and orientation of the grains making up the 
aggregate. Whether these grains are large or small, strained or unstrained, 
oriented at random or in some preferred direction, frequently has very 
important effects on the properties of the material. 

If the aggregate contains more than one phase, its properties naturally 
depend on the properties of each phase considered separately and on the 
way these phases occur in the aggregate. Such a material offers wide 
structural possibilities since, in general, the size, perfection, and orienta- 
tion of the grains of one phase may differ from those of the other phase or 
phases. 

CRYSTAL SIZE 

9-2 Grain size. The size of the grains in a polycrystalline metal or 
alloy has pronounced effects on many of its properties, the best known 
being the increase in strength and hardness which accompanies a decrease 
in grain size. This dependence of properties on grain size makes the meas- 
urement of grain size a matter of some importance in the control of most 
mjetal forming operations. 

The grain sizes encountered in commercial metals and alloys range from 
about 10"" 1 to 10~ 4 cm. These limits are, of course, arbitrary and repre- 

259 



260 THE STRUCTURE OF POLYCRYSTALLINE AGGREGATES [CHAP. 9 

sent rather extreme values; typical values fall into a much narrower range, 
namely, about 10~ 2 to 10~ 3 cm The most accurate method of measuring 
grain size in this range is by microscopic examination; the usual procedure 
is to determine the average number of grains per unit area of the polished 
section and report this in terms of an "index number" established by the 
American Society for Testing Materials. The equation 



n = 



relates n, the number of grains per square inch when viewed at a magnifi- 
cation of 100 X, and TV, the ASTM "index number" or "grain-size 
number." 

Although x-ray diffraction is decidedly inferior to microscopic examina- 
tion in the accurate measurement of grain size, one diffraction photograph 
can yield semiquantitative information about grain size, together with infor- 
mation about crystal perfection and orientation. A transmission or back- 
reflection pinhole photograph made with filtered radiation is best. If the 
back-reflection method is used, the surface of the specimen (which need 
not be polished) should be etched to remove any disturbed surface layer 
which might be present, because most of the diffracted radiation originates 
in a thin surface layer (see Sees. 9-4 and 9-5). 

* The nature of the changes produced in pinhole photographs by progres- 
sive reductions in specimen grain size is illustrated in Fig. 9-1. The gov- 
erning effect here is the number of grains which take part in diffraction. 
This number is in turn related to the cross-sectional area of the incident 
beam, and its depth of penetration (in back reflection) or the specimen 
thickness (in transmission). When the grain size is quite coarse, as in 
Fig. 9-1 (a), only a few crystals diffract and the photograph consists of a 
set of superimposed Laue patterns, one from each crystal, due to the white 
radiation present. A somewhat finer grain size increases the number of 
Laue spots, and those which lie on potential Debye rings generally are 
more intense than the remainder, because they are formed by the strong 
characteristic component of the incident radiation. Thus, the suggestion 
of a Debye ring begins to appear, as in (b). When the grain size is further 
reduced, the Laue spots merge into a general background and only Debye 
rings are visible, as in (c). These rings are spotty, however, since not 
enough crystals are present in the irradiated volume of the specimen to 
reflect to all parts of the ring. A still finer grain size produces- the smooth, 
continuous Debye rings shown in (d). 

Several methods have been proposed for the estimation of grain size 
purely in terms of various geometrical factors. For example, an equation 
may be derived which relates the observed number of spots on a Debye 
ring to the grain size and other such variables as incident-beam diameter, 
multiplicity of the reflection, and specimen-film distance. However, many 
approximations are involved and the resulting equation is not very accu- 



9-3] 



PARTICLE SIZE 



261 





(a) 



(b) 





(c) 



(d) 



FIG. 9-1. Back-reflection pinhole patterns of recrystallized aluminum specimens; 
grain size decreases in the order (a), (b), (c), (d). Filtered copper radiation. 

rate. The best way to estimate grain size by diffraction is to obtain a set 
of specimens having known ASTM grain-size numbers, and to prepare 
from these a standard set of photographs of the kind shown in Fig. 9-1. 
The grain-size number of an unknown specimen of the same material is 
then obtained simply by matching its diffraction pattern with one of the 
standard photographs, provided both are made under identical conditions. 
I 32ia, the g ra i n s i ze reaches a value somewhere in the range 10~ 3 to 
10"" 4 cm, the exact value depending on experimental conditions, the Debye 
rings lose their spotty character and become continuous. Between this 
value and 10~ 5 cm (1000A), no change occurs in the diffraction pattern. 
At about 10~ 5 cm the first signs of line broadening, due to small crystal 
size, begin to be detectable. There is therefore a size range, from 10~~ 3 
(or 10""" 4 ) to 10~~ 5 cm, where x-ray diffraction is quite insensitive to varia- 
tions in grain size. I 

9-3 Particle size. When the size of the individual crystals is less than 
about 10~~ 6 cm (1000A), the term "particle size" is usually used. As we 



262 THE STRUCTURE OF POLYCRYSTALLINE AGGREGATES [CHAP. 9 

saw in Sec. 3-7, crystals in this size range cause broadening of the Debye 
rings, the extent of the broadening being given by Eq. (3-13) : 

B = ^L, (3-13) 

*COS0 

where B = broadening of diffraction line measured at half its maximum 
intensity (radians) and t = diameter of crystal particle. All diffraction 
lines have a measurable breadth, even when the crystal size exceeds 1000A, 
due to such causes as divergence of the incident beam and size of the sam- 
ple (in Debye cameras) and width of the x-ray source (in diffractometers). 
The breadth B in Eq. (3-13) refers, however, to the extra breadth, or 
broadening, due to the particle-size effect alone. In other words, B is 
essentially zero when the particle size exceeds about 1000A. 

The chief problem in determining particle size from line breadths is to 
determine B from the measured breadth B M of the diffraction line. Of the 
many methods proposed, Warren's is the simplest. The unknown is mixed 
with a standard which has a particle size greater than 1000A, and which 
produces a diffraction line near that line from the unknown which is to be 
used in the determination. A diffraction pattern is then made of the mix- 
ture in either a Debye camera or, preferably, a diffractometer. This pat- 
tern will contain sharp lines from the standard and broad lines from the 
unknown, assumed to consist of very fine particles. Let B$ be the meas- 
ured breadth, at half maximum intensity, of the line from the standard. 
Then B is given, not simply by the difference between B M and 5$, but by 
the equation R2 _ r> 2 _ r> 2 



(This equation results from the assumption that the diffraction line has 
the shape of an error curve.) Once B has been obtained from Eq. (9-1), 
it can be inserted into Eq. (3-13) to yield the particle size /. There are 
several other methods of finding B from BM', compared with Warren's 
method, they are somewhat more accurate and considerably more intricate. 

The experimental difficulties involved in measuring particle size from 
line broadening increase with the size of the particle measured. Roughly 
speaking, relatively crude measurements suffice in the range 0-500A, but 
very good experimental technique is needed in the range 500-1000A. The 
maximum size measurable by line broadening has usually been placed at 
1000A, chiefly as a result of the use of camera techniques. Recently, 
however, the diffractometer has been applied to this problem and the upper 
limit has been pushed to almost 2000A. Very careful work was jgcmired 
and back-reflection lines were -employed, since such lines exhibit the largest 
pSrtictePSize broadening, as shown by Eq*, (SHIS). 

From the above discussion it might be inferred tha^ line broadening is 
chiefly used to measure the particle size of loose powders rather than the 



9-4] CRYSTAL PERFECTION 263 

size of the individual crystals in a solid aggregate.! That is correct. At- 
tempts have been made to apply Eq. (3-13) to the broadened diffraction 
lines from very fine-grained metal specimens and so determine the size of 
the individual grains. Such determinations are never very reliable, how- 
ever, because the individual grains of such a material are often nonuni- 
formly strained, and this condition, as we shall see in the next section, 
can also broaden the diffraction lines; an uncertainty therefore exists as 
to the exact cause of the observed broadening. On the other hand, the 
individual crystals which make up a loose powder of fine particle size can 
often be assumed to be strain-free, provided the material involved is a 
brittle (nonplastic) one, and all the observed broadening can confidently 
be ascribed to the particle-size effect. (But note that loose, unannealed 
metal powders, produced by filing, grinding, ball milling, etc., almost 
always contain nonuniform strain.) The. chief applications of the line- 
broadening method have been in the measurement of the particle size of 
such materials as carbon blacks, catalysts, and industrial dusts. 

JAnother x-ray method of measuring the size of small particles deserves 
some mention, although a complete description is beyond the scope of this 
book. This is the method of small-angle scattering. It is a form of diffuse 
scattering very near the undeviated transmitted beam, i.e., at angles 20 
ranging from up to roughly 2 or 3. From the observed variation of 
the scattered intensity vs. angle 20, the size, and to some extent the shape, 
of small particles can be determined, whether they are amorphous or crys- 
talline. Small-angle scattering has also been used to study precipitation 
effects in metallic solid solutions. | 

CRYSTAL PERFECTION 

9-4 Crystal perfection. Of the many kinds of crystal imperfection, the 
one we are concerned with here is nonuniform strain because it is so charac- 
teristic of the cold-worked state of metals and alloys. When a polycrystal- 
line piece of metal is plastically deformed, for example by rolling, slip 
occurs in each grain and the grain changes its shape, becoming flattened 
and elongated in the direction of rolling. The change in shape of any one 
grain is determined not only by the forces applied to the piece as a whole, 
but also by the fact that each grain retains contact on its boundary sur- 
faces with all its neighbors. Because of this interaction between grains, 
a single grain in a polycrystalline mass is not free to deform in the same 
way as an isolated single crystal would, if subjected to the same deforma- 
tion by rolling. As a result of this restraint by its neighbors, a plastically 
deformed grain in a solid aggregate usually has regions of its lattice left 
in an elastically bent or twisted condition or, more rarely, in a state of 
uniform tension or compression. The metal is then said to contain residual 



264 



CRYSTAL LATTICE 



DIFFRACTION 
LINE 



NO STRAIN 



(a) 



THE STRUCTURE OF POLYCRYSTALLINE AGGREGATES [CHAP. 9 

(Such stress is often called "internal stress" but the term is not 
very informative since all stresses, residual or externally imposed, are 
internal. The term "residual stress" emphasizes the fact that the stress 
remains after all external forces are removed.) Stresses of this kind are 
also called microstresses since they vary from one grain to another, or from 
one part of a grain to another part, on a microscopic scale. On the other 
hand, the stress may be quite uniform over large distances; it is then re- 
ferred to as macrostress. 

The effect of strain, both uniform 
and nonuniform, on the direction of 
x-ray reflection is illustrated in Fig. 
9-2. A portion of an unstrained grain 
appears in (a) on the left, and the set 
of transverse reflecting planes shown 
has everywhere its equilibrium spac- 
ing d . The diffraction line from these 
planes appears on the right. If the 
grain is then given a uniform tensile 
strain at right angles to the reflecting 
planes, their spacing becomes larger 
than d > and the corresponding dif- 
fraction line shifts to lower angles but 
does not otherwise change, as shown 
in (b). This line shift is the basis of 
the x-ray method for the measurement 
of macrostress, as will be described 
in Chap. 17. In (c) the grain is bent 
and the strain is nonuniform ; on the 
top (tension) side the plane spacing 
exceeds d , on the bottom (compres- 
sion) side it is less than d , and some- NONTNIFORM STRAIN 
where in between it equals d . We 
may imagine this grain to be com- 
posed of a number of small regions in 
jach of which the plane spacing is substantially constant but different 
from the spacing in adjoining regions. These regions cause the various 
sharp diffraction lines indicated on the right of (c) by the dotted curves. 
The sum of these sharp lines, each slightly displaced from the other, is the 
broadened diffraction line shown by the full curve and, of course, the 
broadened line is the only one experimentally observable. We can find a 
relation between the broadening produced and the nonuniformity of the 
strain by differentiating the Bragg law. We obtain 




UNIFORM STRAIN 




(c) 

FIG. 9-2. Effect of lattice strain 
on Debye-line width and position. 



A20 



-2 tan0, 
d 



(9-2) 



9-4] CRYSTAL PERFECTION 265 

where b is the broadening due to a fractional variation in plane spacing 
Ad/d. This equation allows the variation in strain, Ad/d, to be calculated 
from the observed broadening. This value of Ad/d, however, includes 
both tensile and compressive strain and must be divided by two to obtain 
the maximum tensile strain alone, or maximum compressive strain alone, 
if these two are assumed equal. The maximum strain so found can then 
be multiplied by the elastic modulus E to give the maximum stress present. 
For example, 

/Ad\ Eb 

(Max. tens, stress) = E - (max. tens, strain) = (E)(?) \~) = "A ~*' 

\ a / 4 tan B 

When an annealed metal or alloy is cold worked, its diffraction lines 
become broader. This is a well-established, easily verified experimental 
fact, but its explanation has been a matter of controversy. Some investi- 
gators have felt that the chief effect of cold work is to fragment the 
grains to a point where their small size alone is sufficient to account for 
all the observed broadening. Others have concluded that the nonuni- 
formity of strain produced by cold work is the major cause of broadening, 
with grain fragmentation possibly a minor contributing cause. Actually, 
it is impossible to generalize, inasmuch as different metals and alloys may 
behave quite differently. By advanced methods of mathematical analysis, 
it is possible to divide the observed change in line shape produced by cold 
work into two parts, one due to fine particle size and the other due to 
nonuniform strain. When this is done, it is found, for example, that in 
alpha brass containing 30 percent zinc the observed broadening is due 
almost entirely to nonuniform strain, while in thoriated tungsten (tung- 
sten containing 0.75 percent thorium oxide) it is due both to nonuniform 
strain and fine particle size. But no example is known where all the 
observed broadening can be ascribed to fine particle size. In fact, it is 
difficult to imagine how cold work could fragment the grains to the 
degree necessary to cause particle-size broadening without at the same 
time introducing nonuniform strains, in view of the very complex forces 
that must act on any one grain of an aggregate no matter how simple the 
forces applied to the aggregate as a whole. 

The broadening of a diffraction line by cold work cannot always be 
observed by simple inspection of a photograph unless some standard is 
available for comparison. However, the separation of the Ka doublet 
furnishes a very good "internal standard." In the back-reflection region, 
an annealed metal produces a well-resolved doublet, one component due 
to Kai radiation and the other to Ka 2 - For a given set of experimental 
conditions, the separation of this doublet on the film is constant and inde- 
pendent of the amount of cold work. But as the amount of cold work 
is increased, the broadening increases, until finally the two components 



266 THE STRUCTURE OF POLYCRYSTALLINE AGGREGATES [CHAP. 9 

of the doublet overlap to such an extent that they appear as one unresolved 
line. An unresolved Ka doublet can therefore be taken as evidence of 
cold work, if the same doublet is resolved when the metal is in the an- 
nealed condition. 

We are now in a position to consider some of the diffraction effects 
associated with the processes of recovery, recrystallization, and grain growth. 
When a cold-worked metal or alloy is annealed at a low temperature, 
recovery takes place; at a somewhat higher temperature, recrystallization; 
and at a still higher temperature, grain growth. Or at a sufficiently high 
constant temperature, these processes may be regarded as occurring con- 
secutively in time. Recovery is usually defined as a process involving 
changes in certain properties without any observable change in micro- 
structure, while recrystallization produces an easily visible structure of 
new grains, which then grow at the expense of one another during the 
grain-growth stage. 

The above is a highly oversimplified description of some very complex 
processes which are not yet completely understood. In particular, the 
exact nature of recovery is still rather obscure. It seems clear, however, 
that some form of polygonization takes place during recovery and may, 
in fact, constitute the most important part of that process. (Polygoniza- 
tion can occur in the individual grains of an aggregate just as in a single 
crystal. The structure so produced is called a substructure, and the 
smaller units into which a grain breaks up are called subgrains. Subgrain 
boundaries can be made visible under the microscope if the proper etching 
technique is used.) In some metals and alloys, recovery appears to overlap 
recrystallization (in temperature or time), while in others it is quite sepa- 
rate. It is usually associated with a partial relief of residual stress, on 
both a microscopic and a macroscopic scale, without any marked change 
in hardness. Since microstress is the major cause of line broadening, we 
usually find that the broad diffraction lines characteristic of cold-worked 
metal partially sharpen during recovery. When recrystallization occurs, 
the lines attain their maximum sharpness and the hardness decreases 
rather abruptly. During grain growth, the lines become increasingly 
spotty as the grain size increases. 

The nature of these changes is illustrated for alpha brass containing 
30 weight percent zinc by the hardness curve and diffraction patterns of 
Fig. 9-3. The hardness remains practically constant, for an annealing 
period of one hour, until a temperature of 200 C is exceeded, and then 
decreases rapidly with increasing temperature, as shown in (a). The dif- 
fraction pattern in (b) exhibits the broad diffuse Debye lines produced by 
the cold-rolled, unannealed alloy. These lines become somewhat narrower 
for specimens annealed at 100 and 200 C, and the Ka doublet becomes 
partially resolved at 250C. At 250, therefore, the recovery process 



9-4] 



CRYSTAL PERFECTION 



267 





(e) 1 hour at 




UK) 200 300 400 500 
ANNEALING TEMPERATURE (O 

(a) Hardness curve (d) 1 houi tit 4f>0"(' 

FIG. 9-3. Changes in hardness and diffraction lines of 70-30 brass specimens, 
reduced in thickness by 90 percent by cold rolling, and annealed foi 1 hour at the 
temperatures indicated in (a), (b), (c), and (d) are poitions of back-reflection 
pinhole patterns of specimens annealed at the temperatures stated (filtered cop- 
per radiation). 

appears to be substantially complete in one hour and recrystallization is 
just beginning, as evidenced by the drop in Rockwell B hardness from 
98 to 90. At 300 C the diffraction lines are quite sharp and the doublets 
completely resolved, as shown in (c). Annealing at temperatures above 
300C causes the lines to become increasingly spotty, indicating that the 
newly recrystallized grains are increasing in size. The pattern of a speci- 
men annealed at 450C, when the hardness had dropped to 37 Rockwell B, 
appears in (d). 

Diffract ometer measurements made on the same specimens disclose 
both more, and less, information. Some automatically recorded profiles 
of the 331 line, the outer ring of the patterns shown in Fig. 9-3, are repro- 
duced in Fig. 9-4. It is much easier to follow changes in line shape by 
means of these curves than by inspection of pinhole photographs. Thus 
the slight sharpening of the line at 200 C is clearly evident in the diffrac- 
tometer record, and so is the doublet resolution which occurs at 250 C. 
But note that the diffractometer cannot "see" the spotty diffraction lines 
caused by coarse grains. There is nothing in the diffractometer records 



268 



THE STRUCTURE OF POLYCRYSTALLINE AGGREGATES [CHAP. 9 



I 

x; 




135 



134 



133 



132 

IS (degrees) 



131 



130 



129 



FIG. 9-4. Diffractometer traces of the 331 line of the cold-rolled and annealed 
70-30 brass specimens referred to in Fig. 9-3. Filtered copper radiation. Loga- 
rithmic intensity scale. All curves displaced vertically by arbitrary amounts. 





9-5| DEPTH OF X-RAY PENETRATION 269 



FIG 9-5. Back-reflection pinhole 
patterns of coarse-grained lecrystal- 
lized copper. Vnfiltered coppei ra- 
diation (a) from surface ground on a 
belt sandei , (h) after removal of 0.003 
in fiom this suiface by etching. 



made at 300 and 450C which would immediately .suggest that the speci- 
men annealed at 450 O had the coarser grain size, hut this fact is quite 
evident in the pinhole patterns shown in Figs. 9-3 (c) and (d). 

It must always he remembered that a hack-reflection photograph is 
representative of only a thin surface layer of the specimen. For example, 
Fig. 9-5 (a) was obtained from a piece of copper and exhibits unresolved 
doublets in the high-angle region. The unexperienced observer might 
conclude that this material was highly cold worked. What the x-ray 
"sees" is cold worked, but it sees only to a limited depth. Actually, the 
bulk of this specimen is in the annealed condition, but the surface from 
which the x-ray pattern was made had had 0.002 in. removed by grinding 
on a belt sander after annealing. This treatment cold worked the surface 
to a considerable depth. By successive etching treatments and diffraction 
patterns made after each etch, the change in structure of the cold-worked 
layer could be followed as a function of depth below the surface. Not 
until a total of 0.003 in. had been removed did the diffraction pattern be- 
come characteristic of the bulk of the material; see Fig. 9-5 (b), where the 
sf>otty lines indicate a coarse-grained, recrystallized structure. 

9-6 Depth of x-ray penetration. Observations of this kind suggest that 
it might be well to consider in some detail the general problem of x-ray 
penetration. Most metallurgical specimens strongly absorb x-rays, and 
the intensity of the incident beam is reduced almost to zero in a very short 
distance below the surface. The diffracted beams therefore originate 
chiefly in a thin surface layer whenever a reflection technique, as opposed 
to a transmission technique,* is used, i.e., whenever a diffraction pattern 

* Not even in transmission methods, however, is the information on a diffrac- 
tion pattern truly representative of the entire cross section of the specimen. Cal- 
culations such as those given in this section show that a greater proportion of the 
total diffracted energy originates in a layer of given thickness on the back side of 
the specimen (the side from which the transmitted beam leaves) than in a layer 
of equal thickness on the front side. If the specimen is highly absorbing, a trans- 
mission method can be just as non-representative of the entire specimen as a back- 
reflection method, in that most of the diffracted energy will originate in a thin 
surface layer.* See Prob. 9-5. 



270 THE STRUCTURE OF POLYCRYSTALLINE AGGREGATES [CHAP. 9 

is obtained in a back-reflection camera of any kind, a Seemann-Bohlin 
camera or a diffractometer as normally used. We have just seen how a 
back-reflection pinhole photograph of a ground surface discloses the cold- 
worked condition of a thin surface layer and gives no information what- 
ever about the bulk of the material below that layer. 

These circumstances naturally pose the following question: what is the 
effective depth of x-ray penetration? Or, stated in a more useful manner, 
to what depth of the specimen does the information in such a diffraction 
pattern apply? This question has no precise answer because the intensity 
of the incident beam does not suddenly become zero at any one depth but 
rather decreases exponentially with distance below the surface. However, 
we can obtain an answer which, although not precise, is at least useful, in 
the following way. Equation (7-2) gives the integrated intensity dif- 
fracted by an infinitesimally thin layer located at a depth x below the 
surface as 

d//> = e-^ (1/8in + 1/8in & dx, (7-2) 

sin a 

where the various symbols are defined in Sec. 7-4. This expression, inte- 
grated over any chosen depth of material, gives the total integrated in- 
tensity diffracted by that layer, but only in terms of the unknown constants 
/o, a, and b. However, these constants will cancel out if we express the 
intensity diffracted by the layer considered as a fraction of the total inte- 
grated intensity diffracted by a specimen of infinite thickness. (As we 
saw in Sec. 7-4, "infinite thickness" amounts to only a few thousandths 
of an inch for most metals.) Call this fraction G x . Then 



[ 

J r i 

JlfrSL - = 1 - e - x(ll * ina + llB{nf . 

Jx 



X-X 

dlD 

G 






This expression permits us to calculate the fraction G x of the total dif- 
fracted intensity which is contributed by a surface layer of depth x. If 
we arbitrarily decide that a contribution from this surface layer of 95 per- 
cent (or 99 or 99.9 percent) of the total is enough so that we can ignore 
the contribution from the material below that layer, then x is the effective 
depth of penetration. We then know that the information recorded on the 
diffraction pattern (or, more precisely, 95 percent of the information) 
refers to the layer of depth x and not to the material below it. 
In the case of the diffractometer, a = = 8, and Eq. (9-3) reduces to 

G x = (1 - 



DEPTH OP X-RAY PENETRATION 



9-5] 

which shows that the effective depth 
of penetration decreases as 6 decreases 
and therefore varies from one diffrac- 
tion line to another. In back-reflec- 
tion cameras, a = 90, and 



271 



G x = [1 - 



(9-5) 




03 1.0 1.5 

x (thousandths of an inch) 

FIG. 9-6. The fraction G x of the 
total diffracted intensity contributed 
by a surface layer of depth x, for 
M = 473 cm" 1 , 26 = 136.7, and nor- 
mal incidence. 



where ft = 20 - 90. 

For example, the conditions appli- 
cable to the outer diffraction ring 
of Fig. 9-5 are M = 473 cm"" 1 and 
26 = 136.7. By using Eq. (9-5), we 
can construct the plot of G r as func- 
tion of x which is shown in Fig. 9-6. 
We note that 95 percent of the infor- 
mation on the diffraction pattern re- 
fers to a depth of only about 0.001 in. 
It is therefore not surprising that the 
pattern of Fig. 9-5 (a) discloses only 
the presence of cold-worked metal, 

since we found by repeated etching treatments that the depth of the cold- 
worked layer was about 0.003 in. Of course, the information recorded on 
the pattern is heavily weighted in terms of material just below the surface; 
thus 95 percent of the recorded information applies to a depth of 0.001 in., 
but 50 percent of that information originates in the first 0.0002 in. (Note 
that an effective penetration of 0.001 in. means that a surface layer only 
one grain thick is effectively contributing to the diffraction pattern if the 
specimen has an ASTM grain-size number of 8.) 

Equation (9-4) can be put into the following form, which is more suitable 
for calculation: 

-^- = In 
sin 6 



1 



K x sin B 



x = 



Similarly, we can rewrite Eq. (9-5) in the form 

M.T (l + -^} = In ( V) = K x , 
\ sin /3/ \1 - Gj 

K x sin ft 



x = 



+ sin/3) 



272 THE STRUCTURE OF POLYCRYSTALLINE AGGREGATES [CHAP. 9 

TABLE 9-1 



G* 0.50 


0.75 


0.90 


0.95 


0.99 


0.999 


K x 0.69 


1.39 


2.30 


3.00 


4.61 


6.91 



Values of K x corresponding to various assumed values of G x are given in 
Table 9-1. 

Calculations of the effective depth of penetration can be valuable in 
many applications of x-ray diffraction. We may wish to make the effective 
depth of penetration as large as possible in some applications. Then a 
and ft in Eq. (9-3) must be as large as possible, indicating the use of high- 
angle lines, and ^ as small as possible, indicating short-wavelength radia- 
tion. Other applications may demand very little penetration, as when we 
wish information, e.g., chemical composition or lattice parameter, from a 
very thin surface layer. Then we must make M large, by using radiation 
which is highly absorbed, and a and small, by using a diffractometer at 
low values of 20.* By these means the depth of penetration can often be 
made surprisingly small. For instance, if a steel specimen is examined in a 
diffractometer with Cu Ka. radiation, 95 percent of the information afforded 
by the lowest angle line of ferrite (the 110 line at 26 = 45) applies to a 
depth of only 9 X 10~ 5 in. There are limits, of course, to reducing the 
depth of x-ray penetration, and when information is required from very 
thin surface films, electron diffraction is a far more suitable tool (see Appen- 
dix 14). 

CRYSTAL ORIENTATION 

9-6 General. Each grain in a polycrystalline aggregate normally has 
a crystallographic orientation different from that of its neighbors. Con- 
sidered as a whole, the orientations of all the grains may be randomly 
distributed in relation to some selected frame of reference, or they may 
tend to cluster, to a greater or lesser degree, about some particular orienta- 
tion or orientations. Any aggregate characterized by the latter condition 
is said to have a preferred orientation, or texture, which may be defined 
simply as a condition in which the distribution of crystal orientations is 
nonrandom. 

There are many examples of preferred orientation. The individual crys- 
tals in a cold-drawn wire, for instance, are so oriented that the same crystal- 
lographic direction [uvw] in most of the grains is parallel or nearly parallel 

* Some of these requirements may be contradictory. For example, in measur- 
ing the lattice parameter of a thin surface layer with a diffractometer, we must 
compromise between the low value of 6 required for shallow penetration and the 
high value of required for precise parameter measurements. 



9-6] CRYSTAL ORIENTATION GENERAL 273 

to the wire axis. In cold-rolled sheet, most of the grains are oriented with 
a certain plane (hkl) roughly parallel to the sheet surface, and a certain 
direction [uvw] in that plane roughly parallel to the direction in which the 
sheet was rolled. These are called deformation textures. Basically, they 
are due to the tendency, already noted in Sec. 8-6, for a grain to rotate 
during plastic deformation. There we considered the rotation of a single 
crystal subjected to tensile forces, but similar rotations occur for each 
grain of an aggregate as a result of the complex forces involved, with the 
result that a preferred orientation of the individual grains is produced by 
the deformation imposed on the aggregate as a whole. 

When a cold-worked metal or alloy, possessed of a deformation texture, 
is recrystallized by annealing, the new grain structure usually has a pre- 
ferred orientation too, often different from that of the cold-worked mate- 
rial. This is called an annealing texture or recrystallization texture, and two 
kinds are usually distinguished, primary and secondary, depending on the 
recrystallization process involved. Such textures are due to the influence 
which the texture of the matrix has on the nucleation and/or growth of 
the new grains in that matrix. 

Preferred orientation can also exist in castings, hot-dipped coatings, 
evaporated films, electrodeposited layers, etc. Nor is it confined to metal- 
lurgical products: rocks, natural and artificial fibers and sheets, and similar 
organic or inorganic aggregates usually exhibit preferred orientation. In 
fact, preferred orientation is generally the rule, not the exception, and the 
preparation of an aggregate with a completely random crystal orientation 
is a difficult matter. To a certain extent, however, preferred orientation 
in metallurgical products can be controlled by the proper operating con- 
ditions. For example, some control of the texture of rolled sheet is possible 
by the correct choice of degree of deformation, annealing temperature, 
and annealing time. 

The industrial importance of preferred orientation lies in the effect, often 
very marked, which it has on the over-all, macroscopic properties of mate- 
rials. Given the fact that most single crystals are anisotropic, i.e., have 
different properties in different directions, it follows that an aggregate 
having preferred orientation must also have directional properties to a 
greater or lesser degree. Such properties are usually objectionable. For 
example, in the deep drawing of sheet the metal should flow evenly in all 
directions, but this will not OCCUF if the metal has a high degree of preferred 
orientation, since the yield point, and in fact the whole flow stress curve 
of the material, will then differ in different directions in the sheet. More 
rarely, the intended use of the material requires directional properties, 
and then preferred orientation is desirable. For example, the steel sheet 
used for transformer cores must undergo repeated cycles of magnetization 
and demagnetization in use, requiring a high permeability in the direction 



274 THE STRUCTURE OF POLYCRYSTALLINE AGGREGATES [CHAP. 9 

of the applied field. Since single crystals of iron are more easily mag- 
netized in the [100] direction than in any other, the rolling and annealing 
treatments given the steel sheet are deliberately chosen to produce a high 
degree of preferred orientation, in which as many grains as possible have 
their [100] directions parallel to a single direction in the sheet, in this case 
the rolling direction. 

It should be noted that preferred orientation is solely a crystallographic 
condition and has nothing to do with grain shape as disclosed by the micro- 
scope. Therefore, the presence or absence of preferred orientation cannot 
be disclosed by microscopic examination. It is true that grain -shape is 
affected by the same forces which produce preferred orientation; thus 
grains become flattened by rolling, and rolling is usually accompanied by 
preferred orientation, but a flattened shape is not in itself direct evidence 
of preferred orientation. Only x-ray diffraction can give such evidence. 
This fact is most apparent in recrystallized metals, which may have an 
equiaxed microstructure and, at the same time, a high degree of preferred 
orientation. 

At various places in this book, we have already noted that a pinhole 
photograph made of a polycrystalline specimen with characteristic radia- 
tion consists of concentric Debye rings. We have more or less tacitly 
assumed that these rings are always continuous and of constant intensity 
around their circumference, but actually such rings are not formed unless 
the individual crystals in the specimen have completely random orienta- 
tions.* If the specimen exhibits preferred orientation, the Debye rings 
are of nonuniform intensity around their circumference (if the preferred 
orientation is slight), or actually discontinuous (if there is a high degree 
of preferred orientation). In the latter case, certain portions of the Debye 
ring are missing because the orientations which would reflect to those 
parts of the ring are simply not present in the specimen. Nonuniform 
Debye rings can therefore be taken as conclusive evidence for preferred 
orientation, and by analyzing the nonuniformity we can determine the 
kind and degree of preferred orientation present. 

Preferred orientation is best described by means of a pole figure. This 
is a stereographic projection which shows the variation in pole density 
with pole orientation for a selected set of crystal planes. This method of 
describing textures was first used by the German metallurgist Wever in 
1924, and its meaning can best be illustrated by the following simple ex- 
ample. Suppose we have a very coarse-grained sheet of a cubic metal 
containing only 10 grains, and that we determine the orientation of each 
of these 10 grains by one of the Laue methods. We decide to represent 
the orientations of all of these grains together by plotting the positions of 



1 See the next section for one exception to this statement. 



9-6] 



CRYSTAL ORIENTATION: GENERAL 

R.D RD 



275 



TD 




T D TDK 




T.D 



(a) 



(b) 



FIG. 9-7. (100) pole figures for sheet material, illustrating (a) random orienta- 
tion and (b) preferred orientation. R.D. (rolling direction) and T.D. (transverse 
direction) are reference directions in the plane of the sheet. 

their {100J poles on a single stereographic projection, with the projection 
plane parallel to the sheet surface. Since each grain has three { 100} poles, 
there will be a total of 3 X 10 = 30 poles plotted on the projection. If 
the grains have a completely random orientation, these poles will be dis- 
tributed uniformly* over the projection, as indicated in Fig. 9-7 (a). But 
if preferred orientation is present, the poles will tend to cluster together 
into certain areas of the projection, leaving other areas virtually unoc- 
cupied. For example, this clustering might take the particular form shown 
in Fig. 9-7(b). This is called the "cube texture/' because each grain is 
oriented with its (100) planes nearly parallel to the sheet surface and the 
[001] direction in these planes nearly parallel to the rolling direction. (This 
simple texture, which may be described by the shorthand notation (100) 
[001], actually forms as a recrystallization texture in many face-centered 
cubic metals and alloys under suitable conditions.) If we had chosen to 
construct a (111) pole figure, by plotting only {111) poles, the resulting 
pole figure would look entirely different from Fig. 9-7 (b) for the same pre- 
ferred orientation; in fact, it would consist of four "high-intensity" areas 
located near the center of each quadrant. This illustrates the fact that 
the appearance of a pole figure depends on the indices of the poles plotted, 
and that the choice of indices depends on which aspect of the texture one 
wishes to show most clearly. 

* If the orientation is random, there will be equal numbers of poles in equal 
areas on the surface of a reference sphere centered on the specimen. There will 
not be equal numbers, however, on equal areas of the pole figure, since the stereo- 
graphic projection is not area-true. This results, for randomly oriented grains, 
in an apparent clustering of poles at the center of the pole figure, since distances 
representing equal angles are much smaller in this central region than in other 
parts of the pole figure. 



276 THE STRUCTURE OF POLYCRYSTALLINE AGGREGATES [CHAP. 9 

Naturally, when the grain size is small, as it normally is, separate deter- 
mination of the orientations of a representative number of grains is out of 
the question, so x-ray methods are used in which the diffraction effects 
from thousands of grains are automatically averaged. The (hkl) pole 
figure of a fine-grained material is constructed by analyzing the distribu- 
tion of intensity around the circumference of the corresponding hkl Debye 
ring. There are two methods of doing this, the photographic and the dif- 
fractometer method. The photographic method is qualitative and, al- 
though affording sufficient accuracy for many purposes, it is rapidly being 
made obsolete by the more accurate diffractometer method. Both methods 
are described in the following sections. 

Although only a pole figure can provide a complete description of pre- 
ferred orientation, some information can be obtained simply by a com- 
parison of calculated diffraction line intensities with those observed with a 
Debye-Scherrer camera or a diffractometer. As stated in Sec. 4-12, rela- 
tive line intensities are given accurately by Eq. (4-12) only when the 
crystals of the specimen have completely random orientations. Therefore 
any radical disagreement between observed and calculated intensities is 
immediate evidence of preferred orientation in the specimen, and, from 
the nature of the disagreement, certain limited conclusions can usually be 
drawn concerning the nature of the texture. For example, if a sheet 
specimen is examined in the diffractometer in the usual way (the specimen 
making equal angles with the incident and diffracted beams), then the 
only grains which can contribute to the hkl reflection are those whose 
(hkl) planes are parallel to the sheet surface. If the texture is such that 
there are very few such grains, the intensity of the hkl reflection will be 
abnormally low. Or a given reflection may be of abnormally high inten- 
sity, which would indicate that the corresponding planes were preferen- 
tially oriented parallel or nearly parallel to the sheet surface. As an 
illustration, the 200 diffractometer reflection from a specimen having the 
cube texture is abnormally high, and from this fact alone it is possible to 
conclude that there is a preferred orientation of (100) planes parallel to 
the sheet surface. However, no conclusion is possible as to whether or not 
there is a preferred direction in the (100) plane parallel to some reference 
direction on the sheet surface. Such information can be obtained only by 
making a pole figure. 

9-7 The texture of wire and rod (photographic method). As mentioned 
in the previous section, cold-drawn wire normally has a texture in which a 
certain crystallographic direction [uvw] in most of the grains is parallel, 
or nearly parallel, to the wire axis. Since a similar texture is found in 
natural and artificial fibers, it is called a fiber texture and the axis of the 
wire is called the fiber axis. Materials having a fiber texture have rota- 



9-7] THE TEXTURE OF WIRE AND ROD (PHOTOGRAPHIC METHOD) 277 



F.A 




reflection 
circle 



"V. reference 
sphere 



.Debye 
ring 



FIG. 9-8. Geometry of reflection from material having a fiber texture. F.A. = 
fiber axis. 

tional symmetry about an axis in the sense that all orientations about this 
axis are equally probable. A fiber texture is therefore to be expected in 
any material formed by forces which have rotational symmetry about a 
line, for example, in wire and rod, formed by drawing, swaging, or extru- 
sion. Less common examples of fiber texture are sometimes found in sheet 
formed by simple compression, in coatings formed by hot-dipping, electro- 
deposition, and evaporation, and in castings among the columnar crystals 
next to the mold wall. The fiber axis in these is perpendicular to the plane 
of the sheet or coating, and parallel to the axis of the columnar crystals. 

Fiber textures vary in perfection, i.e., in the scatter of the direction 
[uvw] about the fiber axis, and both single and double fiber textures have 
been observed. Thus, cold-drawn aluminum wire has a single [111] texture, 
but copper, also face-centered cubic, has a double [111] + [100] texture; 
i.e., in drawn copper wire there are two sets of grains, the fiber axis of one 
set being [111] and that of the other set [100]. 

The only crystallographic problem presented by fiber textures is that 
of determining the indices [uvw] of the fiber axis, and that problem is best 
approached by considering the diffraction effects associated with an ideal 
case, for example, that of a wire of a cubic material having a perfect [100] 
fiber texture. Suppose we consider only the 111 reflection. In Fig. 9-8, 
the wire specimen is at C with its axis along NS, normal to the incident 
beam 1C. CP is the normal to a set of (111) planes. Diffraction from 
these planes can occur only when they are inclined to the incident beam 



278 THE STRUCTURE OF POLYCRYSTALLINE AGGREGATES [CHAP. 9 

F A. F.A 




reflect ion circle 




(b) 



FIG. 9-9. Perfect [100] fiber texture: (a) (1 11) pole figure; (b) location of reflect- 
ing plane normals. 

at an angle which satisfies the Bragg law, and this requires that the (111) 
pole lie somewhere on the circle PUV, since then the angle between the 
plane normal and the incident beam will always be 90 6. For this 
reason, PUQV is called the reflection circle. If the grains of the wire had 
completely random orientations, then (111) poles would lie at all positions 
on the reflection circle, and the 111 reflection would consist of the com- 
plete Debye ring indicated in the drawing. But if the wire has a perfect 
[100] fiber texture, then the diffraction pattern produced by a stationary 
specimen is identical with that obtained from a single crystal rotated about 
the axis [100], because of the rotational symmetry of the wire. During 
this rotation, the (111) pole is confined to the small circle PAQB, all points 
of which make a constant angle p = 54.7 with the [100] direction N. Dif- 
fraction can now occur only when the (111) pole lies at the intersections 
of the reflection circle and the circle PAQB. These intersections are located 
at P and Q, and the corresponding diffraction spots at /? and T, at an 
azimuthal angle a from a vertical line through the center of the film. Two 
other spots, not shown, are located in symmetrical positions on the lower 
half of the film. If the texture is not perfect, each of these spots will 
broaden peripherally into an arc whose length is a function of the degree 
^f scatter in the texture. 

By solving the spherical triangle IPN, we can find the following general 
relation between the angles p, 0, and a: 



cos p = cos B cos a. 



(9-6) 



These angles are shown stereographically in Fig. 9-9, projected on a plane 
lormal to the incident beam. The (111) pole figure in (a) consists simply 



9-7] THE TEXTURE OF WIRE AND ROD (PHOTOGRAPHIC METHOD) 279 






of two arcs which are the paths traced out by fill} poles during rotation 
of a single crystal about [100]. In (b), this pole figure has been superposed 
on a projection of the reflection circle in order to find the locations of the 
reflecting plane normals. Radii drawn through these points (P, Q, P', 
and Q') then enable the angle a to be measured and the appearance of the 
diffraction pattern to be predicted. 
An unknown fiber axis is identified 
by measuring the angle a on the 
film and obtaining p from Eq. (9-6). 
When this is done for a number of dif- 
ferent hkl reflections, a set of p values 
is obtained from which the indices 
[uvw] of the fiber axis can be deter- 
mined. The procedure will be illus- 
trated with reference to the diffraction 
pattern of drawn aluminum wire 
shown in Fig. 9-10. The first step is 
to index the incomplete Debye rings. 
Values of 6 for each ring are calculated 
from measurements of ring diameter, 
and hkl indices are assigned by the use 
of Eq. (3-10) and Appendix 0. In 
this way the inner ring is identified as 
a 111 reflection and the outer one as 
200. The angle a is then measured 
from a vertical line through the center 
of the film to the center of each strong Debye arc. The average values of 
these angles are given below, together with the calculated values of p: 




FIG. 9-10. Transmission pinhole 
pattern of cold-drawn aluminum wire, 
wire axis vertical. Filtered copper 
radiation, (The radial streaks near 
the center are formed by the white 
radiation in the incident beam.) 



Line 

Inner 
Outer 



hkl 

111 
200 



69 
52 



19.3 
22.3 



70 
55 



The normals to the (111) and (200) planes therefore make angles of 70 
and 55, respectively, with the fiber axis. We can determine the indices 
[uvw] of this axis either by the graphical construction shown in Fig. 8-8 or 
by inspection of a table of interplanar angles. In this case, inspection of 
Table 2-3 shows that [uvw] must be [111], since the angle between (111) 
and (111) is 70.5 and that between (111) and (100) is 54.7, and these 
values agree with the values of p given above within experimental error. 
The fiber axis of drawn aluminum wire is therefore [111]. There is some 
scatter of the [111] direction about the wire axis, however, inasmuch as 
the reflections on the film are short arcs rather than sharp spots. If we 



280 THE STRUCTURE OF POLYCRYSTALLINE AGGREGATES [CHAP. 9 

wish, this can be taken into account by measuring the angular range of 
a for each arc and calculating the corresponding angular range of p. A 
(111) pole figure of the wire would then resemble Fig. 9-9 (a) except that 
the two curved lines would be replaced by two curved bands, each equal 
in width to the calculated range of p for the (111) poles. 

One other aspect of fiber textures should be noted. In materials having 
a fiber texture, the individual grains have a common crystallographic 
direction parallel to the fiber axis but they can have any rotational posi- 
tion about that axis. It follows that the diffraction pattern of such mate- 
rials will have continuous Debye rings if the incident x-ray beam is parallel 
to the fiber axis. However, the relative intensities of these rings will not 
be the same as those calculated for a specimen containing randomly oriented 
grains. Therefore, continuous Debye rings are not, in themselves, evi- 
dence for a lack of preferred orientation. 

9-8 The texture of sheet (photographic method). The texture of rolled 
sheet, either as rolled or after recrystallization, differs from that of drawn 
wire in having less symmetry. There is no longer a common crystallo- 
graphic direction about which the grains can have any rotational position. 
Sheet textures can therefore be described adequately only by means of a 
pole figure, since only this gives a complete map of the distribution of 
crystal orientation. 

The photographic method of determining the pole figure of sheet is quite 
similar to the method just described for determining wire textures. A 
transmission pinhole camera is used, together with general radiation con- 
taining a characteristic component. The sheet specimen, reduced in thick- 
ness by etching to a few thousandths of an inch, is initially mounted per- 
pendicular to the incident beam with the rolling direction vertical. The 
resulting photograph resembles tha,t of a drawn wire: it contains Debye 
rings of nonuniform intensity and the pattern is symmetrical about a 
vertical line through the center of the film. However, if the sheet is now 
rotated by, say, 10 about the rolling direction and another photograph 
made, the resulting pattern .will differ from the first, because the texture 
of sheet does not have rotational symmetry about the rolling direction. 
This new pattern will not be symmetrical about a vertical line, and the 
regions of high intensity on the Debye rings will not have the same azi- 
muthal positions as they had in the first photograph. Figure 9-11 illus- 
trates this effect for cold-rolled aluminum. To determine the complete 
texture of sheet, it is therefore necessary to measure the distribution of 
orientations about the rolling direction by making several photographs 
with the sheet normal at various angles to the incident beam. 

Figure 9-12 shows the experimental arrangement and defines the angle 
ft between the sheet normal and the incident beam. The intensity of the 



9-8] 



THE TEXTURE OF SHEET (PHOTOGRAPHIC METHOD) 



281 



%/miffim 







f4?if 
VHP j [ji 

. fr 

>: ! v 



^ ,, i( i 

'; 

,v \ I ^^;*^/^K 

% r , , ^"MJ/I \A/ 

''/^ l"^" 



'^1 

;igvj 




FIG. 9-11. Transmission pinhole patterns of cold-rolled aiummum sneet, roiling 
direction vertical: (a) sheet normal parallel to incident beam; (b) sheet normal at 
30 to incident beam (the specimen has been rotated clockwise about the rolling 
direction, as in Fig. 9-12). Filtered copper radiation. 

diffracted rays in any one Debye cone is decreased by absorption in the 
specimen by an amount which depends on the angle 0, and when ft is not 
zero the rays going to the left side of the film undergo more absorption 
than those going to the right. For this reason it is often advisable to make 
measurements only on the right side of the film, particularly when ft is 
large. 

The usual practice is to make photographs at about 10 intervals from 
ft = to ft = 80, and to measure the intensity distribution around a par- 



film 



RD 



TD 




sheet 
normal 




TD 




FIG. 9-12. Section through sheet 

specimen and incident beam (specimen FIG. 9-13. Measurement of azimuthal 

thickness exaggerated). Rolling direc- position of high-intensity arcs on a 

tion normal to plane of drawing. Debye ring, ft = 40, R.D. = rolling 

T.D. = transverse direction. direction. 



282 



THE STRUCTURE OF POLYCRYSTALLINE AGGREGATES 

R.D. 



[CHAP. 9 



T.D. + 




==i TD. 



FIG. 9-14. Method of plotting reflecting pole positions for nonzero values of 
ft. Drawn for 6 = 10 and ft = 40. 

ticular Debye ring on each photograph. The procedure for plotting the 
pole figure from these measurements will be illustrated here for an idealized 
case like that shown in Fig. 9-13, where the intensity of the Debye ring is 
constant over certain angular ranges and zero between them. The range 
of blackening of the Debye arcs is plotted stereographically as a range of 
reflecting pole positions along the reflection circle, the azimuthal angle a 
on the film equal to the azimuthal angle a on the projection. Although 
the reflection circle is fixed in space (see Fig. 9-8 where SCN is now the 
rolling direction of the sheet specimen), its position on the projection 
varies with the rotational position of the specimen, since the projection 
plane is parallel to the surface of the sheet and rotates with it. 

When ft = 0, the reflection circle is concentric with the basic circle of 
the projection and degrees inside it, as shown in Fig. 9-14, which is 
drawn for = 10. When the specimen is then rotated, for example by 
40 in the sense shown in Fig. 9-12, the new position of the reflection circle 
is found by rotating two or three points on the .reflection circle bv 40 



9-8] 



THE TEXTURE OF 6HEET (PHOTOGRAPHIC METHOD) 



283 



to the right along latitude lines and drawing circle arcs, centered on the 
equator or its extension, through these points. This new position of the 
reflection circle is indicated by the arcs ABC DA in Fig. 9-14; since in this 
example exceeds 0, part of the reflection circle, namely CD A, lies in the 
back hemisphere. The arcs in Fig. 9-13 are first plotted on the reflec- 
tion circle, as though the projection plane were still perpendicular to the 
incident beam, and then rotated to the right along latitude circles onto 
the 40 reflection circle. Thus, arc M\N\ in Fig. 9-13 becomes M 2 A^2 and 
then, finally, M 3 7V 3 in Fig. 9-14. Similarly, Debye arc U\Vi is plotted as 
U^Vz, lying on the back hemisphere. 

The texture of sheet is normally such that two planes of symmetry exist, 
one normal to the rolling direction (R.D.) and one normal to the trans- 
verse direction (T.D.). For this reason, arc -M 3 W 3 may be reflected in 
the latter plane to give the arc M^N^ thus helping to fill out the pole 
figure. These symmetry elements are also the justification for plotting 
the arc t T 3 F 3 as though it were situated on the front hemisphere, since 
reflection in the center of the projection (to bring it to the front hemi- 
sphere) and successive reflections in the two symmetry planes will bring it 
to this position anyway. If the diffraction patterns indicate that these 
symmetry planes are not present, then these short cuts in plotting may 
not be used. 

By successive changes in 0, the reflection circle can be made to move 
across the projection and so disclose the positions of reflecting poles. With 
the procedure described, however, the regions near the N and S poles of 
the projection will never be cut by a reflection circle. To explore these 
regions, we must rotate the specimen 90 in its own plane, so that the 
transverse direction is vertical, and take a photograph with @ ~ 5. 

Figure 9-15 shows what might result from a pole figure determination 
involving measurements at = 0, 20, 40, 60, and 80 (R.D. vertical) and 



R.D 



R.D 




T.D. 



T.D. 



FIG. 9-15. Plotting a pole figure. 



FIG. 9-16. Hypothetical pole figure 
derived from Fig. 9-15. 



284 



THE STRUCTURE OF POLYCRYSTALLINE AGGREGATES [CHAP, 9 



R.D 



= 5 (T.D. vertical). The arcs in Fig. 9-14 are replotted here with the 
same symbols, and the arcs E\Fi and E 2 F 2 lie on the 5 reflection circle 
with the transverse direction vertical. The complete set of arcs defines 
areas of high pole density and, by reflecting these areas in the symmetry 
planes mentioned above, we arrive at the complete pole figure shown in 
Fig. 9-16. 

In practice, the variation of inten- 
sity around a Debye ring is not abrupt 
but gradual, as Fig. 9-11 demon- 
strates. This is taken into account 
by plotting ranges in which the in- 
tensity is substantially constant, and 
no more than four such ranges are 
usually required, namely, zero, weak, 
medium, and strong. The result is 
a pole figure in which various areas, 
distinguished by different kinds of 
cross-hatching, represent various de- 
grees of pole density from zero to a 
maximum. Figure 9-17 is a photo- 
graphically determined pole figure in 
which this has been done. It repre- 




T.D 



FIG. 9-17. (Ill) pole figure of re- 
crystallized 70-30 brass, determined 
by the photographic method. (R. M. 
Brick, Trans. A.I.M.E. 137, 193, 1940.) 



sents the primary recrystallization texture of 70-30 brass which has been 
cold-rolled to a 99 percent reduction in thickness and then annealed at 
400C for 30 minutes. 

The texture of sheet is often described in terms of an "ideal orientation," 
i.e., the orientation of a single crystal whose poles would lie in the high- 
density regions of the pole figure. For example, in Fig. 9-17 the solid 
triangular symbols mark the positions of the Jill} poles of a single crys- 
tal which has its (113) plane parallel to the plane of the sheet and the 
[211] direction in this plane parallel to the rolling direction. This orienta- 
tion, when reflected in the two symmetry planes normal to the rolling and 
transverse directions, will approximately account for all the high-density 
regions on the pole figure. Accordingly, this texture has been called a 
(113) [2ll] texture. The actual pole figure, however, is a far better de- 
scription of the texture than any statement of an ideal orientation, since 
the latter is frequently not very exact and gives no information about the 
degree of scatter of the actual texture about the ideal orientation. 

The inaccuracies of photographically determined pole figures are due 
to two factors: 

(1) intensity "measurements" made on the film are usually only visual 
estimates, and 



9-9] 



THE TEXTURE OF SHEET (DIFFRACTOMETER METHOD) 



285 



(2) no allowance is made for the change in the absorption factor with 
changes in ft and a. This variation in the absorption factor makes it very 
difficult to relate intensities observed on one film to those observed on 
another, even when the exposure time is varied for different films in an 
attempt to allow for changes in absorption. 

9-9 The texture of sheet (diffractometer method). In recent years 
methods have been developed for the determination of pole figures with 
the diffractometer. These methods are capable of quite high precision 
because 

(1) the intensity of the diffracted rays is measured quantitatively with 
a counter, and 

(2) either the intensity measurements are corrected for changes in ab- 
sorption, or the x-ray optics are so designed that the absorption is constant 
and no correction is required. 

For reasons given later, two different methods must be used to cover 
the whole pole figure. 

The first of these, called the transmission method, is due to Decker, 
Asp, and Harker, and Fig. 9-18 illustrates its principal features. To deter- 
mine an (hkl) pole figure, the counter is fixed in position at the correct 
angle 26 to receive the hkl reflection. The sheet specimen, in a special 
holder, is positioned initially with the 
rolling direction vertical and coinci- 
dent with the diffractometer axis,* 
and with the plane of the specimen 
bisecting the angle between the inci- 
dent and diffracted beams. The speci- 
men holder allows rotation of the 
specimen about the diffractometer 
axis and about a horizontal axis nor- 
mal to the specimen surface. Al- 
though it is impossible to move the 
counter around the Debye ring and so 
explore the variation in diffracted in- 
tensity around this ring, we can ac- 
complish essentially the same thing 
by keeping the counter fixed and ro- 
tating the specimen in its own plane. 
This rotation, combined with the 
other rotation about the diffractom- 
eter axis, moves the pole of the (hkl) 



specimen 
I normal / 




diffractometer 
axis 



counter 



FIG. 9-18. Transmission method 
for pole-figure determination. (After 
A. H. Geisler, "Crystal Orientation 
and Pole Figure Determination" in 
Modern Research Techniquesin Physical 
Metallurgy, American Society for Met- 
als, Cleveland, 1953.) 



* For simplicity, the method is described here only in terms of a vertical-axis 
diffractometer. 



286 



THE STRUCTURE OF POLYCRY8TALLINE AGGREGATES 



[CHAP. 9 




FIG. 9-19. Specimen holder used in the transmission method, viewed from trans- 
mitted-beam side. (Courtesy of Paul A. Beck.) 

reflecting plane over the surface of the pole figure, which is plotted on a 
projection plane parallel to the specimen plane, as in the photographic 
method. At each position of the specimen, the measured intensity of the 
diffracted beam, after correction for absorption, gives a figure which is pro- 
portional to the pole density at the corresponding point on the pole figure. 
Figure 9-19 shows the kind of specimen holder used for this method. 

The method of plotting the data is indicated in Fig. 9-20. The angle a 
measures the amount of rotation about the diffract ometer axis;* it is 
zero when the sheet bisects the angle between incident and diffracted 
beams. The positive direction of a is conventionally taken as counter- 
clockwise. The angle 6 measures the amount by which the transverse 
direction is rotated about the sheet normal out of the horizontal plane and 

* a is the conventional symbol for this angle, which is measured in a horizontal 
plane. It should not be confused with the angle a used in Sec. 9-8 to measure 
azimuthal positions in a vertical plane. 



9-9] THE TEXTURE OF SHEET (DIFFRACTOMETER METHOD) 

R.D. 



287 



reflecting- 
plane *-) / 




T.D. 



diffrartometer 
axis 




(a) 



(b) 



FIG. 9-20. Angular relationships in the transmission pole-figure method (a) in 
space and (b) on the stereographic projection. (On the projection, the position of 
the reflecting plane normal is shown for 5 = 30 and a = 30.) 

is zero when the transverse direction is horizontal. The reflecting plane 
normal bisects the angle between incident and diffracted beams, and re- 
mains fixed in position whatever the orientation of the specimen. To plot 
the pole of the reflecting plane on the pole figure, we note that it coincides 
initially, when a and 6 are both zero, with the left transverse direction. A 
rotation of the specimen by d degrees in its own plane then moves the pole 
of the reflecting plane 8 degrees around the circumference of the pole figure, 
and a rotation of a degrees about the diffractometer axis then moves 
it a degrees from the circumference along a radius. To explore the pole 
figure, it is convenient to make intensity readings at intervals of 5 or 10 
of a for a fixed value of d: the pole figure is thus mapped out along a 
series of radii.* By this procedure the entire pole figure can be deter- 
mined except for a region at the center extending from about a = 50 
to a = 90; in this region not only does the absorption correction be- 
come inaccurate but the frame of the specimen holder obstructs the dif- 
fracted x-ray beam. 

An absorption correction is necessary in this method because variations 
in a cause variations in both the volume of diffracting material and the 
path length of the x-rays within the specimen. Variations in 6 have no 
effect. We can determine the angular dependence of the absorption factor 

* The chart shown in skeleton form in Fig. 9-20(b) is useful for this purpose. 
It is called a polar stereographic net, because it shows the latitude lines (circles) 
and longitude lines (radii) of a ruled globe projected on a plane normal to the polar 
NS-axis. In the absence of such a net, the equator or central meridian of a Wulff 
net can be used to measure the angle a. 



288 THE STRUCTURE OF POLYCRYSTALLINE AGGREGATES [CHAP. 9 

by a method similar to that used for the reflection case considered in Sec. 
7-4. The incident beam in Fig. 9-21 has intensity 7 (ergs/cm 2 /sec) and 
is 1 cm square in cross section. It is incident on a sheet specimen of thick- 
ness t and linear absorption coefficient p, and the individual grains of this 
specimen are assumed to have a completely random orientation. Let a 
be the volume fraction of the specimen containing grains correctly oriented 
for reflection of the incident beam, and b the fraction of the incident energy 
diffracted by unit volume. Then the total energy per second in the dif- 
fracted beam outside the specimen, originating in a layer of thickness dx 
located at a depth x, is given by 

dI D = ab(DB)I Q e- (AB+BC} dx (ergs/sec), 

where 

1 x t x 

AB = . and BC = 



COS (0 a) COS (0 a) COS (0 + a) 

By substitution, we obtain 

a ^o ,^ Q ffi ,>, _ 



= C (0-a)-l/cos (0+a)J J^. 

COS (0 a) 

(Only clockwise rotation of the specimen about the diffractometer axis, 
i.e., rotation in the sense usually designated by a, is considered here. 
However, in these equations and in Fig. 9-21, the proper sign has already 
been inserted, and the symbol a stands for the absolute value of this angle.) 
If we put a = in Eq. (9-7) and integrate from x = to x = /, we obtain 
the total diffracted energy per second, the integrated intensity, for this 
position of the specimen:* 



I D ( a = 0) = - - e- tlco '. (9-8) 

COS0 



When a is not zero, the same integration gives 



(0 a) _ e n 

I D ( a = a ) = - 1 - . (9-9) 

M[COS (0 - a)/COS (0 + a) ~ 1] 



* In Sec. 6-9 mention was made of the fact that the diffracted beams in any 
transmission method were of maximum intensity when the thickness of the speci- 
men was made equal to I/M. This result follows from Eq. (9-8). If we put = a 
= 0, then the primary beam will be incident on the specimen at right angles (see 
Fig. 9-21), as in the usual transmission pinhole method, and our result will apply 
approximately to diffracted beams formed at small angles 20. The intensity of 
such a beam is given by 

ID = 



By differentiating this expression with respect to t and setting the result equal to 
zero, we can find that ID is a maximum when t = 1 //*. 



9-9] 



THE TEXTURE OP SHEET (DIFFRACTOMETER METHOD) 



289 





-10 -20 -30 -40 -50 -60 -70 -80 
ROTATION ANGLE a (degrees) 



FIG. 9-21. Path length and irradi- 
ated volume in the transmission method. 



FIG. 9-22. Variation of the correc- 
tion factor R with a for clockwise rota- 
tion from the zero position, pi = 1.0, 
6 = 19.25. 



We are interested only in the ratio of these two integrated intensities, 
namely, 

R = D a ~ a = COB * e .. ^ : : (9-10) 



J D (a = 0) 



'[cos (6 - a) /cos (6 + a) - 1] 



A plot of R vs. a is given in Fig. 9-22 for typical values involved in the 111 
reflection from aluminum with Cu Ka radiation, namely, pi = 1.0 and 
6 = 19.25. This plot shows that the integrated intensity of the reflection 
decreases as a increases in the clockwise direction from zero, even for a 
specimen containing randomly oriented grains. In the measurement of 
preferred orientation, it is therefore necessary to divide each measured in- 
tensity by the appropriate value of the correction factor 7? in order to 
arrive at a figure proportional to the pole density. From the way in which 
the correction factor R was derived, it follows that we must measure the 
integrated intensity of the diffracted beam. To do this with a fixed counter, 
the counter slits must be as wide as the diffracted beam for all values of a 
so that the whole width of the beam can enter the counter. The ideal 
incident beam for this method is a parallel one. However, a divergent 
beam may be used without too much error, provided the divergence is not 
too great. There is no question of focusing here: if the incident beam is 
divergent, the diffracted beam will diverge also and very wide counter 
slits will be required to admit its entire width. 

The value of pt used in Eq. (9-10) must be obtained by direct measure- 
ment, since it is not sufficiently accurate to use a tabulated value of M 
together with the measured thickness t of the specimen. To determine 
pi we use a strong diffracted beam from any convenient material and meas- 
ure its intensity when the sheet specimen is inserted in the diffracted beam 



290 



THE STRUCTURE OP POLYCRYSTALLINE AGGREGATES [CHAP. 9 



counter 




FIG. 9-23. Reflection method for pole-figure determination. 

and again when it is not. The value of pt is then obtained from the general 
absorption equation, I t = /o^~" M ', where 7 and // are the intensities inci- 
dent on and transmitted by the sheet specimen, respectively. 

As already mentioned, the central part of the pole figure cannot be cov- 
ered by the transmission method. To explore this region we must use a 
reflection method, one in which the measured diffracted beam issues from 
that side of the sheet on which the primary beam is incident. The reflec- 
tion method here described was developed by Schulz. It requires a special 
holder which allows rotation of the specimen in its own plane about an 
axis normal to its surface and about a horizontal axis; these axes are shown 
as BB' and A A 1 in Fig. 9-23. The horizontal axis A A' lies in the specimen 
surface and is initially adjusted, by rotation about the diffractometer axis, 
to make equal angles with the incident and diffracted beams. After this 
is done, no further rotation about the diffractometer axis is made. Since 
the axis A A' remains in a fixed position during the other rotations of the 
specimen, the irradiated surface of the specimen is always tangent to a 
focusing circle passing through the x-ray source and counter slits. A 
divergent beam may therefore be used since the diffracted beam will con- 
verge to a focus at the counter slits. Figure 9-24 shows a specimen holder 
for the reflection method. 

When the specimen is rotated about the axis A A', the axis BB' normal 
to the specimen surface rotates in a vertical plane, but CAT, the reflecting 
plane normal, remains fixed in a horizontal position normal to A A'. The 
rotation angles a and 6 are defined in Fig. 9-23. The angle a is zero when 



9-9] 



THE TEXTURE OF SHEET (DIFFRACTOMETER METHOD) 



291 




FIG. 9-24. Specimen holder used in the reflection method, viewed from re- 
flected-beam side. (Courtesy of Paul A. Beck.) 

the sheet is horizontal and has a value of 90 when the sheet is in the 
vertical position shown in the drawing. In this position of the specimen, 
the reflecting plane normal is at the center of the projection. The angle 5 
measures the amount by which the rolling direction is rotated away from 
the left end of the axis A A' and has a value of +90 for the position illus- 
trated. With these conventions the angles a and 5 may be plotted on the 
pole figure in the same way as in the transmission method [Fig. 9-20(b)]. 

The great virtue of the reflection method is that no absorption correc- 
tion is required for values of a between 90 and about 40, i.e., up to 
about 50 from the center of the pole figure. In other words, a specimen 
whose grains have a completely random orientation can be rotated over 
this range of a values without any change in the measured intensity of the 
diffracted beam. Under these circumstances, the intensity of the dif- 
fracted beam is directly proportional to the pole density in the specimen, 
without any correction. The constancy of the absorption factor is due 
essentially to the narrow horizontal slit placed in the primary beanr at D 
(Fig. 9-23). The vertical opening in this slit is only about 0.020 in. in 
height, which means that the specimen is irradiated only over a long nar- 
row rectangle centered on the fixed axis A A'. It can be shown that a 



292 THE STRUCTURE OF POLYCRYSTALLINE AGGREGATES [CHAP. 9 

RD. 




FIG. 9-25. (Ill) pole figure of cold-rolled 70-30 brass, determined by the dif- 
fractometer method. (H. Hu, P. R. Sperry, and P. A. Beck, Trans. A.LM.E. 
194,76, 1952.) 

change in absorption does occur, as the specimen is rotated about A A', 
but it is exactly canceled by a change in the volume of diffracting material, 
the net result being a constant diffracted intensity for a random specimen 
when a lies between 90 and about 40. To achieve this condition, 
the reflecting surface of the specimen must be adjusted to accurately coin- 
cide with the axis A A' for all values of a and 5. This adjustment is ex- 
tremely important. 

It is evident that the transmission and reflection methods complement 
one another in their coverage of the pole figure. The usual practice is to 
use the transmission method to cover the range of a from to 50 and 
the reflection method from 40 to 90. This produces an overlap of 
10 which is useful in checking the accuracy of one method against the 
other, and necessary in order to find a normalizing factor for one set of 
readings which will make them agree with the other set in the region of 
overlap. 

When this is done, the numbers which are proportional to pole density 
can then be plotted on the pole figure at each point at which a measure- 
ment was made. Contour lines are then drawn at selected levels con- 
necting points of the same pole density, and the result is a pole figure such 
as that shown in Fig. 9-25, which represents the deformation texture of 
70-30 brass cold-rolled to a reduction in thickness of 95 percent. The 
numbers attached to each contour line give the pole density in arbitrary 



9-9] THE TEXTURE OF SHEET (DIFFRACTOMETER METHOD) 293 

units. A pole figure such as this is far more accurate than any photo- 
graphically determined one, and represents the best description available 
today of the kind and extent of preferred orientation. The accuracy ob- 
tainable with the diffractometer method is sufficient to allow investigation, 
with some confidence, of possible asymmetry in sheet textures. In most 
sheet, no asymmetry of texture is found (see Fig. 9-25), but it does occur 
when sheet is carefully rolled in the same direction, i.e., without any 
reversal end for end between passes. In such sheet, the texture has only 
one reflection plane of symmetry, normal to the transverse direction; the 
plane normal to the rolling direction is no longer a symmetry plane. 

In Fig. 9-25, the solid triangular symbols representing the ideal orienta- 
tion (110) [lT2] lie approximately in the high-density regions of the pole 
figure. But here again the pole figure itself must be regarded as a far 
better description of the texture than any bare statement of an ideal orien- 
tation. A quantitative pole figure of this kind has about the same relation 
to an ideal orientation as an accurate contour map of a hill has to a state- 
ment of the height, width, and length of the hill. 

Geisler has recently pointed out two sources of error in the diffractometer 
method, both of which can lead to spurious intensity maxima on the pole 
figure if the investigator is not aware of them: 

(1) When an (AiMi) pole figure is being determined, the counter is set 
at the appropriate angle 26 to receive Ka radiation reflected from the 
(hikili) planes. But at some position of the specimen, there may be another 
set of planes, (/^tt), so oriented that they can reflect a component of the 
continuous spectrum at the same angle 26. If the (hjtj,^) planes have a 
high reflecting power, this reflection may be so strong that it may be taken 
for an fcjJMi reflection of the Ka wavelength. Apparently the only sure 
way of eliminating this possibility is to use balanced filters. 

(2) The crystal structure of the material being investigated may be such 
that a set of planes, (h 3 kM, has very nearly the same spacing as the 
(hikili) planes. The Ka reflections of these two sets will therefore occur 
at very nearly the same angle 26. If the counter is set to receive the hik^i 
reflection, then there is a possibility that some of the feaMs reflection may 
also be received, especially in the transmission method for which a wide 
receiving slit is used. The best way out of this difficulty is to select another 
reflection, A 4 fc 4 / 4 , well separated from its neighbors, and construct an 
A 4 fc 4 / 4 pole figure instead of an ftiMi- (It is not advisable to attempt to 
exclude the unwanted hjc^ reflection by narrowing the slits. If this is 
done, then the counter may not receive the entire hik^i diffracted beam, 
and if all of this beam is not received, Eq. (9-10) will no longer give the 
correct value of R. If a narrow receiving slit must be used, then the varia- 
tion of R with a must be determined experimentally. This determination 
requires a specimen of the same material as that under investigation, with 



294 THE STRUCTURE OF POLYCRYSTALLINE AGGREGATES [CHAP. 9 

the same value of \d and a perfectly random orientation of its constituent 
grains.) 

One other point about pole-figure determinations should be mentioned, 
and that is the necessity for integrating devices when the grain size of the 
specimen is large, as in recrystallized metals and alloys. With such speci- 
mens, the incident x-ray beam will not strike enough grains to give a good 
statistical average of the orientations present. This is true of both methods, 
the photographic and the diffractometer. With coarse-grained specimens 
it is therefore necessary to use some kind of integrating device, which will 
move the specimen back and forth, or in a spiral, in its own plane and so 
expose a larger number of grains to the incident beam. 

Pole-figure determination is by no means a closed subject, and varia- 
tions and improvements are constantly being described in the technical 
literature. The most interesting among these are devices for the auto- 
matic plotting of pole figures by the diffractometer method. Jn these de- 
vices, the specimen is slowly rotated about the various axes by a mechan- 
ical drive, and the output of the counter-ratemeter circuit is fed to a 
recorder whose chart is driven in synchronism with the rotation of the 
specimen. The chart may be either of the simple strip variety, or even a 
circular pole-figure chart on which the recorder prints selected levels of 
pole density at the proper positions. The time is probably not far off when 
most pole figures will be determined in an automatic or semi-automatic 
manner, at least in the larger laboratories. 

TABLE 9-2 



Appearance of diffraction lines 



Condition of specimen 



Continuous 



Spotty 
Narrow (1) 
Broad (1) 



Uniform intensity 
Nonuniform intensity 



Fine-grained (or coarse-grained and 
cold-worked) 

Coarse-grained 
Strain -free 

Residual stress and possibly small particle 
size (if specimen is a solid aggregate) 

Small particle size (if specimen is a 
brittle powder) 

Random orientation (2) 
Preferred orientation 



Notes: 

(1) Best judged by noting whether or not the Ka doublet is resolved in back re- 
flection. 

(2) Or possibly presence of a fiber texture, if the incident beam is parallel to the 
fiber axis. 



9-10] SUMMARY; PROBLEMS 295 

9-10 Summary. In this chapter we have 'considered various aspects 
of the structure of polycrystalline aggregates and the quantitative effects 
of variations in crystal size, perfection, and orientation on the diffraction 
pattern. Although a complete investigation of the structure of an aggre- 
gate requires a considerable amount of time and rather complex apparatus, 
the very great utility of the simple pinhole photograph should not be over- 
looked. It is surprising how much information an experienced observer 
can obtain simply by inspection of a pinhole photograph, without any 
knowledge of the specimen, i.e., without knowing its chemical identity, 
crystal structure, or even whether it is amorphous or crystalline. The 
latter point can be settled at a glance, since diffraction lines indicate crys- 
tallinity and broad haloes an amorphous condition. If the specimen is 
crystalline, the conclusions that can be drawn from the appearance of the 
lines are summarized in Table 9-2. 



PROBLEMS 

9-1. A cold-worked polycrystalline piece of metal, having a Young's modulus of 
30,000,000 psi, is examined with Cu Ka radiation. A diffraction line occurring at 
28 = 150 is observed to be 1.28 degrees 28 broader than the same line from a 
recrystallized specimen. If this broadening is assumed to be due to residual micro- 
stresses varying from zero to the yield point both in tension and compression, 
what is the yield point of the material? 

9-2. If the observed broadening given in Prob. 9-1 is ascribed entirely to a frag- 
mentation of the grains into small crystal particles, what is the size of these par- 
ticles? 

9-3. For given values of 6 and /x, which results in a greater effective depth of 
x-ray penetration, a back-reflection pinhole camera or a diffractometer? 

9-4. Assume that the effective depth of penetration of an x-ray beam is that 
thickness of material which contributes 99 percent of the total energy diffracted 
by an infinitely thick specimen. Calculate the penetration depth in inches for a 
low-carbon steel specimen under the following conditions: 

(a) Diffractometer; lowest-angle reflection; Cu Ka radiation. 

(6) Diffractometer; highest-angle reflection; Cu Ka radiation. 

(c) Diffractometer; highest-angle reflection; Cr Ka radiation. 

(d) Back-reflection pinhole camera; highest-angle reflection; Cr Ka radiation. 
9-6. (a) A transmission pinhole photograph is made of a sheet specimen of 

thickness t and linear absorption coefficient p. Show that the fraction of the total 
diffracted energy in any one reflection contributed by a layer of thickness w is 
given by 

_ tt(x+(t x)/6O6 2ff\T0 nw(l l/cos 29) I] 

TTT I? J 

w = 



where x is the distance to the side of the layer involved, measured from the side 
of the specimen on which the primary beam is incident. 



296 THE STRUCTURE OF POLYCRYSTALLINE AGGREGATES [CHAP. 9 

(b) A transmission pinhole photograph is made of a sheet of aluminum 0.5 mm 
thick with Cu Ka radiation. Consider only the 111 reflection which occurs at 
26 = 38.4. Imagine the sheet to be divided into four layers, the thickness of 
each being equal to one-fourth of the total thickness. Calculate W for each layer. 

9-6. A transmission pinhole pattern is made with Co Ka radiation of an iron 
wire having an almost perfect [110] fiber texture. The wire axis is vertical. How 
many high-intensity maxima will appear on the lowest-angle 110 Debye ring and 
what are their azimuthal angles on the film? 



CHAPTER 10 
THE DETERMINATION OF CRYSTAL STRUCTURE 

10-1 Introduction. Since 1913, when W. L. Bragg solved the struc- 
ture of NaCl, the structures of some five thousand crystals, organic and 
inorganic, have been determined. This vast body of knowledge is of funda- 
mental importance in such fields as crystal chemistry, solid-state physics, 
and the biological sciences because, to a large extent, structure determines 
properties and the properties of a substance are never fully understood 
until its structure is known. In metallurgy, a knowledge of crystal struc- 
ture is a necessary prerequisite to any understanding of such phenomena 
as plastic deformation, alloy formation, or phase transformations. 

The work of structure determination goes on continuously since there 
is no dearth of unsolved structures. New substances are constantly being 
synthesized, and the structures of many old ones are still unknown. In 
themselves crystal structures vary widely in complexity: the simplest can 
be solved in a few hours, while the more complex may require months or 
even years for their complete solution. (Proteins form a notable example 
of the latter kind; despite intensive efforts of many investigators, their 
structure has not yet been completely determined.) Complex structures 
require complex methods of solution, and structure determination in its 
entirety is more properly the subject of a book than of a single chapter. 
All we can do here is to consider some of the principles involved and how 
they can be applied to the solution of fairly simple structures. Moreover, 
we will confine our attention to the methods of determining structure from 
powder patterns alone, because such patterns are the kind most often en- 
countered by the metallurgist. 

The basic principles involved in structure determination have already 
been introduced in Chaps. 3 and 4. We saw there that the crystal struc- 
ture of a substance determines the diffraction pattern of that substance or, 
more specifically, that the shape and size of the unit cell determines the 
angular positions of the diffraction lines, and the arrangement of the atoms 
within the unit cell determines the relative intensities of the lines. It may 
be worthwhile to state this again in tabular form : 

Crystal structure Diffraction pattern 

Unit cell <- Line positions 

Atom positions <-> Line intensities 
297 



298 THE DETERMINATION OF CRYSTAL STRUCTURE [CHAP. 10 

Since structure determines the diffraction pattern, it should be possible to 
go in the other direction and deduce the structure from the pattern. It is 
possible, but not in any direct manner. Given a structure, we can calculate 
its diffraction pattern in a very straightforward fashion, and examples of 
such calculations were given in Sec. 4-13; but the reverse problem, that 
of directly calculating the structure from the observed pattern, has never 
been solved, for reasons to be discussed in Sec. 10-8. The procedure 
adopted is essentially one of trial and error. On the basis of an educated 
guess, a structure is assumed, its diffraction pattern calculated, and the 
calculated pattern compared with the observed one. If the two agree in 
all detail, the assumed structure is correct; if not, the process is repeated 
as often as is necessary to find the correct solution. The problem is not 
unlike that of deciphering a code, and requires of the crystallographer the 
same qualities possessed by a good cryptanalyst, namely, knowledge, 
perseverance, and not a little intuition. 

The determination of an unknown structure proceeds in three major 
steps: 

(1) The shape and size of the unit cell are deduced from the angular 
positions of the diffraction lines. An assumption is first made as to which 
of the seven crystal systems the unknown structure belongs to, and then, 
on the basis of this assumption, the correct Miller indices are assigned to 
each reflection. This step is called "indexing the pattern" and is only 
possible when the correct choice of crystal system has been made. Once 
this is done, the shape of the unit cell is known (from the crystal system), 
and its size is calculable from the positions and Miller indices of the dif- 
fraction lines. 

(2) The number of atoms per unit cell is then computed from the shape 
and size of the unit cell, the chemical composition of the specimen, and its 
measured density. 

(3) Finally, the positions of the atoms within the unit cell are deduced 
from the relative intensities of the diffraction lines. 

Only when these three steps have been accomplished is the structure 
determination complete. The third step is generally the most difficult, 
and there are many structures which are known only incompletely, in the 
sense that this final step has not yet been made. Nevertheless, a knowl- 
edge of the shape and size of the unit cell, without any knowledge of atom 
positions, is in itself of very great value in many applications. 

The average metallurgist is rarely, if ever, called upon to determine an 
unknown crystal structure. If the structure is at all complex, its deter- 
mination is a job for a specialist in x-ray crystallography, who can bring 
special techniques, both experimental and mathematical, to bear on the 
problem. The metallurgist should, however, know enough about structure 



10-2] PRELIMINARY TREATMENT OF DATA 299 

determination to unravel any simple structures he may encounter and, 
what is more important, he must be able to index the powder patterns of 
substances of known structure, as this is a routine problem in almost all 
diffraction work. The procedures given below for indexing patterns are 
applicable whether the structure is known or not, but they are of course 
very much easier to apply if the structure is known beforehand. 

10-2 Preliminary treatment of data. The powder pattern of the un- 
known is obtained with a Debye-Scherrer camera or a diffractometer, the 
object being to cover as wide an angular range of 26 as possible. A camera 
such as the Seemann-Bohlin, which records diffraction lines over only a 
limited angular range, is of very little use in structure analysis. The speci- 
men preparation must ensure random orientation of the individual par- 
ticles of powder, if the observed relative intensities of the diffraction lines 
are to have any meaning in terms of crystal structure. After the pattern 
is obtained, the value of sin 2 6 is calculated for each diffraction line; this 
set of sin 2 6 values is the raw material for the determination of cell size 
and shape. 

Since the problem of structure determination is one of finding a struc- 
ture which will account for all the lines on the pattern, in both position 
and intensity, the investigator must make sure at the outset that the ob- 
served pattern does not contain any extraneous lines. The ideal pattern 
contains lines formed by x-rays of a single wavelength, diffracted only by 
the substance whose structure is to be determined. There are therefore 
two sources of extraneous lines: 

(1) Diffraction of x-rays having wavelengths different from that of the prin- 
cipal component of the radiation. If filtered radiation is used, then Ka 
radiation is the principal component, and characteristic x-rays of any 
other wavelength may produce extraneous lines. The chief offender is 
Kf$ radiation, which is never entirely removed by a filter and may be a 
source of extraneous lines when diffracted by lattice planes of high reflect- 
ing power. The presence of K0 lines on a pattern can usually be revealed 
by calculation, since if a certain set of planes reflect K/3 radiation at an 
angle fy, they must also reflect Ka radiation at an angle a (unless a ex- 
ceeds 90), and one angle may be calculated from the other. It follows 
from the Bragg law that 



( 



X 2 



sin 2 a , (10-1) 



where X# a 2 /Xx/3 2 has a value near 1.2 for most radiations. If it is sus- 
pected that a particular line is due to K$ radiation, multiplication of its 
sin 2 value by X/r a 2 /A#0 2 will give a value equal, or nearly equal, to the 



300 



THE DETERMINATION OF CRYSTAL STRUCTURE 



[CHAP. 10 



value of sin 2 8 for some Ka line on the pattern, unless the product exceeds 
unity. The K0 line corresponding to a given Ka line is always located at a 
smaller angle 26 and has lower intensity. However, since Ka and Kfl 
lines (from different planes) may overlap on the pattern, Eq. (10-1) alone 
can only establish the possibility that a given line is due to Kft radiation, 
but it can never prove that it is. Another possible source of extraneous 
lines is L characteristic radiation from tungsten contamination on the 
target of the x-ray tube, particularly if the tube is old. If such contamina- 
tion is suspected, equations such as (10-1) can be set up to test the possi- 
bility that certain lines are due to tungsten radiation. 

(2) Diffraction by substances other than the unknown. Such substances 
are usually impurities in the specimen but may also include the specimen 
mount or badly aligned slits. Careful specimen preparation and good ex- 
perimental technique will eliminate extraneous lines due to these causes. 

For reasons to be discussed in Chap. 11, the observed values of sin 2 
always contain small systematic errors. These errors are not large enough 
to cause any difficulty in indexing patterns of cubic crystals, but they can 
seriously interfere with the determination of some noncubic structures. 
The best method of removing such errors from the data is to calibrate the 
camera or diffractometer with a substance of known lattice parameter, 
mixed with the unknown. The difference between the observed and calcu- 



0.008 - 




2 0.4 6 0.8 1 
sm 2 (observed) 

FIG. 10-1. An example of a correction curve for sin 2 6 values. 

lated values of sin 2 for the standard substance gives the error in sin 2 6, 
and this error can be plotted as a function of the observed values of sin 2 6. 
Figure 10-1 shows a correction curve of this kind, obtained with a par- 
ticular specimen and a particular Debye-Scherrer camera.* The errors 
represented by the ordinates of such a curve can then be applied to each 
of the observed values of sin 2 6 for the diffraction lines of the unknown 
substance. For the particular determination represented by Fig. 10-1, 
the errors shown are to be subtracted from the observed values. 

* For the shape of this curve, see Prob. 11-5. 



10-3] INDEXING PATTERNS OF CUBIC CRYSTALS 301 

10-3 Indexing patterns of cubic crystals. A cubic crystal gives dif- 
fraction lines whose sin 2 6 values satisfy the following equation, obtained 
by combining the Bragg law with the plane-spacing equation for the cubic 
system: 

sin 2 B sin 2 B X 2 



Since the sum s = (h? + k 2 + I 2 ) is always integral and A 2 /4a 2 i s a con- 
stant for any one pattern, the problem of indexing the pattern of a cubic 
substance is one of finding a set of integers s which will yield a constant 
quotient when divided one by one into the observed sin 2 6 values. (Certain 
integers, such as 7, 15, 23, 28, 31, etc., are impossible because they cannot 
be formed by the sum of three squared integers.) Once the proper integers 
s are found, the indices hkl of each line can be written down by inspection 
or from the tabulation in Appendix 6. 

The proper integers s can be determined by means of the C and D scales 
of an ordinary slide rule, which permit simultaneous division of one set of 
numbers by another, if the quotient is constant. Pencil marks correspond- 
ing to the sin 2 values of the first five or six lines on the pattern are placed 
on the D scale. A single setting of the C scale is then sought which will 
bring a set of integers on the C scale into coincidence with all the pencil 
marks on the D scale. Because of the systematic errors mentioned earlier, 
these coincidences are never exact, but they are usually close enough to per- 
mit selection of the proper integer, particularly if the C scale is shifted 
slightly from line to line to compensate for the systematic errors in sin 2 6. If 
a set of integers satisfying Eq. (10-2) cannot be found, then the substance 
involved does not belong to the cubic system, and other possibilities (tetrag- 
onal, hexagonal, etc.) must be explored. 

The following example will illustrate the steps involved in indexing the 
pattern of a cubic substance and finding its lattice parameter. In this 
particular example, Cu Ka radiation was used and eight diffraction lines 
were observed. Their sin 2 values are listed in the second column of 
Table 10-1. By means of a slide rule, the integers s listed in the third 
column were found to produce the reasonably constant quotients listed in 
the fourth column, when divided into the observed sin 2 values. The 
fifth column lists the lattice parameter calculated from each line position, 
and the sixth column gives the Miller indices of each line. The systematic 
error in sin 2 6 shows up as a gradual decrease in the value of X 2 /4a 2 , and a 
gradual increase in the value of a, as 8 increases. We shall find in Chap. 11 
that the systematic error decreases as increases; therefore we can select 
the value of a for the highest-angle line, namely, 3.62A, as being the most 
accurate of those listed. Our analysis of line positions therefore leads to 



302 



THE DETERMINATION OF CRYSTAL STRUCTURE [CHAP. 10 

TABLE 10-1 



1 


2 


3 


4 


5 


6 




2 




\2 






Line 


sm 


8 = (h'2 + & + /2) 




a (A) 


hkl 


1 


0.140 


3 


0.0466 


3.57 


111 


2 


0.185 


4 


0.0463 


3.58 


200 


3 


0.369 


8 


0.0462 


3.59 


220 


4 


0.503 


11 


0.0457 


3.61 


311 


5 


0.548 


12 


0.0456 


3.61 


222 


6 


0.726 


16 


0.0454 


3.62 


400 


7 


0.861 


19 


0.0453 


3.62 


331 


8 


0.905 


20 


0.0453 


3.62 


420 



the conclusion that the substance involved, copper in this case, is cubic in 
structure with a lattice parameter of 3.62A. 

We can also determine the Bravais lattice of the specimen by observing 
which lines are present and which absent. Examination of the sixth col- 
umn of Table 10-1 shows that all lines which have mixed odd and even 
indices, such as 100, 110, etc., are absent from the pattern. Reference to 
the rules relating Bravais lattices to observed and absent reflections, given 
in Table 4-1, shows that the Bravais lattice of this specimen is face- 
centered. We now have certain information about the arrangement of 
atoms within the unit cell, and it should be noted that we have had to make 
use of observed line intensities in order to obtain this information. In 
this particular case, the observation consisted simply in noting which 
lines had zero intensity. 

Each of the four common cubic lattice types is recognizable by a charac- 
teristic sequence of diffraction lines, and these in turn may be described 
by their sequential s values: 

Simple cubic: 1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 14, 16, ... 

Body-centered cubic: 2, 4, 6, 8, 10, 12, 14, 16, ... 
Face-centered cubic: 3, 4, 8, 11, 12, 16, ... 
Diamond cubic: 3, 8, 11, 16, ... 

The same information is tabulated in Appendix 6 and shown graphically 
in Fig. 10-2, in the form of calculated diffraction patterns. The calcula- 
tions are made for Cu Ka radiation and a lattice parameter a of 3.50A. 
The positions of all the diffraction lines which would be formed under 
these conditions are indicated as they would appear on a film or chart of 
the length shown. (For comparative purposes, the pattern of a hexagonal 
close-packed structure is also illustrated, since this structure is frequently 



)-3] 

f 




INDEXING PATTERNS OF CUBIC CRYSTALS 303 

CUBIC 1 HEXAGONAL 

Vv x 


simple 


hU 
100 

110 

111 
200 
210 
211 

220 
300, 221 
310 
311 
222 
320 
321 

400 
410, 322 
411, 330 
331 

420 


body- 
centered 




face- 
centered 


^ f 
diamond 


close- 
packed 


"\ 

hkl 
10-0 
00-2 
101 

102 

110 

103 

200 
112 
201 
004 

202 
104 

203 

21-0 
211 
11-4 


,S 
1 

2 

3 
4 

i 

s 

* 

_ 9 

<-*< 

S 10 

* n 

12 
^ 13 

* H 

H 

16 

17 
IS 
19 

20 
- in ^ 

































































































































FIG. 10-2. Calculated diffraction patterns for various lattices, s ti 2 + k 2 + I 2 . 

encountered among metals and alloys. The line positions are calculated 
for CuKa radiation, a = 2.50A, and c/a = 1.633, which corresponds to 
the close packing of spheres.) 

Powder patterns of cubic substances can usually be distinguished at a 
glance from those of noncubic substances, since the latter patterns nor- 



304 THE DETERMINATION OF CRYSTAL STRUCTURE [CHAP. 10 

mally contain many more lines. In addition, the Bravais lattice can usu- 
ally be identified by inspection: there is an almost regular sequence of 
lines in simple cubic and body-centered cubic patterns, but the former 
contains almost twice as many lines, while a face-centered cubic pattern 
is characterized by a pair of lines, followed by a single line, followed by a 
pair, another single line, etc. 

The problem of indexing a cubic pattern is of course very much sim- 
plified if the substance involved is known to be cubic and if the lattice 
parameter is also known. The simplest procedure then is to calculate the 
value of (\ 2 /4a 2 ) and divide this value into the observed sin 2 6 values to 
obtain the value of s for each line. 



There is one difficulty that may arise in the interpretation of cubic powder pat- 
terns, and that is due to a possible ambiguity between simple cubic and body- 
centered cubic patterns. There is a regular sequence of lines in both patterns up 
to the sixth line; the sequence then continues regularly in body-centered cubic 
patterns, but is interrupted in simple cubic patterns since s = 7 is impossible. 
Therefore, if X is so large, or a so small, that six lines or less appear on the pattern, 
the two Bravais lattices are indistinguishable. For example, suppose that the 
substance involved is actually body-centered cubic but the investigator mistakenly 
indexes it as simple cubic, assigning the value s = 1 to the first line, s = 2 to the 
second line, etc. He thus obtains a value of X 2 /4a 2 twice as large as the true one, 
and a value of a which is l/\/2 times the true one. This sort of difficulty can be 
avoided simply by choosing a wavelength short enough to produce at least seven 
lines on the pattern. 

10-4 Indexing patterns of noncubic crystals (graphical methods). The 
problem of indexing powder patterns becomes more difficult as the number 
of unknown parameters increases. There is only one unknown parameter 
for cubic crystals, the cell edge a, but noncubic crystals have two or more, 
and special graphical and analytical techniques have had to be devised in 
order to index the patterns of such crystals. 

The tetragonal system will be considered first. The plane-spacing equa- 
tion for this system involves two unknown parameters, a and c: 

I h 2 + k 2 I 2 

-5- + T (10-3) 

d 2 a 2 c 2 

This may be rewritten in the form 

i -![(* + *") + _ 

d 2 a 2 L (c/o) 



10-4] NONCUBIC CRYSTALS (GRAPHICAL METHODS) 305 

or 

r i 2 i 

2 log d = 2 log a - log (h 2 + k 2 ) + (10-4) 

L (c/a) 2 J 

Suppose we now write Eq. (10-4) for any two planes of a tetragonal crys- 
tal, distinguishing the two planes by subscripts 1 and 2, and then subtract 
the two equations. We obtain 



= - log [ 
L 



2 log d, - 2 log d 2 = - log (V + fc, 2 ) + 



(c/a) 2 

2 



r ^ + fc 2 2 ) + -A- 1 . 

L (c/arj 



This equation shows that the difference between the 2 log d values for any 
two planes is independent of a and depends only on the axial ratio c/a and 
the indices hkl of each plane. This fact was used by Hull and Davey as 
the basis for a graphical method of indexing the powder patterns of tetrag- 
onal crystals. 

The construction of a Hull-Davey chart is illustrated in Fig. 10-3. First, 
the variation of the quantity [(/i 2 + k 2 ) + l 2 /(c/a) 2 ] with c/a is plotted 
on two-range semilog paper for particular values of hkl. Each set of indices 
hkl, as long as they correspond to planes of different spacing, produces a 
different curve, and when I = the curve is a straight line parallel to the 
c/a axis. Planes of different indices but the same spacing, such as (100) 
and (010), are represented by the same curve on the chart, which is then 
marked with the indices of either one of them, in this case (100). [The 
chart shown is for a simple tetragonal lattice; one for a body-centered 
tetragonal lattice is made simply by omitting all curves for which 
(h + k + I) is an odd number.] A single-range logarithmic d scale is then 
constructed; it extends over two ranges of the [(h 2 + k 2 ) + Z 2 /(c/a) 2 ] 
scale and runs in the opposite direction, since the coefficient of logd in 
Eq. (10-4) is -2 times the coefficient of log [(h 2 + k 2 ) + I 2 /(c/a) 2 ]. This 
means that the d values of two planes, for a given c/a ratio, are separated 
by the same distance on the scale as the horizontal separation, at the same 
c/a ratio, of the two corresponding curves on the chart. 

The chart and scale are used for indexing in the following manner. The 
spacing d of the reflecting planes corresponding to each line on the diffrac- 
tion pattern is calculated. Suppose that the first seven of these values for 
a particular pattern are 6.00, 4.00, 3.33, 3.00, 2.83, 2.55, and 2.40A. A 
strip of paper is then laid alongside the d scale in position I of Fig. 10-3, 
and the observed d values are marked off on its edge with a pencil. The 



306 



THE DETERMINATION OF CRYSTAL STRUCTURE 



[CHAP. 10 



s 

U: 
bC 







- e 



. 














I 

03 



68 

-C 



I 



+ J, 



I 



10-4 NONCUBIC CRYSTALS (GRAPHICAL METHODS) 307 

paper strip is then placed on the chart and moved about, both vertically 
and horizontally, until a position is found where each mark on the strip 
coincides with a line on the chart. Vertical and horizontal movements 
correspond to trying various c/a and a values, respectively, and the only 
restriction on these movements is that the edge of the strip must always 
be horizontal. When a correct fit has been obtained, as shown by posi- 
tion II of Fig. 10-3, the indices of each line are simply read from the corre- 
sponding curves, and the approximate value of c/a from the vertical 
position of the paper strip. In the present example, the c/a ratio is 1.5 
and the first line on the pattern (formed by planes of spacing 6.00A) is a 
001 line, the second a 100 line, the third a 101 line, etc. After all the lines 
have been indexed in this way, the d values of the two highest-angle lines 
are used to set up two equations of the form of Eq. (10-3), and these are 
solved simultaneously to yield the values of a and c. From these values, 
the axial ratio c/a may then be calculated with more precision than it can 
be found graphically. 

Figure 10-3 is only a partial Hull-Davey chart. A complete one, show- 
ing curves of higher indices, is reproduced on a small scale in Fig. 10-4, 
which applies to body-centered tetragonal lattices. Note that the curves 
of high indices are often so crowded that it is difficult to assign the proper 
indices to the observed lines. It then becomes necessary to calculate the 
indices of these high-angle lines on the basis of a and c values derived from 
the already indexed low-angle lines. 

Some Hull-Davey charts, like the one shown in Fig. 10-4, are designed 
for use with sin 2 6 values rather than d values. No change in the chart 
itself is involved, only a change in the accompanying scale. This is possible 
because an equation similar to Eq. (10-4) can be set up in terms of sin 2 8 
rather than d, by combining Eq. (10-3) with the Bragg law. This equa- 
tion is 

log sin 2 = log 2 + log [ (h 2 + k 2 ) + -^T 
4a 2 L (c/a 

The sin 2 6 scale is therefore a two-range logarithmic one (from 0.01 to 1.0), 
equal in length to the two-range [(h 2 + fc 2 ) + I 2 /(c/a) 2 ] scale on the charl 
and running in the same direction. A scale of this kind appears at the top 
of Fig. 10-3. 

When the c/a ratio becomes equal to unity, a tetragonal cell becomes 
cubic. It follows that a cubic pattern can be indexed on a tetragonal Hull- 
Davey chart by keeping the paper strip always on the horizontal line corre- 
sponding to c/a = 1. This is seldom necessary because a slide rule wil 
serve just as well. However, it is instructive to consider a tetragonal eel 
as a departure from a cubic one and to examine a Hull-Davey chart ii 



308 



THE DETERMINATION OF CRYSTAL STRUCTURE [CHAP. 10 



0.01 - I IM|immil|IIM|MM[IIM|IIM|IIM[llll|IMI[IIM[UII|Mll|IIMI|MII[llll|llll|llll|llll| 



. ..I.. ..I I.. ..I.. ,,!..,. I ,,,!, ,,,!,,,, I,,,, I,,,, , ,,,l.i i.li ml. 1I.J 




FIG. 



AXIAL RATIO 
10-4. Complete Hull-Davey chart for body-centered tetragonal lattices. 



10-4] NONCUBIC CRYSTALS (GRAPHICAL METHODS) 309 

that light, since the chart shows at a glance how the powder pattern 
changes for any given change in the c/a ratio. It shows, for example, how 
certain lines split into two as soon as the c/a ratio departs from unity, and 
how even the order of the lines on the pattern can change with changes 
in c/a. 

Another graphical method of indexing tetragonal patterns has been de- 
vised by Bunn. Like the Hull-Davey chart, a Bunn chart consists of a 
network of curves, one for each value of hkl, but the curves are based on 
somewhat different functions of hkl and c/a than those used by Hull and 
Davey, with the result that the curves are less crowded in certain regions 
of the chart. The Bunn chart is accompanied by a logarithmic scale of d 
values, and the combination of chart and scale is used in exactly the same 
way as a Hull-Davey chart and scale. 

Patterns of hexagonal crystals can also be indexed by graphical methods, 
since the hexagonal unit cell, like the tetragonal, is characterized by two 
variable parameters, a and c. The plane-spacing equation is 

1 _ 4 h 2 + hk + k 2 I 2 
d* = 3 tf + 7 2 ' 

After some manipulation, this becomes 

21ogd = 21oga - log \- (h 2 + hk + k 2 ) + -4- 
L3 (c/a 

which is of exactly the same form as Eq. (10-4) for the tetragonal system. 
A Hull-Davey chart for the hexagonal system can therefore be constructed 
by plotting the variation of log [ (h 2 + hk + k 2 ) + I 2 /(c/a) 2 ] with c/a. 
A Bunn chart may also be constructed for this system. Special charts for 
hexagonal close-packed lattices may also be prepared by omitting all 
curves for which (h + 2k) is an integral multiple of 3 and I is odd. 

Figure 3-13(c), the powder pattern of zinc made with Cu Ka radiation, 
will serve to illustrate how the pattern of a hexagonal substance is indexed. 
Thirteen lines were observed on this pattern ; their sin 2 6 values and rela- 
tive intensities are listed in Table 10-2, A fit was obtained on a Hull- 
Davey chart for hexagonal close-packed lattices at an approximate c/a 
ratio of 1.87. The chart lines disclosed the indices listed in the fourth 
column of the table. In the case of line 5, two chart lines (10-3 and 11-0) 
almost intersect at c/a = 1.87, so the observed line is evidently the sum 
of two lines, almost overlapping, one from the (10-3) planes and the other 
from (11 -0) planes. The same is true of line 11. Four lines on the chart, 
namely, 20-0, 10-4, 21-0, and 20-4, do not appear on the pattern, and it 
must be inferred that these are too weak to be observed. On the other 
hand, all the observed lines are accounted for, so we may conclude that 



310 



THE DETERMINATION OF CRYSTAL STRUCTURE 
TABLE 10-2 



[CHAP. 10 



Line 


Intensity 


sin 2 


hkl 


1 


s 


0.097 


00-2 


2 


s 


0.112 


10-0 


3 


vs 


0.136 


10-1 


4 


m 


0.209 


10-2 


5 


s 


0.332 


10-3, 11-0 


6 


vw 


0.390 


00-4 


7 


m 


0.434 


11-2 


8 


m 


0.472 


20-1 


9 


vw 


0.547 


20-2 


10 


w 


0.668 


20-3 


11 


m 


0.722 


11-4, 10-5 


12 


m 


0.806 


21-1 


13 


w 


0.879 


21-2 



the lattice of zinc is actually hexagonal close-packed. The next step is to 
calculate the lattice parameters. Combination of the Bragg law and the 
plane-spacing equation gives 

(h 2 + hk + k 2 ) 1 2 ~ 



20 = _ - 



sm" = 



-1- 

C 2 \ 



where X 2 /4 has a value of 0.595A 2 for Cu Ka radiation. Writing this 
equation out for the two highest-angle lines, namely, 12 and 13, we obtain: 



. , 7 
0.806 = 0.595 ( r i + - 

0.879 = 0.595 {--= + -; 



Simultaneous solution of these two equations gives a = 2.66A, c = 4.95A, 
and c/a = 1.86. 

Rhombohedral crystals are also characterized by unit cells having two 
parameters, in this case a and a. No new chart is needed, however, to 
index the patterns of rhombohedral substances, since, as mentioned in 
Sec. 2-4, any rhombohedral crystal may be referred to hexagonal axes. A 
hexagonal Hull-Davey or Bunn chart may therefore be used to index the 
pattern of a rhombohedral crystal. The indices so found will, of course, 
refer to a hexagonal cell, and the method of converting them to rhombo- 
hedral indices is described in Appendix 2. 

We can conclude that the pattern of any two-parameter crystal (tetrag- 
onal, hexagonal, or rhombohedral) can be indexed on the appropriate Hull- 
Davey or Bunn chart. If the structure is known, the procedure is quite 
straightforward. The best method is to calculate the c/a ratio from the 



10-5] NONCUBIC CRYSTALS (ANALYTICAL METHODS) 311 

known parameters, lay a straightedge on the chart to discover the proper 
line sequence for this value of c/a, calculate the value of sin 2 6 for each line 
from the indices found on the chart, and then determine the indices of 
the observed lines by a comparison of calculated and observed sin 2 6 
values. 

If the structure is unknown, the problem of indexing is not always so 
easy as it seems in theory. The most common source of trouble is the 
presence of extraneous lines, as defined in Sec. 10-2, in the observed pat- 
tern. Such lines can be very confusing and, if any difficulty in indexing 
is encountered, every effort should be made to eliminate them from the 
pattern, either experimentally or by calculation. In addition, the ob- 
served sin 2 6 values usually contain systematic errors which make a simul- 
taneous fit of all the pencil marks on the paper strip to curves on the chart 
impossible, even when the paper strip is at the correct c/a position. Be- 
cause of these errors, the strip has to be shifted slightly from line to line 
in order to make successive pencil marks coincide with curves on the chart. 
Two important rules must always be kept in mind when using Hull-Davey 
or Bunn charts: 

(1) Every mark on the paper strip must coincide with a curve on the 
chart, except for extraneous lines. A structure which accounts for only a 
portion of the observed lines is not correct : all the lines in the pattern must 
be accounted for, either as due to the structure of the substance involved 
or as extraneous lines. 

(2) There need not be a mark on the paper strip for every curve on the 
chart, because some lines may have zero intensity or be too weak to be 
observed. 

Orthorhombic, monoclinic, and triclinic substances yield powder pat- 
terns which are almost impossible to index by graphical methods, although 
the patterns of some orthorhombic crystals have been indexed by a com- 
bination of graphical and analytical methods. The essential difficulty is 
the large number of variable parameters involved. In the orthorhombic 
system there are three such parameters (a, b, c), in the monoclinic four 
(a, b, c, 0), and in the triclinic six (a, b, c, a, 0, 7). If the structure is known, 
patterns of substances in these crystal systems can be indexed 6y com- 
parison of the observed sin 2 B values with those calculated for all possible 
values of hkl. 

10-5 Indexing patterns of noncubic crystals (analytical methods). 
Analytical methods of indexing involve arithmetical manipulation of the 
observed sin 2 6 values in an attempt to find certain relationships between 
them. Since each crystal system is characterized by particular relation- 
ships between sin 2 values, recognition of these relationships identifies 
the crystal system and leads to a solution of the line indices. 



312 THE DETERMINATION OF CRYSTAL STRUCTURE [CHAP. 10 

For example, the sin 2 6 values in the tetragonal system must obey the 
relation: 

sin 2 = A(h 2 + k 2 ) + Cl 2 , (10-7) 

where A ( = X 2 /4a 2 ) and C ( = X 2 /4c 2 ) are constants for any one pattern. 
The problem is to find these constants, since, once found, they will disclose 
the cell parameters a and c and enable the line indices to be calculated. 
The value of A is obtained from the hkO lines. When / = 0, Eq. (10-7) 
becomes 

sin 2 - A(h 2 + k 2 ). 

The permissible values of (h 2 + k 2 ) are 1, 2, 4, 5, 8, etc. Therefore the 
hkO lines must have sin 2 6 values in the ratio of these integers, and A will 
be some number which is 1, ^, f , ^, , etc., times the sin 2 6 values of these 
lines. C is obtained from the other lines on the pattern and the use of 
Eq. (10-7) in the form 

k 2 ) = Cl 2 . 



Differences represented by the left-hand side of the equation are set up, 
for various assumed values of h and k, in an attempt to find a consistent 
set of Cl 2 values, which must be in the ratio 1, 4, 9, 16, etc. Once these 
values are found, C can be calculated. 

For hexagonal crystals, an exactly similar procedure is used. In this 
case, sin 2 8 values are given by 



where A = X 2 /3a 2 and C = X 2 /4c 2 . Permissible values of (h 2 + hk + k 2 ) 
are tabulated in Appendix 6; they are 1, 3, 4, 7, 9, etc. The indexing pro- 
cedure is best illustrated by means of a specific example, namely, the pow- 
der pattern of zinc, whose observed sin 2 8 values are listed in Table 10-2. 
We first divide the sin 2 8 values by the integers 1, 3, 4, etc., and tabulate 
the results, as shown by Table 10-3, which applies to the first six lines of 
the pattern. We then examine these numbers, looking for quotients which 
are equal to one another or equal to one of the observed sin 2 8 values. In 

TABLE 10-3 



Line 


sin 2 9 


sin 2 9 


sin 2 


sin 2 Q 


hkl 


1 
2 
3 
4 
5 
6 


0.097 
0.112* 
0.136 
0.209 
0.332 
0.390 


0.032 
0.037 
0.045 
0.070, 
0.111 
0.130 


0.024 
0.028 
0.034 
0.052 
0.063 
0.098 


0.014 
0.016 
0.019 
0.030 
0.047 
0.056 


100 
110 



10-5] 



NONCUBIC CRYSTALS (ANALYTICAL METHODS^ 
TABLE 10-4 



313 



Line 


s!n 2 e 


sin 2 9-;l 


fin 2 9 -34 


hkl 


1 
2 
3 
4 
5 
6 


0.097* 
0.112 
0.136 
0.209 
0.332^ 
0.390 


0.000 
0.024 
0.097* 
0.221 
0.278 


0.054 


002 
100 
101 
102 
110, 103 
004 



this case, the two starred entries, 0.112 and 0.111, are the most nearly 
equal, so we assume that lines 2 and 5 are hkO lines. We then tentatively 
put A =0.112 which is equivalent to saying that line 2 is 100. Since the 
sin 2 6 value of line 5 is very nearly 3 times that of line 2, line 5 should be 
1 10. To find the value of C, we must use the equation 

sin 2 0- A(h 2 + hk + k 2 ) = Cl 2 . 

We now subtract from each sin 2 6 value the values of A (= 0.112), 
34 (= 0.336), 4A (= 0.448), etc., and look for remainders (Cl 2 ) which 
are in the ratio of 1, 4, 9, 16, etc. These figures are given in Table 10-4. 
Here the five starred entries are of interest, because these numbers (0.024, 
0.097, 0.221, and 0.390) are very nearly in the ratio 1, 4, 9, and 16. We 
therefore put 0.024 = C(l) 2 , 0.097 = C(2) 2 , 0.221 = C(3) 2 , and 0.390 = 
C(4) 2 . This gives C = 0.024 and immediately identifies line 1 as 002 and 
line 6 as 004. Since line 3 has a sin 2 value equal to the sum of A and C, 
its indices must be 101. Similarly, the indices of lines 4 and 5 are found 
to be 102 and 103, respectively. In this way, indices are assigned to all 
the lines on the pattern, and a final check on their correctness is made in 
the usual manner, by a comparison of observed and calculated sin 2 values. 
In the orthorhombic system, the basic equation governing the sin 2 
values is 

sin 2 6 = Ah 2 + Bk 2 + Cl 2 . 

The indexing problem is considerably more difficult here, in that three 
unknown constants, A, B, and C, have to be determined. The general 
procedure, which is too lengthy to illustrate here, is to search for signifi- 
cant differences between various pairs of sin 2 6 values. For example, con- 
sider any two lines having indices hkO and hkl, with hk the same for each, 
such as 120 and 121 ; the difference between their sin 2 values is C. Sim- 
ilarly, the difference between the sin 2 values of two lines such as 310 and 
312 is 4C, and so on. If the structure is such that there are many lines 
missing from the pattern, because of a zero structure factor for the corre- 
sponding planes, then the difficulties of indexing are considerably increased, 
inasmuch as the missing lines may be the very ones which would supply 



314 THE DETERMINATION OF CRYSTAL STRUCTURE [CHAP. 10 

the most easily recognized clues if they were present. Despite such diffi- 
culties, this analytical method has been applied successfully to a number 
of orthorhombic patterns. One requisite for its success is fairly high accu- 
racy in the sin 2 6 values (at least 0.0005), and the investigator should 
therefore correct his observations for systematic errors before attempting 
to index the pattern. 

Monoclinic and triclinic substances yield powder patterns of great com- 
plexity because the number of independent constants involved is now four 
and six, respectively. No generally successful method, either analytical 
or graphical, of indexing such patterns has yet been devised. 

We can therefore conclude that the powder pattern of a substance hav- 
ing more than two independently variable cell parameters is extremely 
difficult, if not impossible, to solve. The structures of such materials are 
almost always determined by the examination of a single crystal, by either 
the rotating-crystal method or one of its variations. With these methods 
it is a relatively easy matter to determine the shape and size of an un- 
known unit cell, no matter how low its symmetry. Many substances, of 
course, are very difficult to prepare in single-crystal form, but, on the 
other hand, if the substance involved is one of low symmetry, the time 
spent in trying to obtain a single crystal is usually more fruitful than the 
time spent in trying to solve the powder pattern. The single-crystal speci- 
men need not be large: a crystal as small as 0.1 mm in any dimension can 
be successfully handled and will give a satisfactory diffraction pattern. 
Readers interested in these single-crystal methods will find them described 
in some of the books listed in Chap. 18. 

10-6 The effect of cell distortion on the powder pattern. At this point 
we might digress slightly from the main subject of this chapter, and exam- 
ine some of the changes produced in a powder pattern when the unit cell 
of the substance involved is distorted in various ways. As we have already 
seen, there are many more lines on the pattern of a substance of low sym- 
metry, such as triclinic, than on the pattern of a substance of high sym- 
metry, such as cubic, and we may take it as a general rule that any distor- 
tion of the unit cell which decreases its symmetry, in the sense of intro- 
ducing additional variable parameters, will increase the number of lines on 
the powder pattern. 

Figure 10-5 graphically illustrates this point. On the left is the calcu- 
lated diffraction pattern of the body-centered cubic substance whose unit 
cell is shown at the top. The line positions are computed for a = 4.00A 
and Cr Ka radiation. If this cell is expanded or contracted uniformly but 
still remains cubic, the diffraction lines merely shift their positions but do 
not increase in number, since no change in cell symmetry is involved. 
However, if the cubic cell is distorted along only one axis, then it becomes 



10-6] THE EFFECT OF CELL DISTORTION ON THE POWDER PATTERN 315 



ft 


1 


/I 


/ / A 




7 . 


* 


i 
X4.00A 


<r <r <r | 

4.16A < - 4.16A 

i v * i 


I 


i 

x 

<' 4.00A 


[010] 


./ -/ X 432A ,/x )A 


UOOj/* 
CUBIC 
n 


TF 

101 


;TRAGONAL ORTHORHOMBIC 


200 
211 
220 

310 

222 







Oil 
110 




i 




110 
002 




r 101 

020 
i r\r\o 














200 

112 
211 

202 




^~ '200 

121 
112 

211 

022 
220 


















h~ 202^ 

03L 
130v\ 
013- 






220 

103 
301 








"103^ 
310- 




310 
222 




301 ' 
222 











26 

FIG. 10-5. Effects of cell distortion on powder patterns, 
position are connected by dashed lines. 



Lines unchanged in 



tetragonal, its symmetry decreases, and more diffraction lines are formed. 
The center pattern shows the effect of stretching the cubic cell by 4 percent 
along its [001] axis, so that c is now 4.16A. Some lines are unchanged in 
position, some are shifted, and new lines have appeared. If the tetragonal 



316 THE DETERMINATION OF CRYSTAL STRUCTURE [CHAP. 10 

cell is now stretched by 8 percent along its [010] axis, it becomes ortho- 
rhombic, with a = 4.00A, b = 4.32A, and c = 4.16A, as shown on the 
right. The result of this last distortion is to add still more lines to the pat- 
tern. The increase in the number of lines is due essentially to the intro- 
duction of new plane spacings, caused by nonuniform distortion. Thus, 
in the cubic cell, the (200), (020), and (002) planes all have the same spac- 
ing and only one line is formed, called the 200 line, but this line splits into 
two when the cell becomes tetragonal, since now the (002) plane spacing 
differs from the other two. When the cell becomes orthorhombic, all three 
spacings are different and three lines are formed. 

Changes of this nature are not uncommon among phase transformations 
and ordering reactions. For example, the powder pattern of slowly cooled 
plain carbon steel shows lines due to ferrite (body-centered cubic) and 
cementite (FeaC, orthorhombic). When the same steel is quenched from 
the austenite region, the phases present are martensite (body-centered 
tetragonal) and, possibly, some untransformed austenite (face-centered 
cubic). The a and c parameters of the martensite cell do not differ greatly 
from the a parameter of the ferrite cell (see Fig. 12-5). The result is that 
the diffraction pattern of a quenched steel shows pairs of martensite lines 
occurring at about the same 20 positions as the individual lines of ferrite 
in the previous pattern. If the quenched steel is now tempered, the mar- 
tensite will ultimately decompose into ferrite and cementite, and each pair 
of martensite lines will coalesce into a single ferrite line. Somewhat similar 
effects can be produced in a copper-gold alloy having the composition repre- 
sented by the formula AuCu. This alloy is cubic in the disordered state 
but becomes either tetragonal or orthorhombic when ordered, depending 
on the ordering temperature (see Sec. 13-3). 

The changes produced in a powder pattern by cell distortion depend, in 
degree, on the amount of distortion. If the latter is small, the pattern re- 
tains the main features of the pattern of the original undistorted cell. Thus, 
in Fig. 10-5, the nineteen lines of the orthorhombic pattern fall into the six 
bracketed groups shown, each group corresponding to one of the single 
lines on the cubic pattern. In fact, an experienced crystallographer, if 
confronted with this orthorhombic pattern, might recognize this grouping 
and guess that the unit cell of the substance involved was not far from 
cubic in shape, and that the Bravais lattice was either simple or body- 
centered, since the groups of lines are spaced in a fairly regular manner. 
But if the distortion of the cubic cell had been much larger, each line of 
the original pattern would split into such widely separated lines that no 
features of the original pattern would remain. 

10-7 Determination of the number of atoms in a unit cell. To return 
to the subject of structure determination, the next step after establishing 



10-8] DETERMINATION OF ATOM POSITIONS 317 

the shape and size of the unit cell is to find the number of atoms in that 
cell, because the number of atoms must be known before their positions 
can be determined. To find this number we use the fact that the volume 
of the unit cell, calculated from the lattice parameters by means of the 
equations given in Appendix 1, multiplied by the measured density of the 
substance equals the weight of all the atoms in the cell. From Eq. (3-9), 

we have 

V 

SA = 



1.66020 

where SA is the sum of the atomic weights of the atoms in the unit cell, 
p is the density (gm/cm 3 ), and V is the volume of the unit cell (A 3 ). If the 
substance is an element of atomic weight A , then 

SA = 

where HI is the number of atoms per unit cell. If the substance is a chem- 
ical compound, or an intermediate phase whose composition can be repre- 
sented by a simple chemical formula, then 

ZA = n 2 M, 

where n 2 is the number of "molecules" per unit cell and M the molecular 
weight. The number of atoms per cell can then be calculated from n 2 and 
the composition of the phase. 

When determined in this way, the number of atoms per cell is always an 
integer, within experimental error, except for a very few substances which 
have "defect structures." In these substances, atoms are simply missing 
from a certain fraction of those lattice sites which they would be expected 
to occupy, and the result is a nonintegral number of atoms per cell. FeO 
and the ft phase in the Ni-Al system are well-known examples. 

10-8 Determination of atom positions. We now have to find the posi- 
tions of a known number of atoms in a unit cell of known shape and size. 
To solve this problem, we must make use of the observed relative inten- 
sities of the diffracted beams, since these intensities are determined by 
atom positions. In finding the atom positions, however, we must again 
proceed by trial and error, because there is no known method of directly 
calculating atom positions from observed intensities. 

To see why this is so, we must consider the two basic equations involved, 
namely, 



318 THE DETERMINATION OF CRYSTAL STRUCTURE [CHAP. 10 

which gives the relative intensities of the reflected beams, and 

N 

F = ^f n e 2 l(hu *+ kv n+ lw n\ (4-11) 

1 

which gives the value of the structure factor F for the hkl reflection in terms 
of the atom positions uvw. Since the relative intensity 7, the multiplicity 
factor p, and the Bragg angle are known for each line on the pattern, we 
can find the value of \F\ for each reflection from Eq. (4-12). But \F\ meas- 
ures only the relative amplitude of each reflection, whereas in order to use 
Eq. (4-11) for calculating atom positions, we must know the value of F, 
which measures both the amplitude and phase of one reflection relative to 
another. This is the crux of the problem. The intensities of two reflected 
beams are proportional to the squares of their amplitudes but independent 
of their relative phase. Since all we can measure is intensity, we can de- 
termine amplitude but not phase, which means that we cannot compute 
the structure factor but only its absolute value. Any method of avoiding 
this basic difficulty would constitute the much-sought-after direct method 
of structure determination. This difficulty appears to be insurmountable, 
however, since no direct method, generally applicable to all structures, has 
yet been devised, despite the large amount of effort devoted to the problem. 
Atom positions, therefore, can be determined only by trial and error. 
A set of atom positions is assumed, the intensities corresponding to these 
positions are calculated, and the calculated intensities are compared with 
the observed ones, the process being repeated until satisfactory agreement 
is reached. The problem of selecting a structure for trial is not as hope- 
lessly broad as it sounds, since the investigator has many aids to guide 
him. Foremost among these is the accumulated knowledge of previously 
solved structures. From these known structures he may be able to select 
a few likely candidates, and then proceed on the assumption that his un- 
known structure is the same as, or very similar to, one of these known ones. 
A great many known structures may be classified into groups according to 
the kind of bonding (ionic, covalent, metallic, or mixtures of these) which 
holds the atoms together, and a selection among these groups is aided by a 
knowledge of the probable kind of atomic bonding in the unknown phase, 
as judged from the positions of its constituent elements in the periodic table. 
For example, suppose the phase of unknown structure has the chemical 
formula AB, where A is strongly electropositive and B strongly electro- 
negative, and that its powder pattern is characteristic of a simple cubic 
lattice. Then the bonding is likely to be ionic, and the CsCl structure is 
strongly suggested. But the FeSi structure shown in Fig. 2-19 is also a 
possibility. In this particular case, one or the other can be excluded by a 
density measurement, since the CsCl cell contains one "molecule" and the 
FeSi cell four. If this were not possible, diffracted intensities would have 



10-8J DETERMINATION OF ATOM POSITIONS 319 

to be calculated on the basis of each cell and compared with the observed 
ones. It is this simple kind of structure determination, illustrated by an 
example in the next section, which the metallurgist should be able to carry 
out unaided. 

Needless to say, many structures are too complex to be solved by this 
simple approach and the crystallographer must turn to more powerful 
methods. Chief among these are space-group theory and Fourier series. 
Although any complete description of these subjects is beyond the scope 
of this book, a few general remarks may serve to show their utility in struc- 
ture determination. The theory of space groups, one of the triumphs of 
mathematical crystallography, relates crystal symmetry, on the atomic 
scale, to the possible atomic arrangements which possess that symmetry. 
For example, if a given substance is known to be hexagonal and to have n 
atoms in its unit cell, then space-group theory lists all possible arrange- 
ments of n atoms which will have hexagonal symmetry. This listing of 
possible arrangements aids tremendously in the selection of trial struc- 
tures. A further reduction in the number of possibilities can then be made 
by noting the indices of the reflections absent from the diffraction pattern. 
By such means alone, i.e., before any detailed consideration is given to 
relative diffracted intensities, space-group theory can often exclude all but 
two or three possible atomic arrangements. 

A Fourier series is a type of infinite trigonometric series by which any 
kind of periodic function may be expressed. Now the one essential prop- 
erty of a crystal is that its atoms are arranged in space in a periodic fashion. 
But this means that the density of electrons is also a periodic function of 
position in the crystal, rising to a maximum at the point where an atom is 
located and dropping to a low value in the region between atoms. To re- 
gard a crystal in this manner, as a positional variation of electron density 
rather than as an arrangement of atoms, is particularly appropriate where 
diffraction is involved, in that x-rays are scattered by electrons and not 
by atoms as such. Since the electron density is a periodic function of posi- 
tion, a crystal may be described analytically by means of Fourier series. 
This method of description is very useful in structure determination be- 
cause it can be shown that the coefficients of the various terms in the series 
are related to the F values of the various x-ray reflections. But such a 
series is not of immediate use, since the structure factors are not usually 
known both in magnitude and phase. However, another kind of series has 
been devised whose coefficients are related to the experimentally observ- 
able |F values and which gives, not electron density, but information re- 
garding the various interatomic vectors in the unit cell. This information 
is frequently enough to determine the phase of the various structure fac- 
tors; then the first kind of series can be used to map out the actual electron 
density throughout the cell and thus disclose the atom positions. 



320 



THE DETERMINATION OP CRYSTAL STRUCTURE 



[CHAP. 10 



10-9 Example of structure determination. As a simple example, we 
will consider an intermediate phase which occurs in the cadmium-tellurium 
system. Chemical analysis of the specimen, which appeared essentially 
one phase under the microscope, showed it to contain 46.6 weight percent 
Cd and 53.4 weight percent Te. This is equivalent to 49.8 atomic percent 
Cd and can be represented by the formula CdTe. The specimen was re- 
duced to powder and a diffraction pattern obtained with a Debye-Scherrer 
camera and Cu Ka radiation. 

The observed values of sin 2 6 for the first 16 lines are listed in Table 10-5, 
together with the visually estimated relative line intensities. This pattern 
can be indexed on the basis of a cubic unit cell, and the indices of the ob- 
served lines are given in the table. The lattice parameter, calculated from 
the sin 2 6 value for the highest-angle line, is 6.46A. 

The density of the specimen, as determined by weighing a quantity of 
the powder in a pyknometer bottle, was 5.82 gm/cm 3 . We then find, from 
Eq. (3-9), that 

^ j (5.82) (6.46) 3 



1.66020 



948. 



Since the molecular weight of CdTe is 240.02, the number of "molecules" 
per unit cell is 948/240.02 = 3.94, or 4, within experimental error. 

At this point, we know that the unit cell of CdTe is cubic and that it 
contains 4 "molecules" of CdTe, i.e., 4 atoms of cadmium and 4 atoms of 
tellurium. We must now consider possible arrangements of these atoms 
in the unit cell. First we examine the indices listed in Table 10-5 for evi- 
dence of the Bravais lattice. Since die indices of the observed lines are all 

TABLE 10-5 



Line 


intensity 


sin 2 9 


hkl 


1 


s 


0.0462 


111 


2 


vs 


0.1198 


220 


3 


v$ 


0.1615 


311 


4 


vw 


0.1790 


222 


5 


m 


0.234 


400 


6 


m 


0.275 


331 


7 


s 


0.346 


422 


8 


m 


0.391 


511, 333 


9 


w 


0.461 


440 


10 


m 


0.504 


531 


11 


m 


0.575 


620 


12 


w 


0.616 


533 


13 


w 


0.688 


444 


14 


m 


0.729 


711, 551 


15 


vs 


0.799 


642 


16 


s 


0.840 


731, 553 



10-9] EXAMPLE OF STRUCTURE DETERMINATION 321 

unmixed, the Bravais lattice must be face-centered. (Not all possible sets 
of unmixed indices are present, however: 200, 420, 600, 442, 622, and 640 
are missing from the pattern. But these reflections may be too weak to be 
observed, and the fact that they are missing does not invalidate our con- 
clusion that the lattice is face-centered.) Now there are two common face- 
centered cubic structures of the AB type, i.e., containing two different 
atoms in equal proportions, and both contain four "molecules" per unit 
cell: these are the NaCl structure [Fig. 2-18(b)] and the zinc-blende form 
of ZnS [Fig. 2-19(b)]. Both of these are logical possibilities even though 
the bonding in NaCl is ionic and in ZnS covalent, since both kinds of bond- 
ing have been observed in telluride structures. 

The next step is to calculate relative diffracted intensities for each struc- 
ture and compare them with experiment, in order to determine whether 
or not one of these structures is the correct one. If CdTe has the NaCl 
structure, then its structure factor for unmixed indices [see Example (e) 
of Sec. 4-6] is given by 

F 2 = 16(/ cd + /Te) 2 , if (h + k + I) is even, 
F 2 = 16(/ cd - /Te) 2 , if (h + k + I) is odd. 

On the other hand, if the ZnS structure is correct, then the structure factor 
for unmixed indices (see Sec. 4-13) is given by 

\F\ 2 = 16(/cd 2 + /Te 2 ), if(h + k + l) is odd, 

\F\ 2 = 16(/ cd - /Te) 2 , if (h + k + I) is an odd multiple of 2, (10-9) 

\F\ 2 = 16(/cd + /Te) 2 , if (h + k + I) is an even multiple of 2. 

Even before making a detailed calculation of relative diffracted inten- 
sities by means of Eq. (4-12), we can almost rule out the NaCl structure 
as a possibility simply by inspection of Eqs. (10-8). The atomic numbers 
of cadmium and tellurium are 48 and 52, respectively, so the value of 
(fed + /Te) 2 is several hundred times greater than the value of (/cd /Te) 2 , 
for all values of sin 0/X. Then, if CdTe has the NaCl structure, the 111 
reflection should be very weak and the 200 reflection very strong. Actu- 
ally, 111 is strong and 200 is not observed. Further evidence that the 
NaCl structure is incorrect is given in the fourth column of Table 10-6, 
where the calculated intensities of the first eight possible lines are listed: 
there is no agreement whatever between these values and the observed in- 
tensities. 

On the other hand, if the ZnS structure is assumed, intensity calcula- 
tions lead to the values listed in the fifth column. The agreement between 
these values and the observed intensities is excellent, except for a few 
minor inconsistencies among the low-angle reflections, and these are due 
to neglect of the absorption factor. In particular, we note that the ZnS 



322 



THE DETERMINATION OF CRYSTAL STRUCTURE 
TABLE 10-6 



[CHAP. 10 



1 


2 


3 


4 


5 


1 


177 


Observed 


Calculate* 


d intensity 


Line 


hkl 


intensity 


NaCI structure 


ZnS structure 


1 

2 
3 

4 
5 
6 

7 


111 
200 
220 
311 

222 

400 
331 
420 

422 


s 
nil 
vs 
vs 

vw 
m 
m 
nil 

s 


0.05 
13.2 
10.0 - 
0.02 

3.5 
1.7 
0.01 
4.6 


12.4 
0.03 
^ 10.0 
6.2 

0.007 
1.7 
2.5 
0.01 

3 4 


8 


511, 333 


m 




1 8 


9 


440 


w 




1 i 


10 


531 


m 




2.0 




600, 442 


nil 




005 


11 


620 


m 




1 .8 


12 


533 


w 




9 




622 


nil 




004 


13 


444 


w 




6 


14 


711, 551 


m 




1 8 




640 


nil 




005 


15 


642 


vs 




4 


16 


731, 553 


s 




3 3 













(N.B. Calculated intensities have been adjusted so that the 220 
line has an intensity of 10.0 for both structures.) 

structure satisfactorily accounts for all the missing reflections (200, 420, 
etc.), since the calculated intensities of these reflections are all extremely 
low. We can therefore conclude that CdTe has the structure of the zinc- 
blende form of ZnS. 

After a given structure has been shown to be in accord with the diffrac- 
tion data, it is advisable to calculate the interatomic distances involved in 
that structure. This calculation not only is of interest in itself, but serves 
to disclose any gross errors that may have been made, since there is obvi- 
ously something wrong with a proposed structure if it brings certain atoms 
impossibly close together. In the present structure, the nearest neighbor 
to the Cd atom at is the Te atom at \ \. The Cd-Te interatomic 
distance is therefore \/3 a/4 = 2.80A. For comparison, we can calcu- 
late a " theoretical" Cd-Te interatomic distance simply by averaging the 
distances of closest approach in the pure elements. In doing this, we re- 
gard the atoms as rigid spheres in contact, and ignore the effects of coordi- 



PROBLEMS 323 

nation number and type of bonding on atom size. These distances of 
closest approach are 2.98A in pure cadmium and 2.87A in pure tellurium, 
the average being 2.93A. The observed Cd-Te interatomic distance is 
2.80A, or some 4.5 percent smaller than the calculated value; this differ- 
ence is not unreasonable and can be largely ascribed to the covalent bond- 
ing which characterizes this structure. In fact, it is a general rule that the 
A-B interatomic distance in an intermediate phase A x Bj, is always some- 
what smaller than the average distance of closest approach in pure A and 
pure B, because the mere existence of the phase shows that the attractive 
forces between unlike atoms is greater than that between like atoms. If 
this were not true, the phase would not form. 

PROBLEMS 

10-1. The powder pattern of aluminum, made with Cu Ka radiation, contains 
ten lines, whose sin 2 6 values are 0.1118, 0.1487, 0.294, 0.403, 0.439, 0.583, 0.691, 
0.727, 0.872, and 0.981 . Index these lines and calculate the lattice parameter. 

10-2. A pattern is made of a cubic substance with unfiltered chromium radia- 
tion. The observed sin 2 6 values and intensities are 0.265(m), 0.321(vs), 0.528(w), 
0.638(8) f 0.793(s), and 0.958(vs). Index these lines and state which are due to 
Ka and which to K0 radiation. Determine the Bravais lattice and lattice param- 
eter. Identify the substance by reference to Appendix 13. 

10-3. Construct a Huil-Davey chart, and accompanying sin 2 6 scale, for hex- 
agonal close-packed lattices. Use two-range semilog graph paper, 8j X 11 in. 
Cover a c/a range of 0.5 to 2.0, and plot only the curves 00-2, 10-0, 10-1, 10-2, 
and 11-0. 

10-4. Use the chart constructed in Prob. 10-3 to index the first five lines on the 
powder pattern of a-titanium. With Cu Ka radiation, these lines have the fol- 
lowing sin 2 B values: 0.091, 0.106, 0.117, 0.200, and 0.268. 

In each of the following problems the powder pattern of an element is represented by 
the observed &in 2 values of the first seven or eight lines on the pattern, made with 
Cu Ka radiation. In each case, index the lines, find the crystal system, Bravais lattice, 
and approximate lattice parameter (or parameters), and identify the ekment from the 
tabulation given in Appendix 18. 

10-5 10-6 10-7 10-8 

0.0806 0.0603 0.1202 0.0768 

0.0975 0.1610 0.238 0.0876 

0.1122 0.221 0.357 0.0913 

0.210 0.322 0.475 0.1645 

0.226 0.383 0.593 0.231 

0.274 0.484 0.711 0.274 

0.305 0.545 0.830 0.308 

0.321 0.645 0.319 



CHAPTER 11 
PRECISE PARAMETER MEASUREMENTS 

11-1 Introduction. Many applications of x-ray diffraction require pre- 
cise knowledge of the lattice parameter (or parameters) of the material 
under study. In the main, these applications involve solid solutions; since 
the lattice parameter of a solid solution varies with the concentration of 
the solute, the composition of a given solution can be determined from a 
measurement of its lattice parameter. Thermal expansion coefficients 
can also be determined, without a dilatometer, by measurements of lattice 
parameter as a function of temperature in a high-temperature camera. Or 
the stress in a material may be determined by measuring the expansion or 
contraction of its lattice as a result of that stress. Since, in general, a 
change in solute concentration (or temperature, or stress) produces only 
a small change in lattice parameter, rather precise parameter measure- 
ments must be made in order to measure these quantities with any accu- 
racy. In this chapter we shall consider the methods that are used to obtain 
high precision, leaving the various applications to be discussed at a later 
time. Cubic substances will be dealt with first, because they are the sim- 
plest, but our general conclusions will also be valid for noncubic materials, 
which will be discussed in detail later. 

The process of measuring a lattice parameter is a very indirect one, and 
is fortunately of such a nature that high precision is fairly easily obtainable. 
The parameter a of a cubic substance 
is directly proportional to the spacing 
d of any particular set of lattice 
planes. If we measure the Bragg 
angle 6 for this set of planes, we can 
use the Bragg law to determine d and, 
knowing d, we can calculates. But 
it is sin 0, not 0, which appears in the 
Bragg law. Precision in d, or a, there- 
fore depends on precision in sin 0, a 
derived quantity, and not on precision 
in 0, the measured quantity. This is 
fortunate because the value of sin0 
changes very slowly with in the 
neighborhood of 90, as inspection of 
Fig. 11-1 or a table of sines will show. 
For this reason, a very accurate value 

324 




20 



40 60 

6 (degrees) 



80 



FIG. 11-1. The variation of sin 
with 0. The error in sin caused by a 
given error in decreases as increases 
(A0 exaggerated). 



H_l] INTRODUCTION 326 

of sin can be obtained from a measurement of 6 which is itself not particu- 
larly precise, provided that 6 is near 90. At = 85, for example, a 1 
percent error in leads to an error in sin B of only 0.1 percent. Stated in 
another way, the angular position of a diffracted beam is much more sensi- 
tive to a given change in plane spacing when 6 is large than when it is small. 
We can obtain the same result directly by differentiating the Bragg law 
with respect to B. We obtain 

= ~ cot0A0. (H-1) 

d 
In the cubic system, 



a = d Vh 2 + k 2 + I 2 . 

Therefore 

Aa Arf , - rts 

_ = _ = - cot0A0. (11-2) 

a d 

Since cot 6 approaches zero as approaches 90, Aa/a, the fractional error 
in a caused by a given error in 0, also approaches zero as approaches 90, 
or as 20 approaches 180. The key to precision in parameter measurements 
therefore lies in the use of backward-reflected beams having 20 values as 
near to 180 as possible. 

Although the parameter error disappears as 20 approaches 180, we can- 
not observe a reflected beam at this angle. But since the values of a calcu- 
lated for the various lines on the pattern approach the true value more 
closely as 20 increases, we should be able to find the true value of a simply 
by plotting the measured values against 20 and extrapolating to 20 = 180. 
Unfortunately, this curve is not linear and the extrapolation of a nonlinear 
curve is not accurate. However, it may be shown that if the measured 
values of a are plotted against certain functions of 0, rather than against 
or 20 directly, the resulting curve is a straight line which may be extrapo- 
lated with confidence. The bulk of this chapter is devoted to showing how 
these functions can be derived and used. Because the exact form of the 
function depends on the kind of camera employed, we shall have to con- 
sider successively the various cameras that are normally used for parameter 
measurements. 

But first we might ask: what sort of precision is possible with such 
methods? Without any extrapolation or any particular attention to good 
experimental technique, simply by selection of the parameter calculated 
for the highest-angle line on the pattern, we can usually obtain an accuracy 
of 0.01A. Since the lattice parameters of most substances of metallurgical 
interest are in the neighborhood of 3 to 4A, this represents an accuracy of 
about 0.3 percent. With good experimental technique and the use of the 
proper extrapolation function, this accuracy can be increased to 0.001A, 



326 



PRECISE PARAMETER MEASUREMENTS 



[CHAP. 11 



or 0.03 percent, without much difficulty. Finally, about the best accuracy 
that can be expected is 0.0001A, or 0.003 percent, but this can be obtained 
only by the expenditure of considerable effort, both experimental and com- 
putational. 

In work of high precision it is imperative that the units in which the 
measured parameter is expressed, kX or A, be correctly stated. In order 
to avoid confusion on this point, the reader is advised to review the dis- 
cussion of these units given in Sec. 3-4. 

11-2 Debye-Scherrer cameras. The general approach in finding an 
extrapolation function is to consider the various effects which can lead to 
errors in the measured values of 6, and to find out how these t errors in 6 
vary with the angle 6 itself. For a Debye-Scherrer camera, the chief 
sources of error in 6 are the following: 

(1) Film shrinkage. 

(2) Incorrect camera radius. 

(3) Off-centering of specimen. 

(4) Absorption in specimen. 

Since only the back-reflection region is suitable for precise measurements, 
we shall consider these various errors in terms of the quantities S' and 0, 
defined in Fig. 11-2. S f is the distance on the film between two correspond- 
ing back-reflection lines; 2<f> is the supplement of 26, i.e., </> = 90 6. 
These quantities are related to the camera radius R by the equation 



S f 



4R 



(H-3) 



Shrinkage of the film, caused by processing and drying, causes an error 
AS' in the quantity S'. The camera radius may also be in error by an 
amount Afl. The effects of these two errors on the value of <t> may be found 
by writing Eq. (11-3) in logarithmic 
form: 

In <f> = In S' - In 4 - In R. 
Differentiation then gives 

A< AS' Aft 

= (11-4) 

4> S' R 

The error in <j> due to shrinkage and 
the radius error is therefore given by 

^AS' &R\ 

U. (11-5) 

R ' 




FIGURE 11-2 



11-2] 



DEBYE-SCHERRER CAMERAS 




(a) v ,,, 

FIG. 11-3. Effect of specimen displacement on line positions. 

The shrinkage error can be minimized by loading the film so that the inci- 
dent beam enters through a hole in the film, since corresponding back- 
reflection lines are then only a short distance apart on the film, and their 
separation S' is little affected by film shrinkage. The method of film load- 
ing shown in Fig. 6-5 (a) is not at all suitable for precise measurements. 
Instead, methods (b) or (c) of Fig. 6-5 should be used. Method (c), the 
unsymmetrical or Straumanis method of film loading, is particularly recom- 
mended since no knowledge of the camera radius is required. 

An off-center specimen also leads to an error in 0. Whatever the dis- 
placement of the specimen from the camera center, this displacement can 
always be broken up into two components, one (Ax) parallel to the incident 
beam and the other (Ay) at right angles to the incident beam. The effect 
of the parallel displacement is illustrated in Fig. 11-3 (a). Instead of being 
at the camera center C", the specimen is displaced a distance Ax to the 
point 0. The diffraction lines are registered at D and C instead of at A 
and B, the line positions for a properly centered specimen. The error in 
S' is then (AC + DB) = 2DB, which is approximately equal to 20 AT, or 

AS' 20N = 2Aaxsin 2<t>. (11-6) 

The effect of a specimen displacement at right angles to the incident beam 
[Fig. ll-3(b)] is to shift the lines from A to C and from B to D. When 
Ay is small, AC is very nearly equal to BD and so, to a good approximation, 
no error in S' is introduced by a right-angle displacement. 

The total error in S' due to specimen displacement in some direction in- 
clined to the incident beam is therefore given by Eq. (11-6). This error 
in S f causes an error in the computed value of . Inasmuch as we are con- 
sidering the various errors one at a time, we can now put the radius error 
A# equal to zero, so that Eq. (11-4) becomes 



* 



S' 



(H-7) 



328 PRECISE PARAMETER MEASUREMENTS [CHAP. 11 

which shows how an error in S' alone affects the value of <t>. By combining 
Eqs. (11-3), (ll-), and (11-7), we find that the error in <t> due to the fact 
that the specimen is off center is given by 

sin 2^>) Ax 

sin </> cos </>. (11-8) 



o 4/i0 it 



It should not be assumed that the centering error is removed when the 
specimen is so adjusted, relative to the rotating shaft of the camera, that 
no perceptible wobble can be detected when the shaft is rotated. This sort 
of adjustment is taken for granted in this discussion. The off-center error 
refers to the possibility that the axis of rotation of the shaft is not located 
at the center of the camera, due to improper construction of the camera. 

Absorption in the specimen also causes an error in <. This effect, often 
the largest single cause of error in parameter measurements, is unfortu- 
nately very difficult to calculate with any accuracy. But we have seen, in 
Fig. 4-18(b), that back-reflected rays come almost entirely from that side 
of the specimen which faces the collimator. Therefore, to a rough approx- 
imation, the effect of a centered, highly absorbing specimen is the same as 
that of a nonabsorbing specimen displaced from the camera center in the 
manner shown in Fig. 11 -3 (a). Consequently we can assume that the 
error in <t> due to absorption, A<fo , is included in the centering error given 
byEq. (11-8). 

Thus, the over-all error in </> due to film shrinkage, radius error, centering 
error, and absorption, is given by the sum of Eqs. (11-5) and (11-8): 

/AS' A#\ Ax 
A<te,/2,c,A = I ) <t> + sm </> cos </>. (1 1-9) 

\ O K / 1 

But 

= 90 0, A0 = A0, sin <t> = cos 0, and cos <f> = sin 0. 

Therefore Eq. (11-2) becomes 

Ad cos sin 

= : A0 = A</> 

d sin cos <t> 

and 

Ad sin^r/AS' A#\ Ax 1 

= ( )<H sin cos (11-10) 

d cos < L \ S' R I R J 



In the back-reflection region, < is small and may be replaced, in the second 
term of Eq. (11-10), by sin< cos<, since sin< <f> and cos</> 1, for 



11-2] DEBYE-SCHERRER CAMERAS 329 

small values of <t>. We then have 



Ad /AS' Aft Ax\ 

= ( h 1 sin 2 <t>. 

d \S' R R 



The bracketed terms are constant for any one film, so that 

= K sin 2 4 = K cos 2 6, (11-11) 

d 

where K is a constant. Accordingly, we have the important result that 
the fractional errors in d are directly proportional to cos 2 0, and therefore 
approach zero as cos 2 6 approaches zero or as 6 approaches 90. In the 
cubic system, 

= = tfcos 2 0. (11-12) 

d a 

Hence, for cubic substances, if the value of a computed for each line on the 
pattern is plotted against cos 2 6, a straight line should result, and a , the 
true value of a, can be found by extrapolating this line to cos 2 6 = 0. (Or, 
since sin 2 0=1 cos 2 0, the various values of a may be plotted against 
sin 2 0, and the line extrapolated to sin 2 0=1.) 

From the various approximations involved in the derivation of Eq. 
(1 1-12), it is clear that this equation is true only for large values of 6 (small 
values of #). Therefore, only lines having 6 values greater than about 60 
should be used in the extrapolation, and the more lines there are with 
greater than 80, the more precise is the value of a () obtained. To increase 
the number of lines in the back-reflection region, it is common practice to 
employ unfiltered radiation so that K/3 as well as Ka can be reflected. If 
the x-ray tube is demountable, special alloy targets can also be used to in- 
crease the number of lines; or two exposures can be made on the same film 
with different characteristic radiations. In any case, it must never be 
assumed that the process of extrapolation can automatically produce a 
precise value of a from careless measurements made on a film of poor 
quality. For high precision, the lines must be sharp and the Ka doublets 
well resolved at high angles, which means in turn that the individual par- 
ticles of the specimen must be strain-free and not too fine. The line posi- 
tions must be determined carefully and it is best to measure each one two 
or three times and average the results. In computing a for each line, the 
proper wavelength must be assigned to each component of the Ka doublet 
when that line is resolved and, when it is not resolved, the weighted mean 
wavelength should be used. 

To illustrate this extrapolation method, we shall consider a powder pat- 
tern of tungsten made in a Debye-Scherrer camera 5.73 cm in diameter 
with unfiltered copper radiation. The data for all lines having values 



330 



PRECISE PARAMETER MEASUREMENTS 
TABLE 11-1 



[CHAP. 11 



Line 


hkl 


Radiation 


9 


s!n 2 e 


a (A) 


6 


400 


Kp 


61.71 


0.7754 


3.162 


5 


321 


Ka 


65.91 


0.8334 


3.160 


4 


411,330 


KP 


69.05 


0.8722 


3.162 


r 












3 


400 


K 0l 


76.73 


0.9473 


3.166 


2 


400 


Ka 2 


77.48 


0.9530 


3.164 


1 


420 


Kp 


79.67 


0.9678 


3.164 



greater than 60 are given in Table 11-1. The drift in the computed a 
values is obvious: in general they increase with 6 and tend to approach the 
true value a at high angles. In Fig. 11-4, these values of a are plotted 
against sin 2 6, and ao is found by extrapolation to be 3.165A. 

Other functions of 0, besides sin 2 or cos 2 0, may be used as a basis for 
extrapolation. For example, if we replace sin < cos <f> in Eq. (11-10) by 
<t>, instead of replacing </> by sin 4> cos <, we obtain 

= K<t> tan 0. 
d 

Therefore, a plot of a against </> tan < will also be linear and will extrapolate 
to a at <t> tan = 0. In practice, there is not much difference between an 
extrapolation against < tan <f> and one against cos 2 (or sin 2 0), and either 
will give satisfactory results. If the various sources of error, particularly 
absorption, are analyzed more rigorously than we have done here, it can 
be shown that the relation ~- 



cos 2 d 



Arf 
T 



/cos 2 

K( 

\ sin0 



cos 2 6\ 



= 3.170 
tf i 3165 







1 



0.2 



0.3 



holds quite accurately down to very 
low values of 6 and not just at high 
angles. The value of ao can be found 
by plotting a against (cos 2 0/sin 6 + 
cos 2 0/6), which approaches zero as 6 



W 



|3 



3 155 




J.O 



0.9 



n <2 



08 



07 



FIG. 11-4. Extrapolation of meas- 
ured lattice parameters against sin 2 6 
(or cos 2 0). 



approaches 90. Although it is doubt- 
ful whether any advantage results 
from using (cos 2 0/sin 6 + cos 2 6/6) 
instead of cos 2 6 in the back-reflection region, the greater range of linearity 
of the former function is an advantage in certain cases. 

Noncubic crystals present additional difficulties, regardless of the par- 
ticular extrapolation function chosen. (In the following discussion, we 



11-2] DEBYE-SCHERRER CAMERAS 331 

shall confine our attention to hexagonal and tetragonal crystals, but the 
methods to be described can be generalized to apply to crystals of still 
lower symmetry.) The difficulty is simply this: the position of a line which 
has indices hkl is determined by two parameters, a and c, and it is impos- 
sible to calculate both of them from the observed sin 2 value of that line 
alone. One way of avoiding this difficulty is to ignore the hkl lines and 
divide the remainder into two groups, those with indices hkO and those 
with indices 001. A value of a is calculated for each hkO line and a value 
of c from each 001 line; two separate extrapolations are then made to find 
a and c . Since there are usually very few hkO and 001 lines in the back- 
reflection region, some low-angle lines have to be included, which means 
that the extrapolations must be made against (cos 2 0/sin + cos 2 0/0) 
and not against cos 2 0. And if there are no lines of the type hkO and 001 
with greater than 80, even the former function will not assure an accu- 
rate extrapolation. 

A better but more laborious method, and one which utilizes all the data, 
is that of successive approximations. In the tetragonal system, for exam- 
ple, the value of a for any line is given by 

I 2 

(11-13) 



x r 

a = ( 
2sm0L 



The first step is to calculate approximate values, a\, and Ci, of the lattice 
parameters from the positions of the two highest-angle lines, as was done 
in Sec. 10-4. The approximate axial ratio Ci/a\ is then calculated and 
used in Eq. (11-13) to determine an a value for each high-angle line on the 
pattern. These values of a are then extrapolated against cos 2 to find a 
more accurate value of a, namely a 2 . The value of c 2 is found in similar 
fashion by use of the relation 



2sm0 

and another extrapolation against cos 2 6. The process is repeated with 
the new value of the axial ratio c 2 /a 2 to yield still more accurate values of 
the parameters, namely c 3 and a 3 . Three extrapolations are usually suffi- 
cient to fix the parameters with high accuracy. In addition, the accuracy 
of each extrapolation can be improved by a suitable choice of lines. For 
example, the value of a calculated from Eq. (11-13) is only slightly affected 
by inaccuracies in c/a when (h 2 + k 2 ) is large compared to Z 2 , since the 
term involving c/a is itself small. Therefore, lines with large h and k in- 
dices and a small I index should be chosen for each determination of a. 
Just the reverse is true in the determination of c, as inspection of Eq. 
(11-14) will show. 



332 



PRECISE PARAMETER MEASUREMENTS 



[CHAP. 11 







cos 2 e (] cos 2 e 

(a) (b) 

FIG. 11-5. Extreme forms of extrapolation curves (schematic): (a) large sys- 
tematic errors, small random errors; (b) small systematic errors, large random 
errors. 

To conclude this section, a few general remarks on the nature of errors 
may not be amiss. In the measurement of a lattice parameter, as in many 
other physical observations, two kinds of error are involved, systematic^ 
and random. A systematic error is one which varies in a regular manner 
with some particular parameter. Thus the fractional errors in a due to 
the various effects considered above (film shrinkage, incorrect radius, off- 
center specimen, absorption) are all systematic errors because they vary in 
a regular way with B, decreasing as B increases. Further, a systematic 
error is always of the same sign: for example, the effect of absorption in a 
Debye-Scherrer camera is always to make the computed value of a less than 
the true value. Random errors, on the other hand, are the ordinary chance 
errors involved in any direct observation. For example, the errors involved 
in measuring the positions of the various lines on a film arc random errors; 
they may be positive or negative and do not vary in any regular manner 
with the position of the line on the film. 

As we have already seen, the systematic errors in a approach zero as B 
approaches 90, and may be eliminated by use of the proper extrapolation 
function. The magnitude of these errors is proportional to the slope of 
the extrapolation line and, if these errors are small, the line will be quite 
flat. In fact, if we purposely increase the systematic errors, say, by using 
a slightly incorrect value of the camera radius in our calculations, the 
slope of the line will increase but the extrapolated value of a will remain 
the same. The random errors involved in measuring line positions show 
up as random errors in a, and are responsible for the deviation of the var- 
ious points from the extrapolation line. The random errors in a also de- 
crease in magnitude as B increases, due essentially to the slow variation of 
sin with at large angles. 

These various effects are summarized graphically in Fig. 1 1-5. In (a) 
the calculated points conform quite closely to the line, indicating small 
random errors, but the line itself is quite steep because of large systematic 



11-4] PINHOLE CAMERAS 333 

errors. The opposite situation is shown in (b) : here the systematic error 
is small, but the wide scatter of the points shows that large random errors 
have been made. Inasmuch as the difficulty of drawing the line increases 
with the degree of scatter, it is obvious that every possible effort should 
be made to minimize random errors at the start. 

11-3 Back-reflection focusing cameras. A camera of this kind is pre- 
ferred for work of the highest precision, since the position of a diffraction 
line on the film is twice as sensitive to small changes in plane spacing with 
this camera as it is with a Debye-Scherrer camera of the same diameter. 
It is, of course, not free from sources of systematic error. The most im- 
portant of these are the following : 

(1) Film shrinkage. 

(2) Incorrect camera radius. 

(3) Displacement of specimen from camera circumference. 

(4) Absorption in specimen. (If the specimen has very low absorption, 
many of the diffracted rays will originate at points outside the camera cir- 
cumference even though the specimen surface coincides with the circum- 
ference.) 

A detailed analysis of these various sources of error shows that they pro- 
duce fractional errors in d which are very closely proportional to < tan 0, 
where </> is again equal to (90 8). This function is therefore the one to 
use in extrapolating lattice parameters measured with this camera. 

11-4 Pinhole cameras. The pinhole camera, used in back reflection, 
is not really an instrument of high precision in the measurement of lattice 
parameters, but it is mentioned here because of its very great utility in met- 
allurgical work. Since both the film and the specimen surface are flat, no 
focusing of the diffracted rays occurs, and the result is that the diffraction 
lines are much broader than is normally desirable for precise measurement 
of their positions. The chief sources of systematic error are the following: 

(1) Film shrinkage. 

(2) Incorrect specimen-to-film distance. 

(3) Absorption in the specimen. 

In this case it may be shown that the fractional error in d is proportional 
to sin 40 tan <, or to the equivalent expression cos 2 8(2 cos 2 6 1), where 
= (90 6). With either of these extrapolation functions a fairly pre- 
cise value of the lattice parameter can be obtained ; in addition, the back- 
reflection pinhole camera has the particular advantage that mounted metal- 
lographic specimens may be examined directly. This means that a param- 
eter determination can be made on the same part of a specimen as that ex- 
amined under the microscope. A dual examination of this kind is quite val- 
uable in many problems, especially in the determination of phase diagrams. 



334 PRECISE PARAMETER MEASUREMENTS [CHAP. 11 

11-6 Diffractometers. The commercial diffractometer is a rather new 
instrument and relatively little use has been made of it for the precise 
measurement of lattice parameters. For that reason, no generally valid 
procedure for use in such measurements has yet been devised, and until 
this is done the back-reflection focusing camera must be recognized as the 
most accurate instrument for parameter measurements. 

One reason for the inferiority of the diffractometer in this respect is the 
impossibility of observing the same back-reflected cone of radiation on both 
sides of the incident beam. Thus, the experimenter has no automatic 
check on the accuracy of the angular scale of the instrument or the pre- 
cision of its alignment. 

When a diffractometer is used to measure plane spacings, the more im- 
portant sources of systematic error in d are the following : 

(1) Misalignment of the instrument. In particular, the center of the 
incident beam must intersect the diffractometer axis and the position 
of the receiving slit. 

(2) Use of a flat specimen instead of a specimen curved to conform to 
the focusing circle. 

(3) Absorption in the specimen. 

(4) Displacement of the specimen from the diffractometer axis. (This 
is usually the largest single source of error.) 

(5) Vertical divergence of the incident beam. 

These sources of error cause the fractional error in d to vary in a compli- 
cated way with 0, so that no simple extrapolation function can be used to 
obtain high accuracy. Because some, but not all, of these sources of error 
cause Ad/d to be approximately proportional to cos 2 0, a fairly accurate 
value of the lattice parameter can be obtained by simple extrapolation 
against cos 2 0, just as with the Debye-Scherrer camera. Therefore, in the 
light of our present knowledge, the suggested procedure is: 

(a) Carefully align the component parts of the instrument in accordance 
with the manufacturer's instructions. 

(b) Adjust the specimen surface to coincide as closely as possible with 
the diffractometer axis. 

(c) Extrapolate the calculated parameters against cos 2 8. 

This procedure will undoubtedly be improved as additional experience 
with this instrument is accumulated. In fact, some investigators feel that 
lattice parameters will one day be measurable with the diffractometer with 
greater accuracy than with any kind of powder camera, but whether this 
is true or not remains to be seen. There is, however, one circumstance in 
which the diffractometer is superior to a camera for parameter measure- 
ments and that is wheij the diffraction lines are abnormally broad; this 
particular application arises in stress measurement and will be described 
in Chap. 17. 



11-6] METHOD OF LEAST SQUARES 335 

11-6 Method of least squares. All the previously described methods of 
accurately measuring lattice parameters depend in part on graphical ex- 
trapolation. Their accuracy therefore depends on the accuracy with which 
a straight line can be drawn through a set of experimental points, each of 
which is subject to random errors. However, different persons will in gen- 
eral draw slightly different lines through the same set of points, so that it is 
desirable to have an objective, analytical method of finding the line which 
best fits the data. This can be done by the method of least squares. Since 
this method can be used in a variety of problems, it will be described here 
in a quite general way; in the next section, its application to parameter 
measurements will be taken up in detail. 

If a number of measurements are made of the same physical quantity 
and if these measurements are subject only to random errors, then the 
theory of least squares states that the most probable value of the measured 
quantity is that which makes the sum of the squares of the errors a mini- 
mum. The proof of this theorem is too long to reproduce here but we can 
at least demonstrate its reasonableness by the following simple example. 
Suppose five separate measurements are made of the same physical quan- 
tity, say the time required for a falling body to drop a given distance, and 
that these measurements yield the following values: 1.70, 1.78, 1.74, 1.79, 
and 1.74 sec. Let x equal the most probable value of the time. Then the 
error in the first measurement is ei = (x - 1.70), the error in the second 
is e 2 = (x - 1.78), and so on. The sum of the squares of the errors is given 
by 

Z(e 2 ) = (x - 1.70) 2 + (x - 1.78) 2 

+ (x - 1.74) 2 + (x - 1.79) 2 + (x - 1.74) 2 . 

We can minimize the sum of the squared errors by differentiating this 
expression with respect to x and equating the result to zero: 

^-t = 2(x - 1.70) + 2(x - 1.78) ,+ 2(x - 1.74) + 2(x - 1.79) 
dx 

+ 2(x - 1.74) - 

whence 

x = 1.75 sec. 

On the other hand, the arithmetic average of the measurements is also 
1.75 sec. This should not surprise us as we know, almost intuitively, that 
the arithmetic average of a set of measurements gives the most probable 
value. This example may appear trivial, in that no one would take the 
trouble to use the method of least squares when the same result can be 
obtained by simple averaging, but at least it illustrates the basic principle 
involved in the least-squares method. 



336 PRECISE PARAMETER MEASUREMENTS [CHAP. 11 

Naturally, there are many problems in which the method of simple 
averaging cannot be applied and then the method of least squares becomes 
particularly valuable. Consider, for example, the problem referred to 
above, that of finding the straight line which best fits a set of experimen- 
tally determined points. If there are only two points, there is no problem, 
because the two constants which define a straight line can be unequivocally 
determined from these two points. But, in general, there will be more 
points available than constants to be determined. Suppose that the vari- 
ous points have coordinates x\y\, X 2 y 2 , #32/3, and that it is known that 
x and y are related by an equation of the form 

y - a + bx. (11-15) 

Our problem is to find the values of the constants a and 6, since these de- 
fine the straight line. In general, the line will not pass exactly through 
any of the points since each is subject to a random error. Therefore each 
point is in error by an amount given by its deviation from the straight line. 
For example, Eq. (11-15) states that the value of y corresponding to x = x\ 
is (a + tei). Yet the first experimental point has a value of y = y\. 
Therefore e^ the error in the first point, is given by 

ei = (a + 6x0 - yi. 

We can calculate the errors in the other points in similar fashion, and then 
write down the expression for the sum of the squares of these errors : 

2(e 2 ) = (a + bx l - yi) 2 + (a + bx 2 - y 2 ) 2 + . (1 1-16) 

According to the theory of least squares, the "best" straight line is that 
which makes the sum of the squared errors a minimum. Therefore, the 
best value of a is found by differentiating Eq. (11-16) with respect to a 
and equating the result to zero: 

= 2(a + bx l - yi ) + 2(a + bx 2 - y 2 ) + - - - = 0, 
da 

or Sa + fcSz - Zy = 0. (11-17) 

The best value of b is found in a similar way: 

= 2xi(a + bx l - yi) + 2x 2 (a + 6a* - 2 ) + = 0> 
d& 

or + &Ss 2 - 2x = 0. (11-18) 



Equations (11-17) and (11-18) are the normal equations. Simultaneous 
solution of these two equations yields the best values of a and 6, which 
can then be substituted into Eq. (11-15) to give the equation of the line. 



11-6] METHOD OF LEAST SQUARES 337 

The normal equations as written above can be rearranged as follows: 

Zt/ = Sa + 62x 



and 



(11-19) 



A comparison of these equations and Eq. (11-15) shows that the following 
rules can be laid down for the formation of the normal equations : 

(a) Substitute the experimental values of x and y into Eq. (11-15). If 
there are n experimental points, n equations in a and b will result. 

(b) To obtain the first normal equation, multiply each of these n equa- 
tions by the coefficient of a in each equation, and add. 

(c) To obtain the second normal equation, multiply each equation by 
the coefficient of b, and add. 

As an illustration, suppose that we determine the best straight line 
through the following four points : 



X 


10 


18 


30 


42 


y 


15 


11 


11 


8 



The normal equations are obtained in three steps : 

(a) Substitution of the given values: 

15 = a + 106 
11 = a + 186 
11 = a + 306 

8 = a + 426 

(b) Multiplication by the coefficient of a: 



15 = 
11 = 
11 = 

8 = 



106 
186 
306 
426 



45 = 4a + 1006 (first normal equation) 
(c) Multiplication by the coefficient of 6 : 

150 = 10a + 1006 

198 = 18a + 3246 

330 = 30a + 9006 

336 = 42a + 17646 



1014 = lOOa + 30886 (second normal equation) 



338 



PRECISE PARAMETER MEASUREMENTS 



[CHAP. 11 



Simultaneous solution of the two 
normal equations gives a = 16.0 and 
6 = -0.189. The required straight 
line is therefore 

y = 16.0 - 0.189*. 

This line is shown in Fig. 11-6, to- 
gether with the four given points. 

The least-squares method is not 
confined to finding the constants of a 
straight line; it can be applied to any 
kind of curve. Suppose, for example, 
that x and y are known to be related 
by a parabolic equation 

y = a + bx + ex 2 . 



20 



15 



10 



10 



20 



30 



40 



50 



FIG. 11-6. Best straight line, de- 
termined by least-squares method. 



Since there are three unknown constants here, we need three normal equa- 
tions. These are 

Si/ = Sa + b2x + cSx 2 , 

(11-20) 



2x 2 y - aZz 2 + blx* + cSx 4 , 

These normal equations can be found by the same methods as were used 
for the straight-line case, i.e., successive multiplication of the n observa- 
tional equations by the coefficients of a, 6, and c, followed by addition of 
the equations in each set. 

It should be noted that the least-squares method is not a way of finding 
the best curve to fit a given set of observations. The investigator must 
know at the outset, from his understanding of the phenomenon involved, 
the kind of relation (linear, parabolic, exponential, etc.) the two quantities 
x and y are supposed to obey. All the least-squares method can do is give 
him the best values of the constants in the equation he selects, but it does 
this in a quite objective and unbiased manner. 

11-7 Cohen's method. In preceding sections we have seen that the 
most accurate value of the lattice parameter of a cubic substance is found 
by plotting the value of a calculated for each reflection against a particular 
function, which depends on the kind of camera used, and extrapolating to 
a value a at 6 = 90. Two different things are accomplished by this pro- 
cedure: (a) systematic errors are eliminated by selection of the proper 
extrapolation function, and (b) random errors are reduced in proportion 
to the skill of the investigator in drawing the best straight line through the 



11-7] COHEN'S METHOD 339 

experimental points. M. U. Cohen proposed, in effect, that the least-squares 
method be used to find the best straight line so that the random errors 
would be minimized in a reproducible and objective manner. 

Suppose a cubic substance is being examined in a Debye-Scherrer camera. 
Then Eq. (11-12), namely, 

Ad Aa 

= = #cos 2 0, (11-12) 

d a 

defines the extrapolation function. But instead of using the least-squares 
method to find the best straight line on a plot of a against cos 2 0, Cohen 
applied the method to the observed sin 2 6 values directly. By squaring 
the Bragg law and taking logarithms of each side, we obtain 

(X 2 \ 
- J - 2 In d. 

Differentiation then gives 

A sin 2 6 2Ad 



sm d 

By substituting this into Eq. (11-12) we find how the error in sin 2 6 varies 
with 6: 

A sin 2 6 = -2K sin 2 6 cos 2 6 = D sin 2 26, (11-22) 

where D is a new constant. [This equation is valid only when the cos 2 
extrapolation function is valid. If some other extrapolation function is 
used, Eq. (11-22) must be modified accordingly.] Now the true value of 
sin 2 6 for any diffraction line is given by 

X 2 

sin 2 9 (true) = - (h 2 + k 2 + I 2 ), 
4a 2 

where a , the true value of the lattice parameter, is the quantity we are 
seeking. But 

sin 2 6 (observed) sin 2 6 (true) = A sin 2 6, 

X 2 

sin 2 e -- - (h 2 + fc 2 + I 2 ) = D sin 2 20, 
4oo 2 

sin 2 = Ca + Ad, (11-23) 

where 

C = X 2 /4a 2 , a = (ft 2 + k 2 + I 2 ), A = D/10, and 6 = 10 sin 2 20. 

(The factor 10 is introduced into the definitions of the quantities A and d 
solely to make the coefficients of the various terms in the normal equations 
of the same order of magnitude.) 



340 PRECISE PARAMETER MEASUREMENTS [CHAP. 11 

The experimental values of sin 2 0, a, and d are now substituted into 
Eq. (11-23) for each of the n back-reflection lines used in the determina- 
tion. This gives n equations in the unknown constants C and A, and these 
equations can be solved for the most probable values of C and A by the 
method of least squares. Once C is found, OQ can be calculated directly 
from the relation given above; the constant A is related to the amount of 
systematic error involved and is constant for any one film, but varies 
slightly from one film to another. The two normal equations we need to 
find C and A are found from Eq. (11-23) and the rules previously given. 
They are 

Sasin 2 = 



26 sin 2 6 = C2a5 + A28 2 . 

To illustrate the way in which such calculations are carried out, we will 
apply Cohen's method to a determination of the lattice parameter of tung- 
sten from measurements made on the pattern shown in Fig. 6-10. Since 
this pattern was made with a symmetrical back-reflection focusing camera, 
the correct extrapolation function is 

Ad 

= K<t> tan <t>. 
d 

Substituting this into Eq. (11-21), we have 

A sin 2 = -2K<t> sin 2 6 tan 
= 2K0cos 2 ^ tan 
= D<t> sin 20, 

where D is a new constant. We can therefore write, for each line on the 
pattern, 

X 2 

sin 2 B = cos 2 -- - (h? + k 2 + I 2 ) + D<t> sin 20, (11-24) 
4a 2 

C0 s 2 = Ca + A5, (11-25) 

where 

C = X 2 /4a 2 , a = (h 2 + k 2 + I 2 ), A = D/10, and 8 = 100 sin 20. 

Equation 11-24 cannot be applied directly because lines due to three 
different wavelengths (Cu Kai, Cu Ka%, and Cu K/3) are present on the 
pattern, which means that X varies from line to line, whereas in Eq. (11-24) 
it is treated as a constant. But the data can be "normalized" to any one 
wavelength by use of the proper multiplying factor. For example, sup- 
pose we decide to normalize all lines to the Kfi wavelength. Then for a 



11-7] 



COHEN'S METHOD 
TABLE 11-2 



341 



Line 


hkl 


X 


a 


* 


Observed 


Normalized to K p 


cos 


6 


cos <(> 


d 


1 


321 


Ka, 


14 


24.518 


0.82779 


3.2 


0.67606 


2.6 


2 


321 


Ka 9 


14 


24.193 


0.83205 


3.2 


0.67617 


2.6 


3 - 


411,330 


Kp 


18 


21.167 


0.86961 


2.5 


0.86961 


2.5 


4 


400 


Ka, 


16 


13.302 


0.94706 


1.0 


0.77346 


0.8 


5 


400 


Ka 2 


16 


12.667 


0.95191 


1.0 


0.77357 


0.8 


6 


420 


Kp 


20 


10.454 


0.96708 


0.7 


0.96708 


0.7 



particular line formed by Kai radiation, for instance, we have 

COS 2 <t>Kai = Ot + A8xai, 

/ X A , 
+ ( 2 ) AS Kai . 
VA/JCai / 



\X - 2 / 

From the Bragg law, 



cos J 4>A- ai = 



COS 2 <t> Ka , = COS 2 <t>Ktl, 



cos 



where (\K0 2 /^K ai 2 )f>Ka } is a normalized 5. Equation (11-26) now refers 
only to the K/3 wavelength. Lines due to Ka^ radiation can be normalized 
in a similar manner. When this has been done for all lines, the quantity 
C in Eq. (11-25) is then a true constant, equal to XA r j3 2 /4a 2 . The values 
of the two normalizing factors, for copper radiation, are 



= 0.816699 and 



= 0.812651. 



Table 11-2 shows the observed and normalized values of cos 2 <t> and 6 
for each line on the tungsten pattern. The values of 6 need not be calcu- 
lated to more than two significant figures, since 6 occurs in Eq. (11-25) 
only in the last term which is very small compared to the other two. From 
the data in Table 11-2, we obtain 



Sa 2 = 1628, 25 2 = 21.6, 2a5 = 157.4, 
Sa cos 2 <t> = 78.6783, 25 cos 2 <f> = 7.6044. 



342 PRECISE PARAMETER MEASUREMENTS [CHAP. 11 

The normal equations are 

78.6783 = 1628C + 157.4A, 

7.6044 = 157.4C + 21. 6A. 
Solving these, we find 

C = X*0 2 /4a 2 = 0.0483654 and a = 3. 1651 A, 
A = -0.000384. 

The constant A, called the drift constant, is a measure of the total sys- 
tematic error involved in the determination. 

Cohen's method of determining lattice parameters is even more valuable 
when applied to noncubic substances, since, as we saw in Sec. 1 1-2, straight- 
forward graphical extrapolation cannot be used when there is more than one 
lattice parameter involved. Cohen's method, however, provides a direct 
means of determining these parameters, although the equations are natu- 
rally more complex than those needed for cubic substances. For example, 
suppose that the substance involved is hexagonal. Then 

X 2 4 h 2 + hk + k 2 \ 2 I 2 
sin 2 6 (true) = - + - ^ 

and 

X 2 X 2 

sin 2 6 (h 2 + hk + k 2 ) (I 2 ) = D sin 2 26, 

3a 2 4c 2 

if the pattern is made in a Debye-Scherrer camera. By rearranging this 
equation and introducing new symbols, we obtain 

sin 2 6 = Ca + By + ,46, (11-27) 

where 

C = X 2 /3a 2 , a = (h 2 + hk + /c 2 ), B = X 2 /4c 2 , 7 = I 2 , 

A = D/10, and 6 = 10 sin 2 26. 

The values of C, #, and A, of which only the first two are really needed, 
are found from the three normal equations: 

Za sin 2 6 = CZa 2 + B2ay + AZat, 
S 7 sin 2 6 = CZay + BZy 2 + AZyd, 
S6 sin 2 6 = CSaS + fiZfry + A28 2 . 

11-8 Calibration method. One other procedure for obtaining accurate 
lattice parameters is worth mentioning, if only for its relative simplicity, 
and that is the calibration method already alluded to in Sec. 6-7. It is 
based on a calibration of the camera film (or diffractometer angular scale) 
by means of a substance of known lattice parameter. 



PROBLEMS 343 

If the specimen whose parameter is to be determined is in the form of a 
powder, it is simply mixed with the powdered standard substance and a 
pattern made of the composite powder. If the specimen is a polycrystal- 
line piece of metal, the standard powder may be mixed with petroleum 
jelly and smeared over the surface of the specimen in a thin film. The 
amount of the standard substance used should be adjusted so that the in- 
tensities of the diffraction lines from the standard and those from the speci- 
men are not too unequal. Inasmuch as the true angle can be calculated 
for any diffraction line from the standard substance, a calibration curve 
can be prepared relating the true angle 6 to distance along the camera film 
(or angular position on the diffractometer scale). This curve is then used 
to find the true angle 6 for any diffraction line from the specimen, since 
it may be assumed that any systematic errors involved in the determina- 
tion will affect the diffraction lines of both substances in the same way. 

This method works best when there is a diffraction line from the stand- 
ard substance very close to a line from the specimen and both lines are in 
the back-reflection region. Practically all systematic errors are thus elim- 
inated. To achieve this condition requires an intelligent choice of the 
standard substance and/or the incident wavelength. The most popular 
standard substances are probably quartz and sodium chloride, although 
pure metals such as gold and silver are also useful. 

One disadvantage of the calibration method is that the accuracy of the 
parameter determination depends on the accuracy with which the param- 
eter of the standard substance is known. If the absolute value of the 
parameter of the standard is known, then the calibration method gives the 
absolute value of the parameter of the specimen quite accurately. If not, 
then only a relative value of the parameter of the specimen can be ob- 
tained, but it is an accurate relative value. And frequently this is no dis- 
advantage at all, since we are often interested only in the differences in the 
parameters of a number of specimens and not in the absolute values of 
these parameters. 

If absolute values are required, the only safe procedure is to measure the 
absolute value of the parameter of the standard substance by one of the 
methods described in the preceding sections. It should not be assumed 
that a particular sample of quartz, for example, has the exact lattice param- 
eters tabulated under "quartz" in some reference book, because this par- 
ticular sample may contain enough impurities in solid solution to make 
its lattice parameters differ appreciably from the tabulated values. 

PROBLEMS 

11-1. The lattice parameter of copper is to be determined to an accuracy of 
dbO.OOOlA at 20C. Within what limits must the temperature of the specimen 
be controlled if errors due to thermal expansion are to be avoided? The linear 
coefficient of thermal expansion of copper is 16.6 X 10~ 6 in./in./C. 



344 PRECISE PARAMETER MEASUREMENTS [CHAP. 11 

11-2. The following data were obtained from a Debye-Scherrer pattern of a 
simple cubic substance, made with copper radiation. The given sin 2 6 values are 
for the KOLI lines only. 

h* + A: 2 + P sin 2 



38 0.9114 

40 0.9563 

41 0.9761 

42 0.9980 

Determine the lattice parameter a, accurate to four significant figures, by graphi- 
cal extrapolation of a against cos 2 6. 

11-3. From the data given in Prob. 11-2, determine the lattice parameter to 
four significant figures by Cohen's method. 

11-4. From the data given in Table 11-2, determine the lattice parameter of 
tungsten to five significant figures by graphical extrapolation of a against <j> tan <t>. 

11-5. If the fractional error in the plane spacing d is accurately proportional to 
the function (cos 2 0/sin 6 + cos 2 6/6) over the whole range of 0, show that a plot 
of A sin 2 6 against sin 2 6 has a maximum, as illustrated for a particular case by 
Fig. 10-1. At approximately what value of 6 does the maximum occur? 



CHAPTER 12 
PHASE-DIAGRAM DETERMINATION 

12-1 Introduction. An alloy is a combination of two or more metals, 
or of metals and nonmetals. It may consist of a single phase or of a mix- 
ture of phases, and these phases may be of different types, depending only 
on the composition of the alloy and the temperature,* provided the alloy 
is at equilibrium. The changes in the constitution of the alloy produced 
by given changes in composition or temperature may be convenieptly shown 
by means of a phase diagram, also called an equilibrium diagram or consti- 
tution diagram. It is a plot of temperature vs. composition, divided into 
areas wherein a particular phase or mixture of phases is stable. As such it 
forms a sort of map of the alloy system involved. Phase diagrams are 
therefore of great importance in metallurgy, and much time and effort have 
been devoted to their determination. In this chapter we will consider how 
x-ray methods can be used in the study of phase diagrams, particularly of 
binary systems. Ternary systems will be discussed separately in Sec. 12-6. 

X-ray methods are, of course, not the only ones which can be used in 
investigations of this kind. The two classical methods are thermal analysis 
and microscopic examination, and many diagrams have been determined 
by these means alone. X-ray diffraction, however, supplements these older 
techniques in many useful ways and provides, in addition, the only means 
of determining the crystal structures of the various phases involved. Most 
phase diagrams today are therefore determined by a combination of all 
three methods. In addition, measurements of other physical properties 
may be used to advantage in some alloy systems: the most important of 
these subsidiary techniques are measurements of the change in length and 
of the change in electric resistance as a function of temperature. 

In general, the various experimental techniques differ in sensitivity, and 
therefore in usefulness, from one portion of the phase diagram to another. 
Thus, thermal analysis is the best method for determining the liquidus and 
solidus, including eutectic and peritectic horizontals, but it may fail to 
reveal the existence of eutectoid and peritectoid horizontals because of the 
sluggishness of some solid-state reactions or the small heat effects involved. 
Such features of the diagram are best determined by microscopic examina- 
tion or x-ray diffraction, and the same applies to the determination of solvus 
(solid solubility) curves. It is a mistake to rely entirely on any one method, 
and the wise investigator will use whichever technique is most appropriate 
to the problem at hand. 

* The pressure on the alloy is another effective variable, but it is usually 
constant at that of the atmosphere and may be neglected. , - 

345 



346 



PHASE-DIAGRAM DETERMINATION 



[CHAP. 12 



liquid 




12-2 General principles. The key to the interpretation of the powder 
patterns of alloys is the fact that each phase produces its own pattern in- 
dependently of the presence or absence of any other phase. Thus a single- 
phase alloy produces a single pattern while the pattern of a two-phase alloy 
consists of two superimposed patterns, one due to each phase. 

Assume, for example, that two metals A and B are completely soluble in 
the solid state, as illustrated by the phase diagram of Fig. 12-1. The 
solid phase a, called a continuous solid solution, is of the substitutional type; 
it varies in composition, but not in crystal structure, from pure A to pure 
B, which must necessarily have the same structure. The lattice parameter 
of a also varies continuously from that of pure A to that of pure B. Since 
all alloys in a system of this kind consist of the same single phase, their 
powder patterns appear quite similar, the only effect of a change in composi- 
tion being to shift the diffraction- 
line positions in accordance with the 
change in lattice parameter. 

More commonly, the two metals A 
and B are only partially soluble in the 
solid state. The first additions of B 
to A go into solid solution in the A 
lattice, which may expand or contract 
as a result, depending on the relative 
sizes of the A and B atoms and the 
type of solid solution formed (substi- 
tutional or interstitial). Ultimately 
the solubility limit of B in A is reached, 
and further additions of B cause the 
precipitation of a second phase. This 
second phase may be a B-rich solid 
solution with the same structure as B, 
as in the alloy system illustrated by Fig. 12-2(a). Here the solid solutions 
a and /3 are called primary solid solutions or terminal solid solutions. Or the 
second phase which appears may have no connection with the B-rich solid 
solution, as in the system shown in Fig. 12-2(b). Here the effect of super- 
saturating a. with metal B is to precipitate the phase designated 7. This 
phase is called an intermediate solid solution or intermediate phase. It usu- 
ally has a crystal structure entirely different from that of either a or 0, and 
it is separated from each of these terminal solid solutions, on the phase di- 
agram, by at least one two-phase region. 

Phase diagrams much more complex than those just mentioned are often 
encountered in practice, but they are always reducible to a combination of 
fairly simple types. When an unknown phase diagram is being investi- 
gated, it is best to make a preliminary survey of the whole system by pre- 



PERCENT B 



B 



FIG. 12-1. Phase diagram of two 
metals, showing complete solid solu- 
bility. 



12-2] 



GENERAL PRINCIPLES 



347 



liquid 





PERCENT B 

(a) 



PERCENT B 

(b) 



FIG. 12-2. Phase diagrams showing (a) partial solid solubility, and (b) partial 
solid solubility together with the formation of an intermediate phase. 

paring a series of alloys at definite composition intervals, say 5 or 10 atomic 
percent, from pure A to pure B. The powder pattern of each alloy and each 
pure metal is then prepared. These patterns may appear quite complex 
but, no matter what the complexities, the patterns may be unraveled and 
the proper sequence of phases across the diagram may be established, if 
proper attention is paid to the following principles : 

(1) Equilibrium. Each alloy must be at equilibrium at the temperature 
where the phase relations are being studied. 

(2) Phase sequence. A horizontal (constant temperature) line drawn 
across the diagram must pass through single-phase and two-phase regions 
alternately. 

(3) Single-phase regions. In a single-phase region, a change in composi- 
tion generally produces a change in lattice parameter and therefore a shift 
in the positions of the diffraction lines of that phase. 

(4) Two-phase regions. In a two-phase region, a change in composition 
of the alloy produces a change in the relative amounts of the two phases 
but no change in their compositions. These compositions are fixed at the 
intersections of a horizontal "tie line" with the boundaries of the two-phase 
field. Thus, in the system illustrated in Fig. 12-2(a), the tie line drawn at 
temperature TI shows that the compositions of a and ft at equilibrium at 
this temperature are x and y respectively. The powder pattern of a two- 
phase alloy brought to equilibrium at temperature TI will therefore consist 
of the superimposed patterns of a of composition x and ft of composition y. 
The patterns of a series of alloys in the xy range will all contain the same 
diffraction lines at the same positions, but the intensity of the lines of the 
a phase relative to the intensity of the lines of the ft phase will decrease in 



348 PHASE-DIAGRAM DETERMINATION [CHAP. 12 

a regular manner as the concentration of B in the alloy changes from x to y, 
since this change in total composition decreases the amount of a relative 
to the amount of ft. 

These principles are illustrated with reference to the hypothetical alloy 
system shown in Fig. 12-3. This system contains two substitutional ter- 
minal solid solutions a and p, both assumed to be face-centered cubic, and 
an intermediate phase 7, which is body-centered cubic. The solubility of 
either A or B in 7 is assumed to be negligibly small: the lattice parameter 
of 7 is therefore constant in all alloys in which this phase appears. On the 
other hand, the parameters of a and ft vary with composition in the manner 
shown by the lower part of Fig. 12-3. Since the B atom is assumed to be 
larger than the A atom, the addition of B expands the A lattice, and the 
parameter of a increases from ai for pure A to a 3 for a solution of composi- 
tion x, which represents the limit of solubility of B in A at room tempera- 
ture. In two-phase (a + 7) alloys containing more than x percent B, the 
parameter of a remains constant at its saturated value a 3 . Similarly, the 
addition of A to B causes the parameter of ft to decrease from a 2 to a 4 at 
the solubility limit, and then remain constant in the two-phase (7 + ft) 
field. 

Calculated powder patterns are shown in Fig. 12-4 for the eight alloys 
designated by number in the phase diagram of Fig. 12-3. It is assumed that 
the alloys have been brought to equilibrium at room temperature by slow 
cooling. Examination of these patterns reveals the following : 

(1) Pattern of pure A (face-centered cubic). 

(2) Pattern of a almost saturated with B. The expansion of the lattice 
causes the lines to shift to smaller angles 20. 

(3) Superimposed patterns of a and 7. The a phase is now saturated 
and has its maximum parameter a 3 . 

(4) Same as pattern 3, except for a change in the relative intensities of 
the two patterns which is not indicated on the drawing. 

(5) Pattern of pure 7 (body-centered cubic). 

(6) Superimposed patterns of 7 and of saturated ft with a parameter of a 4 . 

(7) Pattern of pure ft with a parameter somewhat greater than a 4 . 

(8) Pattern of pure B (face-centered cubic). 

When an unknown phase diagram is being determined, the investigator 
must, of course, work in the reverse direction and deduce the sequence of 
phases across the diagram from the observed powder patterns. This is 
done by visual comparison of patterns prepared from alloys ranging in 
composition from pure A to pure B, and the previous example illustrates 
the nature of the changes which can be expected from one pattern to an- 
other. Corresponding lines in different patterns are identified by placing 
the films side by side as in Fig. 12-4 and noting which lines are common to 



12-2] 



GENERAL PRINCIPLES 



349 




PERCENT B *- 
FIG. 12-3. Phase diagram and lattice constants of a hypothetical alloy system. 

26 = 26 = 180 



(2) 



















1 

















FIG. 12-4. Calculated powder patterns of alloys 1 to 8 in the alloy system shown 
in Fig. 12-3. 



350 PHASE-DIAGRAM DETERMINATION [CHAP. 12 

the two patterns. * This may be difficult in some alloy systems where the 
phases involved have complex diffraction patterns, or where it is suspected 
that lines due to K$ radiation may be present in some patterns and not in 
others. It is important to remember that a diffraction pattern of a given 
phase is characterized not only by line positions but also by line intensities. 
This means that the presence of phase X in a mixture of phases cannot be 
proved merely by coincidence of the lines of phase X with a set of lines in 
the pattern of the mixture; the lines in the pattern of the mixture which 
coincide with the lines of phase X must also have the same relative intensities 
as the lines of phase X. The addition of one or more phases to a particular 
phase weakens the diffraction lines of that phase, simply by dilution, but it 
cannot change the intensities of those lines relative to one another. Finally, 
it should be noted that the crystal structure of a phase need not be known 
for the presence of that phase to be detected in a mixture : it is enough to 
know the positions and intensities of the diffraction lines of that phase. 

Phase diagram determination by x-ray methods usually begins with a 
determination of the room-temperature equilibria. The first step is to 
prepare a series of alloys by melting and casting, or by melting and solidifi- 
cation in the melting crucible. The resulting ingots are homogenized at a 
temperature just below the solidus to remove segregation, and very slowly 
cooled to room temperature, t Powder specimens are then prepared by 
grinding or filing, depending on whether the alloy is brittle or not. If the 
alloy is brittle enough to be ground into powder, the resulting powder is 
usually sufficiently stress-free to give sharp diffraction lines. Filed pow- 
ders, however, must be re-annealed to remove the stresses produced by 
plastic deformation during filing before they are ready for x-ray examina- 
tion. Only relatively low temperatures are needed to relieve stresses, but 
the filings should again be slowly cooled, after the stress-relief anneal, to 
ensure equilibrium at room temperature. Screening is usually necessary 
to obtain fine enough particles for x-ray examination, and when two-phase 
alloys are being screened, the precautions mentioned in Sec. 6-3 should be 
observed. 

After the room-temperature equilibria are known, a determination of 
the phases present at high temperatures can be undertaken. Powder 



* Superposition of the two films is generally confusing and may make some of 
the weaker lines almost invisible. A better method of comparison consists in slit- 
ting each Debye-Scherrer film lengthwise down its center and placing the center 
of one film adjacent to the center of another. The curvature of the diffraction 
lines then does not interfere with the comparison of line positions. 

t Slow cooling alone may not suffice to produce room-temperature equilibrium, 
which is often very difficult to achieve. It may be promoted by cold working and 
recrystallizing the cast alloy, in order to decrease its grain size and thus accelerate 
diffusion, prior to homogenizing and slow cooling. 



12-3] SOLID SOLUTIONS 351 

specimens are sealed in small evacuated silica tubes, heated to the desired 
temperature long enough for equilibrium to be attained, and rapidly 
quenched. Diffraction patterns of the quenched powders are then made at 
room temperature. This method works very well in many alloy systems, 
in that the quenched powder retains the structure it had at the elevated 
temperature. In some alloys, however, phases stable at high-temperature 
will decompose on cooling to room temperature, no matter how rapid the 
quench, and such phases can only be studied by means of a high-tempera- 
ture camera or diffractometer. 

The latter instrument is of particular value in work of this kind because 
it allows continuous observation of a diffraction line. For example, the 
temperature below which a high-temperature phase is unstable, such as a 
eutectoid temperature, can be determined by setting the diffractometer 
counter to receive a prominent diffracted beam of the high-temperature 
phase, and then measuring the intensity of this beam as a function of tem- 
perature as the specimen is slowly cooled. The temperature at which the 
intensity falls to that of the general background is the temperature re- 
quired, and any hysteresis in the transformation can be detected by a simi- 
lar measurement on heating. 

12-3 Solid solutions. Inasmuch as solid solubility, to a greater or 
lesser extent, is so common between metals, we might digress a little at 
this point to consider how the various kinds of solid solutions may be dis- 
tinguished experimentally. Irrespective of its extent or its position on the 
phase diagram, any solid solution may be classified as one of the following 
types, solely on the basis of its crystallography : 

(1) Intersitial. 

(2) JSubstitutional. 

(a) Random. 

(b) Ordered. (Because of its special interest, this type is described 
separately in Chap. 13.) 

(c) Defect. (A very rare type.) 

An interstitial solid solution of B in A is to be expected only when the 
B atom is so small compared to the A atom that it can enter the interstices 
of the A lattice without causing much distortion. As a consequence, about 
the only interstitial solid solutions of any importance in metallurgy are 
those formed between a metal and one of the elements, carbon, nitrogen, 
hydrogen, and boron, all of which have atoms less than 2A in diameter. 
The interstitial addition of B to A is always accompanied by an increase in 
the volume of the unit cell. If A is cubic, then the single lattice parameter 
a must increase. If A is not cubic, then one parameter may increase and 
the other decrease, as long as these changes result in an increase in cell 



352 



PHASE-DIAGRAM DETERMINATION 



[CHAP. 12 



3.10 

g 305 
j w 3.00 
i H 2.95 
! 3 290 

:S 

Si 2.85 
280 



a (austenite) 




_l L_ 



3.65 B 3 



3.60 
3.55 



1.0 



15 



20 



WEIGHT PERCENT CARBON 



FIG. 12-5. Variation of martensite and austenite lattice parameters with 
carbon content. (After C. S. Roberts, Trans. A.I.M.E. 197, 203, 1953.) 

volume. Thus, in austenite, which is an interstitial solid solution of car- 
bon in face-centered cubic -y-iron, the addition of carbon increases the cell 
edge a. But in martensite, a supersaturated interstitial solid solution of 
carbon in a-iron, the c parameter of the body-centered tetragonal cell in- 
creases while the a parameter decreases, when carbon is added. These 
effects are illustrated in Fig. 12-5. 

The density of an interstitial solid solution is given by the basic density 

equation 

1.660202^1 , ^ 

p . (3-9) 



where 



n l A l ] 



(12-1) 



n 8 and n l are numbers of solvent and interstitial atoms, respectively, per 
unit cell; and A 8 and A t are atomic weights of solvent and interstitial 
atoms, respectively. Note that the value of n 8 is constant and independent 
of the concentration of the interstitial element, and that n t is normally a 
small fraction of unity. 

The formation of a random substitutional solid solution of B and A 
may be accompanied either by an increase or decrease in cell volume, de- 
pending on whether the B atom is larger or smaller than the A atom. In 
continuous solid solutions of ionic salts, the lattice parameter of the solu- 
tion is directly proportional to the atomic percent solute present. This 
relationship, known as Vegard's law, is not strictly obeyed by metallic 
solid solutions and, in fact, there is no reason why it should be. However, 
it is often used as a sort of yardstick by which one solution may be com- 
pared with another. Figure 12-6 shows examples of both positive and 
negative deviations from Vegard's law among solutions of face-centered 
cubic metals, and even larger deviations have been found in hexagonal close- 



12-3] 



SOLID SOLUTIONS 



353 




Ni 



40 (>() 80 

ATOMIC PERCENT 



l(H) 



FIG. 12-6. Lattice parameters of some continuous solid solutions. Dot-dash 
lines indicate Vegard's law. (From Structure of Metals, by C. S. Barrett, 1952, 
McGraw-Hill Book Company, Inc.) 

packed solutions. In terminal and intermediate solid solutions, the lattice 
parameter may or may not vary linearly with the atomic percent solute 
and, when the variation is linear, the parameter found by extrapolating to 
100 percent solute does not usually correspond to the atom size deduced 
from the parameter of the pure solute, even when allowance is made for a 
possible change in coordination number. 

The density of a random substitutional solid solution is found from Eq. 
(3-9) with the 2A factor being given by 



^solvent^solvent I 



(12-2) 



where n again refers to the number of atoms per cell and A to the atomic 
weight. Whether a given solution is interstitial or substitutional may be 
decided by determining whether the x-ray density calculated according to 
Eq. (12-1) or that calculated according to Eq. (12-2) agrees with the di- 
rectly measured density. 

Defect substitutional solid solutions are ones in which some lattice 
sites, normally occupied by atoms at certain compositions, are simply 
vacant at other compositions. Solutions of this type are rare among metals ; 
the best-known example is the intermediate ft solution in the nickel-alu- 
minum system. A defect solution is disclosed by anomalies in the curves 
of density and lattice parameter vs. composition. Suppose, for example, 
that the solid solution of B and A is perfectly normal up to x percent B, 



354 



PHASE-DIAGRAM DETERMINATION 



[CHAP. 12 



but beyond that point a defect lattice is formed; i.e., further increases in 
B content are obtained, not by further substitution of B for A, but by 
dropping A atoms from the lattice to leave vacant sites. Under these cir- 
cumstances, the density and parameter curves will show sudden changes 
in slope, or even maxima or minima, at the composition x. Furthermore, 
the x-ray density calculated according to Eq. (12-2) will no longer agree 
with the direct density simply because Eq. (12-2), as usually used, applies 
only to normal solutions where all lattice sites are occupied; i.e., it is tacitly 
assumed there that (n 80 i vent + n so i ute ) equals the total number of lattice sites 
in the structure involved. The actual structure of a defect solid solution, 
including the proportion of vacant lattice sites at any given composition, 
can be determined by a comparison of the direct density with the x-ray 
density, calculated according to Eq. (12-2), and an analysis of the dif- 
fracted intensities. 

12-4 Determination of solvus curves (disappearing-phase method). To 

return to the main subject of this chapter, we might now consider the 
methods used for determining the position of a solvus curve on a phase 
diagram. Such a curve forms the boundary between a single-phase solid 
region and a two-phase solid region, and the single-phase solid may be a 
primary or intermediate solid solution. 

One method of locating such curves is based on the "lever law." This 
law, with reference to Fig. 12-7 for example, states that the relative propor- 
tions of a. and ft in an alloy of composition ^ in equilibrium at temperature 
TI is given by the relative lengths of the lines zy and zx, or that 



W a (z - x) = 

where W a and W& denote the relative 
weights of a and ft if x, y, and z are 
expressed in weight percent. It fol- 
lows from Eq. (12-3) that the weight 
fraction of ft in the alloy varies line- 
arly with composition from at point 
x to 1 at point y. The intensity of 
any diffraction line from the ft phase 
also varies from zero at x to a maxi- 
mum at y, but the variation with 
weight percent B is not generally 
linear. * Nevertheless, this variation 
may be used to locate the point x. A 
series of alloys in the two-phase region 



(12-3) 



PS 
w 




WEIGHT PERCENT B *> 

FIG. 12-7. Lever-law construction 
for finding the relative amounts of two 
phases in a two-phase field. 



* The reasons for nonlinearity are discussed in Sec. 14-9. 



12-4] SOLVUS CURVES (DISAPPEARING-PHASE METHOD) 355 

is brought to equilibrium at temperature T\ and quenched. From diffrac- 
tion patterns made at room temperature, the ratio of the intensity /# of a 
prominent line of the ft phase to the intensity I a of a prominent line of the 
a phase is plotted as a function of weight percent B. The composition at 
which the ratio /0// a extrapolates to zero is taken as the point x. (Use of 
the ratio I$/I a rather than /# alone eliminates the effect of any change 
which may occur in the intensity of the incident beam from one diffraction 
pattern to another. However, this ratio also varies nonlinearly with weight 
percent B.) Other points on the solvus curve are located by similar experi- 
ments on alloys quenched from other temperatures. This method is known, 
for obvious reasons, as the disappearing-phase method. 

Since the curve of Ip/I a vs. weight percent B is not linear, high accuracy 
in the extrapolation depends on having several experimental points close 
to the phase boundary which is being determined. The accuracy of the 
disappearing-phase method is therefore governed by the sensitivity of the 
x-ray method in detecting small amounts of a second phase in a mixture, 
and this sensitivity varies widely from one alloy system to another. The 
intensity of a diffraction line depends on, among other things, the atomic 
scattering factor /, which in turn is almost directly proportional to the 
atomic number Z. Therefore, if A and B have nearly the same atomic 
number, the a. and ft phases will consist of atoms having almost the same 
scattering powers, and the intensities of the a and ft diffraction patterns 
will also be roughly equal when the two phases are present in equal amounts. 
Under favorable circumstances such as these, an x-ray pattern can reveal 
the presence of less than 1 percent of a second phase. On the other hand, 
if the atomic number of B is considerably less than that of A, the intensity 
of the ft pattern may be so much lower than that of the a pattern that a 
relatively large amount of ft in a two-phase mixture will go completely un- 
detected. This amount may exceed 50 percent in extreme cases, where the 
atomic numbers of A and B differ by some 70 or 80 units. ' Under such cir- 
cumstances, the disappearing-phase x-ray method is practically worthless. 
On the whole, the microscope is superior to x-rays when the disappearing- 
phase method is used, inasmuch as the sensitivity of the microscope in de- 
tecting the presence of a second phase is generally very high and independ- 
ent of the atomic numbers of the elements involved. However, this sensi- 
tivity does depend on the particle size of the second phase, and if this is 
very small, as it often is at low temperatures, the second phase may not be 
detectable under the microscope. Hence the method of microscopic ex- 
amination is not particularly accurate for the determination of solvus 
curves at low temperatures. 

Whichever technique is used to detect the second phase, the accuracy of 
the disappearing-phase method increases as the width of the two-phase re- 
gion decreases. If the (a + ft) region is only a few percent wide, then the 



356 



PHASE-DIAGRAM DETERMINATION 



[CHAP. 12 



relative amounts of a and ft will vary rapidly with slight changes in the 
total composition of the alloy, and this rapid variation of W a /Ws will 
enable the phase boundary to be fixed quite precisely. This is true, for the 
x-ray method, even if the atomic numbers of A and B are widely different, 
because, if the (a + ft) region is narrow, the compositions of a and ft do not 
differ very much and neither do their x-ray scattering powers. 

12-6 Determination of solvus curves (parametric method). As we have 
just seen, the disappearing-phase method of locating the boundary of the 
a field is based on a determination of the composition at which the ft phase 
just disappears from a series of (a + ft) alloys. The parametric method, on 
the other hand, is based on observations of the a solid solution itself. This 
method depends on the fact, previously mentioned, that the lattice pa- 
rameter of a solid solution generally changes with composition up to the 
saturation limit, and then remains constant beyond that point. 

Suppose the exact location of the solvus curve shown in Fig. 12-8(a) is 
to be determined. A series of alloys, 1 to 7, is brought to equilibrium at 
temperature T\, where the a field is thought to have almost its maximum 
width, and quenched to room temperature. The lattice parameter of a is 
measured for each alloy and plotted against alloy composition, resulting in 
a curve such as that shown in Fig. 12-8(b). This curve has two branches: 
an inclined branch 6c, which shows how the parameter of a varies with the 
composition of a, and a horizontal branch de, which shows that the a phase 
in alloys 6 and 7 is saturated, because its lattice parameter does not change 
with change in alloy composition. In fact, alloys 6 and 7 are in a two- 
phase region at temperature T\, and the only difference between them is in 
the amounts of saturated a they contain. The limit of the a field at tem- 
perature TI is therefore given by the intersection of the two branches of 



ID 

H 



12345' 



6 7 



A y x 

WEIGHT PERCENT B -* 

(a) 




WEIGHT PERCENT B 

(b) 



FIG. 12-8. Parametric method tor determining a solvus curve. 



12-5] SOLVUS CURVES (PARAMETRIC METHOD) 357 

the parameter curve. In this way, we have located one point on the solvus 
curve, namely x percent B at T\. 

Other points could be found in a similar manner. For example, if the 
same series of alloys were equilibrated at temperature T 2 , a parameter 
curve similar to Fig. 12-8(b) would be obtained, but its inclined branch 
would be shorter and its horizontal branch lower. But heat treatments 
and parameter measurements on all these alloys are unnecessary, once the 
parameter-composition curve of the solid solution has been established. 
Only one two-phase alloy is needed to determine the rest of the solvus. 
Thus, if alloy 6 is equilibrated at T 2 and then quenched, it 'will contain a 
saturated at that temperature. Suppose the measured parameter of a in 
this alloy is a y . Then, from the parameter-composition curve, we find that 
a of parameter a y contains y percent B. This fixes a point on the solvus at 
temperature T 2 . Points on the solvus at other temperatures may be found 
by equilibrating the same alloy, alloy 6, at various temperatures, quench- 
ing, and measuring the lattice parameter of the contained a. 

The parameter-composition curve, branch be of Fig. 12-8(b), thus serves 
as a sort of master curve for the determination of the whole solvus. For a 
given accuracy of lattice parameter measurement, the accuracy with which 
the solvus can be located depends markedly on the slope of the parameter- 
composition curve. If this curve is nearly flat, i.e., if changes in the com- 
position of the solid solution produce very small changes in parameter, then 
the composition, as determined from the parameter, will be subject to con- 
siderable error and so will the location of the solvus. However, if the curve 
is steep, just the opposite is true, and relatively crude parameter measure- 
ments may suffice to fix the location of the solvus quite accurately. In 
either case, relative parameter measurements are just as good as absolute 
parameter measurements of the same accuracy. 

Figure 12-9 illustrates the use of the parametric method in determining 
the solid solubility of antimony in copper as a function of temperature. 
The sloping curve in (a) was found from parameter measurements made 
on a series of alloys, containing from to about 12 weight percent Sb, equi- 
librated at 630C. The horizontal lines represent the parameters of two- 
phase alloys, containing about 12 weight percent Sb, equilibrated at the 
temperatures indicated. The solvus curve constructed from these data is 
given in (b), together with adjoining portions of the phase diagram. 

In most cases, the parametric method is more accurate than the disap- 
pearing-phase method, whether based on x-ray measurements or micro- 
scopic examination, in the determination of solvus curves at low tempera- 
tures. As mentioned earlier, both x-ray diffraction and microscopic ex- 
amination may fail to disclose the presence of small amounts of a second 
phase, although for different reasons. When this occurs, the disappearing- 
phase method always results in a measured extent of solubility higher than 



358 



PHASE-DIAGRAM DETERMINATION 



[CHAP. 12 



X 

M 



a 
% 

$ 
2 

8 

14 

& 

3 



3.6800 
3.6700 

3.6600 
36500 
3.6400 
3.6300 
3.6200 

3.6100 
3.6000 












^ 


630 C 

^GOO 1 
rr. c 














1 ^500 
^45 c 








/ 


^ 
















/ 


/ 












/ 












/ 












/ 





























W 

H 
tf 
a. 



G 8 10 12 14 

WEIGHT PERCENT ANTIMONY 

(a) 



11UU 

, 800 

^700 

4 

4 

-t 

H 

H 

300 
100 


v~ 








"*- 


< ~^^_ 


L 




\ 










' -~~ 


"* ~^ 




\ 




a 


+ L 










\ 
s 


V. 














>* 

x 


^ 








/ 








a 






/ 


+ t> 












/ 




^ 










S 


^ 




a -f 


y 




^/ 


























) 2 4 6 8 10 12 1< 



^yEIGHT PERCENT ANTIMONY 

(h) 



FIG. 12-9. Solvus curve determination in the copper-antimony system by the 
parametric method: (a) parameter vs. composition curve; (b) solubility vs. tempera- 
ture curve. (J. C. Mertz and C. H. Mathevvson, Trans. A.I.M.E. 124, 59, 1937.) 

the actual extent. But the parametric method, since it is based on measure- 
ments made on the phase whose range of solubility is being determined 
(the a phase), is not influenced by any property of the second phase (the 
phase). The ft phase may have an x-ray scattering power much higher 
or lower than that of the a phase, and the phase may precipitate in the 
form of large particles or small ones, without affecting the parameter 
measurements made on the a phase. 

Note that the parametric method is not confined to determining the 
extent of primary solid solutions, as in the examples given above. It may 
also be used to determine the solvus curves which bound an intermediate 
solid solution on the phase diagram. Note also that the parametric method 
may be employed even when the crystal structure of the a phase is so com- 
plex that its diffraction lines cannot be indexed. In this case, the plane 
spacing d corresponding to some high-angle line, or, even more directly, 
the 28 value of the line, is plotted against composition and the resulting 
curve used in exactly the same way as a parameter-composition curve. In 
fact, the "parametric" method could be based on the measurement of any 
property of the solid solution which changes with the composition of the 
solid solution, e.g., its electric resistivity. 



12-6] 



TERNARY SYSTEMS 



359 



12-6 Ternary systems. The determination of a ternary phase diagram 
is naturally more complicated than that of a binary diagram, because of 
the extra composition variable involved, but the same general principles 
can be applied. The x-ray methods described above, based on either the 
disappearing-phase or the parametric technique, can be used with very 
little modification and have proved to be very helpful in the study of ter- 
nary systems. 

Phase equilibria in a ternary system can only be represented completely 
in three dimensions, since there are three independent variables (two com- 
positions and the temperature). The composition is plotted in an equi- 
lateral triangle whose corners represent the three pure components, A, B, 
and C, and the temperature is plotted at right angles to the plane of the 
composition triangle. Any isothermal section of the three-dimensional 
model is thus an equilateral triangle on which the phase equilibria at that 
temperature can be depicted in two dimensions. For this reason we usually 
prefer to study ternary systems by determining the phase equilibria at a 



one phase 
two phases 
three phases 




number of selected temperatures. 

The study of a ternary system of 
components A, B, and C begins with 
a determination of the three binary 
phase diagrams AB, BC, and CA, if 
these are not already known. We then 
make up a number of ternary alloys, 
choosing their compositions almost 
at random but with some regard for 
what the binary diagrams may sug- 
gest the ternary equilibria to be. The 
diffraction patterns of these explora- 
tory alloys will disclose the number 
and kind of phases at equilibrium in 
each alloy at the temperature selected. 
These preliminary data will roughly delineate the various phase fields on 
the isothermal section, and will suggest what other alloys need be prepared 
in order to fix the phase boundaries more exactly. 

Suppose these preliminary results suggest an isothermal section of the 
kind shown in Fig. 12-10, where the phase boundaries have been drawn to 
conform to the diffraction results represented by the small circles. This 
section shows three terminal ternary solid solutions, a, /3, and 7, joined in 
pairs by three two-phase regions, (a + 0), (ft + 7), and (a + 7), and in 
the center a single region where the three phases, a, 0, and 7, are in equi- 
librium. 

In a single-phase region the composition of the phase involved, say a, is 
continuously variable. In a two-phase region tie lines exist, just as in 



A c 

FIG. 12-10. Isothermal section of 
hypothetical ternary diagram. 



360 



PHASE-DIAGRAM DETERMINATION 



[CHAP. 12 



binary diagrams, along which the relative amounts of the two phases change 
but not their compositions. Thus in the (a + 7) field of Fig. 12-10, tie 
lines have been drawn to connect the single-phase compositions which are 
in equilibrium in the two-phase field. Along the line de, for example, a of 
composition d is in equilibrium with y of composition e, and the relative 
amounts of these two phases can be found by the lever law. Thus the con- 
stitution of alloy X is given by the relation 

W a (Xd) = W y (Xe). 

Both the relative amounts and the compositions of the two phases will vary 
along any line which is not a tie line. 

In a three-phase field, the compositions of the phases are fixed and are 
given by the corners of the three-phase triangle. Thus the compositions 
of a, 0, and 7 which are at equilibrium in any alloy within the three-phase 
field of Fig. 12-10 are given by a, 6, and c, respectively. To determine the 




8 

fa 






< 



along nhc 




PERCENT A 




PERCENT A 

(c) 



FIG. 12-11. Parametric method of locating phase boundaries in ternary diagrams. 



PROBLEMS 361 

relative amounts of these phases, say in alloy Y, we draw a line through Y 
to any corner of the triangle, say 6, and apply the lever law: 



and 

W a (ag) = W y (ge). 

These relations form the basis of the disappearing-phase method of locat- 
ing the sides and corners of the three-phase triangle. 

Parametric methods are very useful in locating phase boundaries on all 
portions of the isothermal section. Suppose, for example, that we wish to 
determine the a /(a. + 7) boundary of the phase diagram in Fig. 12-11 (a). 
Then we might prepare a series of alloys along the line abc, where be is a 
tie line in the (a + 7) field, and measure the parameter of a in each one. 
The resulting parameter-composition curve would then look like Fig. 
12-ll(b), since the composition and parameter of a in alloys along be is 
constant. However, we do not generally know the direction of the line be 
at this stage, because tie lines cannot be located by any geometrical con- 
struction but must be determined by experiment. But suppose we measure 
the parameter of a along some arbitrary line, say the line Abd. Then we 
can expect the parameter-composition curve to resemble Fig. 12-1 l(c). 
The parameter of a along the line bd is not constant, since bd is not a tie 
line, but in general it will change at a different rate than along the line Ab 
in the one-phase field. This allows us to locate the point b on the phase 
boundary by the point of inflection on the parameter curve. 

The point / on the (a + 7) /(a + & + 7) boundary can be located in 
similar fashion, along a line such as efg chosen at random. Along ef the 
parameter of a will change continuously, because ef crosses over a series of 
tie lines, but along fg in the three-phase field the parameter of a will be 
constant and equal to the parameter of saturated a of composition h. The 
parameter-composition curve will therefore have the form of Fig. 12-ll(b). 

PROBLEMS 

12-1. Metals A and B form a terminal solid solution a, cubic in structure. The 
variation of the lattice parameter of a with composition, determined by quench- 
ing single-phase alloys from an elevated temperature, is found to be linear, the 
parameter varying from 3.6060A for pure A to 3.6140A in a containing 4.0 weight 
percent B. The solvus curve is to be determined by quenching a two-phase alloy 
containing 5.0 weight percent B from a series of temperatures and measuring the 
parameter of the contained a. How accurately must the parameter be measured 
if the solvus curve is to be located within 0.1 weight percent B at any tempera- 
ture? 

12-2. The two-phase alloy mentioned in Prob. 12-1, after being quenched from 
a series of temperatures, contains a having the following measured parameters: 



362 PHASE-DIAGRAM DETERMINATION [CHAP. 12 

Temperature Parameter 

100C 3.6082A 

200 3.6086 

300 3.6091 

400 3.6098 

500 3.6106 

600 3.6118 

Plot the solvus curve over this temperature range. What is the solubility of B in 
A at 440C? 



CHAPTER 13 
ORDER-DISORDER TRANSFORMATIONS 

13-1 Introduction. In most substitutional solid solutions, the two 
kinds of atoms A and B are arranged more or less at random on the atomic 
sites of the lattice. In solutions of this kind the only major effect of a 
change in temperature is to increase or decrease the amplitude of thermal 
vibration. But, as noted in Sec. 2-7, there are some solutions which have 
this random structure only at elevated temperatures. When these solu- 
tions are cooled below a certain critical temperature TV, the A atoms 
arrange themselves in an orderly, periodic manner on one set of atomic 
sites, and the B atoms do likewise on another set. The solution is then 
said to be ordered or to possess a superlattice. When this periodic arrange- 
ment of A and B atoms persists over very large distances in the crystal, it 
is known as long-range order. If the ordered solution is heated above T c , 
the atomic arrangement becomes random again and the solution is said to 
be disordered. 

The change in atom arrangement which occurs on ordering produces 
changes in a large number of physical and chemical properties, and the 
existence of ordering may be inferred from some of these changes. How- 
ever, the only conclusive evidence for a disorder-order transformation is a 
particular kind of change in the x-ray diffraction pattern of the substance. 
Evidence of this kind was first obtained by the American metallurgist Bain 
in 1923, for a gold-copper solid solution having the composition AuCua. 
Since that time, the same phenomenon has been discovered in many other 
alloy systems. 

13-2 Long-range order in AuCua. The gold and copper atoms of 
AuCu 3 , above a critical temperature of about 395C, are arranged more or 
less at random on the atomic sites of a face-centered cubic lattice, as illus- 
trated in Fig. 13-1 (a). If the disorder is complete, the probability that a 
particular site is occupied by a gold atom is simply f , the atomic fraction 
of gold in the alloy, and the probability that it is occupied by a copper atom 
is f , the atomic fraction of copper. / These probabilities are the same for 
every site and, considering the structure as a whole, we can regard each 
site as being occupied by a statistically "average" gold-copper atom. Be- 
low the critical temperature, the gold atoms in a perfectly ordered alloy 
occupy only the corner positions of the unit cube and the copper atoms the 
face-centered positions, as illustrated in Fig. 13-1 (b). Both structures are 
cubic and have practically the same lattice parameters. Figure 13-2 shows 




ORDER-DISORDER TRANSFORMATIONS 




[CHAP. 13 

gold atom 
copper atom 



V_y ' 'average" 
gold-copper atom 



(a) Disordered 



(b) Ordered 



FIG. 13-1. Unit cells of the disordered and ordered forms of AuCu 3 . 

how the two atomic arrangements differ on a particular lattice plane. The 
same kind of ordering has been observed in PtCu 3 , FeNi 3 , MnNi 3 , and 
(MnFe)Ni 3 . 

What differences will exist between the diffraction patterns of ordered 
and disordered AuCu 3 ? Since there is only a very slight change in the size 
of the unit cell on ordering, and none in its shape, there will be practically 
no change in the positions of the diffraction lines. But the change in the 
positions of the atoms must necessarily cause a change in line intensities. 
We can determine the nature of these changes by calculating the structure 
factor F for each atom arrangement: 

(a) Complete disorder. The atomic scattering factor of the "average" 
gold-copper atom is given by 

/av = (atomic fraction Au) / Au + (atomic fraction Cu) /c u , 

/av = 4/Au + f/Cu- 

There are four "average" atoms per unit cell, at 0, f \ 0, \ \, and 
\ \. Therefore the structure factor is given by 

F = 2f Q 2 * i (k u + kv +i w ) 
F = Av[l + e 





Disordered Ordered 

( j gold ^B copper 

FIG. 13-2. Atom arrangements on a (100) plane, disordered and ordered AuCu 3. 



13-2] LONG-RANGE ORDER IN AuCu 3 365 

By example (d) of Sec. 4-6, this becomes 

F = 4/ av = (/ Au + 3/cu), for hkl unmixed, 
F = 0, for hkl mixed. 

We therefore find, as might be expected, that the disordered alloy produces 
a diffraction pattern similar to that of any face-centered cubic metal, say 
pure gold or pure copper. No reflections of mixed indices are present. 

(b) Complete order. Each unit cell now contains one gold atom, at 0, 
and three copper atoms, at ^ ^ 0, ^ f , and ^ f . 



F = /A 

F = (/AU + 3/cu), for hkl unmixed, 

(13-1) 
F = (/AU - /Cu), for hkl mixed. 

The ordered alloy thus produces diffraction lines for all values of hkl, and 
its diffraction pattern therefore resembles that of a simple cubic substance. 
In other words, there has been a change of Bravais lattice on ordering; the 
Bravais lattice of the disordered alloy is face-centered cubic and that of the 
ordered alloy simple cubic. 

The diffraction lines from planes of unmixed indices are called fundamen- 
tal lines, since they occur at the same positions and with the same intensi- 
ties in the patterns of both ordered and disordered alloys. The extra lines 
which appear in the pattern of an ordered alloy, arising from planes of 
mixed indices, are called superlattice lines, and their presence is direct evi- 
dence that ordering has taken place. The physical reason for the forma- 
tion of superlattice lines may be deduced from an examination of Fig. 13-1. 
Consider reflection from the (100) planes of the disordered structure, and 
let an incident beam of wavelength X make such an angle of incidence B 
that the path difference between rays scattered by adjacent (100) planes is 
one whole wavelength. But there is another plane halfway between these 
two, containing, on the average, exactly the same distribution of gold and 
copper atoms. This plane scatters a wave which is therefore X/2 out of 
phase with the wave scattered by either adjacent (100) plane and of ex- 
actly the same amplitude. Complete cancellation results and there is no 
100 reflection. In the ordered alloy, on the other hand, adjacent (100) 
planes contain both gold and copper atoms, but the plane halfway between 
contains only copper atoms. The rays scattered by the (100) planes and 
those scattered by the midplanes are still exactly out of phase, but they now 
differ in amplitude because of the difference in scattering power of the gold 
and copper atoms. The ordered structure therefore produces a weak 100 
reflection. And as Eqs. (13-1) show, all the superlattice lines are much 
weaker than the fundamental lines, since their structure factors involve 



366 



ORDER-DISORDER TRANSFORMATIONS 



[CHAP. 13 





/ 1 

111 200 220 
/ / / 




/ I /\ 

KM) 110 210 211 



FIG. 13-3. Powder patterns of AuCiis (very coarse-grained) made with filtered 
copper radiation: (a) quenched from 440C (disordered); (b) held 30 min at 360C 
and quenched (partially ordered) ; (c) slowly cooled from 360C to room tempera- 
ture (completely ordered). 

the difference, rather than the sum, of the atomic scattering factors of each 
atom. This effect is shown quite clearly in Fig. 13-3, where / and s are 
used to designate the fundamental and superlattice lines, respectively. 

At low temperatures, the long-range order in AuCua is virtually perfect 
but, as T c is approached, some randomness sets in. This departure from 
perfect order can be described by means of the long-range order parameter 
S, defined as follows: 



S = 



i -F 



(13-2) 



where TA = fraction of A sites occupied by the "right" atoms, i.e., A atoms, 
and FA = fraction of A atoms in the alloy. When the long-range order is 
perfect, r A = 1 by definition, and therefore $ = 1. When the atomic 
arrangement is completely random, r A = F A and S = 0. For example, 
consider 100 atoms of AuCus, i.e., 25 gold atoms and 75 copper atoms. 
Suppose the ordering is not perfect and only 22 of these gold atoms are on 
"gold sites," i.e., cube corner positions, the other 3 being on "copper sites." 
Then, considering the gold atom as the A atom in Eq. (13-2), we find that 
r A = f| = 0.88 and F A = -fifc = 0.25. Therefore, 



S 



0.88 - 0.25 
1.00 - 0.25 



= 0.84 



describes the degree of long-range order present. The same result is ob- 
tained if we consider the distribution of copper atoms. 



13-2] 



LONG-RANGE ORDER IN AuCu 3 



367 



Any departure from perfect long-range order in a superlattice causes the 
superlattice lines to become weaker. It may be shown that the structure 
factors of partially ordered AuCua are given by 



F = (/AU + 3/cu), for hkl unmixed, 
F = S(/Au - /cu), for hkl mixed. 



(13-3) 



i o 



08 



Of) 



04 



02 



s AuOus 



Comparing these equations with Eqs. (13-1), we note that only the super- 
lattice lines are affected. But the effect is a strong one, because the inten- 
sity of a superlattice line is proportional to \F\ 2 and therefore to S 2 . For 
example, a decrease in order from K = 1 .00 to S = 0.84 decreases the in- 
tensity of a superlattice line by about 30 percent. The weakening of super- 
lattice lines by partial disorder is illustrated in Fig. 13-3. By comparing 
the integrated intensity ratio of a superlattice and fundamental line, we 
can determine S experimentally. 

Values of S obtained in this way are 
shown in Fig. 13-4 as a function of 
the absolute temperature T, expressed 
as a fraction of the critical tempera- 
ture T e . For AuCu 3 the value of S 
decreases gradually, with increasing 
temperature, to about 0.8 at T c and 
then drops abruptly to zero. Above 
T c the atomic distribution is random 
and there are no superlattice lines. 
Recalling the approximate law of con- 
servation of diffracted energy, already 
alluded to in Sec. 4-12, we might ex- 
pect that the energy lost from the su- 
perlattice lines should appear in some 
form in the pattern of a completely 
disordered alloy. As a matter of fact 
it does, in the form of a weak diffuse 
background extending over the whole 
range of 26. This diffuse scattering is due to randomness, and is another 
illustration of the general law that any departure from perfect periodicity 
of atom arrangement results in some diffuse scattering at non-Bragg angles. 

Von Laue showed that if two kinds of atoms A and B are distributed 
completely at random in a solid solution, then the intensity of the diffuse 
scattering produced is given by 



o 




4 0.5 G 



08 09 1.0 



07 
T/T C 

FIG. 13-4. Variation of the long- 
range order parameter with temper- 
ature, for AuCu 3 and CuZn. (AuCu 3 
data from D. T. Keating and B. E. 
Warren, J. Appl. P%s. 22, 286, 1951; 
CuZn data from D. Chipman and 
B. E. Warren, J. Appl. Phys. 21, 696, 
1950.) 



where k is a constant for any one composition, and /A and /B are atomic 
scattering factors. Both /A and /B decrease as (sin 0)/\ increases, and so 



368 

1100 

1000 

900 

800 



W 700 

t> 



g 600 

I 

500 - 



400 



300 



ORDER-DISORDER TRANSFORMATIONS 



[CHAP. 13 




200 




AuOu 




Ou 



10 20 30 40 50 60 

ATOMIC 1 PERC 1 KNT Au 



70 



90 



KK) 
Au 



FIG. 1 3-5. Phase diagram of the gold-copper system. Two-phase fields not 
labeled for lack of room. (Compiled from Metals Handbook, American Society 
for Metals, 1948; J. B. Newkirk, Trans. A.I.M.E. 197, 823, 1953; F. N. Rhines, 
W. E. Bond, and R. A. Rummel, Trans, A.S.M, 47, 1955; R. A. Onani, Ada Metal- 
lurgica 2, 608, 1954; and G. C. Kuczynski, unpublished results.) 

does their difference; therefore I D is a maximum at 20 = and decreases 
as 20 increases. This diffuse scattering is very difficult to measure experi- 
mentally. It is weak to begin with and is superimposed on other forms of 
diffuse scattering that may also be present, namely, Compton modified 
scattering, temperature-diffuse scattering, etc. It is worth noting, how- 
ever, that Eq. (13-4) is quite general and applies to any random solid solu- 
tion, whether or not it is capable of undergoing ordering at low tempera- 
tures. We will return to this point in Sec. 13-5. 



13-3] 



OTHER EXAMPLES OF LONG-RANGE ORDER 



369 



Another aspect of long-range order that requires some mention is the 
effect of change in composition. Since the ratio of corner sites to face- 
centered sites in the AuCu 3 lattice is 1:3, it follows that perfect order can 
only be attained when the ratio of gold to copper atoms is also exactly 
1 :3. But ordering can also take place in alloys containing somewhat more, 
or somewhat less, than 25 atomic percent gold, as shown by the phase dia- 
gram of Fig. 13-5. (Here the ordered phase is designated ' to distinguish 
it from the disordered phase a stable at high temperatures.) In an ordered 
alloy containing somewhat more than 25 atomic percent gold, all the corner 
sites are occupied by gold atoms, and the remainder of the gold atoms 
occupy some of the face-centered sites normally occupied by copper atoms. 
Just the reverse is true for an alloy containing less than 25 atomic percent 
gold. But, as the phase diagram shows, there are limits to the variation in 
composition which the ordered lattice will accept without becoming un- 
stable. In fact, if the gold content is increased to about 50 atomic per- 
cent, an entirely different ordered alloy, AuCu, can be formed. 

13-3 Other examples of long-range order. Before considering the or- 
dering transformation in AuCu, which is rather complex, we might examine 
the behaviour of /3-brass. This alloy is stable at room temperature over a 
composition range of about 46 to almost 50 atomic percent zinc, and so 
may be represented fairly closely by the formula CuZn. At high tempera- 
tures its structure is, statistically, body-centered cubic, with the copper and 
zinc atoms distributed at random. Below a critical temperature of about 
465C, ordering occurs; the cell corners are then occupied only by copper 
atoms and the cell centers only by zinc atoms, as indicated in Fig. 13-6. 
The ordered alloy therefore has the CsCl structure and its Bravais lattice 
is simple cubic. Other alloys which have the same ordered structure are 
CuBe, CuPd, AgZn, FeCo, NiAl,* etc. Not all these alloys, however, 



( j zinc atom 

copper atom 





f j "average" 

copper-zinc atom 



(a) Disordered (b) Ordered 

FIG. 13-6. Unit cells of the disordered and ordered forms of CuZn. 



* NiAl is the ft phase referred to in Sec. 12-3 as having a defect lattice at certain 
compositions. 



370 ORDER-DISORDER TRANSFORMATIONS [CHAP. 13 

undergo an order-disorder transformation, since some of them remain 
ordered right up to their melting points. 

By calculations similar to those made in the previous section, the struc- 
ture factors of 0-brass, for the ideal composition CuZn, can be shown to be 

F = (/cu + /zn), for (h + k + l) even, 
F = S(fcu ~ /zn), for (h + k + I) odd. 

In other words, there are fundamental lines, those for which (h + k + l) 
is even, which are unchanged in intensity whether the alloy is ordered or 
not. And there are superlattice lines, those for which (h + k +'l) is odd, 
which are present only in the pattern of an alloy exhibiting some degree 
of order, and then with an intensity which depends on the degree of order 
present. 

Figure 13-4 indicates how the degree of long-range order in CuZn varies 
with the temperature. The order parameter for CuZn decreases continu- 
ously to zero as T approaches T e , whereas for AuCu 3 it remains fairly high 
right up to T c and then drops abruptly to zero. There is also a notable dif- 
ference in the velocity of the disorder-order transformation in these two 
alloys. The transformation in AuCu 3 is relatively so sluggish that the 
structure of this alloy at any temperature can be retained by quenching to 
room temperature, as evidenced by the diffraction patterns in Fig. 13-3. 
In CuZn, on the other hand, ordering is so rapid that disorder existing at 
an elevated temperature cannot be retained at room temperature, no mat- 
ter how rapid the quench. Therefore, any specimen of CuZn at room tem- 
perature can be presumed to be completely ordered. (The S vs. T/T C 
curve for CuZn, shown in Fig. 13-4, was necessarily based on measure- 
ments made at temperature with a high-temperature diffract ometer.) 

Not all order-disorder transformations are as simple, crystallographically 
speaking, as those occurring in AuCu 3 and CuZn. Complexities are en- 
countered, for example, in gold-copper alloys at or near the composition 
AuCu; these alloys become ordered below a critical temperature of about 
420C or lower, depending on the composition (see Fig. 13-5). Whereas 
the ratio of gold to copper atoms in AuCu 3 is 1 :3, this ratio is 1 : 1 for AuCu, 
and the structure of ordered AuCu must therefore be such that the ratio 
of gold sites to copper sites is also 1:1. Two ordered forms are produced, 
depending on the ordering temperature, and these have different crystal 
structures: 

(a) Tetragonal AuCu, designated a" (I), formed by slow cooling from 
high temperatures or by isothermal ordering below about 380C. The unit 
cell is shown in Fig. 13-7 (a). It is almost cubic in shape, since c/a equals 
about 0.93, and the gold and copper atoms occupy alternate (002) planes. 

(b) Orthorhombic AuCu, designated a" (II), formed by isothermal 
ordering between about 420 and 380C. Its very unusual unit cell, shown 



13-3] 



OTHER EXAMPLES OF LONG-RANGE ORDER 



371 




(a) "(I)-Utragonal 




(h) a" ( 1 1 l-oithorhombic 
FIG. 13-7. Unit cells of the two ordered forms of AuCu. 

in Fig. 13-7 (b), is formed by placing ten tetragonal cells like that of a"(I) 
side by side and then translating five of them by the vectors c/2 and a/2 
with respect to the other five. (Some distortion occurs, with the result that- 
each of the ten component cells, which together make up the true unit cell, 
is not tetragonal but orthorhombic; i.e., b is not exactly ten times a, but 
equal to about 10.02a. The c/a ratio is about 0.92.) The result is a struc- 
ture in which the atoms in any one (002) plane are wholly gold for a dis- 
tance of 6/2, then wholly copper for a distance of 6/2, and so on. 

From a crystallographic viewpoint, there is a fundamental difference 
between the kind of ordering which occurs in AuCu 3 or CuZn, on the one 
hand, and that which occurs in AuCu, on the other. In AuCu 3 there is a 
change in Bravais lattice, but no change in crystal system, accompanying 
the disorder-order transformation: both the disordered and ordered forms 
are cubic. In AuCu, the ordering process changes both the Bravais lattice 
and the crystal system, the latter from cubic to tetragonal, AuCu(I), or 
orthorhombic, AuCu(II). These changes are due to changes in the sym- 
metry of atom arrangement, because the crystal system to which a given 
structure belongs depends ultimately on the symmetry of that structure 
(see Sec. 2-4). In the gold-copper system, the disordered phase a is cubic, 
because the arrangement of gold and copper atoms on a face-centered lat- 
tice has cubic symmetry, in a statistical sense, at any composition. In 
3 , the ordering process puts the gold and copper atoms in definite 



372 ORDER-DISORDER TRANSFORMATIONS [CHAP. 13 

positions in each cell (Fig. 13-1), but this arrangement still has cubic sym- 
metry so the cell remains cubic. In ordered AuCu, on the other hand, to 
consider only the tetragonal modification, the atom arrangement is such 
that there is no longer three-fold rotational symmetry about directions of 
the form (111). Inasmuch as this is the minimum symmetry requirement 
for the cubic system, this cell [Fig. 13-7 (a)] is not cubic. There is, how- 
ever, four-fold rotational symmetry about [001], but not about [010] or 
[100]. The ordered form is accordingly tetragonal. The segregation of 
gold and copper atoms on alternate (002) planes causes c to differ from a, 
in this case in the direction of a small contraction of c relative to a, because 
of the difference in size between the gold and copper atoms. But even if 
c were equal to a, the cell shown in Fig. 13-7 (a) would still be classified as 
tetragonal on the basis of its symmetry. 

13-4 Detection of superlattice lines. We have already seen that the 
intensity of a superlattice line from an ordered solid solution is much lower 
than that of a fundamental line. Will it ever be so low that the line cannot 
be detected? We can make an approximate estimate by ignoring the varia- 
tion in multiplicity factor and Lorentz-polarization factor from line to line, 
and assuming that the relative integrated intensities of a superlattice and 
fundamental line are given by their relative \F\ 2 values. For fully ordered 
AuCu 3 , for example, we find from Eqs. (13-1) that 

Intensity (superlattice line) \F\ 8 2 _ (/AU ~ /GU)" 
Intensity (fundamental line) |F|/ 2 (/A U + 3/cJ 

At (sin 0)/X = we can put / = Z and, since the atomic numbers of gold 
and copper are 79 and 29, respectively, Eq. (13-6) becomes, for small 
scattering angles, _ ^ 

zz 0.09. 

I f [79 + 3(29)] 2 

Superlattice lines are therefore only about one-tenth as strong as fundamen- 
tal lines, but they can still be detected without any difficulty, as shown by 
Fig. 13-3. 

But in CuZn, even when fully ordered, the situation is much worse. The 
atomic numbers of copper and zinc are 29 and 30, respectively, and, mak- 
ing the same assumptions as before, we find that 

I, (/cu - /zn) 2 (29 - 30) 2 



//~(/Cu+/Zn) 2 (29 + 



0.0003. 



This ratio is so low that the superlattice lines of ordered CuZn can be de- 
tected by x-ray diffraction only under very special circumstances. The 
same is true of any superlattice of elements A and B which differ in atomic 



DETECTION OF SUPERLATTICE LINES 




02 



04 



0.6 8 



FIG. 13-8. Variation of A/ with X/X/t. (Data from R. W. James, The Optical 
Principles of the Diffraction of X-Rays, G. Bell and Sons, Ltd., London, 1948, p. 608.) 

number by only one or two units, because the superlattice-line intensity is 
generally proportional to (/A /e) 2 - 

There is one way, however, of increasing the intensity of a superlattice 
line relative to that of a fundamental line, when the two atoms involved 
have almost the same atomic numbers, and that is by the proper choice of 
the incident wavelength. In the discussion of atomic scattering factors 
given in Sec. 4-3 it was tacitly assumed that the atomic scattering factor 
was independent of the incident wavelength, as long as the quantity 
(sin 0)/X was constant. This is not quite true. When the incident wave- 
length X is nearly equal to the wavelength \K of the K absorption edge of 
the scattering element, then the atomic scattering factor of that element 
may be several units lower than it is when X is very much shorter than X#. 
If we put / = atomic scattering factor for X \K (this is the usual value 
as tabulated, for example, in Appendix 8) and A/ = change in / when X is 
near XA, then the quantity /' = / + A/ gives the value of the atomic scat- 
tering factor when X is near XA- Figure 13-8 shows approximately how 
A/ varies with X/XA, and this curve may be used to estimate the correction 
A/ which must be applied for any particular combination of wavelength and 
scattering element.* 

* Strictly speaking, A/ depends also on the atomic number of the scattering ele- 
ment, which means that a different correction curve is required for every element. 
But the variation of A/ with Z is not very large, and Fig. 13-8, which is computed 
for an element of medium atomic number (about 50), can be used with fairly good 
accuracy as a master correction curve for any element. 



374 



ORDER-DISORDER TRANSFORMATIONS 



[CHAP. 13 








FIG. 13 9. 



04 06 
sin 6 

^'' 

Atomic scattering factors of copper for two different wavelengths. 



When A/AA- is less than about 0.8, the correction is practically negligible. 
When A/A A- exceeds about 1.6, the correction is practically constant and 
independent of small variations in AA. But when A is near AA, the slope 
of the correction curve is quite steep, which means that the A/ correction 
can be quite different for two elements of nearly the same atomic number. 
By taking advantage of this fact, we can often increase the intensity of a 
superlattice line above its normal value. 

For example, if ordered CuZn is examined with Mo Ka radiation, \/\K 
is 0.52 for the copper atom and 0.55 for the zinc atom. The value of A/ is 
then about +0.3 for either atom, and the intensity of a superlattice line 
would be proportional to [(29 + 0.3) - (30 + 0.3)] 2 = 1 at low values of 
20. Under these circumstances the line would be invisible in the presence 
of the usual background. But if Zn Ka radiation is used, A/AA becomes 
1.04 and 1.11 for the copper and zinc atoms, respectively, and Fig. 13-8 
shows that the corrections are 3.6 and 2.7, respectively. The super- 
lattice-line intensity is now proportional to [(29 3.6) (30 2.7)] 2 = 
3.6, which is large enough to permit detection of the line. Cu Ka radia- 
tion also offers some advantage over Mo Ka, but not so large an advantage 
as Zn /fa, and order in CuZn can be detected with Cu Ka only if crystal- 
monochromated radiation is used. 

To a very good approximation, the change in atomic scattering factor 
A/ is independent of scattering angle and therefore a constant for all lines 
on the diffraction pattern. Hence, we can construct a corrected /' curve 
by adding, algebraically, the same value A/ to all the ordinates of the usual 
/ vs. (sin 0)/A curve, as in Fig. 13-9. 



13-5] SHORT-RANGE ORDER AND CLUSTERING 375 

By thus taking advantage of this anomalous change in scattering factor 
near an absorption edge, we are really pushing the x-ray method about as 
far as it will go. A better tool for the detection of order in alloys of metals 
of nearly the same atomic number is neutron diffraction (Appendix 14). 
Two elements may differ in atomic number by only one unit and yet their 
neutron scattering powers may be entirely different, a situation conducive 
to high superlattice-line intensity. 

13-5 Short-range order and clustering. Above the critical tempera- 
ture T c long-range order disappears and the atomic distribution becomes 
more or less random. This is indicated by the absence of superlattice lines 
from the powder pattern. But careful analysis of the diffuse scattering 
which forms the background of the pattern shows that perfect randomness 
is not attained. Instead, there is a greater than average tendency for un- 
like atoms to be nearest neighbors. This condition is known as short-range 
order. 

For example, when perfect long-range order exists in AuCu 3 , a gold atom 
located at is surrounded by 12 copper atoms at f \ and equivalent 
positions (see Fig. 13-1), and any given copper atom is likewise surrounded 
by 12 gold atoms. This kind of grouping is a direct result of the existing 
long-range order, which also requires that gold atoms be on corner sites 
and copper atoms on face-centered sites. Above T c this order breaks down 
and, if the atomic distribution became truly random, a given gold atom 
might be found on either a corner or face-centered site. It would then 
have only f (12) = 9 copper atoms as nearest neighbors, since on the aver- 
age 3 out of 4 atoms in the solution are copper. Actually, it is observed 
that some short-range order exists above T c : at 460C, for example, which 
is 65C above T C1 there are on the average about 10.3 copper atoms around 
any given gold atom. 

This is a quite general effect. Any solid solution which exhibits long- 
range order below a certain temperature exhibits some short-range order 
above that temperature. Above T c the degree of short-range order de- 
creases as the temperature is raised; i.e., increasing thermal agitation tends 
to make the atomic distribution more and more random. One interesting 
fact about short-range order is that it has also been found to exist in solid 
solutions which do not undergo long-range or4ering at low temperatures, 
such as gold-silver and gold-nickel solutions. 

We can imagine another kind of departure from randomness in a solid 
solution, namely, a tendency of like atoms to be close neighbors. This 
effect is known as clustering, and it has been observed in aluminum-silver 
and aluminum-zinc solutions. In fact, there is probably no such thing as 
a perfectly random solid solution. All real solutions probably exhibit either 
short-range ordering or clustering to a greater or lesser degree, simply be- 



376 



ORDER-DISORDER TRANSFORMATIONS 



[CHAP. 13 



04 0.8 12 Hi 20 24 2 S 3.2 3 (> 




FIG. 13-10. Calculated intensity /D of diffuse scattering in powder patterns of 
solid solutions (here, the face-centered cubic alloy Xi 4 Au) which exhibit complete 
randomness, short-range order, and clustering. The short-range order curve is 
calculated on the basis of one additional unlike neighbor ovei the random con- 
figuration, and the clustering curve on the basis of one less unlike neighbor. 
(B. E. Warren and B. L. Averbach, Modern Research Techniques in Physical Metal- 
lurgy, American Society for Metals, Cleveland, 1953, p. 95.) 

cause they are composed of unlike atoms with particular forces of attrac- 
tion or repulsion operating between them. 

The degree of short-range order or clustering may be defined in terms of 
a suitable parameter, just as long-range order is, and the value of this 
parameter may be related to the diffraction effects produced. The general 
nature of these effects is illustrated in Fig. 13-10, where the intensity of the 
diffuse scattering is plotted, not against 26, but against a function of sin B. 
(The fundamental lines are not included in Fig. 13-10 because their in- 
tensity is too high compared with the diffuse scattering shown, but the 
positions of two of them, 111 and 200, are indicated on the abscissa.) If 
the atomic distribution is perfectly random, the scattered intensity de- 
creases gradually as 20 or sin 6 increases from zero, in accordance with 
Eq. (13-4). If short-range order exists, the scattering at small angles be- 
comes less intense and low broad maxima occur in the scattering curve; 
these maxima are usually located at the same angular positions as the sharp 
superlattice lines formed by long-range ordering. Clustering causes strong 
scattering at low angles. 

These effects, however, are all very weak and are masked by the other 
forms of diffuse scattering which are always present. As a result, the de- 



PROBLEMS 377 

tails shown in Fig. 13-10 are never observed in an ordinary powder pattern 
made with filtered radiation. To disclose these details and so learn some- 
thing about the structure of the solid solution, it is necessary to use strictly 
monochromatic radiation and to make allowances for the other, forms of 
diffuse scattering, chiefly temperature-diffuse and Compton modified, 
which are always present. 

PROBLEMS 

13-1. A Debye-Scherrer pattern is made with Cu Ka radiation of AuCu 3 
quenched from a temperature TV The ratio of the integrated intensity of the 420 
line to that of the 421 line is found to be 4.38. Calculate the value of the long- 
range order parameter S at temperature T\. (Take the lattice parameter of AuCua 
as 3.75A. Ignore the small difference between the Lorentz-polarization factors 
for these two lines and the corrections to the atomic scattering factors mentioned 
in Sec. 13-4.) 

13-2. Calculate the ratio of the integrated intensity of the 100 superlattice line 
to that of the 110 fundamental line for fully ordered #-brass, if Cu Ka radiation 
is used. Estimate the corrections to the atomic scattering factors from Fig. 13-8. 
The lattice parameter of /3-brass (CuZn) is 2.95A. 

13-3. (a) What is the Bravais lattice of AuCu(I), the ordered tetragonal 
modification? 

(b) Calculate the structure factors for the disordered and ordered (tetragonal) 
forms of AuCu. 

(c) On the basis of the calculations made in (6) and a consideration of the change 
in the c/a ratio, describe the differences between the powder patterns of the or- 
dered and disordered (tetragonal) forms of AuCu. 



CHAPTER 14 
CHEMICAL ANALYSIS BY DIFFRACTION 

14-1 Introduction. A given substance always produces a characteris- 
tic diffraction pattern, whether that substance is present in the pure state 
or as one constituent of a mixture of substances. This fact is the basis for 
the diffraction method of chemical analysis. Qualitative analysis for a par- 
ticular substance is accomplished by identification of the pattern of that 
substance. Quantitative analysis is also possible, because the intensities 
of the diffraction lines due to one constituent of a mixture depend on the 
proportion of that constituent in the specimen. 

The particular advantage of diffraction analysis is that it discloses the 
presence of a substance as that substance actually exists in the sample, and 
not in terms of its constituent chemical elements. For example, if a sample 
contains the compound A^By, the diffraction method will disclose the pres- 
ence of A X E V as such, whereas ordinary chemical analysis would show only 
the presence of elements A and B. Furthermore, if the sample contained 
both AxBy and A X B 2 |/, both of these compounds would be disclosed by the 
diffraction method, but chemical analysis would again indicate only the 
presence of A and B.* To consider another example, chemical analysis 
of a plain carbon steel reveals only the amounts of iron, carbon, man- 
ganese, etc., which the steel contains, but gives no information regarding 
the phases present. Is the steel in question wholly martensitic, does it 
contain both martensite and austenite, or is it composed only of ferrite 
and cementite? Questions such as these can be answered by the diffrac- 
tion method. Another rather obvious application of diffraction analysis 
is in distinguishing between different allotropic modifications of the same 
substance: solid silica, for example, exists in one amorphous and six crys- 
talline modifications, and the diffraction patterns of these seven forms are 
all different. 

Diffraction analysis is therefore useful whenever it is necessary to know 
the state of chemical combination of the elements involved or the par- 
ticular phases in which they are present. As a result, the diffraction method 

* Of course, if the sample contains only A and B, and if it can be safely assumed 
that each of these elements is wholly in a combined form, then the presence of 
AJB,, and A^B^ can be demonstrated by calculations based on the amounts of 
A and B in the sample. But this method is not generally applicable, and it usually 
involves a prior assumption as to the constitution of the sample. For example, a 
determination of the total amounts of A and B present in a sample composed of 
A, AjBy, and B cannot, in itself, disclose the presence of A x B y , either qualitatively 
or quantitatively. 

378 



14-3] QUALITATIVE ANALYSIS: THE HANAWALT METHOD 379 

has been widely applied for the analysis of such materials as ores, clays, 
refractories, alloys, corrosion products, wear products, industrial dusts, 
etc. Compared with ordinary chemical analysis, the diffraction method 
has the additional advantages that it is usually much faster, requires only 
a very small sample, and is nondestructive. 



QUALITATIVE ANALYSIS 

14-2 Basic principles. The powder pattern of a substance is charac- 
teristic of that substance and forms a sort of fingerprint by which the sub- 
stance may be identified. If we had on hand a collection of diffraction pat- 
terns for a great many substances, we could identify an unknown by pre- 
paring its diffraction pattern and then locating in our file of known patterns 
one which matched the pattern of the unknown exactly. The collection 
of known patterns has to be fairly large, if it is to be at all useful, and then 
pattern-by-pattern comparison in order to find a matching one becomes 
out of the question. 

What is needed is a system of classifying the known patterns so that the 
one which matches the unknown can be located quickly. Such a system 
was devised by Hanawalt in 1936. Any one powder pattern is charac- 
terized by a set of line positions 26 and a set of relative line intensities I. 
But the angular positions of the lines depend on the wavelength used, and 
a more fundamental quantity is the spacing d of the lattice planes forming 
each line. Hanawalt therefore decided to describe each pattern by listing 
the d and / values of its diffraction lines, and to arrange the known pat- 
terns in decreasing values of d for the strongest line in the pattern. This 
arrangement made possible a search procedure which would quickly locate 
the desired pattern. In addition, the problem of solving the pattern was 
avoided and the method could be used even when the crystal structure 
of the substance concerned was unknown. 

14-3 The Hanawalt method. The task of building up a collection of 
known patterns was initiated by Hanawalt and his associates, who ob- 
tained and classified diffraction data on some 1000 different substances. 
This work was later extended by the American Society for Testing Mate- 
rials with the assistance, on an international scale, of a number of other 
scientific societies. The ASTM first published a collection of diffraction 
data in 1941 in the form of a set of 3 X 5" cards which contained data on 
some 1300 substances. Various supplementary sets have appeared from 
time to time, the most recent in 1955, and all the sets taken together now 
cover some 5900 substances. Most of these are elements and inorganic 
compounds, although some organic compounds and minerals are also in- 
cluded. 



380 CHEMICAL ANALYSIS BY DIFFRACTION [CHAP. 14 

The original set (1941) and the first supplementary set (1944) have been out 
of print since 1947. Both of these sets were revised and reissued in 1949. The fol- 
lowing sets are currently available: 

Year Approx. number 
Name of set Section issued of substances 

Revised original 1 1949 1300 

Revised first supplementary 2 1949 1300 

Second supplementary 3 1949 1300 

Fourth * 4 1952 700 

Fifth 5 1954 700 

Sixth 6 1955 600 

Each card contains a five-digit code number: x-xxxx. The digit before the hyphen 
is the section number and the digits after the hyphen form the number of that 
card in the section. Thus, card 3-0167 is the 167th card in Section 3 (the second 
supplementary set). 

Since more than one substance can have the same, or nearly the same, 
d value for its strongest line and even its second strongest line, Hanawalt 
decided to characterize each substance by the d values of its three strongest 
lines, namely di, d 2 , and c? 3 for the strongest, second-strongest, and third- 
strongest line, respectively. The values of di, d 2 , and d 3 , together with 
relative intensities, are usually sufficient to characterize the pattern of an 
unknown and enable the corresponding pattern in the file to be located. 
In each section of the ASTM file, the cards are arranged in groups charac- 
terized by a certain range of d\ spacings. Within each group, e.g., the 
group covering d\ values from 2.29 to 2.25A, the cards are arranged in de- 
creasing order of d 2 values, rather than di values. When several sub- 
stances in the same group have identical d 2 values, the order of decreasing 
d 3 values is followed. The groups themselves are arranged in decreasing 
order of their d\ ranges. 

A typical card from the ASTM file is reproduced in Fig. 14-1. At the 
upper left appear the d valties for the three strongest lines (2.28, 1.50, 
1.35A) and, in addition, the largest d value (2.60A) for this structure. 
Listed below these d values are the relative intensities ///i, expressed as 
percentages of the strongest line in the pattern. Immediately below the 
symbol I/I\ is the serial number of the card, in this case 1-1188. Below 
the intensity data are given details of the method used for obtaining the 
pattern (radiation, camera diameter, method of measuring intensity, etc.), 
and a reference to the original experimental work. The rest of the left- 
hand portion of the card contains room for various crystallographic, opti- 
cal, and chemical data which are fully described on introductory cards of 
the set. The lower right-hand portion of the card lists the values of d and 
///i for all the observed diffraction lines. 



14-3] 



QUALITATIVE ANALYSIS! THE HANAWALT METHOD 



381 



)27B 
d 


2.28 


1.50 


1.3S 


2.60 


MO.C 


1-1194 












I/I, 

i3ia 


100 


35 


35 


29 


MOLYBDENUM CARBIDE 


Had. 




A 0.709 


1 


filter 2ND. 


d A 


I/Is 


hkl 


dA 


Via 


hkl 


Dte. 16 INCHES 


Cutoff 


con. " 


2.60 


29 




0.93 


9 




I/It CALIBRATED STRIPS 


d( 


sotr.ate.7 


No 


2.36 


24 




.91 


5 




B*. H 


2.28 

,75 


100 
24 




.89 

.87 


5 

4 




fl{jrs. HEXAGONAL 


8X3. 






.50 


35 




.84 


8 




2.994 b. 
f 

BflCt W**" 


0.4.722 A 
Y Z 


C 1 
2 


.576 


.35 

.30 


35 
3 




.82 


5 
















.27 


35 










! 




Dm ft 


*Y 


aim 




.18 


4 










IV 


D 


mp 


Color 


















B*l 












1.14 


6 






















1.08 


4 












1.01 


7 










0.98 


3 










.97 


19 











FIG. 14-1. Standard 3 X 5" ASTM diffraction data card tor molybdenum 
carbide. (Courtesy of American Society for Testing Materials.) 

Although a particular pattern can be located by a direct search of the 
card file, a great saving in time can usually be effected by use of the index 
books which accompany the file. Each book contains two indexes: 

(1) An alphabetical index of each substance by name. After the name 
are given the chemical formula, the d values and relative intensities of the 
three strongest lines, and the serial number of the card in the file for the 
substance involved. All entries are fully cross-indexed; i.e., both "sodium 
chloride" and "chloride, sodium" are listed. This index is to be used if 
the investigator has any knowledge of one or more chemical elements in 
the sample. 

(2) A numerical index, which gives the spacings and intensities of the 
three strongest lines, the chemical formula, name, and card serial number. 
Each substance is listed three times, once with the three strongest lines 
listed in the usual order did^d^ again in the order d^d\d^ and finally in 
the order d^did 2 . All entries are divided into groups according to the 
first spacing listed; the arrangement within each group is in decreasing 
order of the second spacing listed. The purpose of these additional listings 
(second-strongest line first and third-strongest line first) is to enable the 
user to match an unknown with an entry in the index even when compli- 
cating factors have altered the relative intensities of the three strongest 
lines of the unknown.* These complicating factors are usually due to the 

* In the original set of cards (1941) and the first supplementary set (1944), this 
threefold method of listing extended to the cards themselves, i.e., there were three 
cards in the file for each substance. Because the resulting card file was too bulky, 
this method was abandoned in all sets issued in 1949 and thereafter. 



382 CHEMICAL ANALYSIS BY DIFFRACTION [CHAP. 14 

presence of more than one phase in the specimen. This leads to additional 
lines and even superimposed lines. Use of the numerical index requires no 
knowledge of the chemical composition of the sample. 

Qualitative analysis by the Hanawalt method begins with the prepara- 
tion of the pattern of the unknown. This may be done with a Debye- 
Scherrer camera or a diffractometer, and any convenient characteristic 
radiation as long as it is so chosen that fluorescence is minimized and an 
adequate number of lines appear on the pattern. (Most of the data in the 
ASTM file were obtained with a Debye-Scherrer camera and Mo Ka radia- 
tion. Since a change in wavelength alters the relative intensities of the 
diffraction lines, this means that a pattern made with Cu Ka radiation, 
for example, may not be directly comparable with one in the file. Factors 
for converting intensities from a Cu Ka to a Mo Ka basis are given on an 
introductory card in the ASTM file.) Specimen preparation should be 
such as to minimize preferred orientation, as the latter can cause relative 
line intensities to differ markedly from their normal values. If the speci- 
men has a large absorption coefficient and is examined in a Debye-Scherrer 
camera, the low-angle lines may appear doubled, and both their positions 
and relative intensities may be seriously in error. This effect may be 
avoided by dilution of the unknown, as described in Sec. 6-3. 

After the pattern of the unknown is prepared, the plane spacing d corre- 
sponding to each line on the pattern is calculated, or obtained from tables 
which give d as a function of 26 for various characteristic wavelengths. 
Alternately, a scale may be constructed which gives d directly as a func- 
tion of line position when laid on the film or diffractometer chart ; the accu- 
racy obtainable by such a scale, although not very high, is generally 
sufficient for identification purposes. If the diffraction pattern has been 
obtained on film, relative line intensities are estimated by eye. The ASTM 
suggests that these estimates be assigned the following numerical values: 

Very, very strong (40 

(strongest line) = 100 1 30 

Very strong = 90 Faint = 20 

80 Very faint = 10 



Strong 

[GO 



Medium . n 

[ OU 

In many cases very rough estimates are all that are needed. If greater 
accuracy is required, relative line intensities may be obtained by com- 
parison with a graded intensity scale, made by exposing various portions 
of a strip of film to a constant intensity x-ray beam for known lengths of 
time. (Many of the intensity data in the ASTM file, including the values 
shown for molybdenum carbide in Fig. 14-1, were obtained in this way.) 



14-4] 



EXAMPLES OF QUALITATIVE ANALYSIS 



383 



If a diffractometer is used to obtain the pattern, automatic recording will 
provide sufficient accuracy, and it is customary to take the maximum in- 
tensity above the background rather than the integrated intensity as a 
measure of the "intensity" of each line, even though the integrated inten- 
sity is the more fundamental quantity. 

After the experimental values of d and I/l\ are tabulated, the unknown 
can be identified by the following procedure : 

(1) Locate the proper d\ group in the numerical index. 

(2) Read down the second column of d values to find the closest match 
to d 2 . (In comparing experimental and tabulated d values, always allow 
for the possibility that either set of values may be in error by 0.01A.) 

(3) After the closest match has been found for d 1? d 2 , and d 3 , compare 
their relative intensities with the tabulated values. 

(4) When good agreement has been found for the three strongest lines 
listed in the index, locate the proper data card in the file, and compare the 
d and 7//i values of all the observed lines with those tabulated. When 
full agreement is obtained, identification is complete. 

14-4 Examples of qualitative analysis. When the unknown is a single 
phase, the identification procedure is relatively straightforward. Con- 
sider, for example, the pattern described by Table 14-1. It was obtained 
with Mo Ka radiation and a Debye-Scherrer camera ; line intensities were 
estimated. The experimental values of di, d 2 , and da are 2.27, 1.50, and 
1.34A, respectively. By examination of the ASTM numerical index we 
find that the strongest line falls within the 2.29 to 2.25A group of di values. 
Inspection of the listed d 2 values discloses four substances having d 2 values 
close to 1.50A. The data on these substances are shown in Table 14-2, in 
the form given in the index. Of these four, only molybdenum carbide has 
a d 3 value close to that of our unknown, and we also note that the relative 
intensities listed for the three strongest lines of this substance agree well 

TABLE 14-1 
PATTERN OF UNKNOWN 



rf(A) 


///i 


d(A) 


///i 


2.59 


30 


1.14 


10 


2.35 


30 


1.07 


5 


2.27 


100 


1.01 


10 


.75 


20 


0.97 


20 


.50 


40 


0.93 


10 


.34 


40 


0.91 


10 


.27 


40 


0.89 


10 


.25 


40 


0.84 


10 


.19 


10 







384 



CHEMICAL ANALYSIS BY DIFFRACTION 



[CHAP. 14 



TABLE 14-2 
PORTION OF ASTM NUMERICAL INDEX 



d(A) 


1/h 


Substance 


Serial 
number 


2.28 


1.50 


1.78 


100 


100 


70 


Cs~Bi(NOJ, Cesium Bismuth 
J l Nitrite 


2-1129 


2.28 


1.50 


1.35 


100 


35 


35 


Mo 2 C Molybdenum Carbide 


1-1188 


2.28 


1.49 


1.10 


100 


100 


100 


Cs Ir(NOJ z Cesium Iridium 
3 2 6 Nitrite 


2-1130 


2.27 


1.49 


1.07 


100 


60 


60 


a-W 7 C Alpha Tungsten 
* Carbide 2:1 


2-1134 



with the observed intensities. We then refer to the data card bearing 
serial number 1-1188, reproduced in Fig. 14-1, and compare the complete 
pattern tabulated there with the observed one. Since the agreement is 
satisfactory for all the observed lines, the unknown is identified as molyb- 
denum carbide, Mo 2 C. 

When the unknown is composed of a mixture of phases, the anal- 
ysis naturally becomes more complex, but not impossible. Consider 
the pattern described in Table 14-3, for which d l = 2.09A, rf 2 = 2.47A, 
and d 3 = 1.80A. Examination of the numerical index in the c/i group 
2.09 to 2.05A reveals several substances having d 2 values near 2.47A, but 
in no case do the three strongest lines, taken together, agree with those of 
the unknown. This impasse suggests that the unknown is actually 
a mixture of phases, and that we are incorrect in assuming that the three 
strongest lines in the pattern of the unknown are all due to the same sub- 
stance. Suppose we assume that the strongest line (d = 2.09A) and the 
second-strongest line (d = 2. 47 A) are formed by two different phases, and 
that the third-strongest line (d = 1.80A) is due to, say, the first phase. 
In other words, we will assume that di = 2.09A and d 2 = 1.80A for one 
phase. A search of the same group of di values, but now in the vicinity of 
d 2 = 1.80 A, discloses agreement between the three strongest lines of the 
pattern of copper, serial number 4-0836, and three lines in the pattern of 
our unknown. Turning to card 4-0836, we find good agreement between 
all lines of the copper pattern, described in Table 14-4, with the starred 
lines in Table 14-3, the pattern of the unknown. 

One phase of the mixture is thus shown to be copper, providing we can 
account for the remainder of the lines as due to some other substance. 
These remaining lines are listed in Table 14-5. By multiplying all the 
observed intensities by a normalizing factor of 1.43, we increase the inten- 
sity of the strongest line to 100. We then search the index and card file 



14-4] 



EXAMPLES OF QUALITATIVE ANALYSIS 



385 



TABLE 14-3 
PATTERN OF UNKNOWN 



TABLE 14-4 
PATTERN OF COPPER 



d(A) 


I/h 


d(A) 


///i 


3.01 


5 


1.28* 


20 


2.47 


70 


1.08* 


20 


2.13 


30 


1.04* 


5 


2.09* 


100 


0.98 


5 


1.80* 


50 


0.91* 


5 


1.50 


20 


0.83* 


10 


1.29 


10 


0.81* 


10 


1.22 


5 







d(A) 


///i 


2.088 


100 


1.806 


46 


1.278 


20 


1.0900 


17 


1.0436 


5 


0.9038 


3 


0.8293 


9 


0.8083 


8 



in the usual way and find that these remaining lines agree with the pattern 
of cuprous oxide, Cu 2 O, which is given at the right of Table 14-5. The 
unknown is thus shown to be a mixture of copper and cuprous oxide. 

The analysis of mixtures becomes still more difficult when a line from 
one phase is superimposed on a line from another, and when this composite 
line is one of the three strongest lines in the pattern of the unknown. The 
usual procedure then leads only to a very tentative identification of one 
phase, in the sense that agreement is obtained for some d values but not 
for all the corresponding intensities. This in itself is evidence of line super- 
position. Such patterns can be untangled by separating out lines which 
agree in d value with those of phase X, the observed intensity of any super- 
imposed lines being divided into two parts. One part is assigned to phase 
X, and the balance, together with the remaining unidentified lines, is 
treated as in the previous example. 

Some large laboratories find it advantageous to use diffraction data cards 
containing a punched code. These are of two kinds, both obtainable from 
the ASTM: Keysort cards, which can be sorted semimechanically, and 

TABLE 14-5 



Remainder of pattern of unknown 


Pattern of Cu 7 O 





///i 


* 


-(A) 


///i 


Observed 


Normalized 


3.01 


5 


7 


3.020 


9 


2.47 


70 


100 


2.465 


100 


2.13 


30 


43 


2.135 


37 








.743 


1 


1.50 


20 


29 


.510 


27 


1.29 


10 


14 


.287 


17 


1.22 


5 


7 


.233 


4 








.0674 


2 


0.98 


5 


7 


0.9795 


4 








0.9548 


3 








0.8715 


3 








0.8216 


3 



386 CHEMICAL ANALYSIS BY DIFFRACTION [CHAP. 14 

standard IBM cards, which can be machine-sorted. A card file of either 
type can be searched on the basis of observed d values, and, in addition, 
particular categories of cards can be removed from the file more rapidly 
than by hand. For example, suppose a complex mixture is to be identified 
and it is known that one particular element, say copper, is present. Then 
the punch coding will permit rapid removal of the cards of all compounds 
containing copper, and the diffraction data on these cards can then be com- 
pared with the pattern of the unknown. 

14-5 Practical difficulties. In theory, the Hanawalt method should 
lead to the positive identification of any substance whose diffraction pat- 
tern is included in the card file. In practice, various difficulties arise, and 
these are usually due either to errors in the diffraction pattern of the un- 
known or to errors in the card file. 

Errors of the first kind, those affecting the observed positions and inten- 
sities of the diffraction lines, have been discussed in various parts of this 
book and need not be reexamined here. There is, however, one point that 
deserves some emphasis and that concerns the diffractometer. It must be 
remembered that the absorption factor for this instrument is independent 
of the angle 20, whereas, in a Debye-Scherrer camera, absorption decreases 
line intensity more at small than at large angles; the result is that the low- 
angle lines of most substances appear stronger, relative to medium- or 
high-angle lines, on a diffractometer chart than on a Debye-Scherrer photo- 
graph. This fact should be kept in mind whenever a diffractometer pattern 
is compared with one of the standard patterns in the ASTM file, because 
practically all of the latter were obtained with a Debye-Scherrer camera. 
On the other hand, it should not be concluded that successful use of the 
Hanawalt method requires relative intensity measurements of extremely 
high accuracy. It is enough, in most cases, to be able to list the lines in 
the correct order of decreasing intensity. 

Errors in the card file itself are generally more serious, since they may 
go undetected by the investigator and lead to mistaken identifications. 
Even a casual examination of the ASTM alphabetical index will disclose 
numerous examples of substances represented in the file by two or more 
cards, often with major differences in the three strongest lines listed. This 
ambiguity can make identification of the unknown quite difficult, because 
the user must decide which pattern in the file is the most reliable. Work 
is now in progress at the National Bureau of Standards to resolve such 
ambiguities, correct other kinds of errors, and obtain new standard pat- 
terns. The results of this work, which is all done with the diffractometer, 
are published from time to time in NBS Circular 539, "Standard X-Ray 
Diffraction Powder Patterns, "* and incorporated in card form in the most 

* Four sections of this circular have been issued to date: Vols. I and II in 1953, 
Vol. Ill in 1954, and Vol. IV in 1955. 



14-6] IDENTIFICATION OF SURFACE DEPOSITS 387 

recently issued sections of the ASTM file. 

Whenever any doubt exists in the investigator's mind as to the validity 
of a particular identification, he should prepare his own standard pattern. 
Thus, if the unknown has been tentatively identified as substance X, the 
pattern of pure X should be prepared under exactly the same experimental 
conditions used for the pattern of the unknown. Comparison of the two 
patterns will furnish positive proof, or disproof, of identity. 

The Hanawalt method fails completely, of course, when the unknown 
is a substance not listed in the card file, or when the unknown is a mixture 
and the component to be identified is not present in sufficient quantity to 
yield a good diffraction pattern. The latter effect can be quite trouble- 
some, and, as mentioned in Sec. 12-4, mixtures may be encountered which 
contain more than 50 percent of a particular component without the pat- 
tern of that component being visible in the pattern of the mixture. 

14-6 Identification of surface deposits. Metal surfaces frequently be- 
come contaminated, either by reaction of some substance with the base 
metal to produce a scale of oxide, sulfide, etc., or by simple adherence of 
some foreign material. Detection and identification of such deposits is 
usually an easy matter if the metal object is examined directly by some 
reflection method of diffraction, without making any attempt to remove 
the surface deposit for separate examination. 

A reflection method is particularly suitable because of the very shallow 
penetration of x-rays into most metals and alloys, as discussed at length 
in Sec. 9-5. The result is that most of the recorded diffraction pattern is 
produced by an extremely thin surface layer, a circumstance favorable to 
the detection of small amounts of surface deposits. The diffractometer is 
an ideal instrument for this purpose, particularly for the direct examination 
of sheet material. Its sensitivity for work of this kind is often surprisingly 
high, as evidenced by strong diffraction patterns produced by surface de- 
posits which are barely visible. 

An example of this kind of surface analysis occurred in the operations 
of a steel plant making mild steel sheet for "tin" cans. The tin coating 
was applied by hot-dipping, and the process was entirely satisfactory ex- 
cept for certain batches of sheet encountered from time to time which were 
not uniformly wetted by the molten tin. The only visible difference be- 
tween the satisfactory and unsatisfactory steel sheet was that the surface 
of the latter appeared somewhat duller than that of the former. Examina- 
tion of a piece of the unsatisfactory sheet in the diffractometer revealed 
the pattern of iron (ferrite) and a strong pattern of some foreign material. 
Reference to the ASTM card file showed that the surface deposit was 
finely divided graphite. 

One difficulty that may be encountered in identifying surface deposits 
from their diffraction patterns is caused by the fact that the individual 



388 CHEMICAL ANALYSIS BY DIFFRACTION [CHAP. 14 

crystals of such deposits are often preferentially oriented with respect to 
the surface on which they lie. The result is a marked difference between 
the observed relative intensities of the diffraction lines and those given on 
the ASTM cards for specimens composed of randomly oriented crystals. 
In the example just referred to, the reflection from the basal planes of the 
hexagonal graphite crystals was abnormally strong, indicating that most 
of these crystals were oriented with their basal planes parallel to the sur- 
face of the steel sheet. 



QUANTITATIVE ANALYSIS (SINGLE PHASE) 

14-7 Chemical analysis by parameter measurement. The lattice pa- 
rameter of a binary solid solution of B in A depends only on the percentage 
of B in the alloy, as long as the solution is unsaturated. This fact can be 
made the basis for chemical analysis by parameter measurement. All 
that is needed is a parameter vs. composition curve, such as curve be of 
Fig. 12-8(b), which can be established by measuring the lattice parameter 
of a series of previously analyzed alloys. This method has been used in 
diffusion studies to measure the change in concentration of a solution with 
distance from the original interface. Its accuracy depends entirely on the 
slope of the parameter-composition curve. In alpha brasses, which can 
contain from to about 40 percent zinc in copper, an accuracy of 1 per- 
cent zinc can be achieved without difficulty. 

This method is applicable only to binary alloys. In ternary solid solu- 
tions, for example, the percentages of two components can be independently 
varied. The result is that two ternary solutions of quite different compo- 
sitions can have the same lattice parameter. 

QUANTITATIVE ANALYSIS (MULTIPHASE) 

14-8 Basic principles. Quantitative analysis by diffraction is based on 
the fact that the intensity of the diffraction pattern of a particular phase 
in a mixture of phases depends on the concentration of that phase in the 
mixture. The relation between intensity and concentration is not gen- 
erally linear, since the diffracted intensity depends markedly on the 
absorption coefficient of the mixture and this itself varies with the con- 
centration. 

To find the relation between diffracted intensity and concentration, we 
must go back to the basic equation for the intensity diffracted by a powder 
specimen. The form of this equation depends on the kind of apparatus 
used, namely, camera or diffractometer; we shall consider only the diffrac- 
tometer here. [Although good quantitative work can be done, and has 
been done, with a Debye-Scherrer camera and microphotometer, the mod- 



14-8] QUANTITATIVE ANALYSIS: BASIC PRINCIPLES 389 

ern trend is toward the use of the diffractometer, because (a) this instru- 
ment permits quicker measurement of intensity and (b) its absorption 
factor is independent of B.] The exact expression for the intensity diffracted 
by a single-phase powder specimen in a diffractometer is: 



/7 e 4 \ / 

- GsO ( 






~ 2M 



where / = integrated intensity per unit length of diffraction line, 7 = 
intensity of incident beam, e, m = charge and mass of the electron, c = 
velocity of light, X = wavelength of incident radiation, r = radius of 
diffractometer circle, A = cross-sectional area of incident beam, v = vol- 
ume of unit cell, F = structure factor, p multiplicity, = Bragg angle, 
e -2M _ temperature factor (a function of 6) (previously referred to quali- 
tatively in Sec. 4-11), and M = linear absorption coefficient (which enters 
as 1/2M, the absorption factor). 

This equation, whose derivation can be found in various advanced texts, 
applies to a powder specimen in the form of a flat plate of effectively in- 
finite thickness, making equal angles with the incident and diffracted beams. 
[The fourth term in Eq. (14-1), containing the square of the structure 
factor, the multiplicity factor, and the Lorentz-polarization factor, will 
be recognized as the approximate equation for relative integrated inten- 
sity used heretofore in this book.] 

We can simplify Eq. (14-1) considerably for special cases. As it stands, 
it applies only to a pure substance. But suppose that we wish to analyze 
a mixture of two phases, a and /3. Then we can concentrate on a particular 
line of the a phase and rewrite Eq. (14-1) in terms of that phase alone. 
/ now becomes /, the intensity of the selected line of the a phase, and 
the right side of the equation must be multiplied by c a , the volume frac- 
tion of a in the mixture, to allow for the fact that the diffracting volume 
of a in the mixture is less than it would be if the specimen were pure a. 
Finally, we must substitute Mm for M, where Mm is the linear absorption 
coefficient of the mixture. In this new equation, all factors are constant 
and independent of the concentration of a except c a and Mm, and we can 
write 



la = (14-2) 

Mm 

where KI is a constant. 

To put Eq. (14-2) in a useful form, we must express M in terms of the 
concentration. From Eq. (1-12) we have 



Ma\ 

) 

Pa/ 



Mm Ma M/3 

= M 

Pm \Pa 



390 CHEMICAL ANALYSIS BT DIFFRACTION [CHAP. 14 

where w denotes the weight fraction and p the density. Consider unit 
volume of the mixture. Its weight is p m and the weight of contained a is 
w a p m . Therefore, the volume of a is w a p m /p a , which is equal to c a , and a 
similar expression holds for cp. Equation (14-3) then becomes 

Mm = CaMa + Cpup = C a /ia + ~ C a )/*/3 
= C a (fJLa - M0) + M/3J 



This equation relates the intensity of a diffraction line from one phase to 
the volume fraction of that phase and the linear absorption coefficients of 
both phases. 

We can put Eq. (14-4) on a weight basis by considering unit mass of 
the mixture. The volume of the contained a is w a /p a and the volume of 
ft is wp/pp. Therefore, 



^L (14-5) 

Wa/Pa + V>P/P0 
77)-. //>_ 

(14-6) 



Pa - 1/P0) 

Combining Eqs. (14-4) and (14-6) and simplifying, we obtain 



/. -- __ (14-7) 

Pa[u>a (Palp* - M0/P0) + M0/P0] 

For the pure a phase, either Eq. (14-2) or (14-7) gives 

I ap = ^- (14-8) 



ap 

Ma 



where the subscript p denotes diffraction from the pure phase. Division 
of Eq. (14-7) by Eq. (14-8) eliminates the unknown constant KI and 
gives 



lap Wa(v-a/Pa ~ M/8/P/?) + M/3/P/3 

This equation permits quantitative analysis of a two-phase mixture, pro- 
vided that the mass absorption coefficients of each phase are known. If 
they are not known, a calibration curve can be prepared by using mixtures 
of known composition. In each case, a specimen of pure a must be avail- 
able as a reference material, and the measurements of I a and I ap must be 
made under identical conditions. 



14-9] 



QUANTITATIVE ANALYSIS: DIRECT COMPARISON METHOD 



391 




In general, the variation of the intensity ratio 7 a //a P with w a is not 
linear, as shown by the curves of Fig. 14-2. The experimental points were 
obtained by measurements on synthetic binary mixtures of powdered 
quartz, cristobalite, beryllium oxide, 
and potassium chloride; the curves 
were calculated by Eq. (14-9). The 
agreement is excellent. The line 
obtained for the quartz-cristobalite 
mixture is straight because these sub- 
stances are two allotropic forms of 
silica and hence have identical mass 
absorption coefficients. When the 
mass absorption coefficients of the 
two phases are equal, Eq. (14-9) be- 
comes simply o 05 1 o 

WK1GHT FRACTION OF 
j QUARTZ W(l 

- - = w a . 

lap FIG. 14-2. Diffractometer meas- 

urements made with Cu Ka radiation 
Fig. 14-2 illustrates very clearly how on binary mixtures. /Q is the iriten- 

the intensity of a particular diffrac- *y of the reflection from the d = 

,. r " i , i 3.34A j)lanes of quartz in a mixture. 

tion lino from one phase depends on ^ w ^ inten j ty ()f ^ flamc ^ 

the absorption coefficient of the other fl e( , tion flom pure quartz. (L. E. 
phase. For Cu Ka radiation, the Alexander ami H. P. Klug, Anal. 
mass absorption coefficient of Be() is Chew. 20, XSG, 194S.) 
8.0, of Si() 2 is 34.9, and of KC1 is 124. 

For various reasons, the analytical procedure just outlined cannot be 
applied to most specimens of industrial interest. A variety of other meth- 
ods, however, has been devised to solve particular problems, and the two 
most important of these, the direct comparison method and the internal 
standard method, will be described in succeeding sections. It is worth noting 
that all these methods of analysis have one essential feature in common: 
the measurement of the concentration of a particular phase depends on the 
measurement of the ratio of the intensity of a diffraction line from that 
phase to the intensity of some standard reference line. In the "single line" 
method described above, the reference line is a line from the pure phase. 
In the direct comparison method, it is a line from another phase in the 
mixture. In the internal standard method, it is a line from a foreign mate- 
rial mixed with the specimen. 

14-9 Direct comparison method. This method is of greatest metallur- 
gical interest because it can be applied directly to massive, poly crystalline 
specimens. It has been widely used for measuring the amount of retained 



392 CHEMICAL ANALYSIS BY DIFFRACTION [CHAP. 14 

austenite in hardened steel and will be described here in terms of that 
specific problem, although the method itself is quite general. 

Many steels, when quenched from the austenite region, do not trans- 
form completely to martensite even at the surface. At room temperature, 
such steels consist of martensite and retained austenite; in addition, undis- 
solved carbides may or may not be present. The retained austenite is 
unstable and may slowly transform while the steel is in service. Since 
this transformation is accompanied by an increase in volume of about 
4 percent, residual stress is set up in addition to that already present, or 
actual dimensional changes occur. For these reasons, the presence of even 
a few percent retained austenite is undesirable in some applications, such 
as gage blocks, closely fitting machine parts, etc. There is therefore con- 
siderable interest in methods of determining the exact amount of austenite 
present. Quantitative microscopic examination is fairly satisfactory as 
long as the austenite content is fairly high, but becomes unreliable below 
about 15 percent austenite in many steels. The x-ray method, on the other 
hand, is quite accurate in this low-austenite range, often the range of 
greatest practical interest. 

Assume that a hardened steel contains only two phases, martensite and 
austenite. The problem is to determine the composition of the mixture, 
when the two phases have the same composition but different crystal 
structure (martensite is body-centered tetragonal and austenite is face- 
centered cubic). The "single line" method could be used if a sample of 
pure austenite or of known austenite content is available as a standard. 
Ordinarily, however, we proceed as follows. In the basic intensity equa- 
turn, Eq. (14-1), we put 



\32 

(14-10) 



The diffracted intensity is therefore given by 

/ = ^, (14-11) 

2n 

where K 2 is a constant, independent of the kind and amount of the diffract- 
ing substance, and R depends on d, hkl, and the kind of substance. Desig- 
nating austenite by the subscript y and martensite by the subscript a, we 
can write Eq. (14-11) for a particular diffraction line of each phase: 

/, = 

7 



14-9] QUANTITATIVE ANALYSIS! DIRECT COMPARISON METHOD 393 

/Y2/t a C a 

7a= ~^r 

Division of these equations yields 

p /. 

(14-12) 



The value of c y /c a can therefore be obtained from a measurement of 7 7 // a 
and a calculation of R y and R a . Once c y /c a is found, the value of C T can 
be obtained from the additional relationship: 



We can thus make an absolute measurement of the austenite content 
of the steel by direct comparison of the integrated intensity of an austenite 
line with the integrated intensity of a martensite line.* By comparing 
several pairs of austenite-martensite lines, we can obtain several inde- 
pendent values of the austenite content; any serious disagreement between 
these values indicates an error in observation or calculation. 

If the steel contains a third phase, namely, iron carbide (cementite), we 
can determine the cementite concentration either by quantitative micro- 
scopic examination or by diffraction. If we measure 7 C , the integrated 
intensity of a particular cementite line, and calculate RC, then we can set 
up an equation similar to Eq. (14-12) from which c 7 /c c can be obtained. 
The value of c 7 is then found from the relation 

c y + c a + c c = 1. 

In choosing diffraction lines to measure, we must be sure to avoid over- 
lapping or closely adjacent lines from different phases. Figure 14-3 shows 
the calculated patterns of austenite and martensite in a 1.0 percent carbon 
steel, made with Co Ka radiation. Suitable austenite lines are the 200, 
220, and 311 lines; these may be compared with the 002-200 and 112-211 
martensite doublets. These doublets are not usually resolvable into sepa- 
rate lines because all lines are usually quite broad, both from the martensite 
and austenite, as shown in Fig. 14-4. (Figure 14-4 also shows how refrig- 
eration, immediately after quenching to room temperature, can decrease 
the amount of retained austenite and how an interruption in the quench, 
followed by air cooling, can increase it.) The causes of line broadening are 
the nonuniform microstrains present in both phases of the quenched steel 
and, in many cases, the very fine grain size. 

* Recalling the earlier discussion of the disappearing-phase x-ray method of lo- 
cating a solvus line (Sec. 12-4), we note from Eq. (14-12) that the intensity ratio 
Iy/Ia is not a linear function of the volume fraction c^, or, for that matter, of the 
weight fraction w y . 



394 



CHEMICAL ANALYSIS BY DIFFRACTION 



[CHAP. 14 



26 (degrees ) 
90 


^> 
180 


111 




austcmte 




222 


200 220 


311 




101 


110 112 
002 200 


211 

202 




103 
220 


801 




310 




11KU teilslte 





FIG. 14-3. Calculated powder patterns of austenite and martensite, each con- 
taining 1.0 percent carbon. Co Ka radiation. 

In calculating the value of R for a particular diffraction line, various fac- 
tors should be kept in mind. The unit cell volume v is calculated from the 
measured lattice parameters, which are a function of carbon and alloy con- 
tent. When the martensite doublets are unresolved, the structure factor 
and multiplicity of the martensite are calculated on the basis of a body- 



austenite 
martensite 200 



220 




tiller-quenched and 
then cooled to -321F 



2 9'V austenite 




v\atei -quenched 



V*^^^ XvHrtv** 



9 3 r (, austenite 





quenched to 125F, 
air-cooled to room temperature 



14 \ c ' (l austenite 



FIG. 14-4. Microphotometer traces of Debye-Scherrer patterns of hardened 
1.07 percent carbon steel. Co Ka. radiation, inonochromated by reflection from 
an XaCl crystal. (B. L. Averbach and M. Colien, Trans. A.I.M.E. 176, 401 , 1948.) 



14-9] 



QUANTITATIVE ANALYSIS: DIRECT COMPARISON METHOD 



395 



centered cubic cell; this procedure, in effect, adds together the integrated 
intensities of the two lines of the doublet, which is exactly what is done 
experimentally when the integrated intensity of an unresolved doublet is 
measured. For greatest accuracy in the calculation of F, the atomic scat- 
tering factor / should be corrected for anomalous scattering by an amount 
A/ (see Fig. 13-8), particularly when Co Ka radiation is used. The Lo- 
rentz-polarization factor given in Eq. (14-10) applies only to unpolarized 
incident radiation; if crystal-monochromated radiation is used, this factor 
will have to be changed to that given in Sec. 6-12. The value of the tem- 
perature factor e~ 2M can be taken from the curve of Fig. 14-5. 



1 U 

09 
OS 
07 
0(i 




""""- 


"^ 


^ 
















^ 


N 


















\ 


















\ 



1 .2 3 4 5 7 8 



FIG. 14-5. Temperature factor e~* M of iron at 20C as a function of (sin 0)/X. 

Specimen preparation involves wet grinding to remove the surface layer, 
which may be decarburized or otherwise nonrepresentative of the bulk of 
the specimen, followed by standard metallographic polishing and etching. 
This procedure ensures a flat, reproducible surface for the x-ray examina- 
tion, and allows a preliminary examination of the specimen to be made 
with the microscope. In grinding and polishing, care should be taken not 
to produce excessive heat or plastic deformation, which would cause par- 
tial decomposition of both the martensite and austenite. 

In the measurement of diffraction line intensity, it is essential that the 
integrated intensity, not the maximum intensity, be measured. Large vari- 
ations in line shape can occur because of variations in microstrain and grain 
size. These variations in line shape will not affect the integrated intensity, 
but they can make the values of maximum intensity absolutely meaning- 



The sensitivity of the x-ray method in determining small amounts of 
retained austenite is limited chiefly by the intensity of the continuous back- 
ground present. The lower the background, the easier it is to detect and 
measure weak austenite lines. Best results are therefore obtained with 
crystal-monochromated radiation, which permits the detection of as little 
as 0.1 volume percent austenite. With ordinary filtered radiation, the 
minimum detectible amount is 5 to 10 volume percent. 



396 CHEMICAL ANALYSIS BY DIFFRACTION [CHAP. 14 

TABLE 14-6 

COMPARISON OF AUSTENITE DETERMINATION BY X-RAY DIFFRACTION AND 

LINEAL ANALYSIS* 



Austenitizing 
temperature (C) 


Volume percent 
carbides 


Volume percent retained 
austenite 


x -ray 


lineal anal /sis 


955 


0.2 


20. 01. 


20.1 1.0 


900 


2.6 


14.0 0.8 


13.8 1.0 


845 


4.0 


7.0 0.4 


6.0 1.0 


790 


10.0 


3.1 0.3 


2.01.0 



* B. L. Averbach and M. Cohen, Trans. A.LM.E. 176, 401 (194X). 

Table 14-6 gives a comparison between retained austenite determina- 
tions made on the same steel (1.0 percent C, 1.5 percent Cr, and 0.2 percent 
V) by x-ray diffraction and by quantitative microscopic examination (lineal 
analysis). The steel was austenitized for 30 minutes at the temperatures 
indicated and quenched in oil. The x-ray results were obtained with a 
Debye-Scherrer camera, a stationary flat specimen, and crystal-monochro- 
mated radiation. The carbide content was determined by lineal analysis. 
Note that the agreement between the two methods is good when the austen- 
ite content is fairly high, and that lineal analysis tends to show lower aus- 
tenite contents than the x-ray method when the austenite content itself is 
low (low austenitizing temperatures). This is not unexpected, in that the 
austenite particles become finer with decreasing austenitizing temperatures 
and therefore more difficult to measure microscopically. Under such cir- 
cumstances, the x-ray method is definitely more accurate. 

14-10 Internal standard method. In this method a diffraction line from 
the phase being determined is compared with a line from a standard sub- 
stance mixed with the sample in known proportions. The internal standard 
method is therefore restricted to samples in powder form. 

Suppose we wish to determine the amount of phase A in a mixture of 
phases A, B, C, . . . , where the relative amounts of the other phases pres- 
ent (B, C, D, . . . ) may vary from sample to sample. With a known 
amount of original sample we mix a known amount of a standard substance 
S to form a new composite sample. Let C A and C A ' be the volume fractions 
of phase A in the original and composite samples, respectively, and let cs 
be the volume fraction of S in the composite sample. If a diffraction pat- 
tern is now prepared from the composite sample, then from Eq. (14-2) 
the intensity of a particular line from phase A is given by 

, K S CA' 



14-10] QUANTITATIVE ANALYSIS: INTERNAL STANDARD METHOD 397 

and the intensity of a particular line from the standard S by 



Mm 

Division of one expression by the other gives 

I A ^3 C A 

= (14-13) 

(Note that Mm, the linear absorption coefficient of the mixture and an un- 
known quantity, drops out. Physically, this means that variations in 
absorption, due to variations in the relative amounts of B, C, D, . . . , 
have no effect on the ratio /A//S since they affect 7 A and 7g in the same 
proportion.) 

By extending Eq. (14-5) to a number of components, we can write 



WA 



VPA + WB'/PB + WC'/PC H h 



and a similar expression for eg. Therefore 



Substitution of this relation into Eq. (14-13) gives 

(14-14) 



if WQ is kept constant in all the composite samples. The relation between 
the weight fractions of A in the original and composite samples is: 



wjj = w A (l - w&). (14-15) 

Combination of Eqs. (14-14) and (14-15) gives 

^ = K,w A . (14-16) 

^s 

The intensity ratio of a line from phase A and a line from the standard S 
is therefore a linear function of WA, the weight fraction of A in the original 
sample. A calibration curve can be prepared from measurements on a 
set of synthetic samples, containing known concentrations of A and a con- 
stant concentration of a suitable standard. Once the calibration curve is 
established, the concentration of A in an unknown sample is obtained 
simply by measuring the ratio I A /I& for a composite sample containing 
the unknown and the same proportion of standard as was used in the cali- 
bration. 



398 



CHEMICAL ANALYSIS BY DIFFRACTION 



[CHAP. 14 






5 

WEIGHT FRACTION OF QUARTZ ITQ 

FIG. 14-6. Calibration curve for 
quartz analysis, with fluorite as inter- 
nal standard. /Q is the intensity of 
the d = 3.34A line of quartz, and 7 F 
is the intensity of the d = 3.16A line 
of fluorite. (L. E. Alexander and 
H. P. King, Anal. Chern. 20, 886, 
1948.) 



The internal standard method has 
been widely used for the measurement 
of the quartz content of industrial 
dusts. (Knowledge of the quartz con- 
tent is important in industrial health 
programs, because inhaled quartz or 
other siliceous material is the cause 
of the lung disease known as silicosis.) 
In this analysis, fluorite (CaF 2 ) has 
been found to be a suitable internal 
standard. Figure 14-6 shows a cali- 
bration curve prepared from mixtures 
of quartz and calcium carbonate, of 
known composition, each mixed with 
enough fluorite to make the weight 
fraction of fluorite in each composite 
sample equal to 0.20. The curve is 
linear and through the origin, as pre- 
dicted by Eq. (14-16). 

Strictly speaking, Eq. (14-16) is valid only for integrated intensities, 
and the same is true of all other intensity equations in this chapter. Yet 
it has been found possible to determine the quartz content of dusts with 
satisfactory accuracy by simply measuring maximum intensities. This 
short cut is permissible here only because the shape of the diffraction lines 
is found to be essentially constant from sample to sample. There is there- 
fore a constant proportionality between maximum and integrated intensity 
and, as long as all patterns are made under identical experimental condi- 
tions, the measurement of maximum intensities gives satisfactory results. 
Quite erroneous results would be obtained by this procedure if the particle 
size of the samples were very small and variable, since then a variable 
amount of line broadening would occur, and this would cause a variation 
in maximum intensity independent of sample composition. 

14-11 Practical difficulties. There are certain effects which can cause 
great difficulty in quantitative analysis because they cause observed in- 
tensities to depart widely from the theoretical. The most important of 
these complicating factors are : 

(1) Preferred orientation. The basic intensity equation, Eq. (14-1), is 
derived on the premise of random orientation of the constituent crystals 
in the sample and is not valid if any preferred orientation exists. It fol- 
lows that, in the preparation of powder samples for the diffractometer, 
every effort should be made to avoid preferred orientation. If the sample 
is a solid polycrystalline aggregate, the analyst has no control over the 



14-11] QUANTITATIVE ANALYSIS: PRACTICAL DIFFICULTIES 399 

distribution of orientations in it, but he should at least be aware of the pos- 
sibility of error due to preferred orientation. 

(2) Microabsorption. Consider diffraction from a given crystal of a in 
a mixture of a and crystals. The incident beam passes through both a 
and |8 crystals on its way to a particular diffracting a crystal, and so does 
the diffracted beam on its way out of the sample. Both beams are de- 
creased in intensity by absorption, and the decrease can be calculated from 
the total path length and /z m , the linear absorption coefficient of the mix- 
ture. But a small part of the total path lies entirely within the diffracting 
a crystal, and for this portion /* is the applicable absorption coefficient. 
If n a is much larger than JL% or if the particle size -of a is much larger than 
that of 0, then the total intensity of the beam diffracted by the a crystals 
will be much less than that calculated, since the effect of microabsorption 
in each diffracting a crystal is not included in the basic intensity equation. 
Evidently, the microabsorption effect is negligible when Ma M/J and both 
phases have the same particle size, or when the particle size of both phases 
is very small. Powder samples should therefore be finely ground before 
analysis. 

(3) Extinction. As mentioned in Sec. 3-7, all real crystals are im- 
perfect, in the sense that they have a mosaic structure, and the degree of 
imperfection can vary greatly from one crystal to another. Equation 
(14-1) is derived on the basis of the so-called "ideally imperfect'' crystal, 
one in which the mosaic blocks are quite small (of the order of 10~ 4 to 10~~ 5 
cm in thickness) and so disoriented that they are all essentially nonparallel. 
Such a crystal has maximum reflecting power. A crystal made up of large 
mosaic blocks, some or all of which are accurately parallel to one another, 
is more nearly perfect and has a lower reflecting power. This decrease in 
the intensity of the diffracted beam as the crystal becomes more nearly 
perfect is called extinction. Extinction is absent for the ideally imperfect 
crystal, and the presence of extinction invalidates Eq. (14-1). Any treat- 
ment which will make a crystal more imperfect will reduce extinction and, 
for this reason alone, powder specimens should be ground as fine as pos- 
sible. Grinding not only reduces the crystal size but also tends to decrease 
the mosaic block size, disorient the blocks, and strain them nonuniformly. 

Microabsorption and extinction, if present, can seriously decrease the 
accuracy of the direct comparison method, because this is an absolute 
method. Fortunately, both effects are negligible in the case of hardened 
steel. Inasmuch as both the austenite and martensite have the same com- 
position and only a 4 percent difference in density, their linear absorption 
coefficients are practically identical. Their average particle sizes are also 
roughly the same. Therefore, microabsorption does not occur. Extinc- 
tion is absent because of the very nature of hardened steel. The change 
in specific volume accompanying the transformation of austenite to mar- 



400 CHEMICAL ANALYSIS BY DIFFRACTION [CHAP. 14 

tensite sets up nonuniform strains in both phases so severe that both kinds 
of crystals can be considered highly imperfect. If these fortunate circum- 
stances do not exist, and they do not in most other alloy systems, the 
direct comparison method should be used with caution and checked by 
some independent method. 

On the other hand, the presence of microabsorption and extinction does 
not invalidate the internal standard method, provided these effects are 
constant from sample to sample, including the calibration samples. Micro- 
absorption and extinction affect only the values of the constants K 3 and 
K 4 in Eq. (14-13), and therefore the constant K Q in Eq. (14-16), and the 
latter constant determines only the slope of the calibration curve. There- 
fore, microabsorption and extinction, if present, will have no effect on the 
accuracy of the internal standard method as long as the crystals of the 
phase being determined, and those of the standard substance, do not vary 
in degree of perfection or particle size from one sample to another. 

PROBLEMS 

The d and l/l\ values tabulated in Probs. 14~1 to 14~4 represent the diffraction pat- 
terns of various unknown substances. Identify the substances involved by reference to 
an ASTM diffraction file. 

14-1. d(A)i I/I rf(A) ///i d(A) ///i 



3.66 ~5(T 1.46 10 1.06 10 

3.17 100 1.42 50 1.01 10 

2.24 80 1.31 30 0.96 10 

1.91 40 1.23 10 0.85 10 

1.83 30 1.12 10 

1.60 20 1.08 10 



14-2. 



5.85 60 2.08 10 1.47 20 

3.05 30 1.95 20 1.42 10 

2.53 100 1.80 60 1.14 20 

2.32 10 1.73 20 1.04 10 



14-3. 



240 5( 1.25 20 0.85 10 

2.09 50 1.20 10 0.81 20 

2.03 100 1.06 20 0.79 20 

1.75 40 1.02 10 

1.47 30 0.92 10 

1.26 10 



14-4. d(A) ///i 

3702 TocT 2AI 10 L46 10 

2.79 10 1.90 20 1.17 10 

2.52 10 1.65 10 

2.31 30 1.62 10 



PROBLEMS 401 

14-6. Microscopic examination of a hardened 1 .0 percent carbon steel shows no 
undissolved carbides. X-ray examination of this steel in a diffractometer with 
filtered cobalt radiation shows that the integrated intensity of the 311 austenite 
line is 2.325 and the integrated intensity of the unresolved 112-211 martensite 
doublet is 16.32, both in arbitrary units. Calculate the volume percent austenite 
in the steel. (Take lattice parameters from Fig. 12-5, A/ corrections from Fig. 
13-8, and temperature factors e~ 23f from Fig. 14-5.) 



CHAPTER 15 
CHEMICAL ANALYSIS BY FLUORESCENCE 

16-1 Introduction. We saw in Chap. 1 that any element, if made the 
target in an x-ray tube and bombarded with electrons of high enough en- 
ergy, would emit a characteristic line spectrum. The most intense lines of 
this spectrum are the Ka and K$ lines. They are always called "charac- 
teristic lines" to emphasize the fact that their wavelengths are fixed and 
characteristic of the emitting element. We also saw that these same lines 
would be emitted if the element were bombarded with x-rays of high enough 
energy (fluorescence). 

In these phenomena we have the basis for a method of chemical analysis. 
If the various elements in the sample to be analyzed are made to emit 
their characteristic lines by electron or x-ray bombardment, then these 
elements may be identified by analyzing the emitted radiation and showing 
that these specific wavelengths are present. The analysis is carried out in 
an x-ray spectrometer by diffracting the radiation from lattice planes of 
known d spacing in a single crystal. In accordance with the Bragg law, 
radiation of only a single wavelength is reflected for each angular setting 
of the crystal and the intensity of this radiation can be measured with a 
suitable counter. The analysis of the sample may be either qualitative, if 
the various characteristic lines in the emitted spectrum are simply identi- 
fied, or quantitative, if the intensities of these lines are compared with the 
intensities of lines from a suitable standard. 

Two kinds of x-ray spectroscopy are possible, depending on the means 
used to excite the characteristic lines : 

(1) The sample is made the target in an x-ray tube and bombarded with 
electrons. Historically, this was the first method. It was employed by 
Moseley in his work on the relation between characteristic wavelength and 
atomic number. It is not used today, except as an occasional research tool, 
because it has certain disadvantages for routine work. For example, the 
specimen must be placed in a demountable x-ray tube, which must then be 
evacuated before the analysis can begin. The same procedure has to be 
repeated for each sample. In addition, the heat produced in the sample by 
electron bombardment may cause some contained elements to vaporize. 

(2) The sample is placed outside the x-ray tube and bombarded with 
x-rays. The primary radiation (Fig. 15-1) causes the sample to emit sec- 
ondary fluorescent radiation, which is then analyzed in a spectrometer. 
This method, commonly known as fluorescent analysis, has come into wide 

402 



15-11 



INTRODUCTION 



403 



spectrometer circle 



x-iay nine 




rountci 



FIG. 15-1. Fluorescent x-rav spectroscopy. 

use in recent years. Tlie phenomenon ot fluorescence, which is just a nui- 
sance in diffraction experiments, is here made to serve a useful purpose. 

It may be helpful to compare some features of x-ray fluorescent analysis 
with those of optical spectroscopy, i.e , spectroscopy in the visible region of 
the spectrum, since the latter method has been used for years as a routine 
analytical tool and its essential features at least are well known. The main 
differences between the two methods are the following: 



Exciting agent 
Emitted radiation 
Analyzer 
Detector 

Nature of spectra 



Optical 
speotroscopy 

arc or spark 
visible light 
prism or grating 
photographic film 

or phototube 
complex 



Fluorescent 
analysis 

x-rays 
x-rays 
crystal 
photographic film 

or counter 
simple 



Both these methods give information about the chemical elements present 
in the sample, irrespective of their state of chemical combination or the 
phases in which they exist. X-ray diffraction, on the other hand, as we 
saw in the previous chapter, discloses the various compounds and phases 
present in the sample. Fluorescent analysis and diffraction analysis there- 
fore complement one another in the kind of information they provide. 

Fluorescent analysis is nondestructive and much more rapid than the 
ordinary wet methods of chemical analysis. It is best suited to determin- 
ing elements present in amounts ranging from a few percent up to 100 
percent, and in this range it is superior to optical spectroscopy. In gen- 
eral, fluorescent analysis is inferior to optical spectroscopy in the concen- 
tration range below 1 percent, but it can be used to advantage in this range 
in special cases. Fluorescent analysis is used today in the analysis of alloys 
(particularly high-alloy steels and high-temperature alloys), ores, oils, gaso- 
line, etc. 



404 CHEMICAL ANALYSIS BY FLUORESCENCE [CHAP. 15 

Chemical analysis by x-ray spectroscopy dates back to the pioneer work 
of von Hevesy and Coster in Germany about 1923. They used photo- 
graphic film to record the spectra. The x-ray method never became popu- 
lar, however, until recent years, when the development of various kinds of 
counters allowed direct measurement of x-ray intensity and thus decreased 
the time required for analysis. The methods of fluorescent analysis are 
still undergoing rapid development, and a wider range of application, to- 
gether with greater speed and accuracy, can be expected in the near future. 

16-2 General principles. Most fluorescent spectrometers, of which 
there are many forms, have the analyzing crystal and counter mechanically 
coupled, as in a diffractometer. Thus, when the crystal is set at a particular 
Bragg angle 0, the counter is automatically set at the corresponding angle 
26. The counter is connected to a sealer, or to a ratemeter and automatic 
recorder. The intensity of individual spectral lines emitted by the sample 
may be measured with the counter-sealer combination, or the whole spec- 
trum may be continuously scanned and recorded automatically. 

Figure 15-2 shows an example of a fluorescent spectrum automatically 
recorded with a commercial spectrometer. The wavelength of each spec- 
tral line is calculable from the corresponding Bragg angle and the inter- 
planar spacing of the analyzing crystal used. The primary radiation was 
supplied by a tungsten-target tube operated at 50 kv, and the sample was 
stainless steel containing 18 percent chromium and 8 percent nickel. The 
K lines of all the major constituents (Fe, Cr, and Ni) and of some of the 
minor constituents (Mn and Co) are apparent. (In addition, tungsten L 
lines can be seen; these will always be present when a tungsten tube is used, 
since they are excited in the tube and scattered by the sample into the 
beam of secondary radiation. The copper K lines are due to copper exist- 
ing as an impurity in the tungsten target.) 

In fluorescent spectrometry, the fluorescent radiation emitted by the 
sample and diffracted by the crystal should be as intense as possible, so 
that it will be accurately measurable in a short counting time. The in- 
tensity of this emitted radiation depends on both the wavelength and the 
intensity of the incident primary radiation from the x-ray tube. Suppose 
that monochromatic radiation of constant intensity and of wavelength X 
is incident on an element which has a K absorption edge at X#, and that we 
can continuously vary X. As we decrease X from a value larger than \K, 
no K fluorescence occurs until X is just shorter than \K- The fluorescent 
intensity is then a maximum. Further decrease in X causes the fluorescent 
intensity to decrease, in much the same manner as the absorption coeffi- 
cient. This is natural since, as mentioned in Sec. 1-5, fluorescence and 
true absorption are but two aspects of the same phenomenon. At any 



15-2] 



GENERAL PRINCIPLES 



405 




406 



CHEMICAL ANALYSIS BY FLUORESCENCE 
100 



[CHAP. 15 



w 
ffl 



80 



60 



20 




normal fluorescent 
analysis range 



JL 



05 1.0 1.5 20 25 3.0 

EMISSION-LINE WAVELENGTH (angstroms) 

FIG. 15-3. Variation with atomic number of the \\avelength of the strongest 
lines of the K and L series. 

one value of X, the fluorescent intensity is directly proportional to the inci- 
dent intensity. 

The best exciting agent would therefore be a strong characteristic line 
of wavelength just shorter than X#. It is clearly impossible to satisfy this 
requirement for more than one fluorescing element at a time, and in prac- 
tice we use a tungsten-target tube with as high a power rating as possible. 
The exciting radiation is then that part of the continuous spectrum and 
such L lines of tungsten as have shorter wavelengths than the absorption 
edge of the fluorescing element. Molybdenum-target tubes are also used. 

The beam of secondary radiation issuing from the sample consists largely 
of fluorescent radiation, but there are some other weak components present 
as well. These are coherent scattered radiation, coherent diffracted radia- 
tion, -and incoherent (Compton modified) radiation. These components 
are partially scattered and diffracted by the analyzing crystal into the 
counter, and appear as a background on which the spectral lines are super- 
imposed. This background is normally low (see Fig. 15-2), but it may 
become rather high if the sample contains a large proportion of elements of 
low atomic number, because the sample will then emit a large amount of 
Compton modified radiation. 

The useful range of fluorescent wavelengths extends from about 0.5 to 
about 2.5A. The lower limit is imposed by the maximum voltage which 
can be applied to the x-ray tube, which is 50 kv in commercial instruments. 
At this voltage the short-wavelength limit of the continuous spectrum from 
the tube is 12,400/50,000 = 0.25A. The maximum intensity occurs at 
about 1.5 times this value, or 0.38A. Incident radiation of this wavelength 



15-3] SPECTROMETERS 407 

would cause K fluorescence in tellurium (atomic number 52), and the 
emitted Ka radiation would have a wavelength of 0.45A. At a tube volt- 
age of 50 kv, little or no K fluorescence is produced in elements with atomic 
numbers greater than about 55, and for such elements the L lines have to 
be used. Figure 15-3 shows how the wavelength of the strongest line in 
each of these series varies with atomic number. 

The upper limit of about 2.5A is imposed by the very large absorption of 
radiation of this wavelength by air and the counter window. This factor 
limits the elements detectable by fluorescence to those with atomic numbers 
greater than about 22 (titanium). Ti Ka radiation (X = 2.75A) is de- 
creased to one-half its original intensity by passage through only 10 cm of 
air. If a path filled with helium is provided for the x-rays traversing the 
spectrometer, absorption is decreased to such an extent that the lower limit 
of atomic number is decreased to about 13 (aluminum). Boron (atomic 
number 5) should be detectable in a vacuum spectrometer. 

Another important factor which limits the detection of light elements is 
absorption in the sample itself. Fluorescent radiation is produced not only 
at the surface of the sample but also in its interior, to a depth depending 
on the depth of effective penetration by the primary beam, which in turn 
depends on the over-all absorption coefficient of the sample. The fluores- 
cent radiation produced within the sample then undergoes absorption on 
its way out. Since long-wavelength fluorescent radiation will be highly 
absorbed by the sample, the fluorescent radiation outside the sample comes 
only from a thin surface skin and its intensity is accordingly low. It fol- 
lows that detection of small amounts of a light element in a heavy-element 
matrix is practically impossible. On the other hand, even a few parts per 
million of a heavy element in a light-element matrix can be detected. 

16-3 Spectrometers. There are various types of fluorescent spectrom- 
eters, differentiated by the kind of analyzing crystal used: flat, curved 
transmitting, or curved reflecting. 

The flat crystal type, illustrated in Fig. 15-4, is the simplest in design. 
The x-ray tube is placed as close as possible to the sample, so that the pri- 
mary radiation on it, and the fluorescent radiation it emits, will be as in- 
tense as possible. For the operator's protection against scattered radiation, 
the sample is enclosed in a thick metal box, which contains a single opening 
through which the fluorescent beam leaves. The sample area irradiated is 
of the order of f in. square. Fluorescent radiation is emitted in all direc- 
tions by this area, which acts as a source of radiation for the spectrometer 
proper. Because of the large size of this source, the beam of fluorescent 
radiation issuing from the protective box contains a large proportion of 
widely divergent and convergent radiation. Collimation of this beam be- 
fore it strikes the analyzing crystal is therefore absolutely necessary, if any 



408 



CHEMICAL ANALYSIS BY FLUORESCENCE [CHAP. 15 

x-rav tube 



sample 




FIG. 15-4. Essential parts of a fluorescent x-ray spectrometer, flat-crystal type 
(schematic). 

resolution at all is to be obtained. This collimation is achieved by passing 
the beam through a Seller slit whose plates are at, right angles to the plane 
of the spectrometer circle, because it is the divergence (and convergence) 
in this plane that we want to eliminate. 

Essentially parallel radiation from the collimator is then incident on the 
flat crystal, and a portion of it is diffracted into the counter by lattice planes 
parallel to the crystal face. Since no focusing occurs, the beam diffracted 
by the crystal is fairly wide and the counter receiving slit must also be wide. 
The analyzing crystal is usually NaCl or LiF, with its face cut parallel to 
the (200) planes. 



x-ray tube 



- sample 




/conn lor 



FIG. 15-5. Fluorescent x-ray spectrometer, curved-transmitting-ciystul type 
(schematic). 



15-3] SPECTROMETERS 409 

Both the commercial diffractometers mentioned in Sec. 7-2 can be 
readily converted into fluorescent spectrometers of this kind. The conver- 
sion involves the substitution of a high-powered (50-kv, 50-ma) tungsten- 
or molybdenum-target tube for the usual tube used in diffraction experi- 
ments, and the addition of an analyzing crystal, a shielded sample box, and 
a different Soller slit. 

The main features of a spectrometer employing a curved transmitting 
crystal are shown in Fig. 15-5. The crystal is usually mica, which is easily 
obtainable in the form of thin flexible sheets. The beam of secondary 
radiation from the sample passes through a baffled tunnel, which removes 
most of the nonconverging radiation. The convergent beam is then re- 
flected by the transverse (33l) planes of the bent mica crystal, and focused 
on the receiving slit of the counter. (The focusing action of such a crystal 
is described in Sec. 6-12.) The beam tunnel is not an essential part of the 
instrument; for a given setting of the crystal, only incident convergent radi- 
ation of a single wavelength will be diffracted into the counter slit. The 
only purpose of the tunnel is to protect the operator by limiting the beam. 

A set of two or three mica crystals of different thicknesses is needed to 
obtain the highest diffraction efficiency over the whole range of wave- 
lengths, inasmuch as thin crystals must be used in analyzing easily ab- 
sorbed long-wavelength radiation and thicker crystals for harder radiation. 
The thickness range is about 0.0006 to 0.004 in. 

Besides the usual two-to-one coupling between the counter and crystal, 
this spectrometer must also have a mechanism for changing the radius of 
curvature of the crystal with every change in 0, in order that the diffracted 
rays be always focused at the counter slit. The necessary relation between 
the radius of curvature 27? (R is the radius of the focusing circle) and the 
crystal-to-focus distance D is given by Eq. (6-15), which we can write in 
the form 

D 
2R = 

COS0 

to emphasize the fact that D is fixed and equal to the radius of the spec- 
trometer circle. The change in 2R with change in 6 is accomplished auto- 
matically in commercial instruments of this type. The General Electric 
diffractometer shown in Fig. 7-2 may be converted into either this kind of 
spectrometer or the flat crystal type. 

The curved reflecting crystal spectrometer is illustrated in Fig. 15-6. 
Radiation from the sample passes through the narrow slit S and diverges 
to the crystal (usually NaCl or LiF), which has its reflecting planes bent 
to a radius of 2R and its surface ground to a radius R. Diffracted radiation 
of a single wavelength is brought to a focus at the counter receiving slit, 
located on the focusing circle passing through S and the face of the crystal, 



410 



CHEMICAL ANALYSIS BY FLUORESCENCE 

x-ray tube 



[CHAP. 15 



crystal 




sample 



\ counter 



FIG. 15-6. Fluorescent x-ray spectrometer, curved-reflecting-crystal type. 

as described in Sec. 6-12. But now the radius R of the focusing circle is 
fixed, for a crystal of given curvature, and the slit-to-crystal and crystal- 
to-focus distances must both be varied as 6 is varied. The focusing relation, 
found from Eq. (6-13), is 

D = 2R sin 0, 

where D stands for both the slit-to-crystal and crystal-to-focus distances, 
which must be kept equal to one another. This is accomplished by rotation 
of both the crystal and the counter about the center of the focusing circle, 
in such a manner that rotation of the crystal through an angle x (about 0) 
is accompanied by rotation of the counter through an angle 2x. At the 
same time the counter is rotated about a vertical axis through its slit, by 
means of another coupling, so that it always points at the crystal. 

D increases as 6 increases and may become inconveniently large, for a 
crystal of given radius of curvature R\, at large values. In order to keep 
D within reasonable limits, it is necessary to change to another crystal, of 
smaller radius 7? 2 , for this high-0 (long-wavelength) range. 

Spectrometers employing curved reflecting crystals are manufactured by 
Applied Research Laboratories. 

15-4 Intensity and resolution. We must now consider the two main 
problems in fluorescent analysis, namely the attainment of adequate in- 
tensity and adequate resolution. The intensity of the fluorescent radiation 



15-4] 



INTENSITY AND RESOLUTION 



411 



emitted by the sample is very much less than that of the primary radiation 
incident on it, and can become very low indeed when the fluorescing ele- 
ment is only a minor constituent of the sample. This fluorescent radiation 
is then diffracted by the analyzing crystal, and another large loss of in- 
tensity occurs, because diffraction is such an inefficient process. The dif- 
fracted beam entering the counter may therefore be very weak, and a long 
counting time will be necessary to measure its intensity with acceptable 
accuracy. Spectrometer design must therefore ensure maximum intensity 
of the radiation entering the counter. At the same time, the spectrometer 
must be capable of high resolution, if the sample contains elements which 
have characteristic lines of very nearly the same wavelength and which 
must be separately ident ified. Both these factors, intensity and resolution, 
are affected by the kind of analyzing crystal used and by other details of 
spectrometer design. 

If we define resolution, or resolving 
power, as the ability to separate 
spectral lines of nearly the same wave- 
length, then we see from Fig. 15-7 
that resolution depends both on A20, 
the dispersion, or separation, of line 
centers, and on B, the line breadth at y 
half-maximum intensity. The resolu- H 
tion will be adequate if A20 is equal to 
or greater than 2B. By differentiat- 
ing the Bragg law, we obtain 



A20 



X 
AX 



2 tan 




A20 



(15-1) 



When the minimum value of A20, 
namely 2B, is inserted, this becomes 

X tan 

= (15-2) 

AX B 



FIG. 15 7. Resolution of closely 
spaced spectral lines. The lines sho\\ n 
have A20 = 2B. Any smaller separa- 
tion might make the two lines appear 
as one. 



The left-hand side of this equation gives the resolution required to separate 
two lines of mean wavelength X and wavelength difference AX. The right- 
hand side gives the resolving power available, and this involves both the 
mean Bragg angle of the lines and their breadth. Note that the available 
resolving power increases rapidly with 0, for a given line breadth. This 
means that, of two crystals producing the same line breadth, the one with 
the smaller plane spacing d will have the greater resolving power, because 
it will reflect to higher 20 angles. The crystals normally used in spectrom- 
eters have the following d values: mica, (33l) planes, 1.5A; LiF, (200) 
planes, 2.01 A; NaCl, (200) planes, 2.82A. For a given crystal, second- 



412 CHEMICAL ANALYSIS BY FLUORESCENCE [CHAP. 15 

order reflections provide greater resolving power than first-order reflections, 
because they occur at larger angles, but their intensity is less than a fifth 
of that of first-order reflections. 

The factors affecting the line width B can be discussed only with refer- 
ence to particular spectrometers. In the flat crystal type (Fig. 15-4), 
the value of B depends partly on the collimation of the beam striking the 
crystal and partly on the perfection of the crystal itself. The beam re- 
flected by the crystal into the counter is fairly wide, in a linear sense, but 
almost parallel; its angular width is measured by its divergence, and this is 
equal, if the crystal is perfect, to the divergence of the beam striking the 
crystal. The latter divergence is controlled by the Soller slit. If I is the 
length of the slit and 5 the spacing between plates, then the maximum di- 
vergence allowed is 2$ 

a = radian. 

For a typical slit with I = 4 in. and s = 0.010 in., a = 0.3. But further 
divergence is produced by the mosaic structure of the analyzing crystal: 
this divergence is related to the extent of disorientation of the mosaic 
blocks, and has a value of about 0.2 for the crystals normally used. The 
line width B is the sum of these two effects and is therefore of the order of 
0.5. The line width can be decreased by increasing the degree of collima- 
tion, but the intensity will also be decreased. Conversely, if the problem 
at hand does not require fine resolution, a more "open" collimator is used 
in order to increase intensity. Normally, the collimation is designed to 
produce a line width of about 0.5, which will provide adequate resolution 
for most work. 

In the curved transmitting crystal spectrometer (Fig. 15-5), the line 
width B depends almost entirely on the degree of focusing of the reflected 
beam at the counter slit. The focusing action of the bent mica crystal, 
although never perfect, can be made good enough to produce extremely fine 
lines if a very narrow slit is used; however, the intensity would then be low, 
so the width of the counter slit is usually made equal to 0.3 to achieve a 
reasonable balance between line width and intensity. Even so, the inten- 
sity is still less than that produced by a flat crystal of NaCl or LiF. 

When a curved reflecting crystal (Fig. 15-6) is used, the line width de- 
pends mainly on the width of the source slit S and the precision with which 
the crystal is ground and bent. The line width is normally about the same 
as that obtained with a flat crystal, namely, about 0.5. 

When intensities are considered, we find tha't a curved reflecting crystal 
provides the greatest intensity and a curved transmitting crystal the least, 
with a flat crystal in an intermediate position. 

Returning to the question of resolution, we can now calculate the resolv- 
ing powers available with typical spectrometers, and compare these values 



15-4] INTENSITY AND RESOLUTION 413 

with the maximum resolution required to separate closely spaced spectral 
lines. The smallest wavelength difference in the K series occurs between 
the K/3 line of an element of atomic number Z and the Ka line of an element 
of atomic number (Z + 1). This difference itself varies with atomic num- 
ber and is least for the K0 line of vanadium (Z = 23) and the Ka line of 
chromium (Z = 24); these two wavelengths are 2.284 and 2.291 A, respec- 
tively, and their difference is only 0.007A. A more common problem is the 
separation of the Kft line of chromium (Z = 24) from the Ka line of man- 
ganese (Z = 25), since both of these elements occur in all stainless steels. 
The wavelength difference here is 0.018A and the mean wavelength 2.094A. 
The required resolution X/AX is therefore 2.094/0.018 or 116. The avail- 
able resolving powers are given by (tan 0)/B, and are equal to 182 for 
curved mica in transmission, 70 for flat or curved LiF in reflection, and 46 
for flat or curved NaCl in reflection, for assumed line widths of 0.3, 0.5, 
and 0.5, respectively, and first-order reflections. Mica would therefore 
provide adequate resolution, but LiF and NaCl would not.* Figure 15-2 
shows the Cr K/3 and Mn Ka lines resolved with a mica crystal in the spec- 
trum of a stainless steel. 

To sum up, flat or curved crystals of either LiF or NaCl produce much 
higher reflected intensities but have lower resolution than curved mica 
crystals. High intensity is desirable in fluorescent analysis in order that 
the counting time required to obtain good accuracy be reasonably short; if 
the element to be detected is present only in small concentrations and a 
crystal of low reflecting power is used, the required counting times will be 
prohibitively long. In the determination of major elements, any of the 
three types of crystals will give adequate intensity. High resolution is de- 
sirable whenever the analysis requires use of a spectral line having very 
nearly the same wavelength as another line from the sample or the x-ray 
tube target. 

There is another point that deserves some consideration, namely, the 
angle 26 at which a particular wavelength is reflected by the analyzing 
crystal. This angle depends only on the d spacing of the crystal. The 
Bragg law shows that the longest wavelength that can be reflected is equal 
to 2d. But wavelengths approaching 2d in magnitude are reflected almost 
backward, and their reflected intensity is low at these large angles. We 
are consequently limited in practice to wavelengths not much longer than d. 
This means that a crystal like gypsum (d = 7. 6 A) must be used to detect a 
light element like aluminum whose Ka wavelength is 8.3A. Some of the 



* An alternative, but equivalent, way of arriving at the same result is to calcu- 
late the dispersion A20 produced by a given crystal and compare it with the dis- 
persion required, namely, 2B. The value of A20 is given by 2 tan 0(AX/X), from 
Eq. (15-1), and is equal to 1.0 for mica, 0.6 for LiF, and 0.4 for NaCl, for first- 
order reflections. The corresponding assumed values of 2B are 0.6, 1.0, and 1.0. 



414 CHEMICAL ANALYSIS BY FLUORESCENCE [CHAP. 15 

other crystals that have been used for light-element detection are oxalic 
acid (d = 6.1A) and mica in reflection (d = 10. 1A). 

15-5 Counters. The reader is advised to review at this point the gen- 
eral discussion of counters given in Chap. 7. Here we are concerned mainly 
with the variation in counter behavior with variation in x-ray wavelength. 
This variation is of no great importance in diffractometer measurements, 
since all diffracted beams have the same wavelength. In spectrometry, 
however, each spectral line has a different wavelength, and variations in 
counter behavior with wavelength must be considered. 

The pulse size is inversely proportional to x-ray wavelength in propor- 
tional and scintillation counters, but independent of wavelength in Geiger 
counters. Of more importance, however, is the variation of counter effi- 
ciency with wavelength. The efficiency of a gas-filled counter (propor- 
tional or Geiger) depends on the gas used; in this respect, krypton is supe- 
rior to argon for fluorescent analysis, in that krypton detects all radiation 
having wavelengths greater than 0.5 A fairly efficiently while argon does 
not (see Fig. 7-17). Below 0.5A, both gases have low efficiency. The 
scintillation counter, on the other hand, is almost 100 percent efficient for 
all wavelengths. The use of scintillation counters in conjunction with 
x-ray tubes operable at higher voltages than those now available would 
permit the detection of heavy elements by their fluorescent A" lines having 
wavelengths below 0.5A. 

Counter speed is another important factor in quantitative analysis, be- 
cause a counter which can operate at high counting rates without losses 
can be used to measure both strong lines and weak lines without correc- 
tions or the use of absorbing foils. In this respect, proportional and scintil- 
lation counters are definitely superior to Geiger counters. 

15-6 Qualitative analysis. In qualitative work sufficient accuracy can 
be obtained by automatic scanning of the spectrum, with the counter out- 
put fed to a chart recorder. Interpretation of the recorded spectrum will 
be facilitated if the analyst has on hand (a) a table of corresponding values 
of X and 26 for the particular analyzing crystal used, and (b) a single table 
of the principal K and L lines of all the elements arranged in numerical 
order of wavelength. 

Since it is important to know whether an observed line is due to an ele- 
ment in the sample or to an element in the x-ray tube target, a preliminary 
investigation should be made of the spectrum emitted by the target alone. 
For this purpose a substance like carbon or plexiglass is placed in the sam- 
ple holder and irradiated in the usual way; such a substance merely scat- 
ters part of the primary radiation into the spectrometer, and does not con- 
tribute any observable fluorescent radiation of its own. The spectrum so 



15-7] 



QUANTITATIVE ANALYSIS 



415 



obtained will disclose the L lines of tungsten, if a tungsten-target tube is 
used, as well as the characteristic lines of whatever impurities happen to 
be present in the target. 

15-7 Quantitative analysis. In determining the amount of element A 
in a sample, the single-line method is normally used: the intensity / u of a 
particular characteristic line of A from the unknown is compared with the 
intensity 7 b of the same line from a standard, normally pure A. The way 
in which the ratio I U /I 8 varies with the concentration of A in the sample 
depends markedly on the other elements present and cannot in general be 
predicted by calculation. It is therefore necessary to establish the varia- 
tion by means of measurements made on samples of known composition. 
Figure 15-8 illustrates typical curves of this kind for three binary mixtures 
containing iron. 

These curves show that the intensity of a fluorescent line from element A 
is not in general proportional to the concentration of A. This nonlinear be- 
havior is due mainly to two effects: 

(1) Matrix absorption. As the composition of the alloy changes, so does 
its absorption coefficient. As a result there are changes both in the absorp- 
tion of the primary radiation traveling into the sample and in the absorp- 
tion of the fluorescent radiation traveling out. The absorption of the pri- 
mary radiation is difficult to calculate, because the part of that radiation 
effective in causing K fluorescence, for example, in A has wavelengths ex- 



l.o 



08 



06 



/u 

/s 



0.4 



0.2 




Fe-Ni 



30 



40 50 60 

ATOMIC PERCENT Fe 



70 



80 



90 



100 



FIG. 15-8. Effect of iron concentration on the intensity of Fe Ka radiation 
fluoresced by various mixtures. 7 U and / B are the Fe Ka intensities from the mix- 
ture and from pure iron, respectively. (H. Friedman and L. S. Birks, Rev. 8ci. 
Inst. 19, 323, 1948.) 



416 CHEMICAL ANALYSIS BY FLUORESCENCE [CHAP. 15 

tending from XSWL, the short-wavelength limit of the continuous spectrum, 
to X#A, the K absorption edge of A. To each of these incident wavelengths 
corresponds a different incident intensity and a different matrix absorption 
coefficient. The absorption of the fluorescent radiation, of wavelength 
X/A, depends only on the absorption coefficient of the specimen for that 
particular wavelength. (Absorption effects are particularly noticeable in 
the Fe-Al and Fe-Ag curves of Fig. 15-8. The absorption coefficient of an 
Fe-Al alloy is less than that of an Fe-Ag alloy of the same iron content, 
with the result that the depth of effective penetration of the incident beam 
is greater for the Fe-Al alloy. A larger number of iron atoms can therefore 
contribute to the fluorescent beam, and this beam itself will undergo less 
absorption than in the Fe-Ag alloy. The over-all result is that the intensity 
of the fluorescent Fe Ka radiation outside the specimen is greater for the 
Fe-Al alloy.) 

(2) Multiple excitation. If the primary radiation causes element B in 
the specimen to emit its characteristic radiation, of wavelength X/B, and if 
X/B is less than \KA, then fluorescent K radiation from A will be excited 
not only by the incident beam but also by fluorescent radiation from B. 
(This effect is evident in the Fe-Ni curve of Fig. 15-8. Ni Ka radiation 
can excite Fe Ka radiation, and the result is that the observed intensity of 
the Fe Ka radiation from an Fe-Ni alloy is closer to that for an Fe-Al alloy 
of the same iron content than one would expect from a simple comparison 
of the absorption coefficients of the two alloys. In the case of an Fe-Ag 
alloy, the observed Fe Ka intensity is much lower, even though Ag Ka 
can excite Fe Ka> because of the very large absorption in the specimen.) 

Because of the complications these effects introduce into any calculation 
of fluorescent intensities, quantitative analysis is always performed on an 
empirical basis, i.e., by the use of standard samples of known composition. 
The greatest use of fluorescent analysis is in control work, where a great 
many samples of approximately the same composition have to be analyzed 
to see if their composition falls within specified limits. For such work, the 
calibration curves need not be prepared over a 0-100 percent range, as in 
Fig. 15-8, but only over quite limited composition ranges. The usual refer- 
ence material for such analyses is one of the standard samples used in the 
calibration, rather than a pure metal. 

Sample preparation for fluorescent analysis is not particularly difficult. 
Solid samples are ground to produce a flat surface but need not be polished; 
however, a standardized method of sample preparation should be adhered 
to for best results. Powder specimens, finely ground and well mixed, can 
be pressed into special holders; adequate mixing is essential, since only a 
thin surface layer is actually analyzed and this must be representative of 
the whole sample. Liquid samples can be contained in various kinds of 
cells. 



16-8] AUTOMATIC SPECTROMETERS 417 

Line intensities should be measured with a sealer rather than taken from 
a recorded chart. For a given line intensity, the accuracy of the analysis 
depends on the time spent in counting, since the relative probable error in 
a measurement of N counts is proportional to l/\/Af. If a line is weak, a 
correction must be made for the background of scattered and diffracted 
radiation. Because of this background, the number of counts required to 
obtain a given accuracy in the measurement of a weak line is larger than 
that required for a strong line (see Eq. 7-7). 

Since the intensity of a particular line from the sample is usually com- 
pared with the intensity of the same line from a standard, the output of the 
x-ray tube must be stabilized or the tube must be monitored. 

The resolution of the spectrometer should be no greater than that re- 
quired by the particular analytical problem involved. The analyzing 
crystal and collimator or counter slit should be chosen to produce this 
minimum amount of resolution and as much intensity as possible, since 
the greater the intensity, the less time required for analysis. 

16-8 Automatic spectrometers. Automatic direct-reading optical spec- 
trometers have been in use for several years and have proved to be of great 
value in industrial process control. A sample is inserted and the concen- 
trations of a number of selected elements are rapidly and directly indicated 
on a chart or set of dials. Because such spectrometers must be preset and 
precalibrated for each particular element determined, they are suitable 
only for control laboratories where large numbers of samples must be ana- 
lyzed for the same set of elements, each of which is variable only over a 
limited range of concentration. 

Recently, x-ray counterparts of these direct-reading optical spectrom- 
eters have become available. There are two types: 

(1) Single-channel type. An instrument of this kind is manufactured by 
North American Philips Co. and called the Autrometer. It uses a flat ana- 
lyzing crystal in reflection and a scintillation counter as a detector. Cor- 
responding to the elements A, B, C, ... to be detected are the wavelengths 
X/A, VB, Vc, of their characteristic spectral lines, and to these corre- 
spond certain diffraction angles 20A, 20B, 20c, ... at which these wave- 
lengths will be diffracted by the crystal. The counter is designed to move 
stepwise from one predetermined angular position to another rather than to 
scan a certain angular range. The various elements are determined in se- 
quence: the counter moves to position 20A, remains there long enough to 
accurately measure the intensity of the spectral line from element A, moves 
rapidly to position 20B, measures the intensity of the line from B, and so on. 
At each step the intensity of the line from the sample is automatically com- 
pared with the intensity of the same line from the standard and the ratio of 
these two intensities is printed on a paper tape. The instrument may also be 



418 



CHEMICAL ANALYSIS BY FLUORESCENCE 



[CHAP. 15 



to control 
channel 



standard 
sample 



crystal 

^ x focusing circle 



receiving sli 




counter 



FIG. 15-9. Relative arrangement of x-ray tube, sample, and one analyzing 
channel of the X-Ray Quantometer (schematic). (The tube is of the "end-on" 
type: the face of the target is inclined to the tube axis and the x-rays produced 
escape through a window in the end of the tube.) 

adjusted so that the actual concentration of the element involved is printed 
on the tape. As many as twelve elements per sample may be determined. 
The curved reflecting crystal spectrometer manufactured by Applied Re- 
search Laboratories (see Sec. 15-3) may also be arranged for this kind of 
automatic, sequential line measurement. 

(2) Multichannel type, manufactured by Applied Research Laboratories 
and called the X-Ray Quantometer. The analyzing crystal is a bent and 
cut LiF or NaCl crystal, used in reflection. Near the sample is a slit which 
acts as a virtual source of divergent radiation for the focusing crystal (Fig. 
15-9). Eight assemblies like the one shown, each consisting of slits, ana- 
lyzing crystal, and counter, are arranged in a circle about the centrally 
located x-ray tube; seven of these receive the same fluorescent radiation 
from the sample, while the eighth receives fluorescent radiation from a 
standard. Each of these seven assemblies forms a separate "channel" 
for the determination of one particular element in the sample. In channel 
A, for example, which is used to detect element A, the positions of the crys- 
tal and counter are preset so that only radiation of wavelength X/ A can be 
reflected into the counter. The components of the other analyzing chan- 
nels are positioned in similar fashion, so that a separate spectral line is 
measured in each channel. The eighth, or control, channel monitors the 
output of the x-ray tube. 

In this instrument each counter delivers its pulses, not to a sealer or rate- 
meter, but to an integrating capacitor in which the total charge delivered 
by the counter in a given length of time is collected. When a sample is 
being analyzed, all counters are started simultaneously. When the control 
counter has delivered to its capacitor a predetermined charge, i.e., a pre- 
determined total number of counts, all counters are automatically stopped. 
Then the integrating capacitor in each analyzing channel discharges in 
turn into a measuring circuit and recorder, and the total charge collected 



L5-9] NONDISPERSIVE ANALYSIS 419 

in each channel is recorded in sequence on a chart. The quantity indi- 
cated on the chart for each element is the ratio of the intensity of a given 
spectral line from the sample to that of a line from the standard, and the 
instrument can be calibrated so that the concentration of each element in 
the sample can be read directly from the chart recording. Because the total 
fluorescent energy received in each analyzing counter is related to a fixed 
amount of energy entering the control counter, variations in the x-ray tube 
output do not affect the accuracy of the results. 

16-9 Nondispersive analysis. Up to this point we have considered only 
methods of dispersive analysis, i.e., methods in which x-ray beams of dif- 
ferent wavelengths are physically separated, or dispersed, in space by an 
analyzing crystal so that the intensity of each may be separately measured. 
But the separate measurement of the intensities of beams of different wave- 
lengths can often be accomplished without the spatial separation of these 
beams. Methods for doing this are 
called nondispersive. No analyzing 
crystal is used and the experimental 

, , , ,, , r x-ray tube 

arrangement takes on the simple torm x"~x 

illustrated in Fig. 15-10. The counter 

receives fluorescent radiation directly 

from the sample, and the filter shown 

may or may not be present.* Three 

methods of nondispersive analysis sample c 

have been used: selective excitation, FlG 15 _ ia Apparatus f or nondis- 

selective filtration, and selective pe rsive analysis. 

counting. 

Selective excitation of a particular spectral line is accomplished simply 
by control of the x-ray tube voltage. Suppose, for example, that a Cu-Sn 
alloy is to be analyzed. If the tube is operated at 28 kv, then Cu Ka will 
be excited (excitation voltage = 9 kv) but not Sn Ka (excitation voltage 
= 29 kv). The L lines of Sn will be excited at 28 kv but their wavelengths 
are so long (about 3A) that this radiation will be almost completely ab- 
sorbed in air. The radiation entering the counter therefore consists almost 
entirely of Cu Ka together with a small amount of white radiation scat- 
tered from the primary beam by the sample; the counter output can there- 
fore be calibrated in terms of the copper concentration of the sample. Evi- 

* The x-ray tube and counter should be as close as possible to the sample but, 
if necessary, a fluorescent spectrometer may be used, with the analyzing crystal 
removed and the counter set at 20 = 0. Or a diffractometer may be used, with 
the sample in the usual position and the counter set almost anywhere except at 
the position of a diffracted beam. In either case, since no focusing of the fluores- 
cent beam occurs, the counter receiving slit should be removed in order to gain 
intensity. 




420 CHEMICAL ANALYSIS BY ^FLUORESCENCE [CHAP. 15 

dently, the selective excitation method works best where the elements in- 
volved differ fairly widely in atomic number. 

When the K radiations of both elements are excited in the sample, se- 
lective filtration can be used to ensure that only one of them enters the 
counter. Consider the analysis of a Cu-Zn alloy. The K excitation voltage 
of copper is 9.0 kv and that of zinc 9.7 kv. Even if the operating voltage 
could be accurately set between these values, the intensity of the fluorescent 
Cu Ka radiation would be very low. It is better to operate at a voltage 
higher than either of these, say 12-15 kv, and use a nickel filter between the 
sample and the counter. This filter will absorb most of the Zn Ka and pass 
most of the Cu Ka radiation. Selective filtration of this kind is most effec- 
tive when the two elements have either nearly the same atomic numbers 
or widely different atomic numbers, because, in either case, a filter material 
can be chosen which will have quite different absorption coefficients for the 
two radiations. (Of course, the air between the sample and counter itself 
acts as a very effective selective filter in many applications. Consider the 
determination of copper in a Cu-Al alloy. The K lines of both elements will 
be excited at any voltage above 9 kv but Al Ka, of wavelength 8.3A, is so 
strongly absorbed by air that practically none of it reaches the counter.) 
Balanced filters do not appear to have been used in nondispersive analysis, 
but there is no reason why they should not be just as effective in this field 
as in diffractometry. 

Finally, the method of selective counting may be used. As mentioned 
in Sec. 7-5, it is possible to measure the intensity of radiation of one wave- 
length in the presence of radiations of other wavelengths by means of a 
proportional counter and a single-channel pulse-height analyzer. Thus the 
counter-analyzer combination can receive two or more characteristic radia- 
tions from the sample and be responsive to only one of them. No filtration 
is needed and the measured intensities are very high. This method works 
best when the elements involved differ in atomic number by at least three. 
If the difference is less, their characteristic radiations will not differ suffi- 
ciently in wavelength for efficient discrimination by the analyzer. 

There is, of course, no reason why any one of these methods cannot be 
combined with any other, or all three may be used together. Thus a par- 
ticular analytical problem may require the use of selective excitation and 
selective filtration, one technique aiding the other. Such combinations 
will usually be necessary when the sample contains more than two elements. 
In general, nondispersive analysis is most effective when applied to binary 
alloys, since the difficulties involved in distinguishing between one charac- 
teristic radiation and another, or in exciting one and not another, increase 
with the number of elements in the sample. These difficulties can be alle- 
viated by a multichannel arrangement, and the X-Ray Quantometer de- 
scribed in the previous section can be used for nondispersive analysis in 



15-10] MEASUREMENT OF COATING THICKNESS 421 

that manner, simply by removing the analyzing crystals and changing the 
counter positions. Each channel contains a different filter material, chosen 
in accordance with the particular element being determined in that channel. 
The main advantage of nondispersive methods of analysis is the very 
large gain in intensity over dispersive methods. The high loss of intensity 
involved in diffraction from an analyzing crystal is completely avoided. 
As a result, the beam entering the counter of a nondispersive system is 
relatively intense, even after passing through the rather thick filters which 
are used to prevent interference from other wavelengths. The greater the 
intensity, the shorter the counting time required to obtain a given accuracy, 
or the higher the accuracy for a given counting time. 

15-10 Measurement of coating thickness. Fluorescent radiation can 
be utilized not only as a means of chemical analysis but also as a method 
for measuring the thickness of surface layers. The following methods, 
both based on fluorescence, have been used to measure the thickness of a 
surface coating of A on B : 

(1) A dispersive system is used and the counter is positioned to receive 
the A Ka line from the sample. The intensity of the A Ka line increases 
with the thickness of the A layer up to the point at which this layer becomes 
effectively of infinite thickness, and then becomes constant. (Effectively 
infinite thickness, which is about 0.001 in. for a metal like nickel, corre- 
sponds to the effective depth of penetration of the primary beam striking 
the sample, and this method is in fact a way of determining this depth.) 
The relation between A Ka intensity and the thickness of A must be ob- 
tained by calibration. The operation of this method is independent of the 
composition of the base material B, which may be either a metal or a non- 
metal. This method may also be used with a nondispersive system, pro- 
vided that B is a nonmetal, or, if B is a metal, provided that the atomic 
numbers of A and B are such that nondispersive separation of A Ka and 
B Ka is practical (see the previous section). 

(2) A dispersive system is used and the intensity of B Ka radiation is 
measured. This intensity decreases as the thickness of A increases, and 
becomes effectively zero at a certain limiting thickness which depends on 
the properties of both A and B. Calibration is again necessary. A non- 
dispersive system may also be used if conditions are favorable, as they are, 
for example, in the measurement of the thickness of tin plate on sheet steel. 
In this case, selective excitation of Fe Ka is the simplest procedure inas- 
much as the operating conditions are exactly similar to those involved in 
the analysis of Cu-Sn alloys described in the previous section. This 
method is used industrially: tinned sheet steel passes continuously beneath 
a nondispersive analyzer, and the thickness of the tin coating is continu- 
ously recorded on a chart. 



422 CHEMICAL ANALYSIS BY FLUORESCENCE [CHAP. 15 

Although they have nothing to do with fluorescence, it is convenient to 
mention here the corresponding diffraction methods for measuring the 
thickness of a coating of A on B : 

(1) The specimen is placed in a diffractometer and the intensity of a 
strong diffraction line from A is measured. The intensity of this line, rela- 
tive to the intensity of the same line from an infinitely thick sample of A, 
is a measure of the thickness of A. The thickness may he directly calcu- 
lated from this intensity ratio by means of Eq. (9-4) and the form of the 
line intensity vs. thickness curve will resemble that of Fig. 9-6. The 
coating A must be crystalline, but B can be any material. 

(2) The intensity of a strong diffraction line from B is measured in a 
diffractometer. The observed intensity 7 depends on the thickness t of the 
A layer in an easily calculable manner. Since the total path length of the 
incident and diffracted beams in the A layer is 2//sin 8, the intensity of a 
diffraction line from B is given by 



where /o = intensity of the same diffraction line from uncoated B, and 
H = linear absorption coefficient of A. In this case B must be crystalline, 
but A can be anything. 

Any one of these methods, whether based on fluorescence or diffraction, 
may be used for measuring the thickness of thin foils, simply by mounting 
the foil on a suitable backing material. 

PROBLEMS 

16-1. Assume that the line breadth B in a fluorescent x-ray spectrometer is 
0.3 for a mica analyzing crystal used in transmission and 0.5 for either a LiF 
or NaCl crystal in reflection. Which of these crystals will provide adequate reso- 
lution of the following pairs of lines? 

(a) Co K$ and Ni Ka (b) Sn K$ and Sb Ka 

Calculate A20 values for each crystal. 

16-2. What operating conditions would you recommend for the nondispersive 
fluorescent analysis of the following alloys with a scintillation counter? 

(a) Cu-Ni (b) Cu-Ag 

15-3. Diffraction method (2) of Sec. 15-10 is used to measure the thickness of 
a nickel electroplate on copper with Cu Ka. incident radiation. What is the maxi- 
mum measurable thickness of nickel if the minimum measurable line intensity is 
1 percent of that from uncoated copper? 



CHAPTER 16 
CHEMICAL ANALYSIS BY ABSORPTION 

16-1 Introduction. Just as the wavelength of a characteristic line is 
characteristic of an emitting element, so is the wavelength of an absorption 
edge characteristic of an absorbing element. Therefore, if a sample con- 
taining a number of elements is used as an absorber and if the absorption 
it produces is measured as a function of wavelength, absorption edges will 
be disclosed, and the wavelengths of these edges will serve to identify the 
various elements in the sample. The method may also be made quantita- 
tive, if the change in absorption occurring at each edge is measured. 

Such measurements require monochromatic radiation of controlled wave- 
length, and this is usually obtained by reflection from a single crystal in a 
diffractometer. The sample whose absorption is to be measured is placed 
in the diffracted beam, as indicated in Fig. 16-1 (a), and x-rays of any de- 
sired wavelength are picked out of the white radiation issuing from the 
tube simply by setting the analyzing crystal at the appropriate angle 6. 
Alternately, the sample may be placed in the beam incident on the crystal. 

Another source of monochromatic radiation of controlled wavelength 
is an element fluorescing its characteristic radiation. The arrangement 
shown in Fig. 16-1(b) is used, with the crystal set to reflect the charac- 
teristic radiation of whatever element is used as radiator. By having on 
hand a set of elements of atomic number Z, (Z + 1), (Z + 2), . . . , we 
have available a discontinuous range of characteristic wavelengths, and 





FIG. 16-1. Experimental arrangement for absorption measurements: (a) with 
diffractometer, (b) with fluorescent spectrometer. 

423 



424 



CHEMICAL ANALYSIS BY ABSORPTION 



[CHAP. 16 



the intensity of this radiation at the sample will be considerably larger 
than that of the white radiation components used in the diffractometer 
method. Even though the wavelengths furnished by fluorescence do not 
form a continuum, they are spaced closely enough to be useful in measuring 
the variation in absorption of the sample with wavelength. In the wave- 
length range from 0.5 to 1.5A, for example, the average difference between 
the Ka wavelengths of an element of atomic number Z and one of (Z + 1) 
is only 0.06A. If a particular element is not available in the pure form, 
its oxide, or some other compound or alloy containing a substantial amount 
of the element, can be used as a radiator of fluorescent radiation. 



17(K) 



8. 

o 

~ 161X) - 



W 
CQ 

Q 
W 



tf 
^ 

O 

^ 
H 

5! 
w 

H 



1500 
1400 
1300 
1200 
1100 
1000 
900 




040 0.45 0.50 055 057 

WAVELENGTH (angstroms) 

FIG. 16-2. Variation of transmitted intensity \\ith wavelength near an absorp- 
tion edge. (For this particular curve, three thicknesses of photographic film were 
used as an absorber and the absorption edge shown is the A' edge of the silver in 
the emulsion.) 

16-2 Absorption-edge method. Suppose we wish to determine the con- 
centration of element A in a sample containing a number of other elements. 
The sample, prepared in the form of a flat plate or sheet of uniform thick- 
ness, is placed in a beam of controllable wavelength, and the intensity / 
of the transmitted radiation is measured for a series of wavelengths on 
either side of an absorption edge of element A. The resulting curve of 
/ vs. X will have the form of Fig. 16-2, since the transmitted intensity will 
increase abruptly on the long wavelength side of the edge. (The exact 



16-2] ABSORPTION-EDGE METHOD 425 

form of the curve depends on the kind of radiation available. The data 
in Fig. 16-2 were obtained with radiation reflected from the continuous 
spectrum in a diffractometer; the upward slope of the curve at wavelengths 
longer than the edge is due to the fact that the intensity of the incident 
beam increases with wavelength in this region of the continuous spectrum 
and this effect more than compensates for the increase in the absorption 
coefficient of the sample with wavelength.) By the extrapolations shown 
we obtain the values of /i and 7 2 , the transmitted intensities for wave- 
lengths just longer and just shorter, respectively, than the wavelength of 
the edge. 

The mass absorption coefficient of the sample is given by 



where w denotes weight fraction, and the subscripts ra, A, and r denote 
the mixture of elements in the sample, element A, and the remaining ele- 
ments in the sample, respectively. At a wavelength not equal to that of 
an absorption edge the transmitted intensity is given by 



where 7 is the intensity of the incident beam, p m is the density of the 
sample, and t is the thickness of the sample. At wavelengths just longer 
and just shorter than that of the absorption edge of A, let the mass absorp- 
tion coefficients of A be (M/P)AI and (M/p)A2> respectively. Then the trans- 
mitted intensities for these two wavelengths will be 



since (M/P)T is the same for both. Division of one equation by the other 
gives 

= e W) Al(M/p)A2- (M/p)Ailpm^ (16-1) 

^2 

If we put [(M/p)A2 ~ WP)AI] = &A and p m t = Af m , then Eq. (16-1) be- 
comes 

(16-2) 



This equation can be used to determine WA from measured and tabulated 
quantities. The constant &A, which measures the change in the mass 
absorption coefficient of A at the absorption edge, is a property of the 
element involved and decreases as the atomic number increases. M m is 



426 



CHEMICAL ANALYSIS BY ABSORPTION 



[CHAP. 16 



the mass of sample per unit area and is given by the mass of the sample 
divided by the area of one face. 

Since M m varies with w\ for samples of constant thickness, and may in 
fact vary independently of w\, it is convenient to lump the two together 
and put w A M m M\ = mass of A per unit area of sample. A plot of 
In (/i // 2 ) vs. M A will then be a straight line through the origin with a slope 
of A. If there is any doubt about the accuracy of the tabulated absorption 
coefficients from which A' A is derived, this curve can be established by 
measurements on samples of known A content. It is important to note 
that the slope of this curve depends only on the clement A being deter- 
mined and is independent, not only of the other elements present, but 
also of any variations in the concentrations of these elements with respect 
to one another. The other elements present affect only M m , which must 
be measured for each sample. The value of w\ is then given by M\/M m . 

The fact that the curve of In (I\/I<z) vs. M A forms a master plot for the 
determination of A whatever the other constituents of the sample repre- 
sents a distinct advantage of the absorption-edge method over fluorescent 
analysis. For example, if element A is being determined by fluorescence 
in samples containing A, B, and C, a calibration curve for the determina- 
tion of A is valid only for samples containing a fixed concentration of B 
or C. 

The main disadvantage of the absorption-edge method, when applied to 
the analysis of alloys, is the very thin sample required to obtain measurable 



JBb 

<N 

& 150 

H 

g 125 



8 



O 

E 



50 



25 







L\ L\\ L\\\ 




I 







0.2 04 0.6 0.8 1.0 12 



WAVELENGTH (angstroms) 

FIG. 16-3. Absorption coefficients of lead, showing K and L absorption edges. 
(Plotted from data in Handbook of Chemistry and Physics, 23rd ed., Chemical Rub- 
ber Publishing Co., Cleveland, 1939.) 



16-3] DIRECT-ABSORPTION METHOD (MONOCHROMATIC BEAM) 427 

transmitted intensity. Many alloy samples have to be ground down to a 
thickness of one or two thousandths of an inch, and this is a tedious and 
time-consuming operation. The method is best suited to the determina- 
tion of a fairly heavy element in a light-element matrix. It is difficult 
to determine light elements, even though they have large values of fc, 
because their absorption edges occur at such long wavelengths that the 
incident radiation is almost completely absorbed even by very thin samples. 
(However, the difficulties involved in preparing thin samples of solid 
materials may be avoided by dissolving the sample, in a known concentra- 
tion, in a suitable liquid. The resulting solution is contained in a flat- 
sided cell of some highly transparent material, and the total sample thick- 
ness may be several millimeters.) 

When the atomic number of the element being determined exceeds about 
50, the LIU rather than the A' absorption edge should be used. Not only 
is k much larger for the Lm edge of such elements, but their K absorption 
edges occur at wavelengths shorter than those available from an x-ray 
tube operated at 50 kv. Figure 16-3 shows the relative size and location 
of the K and L absorption edges of lead. 

16-3 Direct-absorption method (monochromatic beam). Absorption 
methods not involving measurements at an absorption edge have also been 
used. The mass absorption coefficient of a mixture of two elements A and 
B, for a wavelength not equal to that of an absorption edge of either, is 
given by 



\P/A B 

The relation between the incident intensity 7 and the transmitted inten- 
sity 7 is therefore 

In ~ = L A (-} + (1 - A) (-) 1 Pmt. (16-) 

I I \p/A Vp/BJ 

This relation can be used for the determination of the amount of A present, 
provided that p m , the density of the sample, is known as a function of com- 
position. A strong characteristic line is normally used for such measure- 
ments: for greatest sensitivity its wavelength should lie between the ab- 
sorption edges of A and B. 

Naturally, if p m is known as a function of composition, density measure- 
ments alone can disclose the composition of an unknown without any 
necessity for absorption measurements. But there are circumstances in 
which an absorption measurement is more convenient than a density 
measurement. Such circumstances arise in diffusion studies. Metals A 
and B are joined together to form a diffusion couple [Fig. 16-4(a)], held 



428 



CHEMICAL ANALYSIS BY ABSORPTION 



[CHAP. 16 



sample for absorption 
measurements 

/ original 
A / interface 



100 



B 

/ 



XI 



7 



H 



50 




or 

(a) 



x +z - 

DISTANCE FROM 
ORIGINAL INTERFACE 

(b) 



FIG. 16-4. Application of the direct-absorption method to diffusion measurements. 

at a constant elevated temperature for a given period of time, and then 
cooled to room temperature. The problem is to determine the penetration 
of one metal into another, i.e., to arrive at a curve like Fig. l(>-4(b), show- 
ing the change in composition of the alloy along a line normal to the original 
interface. This is usually done by cutting the couple into a number of thin 
slices, parallel to the //z-plane of the original interface, and determining 
the composition of each slice. In the absorption method, a single slice is 
taken parallel to the zz-plane, normal to the original interface. This slice 
is then placed between the diffract ometer counter and a narrow fixed slit 
which defines the x coordinate of the area irradiated. If the sample is then 
moved stepwise relative to the slit in the x direction, a series of measure- 
ments can be made from which the composition vs. distance curve can be 
plotted. 

Another way in which the direct absorption method can be made useful 
involves making measurements at two wavelengths, \i and X 2 , since no 
knowledge of the density or thickness of the sample is then required. Desig- 
nating measurements made at each of these wavelengths by subscripts 1 
and 2, we find from Eq. (16-3) that 



In 



- (M/P)BI] + (M/P)BI 



In (1 Q2/1 2) WA[(M/P)A2 ~ (M/p)B2l + Wp)fi2 



(16-4) 



The wavelengths Xi and X 2 should be chosen to lie near, and on either side 
of, an absorption edge of A. One way of applying this method to routine 
analyses is to use a multichannel nondispersive fluorescent analyzer. Three 
channels are required: channel 1 contains an element which fluoresces 
characteristic radiation of wavelength Xi, channel 2 contains another ele- 
ment producing radiation of wavelength X 2 , and channel 3 is used for 
control. The absorption produced by the sample is measured first in 
channel 1 and then in channel 2, and the ratio of the intensities /i and 7 2 



16-5] APPLICATIONS 429 

transmitted in these two channels is taken as a measure of the A content 
of the sample. The control channel is used to ensure that all samples 
receive the same total energy of incident radiation. No use is made of 
Eq. (16-4). Instead, a calibration curve showing the relation between 
WA and In (/i//2) is prepared from samples of known composition. 

The methods outlined in this section are normally used only for the 
analysis of two-component samples. If more than two components are 
present and equations of the form of (16-3) and (16-4) are used, then all 
but two components in the sample must have known concentrations. If 
the concentration of a particular element is obtained from a calibration 
curve rather than from these equations, the calibration curve applies only 
to samples containing fixed concentrations of all but two components. 

16-4 Direct-absorption method (polychromatic beam). The absorp- 
tion of a polychromatic beam, made up of the sum total of continuous and 
characteristic radiation issuing from an x-ray tube, may also be made the 
basis for chemical analysis. The experimental arrangement is very simple: 
the sample is merely placed in the direct beam from the x-ray tube, and a 
counter behind the sample measures the transmitted intensity. Because 
of the multiplicity of wavelengths present, no exact calculation of trans- 
mitted intensity as a function of sample composition can be made. How- 
ever, a calibration curve can be set up on the basis of measurements made 
on samples of known composition, and this curve will be valid for the 
determination of a particular element in a series of samples, provided all 
samples have the same thickness and the concentrations of all but two 
components are fixed. 

The chief advantage of this method is the very large gain in intensity 
over methods involving monochromatic beams. A monochromatic beam, 
whether produced by diffraction or fluorescence, is quite feeble in com- 
parison to the direct beam from an x-ray tube. The higher the incident 
intensity, the thicker the sample that can be used; or, for the same sample 
thickness, the higher the intensity, the shorter the analysis time for a 
given accuracy of counting. 

16-5 Applications. Absorption methods of analysis are limited to sam- 
ples whose total absorption is low enough to produce a transmitted beam 
of accurately measurable intensity. This means that samples of most 
metallic alloys have to be made extremely thin, at least for methods involv- 
ing low-intensity monochromatic beams, or they have to be dissolved in 
a liquid. 

In industry today, absorption methods are almost entirely confined to 
the analysis of organic liquids and similar materials of low absorption 
coefficient. A typical example of such analyses is the determination of 
tetraethyl lead in gasoline. 



430 CHEMICAL ANALYSIS BY ABSORPTION [CHAP. 16 

PROBLEMS 

16-1. The values of I\ and /2, taken from Fig. 16-2, for the absorption pro- 
duced by three thicknesses of unprocessed x-ray film are 1337 and 945 cps, re- 
spectively. The upper and lower values of the mass absorption coefficient of sil- 
ver at its K absorption edge are 62.5 and 9.8 cm 2 /gm, respectively. Calculate 
the silver content (in mg/cm 2 ) of a single piece of this film. 

16-2. If the minimum detectable increase in transmitted intensity at an absorp- 
tion edge is 5 percent, what is the minimum detectable amount of copper (in weight 
percent) in an Al-Cu alloy if the absorption-edge method is used on a sample 1 mm 
thick? Assume that the density of the sample is the same as that of pure alumi- 
num. The upper and lower values of the mass absorption coefficient of copper 
at its K absorption edge are 307 and 37 cm 2 /gm, respectively. 

16-3. The composition of an Fe-Ni alloy, known to contain about 50 weight 
percent iron, is to be determined by the direct absorption method with Cu Ka 
radiation. Its density is 8.3 gm/cm 3 . The maximum available incident intensity 
is 10,000 cps. The minimum transmitted intensity accurately measurable in a 
reasonable length of time in the presence of the background is 30 cps. What is 
the maximum specimen thickness? 



CHAPTER 17 
STRESS MEASUREMENT 

17-1 Introduction. When a polycrystalline piece of metal is deformed 
elastically in such a manner that the strain is uniform over relatively large 
distances, the lattice plane spacings in the constituent grains change from 
their stress-free value to some new value corresponding to the magnitude 
of the applied stress, this new spacing being essentially constant from one 
grain to another for any particular set of planes. This uniform macro- 
strain, as we saw in Sec. 9-4, causes a shift of the diffraction lines to new 
26 positions. On the other hand, if the metal is deformed plastically, the 
lattice planes usually become distorted in such a way that the spacing of 
any particular (hkl) set varies from one grain to another or from one part 
of a grain to another. This nonuniform microstrain causes a broadening 
of the corresponding diffraction line. Actually, both kinds of strain are 
usually superimposed in plastically deformed metals, and diffraction lines 
are both shifted and broadened, because not only do the plane spacings 
vary from grain to grain but their mean value differs from that of the 
undeformed metal. 

In this chapter we will be concerned with the line shift due to uniform 
strain. From this shift the strain may be calculated and, knowing the 
strain, we can determine the stress present, either by a calculation involving 
the mechanically measured elastic constants of the material, or by a cali- 
bration procedure involving measurement of the strains produced by 
known stresses. X-ray diffraction can therefore be used as a method of 
"stress" measurement. Note, however, that stress is not measured directly 
by the x-ray method or, for that matter, by any other method of "stress" 
measurement. It is always strain that is measured; the stress is deter- 
mined indirectly, by calculation or calibration. 

The various methods of "stress" measurement differ only in the kind of 
strain gauge used. In the common electric-resistance method, the gauge 
is a short length of fine wire cemented to the surface of the metal being 
tested; any strain in the metal is shared by the wire, and any extension 
or contraction of the wire is accompanied by a change in its resistance, 
which can therefore be used as a measure of strain. In the x-ray method, 
the strain gauge is the spacing of lattice planes. 

17-2 Applied stress and residual stress. Before the x-ray method is 
examined in any detail, it is advisable to consider first a more general 
subject, namely, the difference between applied stress and residual stress, 

431 



432 



STRESS MEASUREMENT 



[CHAP. 17 



and to gain a clear idea of what these terms mean. Consider a metal bar 
deformed elastically, for example in uniform tension. The applied stress 
is given simply by the applied force per unit area of cross section. If the 
external force is removed, the stress disappears, and the bar regains its 
initial stress-free dimensions. On the other hand, there are certain opera- 
tions which can be performed on a metal part, which will leave it in a 
stressed condition even after all external forces have been removed. This 
stress, which persists in the absence of external force, is called residual 
stress. 

For example, consider the assembly shown in Fig. 17-1 (a). It consists 
of a hollow section through which is passed a loosely fitting bolt with 
threaded ends. If nuts are screwed on these ends and tightened, the sides 
of the assembly are compressed and the bolt is placed in tension. The 
stresses present are residual, inasmuch as there are no external forces 
acting on the assembly as a whole. Notice also that the tensile stresses in 
one part of the assembly are balanced by compressive stresses in other 
parts. This balance of opposing stresses, required by the fact that the 
assembly as a whole is in equilibrium, is characteristic of all states of 
residual stress. 

An exactly equivalent condition of residual stress can be produced by 
welding a cross bar into an open section, as shown in Fig. 17-1 (b). We 
can reasonably assume that, at the instant the second weld is completed, 
a substantial portion of the central bar is hot but that the two side mem- 
bers are far enough from the heated zone to be at room temperature. On 
cooling, the central bar tries to contract thermally but is restrained by the 
side members. It does contract partially, but not as much as it would if 
it were free, and the end result is that the side members are placed in com- 
pression and the central rod in tension when the whole assembly is at 



weld 



(a) (b) 

FIG. 17-1. Examples of residual stress. T = tension, C = compression. 



17-2] 



APPLIED STRESS AND RESIDUAL STRESS 



433 



room temperature. Residual stress is quite commonly found in welded 
structures. 

Plastic flow can also set up residual stresses. The beam shown in Fig. 
17-2 (a) is supported at two points and loaded by two equal forces F applied 
near each end. At any point between the two supports the stress in the 
outside fibers is constant, tensile on the top of the beam and compressive 
on the bottom. These stresses are a maximum on the outside surfaces and 
decrease to zero at the neutral axis, as indicated by the stress diagram at 
the right of (a). This diagram shows how the longitudinal stress varies 
across the section A A', when all parts of the beam are below the elastic 
limit. Suppose the load on the beam is now increased to the point where 
the elastic limit is exceeded, not only in the outer fibers but to a consider- 
able depth. Then plastic flow will take place in the outer portions of the 
beam, indicated by shading in (b), but there will be an inner region still 
only elastically strained, because the stress there is still below the elastic 
limit. The stresses above the neutral axis are still entirely tensile, both 
in the elastically and plastically strained portions, and those below entirely 
compressive. If the load is now removed, these stresses try to relieve 
themselves by straightening the beam. Under the action of these internal 
forces, the beam does partially straighten itself, and to such an extent that 



/' 




A' 



FIG. 17-2. Residual stress induced by plastic flow in bending: (a) loaded below 
elastic limit; (b) loaded beyond elastic limit; (c) unloaded. Shaded regions have 
been plastically strained. 



434 STRESS MEASUREMENT [CHAP. 17 

the stress in the outer regions is not only reduced to zero but is actually 
changed in sign, as indicated in (c). The end result is that the unloaded 
beam contains residual compressive stress in its top outside portion and 
residual tensile stress in its lower outside portion. It is quite common to 
find residual stress in metal parts which have been plastically deformed, 
not only by bending but by drawing, swaging, extrusion, etc. 

17-3 Uniaxial stress. With these basic ideas in mind, we can now go 
on to a consideration of the x-ray method of stress measurement. The 
simplest way to approach this method is through the case of pure tension, 
where the stress acts only in a single direction. Consider a cylindrical rod 
of cross-sectional area A stressed elastically in tension by a force F (Fig. 
17-3). There is a stress v y F/A in the ^/-direction but none in the x- 
or 2-directions. (This stress is the only normal stress acting; there are also 
shear stresses present, but these are not measurable by x-ray diffraction.) 
The stress v y produces a strain e v in the ?/-direction given by 

AL Lf L 

y = - = - ' 

L Lo 

where L and L/ are the original and final lengths of the bar. This strain 
is related to the stress by the fundamental elastic equation 

v = E*y, (17-1) 

where E is Young's modulus. The elongation of the bar is accompanied 
by a decrease in its diameter D. The strains in the r- and ^-directions 
are therefore given by 



D 

where Z) and D/ are the original and final diameters of the bar. If the 
material of the bar is isotropic, these strains are related by the equation 

.-, -? (17-2) 

where v is Poisson's ratio for the material of the bar. The value of v 
ranges from about 0.25 to about 0.45 for most metals and alloys. 

To measure * y by x-rays would require diffraction from planes perpen- 
dicular to the axis of the bar. Since this is usually physically impossible, 
we utilize instead reflecting planes which are parallel, or nearly parallel, 
to the axis of the bar by taking a back-reflection photograph at normal 
incidence, as shown in Fig. 17-3. (It is essential that a back-reflection 
technique be used, in order to gain sufficient precision in the measurement 
of plane spacing. Even quite large stresses cause only a very small change 



17-3] 



UNIAXIAL STRESS 



435 



surface - - 



buck -reflect ion 
pinliole caineia 




m 



J_ 




FIG. 17-3. Pure tension. FIG. 17-4. Diffraction from strained 

aggregate, tension axis vertical. Lat- 
tice planes shown belong; to the same 
(hkl) set. A' = reflecting-plane normal. 

in rf.) In this way we obtain a measurement of the strain in the z direction 
since this is given by 

,- - '^'. (.7-3) 



where d n is the spacing of the planes reflecting at normal incidence under 
stress, and rfo is the spacing of the same planes in the absence of stress. 
Combining Eqs. (17-1), (17 2), and (17-3), we obtain the relation 



77T / 

V \ 



d n - 



(17-4) 



which gives the required stress in terms of known and observed quantities. 
It should be noted that only a particular set of grains contributes to a 
particular hkl reflection. These are grains whose (hkl) planes are almost 
parallel to the surface of the bar, as indicated in Fig. 17-4, and which are 
compressed by the applied stress, that is, d n is less than d () . Grains whose 
(hkl) planes are normal to the surface have these planes extended, as shown 
in an exaggerated fashion in the drawing. The spacing dhki therefore 
varies with crystal orientation, and there is thus no possibility of using 
any of the extrapolation procedures described in Chap. 11 to measure 



436 



STRESS MEASUREMENT 



[CHAP. 17 




line from specimen 

line from 
reference material 



FIG. 17-5. Back-reflection method at normal incidence. 

accurately. Instead we must determine this spacing from the position 
of a single diffraction line on the film. 

A direct comparison method is usually used. A powder of some reference 
material of known lattice parameter is smeared on the surface of the speci- 
men, and the result is a photograph like that illustrated in Fig. 17-5, 
where the Ka lines are shown unresolved for greater clarity. Since the 
line from the reference material calibrates the film, it is unnecessary to 
know the specimen-to-film distance /). The plane spacings of the specimen 
are determined simply by measuring the diameters of the Debye rings 
from the specimen (2S 8 ) and from the reference material (2S r ). 

Equation (17-4) shows that a measurement of d on the unstressed ma- 
terial must be made. If the specimen contains only applied stress, then 
d is obtained from a measurement on the unloaded specimen. But if 
residual stress is present, d must be measured on a small stress-free por- 
tion cut out of the specimen. 

17-4 Biaxial stress. In a bar subject to pure tension the normal stress 
acts only in a single direction. But in general there will be stress com- 
ponents in two or three directions at right angles to one another, forming 
so-called biaxial or triaxial stress systems. However, the stress at right 
angles to a free surface is always zero, so that at the surface of a body, 
which is the only place where we can measure stress, we never have to deal 
with more than two stress components and these lie in the plane of the 
surface. Only in the interior of a body can the stresses be triaxial. 

Consider a portion of the surface of a stressed body, shown in Fig. 17-6. 
We set up a rectangular coordinate system xyz, with x and y lying in the 
plane of the surface in any convenient orientation. Whatever the stress 
system, three mutually perpendicular directions (1,2, and 3) can be found 
which are perpendicular to planes on which no shear stress acts. These 
are called the principal directions, and the stresses acting in these direc- 
tions, ffiy a 2 , and <7 3 , are called the principal stresses. At the free surface 
shown, 0-3, like <r, is equal to zero. However, 3 , the strain normal to the 



17-4] 



BIAXIAL STRESS 

"3 



437 




FIG. 17-6. Angular relations between stress to be measured (o0), principal 
stresses (<TI, 0-2, and 0-3), and arbitrary axes (x, y, z). 



surface, is not zero. It is given by 



(17-5) 



The value of 3 can be measured by means of a diffraction pattern made 
at normal incidence and is given by Eq. (17-3). Substituting this value 
into (17-5), we obtain 



d n d Q 



(17-6) 



Therefore, in the general case, only the sum of the principal stresses can 
be obtained from a pattern at normal incidence. [If only a single stress is 
acting, say a tensile stress in direction 1, then 0-2 = and Eq. (17-6) re- 
duces to Eq. (17-4).] 

Normally, however, we want to measure the stress <r+ acting in some 
specified direction, say the direction OB of Fig. 17-6, where OB makes an 
angle <t> with principal direction 1 and an angle /3 with the z-axis. This is 
done by making two photographs, one with the incident beam normal to 
the surface and one with it inclined along OA at some angle ^ to the sur- 
face normal. OA lies in a vertical plane through the direction OB in which 
it is desired to measure the stress, and \l/ is usually made equal to 45. The 
normal-incidence pattern measures the strain approximately normal to the 
surface, and the inclined-incidence pattern measures the strain approx- 
imately parallel to OA. These measured strains are therefore approx- 
imately equal to 3 and ^, respectively, where ^ is the strain in a direction 



438 STRESS MEASUREMENT [CHAP. 17 

at an angle ^ to the surface normal. Elasticity theory gives the following 
relation for the difference between these two strains: 

^- 3 = ^(l + ")sin 2 ^ (17-7) 

E 

But 

(17-8) 



do 

where di is the spacing of the inclined reflecting planes, approximately 
normal to OA, under stress, and d is their stress-free spacing. Combining 
Eqs. (17-3), (17-7), and (17-8), we obtain 

di -d d n - d d l - d n or^ . 

(i _)- ,,) sin ^ ^/. (17-9) 



d d d 

Since d , occurring in the denominator above, can be replaced by d n with 
very little error, Eq. (17-9) can be written in the form 

E '"' ~" 1 (17-10) 



(1 + v) sin 2 $\ d n 

This equation allows us to calculate the stress in any chosen direction from 
plane spacings determined from two photographs, one made at normal 
incidence and the other with the incident beam inclined at an angle \f/ to 
the surface normal. Notice that the angle <t> does not appear in this equa- 
tion and fortunately so, since we do not generally know the directions of 
the principal stresses a priori. Nor is it necessary to know the unstressed 
plane spacing d ; the measurement is therefore nondestructive, because 
there is no necessity for cutting out part of the specimen to obtain a stress- 
free sample. 

The direct comparison method is again used to obtain an accurate meas- 
urement of the spacings, and Fig. 17-7 illustrates the appearance of the 
film in the inclined-incidence exposure. The Debye ring from the speci- 
men is no longer perfectly circular. The reason lies in the fact that the 
strain along the normal to reflecting planes varies with the angle \f/ between 
these plane normals and the surface normal, as shown by Eq. (17-7). 
There will therefore be slightly different diffraction angles 26 for planes 
reflecting to the "low" side of the film (point 1) and those reflecting to the 
"high" side (point 2). These planes therefore form two sets of slightly 
different orientation, sets 1 and 2, having normals NI and N% at angles of 
i and 2 to the incident beam, (cq and a 2 are nearly equal to one another 
and to 90 0.) Measurements of the specimen Debye-ring radii Si and 
82 therefore give information about strains in directions at angles of 
(^ + e*i) and (\l/ 2 ) to the surface normal. The usual practice is to 



17-4] 



BIAXIAL STRESS 



439 



lino from 
reference 
material 




specimen 
surface 

* 



line from 
specimen 



FIG. 1 7-7. Back-reflection method at inclined incidence. 

measure only Si, since the position of this side of the ring is more sensitive 
to strain.* 

To save time in calculation, we can put Eq. (17-10) in more usable 
form. By differentiating the Bragg law, we obtain 



= cot 6 A0. 
d 



(17-11) 



The Debye-ring radius S, in back reflection, is related to the specimen- 
to-film distance D by 

S = D tan (180 - 26) = -D tan 26, 
AS = -2Dsec 2 20A0. 
Combining Eqs. (17-11) and (17-12), we obtain 

Ad 

AS = 2D sec 2 26 tan 
d 



(17-12) 



* S\ and Sz cannot be measured directly because of the hole in the center of the 
film, but they can be found indirectly. If the measured diameter of the Debye 
ring from the reference material is 2S r , then the point where the incident beam 
passed through the film is located at a distance S r from any point on the reference 
ring. If xi and z 2 are the measured distances between the specimen and reference 
rings on the "low" side and "high" side, respectively, of the film [see Fig. 17-8(c)], 
then Si = S r x\ and 2 = S r 2- 



440 STRESS MEASUREMENT [CHAP. 17 

Ad di d n 

Put = - 

a d n 

and 

AS = S t - S n , 

where S t is the Debye-ring radius in the inclined-incidence photograph, 
usually taken as the radius S\ in Fig. 17-7, and S n is the ring radius in the 
normal incidence photograph. Combining the last three equations with 
Eq. (17-10), we find 



~ 2D(l + v) sec 2 20 tan 6 sin 2 f 
Put 

E 
K = - - - (17-13) 

2D(l + v) sec 2 26 tan 6 sin 2 ^ 
Then 

cr* = K,(S, - S n ). (17-14) 

This forms a convenient working equation. K\ is known as the stress 
factor, and it can be calculated once and for all for a given specimen, 
radiation, and specimen-to-film distance. To ensure that the specimen-to- 
film distance D is effectively equal for both the inclined- and normal-inci- 
dence exposures, it is enough to adjust this distance to within 1 mm of its 
nominal value with a distance gauge and make the final correction by 
means of the measured diameter 2S r of the Debye ring from the reference 
material. For example, with tungsten powder as a reference material, 
Co Ka radiation, and a specimen-to-film distance of 57.8 mm, 2S r is 50 mm 
for the 222 line from tungsten. Then for each film a multiplying factor is 
found which will make the tungsten-ring diameter equal to exactly 50.00 
mm; this same factor is then applied to the measured radii S and S n of 
the specimen rings before inserting them in Eq. (17-14). All measure- 
ments are best made on the a\ component of the Ka doublet. 

What sort of accuracy can be expected in the measurement of stress by 
x-rays? For a steel specimen examined with Co Ka. radiation, the highest- 
angle reflection is the 310 line, which occurs at about 160 20. Then E = 
30 X 10 6 psi, v = 0.28, D = 57.8 mm, = 80, and ^ = 45 + (90 - 0) 
= 55, if the incident beam is inclined at an angle of 45 to the surface 
normal and we measure the radius Si, rather than S 2 , in the inclined- 
incidence photograph. Putting these values into Eq. (17-13), we find 
the stress factor KI to be 47,000 psi/mm. If the quantity (S t - S n ) is 
measured to an accuracy of 0.1 mm, which requires an accuracy of 0.05 
mm in the measurement of the separate quantities Si and S n , then the 
stress can be determined with an accuracy of 4700 psi. Accuracies some- 
what better than this can sometimes be obtained in practice, but a prob- 



17-5] EXPERIMENTAL TECHNIQUE (PINHOLE CAMERA) 441 

able error of 4000 to 5000 psi is probably typical of most measurements 
made on steel specimens. Higher accuracies are attainable on materials 
having substantially lower elastic moduli, such as aluminum-base alloys, 
since the stress factor is directly proportional to the modulus. 



The stress o^ acting in any specified direction may also be measured by a single 
inclined exposure like that shown in Fig. 17-7, the normal-incidence exposure be- 
ing omitted. Both Debye-ring radii, Si and $2, are measured, the former being 
used to calculate the strain at an angle (^ + i) to the surface normal and the 
latter the strain at an angle (^ o^)- Equation (17-9) is then applied separately 
to each measurement: 

E /d t i - d n \ 

Q __ I J, 

(1 + v) sin 2 (\// + oil) \ do / 



(1 + v) sin 2 (^ #2) \ do 

where d l i and d l % are the plane spacings calculated from S\ and $2, respectively. 
Putting a\ 2 = = (90 0), and eliminating d n from the two equations 
above, we find 

/ E \ Mi - d l2 \ ( 1 \ 

> = ( T~T~ ) ( ^ ) ( ~ r ~^r^T ) 

\1 + v/ \ do / \sm 2^ sin 2a/ 

In this equation do need not be known accurately. Since only one exposure is 
required, this method is twice as fast as the usual two-exposure method, but it 
entails a probable error two or three times as large. 

17-6 Experimental technique (pinhole camera). In this and the next 
section we shall consider the techniques used in applying the tw r o-exposure 
method to the measurement of stress. 

Pinhole cameras of special design are used for stress measurement. The 
design is dictated by two requirements not ordinarily encountered : 

(1) Since the specimens to be examined are frequently large and un- 
wieldy, it is necessary to bring the camera to the specimen rather than the 
specimen to the camera. 

(2) Since the highest accuracy is required in the measurement of diffrac- 
tion-line positions, the lines must be smooth and continuous, not spotty. 
This is achieved by rotating or oscillating the film about the incident-beam 
axis. (Complete rotation of the film is permissible in the normal-incidence 
exposure but not in the one made at inclined incidence. In the latter case 
the Debye ring is noncircular to begin with, and complete rotation of the 
film would make the line very broad and diffuse. Instead, the film is 
oscillated through an angle of about 10. If the specimen grain size is 
extremely coarse, the specimen itself should be oscillated, if possible, 
through an angle of 2 or 3 about an axis normal to the incident beam.) 



442 



STRESS MEASUREMENT 



[CHAP. 17 



These two requirements are satisfied by the camera design illustrated 
in Fig. 17-8(a). The camera is rigidly attached to a portable x-ray tube 
(a shockproof tube energized through a flexible shockproof cable), which 
is held in an adjustable support, permitting the camera to be oriented in 
any desired way relative to the specimen. The film is held in a circular 
cassette which can be oscillated or rotated by means of a gear-and-worm 
arrangement. Both the normal-incidence and inclined-incidence photo- 
graphs may be registered on one film by using the opaque metal film cover 
shown in (b). It has two openings diametrically opposite; after one ex- 
posure is made, the film holder is rotated 90 in its own plane with respect 
to the cover, and the other exposure made. The resulting film has the 
appearance of (c). Figure 17-9 shows a typical camera used for stress 
measurements. 

Some investigators like to use a well-collimated incident beam, like the 
one indicated in Fig. 17-8(a). Others prefer to use a divergent beam and 
utilize the focusing principle shown in Fig. 17-10. A fine pinhole is located 
behind the film at the point A and a larger one, to limit the divergence, at 
point B. Then a circle passing through A and tangent to the specimen 



worm 
drive 



(asset tf 



him cover 





(a) 




(b) 



line from 
specimen 



inclined- 
incidence 
exposure 



(0 



exposure 



FIG. 17-8. Pinhole camera for stress measurement (schematic): (a) section 
through incident beam; (b) front view of cassette; (c) appearance of exposed film. 



17-5] 



EXPERIMENTAL TECHNIQUE (PINHOLE CAMERA) 



443 




FIG. 17-9. Stress camera in position for the measurement of stress in a welded 
steel plate. A combined distance gauge and beam indicator has been temporarily 
attached to the collimator to aid in adjusting the specimen-film distance and the 
angle between the incident beam arid the specimen surface. The latter adjust- 
ment may be quickly made with the protractor shown. (H. J. Isenburger, Machin- 
ery, July 1947, p. 167.) 



focusing circle 



film 



x-ray tube 




specimen 



FIG. 17-10. Back-reflection pinhole camera used under semifocusing conditions. 



444 STRESS MEASUREMENT [CHAP. 17 

will intersect the film at the point where a reflected beam will focus. The 
result is a sharper line and reduced exposure time. Note, however, that 
the focusing condition can only be satisfied for one line on the pattern. 

But line sharpness and exposure time are not the only criteria to be con- 
sidered in deciding between the collimated and divergent beam techniques. 
One of the real advantages of the x-ray method over all other methods of 
stress measurement is the ability to measure the stress almost at a point 
on the specimen. This can be done with a collimated beam, which can be 
made very narrow, but not with a divergent beam, which covers a fairly 
wide area of the specimen. The collimated-beam technique is therefore 
to be preferred when the stress in the specimen varies rapidly from point 
to point on the surface, and when it is important that the existing stress 
gradient be evaluated. 

When the stress gradient normal to the surface is large, errors of interpre- 
tation may arise unless it is realized that the effective depth of x-ray pene- 
tration varies with the angle of incidence of the x-rays. Suppose, for 
example, that 6 = 80 and \l/ = 45. Then it may be shown, by means of 
Eq. (9-3), that the effective penetration depth is 83 percent greater in the 
normal-incidence exposure than it is in the inclined one. 

Correct specimen preparation is extremely important. If dirt and scale 
are present, they may be ground off, but the grinding must be followed by 
deep etching to remove the surface layer left in compression by the grinding. 
The surface is then lightly polished with fine emery paper, to remove the 
roughness caused by deep etching, and lightly re-etched. Surface rough- 
ness must be strictly avoided, because the high points in a rough surface 
are not stressed in the same way as the bulk of the material and yet they 
contribute most to the diffraction pattern, especially the one made at 
inclined incidence, as indicated in Fig. 
17-11. Of course, the surface should 
not be touched at all prior to the 
stress measurement, if the object is 
to measure residual surface stresses 
caused by some treatment such as 
machining, grinding, shot peening, 
etc. Such treatments produce steep 
stress gradients normal to the surface, 
and the removal of any material by FlQ 1? _ n Diffraction from a 
polishing or etching would defeat the rough surface when the indent beam 
purpose of the measurement. is inclined. 

17-6 Experimental technique (diffractometer). The diffractometer may 
also be used for stress measurement, and many details of the diffractometer 
technique, e.g., specimen preparation, are identical to those mentioned in 




17-6] 



EXPERIMENTAL TECHNIQUE (DIFFRACTOMETER) 



445 



the preceding section. The only instrumental changes necessary are the 
addition of a specimen holder which will allow independent rotation of the 
specimen about the diffractometer axis, and a change in the position of the 
receiving slit. 

Figure 17-12 illustrates the angular relationships involved. In (a), the 
specimen is equally inclined to the incident and diffracted beams ; \l/ is zero 
and the specimen normal N* coincides with the reflecting plane normal 
N p . Radiation divergent from the source S is diffracted to a focus at F 
on the diffractometer circle. Even though the primary beam is incident 
on the surface at an angle 8 rather than at 90, a diffraction measurement 
made with the sample in this position corresponds to a normal-incidence 
photograph made with a camera, except that the reflecting planes are now 
exactly parallel to the surface and the strain is measured exactly normal 
to the surface. In (b) the specimen has been turned through an angle \l/ 
for the inclined measurement. Since the focusing circle is always tangent 
to the specimen surface, rotation of the specimen alters the focusing circle 
both in position and radius, and the diffracted rays now come to a focus 
at F', located a distance r from F. If R is the radius of the diffractometer 
circle, then it may be shown that 



r _ cos 
R ~ cos 



+ (90 - 0)] 



- (90 - 0)] 
If ^ = 45, then r/R is 0.30 for 6 = 80 and 0.53 for 8 = 70 



specimen 



diffractometcr 
* S " circle 

counter 




(a) (b) 

FIG. 17-12. Use of a diffractometer for stress measurement: (a) ^ = 0; (b) ^ ^. 



446 STRESS MEASUREMENT [CHAP. 17 

When ^ is not zero, the focal point of the diffracted beam therefore lies 
between F, the usual position of the counter receiving slit, and the speci- 
men. If the receiving slit is kept at F, the intensity of the beam entering 
the counter will be very low. On the other hand, if a wide slit is used at F, 
resolution will suffer. The proper thing to do is to put a narrow slit at 
F' and a wide slit at F, or put a narrow slit at F' and move the counter to a 
position just behind it. In theory, different slit arrangements are therefore 
necessary for the measurement made at \l/ = and the one made at ^ = 
45. In practice, a change in slit position between each of these measure- 
ments is avoided by making a compromise between intensity and resolution, 
and placing a narrow slit at some point between F and F' where experiment 
indicates that satisfactory results are obtained. The slit is then left in 
this position for both measurements. 

Since the angular position 26 of the diffracted beam is measured directly 
with a diffractometer, it is convenient to write the stress equation in terms 
of 26 rather than plane spacings. Differentiating the Bragg law, we obtain 

Ad _ cot 6 A20 

~d ~ 2 

Combining this relation with Eq. (17-10) gives 

_ E cot 6(26 n - 20 t ) 

'* 2(1 + iOsin 2 * 
Put 

E cot6 

2 "" 



2(1 + v) sin 2 ^ 
Then 

er, = K 2 (26 n - 20 t ), (17-15) 

where 20 n is the observed value of the diffraction angle in the "normal" 
measurement ($ = 0) and 20, its value in the inclined measurement 
(^ = ^). For measurements made on the 310 line of steel with Co Ka 
radiation, putting E = 30 X 10 6 psi, v = 0.28, = 80, and ^ = 45, we 
obtain for the stress factor K 2 a value of 720 psi/0.01 20. If 20 n and 20 t 
are both measured to an accuracy of 0.02, then the probable error in the 
stress measured is 2880 psi. 

Essentially, the quantity measured in the diffractometer method is 
A20 = (20 n 20), the shift in the diffraction line due to stress as the 
angle \l/ is changed. But certain geometrical effects, particularly the com- 
promise position of the receiving slit, introduce small errors which cause 
a change in 20 even for a stress-free specimen, when \l/ is changed from 
to 45. It is therefore necessary to determine this change experimentally 
and apply it as a correction (A20) to all A20 values measured on stressed 



17-7] 



SUPERIMPOSED MACROSTRESS AND MICROSTRESS 



447 



specimens. The correction is best determined by measurements on a 
sample of fine powder, which is necessarily free of macrostrain, at ^ = 
and ^ = 45. The powder should have the same composition as the mate- 
rial in which stress is to be measured in order that its diffraction line occur 
at the same position 20, since the correction itself, (A20) , depends on 26. 

17-7 Superimposed macrostress and microstress. As mentioned in the 
introduction, a specimen may contain both a uniform macrostress and a 
nonuniform microstress. The result is a diffraction line which is both 
shifted and broadened. This effect occurs quite commonly in hardened 
steel parts: nonuniform microstress is set up by the austenite-to-martensite 
transformation and on this is superimposed a uniform residual macro- 
stress, due to any one of a number of causes, such as quenching, prior 
plastic deformation, or grinding. 

Stress measurement by x-rays requires the measurement of diffraction- 
line shift. If the lines are sharp, it is relatively easy to measure this shift 
visually with a device such as shown in Fig. 6-18. But if the lines are 
broad (and a breadth at half-maximum intensity of 5 to 10 26 is not 
uncommon in the case of hardened steel), an accurate visual measurement 
becomes impossible. It is then necessary to determine the profile of the 
line, either from a microphotometer record of the film if a camera was used, 
or by directly measuring the intensity at various angles 28 with the dif- 
fractometer. 

After the line profile is obtained, the problem still remains of locating 
the "center" of the line. Since the line may be, and frequently is, unsym- 
metrical, "center" has no precise meaning but is usually taken as the peak 
of the line, i.e., the point of maximum intensity. But the top of a broad 
line is often almost flat so that direct determination of the exact point of 
maximum intensity is extremely difficult. 

Two methods have been used to fix the positions of broadened lines. 
The first is illustrated in Fig. 17-13 (a) and may be used whenever the lines 



line center 
line profile 




line center 




line profile 



parabola 



parabola 



20 26 

(a) (b) 

FIG. 17-13. Methods of locating the centers of broad diffraction lines. 



448 STRESS MEASUREMENT [CHAP. 17 

involved have straight sides. The linear portions are simply extrapolated 
and their point of intersection taken as the "center" of the line. If the 
line is unsymmetrical, the point so found will not have the same 26 value 
as the point of maximum intensity. But this is of no consequence in the 
measurement of stress, as long as this "center" is reproducible, since all 
that is required is the difference between two values of 26 and not the abso- 
lute magnitude of either. 

The other method depends on the fact that the profile of a broad line 
near its peak has the shape of a parabola with a vertical axis, as shown in 
Fig. 17-13(b), even when the over-all shape of the line is unsymmetrical. 
Now the equation 

y = ax 2 + bx + c (17-16) 

is the general equation of a parabola whose axis is parallel to the y axis. 
The maximum on this curve occurs when 

= 2ax + b = 0, 
dx 

x = - - (17-17) 

2a 

If we put x = 26 and y = /, then Eq. (17-16) represents the shape of the 
diffraction line near its peak. We then substitute several pairs of observed 
26, 1 values into this equation and solve for the best values of the constants 
a and b by the method of least squares. Equation (17-17) then gives the 
exact value of x ( = 20) at which the maximum occurs. Only two or three 
points on either side of the peak near its maximum are sufficient to locate 
the parabola with surprising accuracy. The positions of diffraction lines 
as broad as 8 26 at half-maximum intensity have been reproducibly deter- 
mined to within 0.02 by this method. 

Choice of the proper radiation is an important matter when the positions 
of broad diffraction lines have to be accurately measured. Every effort 
should be made to reduce the background, since the accurate measurement 
of a broad, diffuse diffraction line superimposed on a high-intensity back- 
ground is very difficult. Thus, cobalt radiation filtered through iron oxide 
is satisfactory for annealed steel, because the diffraction lines are sharp. 
However, the background is high, since the short wavelength components 
of the continuous spectrum cause fluorescence of iron K radiation by the 
specimen. For this reason cobalt radiation is completely unsuitable for 
stress measurements on hardened steel, where very broad lines have to be 
measured. For such specimens chromium radiation should be used, in 
conjunction with a vanadium filter between the specimen and the photo- 
graphic film or diffractometer counter. The vanadium filter suppresses 
not only the Cr Kfl component of the incident radiation but also the 



17-8] CALIBRATION 449 

fluorescent iron K radiation from the specimen, since the K edge of vana- 
dium lies between the wavelengths of Fe Ka and Cr Ka. The tube voltage 
should also be kept rather low, at about 30 to 35 kv, to minimize the inten- 
sity of the fluorescent radiation. The large gain in the line-to-background 
intensity ratio obtained by using chromium instead of cobalt radiation 
more than compensates for the fact that the diffraction lines occur at 
smaller 26 values with the former. 

17-8 Calibration. For the measurement of stress by x-rays we have 
developed two working equations, Eqs. (17-14) and (17-15), one for the 
pinhole camera and one for the diffractometer. Each of them contains an 
appropriate stress factor K, by which diffraction line shift is converted to 
stress. Furthermore each was derived on the assumption that the material 
under stress was an isotropic body obeying the usual laws of elasticity. 
This assumption has to be examined rather carefully if a calculated value 
of K is to be used for stress measurement. 

The stress factor K contains the quantity E/(l + p), and we have 
tacitly assumed that the values of E and v measured in the ordinary way 
during a tensile test are to be used in calculating the value of K. But 
these mechanically measured values are not necessarily the correct ones to 
apply to a diffraction measurement. In the latter, strains are measured in 
particular crystallographic directions, namely, the directions normal to the 
(hkl) reflecting planes, and we know that both E and v vary with crystal- 
lographic direction. This anisotropy of elastic properties varies from one 
metal to another: for example, measurements on single crystals of a-iron 
show that E has a value of 41.2 X 10 6 psi in the direction [111] and 19.2 X 
10 6 psi in [100], whereas the values of E for aluminum show very little 
variation, being 10.9 X 10 6 psi in [111] and 9.1 X 10 6 psi in [100]. The 
mechanically measured values are 30 X 10 6 and 10 X 10 6 psi for poly- 
crystalline iron and aluminum, respectively. These latter values are evi- 
dently average values for aggregates of contiguous grains having random 
orientation. In the x-ray method, however, only grains having a particular 
orientation relative to the incident beam, and therefore a particular orien- 
tation with respect to the measured stress, are able to reflect. There is 
therefore no good reason why the mechanically measured values of E and 
v should be applied to these particular grains. Stated alternately, an 
aggregate of randomly oriented grains may behave isotropically but indi- 
vidual grains of particular orientations in that aggregate may not. 

These considerations are amply supported by experiment. By making 
x-ray measurements on materials subjected to known stresses, we can 
determine the stress factor K experimentally. The values of K so obtained 
differ by as much as 40 percent from the values calculated from the mechan- 
ically measured elastic constants. Moreover, for the same material, the 



450 



STRESS MEASUREMENT 



[CHAP. 17 



x-ray 





-ray 



(a) 



(b) 



FIG. 17-14. Specimens used for calibrating x-ray method. 

measured values of K vary with the wavelength of the radiation used and 
the Miller indices of the reflecting planes. With steel, for example, the 
calculated value of K happens to be in good agreement with the measured 
value if CoKa radiation is reflected from the (310) planes but not if 
some other combination of X and (hkl) is employed. 

Methods have been proposed for calculating the proper values of E 
and v to use with x-ray measurements from the values measured in various 
directions in single crystals, but such calculations are not very accurate. 
The safest procedure is to measure K on specimens subjected to known 
stresses. We will consider this calibration in terms of the diffractometer 
method, but the same procedure may also be used for calibrating the 
camera method. 

The usual practice is to set up known stresses in a body by bending. 
Both flat beams and heavy split rings have been used, as illustrated in 
Fig. 17-14. The beam shown in (a) is supported at two points and loaded 
by the two forces F\ ; tensile stress is therefore produced in the top surface 
on which the x-ray measurements are made. The split ring shown in (b) 
may be either expanded by the forces F 2 , producing compressive stress at 
the point of x-ray measurement, or compressed by the forces F 3 , producing 
tensile stress at the same point. If the applied forces and the dimensions 
and over-all elastic properties of the stressed member are known, then the 
stress at the point of x-ray measurement may be computed from elasticity 
theory. If not, the stress must be measured by an independent method, 
usually by means of electric-resistance strain gauges placed at the points 
marked X. At no time during the calibration should the elastic limit of 
the material be exceeded. 

A typical calibration curve might have the appearance of Fig. 17-15, 
where the known stress o> is plotted against the observed value of A20 = 
(26 n - 20 t ), in this case for an applied positive (tensile) stress. The slope 
of this line is the stress factor K 2 . However, the experimental curve must 



17-9] 



APPLICATIONS 



451 



be corrected by an amount (A20) , 
measured on a stress-free sample in 
the manner previously described. The 
corrected working curve is therefore a 
line of the same slope as the experi- 
mental curve but shifted by an 
amount (A20) . The working curve 
may or may not pass through zero, 
depending on whether or not the cali- 
brating member contains residual 
stress. In the example shown here, 
a small residual tensile stress was 
present. 




FIG. 17-15. Calibration curve for 
stress measurement. 



17-9 Applications. The proper field of application of the x-ray method 
will become evident if we compare its features with those of other methods 
of stress, or rather strain, measurement. If a camera with a pinhole col- 
limator is used, the incident x-ray beam can be made quite small in diam- 
eter, say T V in., and the strain in the specimen may therefore be measured 
almost at a point. On the other hand, strain gauges of the electrical or 
mechanical type have a length of an inch or more, and they therefore 
measure only the average strain over this distance. Consequently, the 
x-ray method is preferable whenever we wish to measure highly localized 
stresses which vary rapidly from point to point, in a macroscopic sense. 

There is a still more fundamental difference between the x-ray method 
and methods involving electrical or mechanical gauges. The latter meas- 
ure the total strain, elastic plus plastic, which has occurred, whereas x-rays 
measure only the elastic portion. The reason for this is the fact that the 
spacing of lattice planes is not altered by plastic flow, in itself, but only 
by changes in the elastic stress to which the grains are subjected. The 
x-ray "strain gauge" can therefore measure residual stress, but an electric- 
resistance gauge can not. Suppose, for example, that an electric-resistance 
gauge is fixed to the surface of a metal specimen which is then deformed 
plastically in an inhomogeneous manner. The strain indicated by the 
gauge after the deforming forces are removed is not the residual elastic 
strain from which the residual stress can be computed, since the indicated 
strain includes an unknown plastic component which is not recovered 
when the deforming force is removed. The x-ray method, on the other 
hand, reveals the residual elastic stress actually present at the time the 
measurement is made. 

However, the x-ray method is not the only way of measuring residual 
stress. There is another widely used method (called mechanical relaxa- 
tion), which involves (a) removing part of the metal by cutting, grinding, 



452 



STRESS MEASUREMENT 



[CHAP. 17 



60 








A 


A 








50 

At\ 










f 








40 


















30 






J 


















I 






\ 






10 




y 




c 


"X 




v^ 













<^ 


^ 


do 







1.5 -10 -05 05 1 1.5 

DISTANCE FROM CENTER 

OF HEATED AREA (in ) 

(a) 




-1.5 -1.0 -05 05 1.0 1.5 

DISTANCE FROM TENTER 

OF HEATED AREA (in ) 

(b) 

FIG. 17-16. Residual stress pattern 
set up by localized heating: (a) trans- 
verse stress; (b) longitudinal stress. 
c? is diameter of heated area. (J. T. 
Norton and D. Rosenthal, Proc. oc. 
Exp. Stress Analysis 1 (2), 77, 1943.) 



etching, etc., and (b) measuring the 
change in shape or dimensions pro- 
duced as a result of this removal. 
For example, the residual stress in 
the weldment discussed earlier [Fig. 
17-1 (b)] could be measured by cut- 
ting through the central rod along the 
line A A' and measuring the length I 
before and after cutting. When the 
rod is cut through, the tensile stress 
in it is relieved and the two side mem- 
bers, originally in compression, are 
free to elongate. The final length If 
is therefore greater than the original 
length i and the strain present before 
the cut was made must have been 
(If l G )/lf. This strain, multiplied 
by the elastic modulus, gives the 
residual compressive stress present in 
the side members before the central 
rod was cut. Similarly, the residual 
stress at various depths of the bent 
beam shown in Fig. 17-2(c) may be 
measured by successive removal of 
layers parallel to the neutral plane, 
and a measurement of the change in 
curvature of the beam produced by 
each removal. 

There are many variations of this 
method and they are all destructive, 
inasmuch as they depend on the par- 
tial or total relaxation of residual 
stress by the removal of a part of the 
stressed metal. The x-ray method, 
on the other hand, is completely non- 
destructive: all the necessary meas- 
urements may be made on the stressed 
metal, which need not be damaged in 
any way. 

We can conclude that the x-ray 
method is most usefully employed for 
the nondestructive measurement of 
residual stress, particularly when the 



PROBLEMS 453 

stress varies rapidly over the surface of the specimen. This latter condi- 
tion is frequently found in welded structures, and the measurement of 
residual stress in and near welds is one of the major applications of the 
x-ray method. For the measurement of applied stress, methods involving 
electrical or mechanical gauges are definitely superior: they are much more 
accurate, faster, and require less expensive apparatus. In fact, they are 
commonly used to calibrate the x-ray method. 

Figure 17-16 shows an example of residual stress measurement by x-rays. 
The specimen was a thin steel bar, 3 in. wide and 10 in. long. A small 
circular area, whose size is indicated on the graph, was heated locally to 
above 1 100 F for a few seconds by clamping the bar at this point between 
the two electrodes of a butt-welding machine. The central area rapidly 
expanded but was constrained by the relatively cold metal around it. As 
a result, plastic flow took place in and near the central region on heating 
and probably also on cooling as the central region tried to contract. Resid- 
ual stresses were therefore set up, and the curves show how these stresses, 
both longitudinal and transverse, vary along a line across the specimen 
through the heated area. In and near this area there is a state of biaxial 
tension amounting to about 55,000 psi, which is very close to the yield 
point of this particular steel, namely, 60,500 psi. There is also a very steep 
stress gradient just outside the heated area: the transverse stress drops 
from 55,000 psi tension to zero in a distance of one inch, and the longi- 
tudinal stress drops from 55,000 psi tension to 25,000 psi compression in 
less than half an inch. Residual stresses of similar magnitude and gradient 
can be expected in many welded structures. 

PROBLEMS 

17-1. Calculate the probable error in measuring stress in aluminum by the 
two-exposure pinhole-camera method. Take E = 10 X 10 6 psi and v = 0.33. 
The highest-angle line observed with Cu Ka radiation is used. For the inclined- 
incidence photograph, the incident beam makes an angle of 45 with the speci- 
men surface, and the radius S\ (see Fig. 17-7) of the Debye ring from the speci- 
men is measured. Assume an accuracy of 0.05 mm in the measurement of line 
position and a specimen-to-film distance of 57.8 mm. Compare your result with 
that given in Sec. 17-4 for steel. 

17-2. A certain aluminum part is examined in the diffractometer, and the 20 
value of the 511,333 line is observed to be 163.75 when ^ = 0, and 164.00 for 
\j/ = 45. The same values for a specimen of aluminum powder are 163.81 and 
163.88, respectively. What is the stress in the aluminum part, if it is assumed 
that the stress factor calculable from the elastic constants given in Prob. 17-1 is 
correct? 

17-3. Verify the statement made in Sec. 17-5 that the effective depth of x-ray 
penetration is 83 percent greater in normal incidence than at an incidence of 45, 
when 6 = 80. 



CHAPTER 18 
SUGGESTIONS FOR FURTHER STUDY 

18-1 Introduction. In the previous chapters an attempt has been made 
to supply a broad and basic coverage of the theory and practice of x-ray 
diffraction and its applications to metallurgical problems. But in a book 
of this scope much fundamental theory and many details of technique 
have had to be omitted. The reader who wishes to go on to advanced work 
in this field will therefore have to turn to other sources for further informa- 
tion. The purpose of the following sections is to point out these sources 
and indicate the sort of material each contains, particularly material which 
is mentioned only briefly or not at all in this book. 

One thing is absolutely necessary in advanced work on diffraction and 
that is familiarity with the concept of the reciprocal lattice. This concept 
provides a means of describing diffraction phenomena quite independently 
of the Bragg law and in a much more powerful and general manner. In 
particular, it supplies a way of visualizing diffuse scattering effects which 
are difficult, if not impossible, to understand in terms of the Bragg law. 
Such effects are due to crystal imperfections of one kind and another, and 
they provide a valuable means of studying such imperfections. These 
faults in the crystal lattice, though seemingly minor in character, can have 
a profound effect on the physical and mechanical properties of metals and 
alloys; for this reason, there is no doubt that much of the metallurgical re- 
search of the future will be concerned with crystal imperfections, and in 
this research the study of diffuse x-ray scattering will play a large role. 
The utility of the reciprocal lattice in dealing with diffuse scattering effects 
is pointed out in Appendix 15, where the interested reader will find the 
basic principles and more important applications of the reciprocal lattice 
briefly described. 

18-2 Textbooks. The following is a partial list of books in English 
which deal with the theory and practice of x-ray diffraction and crystal- 
lography. 

(1) Structure of Metals, 2nd ed., by Charles S. Barrett. (McGraw-Hill 
Book Company, Inc., New York, 1952.) Deservedly the standard work 
in the field, it has long served as a text and reference book in the crystallo- 
graphic aspects of physical metallurgy. Really two books in one, the first 
part dealing with the theory and methods of x-ray diffraction, and the 
second part with the structure of metals in the wider sense of the word. 

454 



18-2] TEXTBOOKS 455 

Includes a very lucid account of the stereographic projection. Contains 
an up-to-date treatment of transformations, plastic deformation, structure 
of cold-worked metal, and preferred orientations. Gives a wealth of refer- 
ences to original papers. 

(2) X-Ray Crystallographic Technology, by Andr6 Guinier. (Hilger and 
Watts Ltd., London, 1952. Translation by f . L. Tippel, edited by Kath- 
leen Lonsdale, of Guinier's Radiocristallographie, Dunod, Paris, 1945.) 
Written with true French clarity, this book gives an excellent treatment of 
the theory and practice of x-ray diffraction. A considerable body of theory 
is presented, although this is not suggested by the title of the English trans- 
lation, and experimental techniques are given in detail. The theory and 
applications of the reciprocal lattice are very well described. Unusual fea- 
tures include a full description of the use of focusing monochromators and 
chapters on small-angle scattering and diffraction by amorphous substances. 
Crystal-structure determination is not included. 

(3) X-Ray Diffraction Procedures, by Harold P. Klug and Leroy E. Alex- 
ander. (John Wiley & Sons, Inc., New York, 1954.) As its title indicates, 
this book stresses experimental methods. The theory and operation of 
powder cameras and diffractometers are described in considerable and use- 
ful detail. (Single-crystal methods, Laue and rotating crystal, are not in- 
cluded.) Particularly valuable for its discussion of quantitative analysis 
by diffraction, a subject to which these authors have made important con- 
tributions. Also includes chapters on particle-size measurement from line 
broadening, diffraction by amorphous substances, and small-angle scatter- 
ing. 

(4) X-Ray Diffraction by Poly crystalline Materials, edited by H. S. 
Peiser, H. P. Rooksby, and A. J. C. Wilson. (The Institute of Physics, 
London, 1955.) This book contains some thirty chapters, contributed by 
some thirty different authors, on the theory and practice of the powder 
method in its many variations. These chapters are grouped into three 
major sections: experimental technique, interpretation of data, and appli- 
cations in specific fields of science and industry. A great deal of useful 
information is presented in this book, which will be of more value to the 
research worker than to the beginning student, in that most of the con- 
tributors assume some knowledge of the subject on the part of the reader. 

(5) Applied X-Ray s, 4th ed., by George L. Clark. (McGraw-Hill Book 
Company, Inc., New York, 1955.) A very comprehensive bodk, devoted 
to the applications of x-rays in many branches of science and industry. 
Besides diffraction, both medical and industrial radiography (and micro- 
radiography) are included, as well as sections on the chemical and biological 
effects of x-rays. The crystal structures of a wide variety of substances, 
ranging from organic compounds to alloys, are fully described. 



456 SUGGESTIONS FOR FURTHER STUDY [CHAP. 18 

(6) X-Rays in Practice, by Wayne T. Sproull. (McGraw-Hill Book 
Company, Inc., New York, 1946.) X-ray diffraction and radiography, 
with emphasis on their industrial applications. 

(7) An Introduction to X-Ray Metallography, by A. Taylor. (John 
Wiley & Sons, Inc., New York, 1945.) Contains extensive material on the 
crystallographic structure of metals and alloys and on methods of deter- 
mining alloy equilibrium diagrams by x-ray diffraction. Sections on radi- 
ography and microradiography also included. 

(8) X-Rays in Theory and Experiment, by Arthur H. Compton and 
Samuel K. Allison. (D. Van Nostrand Company, Inc., New York, 1935.) 
A standard treatise on the physics of x-rays and x-ray diffraction, with 
emphasis on the former. 

(9) The Crystalline State. Vol. I: A General Survey, by W. L. Bragg. 
(The Macmillan Company, New York, 1934.) This book and the two listed 
immediately below form a continuing series, edited by W. L. Bragg, to 
which this book forms an introduction. It is a very readable survey of the 
field by the father of structure analysis. Contains very clear accounts in 
broad and general terms of crystallography (including space-group theory), 
diffraction, and structure analysis. An historical account of the develop- 
ment of x-ray crystallography is also included. 

(10) The Crystalline State. Vol. II: The Optical Principles of the Diffrac- 
tion of X-Rays, by R. W. James. (George Bell & Sons, Ltd., London, 1948.) 
Probably the best book available in English on advanced theory of x-ray 
diffraction. Includes thorough treatments of diffuse scattering (due to 
thermal agitation, small particle size, crystal imperfections, etc.), the use 
of Fourier series in structure analysis, and scattering by gases, liquids, and 
amorphous solids. 

(11) The Crystalline State. Vol. Ill: The Determination of Crystal Struc- 
tures, by H. Lipson and W. Cochran. (George Bell & Sons, Ltd., London, 
1953.) Advanced structure analysis by means of space-group theory and 
Fourier series. Experimental methods are not included; i.e., the problem 
of structure analysis is covered from the point at which \F\ 2 values have 
been determined by experiment to the final solution. Contains many illus- 
trative examples. 

(12) The Interpretation of X-Ray Diffraction Photographs, by N. F. M. 
Henry, H. Lipson, and W. A. Wooster. (The Macmillan Company, Lon- 
don, 1951.) Rotating and oscillating crystal methods, as well as powder 
methods, are described. Good section on analytical methods of indexing 
powder photographs. 

(13) X-Ray Crystallography, by M. J. Buerger. (John Wiley & Sons, 
Inc., New York, 1942.) Theory and practice of rotating and oscillating 
crystal methods. Space-group theory. 

(14) Small-Angle Scattering of X-Rays, by Andrg Guinier and Gerard 
Fournet. Translated by Christopher B. Walker, and followed by a bibli- 



18-3] REFERENCE BOOKS 457 

ography by Kenneth L. Yudowitch. (John Wiley & Sons, Inc., New York, 
1955.) A full description of small-angle scattering phenomena, including 
theory, experimental technique, interpretation of results, and applications. 

18-3 Reference books. Physical and mathematical data and informa- 
tion on specific crystal structures may be found in the following books: 

(1) Internationale Tabellen zur Bestimmung von Kristallstrukturen [Inter- 
national Tables for the Determination of Crystal Structures]. (Gebriider 
Borntraeger, Berlin, 1935. Also available from Edwards Brothers, Ann 
Arbor, Mich., 1944.) 

Vol. 1 . Space-group tables. 

Vol. 2. Mathematical and physical tables (e.g., values of sin 2 0, atomic 
scattering factors, absorption coefficients, etc.). 

(2) International Tables for X-ray Crystallography. (Kynoch Press, 
Birmingham, England.) These tables are published by the International 
Union of Crystallography and are designed to replace the Internationale 
Tabellen (1935), much of which was in need of revision. 

Vol. I. Symmetry groups (tables of point groups and space groups) 
(1952). The reader should not overlook the interesting Historical Intro- 
duction written by M. von Laue. 

Vol. II. Mathematical tables (in preparation). 

Vol. II L Physical and chemical tables (in preparation). 

(3) Absorption coefficients and the wavelengths of emission lines and 
absorption edges, not included in the Internationale Tabellen (1935), can 
generally be found in the book by Compton and Allison (item 8 of the 
previous section) or in the Handbook of Chemistry and Physics (Chemical 
Rubber Publishing Co., Cleveland). Wavelengths are given in kX units. 

(4) Longueurs d'Onde des Emissions X et des Discontinuity d 1 Absorption 
X [Wavelengths of X-Ray Emission Lines and Absorption Edges], by 
Y. Caiichois and H. Hulubei. (Hermann & Cie, Paris, 1947.) Wavelengths 
of emission lines and absorption edges in X units, listed both in numerical 
order of wavelength (useful in fluorescent analysis) and in order of atomic 
number. 

(5) Strukturbericht. (Akademische Verlagsgesellschaft, Leipzig, 1931- 
1943. Also available from Edwards Brothers, Ann Arbor, Mich., 1943.) 
A series of seven volumes describing crystal structures whose solutions 
were published in the years 1913 to 1939, inclusive. 

(6) Structure Reports. (Oosthoek, Utrecht, 1951 to date.) A continua- 
tion, sponsored by the International Union of Crystallography, of Struk- 
turbericht. The volume numbers take up where Strukturbericht left off: 

Vol. 8. (In preparation.) 

Vol. 9. (1956) Structure results published from 1942 to 1944. 
Vol. 10. (1953) Structure results published in 1945 and 1946. 
Vol. 11. (1952) Structure results published in 1947 and 1948. 



458 SUGGESTIONS FOR FURTHER STUDY [CHAP. 18 

Vol. 12. (1951) Structure results published in 1949. 

Vol. 13. (1954) Structure results published in 1950. 
The results of structure determinations are usually given in sufficient 
detail that the reader has no need to consult the original paper. 

(7) The Structure of Crystals, 2nd ed., by Ralph W. G. Wyckoff. (Chem- 
ical Catalog Company, New York, 1931. Supplement for 1930-34, Rein- 
hold Publishing Corporation, New York, 1935.) Crystallography (includ- 
ing space-group theory) and x-ray diffraction. In addition, full descrip- 
tions are given of a large number of known crystal structures. 

(8) Crystal Structures, by Ralph W. G. Wyckoff. (Interscience Pub- 
lishers, Inc., New York.) A continuation of Wyckoff 's work (see previous 
item) of classification and presentation of crystal structure data. Three 
volumes have been issued to date (Vol. I, 1948; Vol. II, 1951; Vol. Ill, 
1953) and more are planned for the future. Each volume is in loose-leaf 
form so that later information on a particular structure can be inserted in 
the appropriate place. 

(9) Lists of known structures and lattices parameters can also be found 
in the Handbook of Chemistry and Physics (organic and inorganic com- 
pounds) and in the book by Taylor, item 7 of the previous section (inter- 
metallic "compounds"). 

18-4 Periodicals. Broadly speaking, technical papers involving x-ray 
crystallography are of two kinds: 

(a) Those in which crystallography or some aspect of x-ray diffraction 
form the central issue, e.g., papers describing crystal structures, crystallo- 
graphic transformations, diffraction theory, diffraction methods, etc. Such 
papers were published in the international journal Zeitschrift fur Kristal- 
lographie, in which each paper appeared in the language of the author (Eng- 
lish, French, or German). Publication of this journal ceased in 1945 and a 
new international journal, Acta Crystallographica, a publication of the In- 
ternational Union of Crystallography, was established to take its place, 
publication beginning in 1948. (Publication of Zeitschrift fur Kristal- 
lographie was resumed in 1954.) Although the bulk of the papers appear- 
ing in Acta Crystallographica are confined to structure results on complex 
organic and inorganic compounds, occasional papers of metallurical interest 
appear. Papers on diffraction theory and methods are also found in jour- 
nals of physics, applied physics, and instrumentation. 

(b) Those in which x-ray diffraction appears in the role of an experimen- 
tal tool in the investigation of some other phenomenon. Much can be 
learned from such papers about the applications of x-ray diffraction. Many 
papers of this sort are to be found in various metallurgical journals. 



APPENDIX 1 
LATTICE GEOMETRY 

Al-1 Plane spacings. The value of d, the distance between adjacent 
planes in the set (hkl), may be found from the following equations. 

1 h 2 + k 2 + I 2 

Cubic: - = 

d 2 cr 

1 h 2 + k 2 I 2 

Tetragonal: = h -5 

d 2 a 2 (? 

1 4 /h 2 + hk + k?\ I 2 



3\ a 2 
Rhombohedral: 



1 _ (h 2 + k 2 + I 2 ) sin 2 a + 2(hk + kl + hi) (cos 2 a - cos a) 
d 2 " a 2 (l - 3 cos 2 a + 2 cos 3 a) 

1 h 2 k 2 I 2 



OrthoMic: 



1 1 /h 2 k 2 siu 2 I 2 2cos0\ 

) 



Monochnic: = - - I H -- -- h -r 

d 2 sm 2 /8\a 2 6 2 c 2 ac / 

TricUnic: ~T 2 = 2 (Snh 2 + S 22 k 2 + S 3 3^ 2 + 2S 12 /ifc + 2S 23 kl + 2S l3 hl) 

In the equation for triclinic crystals 

V = volume of unit cell (see below), 
Sn = 6 2 c 2 sin 2 a, 
2 ft 

S 33 = a 2 6 2 sin 2 7, 
Si2 = abc 2 (cos a cos )S cos 7), 
^23 = a 2 6c(cos ft cos 7 cos a), 

<Si3 = ob 2 c(cos 7 cos a cos ft). 
459 



a 2 c 2 sin 



460 LATTICE GEOMETRY [APP. 1 

Al-2 Cell volumes. The following equations give the volume V of the 
unit cell. 

Cubic: V = a 3 

Tetragonal: V = a 2 c 



Hexagonal: V = - = 0.866a 2 c 



Rhombohedral: V = a 3 VI 3 cos 2 a + 2 cos 3 a 

Orthorhombic: V = abc 

Monoclinic: V = abc sin ft 



Tridinic: V abc V 1 cos 2 a cos 2 ft cos 2 7 + 2 cos a cos cos 7 



Al-3 Interplanar angles. The angle </> between the plane (AiA'i/i), of 
spacing dj, and the plane (/i 2 /c 2 fe), of spacing rf 2 , may be found from the 
following equations. (F is the volume of the unit cell.) 



Cubic: cos <t> = 



Tetragonal: cos< = 



, 2 + fc, 2 + /I W + *2 2 ""+ 



cos <t> = 



Rhombohedral: 



3a 2 

Z 

4c 2 



fc 2 2 + * 2 fc 2 + 

4c 2 



cos </> = [sin 2 a(/ii/i2 + fc^g + 



+ (cos 2 a - cos a)(*!fe + fc 2 ^i + hh* + fefci + ftifc 2 + 



Al-3] INTERPLANAR ANGLES 461 



Orthorhombic: cos </> = / 2 2 2 2 iT2 

Monocfo'm'c: 

cos ^> = - ^ I TT I ~ ~ 

sin 2 18 L a 2 6 2 c 2 ac 

TricKrac: 

^1^2 077 Q 1 1 



APPENDIX 2 
THE RHOMBOHEDRAL-HEXAGONAL TRANSFORMATION 

The lattice of points shown in Fig. A2-1 is rhombohedral, that is, it 
possesses the symmetry elements characteristic of the rhombohedral sys- 
tem. The primitive rhombohedral cell has axes ai(R), a 2 (R), and aa(R). 
The same lattice of points, however, may be referred to a hexagonal cell 
having axes ai(H), a 2 (H), and c(H). The hexagonal cell is no longer primi- 
tive, since it contains three lattice points per unit cell (at 000, ^ ^, and 
f f f), and it has three times the volume of the rhombohedral cell. 

If one wishes to know the indices (HK-L), referred to hexagonal axes, 
of a plane whose indices (/i/c/), referred to rhombohedral axes, are known, 
the following equations may be used : 



H = h - k, 



K = 



k-l, 



L = h + k + l. 




FIG. A2-1. Rhombohedral and hexagonal unit cells in a rhombohedral attice. 

462 



APP. 2] RHOMBOHEDRAL-HEXAGONAL TRANSFORMATION 463 

Thus, the (001) face of the rhombohedral cell (shown shaded in the figure) 
has indices (01 1) when referred to hexagonal axes. 

Since a rhombohedral lattice may be referred to hexagonal axes, it fol- 
lows that the powder pattern of a rhombohedral substance can be indexed 
on a hexagonal Hull-Davey or Bunn chart. How then can we recognize 
the true nature of the lattice? From the equations given above, it follows 
that 

-H + K + L = 3/r. 

If the lattice is really rhombohedral, then k is an integer and the only lines 
appearing in the pattern will have hexagonal indices (HK L) such that the 
sum ( H + K + L) is always an integral multiple of 3. If this condition 
is not satisfied, the lattice is hexagonal. 

When the pattern of a rhombohedral substance has been so indexed, 
i.e., with reference to hexagonal axes, and the true nature of the lattice de- 
termined, we usually want to know the indices (hkl) of the reflecting planes 
when referred to rhombohedral axes. The transformation equations are 

h = J(2H + K + L), 



I = (-//- 2K + L). 

There is then the problem of determining the lattice parameters an and a 
of the rhombohedral unit cell. But the dimensions of the rhombohedral 
cell can be determined from the dimensions of the hexagonal cell, and this 
is an easier process than solving the rather complicated plane-spacing equa- 
tion for the rhombohedral system. The first step is to index the pattern 
on the basis of hexagonal axes. Then the parameters an and c of the 
hexagonal cell are calculated in the usual way. Finally, the parameters of 
the rhombohedral cell are determined from the following equations: 



+ c 2 , 



Finally, it should be noted that if the c/a ratio of the hexagonal cell in 
Fig. A2-1 takes on the special value of 2.45, then the angle a of the rhom- 
bohedral cell will equal 60 and the lattice of points will be face-centered 
cubic. Compare Fig. A2-1 with Figs. 2-7 and 2-16. 

Further information on the rhombohedral-hexagonal relationship and on 
unit cell transformations in general may be obtained from the International 
Tables jor X-Ray Crystallography (1952), Vol. 1, pp. 15-21. 



APPENDIX 3 



WAVELENGTHS (IN ANGSTROMS) OF SOME CHARACTERISTIC 
EMISSION LINES AND ABSORPTION EDGES 







Ka 


Kaz 


A'ai 


Kfti 


K 


Lai 


L\u 


Element 


Z 


(weighted^ 
average) 


strong 


very 
strong 


weak 


edge 


very 
strong 


edge 


No 


11 




11.909 


11.909 


11.617 








Mg 


12 




9.8889 


9.8889 


9.558 


9.5117 






Al 


13 




8.33916 


8.33669 


7.981 


7.9511 






Si 


14 




7.12773 


7.12528 


6.7681 


6.7446 






P 


15 




6.1549 


6.1549 


5.8038 


5.7866 






s 


16 




5.37471 


5.37196 


5.03169 


5.0182 






Cl 


17 




4.73050 


4.72760 


4.4031 


4.3969 






A 


18 




4.19456 


4.19162 




3.8707 






K 


19 




3.74462 


3.74122 


3.4538 


3.43645 






Co 


20 




3.36159 


3.35825 


3.0896 


3.07016 






Sc 


21 




3.03452 


3.03114 


2.7795 


2.7573 






Ti 


22 




2 75207 


2.74841 


2.51381 


2.49730 






V 


23 




2.50729 


2.50348 


2.28434 


2.26902 






Cr 


24 


2.29092 


2.29351 


2.28962 


2.06480 


2.07012 






Mn 


25 




2.10568 


2.10175 


1.91015 


1.89636 






Fe 


26 


1 93728 


.93991 


1 .93597 


1.75653 


1.74334 






Co 


27 


1 .79021 


79278 


.78892 


.62075 


.60811 






Ni 


28 




.66169 


.65784 


1.50010 


.48802 






Cu 


29 


1.54178 


.54433 


.54051 


.39217 


.38043 


13.357 


13.2887 


Zn 


30 




.43894 


.43511 


.29522 


.28329 


12.282 


12.1309 


Go 


31 




.34394 


.34003 


.20784 


.19567 


11.313 




Ge 


32 




.25797 


.25401 


.12889 


.11652 


10.456 




As 


33 




1.17981 


.17581 


.05726 


.04497 


9.671 


9.3671 


Se 


34 




1.10875 


.10471 


0.99212 


0.97977 


8.990 


8.6456 


Br 


35 




1.04376 


.03969 


0.93273 


.91224 


8.375 




Kr 


36 




0.9841 


0.9801 


0.87845 


0.86546 






Rb 


37 




0.92963 


0.92551 


0.82863 


0.81549 


7.3181 


6.8633 


Sr 


38 




0.87938 


0.875214 


0.78288 


0.76969 


6.8625 


6.3868 


Y 


39 




0.83300 


0.82879 


0.74068 


0.72762 


6.4485 


5.9618 


Zr 


40 




0.79010 


0.78588 


0.701695 


0.68877 


6.0702 


5.5829 


Nb 


41 




0.75040 


0.74615 


0.66572 


0.65291 


5.7240 


5.2226 


Mo 


42 


0.71069 


0.713543 


0.70926 


0.632253 


0.61977 


5.40625 


4.9125 


Tc 


43 




0.676 


0.673 


0.602 








Ru 


44 




0.64736 


0.64304 


0.57246 


0.56047 


4.84552 


4.3689 


Rh 


45 




0.617610 


0.613245 


0.54559 


0.53378 


4.59727 


4.1296 


Pd 


46 




0.589801 


0.585415 


0.52052 


0.50915 


4.36760 


3.9061 


Ag 


47 




0.563775 


0.559363 


0.49701 


Q44&22 


4.15412 


3.6983 


Cd 


48 




0.53941 


0.53498 


0.475078 


0.46409 


3.95628 


3.5038 


In 


49 




0.51652 


0.51209 


0.454514 


0.44387 


3.77191 


3.3244 


Sn 


50 




0.49502 


0.49056 


0.435216 


0.42468 


3.59987 


3.1559 


Sb 


51 




0.47479 


0.470322 


0.417060 


0.40663 


3.43915 


2.9999 


To 


52 




0.455751 


0.451263 


0.399972 


0.38972 


3.28909 


2.8554 


I 


53 




0.437805 


0.433293 


0.383884 


0.37379 


3.14849 


2.7194 


Xe 


54 




0.42043 


0.41596 


0.36846 


0.35849 




2.5924 


Cs 


55 




0.404812 


0.400268 


0.354347 


0.34473 


2.8920 


2.4739 



In averaging, A'ai is given twice the weight of A~e* 2 . 

464 



(cont.) 



APP. 3] 



CHARACTERISTIC EMISSION LINES 



405 







Ka 


Kot2 


Ka\ 


Kfr 


X 


La, 


^ni 


Element 


Z 


(weighted 
average) 


strong 


very 
strong 


weak 


edge 


very 
strong 


edge 


Ba 


56 




0.389646 


0.385089 


0.340789 


0.33137 


2.7752 


2.3628 


La 


57 




0.375279 


0.370709 


0.327959 


0.31842 


2.6651 


2.2583 


Ce 


58 




0.361665 


0.357075 


0.315792 


0.30647 


2.5612 


2.1639 


Pr 


59 




0.348728 


0.344122 


0.304238 


0.29516 


2.4627 


2.0770 


Nd 


60 




0.356487 


0.331822 


0.293274 


0.28451 


2.3701 


1.9947 


II 


61 




0.3249 


0.3207 


0.28209 




2. 28 27 




Sm 


62 




0.31365 


0.30895 


0.27305 


0.26462 


2.1994 


.8445 


Eu 


63 




0.30326 


0.29850 


0.26360 


0.25551 


2.1206 


.7753 


Gd 


64 




0.29320 


0.28840 


0.25445 


0.24680 


2.0460 


.7094 


Tb 


65 




0.28343 


0.27876 


0.24601 


0.23840 


1 .9755 


.6486 


Dy 


66 




0.27430 


0.26957 


0.23758 


0.23046 


1 .90875 


.579 


Ho 


67 




0.26552 


0.26083 




0.22290 


1.8447 


.5353 


Er 


66 




0.25716 


0.25248 


0.22260 


0.21565 


.78428 


.48218 


Tu 


69 




0.24911 


0.24436 


0.21530 


0.2089 


.7263 


.4328 


Yb 


70 




0.24147 


0.23676 


0.20876 


0.20223 


.6719 


.38608 


Lu 


71 




0.23405 


0.22928 


0.20212 


0.19583 


.61943 


.34135 


Hf 


72 




0.22699 


0.22218 


0.19554 


0.18981 


.56955 


.29712 


Ta 


73 




0.220290 


0.215484 


0.190076 


0.18393 


.52187 


.25511 


W 


74 




0.213813 


0.208992 


0.184363 


0.17837 


.47635 


.21546 


Re 


75 




0.207598 


0.202778 


0.178870 


0.17311 


.43286 


.17700 


Os 


76 




0.201626 


0.196783 


0.173607 


0.16780 


.39113 


.14043 


Ir 


77 




0.195889 


0.191033 


0.168533 


0.16286 


.35130 


.10565 


Pt 


78 




0.190372 


0.185504 


0.163664 


0.15816 


.31298 


.07239 


Au 


79 




0.185064 


0.180185 


0.158971 


0.15344 


.27639 


.03994 


Hg 


80 










0.14923 


.24114 


.00898 


' 
Tl 


81 




0.175028 


0.170131 


0.150133 


0.14470 


.20735 


0.97930 


Pb 


82 




0.170285 


0.165364 


0.145980 


0.14077 


.17504 


0.95029 


Bi 


83 




0.165704 


0.160777 


0.141941 


0.13706 


1 . 14385 


0.92336 


Th 


90 




0.137820 


0.132806 


0.117389 


0.11293 


0.95598 


0.76062 


U 


92 




0.130962 


125940 


0.111386 


0.1068 


0.91053 


0.72216 



CHARACTERISTIC L LINES OF TUNGSTEN 



Line 


Relative intensity 


Wavelength (A) 


La! 


Very strong 


1 .47635 


La? 


Weak 


1 .48742 


Lfr 


Strong 


1.28176 


Lfo 


Medium 


1.24458 


Lfa 


Weak 


1.26285 


Ly } 


Weak 


1 .09852 



The above wavelengths are based on those in Longueurs d'Onde des Emissions X 
et des Discontinuity d' Absorption X by Y. Cauchois and H. Hulubei (Hermann, 
Paris, 1947). The Cauchois-Hulubei values have been multiplied by 1.00202 X 
10~ 3 to convert them from X units to angstroms. Values, in angstroms, for the 
K lines and K absorption edge were kindly furnished by G. 1). Rieck prior to 
publication in Vol. Ill of the International Tables for X-Ray Crystallography, and 
are published here with the permission of the Editorial Commission of the Inter- 
national Tables. 



APPENDIX 4 
MASS ABSORPTION COEFFICIENTS (|t/p) AND DENSITIES (p) 



a < 


M3 rx 




CO (X "* 


> CM 


CM - <X 


rxN.O CM CMOOXCMCO COOp'-'OOD 

-CM^IOIX <XCM^-O*CM rxOjjcMO 




^ 








k eg 


sss 


co oo CM M> co CM 


o o a. 


^ ii 


O co ix 


, *. p_ CMCMCMCO"^ 


i$e* S82B5 


is 


3338 


IXCOCMOO OvlXOO 


M3 OO -O OO 


27 


o co o 


2-JQS4 SIQS55 Rfc^|b 


OOtxCMQCO CMOCOOOO> 

rtrxcMo^^3 ixooovOoO 


? 


CM co CM rx 
n M5 


OM3CMM3CM CMO^T^- 


lO CK 00 U"> 


~x 


<N ^ 


"- RS " :?s "'- -- s 


8 s^ 5 fts^s 


.$ 


CO IX O Ox 
^ OO 00 K 


rxIXCMlOCM OvO-tODiO 


^oo^ 


x 


O O CO 


O O >O Q IXIXOO'OO CM O O^ O 

CM co co^iorxo* CM -^ rx 


CMIXIOMDOO rx ^f lO CM 

^Icti^oS g;^^^^ 


a * 

^ ii 


IX 00 O M5 
CO M5 CO O 

O O "- CO 


lOiOrxiOMD O"OIXCOO CO 
lOCOCMIXJ* OOOOOCO CO CO CO CM 


co rx o 

OOOCMiOgo CMiO^iOiO 
^CMCMCMCM COCO 


.i 


CO O O lO 


CO U"> 
^OO'^-'^-CM M3OOiO-t OOO 


O CO 


c "- 


O O CM 


^z2g aS9$ft RSS-! 


COrx^OCOTf OCMiOCMO* 
V O S OOO'~~JJO fXJXCM^f 1 ^ 










s^ 


2883 


OOg.K .^0000 




JDO 


O O O O 


o CM co-^ioorx o CM <o o- 


CO MD O CO 00 IXOs-^ 


^ 








!< 

*J 


CO 




^CO CO CO CO 

.-* ' ' CO 1 -^ ^>CO 1 
!Z O O 1 O > > 1 O 
~ "- O ^ O OO"" 

k-XX^X oixixco"a> I^Xooio 

^-'^Som 'CMCM^ ^^.^0 
CMMjCO^CO CM N. MD 

CM * co *o co oo OCM^O 


CM Tt <> rx' rx rx' oo co oo rx' 







CM_,-_ ^_ CMCO 




c 

i 

LU 


> ._ 0) 

X -i <2 en 


0) O O)__ __ D 

uZOu.Z Zl^^ioa. toU^i^U 


y._ wC 0,0-SC 
uoH->U^ a!uZuN 


il 


CM CO ^t lO 


M3IXOOOO CM CO -^ lO OrxCOO^O 






< c 









(coni.) 



466 



APP. 4] MASS ABSORPTION COEFFICIENTS AND DENSITIES 



467 



^ 


ESS35 




IX fx 00 00 00 00 CN 


II 


'il 


5 Jo rx oo o 




O cp O O O fx O 

oo CX *o oo ^ rx ix 
o o o HJ ix -o *o 


il 


* 


^ 8 ^ vo ^ 


CNtx-^-mO ^fc'^CS "OCNIXQIX 


CN 00 IX CN CX CX *O 


88 

CO CO 


*i 










$ 


CO 


00 , <> CN to CN CN CO aoCNCNXiCN 

^5P^tvcX'~ ^^^Sfc Oco'O'Ooo 




43 


" i 










*s 


CX CN 00 

tx 00 CX O " 


CNCOOpOCO CO IX Q j*OorxCN 


CN r^ CN O CTO 

^ ^jt rx <x CN fx 















^ 


CO "t O CO 'O 

CO CX O CN CN 

O *O IX OO O 


88*R9 S3SS ^SS S 3 


g=^ ftl 


*N 


^ ii 










4 


^ *o in " o 


o o 






c ,-.* 


^t^S^P: 


S5;8o2 JNcoJoig ^^8^ 


ggggg Sc? 


0? 


^ II 










8- 










,2 o 


IX CO (X ^t CN 

ir> *o -O ix oo 


OOCXOO CNCNCN CNCNCNCOCO 


tO "O CX CO lO CN 


8 


rl 


CX CO ix 00 OT 


CO 


CO 

i> 

Jo ctl ex X ex n <x 


o co 


ol 


O O T) < ' 
CN 

CO 


CO 


J. 


NO OK 


? 
j 

UJ 


oo^Jli 


> tSlf S*3^ 


-*a 


^ 


11 


CO CO CO CO CO 




^ZSJ8 


RS 



(conf.) 



468 



MASS ABSORPTION COEFFICIENTS AND DENSITIES [APP. 4 



- U CM 


O GO CO f\ CM CO IO CM 


gss 


II 






| 


OCMCOOrN SfXtM 
^^^^^ ^TfiO 


00 CO CN 

^52 


* a i 


**m 5*3 


O CO ^O 

Co 10 10 


^ H 






_' 


K O O ' CO 'St^'^ 
CMCMCOCOCO COCOCO 


ill 


"" II 






-4 


ag s 


co co ^ 


^ n 






4 


^o^io^co " co 

2 CM CM CM CMC^ICM 


3 CM iO 
CO CO CO 


^ n 






33 


CMOCMO^O ^-CM^- 

iox>rxKoo ^JS 1 " 




^ n 






UJ fx 
c o 


22RSS S5S 


CM CO CO 
IX ^ 10 


"< II 






rl 


CM IO IO ^ O 
lO'O'^'COiO 00 CO 00 


10 IX 


-'I 


ass*" ==* 


IO 00 


! 


6 ~ < p:s 


*=. 


UJ 






.- W 






i 


*OjXpOCXO " CM CO 


g CN 


< c 







* C 



JO ^ 
C3 



g 



o .% 
* 



O C 



O -3 

-bl 
s s 

.Si ^ o 
c > 

E^ >. 



S 



APPENDIX 5 
VALUES OF sin 2 9 



6 


.0 .1 .2 .3 


.4 .5 .6 


.7 .8 .9 


Differences 
.01 .02 .03 .04 .05 


00 


,0000 0000 0000 0000 


0000 0001 0001 


0001 0002 0002 




1 


.0003 0004 0004 0005 


0006 0007 0008 


0009 0010 0011 




2 


.0012 0013 0015 0016 


0018 0019 0021 


0022 0024 0026 




3 


.0027 0029 0031 0033 


0035 0037 0039 


0042 0044 0046 




4 


.0049 0051 0054 0056 


0059 0062 0064 


0067 0070 0073 












Interpolate 


5 


.0076 0079 0082 0085 


0089 0092 0095 


0099 0102 0106 




6 


.0109 0113 0117 0120 


0124 0128 0132 


0136 0140 0144 




7 


.0149 0153 0157 0161 


0166 0170 0175 


0180 0184 0189 




8 


.0194 0199 0203 0208 


0213 0218 0224 


0229 0234 0239 




9 


.0245 0250 0256 0261 


0267 0272 0278 


0284 0290 0296 




10 


.0302 0308 0314 0320 


0326 0332 0338 


0345 0351 0358 


11223 


1 


.0364 0371 0377 0384 


0391 0397 0404 


0411 0418 0425 


11233 


2 


.0432 0439 0447 0454 


0461 0468 0476 


0483 0491 0498 


11234 


3 


.0506 0514 0521 0529 


0537 0545 0553 


0561 0569 0577 


12234 


4 


0585 0593 0602 0610 


0618 0627 0635 


0644 0653 0661 


12334 


15 


.0670 0679 0687 0696 


0705 0714 0723 


0732 0741 0751 


12344 


6 


.0760 0769 0778 0788 


0797 0807 0816 


0826 0835 0845 


12345 


7 


.0855 0865 0874 0884 


0894 0904 0914 


0924 0934 0945 


12345 


8 


0955 0965 0976 0986 


0996 1007 1017 


1028 1039 1049 


12345 


9 


.1060 1071 1082 1092 


1103 1114 1125 


1136 1147 1159 


12346 


20 


.1170 1181 1192 1204 


1215 1226 1238 


1249 1261 1273 


12356 


1 


1284 1296 1308 1320 


1331 1343 1355 


1367 1379 1391 


12456 


2 


.1403 1415 1428 1440 


1452 1464 1477 


1489 1502 1514 


12456 


3 


.1527 1539 1552 1565 


1577 1590 1603 


1616 1628 1641 


13456 


4 


.1654 1667 1680 1693 


1707 1720 1733 


1746 1759 1773 


13457 


25 


.1786 1799 1813 1826 


1840 1853 1867 


1881 1894 1908 


13457 


6 


1922 1935 1949 1963 


1977 1991 2005 


2019 2033 2047 


13467 


7 


.2061 2075 2089 2104 


2118 2132 2146 


2161 2175 2190 


13467 


8 


.2204 2219 2233 2248 


2262 2277 2291 


2306 2321 2336 


13467 


9 


2350 2365 2380 2395 


2410 2425 2440 


2455 2470 2485 


23568 


30 


.2500 2515 2530 2545 


2561 2576 2591 


2607 2622 2637 


23568 


1 


.2653 2668 2684 2699 


2715 2730 2746 


2761 2777 2792 


23568 


2 


2808 2824 2840 2855 


2871 2887 2903 


2919 2934 2950 


23568 


3 


.2966 2982 2998 3014 


3030 3046 3062 


3079 3095 3111 


23568 


4 


.3127 3143 3159 3176 


3192 3208 3224 


3241 3257 3274 


23578 


35 


.3290 3306 3323 3339 


3356 3372 3389 


3405 3422 3438 


23578 


6 


.3455 3472 3488 3505 


3521 3538 3555 


3572 3588 3605 


23578 


7 


.3622 3639 3655 3672 


3689 3706 3723 


3740 3757 3773 


23578 


8 


.3790 3807 3824 3841 


3858 3875 3892 


3909 3926 3943 


23578 


9 


.3960 3978 3995 4012 


4029 4046 4063 


4080 4097 4115 


23579 


40 


.4132 4149 4166 4183 


4201 4218 4235 


4252 4270 4287 


23579 


1 


.4304 4321 4339 4356 


4373 4391 4408 


4425 4443 4460 


23579 


2 


.4477 4495 4512 4529 


4547 4564 4582 


4599 4616 4634 


23579 


3 


.4651 4669 4686 4703 


4721 4738 4756 


4773 4791 4808 


23579 


4 


.4826 4843 4860 4878 


4895 4913 4930 


4948 4965 4983 


23579 



(cont.) 



469 



470 



VALUES OP sin 2 6 



[APP. 5 



6 


.0 .1 .2 .3 


.4 .5 .6 


.7 .8 * .9 


Differences 
.01 .02 .03 .04 .05 


45 


.5000 5017 5035 5052 


5070 5087 5105 


5122 5140 5157 


23579 


6 


.5174 5192 5209 5227 


5244 5262 5279 


5297 5314 5331 


23579 


7 


.5349 5366 5384 5401 


5418 5436 5453 


5471 5488 5505 


23579 


8 


.5523 5540 5557 5575 


5592 5609 5627 


5644 5661 5679 


23579 


9 


.5696 5713 5730 5748 


5765 5782 5799 


5817 5834 5851 


23579 


50 


.5868 5885 5903 5920 


5937 5954 5971 


5988 6005 6022 


23579 


1 


.6040 6057 6074 6091 


6108 6125 6142 


6159 6176 6193 


23579 


2 


.6210 6227 6243 6260 


6277 6294 6311 


6328 6345 6361 


23578 


3 


.6378 6395 6412 6428 


6445 6462 6479 


6495 6512 6528 


23578 


4 


.6545 6562 6578 6595 


6611 6628 6644 


6661 6677 6694 


23578 


55 


.6710 6726 6743 6759 


6776 6792 6808 


6824 6841 6857 


23578 


6 


.6873 6889 6905 6921 


6938 6954 6970 


6986 7002 7018 


23578 


7 


.7034 7050 7066 7081 


7097 7113 7129 


7145 7160 7176 


23568 


8 


.7192 7208 7223 7239 


7254 7270 7285 


7301 7316 7332 


23568 


9 


.7347 7363 7378 7393 


7409 7424 7439 


7455 7470 7485 


23568 


60 


.7500 7515 7530 7545 


7560 7575 7590 


7605 7620 7635 


23568 


1 


.7650 7664 7679 7694 


7709 7723 7738 


7752 7767 7781 


23568 


2 


.7796 7810 7825 7839 


7854 7868 7882 


7896 7911 7925 


3467 


3 


.7939 7953 7967 7981 


7995 8009 8023 


8037 8051 8065 


3467 


4 


.8078 8092 8106 8119 


8133 8147 8160 


8174 8187 8201 


3467 


65 


.8214 8227 8241 8254 


8267 8280 8293 


8307 8320 8333 


3457 


6 


.8346 8359 8372 8384 


8397 8410 8423 


8435 8448 8461 


3457 


7 


.8473 8486 8498 8511 


8523 8536 8548 


8560 8572 8585 


3456 


8 


.8597 8609 8621 8633 


8645 8657 8669 


8680 8692 8704 


2456 


9 


.8716 8727 8739 8751 


8762 8774 8785 


8796 8808 8819 


2456 


70 


.8830 8841 8853 8864 


8875 8886 8897 


8908 8918 8929 


2356 


1 


.8940 8951 8961 8972 


8983 8993 9004 


9014 9024 9035 


2346 


2 


.9045 9055 9066 9076 


9086 9096 9106 


9116 9126 9135 


2345 


3 


.9145 9155 9165 9174 


9184 9193 9203 


9212 9222 9231 


2345 


4 


.9240 9249 9259 9268 


9277 9286 9295 


9304 9313 9321 


2345 


75 


.9330 9339 9347 9356 


9365 9373 9382 


9390 9398 9407 


2344 


6 


.9415 9423 9431 9439 


9447 9455 9463 


9471 9479 9486 


2334 


7 


.9494 9502 9509 9517 


9524 9532 9539 


9546 9553 9561 


2234 


8 


.9568 9575 9582 9589 


9596 9603 9609 


9616 9623 9629 


1234 


9 


.9636 9642 9649 9655 


9662 9668 9674 


9680 9686 9692 


1233 


80 


.9698 9704 9710 9716 


9722 9728 9733 


9739 9744 9750 


11223 


1 


.9755 9761 9766 9771 


9776 9782 9787 


9792 9797 9801 




2 


.9806 9811 9816 9820 


9825 9830 9834 


9839 9843 9847 




3 


.9851 9856 9860 9864 


9868 9872 9876 


9880 9883 9887 




4 


.9891 9894 9898 9901 


9905 9908 9911 


9915 9918 9921 












Interpolate 


85 


.9924 9927 9930 9933 


9936 9938 9941 


9944 9946 9949 




6 


.9951 9954 9956 9958 


9961 9963 9965 


9967 9969 9971 




7 


.9973 9974 9976 9978 


9979 9981 9982 


9984 9985 9987 




8 


.9988 9989 9990 9991 


9992 9993 9994 


9995 9996 9996 




9 


.9997 9998 9998 9999 


9999 9999 1.00 


1.00 1.00 1.00 





From The Interpretation of X-Ray Diffraction Photographs, by N. F. M. Henry, 
H. Lipson, and W. A, Wooster (Macmillan, London, 1951). 



APPENDIX 6 
QUADRATIC FORMS OF MILLER 



INDICES 



Cubic 


Hexagonal 


/|2 -f- A 2 + /- 


/(A/ 


A 2 + /(A -f A 2 


lik 


Simple 


Face- 
centered 


Bod/- 
centered 


Diamond 


1 


100 








1 


10 


2 


110 




110 




2 




3 


111 


111 




111 


3 


11 


4 


200 


200 


200 




4 


20 


5 


210 








5 




6 


211 




211 




6 




7 










7 


21 


8 


220 


220 


220 


220 


8 




9 


300, 221 








9 


30 


10 


310 




310 




10 




11 


311 


311 




311 


11 




12 


222 


222 


222 




12 


22 


13 


320 








13 


31 


14 


321 




321 




14 




15 










15 




16 


400 


400 


400 


400 


16 


40 


17 


410, 322 








17 




18 


411, 330 




411, 330 




18 




19 


331 


331 




331 


19 


32 


20 


420 


420 


420 




20 




21 


421 








21 


41 


22 


332 




332 




22 




23 










23 




24 


422 


422 


422 


422 


24 




25 


500, 430 








25 


50 


26 


510,431 




510,431 




26 




27 


511, 333 


511, 333 




511, 333 


27 


33 


28 










28 


42 


29 


520, 432 








29 




30 


521 




521 




30 




31 










31 


51 


32 


440 


440 


440 


440 


32 




33 


522, 441 








33 




34 


530, 433 




530, 433 




34 




35 


531 


531 




531 


35 




36 


600,442 


600,442 


600,442 




36 


60 


37 


610 








37 


43 


38 


611, 532 




611,532 




38 




39 










39 


52 


40 


620 


620 


620 


620 


40 




41 


621,540,443 








41 




42 


541 




541 




42 




43 


533 


533 




533 


43 


61 


44 


622 


622 


622 




44 




45 


630, 542 








45 




46 


631 


. 


631 




46 




47 










47 




48 


444 


444 


444 


444 


48 


44 


49 


700, 632 








49 


70,53 



(cont.) 



471 



472 



VALUES OF (sin 0)/X 



[APP. 7 



Cubic 


Hexagonal 


)i 2 + A* -f- / 2 


//A/ 


li 2 + M + A- 2 


//A 


Simple 


Face- 
centered 


Body- 
centered 


Diamond 


50 
51 


710,550,543 
711,551 


711,551 


710, 550, 543 


711,551 


50 
51 




52 


640 


640 


640 




52 


62 


53 


720,641 








53 




54 
55 


721,633,552 




721,633,552 




54 
55 




56 


642 


642 


642 


642 


56 




57 


722,544 








57 


71 


58 


730 




730 




58 




59 


731,553 


731,553 




731,553 


59 





APPENDIX 7 



VALUES OF (sin 6)/X (A~') 



e 


Radiation 


Mo Aa 
(0.711 A) 


(\1 A a 

(1.542A) 


Co Aa 
(1.790 A) 


F(> A'a 

(1.937 A) 


( 'i A 
(2. 291 A) 





0.00 


0.00 


0.00 


0.00 


0.00 


i 


0.02 


0.01 


0.01 


0.01 


0.01 


2 


0.05 


0.02 


0.02 


0.02 


0.02 


3 


0.07 


0.03 


0.03 


0.03 


0.02 


4 


0.10 


0.05 


0.04 


0.04 


0.03 


5 


0.12 


0.06 


0.05 


0.04 


0.04 


6 


0.15 


0.07 


0.06 


0.05 


0.05 


7 


0.17 


0.08 


0.07 


0.06 


0.05 


8 


0.20 


0.09 


0.08 


0.07 


0.06 


9 


0.22 


0.10 


0.09 


0.08 


0.07 


10 


0.24 


0.11 


0.10 


0.09 


0.08 


11 


0.27 


0.12 


0.11 


0.10 


0.06 


12 


0.29 


0.13 


0.12 


0.11 


0.09 


13 


0.32 


0.15 


0.13 


0.12 


0.10 


14 


0.34 


0.16 


0.14 


0.12 


0.11 


15 


0.36 


0.17 


0.14 


0.13 


0.11 


16 


0.39 


0.18 


0.15 


0.14 


0.12 


17 


0.41 


0.19 


0.16 


0.15 


0.13 


18 


0.43 


0.20 


0.17 


0.16 


0.13 


19 


0.46 


0.21 


0.18 


0.17 


0.14 


20 


0.48 


0.22 


0.19 


0.18 


0.15 


21 


0.51 


0.23 


0.20 


0.18 


0.15 


22 


0.53 


0.24 


0.21 


0.19 


0.16 


23 


0.55 


0.25 


0.22 


0.20 


0.17 


24 


57 


0.26 


0.23 


0.21 


0.18 


25 


0.60 


0.27 


0.24 


0.22 


0.18 


26 


0.62 


28 


0.24 


0.23 


0.19 


27 


0.64 


0.29 


0.25 


0.23 


0.20 


28 


0.66 


0.30 


0.26 


0.24 


0.20 


29 


0.68 


0.31 


0.27 


0.25 


0.21 



(con*.) 



APP. 7] 



VALUES OF (sin 0)/X 



473 









Radiation 






a 


Mo Ka 

(0.711 A) 


Cu Att 
(1.542A) 


( 'o A'o 
(1.790 A) 


1'V A' 
(1.937 A) 


( 'r Ka 
(2. 291 A) 


30 


0.70 


32 


0.28 


0.26 


0.22 


31 


0.72 


0.33 


0.29 


0.27 


0.22 


32 


0.75 


0.34 


0.30 


0.27 


0.23 


33 


0.77 


0.35 


0.30 


0.28 


0.24 


34 


0.79 


0.36 


0.31 


0.29 


0.24 


35 


0.81 


0.37 


0.32 


0.29 


0.25 


36 


0.83 


0.38 


0.33 


0.30 


0.26 


37 


0.85 


0.39 


0.34 


0.31 


0.26 


38 


0.87 


0.40 


0.34 


0.32 


0.27 


39 


0.89 


0.41 


0.35 


0.32 


0.27 


40 


0.91 


0.42 


0.36 


0.33 


0.28 


41 


0.93 


0.43 


0.37 


0.34 


0.29 


42 


0.94 


0.43 


0.37 


35 


0.29 


43 


0.96 


0.44 


0.38 


0.35 


0.30 


44 


0.98 


0.45 


0.39 


0.36 


0.30 


45 


0.99 


46 


40 


0.36 


0.31 


46 


1 01 


0.47 


0.40 


0.37 


0.31 


47 


1 03 


0.47 


0.41 


0.38 


0.32 


48 


1 05 


0.48 


0.42 


0.38 


0.32 


49 


1.06 


0.49 


0.42 


0.39 


0.33 


50 


1.08 


0.50 


0.43 


0.39 


0.33 


52 


1 11 


0.51 


44 


0.41 


0.34 


54 


14 


0.52 


0.45 


0.42 


0.35 


56 


.17 


54 


0.46 


0.43 


0.36 


58 


.20 


0.55 


0.47 


0.44 


0.37 


60 


.22 


0.56 


0.48 


0.45 


0.38 


62 


.24 


0.57 


0.49 


0.46 


0.39 


64 


.26 


58 


0.50 


0.46 


0.39 


66 


.28 


0.59 


0.51 


47 


0.40 


68 


.30 


0.60 


0.52 


0.48 


0.40 


70 


.32 


61 


0.53 


0.48 


0.41 


72 


.34 


0.62 


0.53 


0.49 


0.41 


74 


.35 


0.62 


0.54 


0.50 


0.42 


76 


.37 


0.63 


0.54 


0.50 


0.42 


78 


38 


0.63 


0.55 


0.50 


0.43 


80 


.39 


0.64 


0.55 


0.51 


0.43 


82 


.39 


0.64 


0.55 


0.51 


0.43 


84 


.40 


0.64 


0.56 


0.51 


0.43 


86 


.40 


0.65 


0.56 


0.51 


0.43 


88 


.41 


0.65 


0.56 


0.52 


0.43 


90 


.41 


0.65 


0.56 


0.52 


0.43 



APPENDIX 8 
ATOMIC SCATTERING FACTORS 



=-v>> 


0.0 0.1 0.2 0.3 


0.4 0.5 0.6 


0.7 0.8 0.9 


1.0 1.1 1.2 


H 
He 
Li 4 
Li 


1 0.81 0.46 0.25 
2 1.88 .46 1.05 
2 1.96 .8 1.5 
3 22 815 


0.13 0.07 0.04 
0.75 0.52 0.35 
1.3 1.0 0.8 
13 0.8 


0.03 0.02 0.01 
0.24 0.18 0.14 
0.6 0.5 0.4 
0.6 0.5 0.4 


00 0.00 
0.11 0.09 
0.3 0.3 
0.3 0.3 


Be '-' 


2 20 917 


16 412 


1.0 09 07 


0.6 0.5 


Be 


4 29 917 


16 4 .2 


.0 0.9 7 


0.6 0.5 


B' 3 
B 


2 1.99 .9 1.8 
5 3.5 2.4 1.9 


1.7 .6 .4 
1.7 .5 .4 


.3 .2 1.0 
.2 .2 1.0 


0.9 0.7 
0.9 0.7 


C 

N 45 


6 4.6 3.0 2.2 
2 2.0 2.0 1.9 


1.9 .7 .6 
1 .9 8 .7 


.4 .3 1.16 
.6 .5 1.4 


1.0 0.9 
1.3 1.16 


N< : > 
N 

0-2 

F 

F ~ 


4 3.7 3.0 2.4 
7 5.8 4.2 3.0 
8 7.1 5.3 3.9 
10 8.0 5.5 3.8 
9 7 8 6.2 4.45 

10 8.7 6 7 4 8 


2.0 .8 .66 
2.3 .9 .65 
2.9 2.2 .8 
2.7 2.1 1.8 
3.35 2.65 2.15 

3.5 2.8 2.2 


.56 .49 1.39 
.54 .49 1.39 
.6 .5 .4 
.5 .5 .4 
.9 .7 .6 

1.9 1.7 .55 


.28 1.17 
.29 1.17 
.35 1.26 
.35 1.26 
.5 1.35 

.5 1.35 


Ne 

No ' 


10 9.3 7.5 5.8 
10 9.5 8.2 6.7 


4.4 3.4 2.65 
5.25 4.05 3.2 


2.2 1.9 .65 
2.65 2.25 .95 


.55 1.5 
.75 1.6 


No 
Mo' 2 

Mg 


11 9.65 8.2 6.7 
10 9.75 8.6 7.25 

12 10.5 8 6 7.25 


5.25 4.05 3.2 
5.95 4.8 3.85 

5.95 4.8 3.85 


2.65 2.25 .95 
3.15 2.55 2.2 

3.15 2.55 2.2 


.75 1.6 
2.0 1.8 

2.0 1.8 


Al' 3 
Al 
Si t4 


10 9.7 8.9 7.8 
13 11.0 8.95 7.75 
10 9.75 9.15 8.25 


6.65 5.5 4.45 
6.6 5.5 4.5 
7.15 6.05 5.05 


3.65 3.1 2.65 
3.7 3.1 2.65 
4.2 3.4 2.95 


2.3 2.0 
2.3 2.0 
2.6 2.3 


Si 

p<5 


14 11.35 9.4 8.2 
10 9.8 9.25 8.45 


7.15 6.1 5.1 
7.5 6.55 5.65 


4.2 3.4 2.95 
4.8 4.05 3.4 


2.6 2.3 
3 2.6 


P 

P- J 

s' 
s 


15 12.4 10.0 8.45 
18 12.7 9.8 8.4 
10 9.85 9.4 8.7 
16 13.6 10.7 8.95 


7.45 6.5 5.65 
7.45 6.5 5.65 
7.85 6.85 6.05 
7.85 6.85 6.0 


4.8 4.05 3.4 
4.85 4.05 3.4 
5.25 4.5 3.9 
5.25 4.5 3.9 


3.0 2.6 
3.0 2.6 
3.35 2.9 
3.35 2.9 


s-' 2 

Cl 

cr 

A 
K ( 

K 


18 14.3 10.7 8.9 
17 14.6 11.3 9.25 
18 15.2 11.5 9.3 
18 15.9 12.6 10.4 
18 16.5 13.3 10.8 

19 16.5 13 3 10 8 


7.85 6.85 6.0 
8.05 7.25 6.5 
8.05 7.25 6.5 
8.7 7.8 7.0 
8.85 7.75 7.05 

92 79 67 


5.25 4.5 3.9 
5.75 5.05 4.4 
5.75 5.05 4.4 
6.2 5.4 4.7 
6.44 5.9 5.3 

59 52 4.6 


3.35 2.9 
3.85 3.35 
3.85 3.35 
4.1 3.6 
4.8 4.2 

42 3.7 33 


Ca (2 


18 16.8 14.0 11.5 


9.3 8.1 7.35 


6.7 6.2 5.7 


5.1 4.6 


Ca 

Sc' 3 
Sc 

Ti' 4 
Ti 
V 
Cr 
Mn 


-20 17.5 14.1 11.4 
18 16.7 14.0 11.4 
21 18.4 14.9 12.1 

18 17.0 14.4 11.9 
22 19.3 15.7 12.8 
23 20.2 16.6 13.5 
24 21.1 17.4 14.2 
25 22.1 18.2 14.9 


9.7 8.4 7.3 
9.4 8.3 7.6 
10 3 8.9 7.7 

9.9 8.5 7.85 
10.9 9.5 8.2 
11.5 10.1 8.7 
12.1 10.6 9.2 
12.7 11.1 9.7 


6.3 5.6 4.9 
6.9 6.4 5.8 
6.7 5.9 5.3 

7.3 6.7 6.15 
7.2 6.3 5.6 
7.6 6.7 5.9 
8.0 7.1 6.3 
8.4 7.5 6.6 


4.5 4.0 3.6 
5.35 4.85 
4.7 4.3 3.9 

5.65 5.05 
5.0 4.6 4.2 
5.3 4.9 4.4 
5.7 5.1 4.6 
6.0 5.4 4.9 



(cont.) 



474 



APP. 8] 



ATOMIC SCATTERING FACTORS 



475 



"fV-M 


0.0 0.1 0.2 0.3 


0.4 0.5 0.6 


0.7 0.8 0.9 


1.0 1.1 1.2 












Fe 
Co 


26 23.1 18.9 15.6 
27 24.1 19.8 16.4 


13.3 11.6 10.2 
14.0 12.1 10.7 


9.3 8.3 7.3 


6.7 6.0 5.5 


Ni 


28 25.0 20.7 17.2 


14.6 12.7 11.2 


9.8 8.7 7 7 


7.0 6.3 5.8 


Cu 


29 25.9 21.6 17.9 


15.2 13.3 11.7 


10.2 9.1 8.1 


7.3 6.6 6.0 


Zn 
Go 


30 26.8 22.4 18.6 
31 27.8 23.3 19.3 


15.8 13.9 12.2 
16.5 14.5 12.7 


11.2 10.0 8.9 


7.9 7.3 6.7 


Ge 


32 28.8 24.1 20.0 


17.1 15.0 13.2 


11.6 10.4 9.3 


8.3 7.6 7.0 


As 


33 29.7 25.0 20.8 


17.7 15.6 13.8 


12.1 10.8 9.7 


8.7 7.9 7.3 


Se 


34 30.6 25.8 21.5 


18.3 16.1 14.3 


12.6 11 2 10.0 


9.0 8.2 7.5 


Br 


35 31.6 26.6 22.3 


18.9 16.7 14.8 


13.1 11.7 10.4 


9.4 8.6 7.8 


Kr 


36 32.5 27.4 23.0 


19.5 17.3 15.3 


13.6 12.1 10.8 


9.8 8.9 8.1 


Rb' 


36 33.6 28.7 24.6 


21 4 18.9 16.7 


14.6 12.8 11.2 


9.9 8.9 


Rb 


37 33.5 28.2 23.8 


20.2 17.9 15.9 


14.1 12.5 11.2 


10.2 9.2 8.4 


Sr 


38 34.4 29.0 24.5 


20.8 18.4 16.4 


14.6 12.9 11.6 


10.5 9.5 8.7 


Y 


39 35.4 29.9 25.3 


21.5 19.0 17.0 


15.1 13/4 12.0 


10.9 9.9 9.0 


Zr 


40 36.3 30.8 26.0 


22.1 19.7 17.5 


15.6 13.8 12.4 


11.2 10.2 9.3 


Nb 


41 37.3 31.7 26.8 


22.8 20.2 18.1 


16.0 14.3 12.8 


11.6 10.6 9.7 


Mo 


42 38.2 32.6 27.6 


23.5 20.8 18.6 


16.5 14.8 13.2 


12.0 10.9 10.0 


Tc 


43 39.1 33.4 28.3 


24.1 21.3 19.1 


17.0 15.2 13.6 


12.3 11.3 10.3 


Ru 


44 40.0 34.3 29.1 


24.7 21.9 19.6 


17.5 15.6 14.1 


12.7 11.6 10.6 


Rh 


45 41.0 35.1 29.9 


25.4 22.5 20.2 


18.0 16.1 14.5 


13.1 12.0 11.0 


Pd 


46 41.9 36.0 30.7 


26.2 23.1 208 


18.5 16.6} 14.9 


13.6 12.3 11.3 


Ag 


47 42.8 36.9 31.5 


26.9 23.8 21.3 


19.0 17.1 15.3 


14.0 12.7 11.7 


Cd 


48 43.7 37.7 32.2 


27.5 24.4 21.8 


19.6 17.6 15.7 


14.3 13.0 12.0 


In 


49 44.7 38.6 33.0 


28.1 25.0 22.4 


20.1 18.0 16.2 


14.7 13.4 12.3 


Sn 


50 45.7 39.5 33.8 


28.7 25.6 22.9 


20 6 18.5 16.6 


15.1 13.7 127 


Sb 


51 46.7 40.4 34.6 


29.5 26.3 23.5 


21.1 19.0 17.0 


15.5 14.1 13.0 


Te 


52 47.7 41.3 35.4 


30.3 26.9 24.0 


21.7 19.5 17.5 


16.0 14.5 13.3 


I 


53 48.6 42.1 36.1 


31.0 27.5 24.6 


22.2 20.0 17.9 


16.4 14.8 13.6 


Xe 


54 49.6 43.0 36.8 


31.6 28.0 25.2 


22.7 20.4 18.4 


16.7 15.2 13.9 


Cs 


55 50.7 43.8 37.6 


32.4 28.7 25.8 


23.2 20.8 18.8 


17.0 15.6 14.5 


Ba 


56 51.7 44.7 38.4 


33.1 29.3 26.4 


23.7 21.3 19.2 


17.4 16.0 14.7 


La 


57 52.6 45.6 39.3 


33.8 29.8 26.9 


24.3 21.9 19.7 


17.9 16.4 15.0 


Ce 


58 53.6 46.5 40.1 


34.5 30.4 27.4 


24.8 22.4 20.2 


18.4 16.6 15.3 


Pr 


59 54.5 47.4 40.9 


35.2 31.1 28.0 


25.4 22.9 20.6 


18.8 17.1 15.7 


Nd 


60 55.4 48.3 41.6 


35.9 31.8 28.6 


25.9 23.4 21.1 


19.2 17.5 16.1 


Pm 


61 56.4 49.1 42.4 


36.6 32.4 29.2 


26.4 23.9 21.5 


19.6 17.9 16.4 


5m 


62 57.3 50.0 43.2 


37.3 32.9 29.8 


26.9 24.4 22.0 


20.0 18.3 16.8 


Eu 


63 58.3 50.9 44.0 


38.1 33.5 30.4 


27.5 24.9 22.4 


20.4 18.7 17.1 


Gd 


64 59.3 51.7 44.8 


38.8 34.1 31.0 


28.1 25.4 22.9 


20.8 19.1 17.5 


Tb 


65 60.2 52.6 45.7 


39.6 34.7 31.6 


28.6 25.9 23.4 


21.2 19.5 17.9 


Dy 


66 61.1 53.6 46.5 


40.4 35.4 32.2 


29.2 26.3 23.9 


21.6 19.9 18.3 


Ho 


67 62.1 54.5 47.3 


41.1 36.1 32.7 


29.7 26.8 24.3 


22.0 20.3 18.6 


Er 


68 63.0 55.3 48.1 


41.7 36.7 33.3 


30.2 27.3 24.7 


22.4 20.7 18.9 


Tm 


69 64.0 56.2 48.9 


42.4 37.4 33.9 


30.8 27.9 25.2 


22.9 21.0 19.3 



(cont.) 



476 



ATOMIC SCATTERING FACTORS 



[APP. 8 



!=-V-> 


0.0 O.I 0.2 0.3 


0.4 0.5 0.6 


0.7 0.8 0.9 


1.0 1J 1.2 


Yb 


70 64.9 57.0 49.7 


43.2 38.0 34.4 


31.3 28.4 25.7 


23.3 21.4 19.7 


Lu 


71 65.9 57.8 50.4 


43.9 38.7 35.0 


31.8 28.9 26.2 


23.8 21.8 20.0 


Hf 


72 66.8 58,6 51.2 


44.5 39.3 35.6 


32.3 29.3 26.7 


24.2 22.3 20.4 


To 


73 67.8 59.5 52.0 


45.3 39.9 36.2 


32.9 29.8 27.1 


24.7 22.6 20.9 


W 


74 68.8 60.4 52.8 


46.1 40.5 36.8 


33.5 30.4 27.6 


25.2 23.0 21.3 


Re 


75 69.8 61.3 53.6 


46.8 41.1 37.4 


34.0 30.9 28.1 


25.6 23.4 21.6 


0$ 


76 70.8 62.2 54.4 


47.5 41.7 38.0 


34.6 31.4 28.6 


26.0 23.9 22.0 


Ir 


77 71.7 63.1 55.3 


48.2 42.4 38.6 


35.1 32.0 29.0 


26.5 24.3 22.3 


Pt 


78 72.6 64.0 56.2 


48.9 43.1 39.2 


35.6 32.5 29.5 


27.0 24.7 22.7 


Au 


79 73.6 65.0 57.0 


49.7 43.8 39.8 


36.2 33.1 30.0 


27.4 25.1 23.1 


Hg 


80 74.6 65.9 57.9 


50.5 44.4 40.5 


36.8 33.6 30.6 


27.8 25.6 23.6 


Tl 


81 75.5 66.7 58.7 


51.2 45.0 41.1 


37.4 34.1 31.1 


28.3 26.0 24.1 


Pb 


82 76.5 67.5 59.5 


51.9 45.7 41.6 


37.9 34.6 31.5 


28.8 26.4 24.5 


Bi 


83 77.5 68.4 60.4 


52.7 46.4 42.2 


38.5 35.1 32.0 


29.2 26.8 24.8 


Po 


84 78.4 69.4 61.3 


53.5 47.1 42.8 


39.1 35.6 32.6 


29.7 27.2 25.2 


At 


85 79.4 70.3 62.1 


54.2 47.7 43.4 


39.6 36.2 33.1 


30.1 27.6 25.6 


Rn 


86 80.3 71.3 63.0 


55.1 48.4 44.0 


40.2 36.8 33.5 


30.5 28.0 26.0 


Fr 


87 81.3 72.2 63.8 


55.8 49.1 44.5 


40.7 37.3 34.0 


31.0 28.4 26.4 


Ra 


88 82.2 73.2 64.6 


56.5 49.8 45.1 


41.3 37.8 34.6 


31.5 28.8 26.7 


Ac 


89 83.2 74.1 65.5 


57.3 50.4 45.8 


41.8 38.3 35.1 


32.0 29.2 27.1 


Th 


90 84.1 75.1 66.3 


58.1 51.1 46.5 


42.4 38.8 35.5 


32.4 29.6 27.5 


Pa 


91 85.1 76.0 67.1 


58.8 51.7 47.1 


43.0 39.3 36.0 


32.8 30.1 27.9 


U 


92 86.0 76.9 67.9 


59.6 52.4 47.7 


43.5 39.8 36.5 


33.3 30.6 28.3 


Np 


93 87 78 69 


60 53 48 


44 40 37 


34 31 29 


Pu 


94 88 79 69 


61 54 49 


44 41 38 


34 31 29 


Am 


95 89 79 70 


62 55 50 


45 42 38 


35 32 30 


Cm 


96 90 80 71 


62 55 50 


46 42 39 


35 32 30 


Bk 


97 91 81 72 


63 56 51 


46 43 39 


36 33 30 


Cf 


98 92 82 73 


64 57 52 


47 43 40 


36 33 31 




99 93 83 74 


65 57 52 


48 44 40 


37 34 31 




100 94 84 75 


66 58 53 


48 44 41 


37 34 31 



From X-Ray Diffraction by Poly crystalline Materials, edited by H. S. Peiser, 
H. P. Rooksby, and A. J. C. Wilson (The Institute of Physics, London, 1955). 



APPENDIX 9 
MULTIPLICITY FACTORS FOR POWDER PHOTOGRAPHS 

Cubic: hkl hhl Okl Okk hhh 001 

48* 24 24* 12 8 ~6~ 

Hexagonal and hk-l hh-l Ok-l hk-0 hh-0 Ok-0 00-1 
Rhombohedral: 04* 19* 12* 12* 6 6 2 

Tetragonal: hkl hhl Okl hkO hhO OkO 001 

16* 8 8 8* 4 4 2 

Orthorhombic: hkl Okl hOl hkO hOO OkO 001 
8444222 

Monodinic: hkl hOl OkO 

T T IT 

Triclinic: hkl 

~2 



* These are the usual multiplicity factors. In some crystals, planes having these 
indices comprise two forms with the same spacing but different structure factor, 
and the multiplicity factor for each form is half the value given above. In the 
cubic system, for example, there are some crystals in which permutations of the 
indices (hkl) produce planes which are not structurally equivalent; in such crys- 
tals (AuBe, discussed in Sec. 2-7, is an example), the plane (123), for example, 
belongs to one form and has a certain structure factor, while the plane (321) be- 
longs to another form and has a different structure factor. There are ~^- = 24 
planes in the first form and 24 planes in the second. This question is discussed 
more fully by Henry, Lipson, and Wooster: The Interpretation of X-Ray Diffraction 
Photographs (MacMillan). 



477 



APPENDIX 10 
LORENTZ-POLARIZATION FACTOR 



/l + cos 2 29\ 
\ sin 2 6 cos 6 / 






.0 


.1 .2 .3 


.4 .5 .6 


.7 .8 .9 


2 


1639 


1486 1354 1239 


1138 1048 968.9 


898.3 835.1 778.4 


3 


727.2 


680.9 638.8 600.5 


565.6 533.6 504.3 


477.3 452.3 429.3 


4 


408.0 


388.2 369.9 352.7 


336.8 321.9 308.0 


294.9 282.6 271.1 


5 


260.3 


250.1 240.5 231.4 


222.9 214.7 207.1 


199.8 192.9 186.3 


6 


180.1 


174.2 168.5 163.1 


158.0 153.1 148.4 


144.0 139.7 135.6 


7 


131.7 


128.0 124.4 120.9 


117.6 114.4 111.4 


108.5 105.6 102.9 


8 


100.3 


97.80 95.37 93.03 


90.78 88.60 86.51 


84.48 82.52 80.63 


9 


78.79 


77.02 75.31 73.66 


72.05 70.49 68.99 


67.53 66.12 64.74 


10 


63.41 


62.12 60.87 59.65 


58.46 57.32 56.20 


55.11 54.06 53.03 


H 


52.04 


51.06 50.12 49.19 


48.30 47.43 46.58 


45.75 44.94 44.16 


12 


43.39 


42.64 41.91 41.20 


40.50 39.82 39.16 


38.51 37.88 37.27 


13 


36.67 


36.08 35.50 34.94 


34.39 33.85 33.33 


32.81 32.31 31.82 


14 


31.34 


30.87 30.41 29.96 


29.51 29.06 28.66 


28.24 27.83 27.44 


15 


27.05 


26.66 26.29 25.92 


25.56 25.21 24.86 


24.52 24.19 23.86 


16 


23.54 


23.23 22.92 22.61 


22.32 22.02 21.74 


21.46 21.18 20.91 


17 


20.64 


20.38 20.12 19.87 


19.62 19.38 19.14 


18.90 18.67 18.44 


18 


18.22 


18.00 17.78 17.57 


17.36 17.15 16.95 


16.75 16.56 16.36 


19 


16.17 


15.99 15.80 15.62 


15.45 15.27 15.10 


14.93 14.76 14.60 


20 


14.44 


14.28 14.12 13.97 


13.81 13.66 13.52 


13.37 13.23 13.09 


21 


12.95 


12.81 12.66 12.54 


12.41 12.28 12.15 


12.03 11.91 11.78 


22 


11.66 


11.54 11.43 11.31 


11.20 11.09 10.98 


10.87 10.76 10.65 


23 


10.55 


10.45 10.35 10.24 


10.15 10.05 9.951 


9.857 9.763 9.671 


24 


9.579 


9.489 9.400 9.313 


9.226 9.141 9.057 


8.973 8.891 8.810 


25 


8.730 


8.651 8.573 8.496 


8.420 8.345 8.271 


8.198 8.126 8.054 


26 


7.984 


7.915 7.846 7.778 


7.711 7.645 7.580 


7.515 7.452 7.389 


27 


7.327 


7.266 7.205 7.145 


7.066 7.027 6.969 


6.912 6.856 6.800 


28 


6.745 


6.692 6.637 6.584 


6.532 6.480 6.429 


6.379 6.329 6.279 


29 


6.230 


6.183 6.135 6.068 


6.042 5.995 5.950 


5.905 5.861 5.817 


30 


5.774 


5.731 5.688 5.647 


5.605 5.564 5.524 


5.484 5.445 5.406 


31 


5.367 


5.329 5.292 5.254 


5.218 5.181 5.145 


5.110 5.075 5.040 


32 


5.006 


4.972 4.939 4.906 


4.873 4.841 4.809 


4.777 4.746 4.715 


33 


4.685 


4.655 4.625 4.595 


4.566 4.538 4.509 


4.481 4.453 4.426 


34 


4.399 


4.372 4.346 4.320 


4.294 4.268 4.243 


4.218 4.193 4.169 


35 


4.145 


4.121 4.097 4.074 


4.052 4.029 4.006 


3.984 3.962 3.941 


36 


3.919 


3.898 3.877 3.857 


3.836 3.816 3.797 


3.777 3.758 3.739 


37 


3.720 


3.701 3.683 3.665 


3.647 3.629 3.612 


3.594 3.577 3.561 


38 


3.544 


3.527 3.513 3.497 


3.481 3.465 3.449 


3.434 3.419 3.404 


39 


3.389 


3.375 3.361 3.347 


3.333 3.320 3.306 


3.293 3.280 3.268 


40 


3.255 


3.242 3.230 3.218 


3.206 3.194 3.183 


3.171 3.160 3.149 


41 


3.138 


3.127 3.117 3.106 


3.096 3.086 3.076 


3.067 3.057 3.048 


42 


3.038 


3.029 3.020 3.012 


3.003 2.994 2.986 


2.978 2.970 2.962 


43 


2.954 


2.946 2.939 2.932 


2.925 2.918 2.911 


2.904 2.897 2.891 


44 


2.884 


2.878 2.872 2.866 


2.860 2.855 2.849 


2.844 2.838 2.833 



(cont.) 



478 



APP. 10] 



LORENTZ-POLARIZATION FACTOR 



479 



6 


.0 


.1 .2 .3 


.4 .5 .6 


.7 .8 .9 


45 


2.828 


2.824 2.819 2.814 


2.810 2.805 2.801 


2.797 2.793 2.789 


46 


2.785 


2.782 2.778 2.775 


2.772 2.769 2.766 


2.763 2.760 2.757 


47 


2.755 


2.752 2.750 2.748 


2.746 2.744 2.742 


2.740 2.738 2.737 


48 


2.736 


2.735 2.733 2.732 


2.731 2.730 2.730 


2.729 2.729 2.726 


49 


2. 728 


2.728 2. 728 2.728 


2.728 2.728 2.729 


2.729 2.730 2.730 


50 


2.731 


2.732 2.733 2.734 


2.735 2.737 2.738 


2.740 2.741 2.743 


51 


2.745 


2.747 2.749 2.751 


2.753 2.755 2.758 


2.760 2.763 2.766 


52 


2.769 


2.772 2.775 2.778 


2.782 2.785 2.788 


2.792 2.795 2.799 


53 


2.803 


2.807 2.811 2.815 


2.820 2.824 2.828 


2.833 2.838 2.843 


54 


2.848 


2.853 2.858 2.863 


2.868 2.874 2.879 


2.885 2.890 2.896 


55 


2.902 


2.908 2.914 2.921 


2.927 2.933 2.940 


2.946 2.953 2.960 


56 


2.967 


2.974 2.981 2.988 


2.996 3.004 3.011 


3.019 3.026 3.034 


57 


3.042 


3.050 3.059 3.067 


3.075 3.084 3.092 


3.101 3.110 3.119 


58 


3.128 


3.137 3.147 3.156 


3.166 3.175 3.185 


3.195 3.205 3.215 


59 


3.225 


3.235 3.246 3.256 


3.267 3.278 3.289 


3.300 3.311 3.322 


60 


3.333 


3.345 3.356 3.368 


3.380 3.392 3.404 


3.416 3.429 3.441 


61 


3.454 


3.466 3.479 3.492 


3.505 3.518 3.532 


3.545 3.559 3.573 


62 


3.587 


3.601 3.615 3.629 


3.643 3.658 3.673 


3.668 3.703 3.718 


63 


3.733 


3.749 3.764 3.780 


3.796 3.812 3.828 


3.844 3.861 3.878 


64 


3.894 


3.911 3.928 3.946 


3.963 3.980 3.998 


4.016 4.034 4.052 


65 


4.071 


4.090 4.106 4.127 


4.147 4.166 4.185 


4.205 4.225 4.245 


66 


4.265 


4. 285 4.306 4.327 


4.348 4.369 4.390 


4.412 4.434 4.456 


67 


4.478 


4.500 4.523 4.546 


4.569 4.592 4.616 


4.640 4.664 4.688 


68 


4.712 


4.737 4.762 4.787 


4.812 4.838 4.864 


4.890 4.916 4.943 


69 


4.970 


4.997 5.024 5.052 


5.080 5.109 5.137 


5.166 5.195 5.224 


70 


5.254 


5.284 5.315 5.345 


5.376 5.408 5.440 


5.471 5.504 5.536 


71 


5.569 


5.602 5.636 5.670 


5.705 5.740 5.775 


5.810 5.846 5.883 


72 


5.919 


5.956 5.994 6.032 


6.071 6.109 6.149 


6.189 6.229 6.270 


73 


6.311 


6.352 6.394 6.437 


6.480 6.524 6.568 


6.613 6.658 6.703 


74 


6.750 


6.797 6.844 6.892 


6.941 6.991 7.041 


7.091 7.142 7.194 


75 


7.247 


7.300 7.354 7.409 


7.465 7.521 7.578 


7.636 7.694 7.753 


76 


7.813 


7.874 7.936 7.999 


8.063 8.128 8.193 


8.259 8.327 8.395 


77 


8.465 


8.536 8.607 8.680 


8.754 8.829 8.905 


8.982 9.061 9.142 


78 


9.223 


9.305 9.389 9.474 


9.561 9.649 9.739 


9.831 9.924 10.02 


79 


10.12 


10.21 10.31 10.41 


10.52 10.62 10.73 


10.84 10.95 11.06 


80 


11.18 


11.30 11.42 11.54 


11.67 11.80 11.93 


12.06 12.20 12.34 


81 


12.48 


12.63 12.78 12.93 


13.06 13.24 13.40 


13.57 13.74 13.92 


82 


14.10 


14.28 14.47 14.66 


14.86 15.07 15.28 


15.49 15.71 15.94 


83 


16.17 


16.41 16.66 16.91 


17.17 17.44 17.72 


18.01 18.31 18.61 


84 


18.93 


19.25 19.59 19.94 


20.30 20.68 21.07 


21.47 21.89 22.32 


85 


22.77 


23.24 23.73 24.24 


24.78 25.34 25.92 


26.52 27.16 27.83 


86 


28.53 


29.27 30.04 30.86 


31.73 32.64 33.60 


34.63 35.72 36.88 


87 


38.11 


39.43 40.84 42.36 


44.00 45.76 47.66 


49.76 52.02 54.50 



From The Interpretation of X-Ray Diffraction Photographs, by N. F. M. Henry, 
H. Lipson, and W. A. Wooster (Macmillan, London, 1951). 



APPENDIX 11 
PHYSICAL CONSTANTS 

Charge on the electron (e) = 4.80 X 10~~ 10 esu 

Mass of electron (m) = 9.11 X 10~ 28 gm 

Mass of neutron = 1.67 X 10~ 24 gm 

Velocity of light (c) = 3.00 X 10 10 cm/sec 

Planck's constant (h) = 6.62 X 10~ 27 erg -sec 

Boltzmann's constant (k) = 1.38 X 10~ 16 erg/A 

Avogadro's number (JV) = 6.02 X 10 23 per mol 

Gas constant (R) = 1.99 cal/A/mol 

1 electron volt = 1.602 X 10~~ 12 erg 
1 cal = 4.182 X 10 7 ergs 
1 kX = 1.00202A 



480 



APPENDIX 12 
INTERNATIONAL ATOMIC WEIGHTS, 1953 





Symbol 


Atomic 
number 


Atomic 
weight 






Symbol 


Atomic 
number 


Atomic 
weight* 


Actinium 


Ac 


89 


227 




Molybdenum 


Mo 


42 


95.95 


Aluminum 


Al 


13 


26.98 




Neodymlum 


Nd 


60 


144.27 


Americium 


Am 


95 


[243] 




Neptunium 


Np 


93 


[237] 


Antimony 


Sb 


51 


121.76 




Neon 


Ne 


10 


20.183 


Argon 


A 


18 


39.944 




Nickel 


Ni 


28 


58.69 


Arsenic 


As 


33 


74.91 




Niobium 


Nb 


41 


92.91 


Astatine 


At 


85 


[2101 




Nitrogen 


N 


7 


14.008 


Barium 


Ba 


56 


137.36 




Osmium 


Os 


76 


190.2 


Berk el i urn 


Bk 


97 


[245] 




Oxygen 


O 


8 


16 


Beryllium 


Be 


4 


9.013 




Palladium 


Pd 


46 


106.7 


Bismuth 


Bi 


83 


209.00 




Phosphorus 


P 


15 


30.975 


Boron 


B 


5 


10.82 




Platinum 


Pt 


78 


195.23 


Bromine 


Br 


35 


79.916 




Plutonium 


Pu 


94 


[242] 


Cadmium 


Cd 


48 


112.41 




Polonium 


Po 


84 


210 


Calcium 


Ca 


20 


40.08 




Potassium 


K 


19 


39.100 


Californium 


Cf 


98 


[246] 




Praseodymium 


Pr 


59 


140.92 


Carbon 


C 


6 


12.011 




Promethium 


Pm 


61 


[145] 


Cerium 


Ce 


58 


140.13 




Protactinium 


Pa 


91 


231 


Caesium 


Cs 


55 


132 91 




Radium 


Ra 


88 


226.05 


Chlorine 


Cl 


17 


35.457 




Radon 


Rn 


86 


222 


Chromium 


Cr 


24 


52.01 




Rhenium 


Re 


75 


186.31 


Cobalt 


Co 


27 


58.94 




Rhodium 


Rh 


45 


102.91 


Copper 


Cu 


29 


63.54 




Rubidium 


Rb 


37 


85.48 


Curium 


Cm 


96 


[243] 




Ruthenium 


Ru 


44 


101.1 


Dysprosium 


Dy 


66 


162.46 




Samarium 


Sm 


62 


150.43 


Erbium 


Er 


68 


167.2 




Scandium 


Sc 


21 


44.96 


Europium 


Eu 


63 


152.0 




Selenium 


Se 


34 


78.96 


Fluorine 


F 


9 


19.00 




Silicon 


Si 


14 


28.09 


Francium 


Fr 


87 


[223] 




Silver 


Ag 


47 


107.880 


Gadolinium 


Gd 


64 


156 9 




Sodium 


Na 


11 


22.991 


Gallium 


Ga 


31 


69.72 




Strontium 


Sr 


38 


87.63 


Germanium 


Ge 


32 


72.60 




Sulfur 


S 


16 


32. 066 t 


Gold 


Au 


79 


197.0 




Tantalum 


Ta 


73 


180.95 


Hafnium 


Hf 


72 


178.6 




Technetium 


Tc 


43 


[99] 


Helium 


He 


2 


4.003 




Tellurium 


Te 


52 


127.61 


Holmium 


Ho 


67 


164.94 




Terbium 


Tb 


65 


158.93 


Hydrogen 


H 


1 


1.0080 




Thallium 


Tl 


81 


204.39 


Indium 


In 


49 


114.76 




Thorium 


Th 


90 


232.05 


Iodine 


I 


53 


126.91 




Thulium 


Tm 


69 


168.94 


Iridium 


Ir 


77 


192.2 




Tin 


Sn 


50 


118.70 


Iron 


Fe 


26 


55.85 




Titanium 


Ti 


22 


47.90 


Krypton 


Kr 


36 


83.80 




Tungsten 


W 


74 


183.92 


Lanthanum 


La 


57 


138.92 




Uranium 


U 


92 


238.07 


Lead 


Pb 


82 


207.21 




Vanadium 


V 


23 


50.95 


Lithium 


Li 


3 


6.940 




Xenon 


Xe 


54 


131.3 


Lutetium 


Lu 


71 


174.99 




Ytterbium 


Yb 


70 


173.04 


Magnesium 


Mg 


12 


24.32 




Yttrium 


Y 


39 


88.92 


Manganese 


Mn 


25 


54.94 




Zinc 


Zn 


30 


65.38 


Mercury 


Hg 


80 


200.61 




Zirconium 


Zr 


40 


91.22 



* A bracketed value is the mass number of the isotope of longest known half-life. 
t Because of natural variations in the relative abundance of its isotopes, the 
atomic weight of sulfur has a range of 0.003. 

481 



APPENDIX 13 



CRYSTAL STRUCTURE DATA 

(N.B. The symbols Al, Bl, etc., in this Appendix are those used in Strukturbericht 
to designate certain common structural types.) 

TABLE A13-1 THE ELEMENTS 



Element and 
modification 


Type of structure 


Lattice 
parameters (A) 


c or axial 
angle 


Temperature 
for which 
constants 
apply 


Distance 
of closest 
approach 
(A) 


a 


b 


Actinium 














Alabamine 


See Francium 








o 






FCC Al 


4.0490 






20C 


2.862 


Aluminum 
Americium 
Antimony 
Argon 
Arsenic 


Rhombohedral, A7 
FCC, Al 
Rhombohedral, A7 


4.5064 
5.43 
4.159 




576.5' 


20C 
-233C 
20C 


2.903 
3.84 
2 51 




5349' 


Astatine 
















BCC A2 


5.025 






20C 


4.35 


Barium 
Beryllium, a* 
p (doubtful) 
Bismuth 


HCP] A3 
Hexagonal 
Rhombohedral, A7 


2.2854 
7.1 
4.7356 




3.5841 
10.8 
57 14. 2' 


20C 
Room 
20C 


2.225 

3.111 


Boron 


Rhombohedral 


9.45 




23.8 


Room 




Bromine 


Orthorhombic 


4.49 


6.68 


8.74 


-150C 


2.27 


Cadmium 


HCP, A3 


2.9787 




5.617 


20C 


2.979 




FCC Al 


5.57 






20C 


3.94 


Calcium, ot 
p(300-450C) 
V(>450C) 


HCP, A3 


3.99 




6.53 


460C 


3.95 


Carbon, diamond* 


Diamond cubic, A4 


3.568 






18C 


1.544 


Graphite, a* 


Hexagonal, A9 


2.4614 




6.7014 


20C 


1.42 


Graphite, p 


Rhombohedral, D : ' :i/ 


2.461 




10.064 






Cerium* 


FCC, Al 


5.140 






Room 


3.64 




FCC, Al 


4.82 






- 180C 


3.40 


MIC f\f\f\ nm 


FCC Al 


4.84 






Room 


3.42 


to, uuu arm 

P^ctum 


BCC A2 


6.06 






- 173C 


5.25 


vesium 
Chlorine, a 


Tetragonal 


8.58 




6.13 


-110C 


1.88 


f*l>rA>Iiim 


BCC A2 


2.8845 






20C 


2.498 


vnromium 
(Transit, at 37C) 
Cobalt, a* 


BCC,' A2 
HCP, A3 


2.8851 
2.507 






38C 
20C 


2.506 




4.069 




FCC Al 


3.552 






Room 


2.511 


Columbian) 


See Niobium 














FCC Al 


3.6153 






20C 


2.556 


Copper 
Dysprosium 
Erbium 


HCP] A3 
HCP, A3 


3.585 
3.539 




5.659 
5.601 


20C 
20C 


3.506 
3.466 


Europium 


BCC, A2 


4.582 






20C 


2.968 


Franclum 
Gadolinium 


(Formerly Alabamine) 
HCP, A3 


3.629 




5.759 


20C 


3.561 


Gallium 


One FC orthorhom- 


3.526 


4.520 


7.660 


20C 


2.442 




blc, All 












r- : 




5.658 






20C 


2.450 


\7ermanium 

ftrtlrl 


FCC Al 


4.0783 






20C 


2.884 


voia 
Hafnium 


HCP' A3 


3.206 




5.087 


20C 


3.15 


Helium 


HCP, A3 (?) 


3.58 




5.84 


-271.5C 


3.58 


Hoi mi urn 


HCP, A3 


3.564 




5.631 


20C 


3.487 



(cont.) 

* Ordinary form of an element that exists (or is thought to exist) in more than 
one form. 

482 



APP. 13] 



CRYSTAL STRUCTURE DATA 



483 



Element and 
modification 


Type of structure 


Lattice 
parameters (A) 


c or axial 
angle 


Temperature 
for which 
constants 
apply 


Distance 
of closest 
approach 
(A) 


a 


b 


Hydrogen, para 
Illinium 
Indium 
Iodine 
Indium 
Iron, * 
y (extrapolated) 
y(908-1403C) 
6(>1403C) 
Krypton 
Lanthanum, en* 

P 
Lead 

Lithium 
(cold worked) 

Lutecium 
Magnesium 
Manganese, a* 
p(727-1095C) 
7(1095-1133C) 
6(>H33C) 
Masurium 
Mercury 
Molybdenum 
Neodymium, * 
Neon 
Neptunium 
Nickel* 
(unstable, with H 2 
or N 3 ?) 
(unstable) (?) 
Niobium 
Nitrogen, a 

P 
Osmium 
Oxygen, 

P 

y 

Palladium 
Phosphorus, white 
Black* 
Platinum 
Plutonium 
Polonium, a 
P (above 75C) 
Potassium 
Praseodymium, a* 

P 
Promethium 
Protactinium 
Radium 
Radon 


Hexagonal 
See Promethium 
FC tetragonal, A6 
Orthorhombic 
FCC, Al 
BCC, A2 
FCC, Al 

BCC, A2 
FCC, Al 
HCP, A3 
FCC, Al 
FCC, Al 
BCC, A2 
FCC, Al 
HCP, A3 (?) 
HCP, A3 
HCP, A3 
Cubic, A12 
Cubic, A13 
FC tetragonal, A6 

(Technetium) 
Rhombohedral, All 
BCC, A2 
HCP, A3(?) 
FCC, Al 

FCC, Al 

HCP, A3 
Tetragonal, D'' 4 /, 
BCC, A2 
Cubic 
Hexagonal 
HCP, A3 
Orthorhombic 
Rhombohedral 
Cubic 
FCC, Al 
Cubic 
Orthorhombic, A16 
FCC, Al 

Simple cubic 
Simple rhombohedral 
BCC, A2 
HCP, A3 (?) 
FCC, Al 


3.76 

4.594 
4.787 
3.8389 
2.3664 
3.571 
3.656 
2.94 
5.69 
3.762 
5.307 
4.9495 
3.5089 
4.40 
3.08 
3.516 
3.2092 
8.912 
6.313 
3.782 

2.006 
3 1466 
3.657 
4.51 

3.5238 

2 66 

4.00 
3.3007 
5.67 
4.04 
2.7333 
5.51 
6.20 
6.84 
3.8902 
7 18 
3.32 
3.9237 

3.345 
3.359 
5.344 
3.669 
5.161 


7 266 


6.13 

4.951 
9.793 


- 271C 

20C 
20C 
20C 
20C 
20C 
950C 
)425C 
-191C 
20C 
Room 
2CPC 
20C 
-195C 
-193?C 
20C 
20C 
20C 
Room 
Room 

-46C 
20C 
20C 
-268C 

20C 

Room 
Room 


3.25 
2.71 
2.714 
2.481 
2.525 
2.585 
2.54 
4.03 
3.74 
3.762 
3.499 
3.039 
3.11 
3.06 
3.446 
3.196 
2.24 
2.373 
2.587 

3.006 
2.7?5 
5.902 
3.21 

2.491 

2.859 
1.06 

2.675 

2.750 

2.17 
2.775 

3.35 
4.40 
4.627 
3.640 
3.649 
























6.075 
















4.82 
5.570 
5.2103 








3.533 

7031 ./ 
















4.32 
3.77 






252C 
-234C 


3.83 


6.60 
4.3191 
3.45 
99.1 


-252PC 
-238C 
-225C 
20C 
-35C 
Room 
20C 










4.39 


10.52 




9813' 




20C 
20C 
Room 




5.920 







(cont.) 

* Ordinary form of an element that exists (or is thought to exist) in more than 
one form. 



484 



CRYSTAL STRUCTURE DATA 



[APP. 13 



EJement and 
modification 


Type of structure 


Lattice 
parameters (A) 


c or axial 
angle 


Temperature 
for which 
constants 
apply 


Distance 
of closest 
approach 
(A) 


a 


b 


Rhenium 
Rhodium, p* 
(electrolytic) 
Rubidium 
Ruthenium, a* 
Samarium 
Scandium, a* 

P 
Selenium* (gray, 
stable, metallic) 

a (red, metastable) 
P (red, metastable) 

Silicon 
Silver 
Sodium 
Strontium 
Sulfur, a, yellow* 

P 

Tantalum 
Tellurium 
Terbium 
Thallium, a* 
P^ 
Thorium 
Thulium 
Tin, , gray 
P, white* 
Titanium, a* 

P 
Tungsten (wolfram), a* 
p (unstable) 
Uranium, *{<665C) 
p(665-775C) 
y(775-1130C) 
Vanadium 
Virginium 
Wolfram 
Xenon 
Ytterbium 
Yttrium 
Zinc 
Zirconium, a* 
P 


HCP, A3 
FCC, Al 
Cubic 
BCC, A2 
HCP, A3 
PC tetragonal (?) 
FCC, Al 
HCP, A3 

Hexagonal, A8 
Monoclinic, P2j 

Monoclinic, C '>/, or 
C';>,,orC 2 5 " 
Diamond cubic, A4 
FCC, Al 
BCC, A2 
FCC, Al 
Orthorhombic, A 17 

Monoclinic 

BCC, A2 
Hexagonal, A8 
HCP, A3 
HCP, A3 
BCC, A2 
FCC, Al 
HCP 
Diamond cubic, A4 
Tetragonal, A5 
HCP, A3 
BCC, A2 
BCC, A2 
Cubic, A15 
Orthorhombic, A20 
Low symmetry 
BCC, A2 
BCC, A2 
See Astatine 
See Tungsten 
FCC, Al 
FCC, Al 
HCP, A3 
HCP, A3 
HCP, A3 
BCC 


2.7609 
3.8034 
9.230 
5.63 
2.7038 

4.541 
3.31 

4.3640 
9.05 
12.76 

5.4282 
4.0856 
4.2906 
6.087 
10.50 

10.92 

3.3026 
4.4559 
3.592 
3.4564 
3.882 
5.088 
3.530 
6.47 
5.8311 
2.9504 
3.33 
3.1648 
5.049 
2.858 

3.49 
3.039 

6.25 
5.488 
3.670 
2.664 
3.230 
3.62 




4.4583 


< C 
20C 

Room 
- 173C 
20C 

20C 
Room 

20C 
Room 

Room 

20C 
20C 
20C 
20C 
20C 

103C 

20C 
20C 
20C 
Room 
26iC 
20C 
20C 
18C 
20C 

^ 
900C 

20C 
20C 
20C 

800C 
20C 

- 185C 


2.740 
2.689 

4.88 
2.649 

3.2110 
3.24 

2.32 
2.34 

2.351 
2.888 
3.715 
4.31 
2.12 

2.860 
2.87 
3.515 
3.407 
3.362 
3.60 
3.453 
2.81 
3.022 
2.89 
2.89 
2.739 
2.524 
2.77 

3.02 
2.632 

4.42 
3.874 
3.60 
2.664 

3.17 
3.13 












4.2816 


9.07 
8.06 


5.24 

4.9594 
|P = 9046' 
111.61 
(P = 934' 
I 9.27 














12.94 
11.04 


24.60 
fp = 8316' 
(J0.98 




5.9268 
5.675 
5.531 








5.575 




3.1817 
4.6833 










5.877 


4.955 
















5.826 
4.945 
5.133 








867C 







* Ordinary form of an element that exists (or is thought to exist) in more than 
one form. 



From Structure of Metals, 2nd edition, by Charles S. Barrett (McGraw-Hill Book 
Company, Inc., New York, 1952). 



APP. 13] CRYSTAL STRUCTURE DATA 

TABLE A13-2. SOME COMPOUNDS AND SOLID SOLUTIONS 



485 



Substance 


Type of structure 


Lattice 
parameters (A) 


Spacing of 
cleavage planes (A) 


NaCI 
KCI 
AgBr 


FCC, Bl 
FCC, Bl 
FCC, Bl 


a = 5.639 
a = 6.290 
a * 5.77 


2.820 


Cap2(fluorite) 


FCC, Cl 


a = 5.46 




CaCO 3 (calcite) 


Rhombohedral, Gl 


a = 6.37 
a = 46.1 


3.036 


SiOjC a -quartz) 


Hexagonal, C8 


a = 4.90 
c = 5.39 




H 2 KAI 2 (Si0 4 ) 3 
(mica, muscovite) 


Monoclinic 


a = 5.18 
b = 8.96 
c = 20.15 
p = 98.6 


10.08 


Fe^C (cementite) 


Orthorhombic 


a = 4.525 
b = 5.088 
c = 6.740 




Austen ite 


FCC, Al 


a = 3.555 + 0.044.T 
( x - weight percent 
carbon) 




Martensite 


BC Tetragonal 


a = 2.867 - 0.013.T 
c = 2.867 + 0.116-r 
(x = weight percent 
carbon) 





APPENDIX 14 
ELECTRON AND NEUTRON DIFFRACTION 

A14-1 Introduction. Just as a beam of x-rays has a dual wave-particle 
character so, inversely, does a stream of particles have certain properties 
peculiar to wave motion. In particular, such a stream of particles can be 
diffracted by a periodic arrangement of scattering centers. This was first 
predicted theoretically by de Broglie in 1924 and demonstrated experimen- 
tally by Davisson and Germer in 1927 (for electrons) and by Von Halban 
and Preiswerk in 1936 (for neutrons). 

If a stream of particles can behave like wave motion, it must have a 
wavelength associated with it. The theory of wave mechanics indicates 
that this wavelength is given by the