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ELEMENTS OF
XRAY DIFFRACTION
ADDISONWESLEY METALLURGY SERIES
MORRIS COHEN, Consulting Editor
Cidlity ELEMENTS OF XRAY DIFFRACTION
Guy ELEMENTS OF PHYSICAL METALLURGY
Norton ELEMENTS OF CERAMICS
Schuhmann METALLURGICAL ENGINEERING
VOL. I: ENGINEERING PRINCIPLES
Wagner THERMODYNAMICS OF ALLOYS
ELEMENTS OF
XRAY DIFFRACTION
by
B. D. CULLITY
Associate Professor of Metallurgy
University of Notre Dame
ADDISONWESLEY PUBLISHING COMPANY, INC.
READING, MASSACHUSETTS
Copyright 1956
ADD1SONWESLEY 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 5610137
PREFACE
Xray 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 xrays, the manner of the diffraction revealing the
structure of the crystal. At first, xray 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 xray 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 xray diffrac
tion purely as a laboratory tool for the sort of problems already mentioned.
Members of this group, unlike xray crystallographers, are not normally
concerned with the determination of complex crystal structures. For this
reason the rotatingcrystal method and spacegroup 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, xray 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 reciprocallattice 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 XRAYS 1
11 Introduction 1
12 Electromagnetic radiation 1
13 The continuous spectrum . 4
14 The characteristic spectrum 6
15 Absorption . 10
16 Filters 16
17 Production of xrays 17
1 8 Detection of xrays 23
1 9 Safety precautions . 25
CHAPTER 2 THE GEOMETRY OF CRYSTALS 29
^21 Introduction . 29
J22 Lattices . 29
23 Crystal systems 30
^24 Symmetry 34
25 Primitive and nonprimitive cells 36
26 Lattice directions and planes * . 37
27 Crystal structure J 42
28 Atom sizes and coordination 52
29 Crystal shape 54
210 Twinned crystals . 55
211 The stereographic projection . . 60
CHAPTER 3 DIFFRACTION I: THE DIRECTIONS OF DIFFRACTED BEAMS 78
31 Introduction . .78
32 Diffraction f . 79
^33 The Bragg law * ' . 84
34 Xray spectroscopy 85
35 Diffraction directions  88
36 Diffraction methods . 89
37 Diffraction under nonideal conditions . 96
CHAPTER 4 DIFFRACTION II: THE INTENSITIES OF DIFFRACTED BEAMS . 104
41 Introduction 104
42 Scattering by an electrons . . 105
43 Scattering by an atom >, . / 108
44 Scattering by a unit cell */ . Ill
CONTENTS
45 Some useful relations . 118
46 Structurefactor calculations ^ 118
47 Application to powder method ' 123
48 Multiplicity factor 124
49 Lorentz factor 124
110 Absorption factor 129
411 Temperature factor 130
412 Intensities of powder pattern lines 132
413 Examples of intensity calculations 132
414 Measurement of xray intensity 136
EXPERIMENTAL METHODS
LPTER 5 LAUE PHOTOGRAPHS 138
51 Introduction 138
52 Cameras . 138
53 Specimen holders 143
54 Collimators . .144
55 The shapes of Laue spots . 146
kPTER 6 POWDER PHOTOGRAPHS . .149
61 Introduction . 149
62 DebyeScherrer method . 149
63 Specimen preparation .... 153
64 Film loading . . 154
65 Cameras for high and low temperatures . 156
66 Focusing cameras ... . 156
67 SeemannBohlin camera . 157
68 Backreflection focusing cameras . . .160
69 Pinhole photographs . 163
610 Choice of radiation . .165
611 Background radiation . 166
612 Crystal monochromators . 168
613 Measurement of line position 173
614 Measurement of line intensity . 173
VPTER 7 DlFFRACTOMETER MEASUREMENTS 177
71 Introduction . . . 177
72 General features .... 177
73 Xray optics . . .  184
74 Intensity calculations . ... 188
75 Proportional counters . . . . 190
76 Geiger counters . . ... .193
77 Scintillation counters .  201
78 Sealers . ... .... .202
79 Ratemeters .  206
710 Use of monochromators 211
CONTENTS XI
APPLICATIONS
CHAPTER 8 ORIENTATION OF SINGLE CRYSTALS . . . 215
81 Introduction . . .... 215
82 Backreflection Laue method . . .215
83 Transmission Laue method .... . 229
84 Diffractometer method ' . ... 237
85 Setting a crystal in a required orientation . 240
86 Effect of plastic deformation . 242
87 Relative orientation of twinned crystals 250
88 Relative orientation of precipitate and matrix . . . 256
CHAPTER 9 THE STRUCTURE OF POLYCRYSTALLINE AGGREGATES . 259
91 Introduction . 259
CRYSTAL SIZE
92 Grain size 259
93 Particle size . 261
CRYSTAL PERFECTION
94 Crystal perfection . .... 263
95 Depth of xray penetration . . 269
CRYSTAL ORIENTATION
96 General . .272
97 Texture of wire and rod (photographic method) . . . 276
98 Texture of sheet (photographic method) 280
99 Texture of sheet (diffractometer method) . . 285
910 Summary . . 295
CHAPTER 10 THE DETERMINATION OF CRYSTAL STRUCTURE . . . 297
101 Introduction . . 297
102 Preliminary treatment of data . . . 299
103 Indexing patterns of cubic crystals 301
104 Indexing patterns of noncubic crystals (graphical methods) 304
105 Indexing patterns of noncubic crystals (analytical methods) . .311
106 The effect of cell distortion on the powder pattern . . . 314
107 Determination of the number of atoms in a unit cell . .316
108 Determination of atom positions . 317
109 Example of structure determination .... . 320
CHAPTER 11 PRECISE PARAMETER MEASUREMENTS . ... 324
111 Introduction .... 324
112 DebyeScherrer cameras .... .... 326
1 13 Backreflection focusing cameras 333
114 Pinhole cameras 333
115 Diffractometers 334
116 Method of least squares .335
Xll CONTENTS
117 Cohen's method .... 338
118 Calibration method . . 342
CHAPTER 12 PHASEDIAGRAM DETERMINATION . . . 345
121 Introduction . 345
122 General principles . . 346
123 Solid solutions . 351
124 Determination of solvus curves (disappearingphase method) 354
125 Determination of solvus curves (parametric method) 356
126 Ternary systems 359
CHAPTER 13 ORDERDISORDER TRANSFORMATIONS 363
131 Introduction . 363
132 Longrange order in AuCus 363
133 Other examples of longrange order 369
134 Detection of superlattice lines 372
135 Shortrange order and clustering 375
CHAPTER 14 CHEMICAL ANALYSIS BY DIFFRACTION 378
141 Introduction 378
QUALITATIVE ANALYSIS
142 Basic principles 379
143 Hanawait method 379
144 Examples of qualitative analysis 383
145 Practical difficulties 386
146 Identification of surface deposits 387
QUANTITATIVE ANALYSIS (SINGLE PHASE)
147 Chemical analysis by parameter measurement 388
QUANTITATIVE ANALYSIS (MULTIPHASE)
148 Basic principles . . . 388
149 Direct comparison method . . . 391
1410 Internal standard method . . . 396
1411 Practical difficulties . . . 398
CHAPTER 15 CHEMICAL ANALYSIS BY FLUORESCENCE 402
151 Introduction . ... 402
152 General principles . . 404
153 Spectrometers ... . 407
154 Intensity and resolution . . . 410
155 Counters .... . 414
156 Qualitative analysis .... ... 414
157 Quantitative analysis ... . . 415
158 Automatic spectrometers . . 417
159 Nondispersive analysis ..... . 419
1510 Measurement of coating thickness 421
CONTENTS xiil
CHAPTER 16 CHEMICAL ANALYSIS BY ABSORPTION . . . 423
161 Introduction . . . ... 423
162 Absorptionedge method . . ... 424
163 Directabsorption method (monochromatic beam) . 427
164 Directabsorption method (polychromatic beam) 429
165 Applications . . 429
CHAPTER 17 STRESS MEASUREMENT . ... 431
171 Introduction . 431
172 Applied stress and residual stress . . 431
173 Uniaxial stress . . 434
174 Biaxial stress . 436
175 Experimental technique (pinhole camera) 441
176 Experimental technique (diffractometer) 444
177 Superimposed macrostress and microstress 447
178 Calibration 449
1 79 Applications 451
CHAPTER 18 SUGGESTIONS FOR FURTHER STUDY . 454
181 Introduction 454
182 Textbooks . 454
183 Reference books . 457
184 Periodicals 458
APPENDIXES
APPENDIX 1 LATTICE GEOMETRY . 459
Al1 Plane spacings 459
Al2 Cell volumes . . 460
Al3 Interplanar angles . . . 460
APPENDIX 2 THE RHOMBOHEDRALHEXAGONAL 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 LORENTZPOLARIZATION 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
A141 Introduction . ... . . 486
A14r2 Electron diffraction ... . 486
A143 Neutron diffraction .... . 487
APPENDIX 15 THE RECIPROCAL LATTICE . . 490
A151 Introduction . .... .490
A152 Vector multiplication . ... 490
A153 The reciprocal lattice . . ... 491
A154 Diffraction and the reciprocal lattice . 496
A155 The rotatingcrystal method . 499
A156 The powder method . 500
A157 The Laue method . . 502
ANSWERS TO SELECTED PROBLEMS . 506
INDEX ... 509
CHAPTER 1
PROPERTIES OF XRAYS
11 Introduction. Xrays 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
xrays 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 xrays 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
xradiation 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
xrays was established. In that year the phenomenon of xray diffraction
by crystals was discovered, and this discovery simultaneously proved the
wave nature of xrays 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 xrays 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 xrays by crystals which follows.
12 Electromagnetic radiation. We know today that xrays are elec
tromagnetic radiation of exactly the same nature as light but of very much
shorter wavelength. The unit of measurement in the xray region is the
angstrom (A), equal to 10~ 8 cm, and xrays used in diffraction have wave
lengths lying approximately in the range 0.52.5A, whereas the wavelength
of visible light is of the order of 6000A. Xrays therefore occupy the
1
PROPERTIES OF XRAYS
[CHAP. 1
Frequency Wavelength
(cycles/sec) in millimicrons
10 23
10 22
_10 5
10 21
Gammarays <
_io 4 i
X unit
10 20
"
_io~ 3
10 19
_10 2
10' 8
_J
xrays
..
__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. ii. 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. 11). Other units sometimes used to measure xray
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 xrays, i.e., xrays of a single
wavelength, is traveling in the x direction (Fig. 12). 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 xyplane 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. 34.
12]
ELECTROMAGNETIC RADIATION
FIG. 12. Electric and magnetic
fields associated with a wave moving
in the jdirection.
t/2plane.) The magnetic field is of
no concern to us here and we need
not consider it further.
In the planepolarized 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 thexaxis.
If both variations are assumed to be sinusoidal, they may be expressed in
the one equation
E = Asin27r(  lA (11)
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 13
shows the variation of E graphically. The wavelength and frequency are
connected by the relation c
X  . (12)
V
where c = velocity of light = 3.00 X 10 10 cm/sec.
Electromagnetic radiation, such as a beam of xrays, 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 xray intensity measurements are made on a relative basis in
+E
E
+E
i
(a) (b)
FIG. 13. 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 XRAYS [CHAP. 1
arbitrary units, such as the degree of blackening of a photographic film
exposed to the xray 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 waveparticle 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.
13 The continuous spectrum. Xrays 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 xray 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. Xrays 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*, (13)
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 onethird 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 xrays.
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).
13]
THE CONTINUOUS SPECTRUM
1.0 2.0
WAVELENGTH (angstroms)
FIG. 14. Xray 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 14
shows the kind of curves obtained. The intensity is zero up to a certain
wavelength, called the shortwavelengthjimit (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 shortwavelength limit and the position of the max
imum shift to shorter wavelengths. We are concerned now with the
smooth curves in Fig. 14, 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 xrays of minimum wavelength. Such electrons
transfer all their energy eV into photon energy and we may write
PROPERTIES OF XRAYS [CHAP. 1
c he
12,400
(14)
This equation gives the shortwavelength 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 xray 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. 14 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 xray energy emitted per second, which is proportional to the
area under one of the curves of Fig. 14, 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 xray
intensity is given by
/cent spectrum = AlZV, (15)
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,
14 The characteristic spectrum. When the voltage on an xray 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 xray diffraction, the longerwavelength lines being
too easily absorbed. There are several lines in the K set, but only the
14] 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.
14. 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. 14. 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 15 shows the spectrum of molybdenum at 35 kv on a
compressed vertical scale relative to that of Fig. 14 ; 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 , (16)
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. 15. The existence of this strong sharp Ka. line is what
makes a great deal of xray 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 XRAYS
[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. 15. Spectrum of Mo at 35 kv (schematic). Line widths not to scale.
The characteristic xray 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),
(17)
where C and <r are constants. This relation is plotted in Fig. 16 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
14]
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. 16. Moseley's relation between \/v and Z for two characteristic lines.
wavelengths of the characteristic xray 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. 17). 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, highenergy state,
FlG ^ Elec tronic transitions in
an at0 m (schematic). Emission proc
esses indicated by arrows.
10 PROPERTIES OF XRAYS [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 Jffshell 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 Kshell vacancy by an electron from
the LOT M shells, respectively. It is possible to fill a 7shell 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 jfshell 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.
16 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 xrays and atoms.
When xrays encounter any form of matter, they are partly transmitted
and partly absorbed. Experiment shows that the fractional decrease in
the intensity 7 of an xray beam as it passes through any homogeneous
substance is proportional to the distance traversed, x. In differential form,
J/.AC, (19)
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 xrays. Integration of Eq. (19) gives
4  /or**, (110)
where /o = intensity of incident xray beam and I x = intensity of trans
mitted beam after passing through a thickness x.
15]
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 (110)
may then be rewritten in a more usable form :
(111)
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 + . . .. (112)
The way in which the absorption
coefficient varies with wavelength
gives the clue to the interaction of
xrays and atoms. The lower curve
of Fig. 18 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.
Shortwavelength xrays 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. 18. Variation with wave
length of the energy per xray quantum
and of the mass absorption coefficient
of nickel.
12 PROPERTIES OF XRAYS [CHAP. 1
termed hard, while longwavelength xrays are easily absorbed and are said
to be soft.
Matter absorbs xrays 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 xrays 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
xrays, 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 xray 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. 18 in light
of the above. Suppose that xrays 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
15] 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. 18, 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 righthand branch of the curve of Fig. 18, for example,
lies between the K and L absorption edges; in this wavelength region inci
dent xrays have enough energy to remove L, M, etc., electrons from nickel
but not enough to remove K electrons. Absorptionedge 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 absorptionedge wavelengths
are given in Appendix 3.
The measured values of the absorption edges can be used to construct
an energylevel diagram for the atom, which in turn can be used in the
calculation of characteristicline 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. (114).
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 energylevel diagram for the atom (Fig. 19).
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 XRAYS [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. 19. Atomic energy levels (schematic). Excitation and emission processes
indicated by arrows. (From Structure of Metals, by C. S. Barrett, McGrawHill
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. 17, which shows the transitions of the
electron. Thus, if a K electron is removed from an atom (whether by an
incident electron or xray), 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 19 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
15]
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
(115)
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. (14).
To excite K radiation, for example, in the target of an xray tube, the bom
barding electrons must have energy equal to WK> Therefore
= W K =
i.
he
'
e\ K
12,400
he
.
*
(116)
where VK is the K excitation voltage (in practical units) and \K is the K
absorption edge wavelength (in angstroms).
Figure 110 summarizes some of the relations developed above. This
curve gives the shortwavelength limit of the continuous spectrum as a
function of applied voltage.
Because of the similarity be
tween Eqs. (14) and (116),
the same curve also enables us
to determine the critical exci
tation voltage from the wave
length of an absorption edge.
FIG. 110. Relation between
the voltage applied to an xray
tube and the shortwavelength
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 XRAYS
[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. 111. 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.
16 Filters. Many xray diffraction experiments require radiation
which is as closely monochromatic as possible. However, the beam from
an xray 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. 111, 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
17]
PRODUCTION OF XRAYS
17
TABLE 11
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 11 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 11.
17 Production of xrays. We have seen that xrays are produced
whenever highspeed electrons collide with a metal target. Any xray
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 watercooled to prevent its melting.
All xray 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. Xray 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 watercooled block of copper con
taining the desired target metal as a small insert at one end. Figure 112
18
PROPERTIES OF XRAYS
[CHAP. 1
17]
PRODUCTION OF XEAY8
19
is a photograph of such a tube, and Fig. 113 shows its internal construc
tion. One lead of the highvoltage 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. Xrays 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 xrays, they are usually made of beryllium,
aluminum, or mica.
Although one might think that an xray 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 halfcycle in which the filament is negative with respect to the
target; during the reverse halfcycle 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. 114 suffices for many installa
tions, although more elaborate circuits, containing rectifying tubes, smooth
ing capacitors, and voltage stabilizers, are often used, particularly when
the xray intensity must be kept constant within narrow limits. In Fig.
114, the voltage applied to the tube is controlled by the autotransformer
which controls the voltage applied to the primary of the highvoltage
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
~
highvoltage transformer
M AK Q000 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. 114. Wiring diagram for selfrectifying filament tube.
20
PROPERTIES OP XRAYS
[CHAP. 1
c
o
18]
DETECTION OF XRAYS
23
electrons
xrays
target
metal
anode
FIG. 116. Reduction in apparent
size of focal spot.
FIG. 117. Schematic drawings of two
types of rotating anode for high power
xrav tubes.
Since an xray tube is less than 1 percent efficient in producing xrays
and since the diffraction of xrays by crystals is far less efficient than this,
it follows that the intensities of diffracted xray 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 xray source. One solution to this
problem is the rotatinganodc tube, in which rotation of the anode con
tinuously brings fresh target metal into the focalspot area and so allows
a greater power input without excessive heating of the anode. Figure 117
shows two designs that have been used successfully; the shafts rotate
through vacuumtight seals in the tube housing. Such tubes can operate
at a power level 5 to 10 times higher than that of a fixedfocus tube, with
corresponding reductions in exposure time.
18 Detection of xrays. The principal means used to detect xray
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
xrays, 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 xrays.
24
PROPERTIES OF XRAYS
[CHAP. 1
(a)
(h)
K edge of
silver
(0.48A).
A' edge of
bromine
(0.92A)
V
1 1 5
X (angstroms)
FIG. 118. 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
xrays in much the same way as by
visible light, and film is the most
widely used means of recording dif
fracted xray beams. However, the
emulsion on ordinary film is too
thin to absorb much of the incident
xradiation, and only absorbed x
rays can be effective in blackening
the film. For this reason, xray 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 xray 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 xray 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 l18(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
19] SAFETY PRECAUTIONS 25
per than to the K radiation from molybdenum, other things being equal.)
Curve (c) of Fig. 118 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 xray beams by the amount
of ionization they produce in a gas. Xray quanta can cause ionization
just as highspeed 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 xray 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 xray intensity, and the magnitude of the current
is a measure of the xray intensity. In the Geiger counter and proportional
counter, this current pulsates, and the number of pulses per unit of time is
proportional to the xray intensity. These devices are discussed more
fully in Chap. 7.
In general, fluorescent screens are used today only for the detection of
xray 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.
19 Safety precautions. The operator of xray 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 xray worker to be continually aware of these hazards.
The danger of electric shock is always present around highvoltage appa
ratus. The anode end of most xray tubes is usually grounded and there
fore safe, but the cathode end is a source of danger. Gas tubes and filament
26 PROPERTIES OF XRAYS [CHAP. 1
tubes of the nonshockproof variety (such as the one shown in Fig. 112)
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
highvoltage parts without automatically disconnecting the high voltage.
Shockproof sealedoff 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 xrays can kill human tis
sue; in fact, it is precisely this property which is utilized in xray therapy
for the killing of cancer cells. The biological effects of xrays include burns
(due to localized highintensity 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 xrays are not
cumulative, but above a certain level called the "tolerance dose," they
do have a cumulative effect and can produce permanent injury. The
xrays 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 xray
workers did through ignorance. There would probably be no accidents if
xrays 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 whitebloodcell 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 leadglass 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
11. What is the frequency (per second) and energy per quantum (in ergs) of
xray beams of wavelength 0.71 A (Mo Ka) and 1.54A (Cu Ka)l
12. Calculate the velocity and kinetic energy with which the electrons strike
the target of an xray tube operated at 50,000 volts. What is the shortwavelength
PROBLEMS 27
limit of the continuous spectrum emitted and the maximum energy per quantum
of radiation?
13. Graphically verify Moseley's law for the K($\ lines of Cu, Mo, and W.
14. 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.
15. Graphically verify Eq. (113) 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.
16. Lead screens for the protection of personnel in xray 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.
17. (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.
18. A sheet of aluminum 1 mm thick reduces the intensity of a monochromatic
xray beam to 23.9 percent of its original value. What is the wavelength of the
xrays?
19. Calculate the K excitation voltage of copper.
110. Calculate the wavelength of the Lm absorption edge of molybdenum.
111. Calculate the wavelength of the Cu Ka\ line.
112. Plot the curve shown in Fig. 110 and save it for future reference.
113. What voltage must be applied to a molybdenumtarget tube in order
that the emitted xrays excite A' fluorescent radiation from a piece of copper placed
in the xray 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 11.
114. 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.)
116. 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.)
116. What is the power input to an xray 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?
117, A coppertarget xray tube is operated at 40,000 volts and 25 ma. The
efficiency of an xray 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 XRAYS [CHAP. 1
pation of heat by watercooling, conduction, radiation, etc., how long would it
take a 100gm 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.)
118. Assume that the sensitivity of xray 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
21 Introduction. Turning from the properties of xrays, 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 xrays. 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."
22 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
spacedividing planes will intersect each other in a set of lines (Fig. 21),
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. 21 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. 21. 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.
22). 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. 22. A unit cell.
23 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.
23]
CRYSTAL SYSTEMS
31
TABLE 21
CRYSTAL SYSTEMS AND BRAVAIS LATTICES
(The symbol ^ implies nonequality by reason of symmetry. Accidental equality
may occur, as shown by an example in Sec. 24.)
System
Axials lengths and angles
Bravais
lattice
Lattice
symbol
r
Cubic
Three equal axes at right angles
a = /, = r , a = p = J = 90
Simple
Bodycentered
Facecentered
P
1
F
Tetragonal
Three axes at right angles, two equal
a = 6 ^ c , a = p = 7 = 90
Simple
Bodycentered
P
I
Orthorhombic
Three unequal axes at right angles
a ^ b i c, a = p = 7 = 90
Simple
Body centered
Basecentered
Facecentered
.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 21.
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)
BODYCENTERED FACEC 'ENTERED
CUBIC (/) CUBIC 1 (F)
SIMPLE BOD Y( CENTERED SIMPLE BODYCENTERED
TETRAGONAL TETRAGONAL ORTHORHOMBIC ORTHORHOMBIC
(P) (/) (P) (/)
BASECENTERED FACECENTERED RHOMBOHEDRAL
ORTHORHOMBIC 1 ORTHORHOMBIC (/?)
(O (F)
^^
c
120
a
W J**
^^
HEXAGONAL
(P)
SIMPLE
MONOCLINIC
BASECENTERED TRICLINIC (P)
(P) MONOCLINIC 1 (C)
FIG. 23. 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
23]
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 21 and illustrated
in Fig. 23, 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
(21 ;
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 facecentered and bodycentered cells, respectively, while A, B,
and C refer tqjmsecentered 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. 23,
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 21 appears incom
plete. Why not, for example, a basecentered tetragonal lattice? The
full lines in Fig. 24 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 basecentered arrange
ment of points is not a new lattice.
/
FIG. 24. Relation of tetragonal C FIG. 25. 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 unitcell 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. 25).
24 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. 26(a).
There are in all four macroscopic* symmetry operations or elements:
reflection, rotation, inversion, and rotationinversion. A body has nfold
rotational symmetry about an axis if a rotation of 360 /n brings it into
selfcoincidence. Thus a cube has a 4fold rotation axis normal to each
face, a 3fold axis along each body diagonal, and 2fold axes joining the
centers of opposite edgesf Some of these are shown in Fig. 26 (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, 26. Some symmetry elements of a cube, (a) Reflection plane. AI be
comes A%. (b) Rotation axes. 4fold axis: A\ becomes A^ 3fold axis: A\ becomes
AZ\ 2fold axis: AI becomes A*, (c) Inversion center. AI becomes A%. (d) Rota
tioninversion axis. 4fold 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 welldeveloped 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.
24] SYMMETRY 35
kinds of axes. In general, rotation axes may be 1, 2, 3, 4, or 6fold. A
1fold axis indicates no symmetry at all, while a 5fold 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. 26(c)]. Finally, a body may have a rotationinversion
axis, either 1, 2, 3, 4, or 6fold. If it has an nfold rotationinversion
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
26(d) illustrates the operation of a 4fold rotationinversion 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 22. { 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. 23J
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 basecentered 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 3fold rotation
axes. Such a lattice would be classified in the tetragonal system, which
has no 3fold axes and in which accidental equality of the a and c axes is
TABLE 22
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. 26) to be expressed in terms of
either set of axes.
25 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. 23 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 facecentered
cubic lattice shown in Fig. 27 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 21 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 unitcell 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. 27, 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. 27. Facecentered cubic point
lattice referred to cubic and rhombo
hedral cells.
26]
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 bodycentered 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 "bodycentering translations," since they will produce the
two point positions characteristic of a bodycentered cell when applied to
a point at the origin. Similarly, the four point positions characteristic of a
facecentered cell, namely 0, \ ^, \ ^, and \ \ 0, are called the
facecentering translations. The basecentering translations depend on
which pair of opposite faces are centered; if centered on the C face, for
example, they are 0, \ \ 0.
26 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. 28.
Direction^ related by symmetry are
called directions of a form, and a set
of these arePepresented 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. 29(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. 29 (b)
as follows :
1A 2A 3A 4A
(a) (b)
FIG. 29. Plane designation by Miller indices.
26]
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. 210.
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. 211 (a)]. The complete lattice is built up, as usual, by
HfeocH
(110)
(110) (111)
FIG. 210. 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. 211. (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 6fold 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 MillerBra 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.
(22)
Since i is determined by h and A;, it is sometimes replaced by a dot and
the plane symbol written (hkl). However, this usage defeats the pur
pose for which MillerBra vais indices were devised, namely, to give similar
indices to similar planes. For example, the side planes of the hexagonal
prism in Fig. 21 l(b) are all similar and symmetrically located, and their
relationship is clearly shown in their full MillerBra vais symbols: (10K)),
(OlTO), (TlOO), (T010), (OTlO), (iTOO). On the other hand, the_abbreviated
symbols of these planes, (100), (010), (110), (100), (010), (110)
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 21 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
26]
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 4fold 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 4fold axis and the last two by a 2fold 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
212 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. (23)
(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, 212, All shaded planes in the
cubic lattice shown are planes of the
zone [001].
(24)
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. 213. Twodimensional 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 213 illustrates this for a twodimensional 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===. (25)
In the tetragonal system the spacing equation naturally involves both
a and c since these are not generally equal :
(Tetragonal) d h ki =
(20)
Interplanar spacing equations for all systems are given in Appendix 1 .
27 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 xray
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
27]
CRYSTAL STRUCTURE
43
BCC FCC
FIG. 214. Structures of some com
mon metals. Bodycentered cubic: a
Fe, Cr, Mo, V, etc.; facecentered
cubic: 7Fe, 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. 214 shows two common structures based on the
bodycentered cubic (BCC) and facecentered cubic (FCC) lattices. The
former has two atoms per unit cell and the latter four, as we can find by
rewriting Eq. (21) 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 closepacked (HCP) structure common to
many metals. This structure is simple hexagonal and is illustrated in
Fig. 215. 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 215(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 closepacked
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 215(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. 215(d). If these spheres are regarded
44
THE GEOMETRY OF CRYSTALS
(a)
(c)
FIG. 215. The hexagonal closepacked structure, shared by Zn, Mg, He, aTi, 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. 215(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 closepacked arrangement. Its rela
tion to the HCP structure is not immediately obvious, but Fig. 216 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
27]
CRYSTAL STRUCTURE
i HID
45
[001]
HEXAGONAL CLOSEPACKED
FIG. 216. Comparison of FCC and HCP structures.
46
THE GEOMETRY OF CRYSTALS
[CHAP. 2
j;
HH
FIG. 217. The structure of auranium.
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. 21 (>.
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, auranium, is illustrated in Fig. 217 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 basecentered
orthorhombic, centered on the C face, and Fig. 217 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:
27]
CRYSTAL STRUCTURE
47
O CB+
[010]
(a) CsCl
(b) NaCl
FIG. 218. The structures of (a) CsCl (common to CsBr, NiAl, ordered /3brass,
ordered CuPd, etc.) and (b) NaCl (common to KC1, CaSe, Pbf e, etc.).
(1) Body, face, or basecentering translations, if present, must begin
and end on atoms of the same kind. For example, if the structure is based
on a bodycentered 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 218 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 xray diffrac
tion pattern of that crystal.
What is the Bravais lattice of CsCl? Figure 21 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 facecentered, but
we note that the bodycentering 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. 218(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 facecentered, and we note that the facecenter
ing translations (0 0, \ \ 0, \ \, \ ^), when applied to the chlorine
ion at \\\, will reproduce all the chlorineion positions. The Bravais
lattice of NaCl is therefore facecentered cubic. The ion positions, inci
dentally, may be written in summary form as:
4 Na 4 " at + facecentering translations
4 Cl~ at \ \ \ + facecentering 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. 218(b), 90 rotation about
the 4fold [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
219 shows the unit cells of diamond and the zincblende form of ZnS.
Both are facecentered cubic. Diamond has 8 atoms per unit cell, lo
cated at
000 + facecentering translations
1 i I + facecentering 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 facecentered cubic Bravais lattice.
To distinguish between these two, the terms "diamond cubic" and "face
centered cubic'' are usually used.
27]
CRYSTAL STRUCTURE
51
O Fe
C position
<
(a)
(b)
FIG. 221. Structure of solid solutions: (a) Mo in Cr (substitutional) ; (b) C in
aFe (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 bodycentered positions of the cube in a random, ir
regular manner, and a small portion of the crystal might have the appear
ance of Fig. 221 (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 righthand 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 xray diffraction effects proper to a BCC lattice. This is not surpris
ing since the xray 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 airon, is a good example. In the unit cell shown in
Fig. 221 (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
28] ATOM SIZES AND COORDINATION 53
the distance of closest approach in the three common metal structures:
BCC =
2 '
V2
2 a > (27)
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
airon. 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
29 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 sixsided
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 welldeveloped 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 welldeveloped 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 welldeveloped prism of natural quartz,
since the essence of crystallinity is a periodicity of inner atomic arrange
ment and not any regularity of outward form.
210] TWINNED CRYSTALS 55
210 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,
abrass, Al, etc.), which have been coldworked and then annealed to
cause recrystallization.
(2) Deformation twins, such as occur in deformed HCP metals (Zn,
Mg, Be, etc.) and BCC metals (aFe, 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.
222 (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 straightline trace on the plane of polish. More common,
however, is the kind shown in Fig. 222 (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. 222. 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. 223. Twin band in FCC lattice. Plane of main drawing is (110).
210]
TWINNED CRYSTALS
59
twinning
shear
[211]
(1012)
twin plane
PLAN OF CRYSTAL
PLAN OF TWIN
FIG. 224. Twin band in HCP lattice. Plane of main drawing is (1210).
60
THE GEOMETRY OF CRYSTALS
[CHAP. 2
are said to be firstorder, secondorder, etc., twins of the parent crystal A.
Not all these orientations are new. In Fig. 222 (b), for example, B may
be regarded as the firstorder twin of AI, and A 2 as the first order twin
of B. 42 is therefore the secondorder twin of AI but has the same orien
tation as A i.
211 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. 225, 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. 226, 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. 225.
crystal.
100
{1001 poles of a cubic
M
FIG. 226. Angle between two planes.
21 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. 226). 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. 227. 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 equalarea 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. 227 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 lefthand hemisphere will project within
this basic circle. Poles on the righthand 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 worthwhile 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. 228), as straight
211]
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. 228 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. 228. Stereographic projection of great and small circles.
64
THE GEOMETRY OP CRYSTALS
[CHAP. 2
FIG. 229. 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. 229. It is the projection of a
sphere ruled with parallels of latitude and longitude on a plane parallel
to the northsouth 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.
211]
THE STEREOGRAPHIC PROJECTION
65
PROJECTION
Wulff net
FIG. 230. Stereographie projection superimposed on Wulff net for measurement
of angle between poles.
These nets are available in various sizes, one of 18cm 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. 226 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. 230, 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 CD equals the angle EF, 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. 231. (a) Stereo
graphic projection of poles
Pi and P 2 of Fig. 226. (b)
Rotation of projection to put
poles on same great circle of Wulff
net. Angle between poles = 30.
(b)
211]
THE STEREOGRAPHIC PROJECTION
67
sired angle measurement can then be made. Figure 231 (a) is a projec
tion of the two poles PI and P 2 shown in perspective in Fig. 226, and the
angle between them is found by the rotation illustrated in Fig. 23 l(b).
This rotation of the projection is equivalent to rotation of the poles on
latitude circles of a sphere whose northsouth axis is perpendicular to the
projection plane.
As shown in Fig. 226, 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.
232. If this is done for two poles, as in Fig. 233, 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 angletrue. This method of an
gle measurement is not as accurate, however, as that shpwn in Fig. 23 l(b).
FIG. 232. Method of finding the trace of a pole (the pole P 2 ' in Fig. 231).
68
THE GEOMETRY OF CRYSTALS
[CHAP. 2
PROJECTION
FIG. 233. Measurement of an angle between two poles (Pi and P 2 of Fig. 226)
by measurement of the angle of intersection of the corresponding traces.
PROJECTION
FIG. 234. Rotation of poles about NS axis of projection.
211] 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 northsouth 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. 234 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. 235). 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. 235. Rotation of a pole about an inclined axis.
211]
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 23, 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 236 (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.
225. To locate the {110} poles we first note from Table 23 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. 236. Standard projections of cubic crystals, (a) on (001) and (b) on (Oil).
72
THE GEOMETRY OF CRYSTALS
[CHAP. 2
TABLE 23
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.
211]
THE STEREOGRAPHIC PROJECTION
73
[112]
zone
mi]
1110]
[001]
zone
[100] //
zone
FIG. 237. Standard (001) projection of a cubic crystal. (From Structure of
Metals, by C. S. Barrett, McGrawHill 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.
(23) 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. 236(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. 26(b), and it will be noted that the projection
itself has the symmetry of the axis perpendicular to its plane, Figs. 236(a)
and (b) having 4fold and 2fold 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. 238. Standard (0001) projection for zinc (hexagonal, c/a = 1.86). (From
Structure of Metals, by C. S. Barrett, McGrawHill Book Company, Inc., 1952.)
Figure 237 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. 238.
It is sometimes necessary to determine the Miller indices of a given
pole on a crystal projection, for example the pole A in Fig. 239(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. 239 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. 239(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
211]
THE STEREOGRAPHIC PROJECTION
75
100
(a) (b)
FIG. 239. Determination of the Miller indices of a pole.
h:k:l = pa:qb:rc. (28)
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. 240 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 xray 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. 240. Stereographic projection
of an FCC crystal and its twin.
76 THE GEOMETRY OP CRYSTALS [CHAP. 2
PROBLEMS
21. Draw the following planes and directions in a tetragonal unit cell: (001),
(Oil), (113), [110], [201], [I01]._
22. 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.
23. In a drawing of a hexagonal prism, indicate the following planes and di
rections: (1210), (1012), (T011), [110], [111), [021].
24. Derive Eq. (22) of the text.
25. Show that the planes (110), (121), and (312) belong to the zone [111]^
26. 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.
27. Prepare a crosssectional 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.
28. Show that c/a for hexagonal close packing of spheres is 1.633.
29. Show that the HCP structure (with c/a = 1.633) and the FCC structure
are equally closepacked, and that the BCC structure is less closely packed
than either of the former.
210. 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.
211. Make a drawing, similar to Fig. 223, of a (112) twin in a BCC lattice
and show the shear responsible for its formation. Obtain the magnitude of the
shear strain graphically.
212. 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.
213. 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 angletrue by measuring With a protractor the angle between
the great circles of A and B.
PROBLEMS 77
214. 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.
216. 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. 236(a) and (b).
216. 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. 236(a).
217. 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. 238.
218. 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 23.
219. Duplicate the operations shown in Fig. 240 and thus find the locations
of the cube poles of a (TTl) reflection twin in a cubic crystal. What are their
coordinates?
220. 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
31 Introduction. After our preliminary survey of the physics of xrays
and the geometry of crystals, we can now proceed to fit the two together
and discuss the phenomenon of xray 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 xrays 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
xrays, and if xrays were electromagnetic waves of wavelength about
equal to the interatomic distance in crystals, then it should be possible to
diffract xrays 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 xrays 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 xrays
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 xrays 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
32]
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 xray 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 crystalstructure determina
tions ever made.
32 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 xrays, such as beam 1 in Fig. 31, proceeding from left to
right. For convenience only, this beam is assumed to be planepolarized
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. 31. 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. 31 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 xrays. Figure 32 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 xrays 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 xrays 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
32 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
xrays 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 xrays 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
xrays. 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 xrays 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 xray 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 (xrays) and a set of periodi
cally arranged scattering centers (the atoms of a crystal).
* For the sake of completeness, it should be mentioned that xrays 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 xray metallography and need not concern us further.
84 DIFFRACTION i: THE DIRECTIONS OF DIFFRACTED BEAMS [CHAP. 3
33 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. (32)
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'. (33)
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. (34)
n
Since the coefficient of X is now unity, we can consider a reflection of any
order as a firstorder 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
(35)
This form will be used throughout this book.
This usage is illustrated by Fig. 33. Consider the secondorder 100 re
flection* shown in (a). Since it is secondorder, 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).
34]
XRAY SPECTROSCOPY
85
(100)
(200)
FIG. 33. Equivalence of (a) a secondorder 100 reflection and (b) a firstorder
200 reflection.
lengths. If there is no real plane of atoms between the (100) planes, we
can always imagine one as in Fig. 33 (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 firstorder 200 reflection. Similarly,
300, 400, etc., reflections are equivalent to reflections of the third, fourth,
etc., orders from the (100) planes. In general, an nthorder reflection
from (hkl) planes of spacing d f may be considered as a firstorder 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.
34 Xray spectroscopy. Experimentally, the Bragg law can be uti
lized in two ways. By using xrays 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 xray spectroscopy.
The essential features of an xray
spectrometer are shown in Fig. 34.
Xrays from the tube T are incident
on a crystal C which may be set at
any desired angle to the incident FIG. 34. The xray 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 xrays; 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. 15 and the characteristic wavelengths tabu
lated in Appendix 3 were obtained. W. H. Bragg designed and used the
first xray 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 xray 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 = , (36)
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. (36), 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 planespacing
equation (Eq. 25), 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
34] XRAY 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 xrays 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:
(38)
1 kX = 1.00202A
This conversion factor was decided on in 1946 by international agreement,
and it was recommended that, in the future, xray wavelengths and the
lattice parameters of crystals be expressed in angstroms. If V in Eq. (36)
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 = (39)
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.
35 Diffraction directions. What determines the possible directions,
i.e., the possible angles 20, in which a given crystal can diffract a beam of
monochromatic xrays? Referring to Fig. 33, 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 planespacing 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 ). (310)
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. (310) 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.
36] 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. (310) and (311).
It is in this sense that equations of this kind predict all possible diffracted
beams.
36 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 xrays 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
Rotatingcrystal 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 xray 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. 35). 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. 35. (a) Transmission and (b) backreflection Laue methods.
method is so called because the diffracted beams are partially transmitted
through the crystal. In the backreflection Laue method the film is placed
between the crystal and the xray 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. 36. 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) backreflection Laue patterns of an alumi
num crystal (cubic). Tungsten radiation, 30 kv, 19 ma.
36]
DIFFRACTION METHODS
91
Z.A.
(b)
FIG. 37. Location of Laue spots (a) on ellipses in transmission method and (b)
on hyperbolas in backreflection method. (C = crystal, F film, Z.A. = zone
axis.)
These curves are generally ellipses or hyperbolas for transmission patterns
[Fig. 36(a)] and hyperbolas for backreflection patterns [Fig. 36(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. 37 (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 semiapex 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 xray source to record the backreflection pat
tern will intersect the cone in a hyperbola, as shown in Fig. 37 (b).
92
DIFFRACTION i: THE DIRECTIONS OF DIFFRACTED BEAMS [CHAP. 3
Z.A.
FIG. 38. Stereographic projection
of transmission Laue method.
FIG. 39. Rotatingcrystal 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. 38, 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
backreflection 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 rotatingcrystal method a single crystal is mounted with one of
its axes, or some important crystallographic direction, normal to a mono
chromatic xray 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. 39). As the crystal rotates,
36] DIFFRACTION METHODS 93
^m^mm
^'S'lililtt
FIG. 310. Rotatingcrystal 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. 310.
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 rotatingcrystal method and its variations is in the
determination of unknown crystal structures, and for this purpose it is
the most powerful tool the xray 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 xray diffraction as a laboratory tool. For this
reason the rotatingcrystal 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 xrays. 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. 311. Formation of a diffracted cone of radiation in the powder method.
Bragg angle for reflection; Fig. 311 (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. 31 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 312 shows four such cones and also illustrates the most common
powderdiffraction method. In this, the DebyeScherrer 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. 312(b). Actual patterns,
produced by various metal powders, are shown in Fig. 313. 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
36]
DIFFRACTION METHODS
95
point where
incident beam
enters (26 = 180) /
(a)
\
26 =
1 t "]
o
I
1
b )
(b)
FIG. 312. DebyeScherrer 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. 313. DebyeScherrer 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 DebyeScherrer 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 xray spectrometer can be used as a tool in diffraction anal
ysis. This instrument is known as a diffractometer when it is used with
xrays 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 DebyeScherrer camera in that the
counter intercepts and measures only a short arc of any one cone of dif
fracted rays.
37 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. 32 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. 32 is that the waves in
volved must differ in path length, that is, in phase, by exactly an integral
37J
DIFFRACTION UNDER NONIDEAL CONDITIONS
97
number of wavelengths. But suppose that the angle 9 in Fig. 32 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. 31, 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 "outofphaseness" 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. 314).
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. 314, 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. 314.
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 onehalf (actually, an
integer plus onehalf) 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. 315(a) in contrast to Fig. 315(b), which illus
trates the hypothetical case of diffraction occurring only at the exact Bragg
angle.
The width of the diffraction curve of Fig. 31 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. 315. Effect of fine particle size on diffraction curves (schematic).
37] 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 pathdifference 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 = (312)
JS cos SB
A more exact treatment of the problem gives
, . _*_. (313)
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. 314, 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. 32 has never been produced in the laboratory. As will
be shown in Sec. 54, any actual beam of xrays 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. 316 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. 3K). The mosaic structure of
a real crystal.
37]
DIFFRACTION UNDER NONIDEAL CONDITIONS
101
(a)
(1))
FIG. 317. (a) Scattering by
atom, (b) Diffraction by a crystal.
crystal
liquid or amorphous solid
90 180
DIFFRAC TION (SCATTERING)
ANGLE 28 (degrees)
FIG. 318. Comparative xray 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 xrays 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)
xrays in relatively few directions, as illustrated schematically in Fig. 317.
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 nonBragg 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 overall 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 xray scattering by solids,
liquids, and gases (Fig. 318). 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 xray 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
31. Calculate the "xray density" [the density given by Eq. (39)] of copper
to four significant figures.
32. A transmission Laue pattern is made of a cubic crystal having a lattice
parameter of 4.00A. The xray beam is horizontal. _ The [OlO] axis of the crystal
points along the beam towards the xray 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?
33. A backreflection Laue pattern is made of a cubic crystal in the orientation
of Prob. 32. By means of a stereographic projection similar to Fig. 38, 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?
34. 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
36. 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.
36. Check the value given in Sec. 37 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
41 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. 41. Both are orthorhombic with two atoms of the same kind per
unit cell, but the one on the left is basecentered and the one on the right
bodycentered. 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. 42. For the basecentered 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 bodycentered 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 bodycentered 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 xrays are
scattered first by a single electron, then by an atom, and finally by all the
,$ (a) (b)
FIG. 41. (a) Basecentered and (b) bodycentered orthorhombic unit cells.
104
42]
SCATTERING BY AN ELECTRON
r i
3
105
(a)
(b)
FIG. 42. Diffraction from the (001) planes of (a) basecentered and (b) body
centered orthorhombir lattices.
atoms in the unit cell. We will apply these results to the powder method
of xray 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 xrays.
42 Scattering by an electron. We have seen in Chap. 1 that aq xray
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 xray 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 xray
tube, where xrays are emitted because of the rapid deceleration of the
electrons striking the target. Similarly, an electron which has been set
into oscillation by an xray 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 xrays, 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 xrays 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,
(41)
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. 43) 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 xray 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. (41) 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
= rrr (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
42]
SCATTERING BY AN ELECTRON
107
\
before impact
FIG. 43. Coherent scattering of x
rays by a single electron.
after impart
FIG. 44. Elastic collision of photon
and electron (Compton effect).
This is the Thomson equation for the scattering of an xray 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. (42) 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
xrays, and that is manifested in the Compton effect. This effect, discovered
by A. H. Compton in 1923, occurs whenever xrays 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 xray 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
43 Scattering by an atom. 1 When an xray 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.
43] 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. 45, 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, 45. Xray scattering by an atom.
FIG. 46. 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. 46. 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.
46. (The resulting curve closely approximates the observed scattered in
tensity per atom of a monatomic gas, as shown in Fig. 318.)
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 xrays strikes an atom,
two scattering processes occur 4 Tightly bound electrons are jet, into pscTP"
lation and radiate xrays of the saiffi wavelength as that of the incident
SCATTERING BY A UNIT CELL
incident beam
absorbing substance
fluorescent xrays
unmodified
(coherent)
Compton modified
(incoherent)
Compton recoil
electrons
photoelectrons
FIG. 47. Effects produced by the passage of xrays through matter. (After
N. F. M. Henry, H. Lipson, and W. A. Wooster, The Interpretation of XRay 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 xrays through
matter. This is done schematically in Fig. 47. The incident xrays 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 xray 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. 48. 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. 42(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. 48. 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.
44] SCATTERING BY A UNIT CELL 113
From the definition of Miller indices,
a
= AC = 
n
How is this reflection affected by xrays 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. 49, 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). (44)
This relation is general and applicable to a unit cell of any shape.
114 DIFFRACTION II : THE INTENSITIES OF DIFFRACTED BEAMS [CHAP. 4
FIG. 49. The threedimensional analogue of Fig. 48.
These two waves may differ, not only in phase, jbut^also in amplitude if
atom B and the atonTstrtrre 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. 410. The addition of sine waves of different phase and amplitude.
44]
SCATTERING BY A UNIT CELL
117
~2
FIG. 411. Vector addition of waves.
FIG. 412. A
complex plane.
wave vector in the
The two waves shown as full lines in Fig. 410 represent the variations
in electric field intensity E with time t of two rays on any given wave front
in a diffracted xray beam. Their equations may be written
EI = A\ sin (2irvt ^i), (45)
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. 411, each component wave is represented
by a vector whose length is equal to the amplitude of the wave and which
is inclined to the :raxis 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 = Vil 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. 412. 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 powerseries expansions of e ix , cos x y and sin x, we
find that
e ix = cos x + i sin x (47)
or
Ae* = A cos <t> + Ai sin 4. (48)
Thus the wave vector may be expressed analytically by either side of
Eq. (48). 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 , (49)
which is the quantity desired. Or, using the other form given by Eq. (48),
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. (44) 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
(410)
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(Au3H;i>sf Iwi) i . . .
44] SCATTERING BY A UNIT CELL 117
This equation may be written more compactly as
N
1 hkl Z^Jn
1
\~* f
1411)
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. (41 1) by its complex con
jugate* Equation (411) is therefore a very important relation in xray
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. 48, 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. (44).
Although it is more unwieldy, the following trigonometric equation may be
used instead of Eq. (411):
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 (411) 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.
45 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.
(47).
(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.
4r6 Structurefactor calculations. Facility in the use of Eq. (411) 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 basecentered cell discussed at the beginning of
this chapter and shown in Fig. 41 (a). It has two atoms of the same kind
per unit cell located at 0,and J J 0.
/[I
46] STRUCTUREFACTOR 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 hodyppntfifpH r,el] ahnwn In Fig. 41 (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
basecentered cell would produce a 001 reflection but that the bodycentered
cell would not. This result is in agreement with the structurefactor 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 facecentered cubic cell, such as that shown in Fig. 214, 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 structurefactor
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 41. 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 facecentered cubic lattice, but it contains eight
TABLE 41
Bravais lattice
Reflections present
Reflections absent
Simple
Base centered
Bodycentered
Facecentered
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.
46] STRUCTUREFACTOR 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 facecentered, 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. 218). 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 structurefactor equation :
F = /Na[l + e
+ e'* 7 + e* lk +
As discussed in Sec?. 27, the sodiumatom positions are related by the
facecentering translations and so are the chlorineatom positions. When
ever a lattice contains common translations, the corresponding terms in
the structurefactor 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. 45. Therefore
Here the terms corresponding to the facecentering 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 facecentered 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 facecentered. The introduction of additional atoms has not elim
inated any reflections present in the case of the fouratom 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 closepacked hexagonal cell shown in Fig. 215 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. 45, 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.
47] APPLICATION TO POWDER METHOD 123
It is by these missing reflections, such as 111, 113, 221, 223, that a
hexagonal structure is recognized as being closepacked. 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
47 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 rotatingcrystal
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
48 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.
49 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. 413 (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. 37, 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. 413 (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. 413 (b). The integrated intensity is
of much more interest than the maximum intensity, since the former is
49]
LORENTZ FACTOR
125
(a)
DIFFRACTION ANGLE 26
(b)
FIG. 413. 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. 413(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. 414(a), the dashed
lines show the position of the crystal after rotation through a small angle
2,
2'
(a) (b)
FIG. 414. 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. 414(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. 37.) 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 backreflection region.
The breadth of the diffraction curve varies in the opposite way, being
larger at large values of 20#, as was shown in Sec. 37, 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 xB, 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.
49]
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 singlecrystal 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. 415
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. 415. 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
DebyeScherrer method, shown in Fig. 416, 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. 416. Intersection of cones of diffracted rays with DebyeScherrer 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. 42 to give the combined Lorentz
polarization factor which, with a con
stant factor of ^ omitted, is given by
Lorentzpolarization 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. 417
as a function of 6. (jhe overall 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. 417. Lorentzpolarization factor.
410]
ABSORPTION FACTOR
129
\
(a)
(h)
FIG. 418. Absorption in DebyeScherrer specimens: (a) general case, (b) highly
absorbing specimen.
410 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 DebyeScherrer
method has the form of a very thin cylinder of powder placed on the camera
axis, and Fig. 41 8 (a) shows the cross section of such a specimen. For
the lowangle 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 highangle
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. 41 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 backwardreflected beams then
undergo very little absorption, but forwardreflected beams have to pass
through the whole specimen and are greatly absorbed.^ Actually, the
forwardreflected 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. 313 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 DebyeScherrer photographs,
or split lowangle 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
high0 and low0 reflections decreases as the linear absorption coefficient
of the specimen decreases, but the absorption is always greater for the
low0 reflections. (These remarks apply only to the cylindrical specimen
used in the DebyeScherrer method. The absorption factor has an entirely
different form for the flatplate specimen used in a diffractometer, as will
be shown in Sec. 74.)
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 DebyeScherrer method
is used. Justification of this omission will be found in the next section.
411 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 illdefined 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 high0 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
411] 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, lowmeltingpoint 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 backreflection 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 backreflection
region scarcely distinguishable from the background.
In the phenomenon of temperaturediffuse scattering we have another
example, beyond those alluded to in Sec. 37, of scattering at nonBragg
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 nonBragg 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 spreadout form as temperaturediffuse scat
tering.
132 DIFFRACTION II : THE INTENSITIES OF DIFFRACTED BEAMS [CHAP. 4
412 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> ) , (412)
\ 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.
42) 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. (412) is valid only for the DebyeScherrer method
and then only for lines fairly close together on the pattern; this latter
restriction is not as serious as it may sound. Equation (412) is also re
stricted to the DebyeScherrer 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. (412)
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
xray diffraction to be changed now.
413 Examples of intensity calculations. The use of Eq. (412) will
be illustrated by the calculation of the position and relative intensities of
EXAMPLES OF INTENSITY CALCULATIONS
133
413]
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 42.
TABLE 42
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 facecentered 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. (310) :
sm"0 = jgC/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, sliderule
accuracy is ample.
Column 6: Needed to determine the Lorentzpolarization factor and (sin 0)/X.
Column 7: Obtained from Appendix 7. Needed to determine / Cu 
Column 8: Read from the curve of Fig. 46.
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. 31 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 zincblende form of ZnS, shown in Fig. 219(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: \ \ \ + facecentering translations,
S: + facecentering translations.
Since the structure is facecentered, we know that the structure factor
will be zero for planes of mixed indices. We also know, from example (e)
of Sec. 46, that the terms in the structurefactor equation corresponding
to the facecentering 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
413] 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; (413)
\F\ 2 = 16(/ s  / Z n) 2 when (h + k + 1} is an odd multiple of 2; (414)
^ 2 = 16(/ s + /zn) 2 when (h + k + I) is an even multiple of 2. (415)
The intensity calculations are carried out in Table 43, with some columns
omitted for the sake of brevity.
TABLE 43
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 scatteringfactor curves plotted
from the data of Appendix 8.
Column 7: \F\~ is obtained by the use of Eq. (413), (414), or (415), depending
on the particular values of hkl involved. Thus, Eq. (413) is used for the 111 re
flection and Eq. (415) 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.
414 Measurement of xray 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 xrays and the other on the ability of xrays to ionize
gases and cause fluorescence of light in crystals. These methods have
already been mentioned briefly in Sec. 18 and will be described more fully
in Chaps. 6 and 7, respectively.
PROBLEMS
41. By adding Eqs. (45) and (46) 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
42. Obtain the same result by solving the vector diagram of Fig. 411 for the
rightangle 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
44. 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?
46. 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 structurefactor equation.
46. In Sec. 49, 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.
47. 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 DebyeScherrer 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. (310) and Appendix 6) and calculate their relative integrated intensities.
49. A DebyeScherrer 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?
410. A DebyeScherrer pattern is made of the intermediate phase InSb with
Cu Ka radiation. This phase has the zincblende 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?
411. Calculate the relative integrated intensities of the first six lines of the
DebyeScherrer pattern of zinc, made with Cu Ka radiation. The indices and ob
served 6 values of these lines are:
Line hkl 6
1
002
18.8
2
100
20.2
3
101
22.3
4
102
27.9
5
110, 103
36.0
6
004
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. 313(b).
CHAPTER 5
LAUE PHOTOGRAPHS
61 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 heavymeta! 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 xray 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 firstorder reflection is made up of radiation of wavelength X, the
secondorder of X/2, the thirdorder 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.
52 Cameras. Laue cameras are so simple to construct that home
made models are found in a great many laboratories. Figure 51 shows
a typical transmission camera, in this case a commercial unit, and Fig.
138
52]
CAMERAS
139
FIG. 51. Transmission Laue camera. Specimen holder not shown. (Courtesy
of General Electric Co., XRay Department.)
52 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 singlecrystal
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 lighttight film holder, or
FIG. 52. 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. 36(a).
The Bragg angle corresponding to any transmission Laue spot is found
very simply from the relation
tan 20 = > (51)
D
where r\ = distance of spot from center of film (point of incidence of trans
mitted beam) and D = specimentofilm distance (usually 5 cm). Adjust
ment of the specimentofilm distance is best made by using a feeler gauge
of the correct length.
The voltage applied to the xray 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 shortwavelength side at a value of the wavelength which varies in
versely as the tube voltage [Eq. (14)]. Laue spots near the center of a
transmission pattern are caused by firstorder reflections from planes in
clined at very small Bragg angles to the incident beam. Only shortwave
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 shortwave
length limit, and such spots will be eliminated by a decrease in tube voltage
no matter how long the exposure.
A backreflection camera is illustrated in Figs. 53 and 54.. 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 backreflection pattern may be
found from the relation
tan (180  20) = > (52)
62]
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 coarsegrained polycrystaJline one
poBitioned so that only a single, selected grain will be struck by the incident beam!
FIG. 54. Backreflection Laue camera (schematic).
where r 2 = distance of spot from center of film and D = specimentofilm
distance (usually 3 cm). In contrast to transmission patterns, backreflec
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 highorder 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 shortwave
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 backreflection 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 backreflection region, as shown clearly in Fig. 36(a) and (b); it is
in this region also that the temperaturediffuse scattering is most intense.
In both Laue methods, the shortwavelength 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 xrays. When such a screen is placed
with its active face in contact with the film (Fig. 55), the film is blackened
not only by the incident xray 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 xray 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. 55. Arrangement of film and
intensifying screen (exploded view).
(a) (b)
FIG. 56. Effect of doublecoated film
on appearance of Laue spot: (a) section
through diffracted beam and film; (b)
front view of doubled spot on film.
53]
SPECIMEN HOLDERS
143
they would ordinarily bo. Each particle of the screen which is struck by
xrays emits light in all directions and therefore blackens the film outside
the region blackened by the diffracted beam itself, as suggested in Fig. 55.
This effect is aggravated by the fact that most xray film is doublecoated,
the two layers of emulsion being separated by an appreciable thickness of
film base. Even when an intensifying screen is not used, doublecoated
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. 50.
53 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 backreflection 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 backreflection 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* 57. Goniometer with
rotation axes, (Courtesy of
Supper Co,)
' See Sec. 95 for further discussion of this point.
144 LAUE PHOTOGRAPHS [CHAP. 5
relative to the xray beam. In this case, a threecircle goniometer is used
(Fig. 57) ; 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 xray beam. The mechanical stage from a
microscope can be easily converted to this purpose.
It is often necessary to know exactly where the incident xray 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 backreflection
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.
64 Collimators. Collimators of one kind or another are used in all
varieties of xray 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. 58), 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. 58. Pinhole collimator and small source.
54]
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. (53)
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 xray tubes which have focal spots of finite size, usually
rectangular in shape. The projected shape of such a spot, at a small target
tobeam angle, is either a small square or a very narrow line (Fig. 116),
depending on the direction of projection. Such sources produce beams
having parallel, divergent, and convergent rays.
Figure 59 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
(54)
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,
(55)
FIG. 59. 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. 59 is given by
/2u \
d(l).
\u /
(56)
In practice, v is very often about twice as large as u, which means that the
conditions illustrated in Fig. 59 are achieved when the pinholes are about
onethird the size of the projected source. If the value of h is smaller than
that given by Eq. (56), then conditions will be intermediate between
those shown in Figs. 58 and 59; as h approaches zero, the maximum
divergence angle decreases from the value given by Eq. (54) to that given
by Eq. (53) 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. (56), none of the conditions depicted in Fig. 59 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. 58, and in a plane parallel to the source by Fig. 59. 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. (56), a large fraction of the xray
energy is wasted with this arrangement of source and collimator.
The extent of the nonparallelism of actual xray beams may be illus
trated by taking, as typical values, d = 0.5 mm, u = 5 cm, and w = 3 cm.
Then Eq. (54) gives 2 = 1.15 and Eq. (55) 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.
65 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. 58 or 59), then a focusing action takes place on diffraction. Figure
510 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
55]
THE SHAPES OF LAUE SPOTS
147
H
FIG. 510. 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. 511. 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. 36(a) is an
example.
In back reflection, no focusing oc
curs and a divergent incident beam
intinues to diverge in all directions
ter diffraction. Backreflection
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
51. A transmission Laue pattern is made of an aluminum crystal with 40kv
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)?
62. A transmission Laue pattern is made of an aluminum crystal with a speci
mentofilm 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?
63. (a) A backreflection 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
61 Introduction. The powder method of xray 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 xrays by a
powder specimen. In this connection, "monochromatic" usually means
the strong K characteristic component of the general radiation from an
xray 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) DebyeScherrer 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 xray source are all placed
on the surface of a cylinder.
(3) Pinhole method. The film is flat, perpendicular to the incident xray
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 DebyeScherrer
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.
62 DebyeScherrer method. A typical Debye camera is shown in
Fig. 61. It consists essentially of a cylindrical chamber with a lighttight
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. 61. DebyeScherrer 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 crystalstructure
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. 62), 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.
62] DEBYESCHERRER METHOD 151
and AS = #A20, (61)
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. (61) 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. , (62)
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, (63,
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 (63) 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 (63) 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. 17 and the curves of Fig. 63 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 lowercase 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 xrays 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. 63. Absorption of Cu Ka and
Cr Ka radiation by air.
eliminates this effect is shown in Fig. 64, 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 xray tube. Back scatter from the
stop is minimized by extending the beamstop tube backward and con
stricting its end B. Another reason for extending the collimator and
beamstop 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
lowangle and highangle 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. 64. Design of collimator and beam stop (schematic).
63] 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.
68 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 325mesh 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 twophase alloys. If a
small, representative sample is selected from an ingot for xray 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 xray
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 lowangle lines (see Sec. 410);
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. 
64 Film loading. Figure 65 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 filmshrink
age error may be allowed for by slipping the ends of the film under metal
knifeedges 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 knifeedges 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 knifeedge shadows on the film.
Figure 65(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.
Knifeedges may also be used in this case as a basis for filmshrinkage cor
rections.
The unsymmetrical, or Straumanis, method of film loading is shown in
Fig. 65 (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 knifeedges
are required to make the filmshrinkage correction. The point X (20 =
64]
FILM LOADING
155
5 4
knifeedge
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. 65. 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 filmshrinkage correction
without calibration of the camera or knowledge of any camera dimension.
The shapes of the diffraction lines in Fig. 65 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 highangle
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. 313.
156 POWDER PHOTOGRAPHS [CHAP. 6
66 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 hightemperature 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 hightemperature 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 hightemperature Debye cameras varies almost from
laboratory to laboratory. They all involve a small furnace, usually of the
electricresistance 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 xray 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 thinwalled silica tube. Because of
the small size of the furnace in a hightemperature 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
hightemperature 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 xray 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.
66 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. 66) : all angles inscribed in a
67]
8EEMANNBOHLIN CAMERA
157
FIG. 66. 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
xrays 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.
67 SeemannBohlin camera. This focusing principle is utilized in the
SeemannBohlin camera shown in Fig. 67. The slit S acts as a virtual
line source of xrays, the actual source being the extended focal spot on
the target T of the xray 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 fineline 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 DebyeScherrer 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. 67. SeemannBohlin focusing camera. Only one hkl reflection is shown.
ticular value of 6 involved. Figure 08 shows a typical powder pattern
made with this camera.
The ends of the film strip are covered by knifeedges 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 lowangle knifeedge N, by use of the relation
46R
*rcSABN.
(64)
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. (64). 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 knifeedge 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. 68. Powder pattern of tungsten, made in a SeemannBohlin camera, 8.4
cm in diameter. This camera covers a 28 range of 92 to 166. Highangle end of
film at left. Filtered copper radiation. (Courtesy of John T. Norton.)
67] SEEMANNBOHLIN 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 (64) 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. 116). 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. (04), we obtain
dU
dd =
4R
This relation may be combined with Eq. ((52) to give
dU 4R
= tan 6.
dd d
d 4R
Resolving power = = tan 6. (65)
M AU
The resolving power, or ability to separate diffraction lines from planes
of almost the same spacing, is therefore twice that of a DebyeScherrer
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. 67 is of the order of 1 cm) and diffracted rays from a considerable
volume of material are all brought to one focus. The SeemannBohlin
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 1in. 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 xrays is obvious. It is
160 POWDER PHOTOGRAPHS [CHAP. 6
worth noting also that both methods of examination, the optical and the
xray, provide information only about the surface layer of the specimen,
since the xray 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 SeemannBohlin camera has the disadvantage that
the reflections registered on the film cover only a limited range of 26 values,
particularly on the lowangle 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 SeemannBohlin cameras, designed to cover practically
the whole range of 26 values in overlapping angular ranges.
Diffraction lines formed in a SeemannBohlin camera are normally
broader than those in a DebyeScherrer 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 xray diffraction. In special cases, it may
pay to use singleemulsion film at the cost of increased exposure time.
68 Backreflection focusing cameras. The most precise measurement
of lattice parameter is made in the backreflection region, as discussed in
greater detail in Chap. 11. The most suitable camera for such measure
ments is the symmetrical backreflection focusing camera illustrated in
Fig. 69.
It employs the same focusing principle as the SeemannBohlin 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. 610. The value of 6
for any diffraction line may be calculated from the relation
(4T  86)R = V, (66)
where V is the distance on the film between corresponding diffraction lines
on either side of the entrance slit.
68]
BACKREFLECTION FOCUSING CAMERAS
161
film
FIG. 69. Symmetrical backreflection focusing camera. Only one hkl reflec
tion is shown.
Differentiation of Eq. (66) gives
'ft
4R \2/
A0
(67)
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. (62) 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
SeemannBohlin camera of the same diameter.
In the pattern shown in Fig. 610, 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. 610. Powder photograph of tungsten made in a symmetrical backreflec
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 backreflection 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. (07) gives
(68)
X 4J?tan0
Resolving power = = (69)
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 (69) 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. 610, which shows a greater
separation of the higherangle 400 reflections as compared to the 321 re
flections.
By use of Eq. (69), we can calculate the resolving power, for the 321
reflections, of the camera used to obtain Fig. 610. The camera radius is
69]
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 backreflection region
can be seen from the Debye photographs reproduced in Fig. 313.
69 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 backreflection camera may be used. A typical transmission
photograph, made of finegrained aluminum sheet, is shown in Fig. 611.
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 lowangle or
highangle reflections may be ob
tained, but not those in the median
FIG. 611. 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. 612. Angular relationships in
the pinhole method.
164 POWDER PHOTOGRAPHS [CHAP. 6
range of 6 (see Fig. 612). In the transmission method, the value of for
a particular reflection is found from the relation
U
tan 21? = . (610)
2D
where U = diameter of the" Debye ring and D = specimentofilm dis
tance. The corresponding relation for the backreflection method is
tan (*  28) = > (611)
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. 99, 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. (110) 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. 613). 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 DebyeScherrer cameras.
The pinhole method is used in studies of preferred orientation, grain
size, and crystal perfection. With a backreflection camera, fairly precise
parameter measurements can be made by this method. Precise knowledge
of the specimentofilm 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.
610]
CHOICE OF RADIATION
165
specimen
(a)
film
(b)
FIG. 613. 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.
610 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 xray 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 latticeparameter measurements require that there be a num
ber of lines in the backreflection 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 FeCo alloy is used as a tar
get, and no filter is used in the xray 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 xray 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 shortwavelength
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 shortwavelength 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
611] 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) Temperaturediffuse 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. 62.
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 fusedquartz 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.
612 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 xray 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 crystalmonochromated 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. 614. The beam from an xray 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.
612]
CRYSTAL MONOCHROMATORS
169
FIG. 614. 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. 615. A
line source of xrays, 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. 615. 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. (012)
The sourcetocrystal distance 8C, which equals the crystaltofocus dis
tance CF, is given by
SC = 2fl cos (  0V (013)
By combining Eqs. (612) and (013), we obtain
SC = R (0M)
d
For reflection of Cu Ka radiation from the (101) 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 unbentcrystal 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 010(a) shows the best arrangement.
A cylindrical camera is used with the specimen and film arranged on the
surface of the cylinder. Lowangle 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. Highangle 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 SeemannBohlin camera, the focal point F of the
612]
CRYSTAL MONOCHROMATORS
171
(h) '/"
FIG. 616. Cameras used with focusing monochromators (a) focusing cameras;
(b) DebyeScherrer and flatfilm cameras Only one diffracted beam is shown in
each case. (After A. (Juinier, Xray 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 DebyeScherrer or flatfilm camera may also be used with a focusing
monochromator, if the incidentbeam collimator is removed. Figure
61 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 abovementioned 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. 617, 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. 617. 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
crystaltofocus distance CF is given by
CF = 2R cos 0.
(015)
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 (412), for example,
was derived for the completely unpolarized incident beam obtained from
the xray 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. (412), 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. 616(b)]. Since the denominator in
this expression is independent of 0, it may be omitted; the combined
Lorentzpolarization factor for a DebyeScherrer camera and crystal
monochromated radiation is therefore (1 + cos 2 2a cos 2 20) /sin 2 cos 6.
614]
MEASUREMENT OP LINE INTENSITY
173
FIG. 618. Filmmeasuring device.
Department.)
(Courtesy of General Electric Co., XRay
613 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. 618 is commonly
used for this purpose. It is essentially a box with an opalglass 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 crosshair; the crosshair is moved over the illuminated
film from one diffraction line to another and their positions noted. The
film is usually measured without magnification. A lowpower 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 xray film.
614 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 xray 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 xray 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 xray 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 xray 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 xray 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.
619(a)]. The resulting galvanometer record [Fig. 61 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 xrays. 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 xray intensity as a function of 26 [Fig.
619(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
614]
MEASUREMENT OF LINE INTENSITY
175
DISTANCE ALONCJ FILM (Yj>)
20
1C)
FIG. 619. Measurement of line intensity with a microphotomctci (schematic)
(a) film; (b) galvanometei recoul, (c) xiay 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 xray
exposure Instead, each film should be calibrated by exposing a strip near
its edge to a constantintensity xray beam for increasing amounts of time
so that a series of stepwise increasing exposures is obtained. The exact
relation between density and xray exposure can then be determined ex
perimentally.
1*10. 620. 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
61. Plot a curve similar to that of Fig. 64 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 1hr exposure in air is required to produce a certain
diffraction line intensity in a 19cmdiameter camera with Fe Ka radiation, what
exposure is required to obtain the same line intensity with the camera evacuated,
other conditions being equal?
62. Derive an equation for the resolving power of a DebyeScherrer camera
for two wavelengths of nearly the same value, in terms of AS, where S is defined
by Fig. 62.
63. For a Debye pattern made in a 5.73crndiameter 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.
64. What is the smallest value of 6 at which the Cr Ka doublet will be resolved
in a 5.73cmdiameter Debye camera? Assume that the line width is 0.03 cm and
that the separation must be twice the width for resolution.
65. A powder pattern of zinc is made in a DebyeScherrer 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. 313(c), which shows these lines unresolved from one another. They
form the fifth line from the lowangle end.)
66. A transmission pinhole photograph is made of copper with Cu Ka radia
tion. The film measures 4 by 5 in. What is the maximum specimentofilm dis
tance which can be used and still have the first two Debye rings completely re
corded on the film?
67. 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?
68. 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 xray 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
71 Introduction. The xray spectrometer, briefly mentioned in Sec.
34, has had a long and uneven history in the field of xray diffraction. It
was first used by W. H. and W. L. Bragg in their early work on xray
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 xray 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 xray spectrometer is really
two instruments:
s (1) An instrument for measuring xray 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 xrays 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.
72 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 xray 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. 15, incident xray quanta can
eject electrons from atoms and thus convert them into positive ions. If
an xray 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 xray 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 xray 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 xray 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 DebyeScherrer
camera, except that a movable counter replaces the strip of film. In both
instruments, essentially monochromatic radiation is used and the xray
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. 71. 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 xray source is S, the line focal
spot on the target T of the xray tube; S is also normal to the plane of the
drawing and therefore parallel to the diffractometer axis 0. Xrays 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 powerdriven at a constant angular velocity
about the diffractometer axis or moved by hand to any desired angular
position.
72]
GENERAL FEATURES
179
FKJ 7 1. Xray difTrartoinetei (schematic)
Figures 72 and 73 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 countingrate meter, calibrated in
such units as counts (pulses) per second. Such a circuit gives a continuous
indication of xray 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 xray 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. 72. General Electric diffractometer. (Courtesy of General Electric Co.,
XRay Department.)
72]
GENERAL FEATURES
181
' ' '
FIG. 73. 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. Xray 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
countingrate meter. The output of this circuit is fed into a fastacting
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. 74, 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 xray 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 incidentbeam
intensity constant when relative line intensities must be measured accu
rately. Since the usual variations in line voltage are quite appreciable,
the xray tube circuit of a diffractometer must include a voltage stabilizer
and a tubecurrent stabilizer, unless a monitoring system is used (see
Sec. 78).
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
72]
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 DebyeScherrer
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 lowangle 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 flatsurfaced compact of very fine powders or a specimen with
a polished surface.
If not enough powder is available for a flat specimen, a thinrod speci
men of the kind used in DebyeScherrer cameras may be used ; it is mounted
on the diffractometer axis and continuously rotated by a small motor
(see Fig. 73). 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.
Singlecrystal specimens may also be examined in a diffractometer by
mounting the crystal on a threecircle goniometer, such as that shown in
Fig. 57, 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 overall 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.
73 Xray optics. The chief reason for using a flat specimen is to take
advantage of the focusing action described in Sec. 66 and so increase the
73]
XRAY OPTICS
iffract omc'tor circle
185
(a)
FIG. 75. 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 75 shows how this is done. For any position of the
counter, the receiving slit F and the xray 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. 75. 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. 75 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. 70), slit A in Fig. 71, 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. 76. 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
73]
XRAY OPTICS 187
incidentbeam slits
specimen
S
* iecei\ ing slit
.^ ^_ to counter
FIG. 77. 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
forwardreflection 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 Sollerslit
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 77 illustrates the relative arrangement of the various
*A number of things besides slit width (e.g., xray 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 receivingslit 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.
74 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 flatplate 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. 78,
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
74] 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). (71)
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. 49) 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. (73) 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 xray 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. (41 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 higherangle
line, relative to that of the lowerangle 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 incidentbeam slits are chosen to produce
a thin, essentially parallel beam. The xray geometry is then entirely
equivalent to that of a DebyeScherrer camera equipped with slits, and
Eq. (412) applies, with exactly the same limitations as mentioned in
Sec. 412.
75 Proportional counters. Proportional, Geiger, and scintillation
counters may be used to detect, not only x and 7radiation, but also
charged particles such as electrons or aparticles, 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
xrays of the wavelengths commonly employed in diffraction.
Consider the device shown in Fig. 79, 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 xrays. Of the xrays 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
75]
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 xray 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 xray 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 xray 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 electricfield 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 xray 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 xray quantum.
We can define a gas amplification factor A as follows : if n is the number
of atoms ionized by one xray quantum, then An is the total number
ionized by the cumulative process described above. Figure 710 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"'
EI
<
^ 10 s
^ 1() 7
O
r< !().
E KM
i io *
^ H) 2
10
1
glow
discharge
corona
discharge
avalanche
region
VOLTAC5E
FIG. 710. 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 xray 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 xrays 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 pulseheight discriminator. If two such circuits are used together, one
76] 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 singlechannel pulseheight
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.
76 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 xrays 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. 711. 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. 711. 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 xray 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 xray 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. 710), 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 xray quantum that caused the primary
ionization. x . ra> (iuantimi
These differences are illustrated absorbed hm
schematically in Fig. 712. The ab
sorption of an xray 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. 712. 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 xray 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 xray quanta. If these quanta are arriving at a very rapid
76]
(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. 713. 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 xray 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. 713 (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. 713(b) on enlarged voltagetime
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 713(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. 714. Here "quanta absorbed per second" are directly
proportional to the xray 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 xray intensity.
The straight line shows the ideal response which can be obtained with a
proportional counter at the rates shown.
76]
GEIGER COUNTERS
197
5000 r
singlechain her
CJcieer counter
1000 2000 3000 4000 5000
QUANTA ABSOHBKI) PKK SECOND
FIG. 714. 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, countingrate 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. 714. 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
counterscaler 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. 715. Calibration curves of a multichamber Geiger counter for two values
of the xray 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. 714. A curve of
this kind is shown in Fig. 715. 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 715 also shows that the range of linearity of a counting rate
curve is dependent on the xray tube voltage and is shorter for lower voltages.
The reason for this dependence is the fact that the xray tube emits charac
teristic xrays not continuously but only in bursts during those times when
76]
GEIGER COUNTERS
199
\ cycle 
I
critical excitation
voltage
TIME
FIG. 716. Variation of tube volt
age with time for a fullwave rectified
xray 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. 716, 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 xray 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
halfcycle 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 Geigercounter 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 selfquenching 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
singlegas counters and prevents the initial avalanche of ionization from
200
DIFFRACTOMETER MEASUREMENTS
[CHAP. 7
becoming a continuous discharge. In an argonchlorine 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
selfquenching 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
xray 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 717 shows how the amount absorbed depends on wavelength for
the two gases most often used in xray counters. Note that a krypton
filled counter has high sensitivity for
all the characteristic radiations nor
mally used in diffraction but that an
argonfilled 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 argonfilled 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
kryptonfilled counter had been
used.
Mo A
c KM)
05 10 1.5
WAVELENGTH
2.0
(A)
FIG. 717. Absorption of xrays in
a 10cm path length of krypton and
argon, each at a pressure of 65 cm
Hg.
77]
SCINTILLATION COUNTERS
201
77 Scintillation counters. This type of counter utilizes the ability of
xrays to cause certain substances to fluoresce visible light. The amount
of light emitted is proportional to the xray 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 xrays is a sodium iodide crystal
activated with a small amount of thallium. It emits blue light under
xray bombardment. The crystal is cemented to the face of a photo
.multiplier tube, as indicated in Fig. 718, and shielded from external light
by means of aluminum foil. A flash of light is produced in the crystal for
every xray 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 caesiumantimony inter
metallic compound. (For simplicity, only one of these electrons is shown
in Fig. 718.) 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 xray 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. 718. 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 xray 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 xray quanta of different energies on the basis of pulse size.
The efficiency of a scintillation counter approaches 100 percent over the
whole range of xray wavelengths, short and long, because all incident
xray quanta are absorbed in the crystal. Its chief disadvantage is its
rather high background count; a socalled "dark current" of pulses is pro
duced even when no xray quanta are incident on the counter. The main
source of this dark current is thermionic emission of electrons from the
photocathode.
78 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
78]
8CALER8
203
must have been
AT = N (2 n ) + a, (75)
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.
719 for a scaleof16 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. 719, 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 10stage 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 shortcircuit
ing the remainder.
Because the arrival of xray 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 xray 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. 719. 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, (76)
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
lowintensity beams.
Equation (76) 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 xray
tube shut off, and not the "diffraction background" at nonBragg angles
due to any of the several causes listed in Sec. 611 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.
78] 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. (76) does not apply. Let TV be the number of
pulses counted in a given time with the xray 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  (77)
(N  N b )
Comparison of Eqs. (70) and (77) 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. 72, 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 lowintensity portions of the diffraction line are measured
with less accuracy than the highintensity 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 fixedtime methods such as these.
The integrating ability of a sealer is also put to use in xray tube moni
tors. In Sec. 72 it was mentioned that the incidentbeam 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 xray 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.
79 Ratemeters. The countingrate meter, as its name implies, is a
device which indicates the average counting rate directly without requir
ing, as in the sealertimer 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.
720(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 shortcircuit
79]
RATEMETERS
207
FIG. 720. The capacitorresistor 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. 720(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 pulseshaping 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. 721, a circuit basically pulse input
similar to that of Fig. 720 (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 xray 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 xray 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 capacitorresistor 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. 722, which shows the
automatically recorded output of a ratemeter when the counter is receiving
a constantintensity xray 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. 78 we saw that the error in a countingrate 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. (7G), namely by
C7
E = ; percent, (78)
\/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. 723,
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
79]
RATEMETERS
209
FIG. 722. Effect of time constant (T.C.) on recorded fluctuations in counting
rate at constant xray 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. 723. Effect of average counting rate on recorded fluctuations in counting
rate, for a fixed time constant (schematic). Xray 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 xray intensity (100, 1000, and 10,000 cps for fullscale
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 xray 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. 78. 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 723 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 ratemeterrecorder combination.
710] USE OF MONOCHROMATORS 211
710 Use of monochromators. Some research problems, notably the
measurement of diffuse scattering at nonBragg 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. 612 may be used in conjunction
with a diffractometer in the manner shown in Fig. 724. Rays from the
physical line source S on the xray 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. 71 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.
(113). 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 xray 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. 724. 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. 725 (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. 725(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 xray 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.
710]
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
71. 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?
72. Prove the statement made in Sec. 74 that the effective irradiated volume
of a flat plate specimen in a diffractometer is constant and independent of 6.
73. 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 xray 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?
74. (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
81 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 xray methods of determining crystal
orientation will be described: the backreflection 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 xray 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.
82 The backreflection Laue method. As mentioned in Sec. 36, 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 backreflection,
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. 81. Intersection of a conical array of diffracted beams with a film placed
in the backreflection 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 backreflection Laue spots on the film. If wo wished, we
could determine the Bragg angle corresponding to each Laue spot from
Eq. (52), 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 somiapox angle is
equal to the angle <t> at which tho zono axis is inclined to the transmitted
beam (Fig. 81). If <t> doos not exceed 45, tho cone will not intersect a
film placed in tho backreflection 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. 81 upside down.)
Diffraction spots on a backreflection Laue film therefore lie on hyper
82]
THE BACKREFLECTION LAUE METHOD
217
film
[ornri cut tor
identification
Fid S 2 Locution oi buckieflertion 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 zaxis 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. 82 that
jc = OK sin /x, U = OS cos M, and OS = OC tan 2a,
218 ORIENTATION OF SINGLE CRYSTALS [CHAP. 8
where OC = D = specimenfilm 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. 83. 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. 82, we can plot the pole of the plane on a stereographic projection.
Imagine a reference sphere centered on the crystal in Fig. 82 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 xray 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 xray 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. 82. When the film is read, this
82]
THE BACKREFLECTION LAUE METHOD
219
6 20
7 = 20
7 = 10
7=0
21)
FIG. (S3. (jreniiifter chait for the solution of backreflection Laue patterns,
reproduced in the correct size for a specimentofilm distance D of 3 cm.
cut corner must therefore be at the upper left, as shown in Fig. 84(a).
The angles 7 and 6, read from the chart, are then laid out on the projection
as indicated in Fig. 84 (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.JIXTION
(b)
P'IG. <S4. 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 BACKREFLECTION LAUE METHOD
82]
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 backreflec
tion Laue pattern is shown in Fig.
3(>(b) Fig. 85 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. 80 by the method
of Fig. 84 and are shown as solid
circles.
221
FIG. X5. Selected diffraction spots
of backreflection Laue pattern of an
aluminum crystal, traced from Fig.
36(b).
FIG. 86. Stereographic projection corresponding to backreflection pattern of
Fig. 85.
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. 86,
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 23). 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. 237, 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 facecentered
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.
82]
THE BACKREFLECTION LAUE METHOD
223
FIG. 87. Stereographic projection of Fig. 86 with poles identified.
Figure 87 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. 87 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. 87 is a com
plete description of the orientation of the crystal. Other methods of
description are also possible. The crystal to which Fig. 87 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. 236(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. 87, the plate normal lies in the (001)(101)(1 Jl) triangle which is
redrawn in Fig. 88 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. 87 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. 88. 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.
88 from Fig. 87 consists simply in
rotating the whole projection, poles
and plate normal together, from the
orientation shown in Fig. 87 to that
of a standard (001) projection.
Similarly, the orientation of a
singlecrystal wire or rod may be de FIG. 88. 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. 87.
g2] THE BACKREFLECTION 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.
S9. 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. 84, the
pole of the plane causing the most important spot or spots on the pattern. (The
latter are, like spot 5 of Fig. S5, 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 S6. 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 MSaxis 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. 89(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. 89. 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.
82]
THE BACKREFLECTION LAUE METHOD
227
plane
FIG. 810. Relation between diffraction spot 8 and stereographic projection P
of the plane causing the spot, for back reflection.
of Fig. 86, 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. 810. 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
(81)
(82)
where D is the specimenfilm 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. 811. 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  )
(83)
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. (Sl) 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. 811,
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. 810 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. (81). 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. 810, it is apparent that the pro
jection must be viewed from the side opposite the xray 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. 811. 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
83]
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 xray source, whereas the Oreninger chart gives a pro
jection of the crystal as seen from the xray 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 EWaxis. Although simple to use and construct, the stereographic ruler
is not as accurate as the Greninger chart in the solution of backreflection 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.
83 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 backreflection 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. 812). 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. 812. 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. 813, 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. 813. 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 righthand corner, viewed from the crystal, is cut off for
identification of its position during the xray 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 jyzplane 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
83]
TRANSMISSION LAUE METHOD
10 20
231
60
FIG. (S14. Leonhardt chart for the solution of transmission Laue patterns, re
produced in the correct size for a specimentofilm 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 crystalfilm
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. 814.
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. 813 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. 815. 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).
83]
TRANSMISSION LAUE METHOD
FILM
233
(a)
Ellipse of spots from
plant' of zone A
10 20 / JO
cut cornel
(b)
FIG. S16. 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 815 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. 816; 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. 817. Relation between diffraction spot S and stereographic projection P
of the plane causing the spot, in transmission.
FIG. 818. 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 .
83] 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 817, 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 818 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 lefthand position.
Whether the chart or the ruler is employed to plot the poles of reflecting
planes, they are indexed in the same way as backreflection patterns. For
example, the transmission Laue pattern shown in Fig. 819 in the form
of a tracing yields the stereographic projection shown in Fig. 820. 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 backreflection pat
terns, as inspection of Fig. 86 will show.) The solution of Fig. 820
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. 819 and the backreflection
pattern shown in Fig. 85 were both obtained from the same crystal in the
same orientation relative to the incident beam. The corresponding pro
jections, Figs. 820 and 87, 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. 819. Transmission Laue pattern of an aluminum crystal, traced from Fig.
36 (a). Only selected diffraction spots are shown.
FIG. 820. Stereographic projection corresponding to transmission pattern of
Fig. 819.
84] DIFFRACTOMETER METHOD 237
84 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. 99, 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. 821 : 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. 821.
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 countingrate 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. 822. The projection is made
on a plane parallel to the specimen
surface, and with the MSaxis 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. 821. Crystal rotation axes
for the diffractometer method of de
termining orientation.
PROJECTION
FIG. 822. 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. 821.)
84] 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. 86, 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
85 Setting a crystal in a required orientation. Some xray 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 threecircle goniometer like
that shown in Fig. 57, whose arcs have been set at zero, and its orienta
tion is determined by, for example, the backreflection 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. 87 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. 236 (b) if the latter were rotated 90
about the center. The initial orientation (Position 1) is shown in Fig. 823
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. 823. 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.
85]
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 incidentbeam axis to bring this point
onto the vertical axis of the underlying Wulff net. (In Fig. 823, 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 .EWaxis, 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 SWaxis;
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 UWaxis), and
neither of the first two rotations dis
FIG. 824. Backreflection 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
specimentofilm 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 backreflection Laue pattern of an aluminum crystal rotated into
the orientation described above is shown in Fig. 824. Note that the
arrangement of spots has 2fold rotational symmetry about the primary
beam, corresponding to the 2fold 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 lighttight 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 rightangle prism. For xray protection, the optical system
should include lead glass, and the observer's hands should be shielded
during manipulation of the crystal.
86 The effect of plastic deformation. Nowhere have xray 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 xray
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
86]
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 xrays 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. 825.
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. 825. 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. 826, which applies to a facecentered 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 xray
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. 827,
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. 826. Lattice rotation during
slip in elongation of FCC metal crys
tal.
86]
THE EFFECT OF PLASTIC DEFORMATION
245
SPECIMEN AXIS
AFTER TWIST OF 2
AFTER TWIST OF 11
FIG. 827. 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 xray 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 xray 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 singlecrystal 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. 828. Laue photographs of a deformed aluminum crystal. Specimento
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 shortwavelength 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 backreflection Laue patterns made from the same
deformed region usually differ markedly in appearance. The photographs
in Fig. 828 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 backreflection
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. 828 (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. 829. Effect of lattice distortion
on the shape of a transmission Laue
spot. CN is the normal to the reflect
ing plane.
FIG. 830. Effect of lattice distortion
on the shape of a backreflection 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. 829 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 backreflection region, the situation is entirely different and the
spot S is roughly circular, as shown in Fig. 830. Both axes of the spot
subtend an angle of approximately 4c at the crystal. We may therefore
conclude that the shape of a backreflection 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 backwardreflected beam but
elliptical motion of the forwardreflected 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. 829,
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. 831. Formation of Debye arcs on Laue patterns of deformed crystals.
One feature of the backreflection pattern of Fig. 828 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. 69). 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. 831. 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. 828(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).
86]
THE EFFECT OF PLASTIC DEFORMATION
249
HKXT
POLYGOMZKD
FlG. 832.
(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 xray 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, reexamination of the patterns shown in Fig.
828 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 backreflection 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. 832. (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. 833. Enlarged transmission Laue
spots from a thin crystal of silicon fer
rite (airon 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 833
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 strainfree 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 xray beam. The appearance of such pat
terns is discussed and illustrated in Sec. 92.
87 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 compositionplane 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. 834(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. 834 (b), the AT and $poles repre
sent the reference line NS and the points A and B, located at an angle a
87]
RELATIVE ORIENTATION OF TWINNED CRYSTALS
251
trace of
composition
plane
(a)
FIG. 834. 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. 835. 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. 835(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. 835(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 836 Backreflection Laue photographs of two parts, A and B, of a twinned
crystal of copper. Tungsten radiation, 30 kv, 20 ma. Film covered with 0.01m.
thick aluminum to reduce the intensity of K fluorescent radiation from specimen.
87J
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 backreflection Lane photographs of Fig. 830 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.
837, 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. 83()(<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. 837. Projection of part A (open symbols) and part B (solid symbols) of
a twin in copper, made from Figs. 836(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. 211, 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 xrays. The trace normals are
87]
RELATIVE ORIENTATION OF TWINNED CRYSTALS
N
255
FIG. 838. 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
xray 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. 838 (a), where twin bands have formed on
two different {111} planes of the parent grain. The trace normals are
plotted in Fig. 838(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.
88 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. 839. 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. 839. Matrixprecipitate rela
tionship.
where the subscripts p and m refer to
precipitate and matrix, respectively.
88] RELATIVE ORIENTATION OF PRECIPITATE AND MATRIX 257
FIG. X40. 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. 840 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 FeC martensite, change with composition.)
The crystallographic problems presented by such structures are very
much the same as those described in Sec. 87, 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 rotatingcrystal method must
be used.
258 ORIENTATION OF SINGLE CRYSTALS [CHAP. 8
PROBLEMS
81. A backreflection Laue photograph is made of an aluminum crystal with
a crystaltofilm distance of 3 cm. When viewed from the xray 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.
82. A transmission Laue photograph is made of an aluminum crystal with a
crystaltofilm distance of 5 cm. To an observer looking through the film toward
the xray 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. 81, but use a stereographic ruler to plot the poles of reflecting
planes.
83. Determine the necessary angular rotations about (a) the incident beam
axis, (6) the eastwest axis, and (c) the northsouth axis to bring the crystal of
Prob. 82 into the "cube orientation/' i.e., that shown by Fig. 236(a).
84. With reference to Fig. 835(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?
86. 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
91 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 xray
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 singlephase
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
92 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 "grainsize
number."
Although xray 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
backreflection 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. 94 and 95).
* The nature of the changes produced in pinhole photographs by progres
sive reductions in specimen grain size is illustrated in Fig. 91. The gov
erning effect here is the number of grains which take part in diffraction.
This number is in turn related to the crosssectional 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. 91 (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 incidentbeam diameter,
multiplicity of the reflection, and specimenfilm distance. However, many
approximations are involved and the resulting equation is not very accu
93]
PARTICLE SIZE
261
(a)
(b)
(c)
(d)
FIG. 91. Backreflection 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 grainsize numbers, and to prepare
from these a standard set of photographs of the kind shown in Fig. 91.
The grainsize 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 xray diffraction is quite insensitive to varia
tions in grain size. I
93 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. 37, crystals in this size range cause broadening of the Debye
rings, the extent of the broadening being given by Eq. (313) :
B = ^L, (313)
*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 xray source (in diffractometers).
The breadth B in Eq. (313) refers, however, to the extra breadth, or
broadening, due to the particlesize 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. (91),
it can be inserted into Eq. (313) 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 0500A, but
very good experimental technique is needed in the range 5001000A. 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 backreflection 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
94] CRYSTAL PERFECTION 263
size of the individual crystals in a solid aggregate.! That is correct. At
tempts have been made to apply Eq. (313) to the broadened diffraction
lines from very finegrained 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 strainfree, provided the material involved is a
brittle (nonplastic) one, and all the observed broadening can confidently
be ascribed to the particlesize 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 xray 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 smallangle 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. Smallangle scattering has also been used to study precipitation
effects in metallic solid solutions. 
CRYSTAL PERFECTION
94 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 coldworked 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
xray reflection is illustrated in Fig.
92. 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 xray 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. 92. Effect of lattice strain
on Debyeline width and position.
A20
2 tan0,
d
(92)
94] 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 wellestablished, 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 particlesize 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 backreflection region,
an annealed metal produces a wellresolved 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 coldworked 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
graingrowth 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 coldworked
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. 93. 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 coldrolled, 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
94]
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. 93. Changes in hardness and diffraction lines of 7030 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 backreflection
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. 93, are repro
duced in Fig. 94. 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. 94. Diffractometer traces of the 331 line of the coldrolled and annealed
7030 brass specimens referred to in Fig. 93. Filtered copper radiation. Loga
rithmic intensity scale. All curves displaced vertically by arbitrary amounts.
95 DEPTH OF XRAY PENETRATION 269
FIG 95. Backreflection pinhole
patterns of coarsegrained 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. 93 (c) and (d).
It must always he remembered that a hackreflection photograph is
representative of only a thin surface layer of the specimen. For example,
Fig. 95 (a) was obtained from a piece of copper and exhibits unresolved
doublets in the highangle region. The unexperienced observer might
conclude that this material was highly cold worked. What the xray
"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 xray 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 coldworked
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. 95 (b), where the
sf>otty lines indicate a coarsegrained, recrystallized structure.
96 Depth of xray penetration. Observations of this kind suggest that
it might be well to consider in some detail the general problem of xray
penetration. Most metallurgical specimens strongly absorb xrays, 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 nonrepresentative 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. 95.
270 THE STRUCTURE OF POLYCRYSTALLINE AGGREGATES [CHAP. 9
is obtained in a backreflection camera of any kind, a SeemannBohlin
camera or a diffractometer as normally used. We have just seen how a
backreflection 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 xray 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 (72) 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, (72)
sin a
where the various symbols are defined in Sec. 74. 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. 74, "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
XX
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. (93) reduces to
G x = (1 
DEPTH OP XRAY PENETRATION
95]
which shows that the effective depth
of penetration decreases as 6 decreases
and therefore varies from one diffrac
tion line to another. In backreflec
tion cameras, a = 90, and
271
G x = [1 
(95)
03 1.0 1.5
x (thousandths of an inch)
FIG. 96. 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. 95 are M = 473 cm"" 1 and
26 = 136.7. By using Eq. (95), we
can construct the plot of G r as func
tion of x which is shown in Fig. 96.
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. 95 (a) discloses only
the presence of coldworked 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 grainsize number of 8.)
Equation (94) 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. (95) 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 91
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 91.
Calculations of the effective depth of penetration can be valuable in
many applications of xray diffraction. We may wish to make the effective
depth of penetration as large as possible in some applications. Then a
and ft in Eq. (93) must be as large as possible, indicating the use of high
angle lines, and ^ as small as possible, indicating shortwavelength 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 xray 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
96 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 colddrawn 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.
96] CRYSTAL ORIENTATION GENERAL 273
to the wire axis. In coldrolled 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. 86, 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 coldworked 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 coldworked 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, hotdipped 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 overall, 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 xray 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 coarsegrained 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.
96]
CRYSTAL ORIENTATION: GENERAL
R.D RD
275
TD
T D TDK
T.D
(a)
(b)
FIG. 97. (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. 97 (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. 97(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 facecentered
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. 97 (b) for the same pre
ferred orientation; in fact, it would consist of four "highintensity" 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 areatrue. 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 xray methods are used in which the diffraction effects
from thousands of grains are automatically averaged. The (hkl) pole
figure of a finegrained 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
DebyeScherrer camera or a diffractometer. As stated in Sec. 412, rela
tive line intensities are given accurately by Eq. (412) 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.
97 The texture of wire and rod (photographic method). As mentioned
in the previous section, colddrawn 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
97] THE TEXTURE OF WIRE AND ROD (PHOTOGRAPHIC METHOD) 277
F.A
reflection
circle
"V. reference
sphere
.Debye
ring
FIG. 98. 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 hotdipping, 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, colddrawn aluminum wire has a single [111] texture,
but copper, also facecentered 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. 98,
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. 99. 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.
(96)
These angles are shown stereographically in Fig. 99, projected on a plane
lormal to the incident beam. The (111) pole figure in (a) consists simply
97] 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. (96).
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. 910. 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. (310) 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. 910. Transmission pinhole
pattern of colddrawn 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. 88 or
by inspection of a table of interplanar angles. In this case, inspection of
Table 23 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. 99 (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 xray 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.
98 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 911 illus
trates this effect for coldrolled 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 912 shows the experimental arrangement and defines the angle
ft between the sheet normal and the incident beam. The intensity of the
98]
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. 911. Transmission pinhole patterns of coldrolled 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. 912). 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. 912. Section through sheet
specimen and incident beam (specimen FIG. 913. Measurement of azimuthal
thickness exaggerated). Rolling direc position of highintensity 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. 914. 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. 913, 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. 98 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. 914, which is
drawn for = 10. When the specimen is then rotated, for example by
40 in the sense shown in Fig. 912, the new position of the reflection circle
is found by rotating two or three points on the .reflection circle bv 40
98]
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. 914; since in this
example exceeds 0, part of the reflection circle, namely CD A, lies in the
back hemisphere. The arcs in Fig. 913 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. 913 becomes M 2 A^2 and
then, finally, M 3 7V 3 in Fig. 914. 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 915 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. 915. Plotting a pole figure.
FIG. 916. Hypothetical pole figure
derived from Fig. 915.
284
THE STRUCTURE OF POLYCRYSTALLINE AGGREGATES [CHAP, 9
R.D
= 5 (T.D. vertical). The arcs in Fig. 914 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. 916.
In practice, the variation of inten
sity around a Debye ring is not abrupt
but gradual, as Fig. 911 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
crosshatching, represent various de
grees of pole density from zero to a
maximum. Figure 917 is a photo
graphically determined pole figure in
which this has been done. It repre
T.D
FIG. 917. (Ill) pole figure of re
crystallized 7030 brass, determined
by the photographic method. (R. M.
Brick, Trans. A.I.M.E. 137, 193, 1940.)
sents the primary recrystallization texture of 7030 brass which has been
coldrolled 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. 917 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 highdensity
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
99]
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.
99 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 xray 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. 918 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. 918. Transmission method
for polefigure 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 verticalaxis
diffractometer.
286
THE STRUCTURE OF POLYCRY8TALLINE AGGREGATES
[CHAP. 9
FIG. 919. Specimen holder used in the transmission method, viewed from trans
mittedbeam 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 919 shows the kind of specimen holder used for this method.
The method of plotting the data is indicated in Fig. 920. 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. 98 to measure
azimuthal positions in a vertical plane.
99] THE TEXTURE OF SHEET (DIFFRACTOMETER METHOD)
R.D.
287
reflecting
plane *) /
T.D.
diffrartometer
axis
(a)
(b)
FIG. 920. Angular relationships in the transmission polefigure 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 xray 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 xrays 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. 920(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
NSaxis. 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.
74. The incident beam in Fig. 921 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 (0a)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. 921, 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. (97) 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 '. (98)
COS0
When a is not zero, the same integration gives
(0 a) _ e n
I D ( a = a ) =  1  . (99)
M[COS (0  a)/COS (0 + a) ~ 1]
* In Sec. 69 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. (98). If we put = a
= 0, then the primary beam will be incident on the specimen at right angles (see
Fig. 921), 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 //*.
99]
THE TEXTURE OP SHEET (DIFFRACTOMETER METHOD)
289
10 20 30 40 50 60 70 80
ROTATION ANGLE a (degrees)
FIG. 921. Path length and irradi
ated volume in the transmission method.
FIG. 922. 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 .. ^ : : (910)
J D (a = 0)
'[cos (6  a) /cos (6 + a)  1]
A plot of R vs. a is given in Fig. 922 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. (910) 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. 923. Reflection method for polefigure 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. 923. 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 xray 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 924 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. 923. The angle a is zero when
99]
THE TEXTURE OF SHEET (DIFFRACTOMETER METHOD)
291
FIG. 924. Specimen holder used in the reflection method, viewed from re
flectedbeam 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. 920(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. 923). 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. 925. (Ill) pole figure of coldrolled 7030 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. 925, which represents the deformation texture of
7030 brass coldrolled to a reduction in thickness of 95 percent. The
numbers attached to each contour line give the pole density in arbitrary
99] 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. 925), 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. 925, the solid triangular symbols representing the ideal orienta
tion (110) [lT2] lie approximately in the highdensity 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. (910) 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 polefigure 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 xray 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 coarsegrained 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.
Polefigure 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 counterratemeter 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 polefigure 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 semiautomatic
manner, at least in the larger laboratories.
TABLE 92
Appearance of diffraction lines
Condition of specimen
Continuous
Spotty
Narrow (1)
Broad (1)
Uniform intensity
Nonuniform intensity
Finegrained (or coarsegrained and
coldworked)
Coarsegrained
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.
910] SUMMARY; PROBLEMS 295
910 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 92.
PROBLEMS
91. A coldworked 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?
92. If the observed broadening given in Prob. 91 is ascribed entirely to a frag
mentation of the grains into small crystal particles, what is the size of these par
ticles?
93. For given values of 6 and /x, which results in a greater effective depth of
xray penetration, a backreflection pinhole camera or a diffractometer?
94. Assume that the effective depth of penetration of an xray 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
lowcarbon steel specimen under the following conditions:
(a) Diffractometer; lowestangle reflection; Cu Ka radiation.
(6) Diffractometer; highestangle reflection; Cu Ka radiation.
(c) Diffractometer; highestangle reflection; Cr Ka radiation.
(d) Backreflection pinhole camera; highestangle reflection; Cr Ka radiation.
96. (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 onefourth of the total thickness. Calculate W for each layer.
96. 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 highintensity maxima will appear on the lowestangle 110 Debye ring and
what are their azimuthal angles on the film?
CHAPTER 10
THE DETERMINATION OF CRYSTAL STRUCTURE
101 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, solidstate 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. 413; 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. 108. 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 xray crystallography, who can bring
special techniques, both experimental and mathematical, to bear on the
problem. The metallurgist should, however, know enough about structure
102] 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.
102 Preliminary treatment of data. The powder pattern of the un
known is obtained with a DebyeScherrer camera or a diffractometer, the
object being to cover as wide an angular range of 26 as possible. A camera
such as the SeemannBohlin, 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 xrays 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 xrays 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 xrays 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 , (101)
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. (101) 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 xray tube, particularly if the tube is old. If such contamina
tion is suspected, equations such as (101) 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. 101. 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 101 shows a correction curve of this kind, obtained with a par
ticular specimen and a particular DebyeScherrer 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. 101,
the errors shown are to be subtracted from the observed values.
* For the shape of this curve, see Prob. 115.
103] INDEXING PATTERNS OF CUBIC CRYSTALS 301
103 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 planespacing 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. (102) 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 101. 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 highestangle 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 101
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 101 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 41, 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, ...
Bodycentered cubic: 2, 4, 6, 8, 10, 12, 14, 16, ...
Facecentered 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. 102, 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
closepacked 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
100
002
101
102
110
103
200
112
201
004
202
104
203
210
211
114
,S
1
2
3
4
i
s
*
_ 9
<*<
S 10
* n
12
^ 13
* H
H
16
17
IS
19
20
 in ^
FIG. 102. 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 bodycentered cubic patterns, but the former
contains almost twice as many lines, while a facecentered 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 bodycentered 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 bodycentered 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.
104 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 planespacing equa
tion for this system involves two unknown parameters, a and c:
I h 2 + k 2 I 2
5 + T (103)
d 2 a 2 c 2
This may be rewritten in the form
i ![(* + *") + _
d 2 a 2 L (c/o)
104] NONCUBIC CRYSTALS (GRAPHICAL METHODS) 305
or
r i 2 i
2 log d = 2 log a  log (h 2 + k 2 ) + (104)
L (c/a) 2 J
Suppose we now write Eq. (104) 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 HullDavey chart is illustrated in Fig. 103. First,
the variation of the quantity [(/i 2 + k 2 ) + l 2 /(c/a) 2 ] with c/a is plotted
on tworange 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 bodycentered
tetragonal lattice is made simply by omitting all curves for which
(h + k + I) is an odd number.] A singlerange 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. (104) 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. 103,
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
104 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. 103, 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 highestangle lines
are used to set up two equations of the form of Eq. (103), 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 103 is only a partial HullDavey chart. A complete one, show
ing curves of higher indices, is reproduced on a small scale in Fig. 104,
which applies to bodycentered 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 highangle lines on the basis of a and c values derived from
the already indexed lowangle lines.
Some HullDavey charts, like the one shown in Fig. 104, 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. (104) can be set up in terms of sin 2 8
rather than d, by combining Eq. (103) 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 tworange logarithmic one (from 0.01 to 1.0),
equal in length to the tworange [(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. 103.
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 HullDavey chart ii
308
THE DETERMINATION OF CRYSTAL STRUCTURE [CHAP. 10
0.01  I IMimmilIIMMM[IIMIIMIIM[llllIMI[IIM[UIIMllIIMIMII[llllllllllllllll
. ..I.. ..I I.. ..I.. ,,!..,. I ,,,!, ,,,!,,,, I,,,, I,,,, , ,,,l.i i.li ml. 1I.J
FIG.
AXIAL RATIO
104. Complete HullDavey chart for bodycentered tetragonal lattices.
104] 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 HullDavey 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 HullDavey 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 planespacing 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. (104) for the tetragonal system.
A HullDavey 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 closepacked lattices may also be prepared by omitting all
curves for which (h + 2k) is an integral multiple of 3 and I is odd.
Figure 313(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 102, A fit was obtained on a Hull
Davey chart for hexagonal closepacked 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 (103 and 110)
almost intersect at c/a = 1.87, so the observed line is evidently the sum
of two lines, almost overlapping, one from the (103) planes and the other
from (11 0) planes. The same is true of line 11. Four lines on the chart,
namely, 200, 104, 210, and 204, 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 102
[CHAP. 10
Line
Intensity
sin 2
hkl
1
s
0.097
002
2
s
0.112
100
3
vs
0.136
101
4
m
0.209
102
5
s
0.332
103, 110
6
vw
0.390
004
7
m
0.434
112
8
m
0.472
201
9
vw
0.547
202
10
w
0.668
203
11
m
0.722
114, 105
12
m
0.806
211
13
w
0.879
212
the lattice of zinc is actually hexagonal closepacked. The next step is to
calculate the lattice parameters. Combination of the Bragg law and the
planespacing 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 highestangle 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. 24, any rhombohedral crystal may be referred to hexagonal axes. A
hexagonal HullDavey 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 twoparameter 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
105] 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. 102, 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 HullDavey
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.
105 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 , (107)
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. (107)
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. (107) in the form
k 2 ) = Cl 2 .
Differences represented by the lefthand 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 102.
We first divide the sin 2 8 values by the integers 1, 3, 4, etc., and tabulate
the results, as shown by Table 103, 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 103
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
105]
NONCUBIC CRYSTALS (ANALYTICAL METHODS^
TABLE 104
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 104.
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 rotatingcrystal 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 singlecrystal 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 singlecrystal 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 singlecrystal methods will find them described
in some of the books listed in Chap. 18.
106 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 105 graphically illustrates this point. On the left is the calcu
lated diffraction pattern of the bodycentered 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
106] 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. 105. 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 (bodycentered cubic) and
cementite (FeaC, orthorhombic). When the same steel is quenched from
the austenite region, the phases present are martensite (bodycentered
tetragonal) and, possibly, some untransformed austenite (facecentered
cubic). The a and c parameters of the martensite cell do not differ greatly
from the a parameter of the ferrite cell (see Fig. 125). 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 coppergold 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. 133).
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. 105, 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.
107 Determination of the number of atoms in a unit cell. To return
to the subject of structure determination, the next step after establishing
108] 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. (39),
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 NiAl system are wellknown examples.
108 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\ (411)
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. (412). But \F\ meas
ures only the relative amplitude of each reflection, whereas in order to use
Eq. (411) 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 muchsoughtafter 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. 219 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
108J 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 spacegroup 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 spacegroup 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, spacegroup 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 xrays 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 xray 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
109 Example of structure determination. As a simple example, we
will consider an intermediate phase which occurs in the cadmiumtellurium
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 DebyeScherrer
camera and Cu Ka radiation.
The observed values of sin 2 6 for the first 16 lines are listed in Table 105,
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 highestangle 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. (39), 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 105 for evi
dence of the Bravais lattice. Since die indices of the observed lines are all
TABLE 105
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
109] EXAMPLE OF STRUCTURE DETERMINATION 321
unmixed, the Bravais lattice must be facecentered. (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 facecentered.) 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. 218(b)] and the zincblende form
of ZnS [Fig. 219(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. 46] 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. 413) 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, (109)
\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. (412), we can almost rule out the NaCl structure
as a possibility simply by inspection of Eqs. (108). 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 106,
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 lowangle 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 106
[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 CdTe interatomic
distance is therefore \/3 a/4 = 2.80A. For comparison, we can calcu
late a " theoretical" CdTe 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 CdTe 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
AB 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
101. 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.
102. 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.
103. Construct a HuilDavey chart, and accompanying sin 2 6 scale, for hex
agonal closepacked lattices. Use tworange semilog graph paper, 8j X 11 in.
Cover a c/a range of 0.5 to 2.0, and plot only the curves 002, 100, 101, 102,
and 110.
104. Use the chart constructed in Prob. 103 to index the first five lines on the
powder pattern of atitanium. 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.
105 106 107 108
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
111 Introduction. Many applications of xray 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 hightemperature 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. 111 or a table of sines will show.
For this reason, a very accurate value
324
20
40 60
6 (degrees)
80
FIG. 111. 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. (H1)
d
In the cubic system,
a = d Vh 2 + k 2 + I 2 .
Therefore
Aa Arf ,  rts
_ = _ =  cot0A0. (112)
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 backwardreflected 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 highestangle 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. 34.
112 DebyeScherrer 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 DebyeScherrer camera, the chief
sources of error in 6 are the following:
(1) Film shrinkage.
(2) Incorrect camera radius.
(3) Offcentering of specimen.
(4) Absorption in specimen.
Since only the backreflection region is suitable for precise measurements,
we shall consider these various errors in terms of the quantities S' and 0,
defined in Fig. 112. S f is the distance on the film between two correspond
ing backreflection 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
(H3)
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. (113) in logarithmic
form:
In <f> = In S'  In 4  In R.
Differentiation then gives
A< AS' Aft
= (114)
4> S' R
The error in <j> due to shrinkage and
the radius error is therefore given by
^AS' &R\
U. (115)
R '
FIGURE 112
112]
DEBYESCHERRER CAMERAS
(a) v ,,,
FIG. 113. 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. 65 (a) is not at all suitable for precise measurements.
Instead, methods (b) or (c) of Fig. 65 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 offcenter 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. 113 (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>. (116)
The effect of a specimen displacement at right angles to the incident beam
[Fig. ll3(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 rightangle displacement.
The total error in S' due to specimen displacement in some direction in
clined to the incident beam is therefore given by Eq. (116). 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. (114) becomes
*
S'
(H7)
328 PRECISE PARAMETER MEASUREMENTS [CHAP. 11
which shows how an error in S' alone affects the value of <t>. By combining
Eqs. (113), (ll), and (117), 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 </>. (118)
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 offcenter 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. 418(b), that backreflected 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. (118).
Thus, the overall error in </> due to film shrinkage, radius error, centering
error, and absorption, is given by the sum of Eqs. (115) and (118):
/AS' A#\ Ax
A<te,/2,c,A = I ) <t> + sm </> cos </>. (1 19)
\ O K / 1
But
= 90 0, A0 = A0, sin <t> = cos 0, and cos <f> = sin 0.
Therefore Eq. (112) becomes
Ad cos sin
= : A0 = A</>
d sin cos <t>
and
Ad sin^r/AS' A#\ Ax 1
= ( )<H sin cos (1110)
d cos < L \ S' R I R J
In the backreflection region, < is small and may be replaced, in the second
term of Eq. (1110), by sin< cos<, since sin< <f> and cos</> 1, for
112] DEBYESCHERRER 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, (1111)
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. (1112)
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.
(1 112), 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 backreflection region, it is common practice to
employ unfiltered radiation so that K/3 as well as Ka can be reflected. If
the xray 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 strainfree 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 DebyeScherrer camera 5.73 cm in diameter
with unfiltered copper radiation. The data for all lines having values
330
PRECISE PARAMETER MEASUREMENTS
TABLE 111
[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 111. 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. 114, 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. (1110) 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. 114. 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 backreflection 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
112] DEBYESCHERRER 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 lowangle 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
(1113)
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 highestangle lines, as was done
in Sec. 104. The approximate axial ratio Ci/a\ is then calculated and
used in Eq. (1113) to determine an a value for each highangle 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. (1113) 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.
(1114) will show.
332
PRECISE PARAMETER MEASUREMENTS
[CHAP. 11
cos 2 e (] cos 2 e
(a) (b)
FIG. 115. 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
DebyeScherrer 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 15. 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
114] 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.
113 Backreflection 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 DebyeScherrer 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.
114 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 specimentofilm 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
116 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 backreflection 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 backreflected 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 DebyeScherrer 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.
116] METHOD OF LEAST SQUARES 335
116 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 leastsquares 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. (1115)
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. (1115) 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 + . (1 116)
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. (1116) 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. (1117)
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. (1118)
Equations (1117) and (1118) 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. (1115) to give the equation of the line.
116] METHOD OF LEAST SQUARES 337
The normal equations as written above can be rearranged as follows:
Zt/ = Sa + 62x
and
(1119)
A comparison of these equations and Eq. (1115) 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. (1115). 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. 116, to
gether with the four given points.
The leastsquares 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. 116. Best straight line, de
termined by leastsquares method.
Since there are three unknown constants here, we need three normal equa
tions. These are
Si/ = Sa + b2x + cSx 2 ,
(1120)
2x 2 y  aZz 2 + blx* + cSx 4 ,
These normal equations can be found by the same methods as were used
for the straightline 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 leastsquares 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 leastsquares 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.
117 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
117] COHEN'S METHOD 339
experimental points. M. U. Cohen proposed, in effect, that the leastsquares
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 DebyeScherrer camera.
Then Eq. (1112), namely,
Ad Aa
= = #cos 2 0, (1112)
d a
defines the extrapolation function. But instead of using the leastsquares
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. (1112) 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, (1122)
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. (1122) 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, (1123)
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. (1123) for each of the n backreflection 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. (1123) 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. 610. Since
this pattern was made with a symmetrical backreflection focusing camera,
the correct extrapolation function is
Ad
= K<t> tan <t>.
d
Substituting this into Eq. (1121), 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, (1124)
4a 2
C0 s 2 = Ca + A5, (1125)
where
C = X 2 /4a 2 , a = (h 2 + k 2 + I 2 ), A = D/10, and 8 = 100 sin 20.
Equation 1124 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. (1124)
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
117]
COHEN'S METHOD
TABLE 112
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 (1126) 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. (1125) 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 112 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. (1125)
only in the last term which is very small compared to the other two. From
the data in Table 112, 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 12, 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 DebyeScherrer camera. By rearranging this
equation and introducing new symbols, we obtain
sin 2 6 = Ca + By + ,46, (1127)
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 .
118 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. 67. 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 backreflection 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
111. 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
112. The following data were obtained from a DebyeScherrer 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.
113. From the data given in Prob. 112, determine the lattice parameter to
four significant figures by Cohen's method.
114. From the data given in Table 112, determine the lattice parameter of
tungsten to five significant figures by graphical extrapolation of a against <j> tan <t>.
115. 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. 101. At approximately what value of 6 does the maximum occur?
CHAPTER 12
PHASEDIAGRAM DETERMINATION
121 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
xray methods can be used in the study of phase diagrams, particularly of
binary systems. Ternary systems will be discussed separately in Sec. 126.
Xray 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. Xray 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 solidstate reactions or the small heat effects involved.
Such features of the diagram are best determined by microscopic examina
tion or xray 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
PHASEDIAGRAM DETERMINATION
[CHAP. 12
liquid
122 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 twophase 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. 121. 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 Brich solid
solution with the same structure as B,
as in the alloy system illustrated by Fig. 122(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 Brich solid
solution, as in the system shown in Fig. 122(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 twophase 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. 121. Phase diagram of two
metals, showing complete solid solu
bility.
122]
GENERAL PRINCIPLES
347
liquid
PERCENT B
(a)
PERCENT B
(b)
FIG. 122. 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 singlephase and twophase regions
alternately.
(3) Singlephase regions. In a singlephase 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) Twophase regions. In a twophase 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 twophase
field. Thus, in the system illustrated in Fig. 122(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 PHASEDIAGRAM 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. 123. This system contains two substitutional ter
minal solid solutions a and p, both assumed to be facecentered cubic, and
an intermediate phase 7, which is bodycentered 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. 123. 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 twophase (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 twophase (7 + ft)
field.
Calculated powder patterns are shown in Fig. 124 for the eight alloys
designated by number in the phase diagram of Fig. 123. 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 (facecentered 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 (bodycentered 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 (facecentered 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. 124 and noting which lines are common to
122]
GENERAL PRINCIPLES
349
PERCENT B *
FIG. 123. Phase diagram and lattice constants of a hypothetical alloy system.
26 = 26 = 180
(2)
1
FIG. 124. Calculated powder patterns of alloys 1 to 8 in the alloy system shown
in Fig. 123.
350 PHASEDIAGRAM 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 xray methods usually begins with a
determination of the roomtemperature 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 stressfree to give sharp diffraction lines. Filed pow
ders, however, must be reannealed to remove the stresses produced by
plastic deformation during filing before they are ready for xray examina
tion. Only relatively low temperatures are needed to relieve stresses, but
the filings should again be slowly cooled, after the stressrelief anneal, to
ensure equilibrium at room temperature. Screening is usually necessary
to obtain fine enough particles for xray examination, and when twophase
alloys are being screened, the precautions mentioned in Sec. 63 should be
observed.
After the roomtemperature 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 DebyeScherrer 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 roomtemperature 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.
123] 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 hightemperature
will decompose on cooling to room temperature, no matter how rapid the
quench, and such phases can only be studied by means of a hightempera
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 hightemperature phase is unstable, such as a
eutectoid temperature, can be determined by setting the diffractometer
counter to receive a prominent diffracted beam of the hightemperature
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.
123 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
PHASEDIAGRAM 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. 125. 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 facecentered cubic yiron, the addition of carbon increases the cell
edge a. But in martensite, a supersaturated interstitial solid solution of
carbon in airon, the c parameter of the bodycentered tetragonal cell in
creases while the a parameter decreases, when carbon is added. These
effects are illustrated in Fig. 125.
The density of an interstitial solid solution is given by the basic density
equation
1.660202^1 , ^
p . (39)
where
n l A l ]
(121)
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 126 shows examples of both positive and
negative deviations from Vegard's law among solutions of facecentered
cubic metals, and even larger deviations have been found in hexagonal close
123]
SOLID SOLUTIONS
353
Ni
40 (>() 80
ATOMIC PERCENT
l(H)
FIG. 126. Lattice parameters of some continuous solid solutions. Dotdash
lines indicate Vegard's law. (From Structure of Metals, by C. S. Barrett, 1952,
McGrawHill 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.
(39) with the 2A factor being given by
^solvent^solvent I
(122)
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 xray density calculated according to
Eq. (121) or that calculated according to Eq. (122) 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 bestknown example is the intermediate ft solution in the nickelalu
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
PHASEDIAGRAM 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 xray density calculated according to Eq. (122) will no longer agree
with the direct density simply because Eq. (122), 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 xray
density, calculated according to Eq. (122), and an analysis of the dif
fracted intensities.
124 Determination of solvus curves (disappearingphase 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 singlephase solid
region and a twophase solid region, and the singlephase 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. 127 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. (123) 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 twophase region
(123)
PS
w
WEIGHT PERCENT B *>
FIG. 127. Leverlaw construction
for finding the relative amounts of two
phases in a twophase field.
* The reasons for nonlinearity are discussed in Sec. 149.
124] SOLVUS CURVES (DISAPPEARINGPHASE 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 disappearingphase 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
disappearingphase method is therefore governed by the sensitivity of the
xray 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 xray 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 twophase 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 disappearingphase xray method is practically worthless.
On the whole, the microscope is superior to xrays 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 disappearingphase method increases as the width of the twophase re
gion decreases. If the (a + ft) region is only a few percent wide, then the
356
PHASEDIAGRAM 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
xray 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 xray scattering powers.
126 Determination of solvus curves (parametric method). As we have
just seen, the disappearingphase 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. 128(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. 128(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. 128. Parametric method tor determining a solvus curve.
125] 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. 128(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
parametercomposition curve of the solid solution has been established.
Only one twophase 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 parametercomposition 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 parametercomposition curve, branch be of Fig. 128(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 129 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
pearingphase method, whether based on xray measurements or micro
scopic examination, in the determination of solvus curves at low tempera
tures. As mentioned earlier, both xray 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
PHASEDIAGRAM 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. 129. Solvus curve determination in the copperantimony 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 xray 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 highangle 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 parametercomposition 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.
126]
TERNARY SYSTEMS
359
126 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 xray methods described above, based on either the
disappearingphase 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 threedimensional
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. 1210, 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 twophase 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 singlephase region the composition of the phase involved, say a, is
continuously variable. In a twophase region tie lines exist, just as in
A c
FIG. 1210. Isothermal section of
hypothetical ternary diagram.
360
PHASEDIAGRAM 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. 1210, tie
lines have been drawn to connect the singlephase compositions which are
in equilibrium in the twophase 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 threephase field, the compositions of the phases are fixed and are
given by the corners of the threephase triangle. Thus the compositions
of a, 0, and 7 which are at equilibrium in any alloy within the threephase
field of Fig. 1210 are given by a, 6, and c, respectively. To determine the
8
fa
<
along nhc
PERCENT A
PERCENT A
(c)
FIG. 1211. 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 disappearingphase method of locat
ing the sides and corners of the threephase 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. 1211 (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 parametercomposition curve would then look like Fig.
12ll(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 parametercomposition curve to resemble Fig. 121 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 onephase 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 threephase field the parameter of a will be
constant and equal to the parameter of saturated a of composition h. The
parametercomposition curve will therefore have the form of Fig. 12ll(b).
PROBLEMS
121. 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 singlephase 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 twophase 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?
122. The twophase alloy mentioned in Prob. 121, after being quenched from
a series of temperatures, contains a having the following measured parameters:
362 PHASEDIAGRAM 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
ORDERDISORDER TRANSFORMATIONS
131 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. 27, 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 longrange 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 disorderorder transformation is a
particular kind of change in the xray diffraction pattern of the substance.
Evidence of this kind was first obtained by the American metallurgist Bain
in 1923, for a goldcopper solid solution having the composition AuCua.
Since that time, the same phenomenon has been discovered in many other
alloy systems.
132 Longrange 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 facecentered cubic lattice, as illus
trated in Fig. 131 (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" goldcopper 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
facecentered positions, as illustrated in Fig. 131 (b). Both structures are
cubic and have practically the same lattice parameters. Figure 132 shows
ORDERDISORDER TRANSFORMATIONS
[CHAP. 13
gold atom
copper atom
V_y ' 'average"
goldcopper atom
(a) Disordered
(b) Ordered
FIG. 131. 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"
goldcopper 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. 132. Atom arrangements on a (100) plane, disordered and ordered AuCu 3.
132] LONGRANGE ORDER IN AuCu 3 365
By example (d) of Sec. 46, 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 facecentered 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,
(131)
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 facecentered 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. 131.
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. (131) show, all the superlattice lines are much
weaker than the fundamental lines, since their structure factors involve
366
ORDERDISORDER TRANSFORMATIONS
[CHAP. 13
/ 1
111 200 220
/ / /
/ I /\
KM) 110 210 211
FIG. 133. Powder patterns of AuCiis (very coarsegrained) 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. 133, where / and s are
used to designate the fundamental and superlattice lines, respectively.
At low temperatures, the longrange 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 longrange order parameter
S, defined as follows:
S =
i F
(132)
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 longrange 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. (132), 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 longrange order present. The same result is ob
tained if we consider the distribution of copper atoms.
132]
LONGRANGE ORDER IN AuCu 3
367
Any departure from perfect longrange 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.
(133)
i o
08
Of)
04
02
s AuOus
Comparing these equations with Eqs. (131), 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. 133. 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. 134 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. 412, 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 nonBragg 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. 134. 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
ORDERDISORDER TRANSFORMATIONS
[CHAP. 13
200
AuOu
Ou
10 20 30 40 50 60
ATOMIC 1 PERC 1 KNT Au
70
90
KK)
Au
FIG. 1 35. Phase diagram of the goldcopper system. Twophase 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, temperaturediffuse scattering, etc. It is worth noting, how
ever, that Eq. (134) 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. 135.
133]
OTHER EXAMPLES OF LONGRANGE ORDER
369
Another aspect of longrange 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. 135. (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 facecentered 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.
133 Other examples of longrange order. Before considering the or
dering transformation in AuCu, which is rather complex, we might examine
the behaviour of /3brass. 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, bodycentered 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. 136.
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"
copperzinc atom
(a) Disordered (b) Ordered
FIG. 136. Unit cells of the disordered and ordered forms of CuZn.
* NiAl is the ft phase referred to in Sec. 123 as having a defect lattice at certain
compositions.
370 ORDERDISORDER TRANSFORMATIONS [CHAP. 13
undergo an orderdisorder 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 0brass, 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 134 indicates how the degree of longrange 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 disorderorder 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. 133.
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. 134, was necessarily based on measure
ments made at temperature with a hightemperature diffract ometer.)
Not all orderdisorder transformations are as simple, crystallographically
speaking, as those occurring in AuCu 3 and CuZn. Complexities are en
countered, for example, in goldcopper 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. 135). 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. 137 (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
133]
OTHER EXAMPLES OF LONGRANGE ORDER
371
(a) "(I)Utragonal
(h) a" ( 1 1 loithorhombic
FIG. 137. Unit cells of the two ordered forms of AuCu.
in Fig. 137 (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 disorderorder 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. 24). In the goldcopper system, the disordered phase a is cubic,
because the arrangement of gold and copper atoms on a facecentered 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 ORDERDISORDER TRANSFORMATIONS [CHAP. 13
positions in each cell (Fig. 131), 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 threefold rotational symmetry about directions of
the form (111). Inasmuch as this is the minimum symmetry requirement
for the cubic system, this cell [Fig. 137 (a)] is not cubic. There is, how
ever, fourfold 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. 137 (a) would still be classified as
tetragonal on the basis of its symmetry.
134 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 Lorentzpolarization 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. (131) 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. (136) becomes, for small
scattering angles, _ ^
zz 0.09.
I f [79 + 3(29)] 2
Superlattice lines are therefore only about onetenth as strong as fundamen
tal lines, but they can still be detected without any difficulty, as shown by
Fig. 133.
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 xray 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. 138. Variation of A/ with X/X/t. (Data from R. W. James, The Optical
Principles of the Diffraction of XRays, G. Bell and Sons, Ltd., London, 1948, p. 608.)
number by only one or two units, because the superlatticeline 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. 43 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 138 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. 138, 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
ORDERDISORDER 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. 138
shows that the corrections are 3.6 and 2.7, respectively. The super
latticeline 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. 139.
135] SHORTRANGE ORDER AND CLUSTERING 375
By thus taking advantage of this anomalous change in scattering factor
near an absorption edge, we are really pushing the xray 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 superlatticeline intensity.
135 Shortrange order and clustering. Above the critical tempera
ture T c longrange 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 shortrange
order.
For example, when perfect longrange order exists in AuCu 3 , a gold atom
located at is surrounded by 12 copper atoms at f \ and equivalent
positions (see Fig. 131), and any given copper atom is likewise surrounded
by 12 gold atoms. This kind of grouping is a direct result of the existing
longrange order, which also requires that gold atoms be on corner sites
and copper atoms on facecentered 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 facecentered 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 shortrange 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 shortrange order
above that temperature. Above T c the degree of shortrange 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 shortrange order is that it has also been found to exist in solid
solutions which do not undergo longrange or4ering at low temperatures,
such as goldsilver and goldnickel 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 aluminumsilver
and aluminumzinc solutions. In fact, there is probably no such thing as
a perfectly random solid solution. All real solutions probably exhibit either
shortrange ordering or clustering to a greater or lesser degree, simply be
376
ORDERDISORDER TRANSFORMATIONS
[CHAP. 13
04 0.8 12 Hi 20 24 2 S 3.2 3 (>
FIG. 1310. Calculated intensity /D of diffuse scattering in powder patterns of
solid solutions (here, the facecentered cubic alloy Xi 4 Au) which exhibit complete
randomness, shortrange order, and clustering. The shortrange 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 shortrange order or clustering may be defined in terms of
a suitable parameter, just as longrange 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. 1310, 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. 1310 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. (134). If shortrange 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 longrange 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. 1310 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 temperaturediffuse and Compton modified,
which are always present.
PROBLEMS
131. A DebyeScherrer 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 Lorentzpolarization factors
for these two lines and the corrections to the atomic scattering factors mentioned
in Sec. 134.)
132. 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. 138.
The lattice parameter of /3brass (CuZn) is 2.95A.
133. (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
141 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
143] 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
142 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
patternbypattern 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.
143 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 fivedigit code number: xxxxx. 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 30167 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, secondstrongest, 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. 141. 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 11188. 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 righthand portion of the card lists the values of d and
///i for all the observed diffraction lines.
143]
QUALITATIVE ANALYSIS! THE HANAWALT METHOD
381
)27B
d
2.28
1.50
1.3S
2.60
MO.C
11194
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. 141. 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 crossindexed; 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
(secondstrongest line first and thirdstrongest 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 DebyeScherrer 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 DebyeScherrer
camera, the lowangle 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. 63.
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 xray beam for known lengths of
time. (Many of the intensity data in the ASTM file, including the values
shown for molybdenum carbide in Fig. 141, were obtained in this way.)
144]
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.
144 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 141. It was obtained
with Mo Ka radiation and a DebyeScherrer 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 142, 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 141
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 142
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
21129
2.28
1.50
1.35
100
35
35
Mo 2 C Molybdenum Carbide
11188
2.28
1.49
1.10
100
100
100
Cs Ir(NOJ z Cesium Iridium
3 2 6 Nitrite
21130
2.27
1.49
1.07
100
60
60
aW 7 C Alpha Tungsten
* Carbide 2:1
21134
with the observed intensities. We then refer to the data card bearing
serial number 11188, reproduced in Fig. 141, 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 143, 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
secondstrongest line (d = 2. 47 A) are formed by two different phases, and
that the thirdstrongest 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 40836, and three lines in the pattern of
our unknown. Turning to card 40836, we find good agreement between
all lines of the copper pattern, described in Table 144, with the starred
lines in Table 143, 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 145. 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
144]
EXAMPLES OF QUALITATIVE ANALYSIS
385
TABLE 143
PATTERN OF UNKNOWN
TABLE 144
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 145. 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 145
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 machinesorted. 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.
145 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 DebyeScherrer 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
highangle lines, on a diffractometer chart than on a DebyeScherrer 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 DebyeScherrer 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 XRay
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.
146] 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. 124, 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.
146 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 xrays into most metals and alloys, as discussed at length
in Sec. 95. 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 hotdipping, 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)
147 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. 128(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 parametercomposition 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)
148 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 DebyeScherrer camera and microphotometer, the mod
148] 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 singlephase 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 = crosssectional 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. 411), 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. (141), containing the square of the structure
factor, the multiplicity factor, and the Lorentzpolarization factor, will
be recognized as the approximate equation for relative integrated inten
sity used heretofore in this book.]
We can simplify Eq. (141) 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. (141) 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 = (142)
Mm
where KI is a constant.
To put Eq. (142) in a useful form, we must express M in terms of the
concentration. From Eq. (112) 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 (143) 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. (144) 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 (145)
Wa/Pa + V>P/P0
77). //>_
(146)
Pa  1/P0)
Combining Eqs. (144) and (146) and simplifying, we obtain
/.  __ (147)
Pa[u>a (Palp*  M0/P0) + M0/P0]
For the pure a phase, either Eq. (142) or (147) gives
I ap = ^ (148)
ap
Ma
where the subscript p denotes diffraction from the pure phase. Division
of Eq. (147) by Eq. (148) eliminates the unknown constant KI and
gives
lap Wa(va/Pa ~ M/8/P/?) + M/3/P/3
This equation permits quantitative analysis of a twophase 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.
149]
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. 142. 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. (149). The
agreement is excellent. The line
obtained for the quartzcristobalite
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. (149) be
comes simply o 05 1 o
WK1GHT FRACTION OF
j QUARTZ W(l
  = w a .
lap FIG. 142. Diffractometer meas
urements made with Cu Ka radiation
Fig. 142 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.
149 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 xray method, on the other
hand, is quite accurate in this lowaustenite 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 bodycentered 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. (141), we put
\32
(1410)
The diffracted intensity is therefore given by
/ = ^, (1411)
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. (1411) for a particular diffraction line of each phase:
/, =
7
149] QUANTITATIVE ANALYSIS! DIRECT COMPARISON METHOD 393
/Y2/t a C a
7a= ~^r
Division of these equations yields
p /.
(1412)
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 austenitemartensite 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. (1412) 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 143 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 002200 and 112211
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. 144. (Figure 144 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 disappearingphase xray method of lo
cating a solvus line (Sec. 124), we note from Eq. (1412) 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. 143. 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
tillerquenched and
then cooled to 321F
2 9'V austenite
v\atei quenched
V*^^^ XvHrtv**
9 3 r (, austenite
quenched to 125F,
aircooled to room temperature
14 \ c ' (l austenite
FIG. 144. Microphotometer traces of DebyeScherrer 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.)
149]
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. 138), particularly when Co Ka radiation is used. The Lo
rentzpolarization factor given in Eq. (1410) applies only to unpolarized
incident radiation; if crystalmonochromated radiation is used, this factor
will have to be changed to that given in Sec. 612. The value of the tem
perature factor e~ 2M can be taken from the curve of Fig. 145.
1 U
09
OS
07
0(i
""""
"^
^
^
N
\
\
1 .2 3 4 5 7 8
FIG. 145. 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 xray 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 xray 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
crystalmonochromated 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 146
COMPARISON OF AUSTENITE DETERMINATION BY XRAY 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 146 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 xray diffraction and by quantitative microscopic examination (lineal
analysis). The steel was austenitized for 30 minutes at the temperatures
indicated and quenched in oil. The xray results were obtained with a
DebyeScherrer camera, a stationary flat specimen, and crystalmonochro
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 xray 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 xray method is definitely more accurate.
1410 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. (142)
the intensity of a particular line from phase A is given by
, K S CA'
1410] 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
= (1413)
(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. (145) 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. (1413) gives
(1414)
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&). (1415)
Combination of Eqs. (1414) and (1415) gives
^ = K,w A . (1416)
^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. 146. 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 146 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. (1416).
Strictly speaking, Eq. (1416) 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.
1411 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. (141), 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
1411] 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. 37, 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
(141) is derived on the basis of the socalled "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. (141). 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. (1413), and therefore the constant K Q in Eq. (1416), 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.
141. 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
142.
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
143.
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
144. 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
146. Microscopic examination of a hardened 1 .0 percent carbon steel shows no
undissolved carbides. Xray 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 112211 martensite
doublet is 16.32, both in arbitrary units. Calculate the volume percent austenite
in the steel. (Take lattice parameters from Fig. 125, A/ corrections from Fig.
138, and temperature factors e~ 23f from Fig. 145.)
CHAPTER 15
CHEMICAL ANALYSIS BY FLUORESCENCE
161 Introduction. We saw in Chap. 1 that any element, if made the
target in an xray 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 xrays 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 xray 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 xray 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 xray spectroscopy are possible, depending on the means
used to excite the characteristic lines :
(1) The sample is made the target in an xray 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 xray 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 xray tube and bombarded with
xrays. The primary radiation (Fig. 151) 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
1511
INTRODUCTION
403
spectrometer circle
xiay nine
rountci
FIG. 151. Fluorescent xrav 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 xray 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
xrays
xrays
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. Xray 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 highalloy steels and hightemperature alloys), ores, oils, gaso
line, etc.
404 CHEMICAL ANALYSIS BY FLUORESCENCE [CHAP. 15
Chemical analysis by xray 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 xray method never became popu
lar, however, until recent years, when the development of various kinds of
counters allowed direct measurement of xray 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.
162 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 countersealer combination, or the whole spec
trum may be continuously scanned and recorded automatically.
Figure 152 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 tungstentarget 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 xray 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. 15, fluorescence and
true absorption are but two aspects of the same phenomenon. At any
152]
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
EMISSIONLINE WAVELENGTH (angstroms)
FIG. 153. 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 tungstentarget 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. Molybdenumtarget 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. 152), 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 xray tube, which is 50 kv in commercial instruments.
At this voltage the shortwavelength 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
153] 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 153 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 onehalf its original intensity by passage through only 10 cm of
air. If a path filled with helium is provided for the xrays 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 overall absorption coefficient of the sample. The fluores
cent radiation produced within the sample then undergoes absorption on
its way out. Since longwavelength 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 heavyelement
matrix is practically impossible. On the other hand, even a few parts per
million of a heavy element in a lightelement matrix can be detected.
163 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. 154, is the simplest in design.
The xray 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
xrav tube
sample
FIG. 154. Essential parts of a fluorescent xray spectrometer, flatcrystal 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.
xray tube
 sample
/conn lor
FIG. 155. Fluorescent xray spectrometer, curvedtransmittingciystul type
(schematic).
153] SPECTROMETERS 409
Both the commercial diffractometers mentioned in Sec. 72 can be
readily converted into fluorescent spectrometers of this kind. The conver
sion involves the substitution of a highpowered (50kv, 50ma) tungsten
or molybdenumtarget 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. 155. 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. 612.) 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 longwavelength radiation and thicker crystals for harder radiation.
The thickness range is about 0.0006 to 0.004 in.
Besides the usual twotoone 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
crystaltofocus distance D is given by Eq. (615), 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. 72 may be converted into either this kind of
spectrometer or the flat crystal type.
The curved reflecting crystal spectrometer is illustrated in Fig. 156.
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
xray tube
[CHAP. 15
crystal
sample
\ counter
FIG. 156. Fluorescent xray spectrometer, curvedreflectingcrystal type.
as described in Sec. 612. But now the radius R of the focusing circle is
fixed, for a crystal of given curvature, and the slittocrystal and crystal
tofocus distances must both be varied as 6 is varied. The focusing relation,
found from Eq. (613), is
D = 2R sin 0,
where D stands for both the slittocrystal and crystaltofocus 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 high0 (longwavelength) range.
Spectrometers employing curved reflecting crystals are manufactured by
Applied Research Laboratories.
154 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
154]
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. 157
that resolution depends both on A20,
the dispersion, or separation, of line
centers, and on B, the line breadth at y
halfmaximum 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
(151)
When the minimum value of A20,
namely 2B, is inserted, this becomes
X tan
= (152)
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 lefthand 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 firstorder reflections,
because they occur at larger angles, but their intensity is less than a fifth
of that of firstorder reflections.
The factors affecting the line width B can be discussed only with refer
ence to particular spectrometers. In the flat crystal type (Fig. 154),
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. 155), 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. 156) 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
154] 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 firstorder reflections. Mica would therefore
provide adequate resolution, but LiF and NaCl would not.* Figure 152
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 xray
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. (151), 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 lightelement detection are oxalic
acid (d = 6.1A) and mica in reflection (d = 10. 1A).
155 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 xray 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 xray 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 gasfilled 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. 717). 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
xray 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.
156 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 xray 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
157]
QUANTITATIVE ANALYSIS
415
obtained will disclose the L lines of tungsten, if a tungstentarget tube is
used, as well as the characteristic lines of whatever impurities happen to
be present in the target.
157 Quantitative analysis. In determining the amount of element A
in a sample, the singleline 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 158 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
FeNi
30
40 50 60
ATOMIC PERCENT Fe
70
80
90
100
FIG. 158. 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 shortwavelength 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 FeAl and FeAg curves of Fig. 158. The absorption coefficient of an
FeAl alloy is less than that of an FeAg alloy of the same iron content,
with the result that the depth of effective penetration of the incident beam
is greater for the FeAl 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 FeAg alloy. The overall result is that the intensity
of the fluorescent Fe Ka radiation outside the specimen is greater for the
FeAl 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 FeNi curve of Fig. 158. Ni Ka radiation
can excite Fe Ka radiation, and the result is that the observed intensity of
the Fe Ka radiation from an FeNi alloy is closer to that for an FeAl 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 FeAg
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 0100 percent range, as in
Fig. 158, 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.
168] 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. 77).
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
xray 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.
168 Automatic spectrometers. Automatic directreading 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, xray counterparts of these directreading optical spectrom
eters have become available. There are two types:
(1) Singlechannel 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. 159. Relative arrangement of xray tube, sample, and one analyzing
channel of the XRay Quantometer (schematic). (The tube is of the "endon"
type: the face of the target is inclined to the tube axis and the xrays 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. 153) may also be arranged for this kind of
automatic, sequential line measurement.
(2) Multichannel type, manufactured by Applied Research Laboratories
and called the XRay 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.
159). Eight assemblies like the one shown, each consisting of slits, ana
lyzing crystal, and counter, are arranged in a circle about the centrally
located xray 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 xray 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
L59] 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 xray tube
output do not affect the accuracy of the results.
169 Nondispersive analysis. Up to this point we have considered only
methods of dispersive analysis, i.e., methods in which xray 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 xray tube
arrangement takes on the simple torm x"~x
illustrated in Fig. 1510. 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 xray tube voltage. Suppose, for example, that a CuSn
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 xray 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 CuZn 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 1215 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 CuAl 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. 75, 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 singlechannel pulseheight analyzer. Thus the
counteranalyzer 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 XRay Quantometer de
scribed in the previous section can be used for nondispersive analysis in
1510] 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.
1510 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 CuSn 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. (94) and the form of the
line intensity vs. thickness curve will resemble that of Fig. 96. 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
161. Assume that the line breadth B in a fluorescent xray 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.
162. What operating conditions would you recommend for the nondispersive
fluorescent analysis of the following alloys with a scintillation counter?
(a) CuNi (b) CuAg
153. Diffraction method (2) of Sec. 1510 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
161 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. 161 (a), and xrays 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. 161(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. 161. 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. 162. 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.)
162 Absorptionedge 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. 162, since the transmitted intensity will
increase abruptly on the long wavelength side of the edge. (The exact
162] ABSORPTIONEDGE METHOD 425
form of the curve depends on the kind of radiation available. The data
in Fig. 162 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^ (161)
^2
If we put [(M/p)A2 ~ WP)AI] = &A and p m t = Af m , then Eq. (161) be
comes
(162)
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 absorptionedge 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 absorptionedge 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. 163. 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.)
163] DIRECTABSORPTION 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
timeconsuming operation. The method is best suited to the determina
tion of a fairly heavy element in a lightelement 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 xray
tube operated at 50 kv. Figure 163 shows the relative size and location
of the K and L absorption edges of lead.
163 Directabsorption 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. 164(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. 164. Application of the directabsorption 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 //zplane of the original interface, and determining
the composition of each slice. In the absorption method, a single slice is
taken parallel to the zzplane, 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. (163) that
In
 (M/P)BI] + (M/P)BI
In (1 Q2/1 2) WA[(M/P)A2 ~ (M/p)B2l + Wp)fi2
(164)
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
165] 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. (164). 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 twocomponent samples. If more than two components are
present and equations of the form of (163) and (164) 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.
164 Directabsorption method (polychromatic beam). The absorp
tion of a polychromatic beam, made up of the sum total of continuous and
characteristic radiation issuing from an xray 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 xray 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 xray 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.
165 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 lowintensity 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
161. The values of I\ and /2, taken from Fig. 162, for the absorption pro
duced by three thicknesses of unprocessed xray 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.
162. 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 AlCu alloy if the absorptionedge 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.
163. The composition of an FeNi 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
171 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 stressfree 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. 94, 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. Xray diffraction can therefore be used as a method of
"stress" measurement. Note, however, that stress is not measured directly
by the xray 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 electricresistance 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 xray method,
the strain gauge is the spacing of lattice planes.
172 Applied stress and residual stress. Before the xray 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 stressfree 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. 171 (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. 171 (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. 171. Examples of residual stress. T = tension, C = compression.
172]
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.
172 (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. 172. 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.
173 Uniaxial stress. With these basic ideas in mind, we can now go
on to a consideration of the xray 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 crosssectional area A stressed elastically in tension by a force F (Fig.
173). There is a stress v y F/A in the ^/direction but none in the x
or 2directions. (This stress is the only normal stress acting; there are also
shear stresses present, but these are not measurable by xray 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, (171)
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
., ? (172)
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 xrays 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 backreflection photograph at normal
incidence, as shown in Fig. 173. (It is essential that a backreflection
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
173]
UNIAXIAL STRESS
435
surface  
buck reflect ion
pinliole caineia
m
J_
FIG. 173. Pure tension. FIG. 174. Diffraction from strained
aggregate, tension axis vertical. Lat
tice planes shown belong; to the same
(hkl) set. A' = reflectingplane normal.
in rf.) In this way we obtain a measurement of the strain in the z direction
since this is given by
,  '^'. (.73)
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. (171), (17 2), and (173), we obtain the relation
77T /
V \
d n 
(174)
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. 174, 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. 175. Backreflection 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. 175,
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 specimentofilm 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 (174) 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 stressfree por
tion cut out of the specimen.
174 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
socalled 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. 176.
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, 03, like <r, is equal to zero. However, 3 , the strain normal to the
174]
BIAXIAL STRESS
"3
437
FIG. 176. Angular relations between stress to be measured (o0), principal
stresses (<TI, 02, and 03), and arbitrary axes (x, y, z).
surface, is not zero. It is given by
(175)
The value of 3 can be measured by means of a diffraction pattern made
at normal incidence and is given by Eq. (173). Substituting this value
into (175), we obtain
d n d Q
(176)
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 02 = and Eq. (176) re
duces to Eq. (174).]
Normally, however, we want to measure the stress <r+ acting in some
specified direction, say the direction OB of Fig. 176, where OB makes an
angle <t> with principal direction 1 and an angle /3 with the zaxis. 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
normalincidence pattern measures the strain approximately normal to the
surface, and the inclinedincidence 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 ^ (177)
E
But
(178)
do
where di is the spacing of the inclined reflecting planes, approximately
normal to OA, under stress, and d is their stressfree spacing. Combining
Eqs. (173), (177), and (178), we obtain
di d d n  d d l  d n or^ .
(i _) ,,) sin ^ ^/. (179)
d d d
Since d , occurring in the denominator above, can be replaced by d n with
very little error, Eq. (179) can be written in the form
E '"' ~" 1 (1710)
(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. 177 illustrates the appearance of the
film in the inclinedincidence 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. (177).
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 Debyering 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
174]
BIAXIAL STRESS
439
lino from
reference
material
specimen
surface
*
line from
specimen
FIG. 1 77. Backreflection 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. (1710) in more usable
form. By differentiating the Bragg law, we obtain
= cot 6 A0.
d
(1711)
The Debyering radius S, in back reflection, is related to the specimen
tofilm distance D by
S = D tan (180  26) = D tan 26,
AS = 2Dsec 2 20A0.
Combining Eqs. (1711) and (1712), we obtain
Ad
AS = 2D sec 2 26 tan
d
(1712)
* 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. 178(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 Debyering radius in the inclinedincidence photograph,
usually taken as the radius S\ in Fig. 177, and S n is the ring radius in the
normal incidence photograph. Combining the last three equations with
Eq. (1710), we find
~ 2D(l + v) sec 2 20 tan 6 sin 2 f
Put
E
K =    (1713)
2D(l + v) sec 2 26 tan 6 sin 2 ^
Then
cr* = K,(S,  S n ). (1714)
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 specimentofilm distance. To ensure that the specimento
film distance D is effectively equal for both the inclined and normalinci
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 specimentofilm 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 tungstenring 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. (1714). 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
xrays? 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. (1713), 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
175] 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 aluminumbase 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. 177, the normalincidence exposure be
ing omitted. Both Debyering 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 (179) 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 twoexposure method, but it
entails a probable error two or three times as large.
176 Experimental technique (pinhole camera). In this and the next
section we shall consider the techniques used in applying the tw r oexposure
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
tionline positions, the lines must be smooth and continuous, not spotty.
This is achieved by rotating or oscillating the film about the incidentbeam
axis. (Complete rotation of the film is permissible in the normalincidence
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. 178(a). The camera is rigidly attached to a portable xray 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 gearandworm
arrangement. Both the normalincidence and inclinedincidence 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 179 shows a typical camera used for stress
measurements.
Some investigators like to use a wellcollimated incident beam, like the
one indicated in Fig. 178(a). Others prefer to use a divergent beam and
utilize the focusing principle shown in Fig. 1710. 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. 178. Pinhole camera for stress measurement (schematic): (a) section
through incident beam; (b) front view of cassette; (c) appearance of exposed film.
175]
EXPERIMENTAL TECHNIQUE (PINHOLE CAMERA)
443
FIG. 179. 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 specimenfilm 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
xray tube
specimen
FIG. 1710. Backreflection 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 xray 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 collimatedbeam 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 xray pene
tration varies with the angle of incidence of the xrays. Suppose, for
example, that 6 = 80 and \l/ = 45. Then it may be shown, by means of
Eq. (93), that the effective penetration depth is 83 percent greater in the
normalincidence 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 reetched. 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.
1711. 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.
176 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
176]
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 1712 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 normalincidence
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. 1712. 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. (1710) 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 ), (1715)
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 stressfree 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
177]
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.
177 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 austenitetomartensite
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 xrays 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. 618. But if the lines are
broad (and a breadth at halfmaximum 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. 1713 (a) and may be used whenever the lines
line center
line profile
line center
line profile
parabola
parabola
20 26
(a) (b)
FIG. 1713. 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. 1713(b), even when the overall shape of the line is unsymmetrical.
Now the equation
y = ax 2 + bx + c (1716)
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 =   (1717)
2a
If we put x = 26 and y = /, then Eq. (1716) 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 (1717) 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 halfmaximum 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 highintensity 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
178] 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 linetobackground
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.
178 Calibration. For the measurement of stress by xrays we have
developed two working equations, Eqs. (1714) and (1715), 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 airon
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 xray 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
xray 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
xray
ray
(a)
(b)
FIG. 1714. Specimens used for calibrating xray 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 xray 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. 1714. 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 xray 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 xray measurement, or compressed by the forces F 3 , producing
tensile stress at the same point. If the applied forces and the dimensions
and overall elastic properties of the stressed member are known, then the
stress at the point of xray measurement may be computed from elasticity
theory. If not, the stress must be measured by an independent method,
usually by means of electricresistance 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. 1715,
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
179]
APPLICATIONS
451
be corrected by an amount (A20) ,
measured on a stressfree 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. 1715. Calibration curve for
stress measurement.
179 Applications. The proper field of application of the xray 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 xray 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
xray 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 xray method
and methods involving electrical or mechanical gauges. The latter meas
ure the total strain, elastic plus plastic, which has occurred, whereas xrays
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
xray "strain gauge" can therefore measure residual stress, but an electric
resistance gauge can not. Suppose, for example, that an electricresistance
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 xray method, on the other
hand, reveals the residual elastic stress actually present at the time the
measurement is made.
However, the xray 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. 1716. 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.
171 (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. 172(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 xray 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 xray
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
xray 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 xray method.
Figure 1716 shows an example of residual stress measurement by xrays.
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 buttwelding 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
171. Calculate the probable error in measuring stress in aluminum by the
twoexposure pinholecamera method. Take E = 10 X 10 6 psi and v = 0.33.
The highestangle 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. 177) 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 specimentofilm distance of 57.8 mm. Compare your result with
that given in Sec. 174 for steel.
172. 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. 171 is
correct?
173. Verify the statement made in Sec. 175 that the effective depth of xray
penetration is 83 percent greater in normal incidence than at an incidence of 45,
when 6 = 80.
CHAPTER 18
SUGGESTIONS FOR FURTHER STUDY
181 Introduction. In the previous chapters an attempt has been made
to supply a broad and basic coverage of the theory and practice of xray
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 xray 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.
182 Textbooks. The following is a partial list of books in English
which deal with the theory and practice of xray diffraction and crystal
lography.
(1) Structure of Metals, 2nd ed., by Charles S. Barrett. (McGrawHill
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 xray diffraction, and the
second part with the structure of metals in the wider sense of the word.
454
182] TEXTBOOKS 455
Includes a very lucid account of the stereographic projection. Contains
an uptodate treatment of transformations, plastic deformation, structure
of coldworked metal, and preferred orientations. Gives a wealth of refer
ences to original papers.
(2) XRay 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 xray 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 smallangle scattering and diffraction by amorphous substances.
Crystalstructure determination is not included.
(3) XRay 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. (Singlecrystal 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 particlesize measurement from line
broadening, diffraction by amorphous substances, and smallangle scatter
ing.
(4) XRay 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 XRay s, 4th ed., by George L. Clark. (McGrawHill Book
Company, Inc., New York, 1955.) A very comprehensive bodk, devoted
to the applications of xrays 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 xrays. 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) XRays in Practice, by Wayne T. Sproull. (McGrawHill Book
Company, Inc., New York, 1946.) Xray diffraction and radiography,
with emphasis on their industrial applications.
(7) An Introduction to XRay 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 xray diffraction. Sections on radi
ography and microradiography also included.
(8) XRays 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 xrays and xray 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 spacegroup theory),
diffraction, and structure analysis. An historical account of the develop
ment of xray crystallography is also included.
(10) The Crystalline State. Vol. II: The Optical Principles of the Diffrac
tion of XRays, by R. W. James. (George Bell & Sons, Ltd., London, 1948.)
Probably the best book available in English on advanced theory of xray
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 spacegroup 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 XRay 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) XRay Crystallography, by M. J. Buerger. (John Wiley & Sons,
Inc., New York, 1942.) Theory and practice of rotating and oscillating
crystal methods. Spacegroup theory.
(14) SmallAngle Scattering of XRays, by Andrg Guinier and Gerard
Fournet. Translated by Christopher B. Walker, and followed by a bibli
183] REFERENCE BOOKS 457
ography by Kenneth L. Yudowitch. (John Wiley & Sons, Inc., New York,
1955.) A full description of smallangle scattering phenomena, including
theory, experimental technique, interpretation of results, and applications.
183 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 . Spacegroup tables.
Vol. 2. Mathematical and physical tables (e.g., values of sin 2 0, atomic
scattering factors, absorption coefficients, etc.).
(2) International Tables for Xray 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 XRay 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 193034, Rein
hold Publishing Corporation, New York, 1935.) Crystallography (includ
ing spacegroup theory) and xray 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 looseleaf
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").
184 Periodicals. Broadly speaking, technical papers involving xray
crystallography are of two kinds:
(a) Those in which crystallography or some aspect of xray 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 xray 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 xray diffraction. Many
papers of this sort are to be found in various metallurgical journals.
APPENDIX 1
LATTICE GEOMETRY
Al1 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
Al2 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
Al3 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 +
Al3] 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 RHOMBOHEDRALHEXAGONAL TRANSFORMATION
The lattice of points shown in Fig. A21 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 (HKL), 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 =
kl,
L = h + k + l.
FIG. A21. Rhombohedral and hexagonal unit cells in a rhombohedral attice.
462
APP. 2] RHOMBOHEDRALHEXAGONAL 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 HullDavey 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 planespacing 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. A21 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 facecentered
cubic. Compare Fig. A21 with Figs. 27 and 216.
Further information on the rhombohedralhexagonal relationship and on
unit cell transformations in general may be obtained from the International
Tables jor XRay Crystallography (1952), Vol. 1, pp. 1521.
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 CauchoisHulubei 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 XRay 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$e* S82B5
is
3338
IXCOCMOO OvlXOO
M3 OO O OO
27
o co o
2JQS4 SIQS55 Rfc^b
OOtxCMQCO CMOCOOOO>
rtrxcMo^^3 ixooovOoO
?
CM co CM rx
n M5
OM3CMM3CM CMO^T^
lO CK 00 U">
~x
<N ^
" RS " :?s "'  s
8 s^ 5 fts^s
.$
CO IX O Ox
^ OO 00 K
rxIXCMlOCM OvOtODiO
^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 M3OOiOt 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""
kXX^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 0,0SC
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 XRay 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
02
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
"fVM
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 XRay 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 hkl hhl Okl hk0 hh0 Ok0 001
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. 27, 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 XRay Diffraction
Photographs (MacMillan).
477
APPENDIX 10
LORENTZPOLARIZATION 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]
LORENTZPOLARIZATION 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 XRay 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 halflife.
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 A131 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(300450C)
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(9081403C)
6(>1403C)
Krypton
Lanthanum, en*
P
Lead
Lithium
(cold worked)
Lutecium
Magnesium
Manganese, a*
p(7271095C)
7(10951133C)
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(665775C)
y(7751130C)
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 (McGrawHill Book
Company, Inc., New York, 1952).
APP. 13] CRYSTAL STRUCTURE DATA
TABLE A132. 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 + 0.116r
(x = weight percent
carbon)
APPENDIX 14
ELECTRON AND NEUTRON DIFFRACTION
A141 Introduction. Just as a beam of xrays has a dual waveparticle
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