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Full text of "Chemical Crystallography"

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g]<OU_160126 >m 



OSMANIA UNIVERSITY LIBRARY 

Call No. ZUtf&lkC- Accession No., 

Author &UHVI i. C.' OO 

Title C*ut~ie*Jl frnjsbULoyrf*y. 

This book should be returned on or before the datr, last marked below. 



CHEMICAL 
CBYSTALLOGRAPHY 

AN. INTRODUCTION TO 
OPTICAL AND X-RAY METHODS 

BY 

C. W. BUNN 



OXFORD 
AT THE CLARENDON PRESS 



Oxford University Press, Amen House, London E.G. 4 

GLASGOW NEW YORK TORONTO MELBOURNE WELLINGTON 
BOMBAY CALCUTTA MADRAS CAPETOWN 

Geoffrey Cumberlege, Publisher to the University 



FIRST PUBLISHED 1045 
REPRINTED WITH CORRECTIONS 1946 



Reprinted lithographically in Great Britain 

at the UNIVERSITY PRESS, OXFORD, 1948, 1949, 

1952, from corrected sheets of the second 

impression 



PREFACE 

CEYSTALLOGRAPHIC methods are used in chemistry for two main pur- 
poses the identification of solid substances, and the determination of 
atomic configurations ; there are also other applications, most of which, 
as far as technique is concerned, may* be said to lie between the two 
main subjects. This baok is intended to be a guide to these methods. 
I have tried to explain the elementary principles involved, and to give 
as much practical information as will enable the reader to start using 
the methods described. I have not attempted to give a rigorous treat- 
ment of the physical principles: the f approach is consistently from the 
chemist's point of view, and physical theory is included only in so far 
as it is necessary for the general comprehension of the principles and 
methods described. Nor have I attempted to give an exhaustive 
account of any subject ; the aim throughout has been to lay the founda- 
tions, and to give sufficient references (either to larger works or to 
original papers) to enable the reader to follow up any subject in greater 
detail if he so desires. 

The treatment of certain subjects is perhaps somewhat unorthodox. 
Crystal morphology, for instance, is described in terms of the concept 
of the unit cell (rather than in terms of the axial ratios of the earlier 
morphologists), and is approached by way of the phenomena of crystal 
growth. The optical properties of crystals are described solely in terms 
of the phenomena observed in the polarizing microscope. X-ray diffrac- 
tion is considered first in connexion with powder photographs; it is 
moj*e usual to start with the interpretation of the diffraction effects of 
single crystals. These methods of treatment are dictated by the form 
and scope of the book ; they also reflect the course of the writer's own 
experience in applying crystallographic methods to chemical problems. 
It is therefore hoped that they may at any rate seem natural to those 
to whom the book is addressed -students of chemistry who wish to 
acquire some knowledge of crystallographic methods, and research 
workers who wish to make practical use of such methods. If the book 
should come to the notice of a more philosophical reader, I can only 
hope that any qualms such a reader may feel about its avoidance of 
formal physical or mathematical treatment may be somewhat offset 
by the interest of a novel, if rather severely practical, viewpoint. 

The difficulties of three-dimensional thinking have, I hope, been 
lightened as much as possible by the provision of a large number of 



vi PREFACE 

diagrams ; but crystallography is emphatically not a subject which can 
be learnt solely from books: solid models should be used freely models 
of crystal shapes, of atomic .nd molecular configurations, of reciprocal 
lattices and of vectorial representations of optical and other physical 
properties. Most of the diagrams are original, but a few have been 
reproduced, by permission, from published books and journals: Figs. 
197, 207-9, 215, and 222 from the Journal of the Chemical Society ; 
Figs. 199, 203, and 217 from the Proceedings of the Royal Society, 
Figs. 102-4 from the Journal of Scientific Instruments ; Fig. 229 from 
the Journal of the American Chemical Society, Fig. 161 from Inter- 
nationale Tabetten ftir Bestimmung von Kristallstrukturen (Berlin: Born- 
traeger); Fig. 192 from the * Strnkturbericht ' of the Zeitschrift fttr 
Kristallographie\ and Figs. 212 and 216 from Bragg's The Crystalline 
State (London: G. Bell & Sons, Ltd.). For Figs. 220-1 I wish to thank 
Messrs. G. Huse and H. M. Powell. Finally I wish to thank Dr. F. C. 
Phillips and his colleagues at the Department of Mineralogy and 
Petrology, Cambridge, for permission to use* their scheme of exhibiting 
the relations between the crystal classes by miniature stereographic 
projections (Fig. 32). 

I have great pleasure in acknowledging the help of my friends and 
colleagues, and proclaiming my gratitude for it. First of all I wish to 
thank Professor C. N. Hinshelwood, at whose suggestion the book was 
written, and whose interest and encouragement stimulated its progress. 
Next I must thank Mr. H. S. Peiser, who read the whole work in manu- 
script, made many valuable suggestions, contributed the geometrical 
proofs of appendixes 2 and 4, and compiled the subject index. Parts of 
the book were read by Mr. R. Brooks, Dr. L. M. Clark, and Mr. T. C. 
Alcock ; their suggestions were gratefully received. I am also indebted 
to Dr. H. Lipson for a discussion on nomenclature. In checking the 
typescript and proofs I have been very much helped by my wife, by 
Mr. C. A. Smale, and Miss A. Turner-Jones. The last-mentioned and 
Mr. H. Emmett kindly drew some of the diagrams. The X-ray photo- 
graphs were, with one exception, taken by Mr. J. L. Matthews and 
Mr. T. C. Alcook, and printed by Mr. W. J. Jackson ; the exception is 
the Weissenberg photograph of Plate VIII, for which I am indebted 
to Messrs. R. C. Evans and H. S. Peiser. The photomicrographs and 
optical diffraction photographs (Plates I, II, V, and XIII) were taken 
by Mr, H. Emmett. 

Finally, I wish to say that the experience on which the book is based 
was gained in the Research Laboratory of I.C.I. Limited (Alkali 



PBEFACE vii 

Division) at Northwich. The support and encouragement of Mr. H. E. 
Cocksedge (formerly Research Manager), of his successor Dr. J. C. 
Swallow, of the present Research Manager, Dr. J. Ferguson, and of 
many of my colleagues more especially Dr. L. M. Clark and Mr. E. A. 
Cooke are gratefully acknowledged. 

C.W.B. 



CONTENTS 

I. INTRODUCTORY SURVEY .... 1 

Anisotropy .... .2 

Identification of crystals under the microscope . . 3 

Origin of anisotropic properties of crystals . . .4 

Molecular type and arrangement deduced from anisotropic 

properties of crystals . . , . .5 

The use of X-rays . . . . .6 

Electron density maps . . . . .6 

Limitations of X-ray methods . . * .7 

Use of X-ray diffraction patterns for identification . . 8 

Information obtainable by partial interpretation of X-ray 

diffraction patterns . , , . 9 

Value of using more than one method . . .9 

Plan of this book . . . . .9 



SECTION 1. IDENTIFICATION 

II. THE SHAPES OF CRYSTALS . . , .11 

Shape varies with conditions of grc vvth . . .12 

The unit of pattern ('unit cell 5 ) . . . .14 

Crystal growth . . . . . .18 

Nomenclature of crystal planes . , . .24 

The law of rational indices . . . .27 

Measurement of interfacial angles, and graphical representation 28 
Deduction of possible unit cell shape from crystal shape. Pre- 
liminary . . . . . .30 

Internal symmetry and crystal shape . . .34 
Nomenclature of symmetry elements and crystal classes . 44 
The thirty -two point-group symmetries or crystal classes . 45 
The unit cell types or crystal systems . . .47 
Deduction of a possible unit cell shape and point-group sym- 
metry from interfacial angles . . . .62 

Identification by shape . . . . .64 

Twinning . . . . . .67 

Cleavage . . . . , .68 

Polymorphism . . . . . .69 

Isomorphism and mixed crystal formation . . .69 

Oriented overgrowth . . . . .60 



CONTENTS ix 

in. THE OPTICAL PROPERTIES OF CRYSTALS . . 62 

Cubic crystals . . . . . .63 

Measurement of refractive index under the microscope . 63 
Tetragonal, hexagonal, and trigonal crystals. Preliminary . 65 

Use of crossed Nicols. Extinction directions. Interference 

colours . . . . . .67 

The indicatrix . , . . . .70 

Orthorhombic crystals . . . . .73 

Monoclinic and triclinic crystals . . . .75 

Use of convergent light . . . . .77 

Use of the quartz wedge . . . . .80 

Dispersion . . . . . .83 

Pleochroism . t . . .85 

Rotation of the plane of polarization . . .87 

Optical properties of twinned crystals . . .89 



IV. IDENTIFICATION OF TRANSPARENT CRYSTALS UNDER 

THE MICROSCOPE . . . . .91 

Cubic crystals and amorphous substances . . .93 

Optically uniaxial crystals . . . . .94 

Optically biaxial crystals . . . . .96 

Mixtures . . . . . .99 

Identification when it is not possible to measure refractive indices 100 



V. IDENTIFICATION BY X-RAY POWDER PHOTOGRAPHS . 103 

The production of X-rays . . . .103 

X-ray wave-lengths . . . . .105 

X-ray powder photographs . . . .108 

Powder cameras . . . . . .109 

General characteristics of X-ray powder photographs , 112 

Diffraction of X-rays by a crystal . . . .114 

Measurement of powder photographs . . .119 

Spacing errors in powder photographs . . .120 

Identification of single substances, and classification of powder 

photographs . . . . . .122 

Identification and analysis of mixtures . . .123 

Non -crystalline substances . . .127 



; CONTENTS 

SECTION 2. STRUCTURE DETERMINATION 

VI. DETERMINATION OF UNIT CELL DIMENSIONS . .128 

Cubic unit cells . . . . . .129 

Tetragonal unit cells . . . . .132 

Hexagonal, trigonal, and rhombohedral unit cells . .134 

Other types of unit cells . . . . .136 

Single-crystal rotation photographs . . .137 

Unit cell dimensions from rotation photographs . .139 

Indexing rotation photographs. Preliminary consideration . 142 

The 'reciprocal lattice' . . . . .144 

Indexing rotation photographs by reciprocal lattice methods*. 

Orthorhombic crystals . n . . . .151 

Monoclinic crystals . . . . .153 

Triclinic crystals . . . . . .156 

Oscillation photographs . . . . .158 

The tilted crystal method . . . .162 

Moving-film goniometers . . . . .166 

The simplest unit cell , . . . .172 

The accurate setting of ill -formed crystals . . .173 

Oriented polycrystalline specimens . . . .175 

Determination of unit cell dimensions with the highest accuracy 1 80 
Application of knowledge of unit cell dimensions : 

1. Identification . . . . .181 

2. Determination of composition in mixed crystal series . 183 

3. Determination of molecular weight . . .185 

4. Shapes of molecules, and orientation in the unit cell . 1 87 

5. Chain-type in crystals of linear polymers . 188 



VII. DETERMINATION OF THE POSITIONS OF THE ATOMS 
IN THE UNIT CELL BY THE METHOD OF TRIAL AND 
ERROR . . . . . . .190 

Measurement of X-ray intensities . . . .192 

Calculation of intensities. Preliminary . . .196 

The diffracting powers of atoms . . . .196 

The structure amplitude, F . . . .196 

The number of equivalent reflections, p . . .199 

Angle factors ...... 200 

Thermal vibrations ..... 204 

Absorption ..... 206 

Complete expression for intensity of reflection. Perfect and 

imperfect crystals. ..... 207 



I 

INTRODUCTORY SURVEY 

MOST solid substances are crystalline, that is to say, the atoms or mole- 
cules of which they are composed are packed together in a regular 
manner, forming a three-dimensional pattern. In some solids many 
minerals, for instance the fact that they are crystalline is obvious to 
the unaided eye ; the plane faces and the more or less symmetrical shape 
of the particles are evidence of an orderly internal structure. In other 
solids all we see is a powder or some irregular lumps ; but with the aid of 
the microscope and the still more dedicate X-ray methods we have come 
to realize that most of the solids with which we are familiar, from rocks 
to sand and soil, from the chemical reagents on our laboratory shelves 
to paint pigments and cleaning powders, from steel and concrete to 
bones and teeth, really consist of small crystals. Even such apparently 
unlikely materials as wood, silk, and hair are at any rate partly crystal- 
line ; the molecules composing them are to some extent packed together 
in an orderly way, though the regularity of arrangement is not main- 
tained throughout the whole of the material. 

The crystalline condition is, in fact, the natural condition in the solid 
state ; at low temperatures atoms and molecules always try to arrange 
themselves in a regular manner. When they do not succeed in doing so 
there is good reason for their failure. Some glasses, for instance, are 
siipercoolcd liquids in which crystals have not been able to grow owing 
to very rapid cooling and the very high viscosity of the liquid ; low- 
temperature decomposition products such as 'amorphous' carbon are 
formed at such temperatures that atomic movements are too sluggish 
to permit crystal growth; some polymers (such as 'bakelite') are com- 
posed of molecules which are large and irregular in structure and cannot 
pack together neatly. 

(Even in these 'amorphous' substances it is by no means certain that 
order is entirely lacking. The word 'amorphous' has to be used with 
caution and inverted commas, for some people consider that glasses 
and low-temperature decomposition products are really composed of 
extremely small crystals only a few atoms across ; moreover, some of 
the macromolecular polymers like rubber, which are 'amorphous' in the 
ordinary condition at room temperature, can be brought to a crystal- 
line condition by stretching. Even in liquids disorder is not complete ; 
there is SOTIIO attempt to form a regular arrangement. An interesting 

4458 B 



2 INTRODUCTORY SURVEY CHAP. I 

account of work on these substances up to 1934 is to be found in J. T. 
Randall's book, The Diffraction of X-rays and Electrons by Amorphous 
Solids, Liquids, and Oases.) 

The fact that in most solid substances the atoms or molecules are 
arranged in an orderly manner is of great significance for the chemist, 
whether he is a philosopher in a university or an analyst in an industrial 
laboratory. The chemist is interested in such things as the structure 
of molecules, the nature of the bonds between dtoms, and the arrange- 
ment of ions ; and he uses every property of a substance which can give 
him any information on these matters. He is also inevitably concerned 
with methods for the identification and analysis of the substances he 
encounters. Crystals, in virtue of the orderly arrangement of the atoms 
or molecules composing them, have very special properties, which not 
only make possible the most precise determinations of molecular struc- 
tures, but also provide powerful and certain methods of identification 
and analysis. 

Anisotropy . To begin with, the properties of a crystal are, in general, 
not the same in all directions. A crystal grows, not as a sphere, but as 

a polyhedron; it dissolves more 
quickly in some directions than in 
others ; its refractive index (except 
in certain special cases) varies with 
the direction of vibration of the 
light waves ; its magnetic suscepti- 
bility, its cohesion, its thermal ex- 
QO GO pansion, its electrical conductivity, 
'QQ Q) a U var y with direction in the 
crystal. This variation of proper- 

FIG. 1. Crystal properties vary with tieg with cryg t a l direction, Or 'ani- 
direetion. , . J _ _ 

sotropy , is a consequence of the 

regular packing of atoms or molecules in a crystal. In a normal liquid 
or a gas the atoms or molecules are oriented at random, and con- 
sequently the properties are the same in all directions; individual 
molecules may be strongly anisotropic, but owing to the random 
orientation of the large numbers of molecules present even in micro- 
scopic samples, the properties are averaged out in all directions. In 
a crystal the atoms are drawn up in ranks ; pass through it in imagina- 
tion, first in one direction and then in another, and (unless you have 
chosen two special directions) you will encounter the constituent atoms 
or molecules at different intervals and perhaps (if there are different 




CHAP, i INTRODUCTORY SURVEY 3 

kinds of atoms) in a different order in the two directions. (See Fig. 1, 
a two-dimensional analogy.) Since the arrangement of the atoms or 
molecules in a crystal varies witjh direction, certain properties of the 
crystal must also vary with direction. 

Crystals thus have a greater wealth and variety of measurable charac- 
teristics than liquids or gases. This circumstance can be turned to good 
account ; we can use these varied directional properties for the identi- 
fication of crystalline siibstances. Since there are more characteristic 
magnitudes to determine, identification by physical methods is im- 
mensely more certain for crystals than it is for liquids or gases. 

Identification of crystals under the microscope. Of the charac- 
teristics which are most useful for identification purposes the most 
readily determined are shape and refractive indices. The determinative 
method which has proved most valuable for microscopic crystals (such 
as those in the average experimental or industrial product) is to measure 
the principal refractive indices (up to three in number, depending on the 
symmetry of the crystal) and, if jjossible, to find the orientation of 
the principal optical directions with respect to the geometrical form of 
the crystal. This information, which can all be obtained by simple and 
rapid microscopic methods, is usually sufficient to identify any crystal- 
line substance whose properties have previously been recorded. Mix- 
tures of two or more crystalline substances can be identified by the same 
method ; in phase equilibrium studies and in industrial research it is not 
uncommon to encounter mixtures of three or four constituents, all of 
which can be identified in this way. 

This method of identification sometimes has certain advantages over 
chemical analysis. A single substance can often be identified in a few 
minutes where a chemical analysis might take hours, and only very small 
quantities of material are required. But in general the method must not 
be regarded as a rival to chemical analysis but as a valuable complement. 
It gives essential information in cases where chemical analysis does not 
tell the whole story or does not even touch the most important part of 
the story. Where substances capable of crystallizing in two or more 
different forms are concerned (for instance, the three forms of calcium 
carbonate calcite, aragonite, and vaterite), chemical analysis cannot 
distinguish between them, and a crystallographic method is essential. 
The greatest advantages, however, are shown in the analysis of mixtures 
of several solid phases. Chemical analysis tells us which atoms or ions 
are present, as well as the proportion of each, but it does not usually tell 
us which of these are linked together. For instance, a solid obtained in a 



4 INTRODUCTORY SURVEY CHAP, i 

phase equilibrium study of the reciprocal salt pair NaN0 3 -KCl-(H 2 0) 
is shown by chemical analysis to contain all four ions, Na, K, N0 3 , 01, 
in certain proportions. But which substances are present ? NaCl, NaN0 3 , 
and KNO 8 , or NaCl, NaNO 3 , and KC1, or perhaps all four possible salts, 
NaCl, NaN0 3 , KC1, KN0 3 ? This question can be most readily settled 
by a ocystallographic method of identification. As another example, 
consider a refractory material whose composition can be represented as 
so much alumina and so much silica; are these present as separate 
constituents or are they combined as an aluminium silicate ? If they 
are combined, which of the several known aluminium silicates is present ? 
And is the material all aluminium silicate, or is there some excess silica 
as well as an aluminium silicate ? <*If there is excess silica, which of the 
several forms of silica is present ? These questions can be settled by 
crystallographic identification. If the crystals are large enough to be 
seen as individuals under the microscope, they can usually be identified 
by refractive index measurements ; the crystals need not have a regular 
geometrical shape, for refractive r index measurements can be made 
quite as well on completely irregular crystal fragments as on well-formed 
crystals. 

This method was first developed by mineralogists, but it is now being 
used to an increasing extent in such problems of inorganic chemistry 
as those just mentioned. In the organic field it has so far made slower 
progress, probably because a rapid and convenient physical method of 
identification (the measurement of melting-points) is already well 
established ; but when the possibilities of the microscopic method are 
realized, there is no doubt that it will find a very large field of usefulness 
in organic chemistry, especially in circumstances in which the estab- 
lished methods are inadequate or inapplicable (for instance, when a 
sample contains a mixture of solid phases). 

Origin of anisotropic properties of crystals. If we inquire a 
little more deeply into the origin of the anisotropic properties of crystals, 
we can distinguish two factors. Consider first crystals composed of 
unionized molecules. The molecules themselves may be anisotropic; 
a long molecule, for instance, has a greater refractivity for light vibrating 
along itf than for light vibrating across it, while a flat molecule has a 
greater refractivity for light vibrating in the plane of the molecule than 
for light vibrating perpendicular to this plane. The same is true for 
polyatomic ions. This is the first factor. The second is the way in which 

f The vibration direction is defined as the direction of the electric vector of the waves. 
(See Chapters III and VIII.) 



CHAP. I INTRODUCTORY SURVEY 5 

the molecules or polyatomic ions are packed. In some crystals all the 
molecules are packed parallel to each other, and these crystals have 
properties which correspond with those of individual molecules. A 
crystal composed of long molecules all packed parallel as in Fig. 2 a 
(a crystal of a long-chain hydrocarbon, for instance) has a high refrac- 
tive index for light vibrating along the molecules, and low refractive 
indices for light vibrating in all directions perpendicular to the mole- 
cules. In other crystals*the molecules are not all parallel to each other ; 
sometimes half the molecules have one orientation and half another 
orientation, as in Fig. 2 6 ; sometimes the arrangement is still more 




00 





no* 
O U O U 

MB 9^9 9 

w 

FIG. 2. a. Long molecules packed parallel, b. Long molecules arranged so 
that there are two different orientations, c. In some crystals composed of 
monatomic ions, anisotropy results from the mode of packing of the ions. 

complex (it depends on the shape of the molecules and the intermole- 
cular forces). The properties of these crystals correspond, not with 
those of a single molecule, but with those of a small group of two or 
more differently oriented molecules. 

To turn now to crystals composed of 'unattached* atoms or monatomic 
ions, which are individually isotropic. Here it is only the second factor 
the effect of arrangement which can be responsible for anisotropy 
in the crystal. It is the orderliness of arrangement itself which, because 
it gives rise to different atomic distributions in different directions 
(Fig. 2 c), confers properties varying with crystal direction. The degree 
of anisotropy is usually far less in these crystals than in crystals con- 
taining molecules or polyatomic ions which are themselves anisotropic. 

Molecular type and arrangement deduced from anisotropic 
properties of crystals. It is evident that, in dealing with crystals of 
unljpiown structure, the anisotropic properties may often be used to 
give direct information about the general shape of the molecules or 
polyatomic ions in the crystals and the way in which the molecules or 
ions are packed. A strongly anisotropic crystal must contain strongly 
anisotropic molecules or polyatomic ions packed in such a way that the 



6 INTRODUCTORY SURVEY CHAP. I 

anisotropies of the different molecules or ions do not neutralize each 
other, and a consideration of the properties of the crystal in all directions 
may lead to a fair idea of the general shape of the molecules or ions and 
the way they are packed. This use of optical and other properties to 
give information about molecular or ionic shape and arrangement is a 
striking example of the advantages conferred by the ordered structure 
of crystals. A molecule is too small to study individually by methods 
available at the present time ; but a crystal, in 'which a large number of 
molecules are packed in a regular manner, is in a sense a vastly enlarged 
model of a molecule or a small group of molecules, and when we observe 
the optical properties of such a crystal under the microscope, we are 
observing in effect the optical properties of a molecule or a small group 
of molecules, and this may tell us something about the shape of the 
molecules and the way they are packed in the crystal. 

The use of X-rays. All the information mentioned hitherto is ob- 
tained by old and well-established methods, of which by far the most 
important and generally useful is the determination of optical properties 
under the microscope. Visible light, however, gives us only a rough idea 
of the internal structure of a crystal ; its waves, being much longer than 
the distances between atoms, are much too coarse to show the details. If 
we want a more detailed picture of the structure of molecules and the 
arrangement of atoms and ions, as well as a yet more powerful method 
of identification, we must use much shorter waves, of about the same 
length as the distances between atoms. The X-rays, produced when 
high-speed electrons hit atoms, happen to be about the right length. 
The discovery of this fact, due to Laue in 1 912, was of course one of the 
most important -discoveries in the present century ; it opened the way, 
not only to an understanding of the nature of X-rays, but also to the 
determination of the exact arrangement of the atoms in crystals. True, 
we cannot get a direct image of the atomic pattern in a crystal ; X-rays 
cannot be focused in the convenient ways used for visible light. What 
we have to do is to study the diffraction effects produced when X-rays 
pass through a crystal, and build up an image of the structure by calcu- 
lation. The diffraction of X-rays by crystals is not essentially different 
from that of visible light by a diffraction grating ; but to synthesize the 
image from the diffracted waves we must use, not lenses, but equations.! 

Electron density maps. Since it is the electrons in the atoms which 

t Recently, W. L. Bragg ( 1 939, 1 942 a) has shown that, starting with the data provided 
by tho X-ray diffraction pattern, an image can be formed experimentally by a method 
employing visible light : the interference of light waves takes the place of calculations. 



CHAP, i INTRODUCTORY SURVEY 7 

are responsible for the diffraction of X-rays, the image we build up by 
calculation is a sort of contour map of electron densities in the crystal. 
Two or three such maps or projections, giving views of the structure 
from two or three different directions, are sufficient to enable us to 
build a complete space model of the crystal structure, showing the exact 
position of every atom. The different sorts of atoms can be identified by 
their different electron densities. The value of such a model is obviously 
enormous. The exact arrangement of ions and their distances apart (giv- 
ing the coordination numbers and "sizes' of the ions) ; the exact spatial 
configuration and interatomic distances in polyatomic ions and 
organic molecules (with all that this tells us about the specific properties 
of these bodies and the nature of \be bonds between the atoms) ; the 
mode of packing of molecules (which depends on the shape and the 
intermolecular forces) these are some of the fundamentals revealed 
at once by such a model. In the words of Bernal and Crowfoot (1933 c), 
the intensive analysis of X-ray diffraction patterns 'is one of the chief 
means of transformation from the classical qualitative, topological 
chemistry of the nineteenth century to the quantum-mechanical, 
metrical chemistry of the present day'. 

Limitations of X-ray methods. If it were possible to find the 
structure of every crystalline substance in this way, chemists would no 
longer have to spend their time in deducing the structures of new 
substances by more or less indirect methods ; they could turn all their 
energies to preparation and synthesis. In the future it may well happen 
that the structures of crystals will be determined by X-ray methods 
without chemical evidence of any sort, but at the present time there are 
certain difficulties which restrict the scope of such methods. 

As may be imagined, the building by calculation of an image of the 
pattern of atoms in a crystal is a complex and lengthy task. Moreover, 
it is not (except in special cases) straightforward ; that is to say, we 
cannot proceed straight from the experimental data (the positions and 
intensities of the diffracted X-ray beams) to the calculation of the image ; 
at one stage it is nearly always necessary to use the procedure of trial 
and error, that is, to think of an atomic arrangement, calculate the 
diffraction effects it would give, and compare these with the actual 
diffraction effects observed ; if they do not agree, another arrangement 
must be tried, and so on. Only when the approximate atomic positions 
have been found in this way is it possible to calculate the final image in 
all its details from the experimental data. For the simpler structures 
this does not present any great difficulties, but for the more complex 



8 INTRODUCTORY SURVEY CHAP. I 

structures much depends on the extent of knowledge available at the 
time for the building of trial structures. In the early days of X-ray 
crystallography the structures of only elements and simple salts could 
be tackled with any hope of success, but with the accumulation of 
knowledge, structures of ever-increasing complexity have been success- 
fully worked out. Up to the present time (1944) many inorganic struc- 
tures of considerable complexity (such as the silicate minerals, the alums, 
and the hetero-polyacids like phosphotungstic acid) have been worked 
out completely. Among organic compounds progress was at first slower, 
but as soon as the structures of the principal fundamental types of 
molecules (normal paraffin chain, benzene ring, naphthalene nucleus) 
were well established, the pace accelerated, and recently, the structures 
of such complex substances as dyestuffs, carbohydrates, sterols, and 
high polymers have been solved, and even substances of extreme com- 
plexity (proteins) are being actively studied by this method. X-ray 
analysis at first merely confirmed the conclusions of organic chemistry, 
but now it plays a useful part in research on chemical constitution. 

Use of X-ray diffraction patterns for identification. Even when 
complete structure determination is not possible, however, much 
valuable information of a less detailed character may be obtained by 
X-ray methods. In the first place, the diffracted beams produced when 
X-rays pass through crystals may be recorded on photographic films or 
plates, and the patterns thus formed may be used quite empirically, 
without any attempt at interpretation, to identify crystalline sub- 
stances, in much the same way as we use optical emission spectra to 
identify elements. Each crystalline substance gives its own character- 
istic pattern, which is different from the patterns of all other substances ; 
and the pattern is of such complexity (that is, it presents so many 
measurable quantities) that in most cases it constitutes by far the most 
certain physical criterion for identification. The X-ray method of 
identification is of greatest value in cases where microscopic methods 
are ir adequate; for instance, when the crystals are opaque or are too 
small to be seen as individuals under the microscope. The X-ray 
diffraction patterns of different substances generally differ so much 
from each other that visual comparison without precise measurement 
is usually sufficient for identification ; but in doubtful cases measure- 
ment of the positions of the recorded diffractions may be necessary. 
Mixtures of two or more different substances which are present as 
separate crystals give X-ray diffraction patterns consisting of the super- 
imposed patterns of the constituents. 



CHAP, i INTRODUCTORY SURVEY 9 

Information obtainable by partial interpretation of X-ray 
diffraction patterns. Between the recording of an X-ray diffraction 
pattern and the elucidation of the complete atomic arrangement there 
are several well-defined stages. Arrival at each stage gives more and 
more intimate information about the substance in question. It may be 
possible to form conclusions about the degree of purity of a substance, 
to determine its molecular weight more accurately than by any other 
method, to discover something about the symmetry of the molecules or 
ions in the crystal, or to determine the overall dimensions of the mole- 
cules. Individual circumstances, the nature of the substance, and the 
size and form of the crystals determine in each case how far it is possible 
or desirable to go. 

Value of using more than one method. It must be emphasized 
that the combination of different lines of evidence is often of much greater 
value than any single method of approach. X-ray methods should never 
be used alone ; the combination of evidence given by X-ray diffraction 
patterns with that given by optical properties, habit, cleavage, and so 
on may lead to valuable conclusions in circumstances where each of 
these lines of evidence taken by itself would leave unresolved ambi- 
guities. 

Plan of this book. It will be evident from the foregoing survey of 
the principal applications of crystallographic methods to chemical prob- 
lems that these applications fall into two classes: firstly, the use of 
crystal properties lor the purpose of identifying substances ; secondly, 
the elucidation of the internal structure of crystals by interpretation 
of their properties. This natural division determines the plan of this 
book, which is in two main sections, on identification and internal 
structure respectively. 

Section I (on identification) comprises four chapters. Chapter II 
is an introduction to the shapes of crystals and the relation between 
shape and structure, and Chapter III is an elementary account of 
crystal optics ; some knowledge of both subjects is essential, not only 
for the identification of crystals by microscopic methods, but also for 
the understanding of the problems of structure determination dealt 
with in Section II. Chapter IV deals with procedure in microscopic 
methods of identification. 

Chapter V, on identification by X-ray methods, is concerned with 
the practical details of taking X-ray powder photographs, and also 
includes elementary diffraction theory, taken as far as is necessary for 
most identification problems. 



10 INTRODUCTORY SURVEY CHAP.! 

Section II deals, in six chapters, with the principles underlying the 
progressive stages in the elucidation of internal structure. Chapters VI 
and VII deal with the principles of structure determination by trial ; 
Chapter VIII with the use of physical properties (such as habit, cleavage, 
and optical, magnetic, pyro- and piezo-electric properties) as auxiliary 
evidence in structure determination. In Chapter IX are to be found 
several examples of the derivation of complete structures. Chapter X 
gives an introductory account of the use c*f direct and semi-direct 
Fourier series methods of building electron density maps and vector 
diagrams from X-ray diffraction data. 

Certain crystals give diffuse X-ray reflections ; there are various 
possible causes for this small costal size, structural irregularities, or 
thermal movements. The consideration of these phenomena in Chapter 
XI leads on to a brief introduction to the interpretation of the very 
diffuse diffraction patterns given by non-crystalline substances. 



SECTION I. IDENTIFICATION 

II 
THE SHAPES OF CRYSTALS 

ANYONE who has seen the well-formed crystals of minerals in our 
museums must have boen impressed by the great variety of shapes 
exhibited : cubes and octahedra, prisms of various kinds, pyramids and 
double pyramids, flat plates of various shapes, rhornbohedra and other 
less symmetrical parallelepipeda, and many other shapes less easy to 
describe in a word or two. These crystal shapes are extremely fascinat- 
ing in themselves ; artists (notably Diirer) have used crystal shapes for 
formal or symbolic purposes, while many a natural philosopher has been 
drawn to the attempt to understand first of all the geometry of crystal 
shapes considered simply as solid figures, and then the manner in which 
these shapes are formed by the anisotropie growth of atomic and mole- 
cular space-patterns. 

But this book has a practical object, as its title proclaims. Our pur- 
pose in this chapter is to inquire to what extent crystal shapes can be 
criteria for identification, and how much they tell us about the atomic 
and molecular space-patterns within them. 

In view of the great variety of crystal shapes and the rich face- 
development on many crystals, it is natural to expect that, on the basis 
of accurate methods of measurement and a sound system of classifi- 
cation, it would be possible to identify crystals by their shapes alone ; 
and indeed, in recent years attempts have been made, first by Fedorov 
and later by Barker and his school, to develop such a method resting on 
the measurement of the angles between face-normals. There is no doubt 
that when well-formed crystals, large enough to be handled individually 
so that they can be mounted on a goniometer, are available, this morpho- 
logical method of identification is a practicable one; Barker (1930) has 
demonstrated this. But as a standard method of identification in a 
chemical laboratory it has very serious limitations. One of them is 
that the crystals formed in laboratory experiments or in industrial 
processes are often too small to be handled individually ; they can only 
be examined under the microscope, and under these conditions angular 
measurements either cannot be made at all, or if they can be made are 
only approximate. Another is that the shapes of such crystals are often 
not sufficiently characteristic; sometimes there are too few faces on 



12 IDENTIFICATION CHAP, n 

each crystal ; or perhaps the substance grows in the form of skeletal 
crystals without definite faces; or, worse still, the crystals may be 
broken into irregular pieces. To identify such materials we need a 
method which does not depend on shape, but on some characteristics of 
the crystal material itself properties of the atomic space-pattern. 
The properties most conveniently measured under the microscope are 
the optical constants, particularly the refractive indices ; and in practice 
the measurement of refractive indices has provfed by far the most useful 
single method of identifying crystalline substances under the micro- 
scope. The technique is described in the next chapter. 

There is no need, however, to ignore crystal shape in identifica- 
tion work. On the contrary, whenever crystals do show good face- 
development their shapes, even if they cannot be measured precisely 
but only observed in a qualitative way, reinforce and implement the 
evidence provided by optical properties, especially if the relations 
between the principal optical and geometrical directions can be 
discovered. 

This is one reason for studying crystal shapes. Another and more 
weighty reason is that crystal shapes tell us a great deal about the 
relative, dimensions and the symmetries of the atomic and molecular 
space -patterns constituting the crystalline material. 

In this chapter, therefore, we make some inquiry into the origins of 
crystal shapes and their classification on the basis of symmetry charac- 
teristics. 

Shape varies with conditions of growth. The shape of a crystal, 
taken as it stands, is not a fixed characteristic of the substance in 
question. In the first place, the shape is controlled to some extent by 
the supply of material round the crystal during growth. In uniform 
surroundings, as in a stirred solution, crystals of sodium chloride grow 
as cubes, but if they grow, well separated, on the bottom of a dish of 
stagnant solution, they grow as square tablets whose thickness is not more 
than half their other dimensions ; the reason is that growth can occur 
only upwards and sideways, not downwards. If the crystals on the 
bottom of the dish are crowded, the tablets formed are not all square ; 
many have unequal edges owing to local variations in the supply of 
solute. As another example, sodium chlorate, NaClO 3 , when grown 
rapidly in a stirred solution, forms cubes, but when grown very slowly in 
a still solution grows in the form of a modified cube showing additional 
facets on the edges and corners (Fig. 3). Crystals which grow in 
rod-like forms such as gypsum, CaS0 4 .2H 2 0, which is also illustrated 



CHAP. II 



SHAPES OF CRYSTALS 



13 



(b) 



in Fig. 3 usually tend to grow longer and thinner when formed 
rapidly than when growth is slow. 

These are, comparatively speaking, minor variations of shape ; but 
the crystal shapes of some substances may be completely altered by the 
presence of certain other substances in the mother liquor. Sodium 
chloride grows from a pure solution in the form of cubes, but if the 
mother liquor contains 10 per 
cent, of urea, the crystals which 
grow (Fig. 3) are octahedra 
(Gille and Spangonberg, 1927). 
Yet the internal structure the 
pattern of atoms of this sub- 
stance is. not changed by such 
differing external conditions ; it 
is only the form of the bounding 
surface of the crystalline material 
which is changed. It is evident 
that if we want to use crystal 
shapes for identification we must, 
so to speak, get behind the shape 
as it stands, and try to deduce 
from the actual shape something 
about the internal structure. 




FIG. 3. Variation of crystal shape with con- 
ditions of growth. Sodium chlorate, NaClO 3 , 
grown (a) rapidly and (b) slowly; gypsum, 
CaSO 4 . 2H 2 O, grown (c) slowly and (d) rapidly; 
sodium chloride, NaCl, grown (c) from pure 
solution and (/) from sohition containing 10 
per cent, of urea. 



The possibility of doing this is 

indicated by the fact that the 

angles between the faces of the 

long thin gypsum crystals in 

the sketch are exactly the same 

as those of the shorter crystals. 

Likewise, all octahedra of sodium chloride, however much they differ 

in size, and however unequal the areas of the different faces of any 

one crystal may be, have exactly the same interfacial angles. The slopes 

of the various faces are in fact controlled by the rigid, precise internal 

structure. The relation between totally different shapes of any one 

substance such as the cubes and octahedra of sodium chloride is 

less obvious ; but it can be shown that the faces of cubes and octahedra 

are oriented in precise but different ways with respect to the internal 

atomic pattern. 

Two pieces of information about the fundamental atomic pattern may 
be deduced from the actual shape of a crystal, provided this crystal 



H IDENTIFICATION OHAP. n 

shows a sufficient variety effaces and is large enough to permit measure- 
ments of the angles between the faces. One is a knowledge of the shape 
and relative dimensions of the unit of pattern. The other is a partial 
knowledge of the symmetries of the atomic arrangement. 

The unit of pattern ('unit cell'). A crystal consists of a large 
number of repetitions of a basic pattern of atoms. Just as in many textile 
materials and wall-papers a pattern is repeated over and over again on 
a surface, so in a crystal a particular grouping of atoms is repeated 



c 



cf 



<- ....... 



FIG. 4. A plane pattern. Lower part divided into identical unit noils Biirh 
as ATtCD. Alternative unit cells RFGJJ and JJKL are also outlined. 

many times in space. The reason for the formation of regular patterns 
is that atoms, ions, or molecules tend to settle down in positions of 
minimum energy; for each atom, ion, or molecule a particular environ- 
ment of neighbours has a lower energy content than any other, and there 
is therefore a tendency for this arrangement to be taken up everywhere. 
The only patterns of exactly repeated environments capable of 
indefinite extension are those in which successions of pattern-units lie 
on straight lines. Consider the shape of the unit of pattern, first of 
all in the simpler case of a plane pattern, such as that shown in Fig. 4. 
Mark any point such as A, and then mark other points whose surround- 
ings are exactly the same (in orientation as well as geometrical character) 
as those of A ; these points fall on straight lines which divide the pattern 
into a number of exactly similar parallelogram-shaped areas. Each area, 
such as A BCD, represents one unit of pattern ; the whole pattern can be 
built up by parallel contiguous repetitions of ABCD. Of course, we 
might have started by choosing a differently situated point E, but we 



CHAP, n SHAPES OF CRYSTALS 15 

should have arrived at the same shape EFGH for the unit of pattern ; 
the position of the origin does not matter. Note that IJKL may equally 
claim to be the unit of pattern, inasmuch as it contains one unit of 
pattern and has exactly the same area as ABCD or EFGH ; and many 
other still more elongated areas, each containing one unit of pattern, 
could be drawn ; in practice, however, it is usually most convenient to 
accept as the unit of pattern the area with the shortest sides, that is to 
say, the area most nearly* approaching rectangular shape. 

All patterns on surfaces can be divided into similar areas in this way, 
and the unit of pattern is always a parallelogram. The shape and dimen- 
sions of the parallelogram vary in different ways ; it is possible to have 
square units, rectangular units with ur^equal sides, and non-rectangular 
units with either equal or unequal sides. 

In a crystal we can do the same thing in three dimensions. Again the 
choice of origin does not matter, and again we can divide the whole 
structure into units (of volume this time) by joining similarly situated 
points by straight lines. Fig. 5 show$ the arrangement of the ions in a 
crystal of caesium bromide. Any caesium ion has exactly the same 
surroundings as any other, and if the centre of each is joined to the 
centres of its nearest neighbours, the whole structure is found to be 
divided into cubes, each of which has caesium ions at its corners and a 
bromine ion at its centre. The centre of a bromine ion might equally 
well have been selected as the origin, and then the cubic units of pattern 
would have bromine ions at their corners and caesium ions at their 
centres. (Note that no bromine ion 'belongs' specifically to any one 
caesium ion ; its relations to the eight caesium ions surrounding it are 
equal. There are thus no 'molecules' of CsBr in the crystal ; the structure 
is simply a stack of positively charged caesium ions and negatively 
charged bromine ions.) 

Fig. 5 also shows an example of a molecular structure that of 
hexamethylbenzene, C 6 (CH 3 ) 6 . The molecules, which can be represented 
as disks, are all stacked parallel to each other, and if the centre of each 
molecule is joined to those of its nearest neighbours, the structure is 
divided into a number of identical units of pattern, each of which is a 
non-rectangular 'box' with all three sets of edges unequal in length. 

The unit of pattern in a crystal is always a 'box' bounded by three 
pairs of parallel sides. The shape and dimensions of the box, that is, 
the lengths of its three different sorts of edges ('axes') and the angles 
between them, are characteristic for each different crystal species ; in 
some crystals the box is a cube, in others it is rectangular with unequal 





plane 
BCHE 



plane ADGF 



plane DCF6 




plane BCF 
(100) 



plane ABCD 




plane AFH 



plane EBO \\. ; / plane AFC 




FIG. 5. a. Caesium bromide, CsBr. Loft, structure. Right, shape of crystal. 6. Hexu- 

methylbenzene, C,(CH,) 6 . Left, structure. Right, sha.po of crystal, c. Coppor. Loft, 

structure. Right, shape ot crystal. 



CHAP. II 



SHAPES OF CRYSTALS 



17 



edges, in others the angles are not right angles, and so on. We shall not 
at this point catalogue the various types of shape ; we merely observe 
that various shapes of pattern-unit are possible ; the crystal structure of 
caesium bromide represents the most highly symmetrical and that of 
hexamethylbenzene the least symmetrical of the possible shapes. 

It is sometimes more appropriate to use for purposes of reference a 
box containing more than one unit of pattern. For instance, in crystals 
of metallic copper the atoms are arranged in the manner shown in 
Figs. 5 c and 6. All the atoms have precisely the same surroundings, and 





FIG. 6. Face-centred cubic unit coll of roj>|x*r (left), and body -centred culm; unit cell of 

a iron (both shown by broken lines). In each case a unit containing one pattern-unit 

(one atom) is heavily outlined. 

the true unit of pattern, formed by joining similarly situated points so as 
to divide the structure into 'boxes' with atoms at the corners only, is the 
heavily outlined rhombohedron in Fig. 6 ; there is one atom, one pattern- 
unit, to each 'box'. (One at each corner of the box makes eight in all ; 
but each one is shared between the eight boxes which meet at the corner ; 
therefore each box has the volume of one pattern-unit .) But it is found 
that atoms A,B,C 9 D, E, F, G, and // fall at the corners of a cube, and 
atoms /, 7, K, L, M, and N in the centres of the faces of the same cube. 
This cube is accepted as the unit cell, in spite of the fact that it contains 
four pattern-units comprising one copper atom each. (The corner atoms 
count as one to each cube ; the six atoms in the face centres are each 
shared between two cubes; thus the number of atoms per unit cube 
is 1 + 3 = 4.) There are two reasons for this. The first and more im- 
portant reason is that the symmetries of the complete arrangement 
are the same as those of crystals in which the shape of the true pat- 
tern-unit is cubic; crystal symmetry will not be discussed here an 

4458 



18 IDENTIFICATION CHAP, n 

introductory account of it is given later in this chapter. The second 
reason for accepting the four-atom cube as the unit cell is that a cube is 
a more convenient frame of reference than a rhombohedron. This parti- 
cular 'compound' unit cell is described as 'face-centred'. Other types of 
'compound' unit cell are the body-centred, with identical pattern-units 
in the centres of the cells as well as at the corners (see the structure of a 
iron in Fig. 6), and the side-centred, with identical pattern-units at the 
centres of one pair of opposite faces in addition to those at the corners. 
The arrangement of the pattern-units, the assemblage of points each of 
which represents one pattern-unit, is called the space-lattice. The points 
of the space-lattice the 'lattice points' are thus corners of the true 
unit of pattern ; the conventionally accepted unit cell may be simple or 
compound; if compound, it may contain two or more space-lattice 
points. 

We now have to consider the faces of crystals and their relation to 
the geometry of the precisely patterned assemblage of atoms which 
constitutes the solid material. This subject is best approached by 
thinking about the manner in which crystals grow. Crystals usually 
have plane faces, firstly because they do not grow at the same rate in all 
directions, and secondly as a result of the specific manner in which 
solid material is deposited. 

Crystal growth . Suppose we had the task of packing a large number 
of atoms or ions or molecules together to form a predetermined arrange- 
ment. We should find that the most convenient way of building up the 
structure is to arrange one layer of building units, then put a second 
layer on top of the first, and so on. But we should have to choose which 
layer to put down first, and there are many different layers which might 
be selected ; there* are very many ways in which a crystal structure 
could be divided into layers by planes passing through it. A few possible 
ways are shown in Fig. 7. In practice we should choose the 'simplest' 
possible plane, that is to say, a plane which is as layer-like as possible, 
a plane in which the building-units atoms in sDme crystals, ions or 
molecules in others are packed closely together. Thus, to build the 
crystal of hexamethylbenzene (Fig. 5 6), it would obviously be more 
convenient to choose planes such as ABCD and DCFG, which are paral- 
lel to the side of the unit cell, rather than a plane such as BDF, which 
is inclined to all the edges of the unit cell, as the basis for our building 
operation. 

This is apparently what happens in nature when a crystal grows from 
a solution or melt. When growing crystals are watched under the micro- 





FIG. 8. Above: layer formation on crystal of cadmium iodide ( X 600). Below, left: layer 

formation on crystal of sodium chloride (X1400). Below, right: skeletal growths of 

ammonium chloride ( X 20). 



n 



SHAPES OF CRYSTALS 



19 



scope, using a high magnification and dark ground illumination,, layers 
can often be seen spreading over the faces one after another (Fig. 8, 
Plate I) ; sometimes it can be seen that relatively thick layers which 
spread at a moderate speed are built up from much thinner, much more 
rapidly spreading, layers ; and it seems likely that the same thing occurs, 
down to the molecular or ionic scale the building units arrange them- 
selves layer by layer. (See also Marcelin, 1918 ; Volmer, 1923 ; Kowarski, 




Fio. 7. Dividing a crystal into layers. A few of the simpler ways. 
(Each dot is a lattice point.) 

1935.) And this process occurs only on certain planes ; most crystals are 
bounded by only a few faces, sometimes all of the same type (for in- 
stance, in cubic crystals), though more frequently of a few different 
types ; and in structurally simple crystals these types are always densely 
packed planes. 

In the hexamethylbenzene crystal the most densely packed planes 
are those parallel to the unit cell edges, and we find that crystals of 
hexamethylbenzene grown from a pure solution in benzene are parallele- 
pipeda with the three pairs of faces parallel to the faces of the unit cell 
(Lonsdale, 1929). In caesium bromide (Fig. 5) the most densely packed 
planes are those such as ACGE which cut two edges of the unit cell at 
equal angles and are parallel to the third, and caesium bromide crystals 
(grown from pure aqueous solution) are rhombic dodecahedra which are 



20 IDENTIFICATION CHAP. II 

bounded entirely by such planes (Groth, 1906-19). In crystalline copper 
(Fig. 5 c) the most densely packed planes are those such as BEG which 
cut the three edges of the unit cell symmetrically (note that atoms K , J, 
and N fall on plane BEG) ; copper crystals grow from the vapour as 
octahedra, the faces of which are just these most densely packed planes 
(Groth, 1906-19). 

For some of the more complex crystals it is not easy to define plane 
density of packing of atoms or molecules : a pfane parallel to a crystal 
face, taken at any level, passes through many atoms, but it cannot pass 
through the centres of more than a small proportion of them. For 
instance, the particular plane of the lead chloride crystal illustrated in 
Fig. 39, if it passes through the cerffcres of the atoms at the corners of the 
marked unit area, does not pass exactly through the centres of any of 
the other atoms, which lie at various distances above or below the plane 
of the paper. It would be difficult to say which of these should 'count* 
in the reckoning of plane density of packing of atoms. (See Niggli, 1920.) 
But plane density of lattice points is a precisely defined magnitude ; 
and it is on this that we must focus our attention for it is found that 
the faces of crystals are always densely packed with lattice points. In 
other words, if we regard the group of atoms associated with a lattice 
point as the building unit, we may say that the faces of crystals are 
planes of high reticular density of building units. 

It will be evident that, since the faces are parallel to definite planes 
of lattice points, the interfacial angles are constant in different crystals 
of the same substance. Variations in local conditions during growth 
may cause some crystals of hexamethylbenzene, for instance, to be 
longer or thinner than others in the same batch ; and the eight faces of a 
copper crystal, which in uniform growth conditions would grow to the 
same size, may in practice be found to have very different sizes ; but 
whatever the variation in the actual dimensions of crystals of any parti- 
cular substance, the interfacial angles are constant, provided that the 
same type of face is present. 

Sparsely packed planes usually do not appear as faces on growing 
crystals, but if we deliberately create such surfaces we can study their 
growth. Fig. 9 illustrates what happens when a cubic crystal of sodium 
chlorate (NaClO 3 ) is partly dissolved to a rounded shape so as to present 
all possible surfaces, and then put into a supersaturated solution. The 
diagram is two-dimensional for the sake of simplicity it is a section 
through the middle of the crystal. At first, small faces appear on the 
corners of the square section ; but it is found that the rate of growth of 



CHAP. II 



SHAPES OF CRYSTALS 



21 



these small faces the thickness of solid deposited in a unit of time is 
greater than that of the cube faces, and as a result of this, the small 
faces ultimately disappear and the final crystal is entirely bounded by 
the most slowly growing faces, the ordinary cube faces. (See also Arte- 
meev, 1910; Spangenberg, 1928.) This experiment brings out the fact 
that the faces which appear on growing crystals are those with the 
smallest rate of thickening. A small rate of thickening, with perhaps a 
great rate of spreading, are the growth characteristics one expects of the 
planes with highest reticular density 
and widest interplanar spacing. 

When crystals grow rapidly in stirred, 
strongly supersaturated solutions (&s 
they often do under the usual condi- 
tions of crystallization in the laboratory 
or in industrial plant) there is a plentiful 
supply of solute round each growing 
crystal; external conditions are fairly 
uniform, and the controlling factor is 
the architecture of the crystal. Under 
these conditions the picture of crystal 
growth given in the previous para- 
graphs adequately represents what 
happens ;f the crystals are bounded by 
very few faces the minimum number 
of the most slowly growing 'simple' 
planes necessary to enclose a solid figure. On the other hand, crystals 
of many minerals, for instance, have grown very slowly in very slightly 
supersaturated solutions in which the supply of solute is very limited 
and may vary locally owing to stagnant conditions, convection currents, 
the proximity of other crystals, and so on. The external conditions thus 
play a large part in determining the shape ; faces which, given equal 
chances, would grow at different rates may actually grow at the same 
rate, &nd vice versa. These crystals therefore often show a variety of 
facets which do not appear on crystals grown rapidly. Subsidiary facets 
may also appear if the temperature of a crystallizing solution fluctuates ; 
partial dissolution rounds off the crystals, and when growth is resumed, 
small facets appear on the rounded corners, and these may not have 
time or opportunity to eliminate themselves by rapid growth as in Fig. 9. 

t Except in extreme conditions (very high supersaturation), when skeletal crystals are 
formed ; and a few substances grow in skeletal form under ordinary conditions. See later. 




FIG. 9. A rounded crystal of sodium 
chlorate, on being put into super- 
saturated solution, develops 110 and 
1 00 faces. The more rapidly growing 
1 10 faces are subsequently eliminated. 



22 IDENTIFICATION CHAP, n 

The production of beautiful, richly faceted crystals by the simple 
method of leaving a dish of solution for several days on a laboratory 
bench without temperature control is undoubtedly often due to such 
temperature fluctuations. It is still true, however, that all the faces on 
such richly faceted crystals are fairly simple planes, in the sense that 
they have a fairly high reticular density of lattice points. It is also true 
that the principal faces are in general simpler than the subsidiary facets. 

The shape of a crystal may be modified, or even completely changed, 
by the presence of certain impurities in the solution (see Fig. 3). The 
reason is that the impurities are strongly adsorbed only on certain faces 
of the crystal, thereby retarding the growth of these faces (Gaubert, 
1906; Bunn, 1933; Royer, 1934).* The impurity may be adsorbed on 
faces which normally grow rapidly (that is, planes which are not the 
simplest and do not normally appear), and in these circumstances the 
rate of growth of these faces may be so much reduced that they become 
the predominant faces on the crystal. The presence of modifying impuri- 
ties may often be unsuspected ; hence we sometimes find crystals exhi- 
biting for no apparent reason faces not of the simplest type. 

Abnormal external conditions may thus be responsible for an ap- 
parent breakdown in the principle of simplicity of faces. However, 
apparent exceptions to the principle cannot always be attributed to ab- 
normal external conditions. It is not justifiable to regard the principle of 
simplicity as more than a broad generalization ; that is to say, even when 
external conditions are normal, the faces on crystals, though always 
simple, are not necessarily the simplest possible. (See also Niggli, 1920.) 
The rates of growth of crystal faces are of course determined by the 
distribution of the forces between the atoms, ions, or molecules, and it 
is not to be expected that a purely geometrical generalization (as the 
principle of simplicity is) would cover adequately such complexities. 
In particular it is to be noted that in ionic crystals the distribution of 
electric charges in the various planes plays an important part (Kossel, 
1927; Stransky, 1928; Brandes and Volmer, 1931). 

Nevertheless the broad generalization is of the greatest value ; for we 
can measure the angles between the faces of a crystal, 'and, assuming 
that these faces are simple that is, they are densely packed with lattice 
points and are either parallel to the unit ceil faces or are related in some 
simple way to the unit cell we can usually deduce the type of unit cell, 
and very often calculate its relative dimensions and angles. 

Not all crystals are solid polyhedra. We may approach the subject of 
irregularities in crystals by remarking that when a crystal is growing 



CHAP, ii SHAPES OF CRYSTALS 23 

from a solution, it sometimes happens that growth in the centres of the 
faces stops, while growth in the outer regions of the faces (near the edges 
and corners) continues, A hollow is thus formed in the centre of each 
face. If, as often happens, the hollow is subsequently closed over, 
mother liquor is included in the crystal. This may be repeated more 
than once, and is a common cause of opacity in crystals, and also of the 
subsequent caking of crystalline products when stored. (Mother liquor 
diffuses out, and deposits solute at the points of contact of crystals, 
cementing them together.) 

If such cavities are not closed over, the final crystals have hollow 
faces; often there is a step-formation down each hollow. In extreme 
cases growth is maintained only towards the corners of crystals, leading 
to skeletal forms, in which successive branching occurs, as in ammonium 
chloride, illustrated in Fig. 8, Plate I ; the directions of growth here are 
the axial directions of the cubic unit cell. When crystals grow in thin 
films or droplets of liquid, distortion may occur ; a familiar example is 
ice, which forms irregular tree-like patterns when it crystallizes from 
liquid on window panes. 

Such tendencies may be reduced by growing crystals very slowly, for 
instance by extremely slow cooling or evaporation. In fact, when it is 
desired to obtain perfect crystals for goniometric or X-ray work, the 
golden rule is to grow them as slowly as possible. Excessive nucleus 
formation in solutions can often be avoided by removing dust particles 
in the following way. A solution saturated at, say, 30 C. is made up 
and allowed to cool without disturbance to room temperature ; it is then 
suddenly disturbed, so that a shower of small crystals is formed ; these 
carry down with them any nucleus-forming particles which were in the 
solution. The solution is then filtered, warmed slightly to destroy any 
now nuclei formed during filtration, and then left undisturbed to eva- 
porate slowly. 

Another method, often useful for organic substances, is to make a solu- 
tion in one solvent and to cover this with a less dense liquid in which 
the substance is much less soluble ; crystals grow at the interface. The 
two solvents must be at least partially miscible. 

Sparingly soluble salts which are conveniently formed by precipita- 
tion reactions may sometimes be induced to form good crystals by a 
diffusion method. Solutions of the reagents are put in two separate 
beakers, both completely filled and standing in a larger vessel ; water is 
carefuDy poured in to cover both beakers, and the arrangement is then 
left undisturbed (L. M. Clark: private communication). 



24 



IDENTIFICATION 



CHAP. II 



The amount of structural information obtainable by the morpholo- 
gical study of skeletal crystals is naturally very limited, especially when 
they are distorted. In order to be able to deduce the shape of the unit 
cell it is necessary to ha^ve well-formed polyhedral crystals. The faces 
of such crystals are, as we have already seen, related in some simple way 
to the unit cells. We must now define more closely what is meant by the 
last phrase 'related in some simple way to the unit cells' and to do 




Flu. 10. Various sots of planes in a crystal. 

this it is necessary to give some account of the accepted nomenclature 
of crystal planes. 

Nomenclature of crystal planes. Attention has already been 
drawn to the many ways of dividing a crystal into layers by sets of planes 
passing through lattice points (Fig. 7). Each of these sets of parallel 
planes is described by three numbers such as 210 or 132, the meaning of 
which is best shown by a few examples. For simplicity, think first of all 
in only two dimensions, that is, look at the crystal along one axis say 
the c axis as in Fig. 10. In this diagram the points, each of which 
represents a row of lattice points one behind the other, are seen to lie on 
sets of straight lines (planes seen edgewise). Every point lies on one of 
these planes. Now along the axial directions count the number of planes 



CHAP. II 



SHAPES OF CRYSTALS 



25 



crossed between one lattice point and the next ; these numbers are the 
index numbers. Thus, for the set of planes in the bottom right-hand 
corner, three planes are crossed in going along a from one lattice point 
to the next, and two planes in going along b from one lattice point to the 
next ; the first two index numbers are therefore 32. The third index 
number is 0, because this set of planes is parallel to the c axis, and there- 
fore no planes are crossed in going along c ; this set of planes is thus the 
320 set. Other sets of planes, with indices 110, 100, 010, and 120 (all 
parallel to the c axis), are also illustrated in this diagram. 




Fia. 11. This set of parallel planes has indices 312. 

A set of planes inclined to all three axes is shown in Fig. 11. Along a, 
three planes are crossed between one lattice point and the next ; along 
6, one plane is crossed at each lattice point, and along c, two planes per 
lattice point : the indices are 312. 

Alternatively, one could say that these planes cut the a axis at inter- 
vals of a/3 (a being the repeat distance in this direction), the b axis at 
intervals of 6/1, and the c axis at intervals of c/2, the indices being 
defined as the reciprocals of these intercepts. This comes to the same 
thing as the definition already given, and corresponds to that found in 
most text-books of crystal morphology ; but it is really simpler to think 
of numbers of planes rather than reciprocals of intercepts ; andmoreover, 
the present definition links up with the method of indexing X-ray 
reflections (see Chapter VI). 

Each type of plane is a possible crystal face, although in actual fact 
only a few simple types of plane usually appear as crystal faces. The 
next sketch, Fig. 12, shows an actual crystal (ammonium sulphate) 
with the indices of its front faces marked. This sketch will also serve to 
illustrate the conventions about crystal set-up and positive and negative 



IDENTIFICATION 



CHAP. II 



directions. In order to show as many faces as possible, crystals are 
drawn as seen from a viewpoint inclined to all three axes and defined in 
the following way. Imagine first of all the crystal with its c axis vertical 
and its 010 plane seen edgewise ; now shift the eye a little to the right 
and upwards. The c axis still appears vertical, the b axis lies left and 
right but not quite in the plane of the paper, and the a axis points a 
little to the left and downwards as it appears to come out above the 




Flo. 13. Indices of planes of hexagonal 
crystals. ABCDEFA'B'C'D'F/F', hexa- 
gonal prism; ABCOA'B'C'O', unit cell. 
FIG. 12. A crystal of ammonium sulphate ACO ^ plane whir}]> in con f orinjty with 
(class rnmm). (After Tutton.) indices of crystals of other systems, is 

called 111. For the sake of treating the 

three equivalent directions O/4, OG, and 

OK equally, this plane i sometimes known 

as 1121. 

paper. Usually perspective drawing is not attempted; most crystal 
drawings are orthogonal projections. Positive directions are upwards 
along c, to the right along 6, and forwards (above the paper) along a. 
Intercepts in the negative directions are represented by minus signs 
above the index numbers, thus: 120, 111. Naturally it is sometimes 
necessary to depart from the conventional viewpoint to illustrate 
particular features of crystals more clearly. 

An extension of this system of nomenclature is sometimes encoun- 
tered in descriptions of crystals of hexagonal type (Fig. 13). The unit 
cell of these crystals has a diamond-shaped base, the a and b axes being 
equal in length and inclined to each other at an angle of 120. The c 
axis is perpendicular to the other two. Although only two horizontal 



CHAP, ii SHAPES OF CRYSTALS 27 

axes are strictly necessary for purposes of description, nevertheless there 
are three horizontal directions, all exactly equivalent, at 120 to each 
other ; any two of them could be taken as the a and b axes. In order to 
bring out this feature, index numbers referring to all three horizontal 
axes, as well as the vertical (c) axis, are given, thus: 1121. The last 
number refers to the c axis, the first three to the horizontal axes. The 
third index, which is always necessarily numerically equal to the sum 
of the first two but of opposite sign, is really redundant. This nomen- 
clature will be found in descriptions of the shapes of hexagonal crystals, 
but for internal crystal planes it is customary to omit the third index. 

The indices of single crystal faces are sometimes enclosed in brackets, 
thus: (100); this distinguishes a face from the corresponding set of 
internal planes 100. Curly brackets signify a set of equivalent faces : for 
a cubic crystal {100} would mean the set 100, TOO, 010, OK), 001, and OOT. 

The law of rational indices. We have seen that the faces of 
structurally simple crystals, the planes on which deposition of solid 
occurs layer by layer, are in general those planes which have a high 
reticular density of lattice points in each plane and wide interplanar 
spacing. Sometimes the faces are the planes with the densest packing 
and the widest interplanar spacing, but there are many exceptions to 
this, for various reasons which have already been mentioned. It re- 
mains true, however, that in all cases the actual faces of a crystal are 
planes of high (though not necessarily the highest) reticular density. 
We may call these the 'simple' planes. 

It is evident from Figs. 10 and 1 1 that these planes have small indices ; 
we may therefore state that the actual faces on crystals are planes with 
small indices. In this form, the generalization is what is known as the 
'law of rational indices', which says simply that all the faces on a crystal 
may be described, with reference to the three axes, by three small whole 
numbers. It is frequently found that all the faces of even richly faceted 
crystals can be described by index numbers not greater than 3 ; numbers 
greater than 5 are very rare. 

It is the recognition of the law of rational indices which makes it 
possible to deduca*probable unit cell shapes from crystal shapes. (It is, 
of course, not possible to discover the absolute dimensions ; X-ray or 
electron diffraction photographs are necessary for this purpose (Chapter 
VI).) The general principle is to find that unit cell (its angles and relative 
dimensions) which will enable us to describe all the faces of the crystal 
by the smallest whole numbers, and, in particular, the largest faces by 
the smallest numbers. There is a further condition: all faces which 



28 



IDENTIFICATION 



CHAP. II 



appear to be equivalent (for instance, all the eight faces of a regular 
octahedron) are given similar indices, that is, are assumed to be related 
in the same way to the most appropriate unit cell ; in other words, the 
directions of unit cell edges are chosen in conformity with 'the symmetry 
of the crystal. We shall return to this subject later in this chapter. 
Meanwhile, the first step in the attempt ta deduce the angles and 
relative dimensions of the unit cell of a crystal from its actual shape 
is the accurate measurement of the angles between all the faces of the 
crystal. 



LIGHT 




FIG. 14. Principle of the reflecting goniometer. The adjusting head comprises two 
iriutually perpendicular arc movements and two cross movements. 

Measurement of interfacial angles, and graphical representa- 
tion. The most accurate method of measuring the angles between 
crystal faces is an optical one, which makes use of the reflection of light 
by the plane faces. The crystal is mounted on the stem of a goniometer 
head (Fig. 14) by means of wax, shellac, or plasticine ; a beam of parallel 
light from the collimator strikes the crystal, which is rotated until 
one of its faces reflects the beam into the telescope, which is at any 
convenient angle to the collimator. A suitable sharply defined aperture 
is provided in the collimator, so that its image can be adjusted accu- 
rately to the cross-wires of the telescope. The crystal is then rotated 
until the light is reflected by the next face ; the angle through which the 
goniometer head has been turned is the angle between the normals of the 
two faces. It is evident that, in order to get reflections from both faces 
into the telescope, the crystal must be adjusted very carefully by means 



CHAP, n SHAPES OF CRYSTALS 29 

of the arc movements of the goniometer head. This is simplest when the 
crystal is mounted on the goniometer head so that one of the face- 
normals is approximately parallel to one of the arc movements ; this arc 
is adjusted until the reflection from this face appears accurately on the 
cross-wires. The crystal is now rotated so that the reflection from 
another face (preferably one which is roughly at right angles to the first) 
enters the telescope ; by a movement of the second arc this reflection is 
brought to the cross-wires. 

It is found that, when the reflections from two faces are registered 
accurately on the cross-wires of the telescope, other faces automatically 
give their reflections when the crystal is rotated further ; for instance, 
all the vertical faces of the ammonium sulphate crystal in Fig. 12 give 
their reflections one after the other as the crystal is rotated round the 
c axis. Such a set of faces is called a 'zone', and the axis of rotation 
parallel to all the faces is called the 'zone axis'. All the faces of any 
crystal fall on one or other of a few zones, and therefore in order to 
measure all the interfacial angles each of these zone axes in turn must 
be set parallel to the axis of rotation of the goniometer head. On a 
single-circle goniometer this must be done by remounting the crystal 
for each zone ; but two-circle goniometers which obviate the necessity 
of such re-setting are also obtainable. 

It is often useful to be able to represent precisely on a flat surface the 
three-dimensional relations between the interfacial angles. The most 
convenient projection for most purposes is the stereographic projection, 
which is derived in the following way. From a point within the crystal 
imagine lines drawn outwards normal to all the faces (Fig. 15). Round 
the crystal describe a sphere having the point as its centre. The positions 
at which the face-normals meet the surface of the sphere are known as 
the poles of the faces. The crystal is thus replaced by a set of points on 
the surface of the sphere, each point representing the orientation of a 
crystal face. In this way we have left behind the actual shape of the 
crystal, with the irregularities arising from the conditions of growth, 
and are now dealing simply with the orientations of faces that is, with 
the orientations of lattice planes, which are related in a simple way to the 
unijj cell. The sphere is now projected on to a selected plane the equa- 
torial plane in Fig. 15 b by joining all points on its upper half to the 
'south pole' and all points on its lower half to the 'north pole'. The great 
advantage of this projection (Fig. 15c) is that all zones of faces fall 
either on arcs of circles or else on straight lines, a circumstance which 
much facilitates graphical construction. (Each such arc or straight line 



30 



IDENTIFICATION 



CHAP. II 




FIG. 15. The stereographic projection. 



passes through opposite points 
on the equatorial circle.) Poles 
in the northern hemisphere are 
denoted by dots, those in the 
southern hemiphere by little 
rings. 

For further information on 
stereographic projections and 
the spherical trigonometry 
necessary for handling gonio- 
metric data, books by Miers 
(1929), Tutton (1922), and 
Barker (1922) may be consulted. 

Deduction of possible unit 
cell shape from crystal 
shape. Preliminary. In this 
book we are concerned chiefly 
with optical and X-ray methods, 
and we shall consider crystal 
morphology only so far as is 
necessary for the full use of such 
methods for identification or for 
structure determination. But 
although it is not intended 
to deal with morphological 
methods in a quantitative way, 
it is very necessary to consider 
in rather more detail the rela- 
tion between the external shape 
of a crystal and that of its unit 
cell ; and this subject is perhaps 
best developed in the guise of 
a consideration of the problem 
of deducing the probable unit 
cell shape from the external 
shape of a crystal. We have al- 
ready seen that the principle on 
which the attempt is based is 
the principle of simplicity of 
indices, coupled with the con- 



CHAP, ii SHAPES OF CRYSTALS 31 

formity of the indices with the symmetry of the crystal. We now see 
how this principle can be applied in practice. First of all, we shall see 
the principle of simplicity in action by itself; and we shall then find 
it necessary to consider crystal symmetry in some detail. 

The planes with the simplest indices 100, 010, and 001 are those 
which are parallel to the sides of the unit cell, and we find that on many 
crystals these form the principal faces, and on some crystals (especially 
those grown rapidly m,sti f ongly supersaturated solutions) the only faces. 
One example, hexamethylbenzene, has already been given; it forms 
non-rectangular parallelepipeda with the three pairs of faces parallel 
to the unit cell faces. Another example is anhydrite, CaS0 4 ; the unit 
cell of this crystal is a rectangular box* with unequal edges, and it grows 
as a rectangular brick with unequal edges, though it must be emphasized 
that the relative dimensions of the crystal itself have no direct con- 
nexion with the dimensions of the unit cell. (The rates of growth of the 
various faces of any crystal depend, in the first place, on the forces 
between the atoms, ions, or molecules in different directions, and these 
forces have no direct connexion with the unit cell dimensions ; moreover, 
these rates of growth are affected by external conditions.) Such crystals 
tell us the angles of the unit cell, but they do not tell us anything about 
the relative dimensions of the unit cell edges. 

If we are to be able to calculate the relative dimensions of the unit 
cell of any crystalline substance, some of the faces on the crystals must 
be inclined to the faces of the unit cell. Suppose we have a crystal of the 
shape shown in Fig. 16 a a rectangular brick with the (unequal) edges 
bevelled (an orthorhombic crystal). We naturally assume that the faces 
which are perpendicular to each other are parallel to the faces of the unit 
cell, which is evidently a rectangular box. The indices of the principal 
faces are thus assumed provisionally to be 100, 010, and 001. The 
simplest indices for the faces which bevel the edges are 110, Oil, and 
101. If we assume that a face is Oil, we are assuming that successive 
identical planes of lattice points parallel to this face are parallel to the 
a axis, and that in passing along either b or c, only one plane is crossed 
in the interval between one lattice point and the next. (See Fig. 16 6.) 
It is evident that c/b = cot 0. In the same way, by assuming that 
another face is 110, we can obtain a/6 ; and this settles the shape of the 
unit cell and the indices of the remaining faces ; thus, the third different 
bevelling face might turn out to be, not 101 as first suggested, but 201 
or 102. If our crystals also have faces cutting off the corners (Fig. 
16 c), the indices of these faces can be found (by slightly more complex 



IDENTIFICATION 



CHAP. II 



trigonometry) from the angles between these 'corner' faces and the 
principal faces. 

Alternatively, it might have been assumed initially that these 'corner* 
faces are 111, III, and so on ; this assumption would have given us a set 
of axial ratios, from which the indices of the bevelling faces could be 
deduced. 

It is always possible to find alternative sets of indices, corresponding 
to different axial ratios, for any crystal. Thus 1 ,' consider the ammonium 




(b) 



c-UNIT 

: CELL 



010 



FIG. 16. Determination of the probable shape of the unit 
cell from interfacial angles. 

sulphate crystal (Fig. 12), which, like the example just given, has a 
rectangular unit cell. Let us call the faces 110, Oil, 130, 021, and 111 
P> ?> P'> q'> and o respectively. If it had been assumed that q' is 01 1 and 
p 110, then this group effaces would be 110, 012, 130, Oil, and 112. 
Or it might have been assumed that p 1 is 110 and q Oil, in which case 
the group of faces would be, 310, Oil, 110, 021, 311. But the sets of 
indices given by the second and third schemes are less simple than those 
resulting from the first assumptions, and therefore the axial ratios 
derived in the first scheme are accepted as the probable relative dimen- 
sions of the unit cell edges. This turns out to be correct. 

Here we have the key to morphological crystallography. The principle 
followed throughout is to find that unit cell shape which, subject to the 
condition that similar faces shall have similar indices, will allow all the 



CHAP, ii SHAPES OF CRYSTALS 33 

faces of a crystal to be indexed by the smallest possible whole numbers, 
the principal faces being given, as a general rule, the simplest indices. 
This method was developed during the last century, long before X-rays 
were discovered, though the term 'unit cell' was not used. The set of 
axes deduced in this way was regarded primarily as the most convenient 
frame of reference for the accurate description and classification of any 
crystal. Nevertheless it is clearly more than a convenient frame of 
reference ; it corresponds'to some fundamental feature of the ultimate 
structure of the crystal. We know now, as the result of the study of the 
atomic structure of crystals by X-ray methods, that the relative axial 
dimensions deduced by morphological methods are in fact very often 
the exact relative dimensions of the u&it cell. Even when they are not 
correct, there is always a very simple relation between the 'morpholo- 
gical' unit and the true unit ; one of the morphological axes is perhaps 
twice as long or half as long (in relation to the other axes) as it should be. 
This obviously means that the principle of simplest indices is not strictly 
true for these crystals ; some of the faces on these crystals are, so to 
speak, not the simplest but the next in order of simplicity. There is no 
doubt about the general soundness of the principle of simplest indices, 
but it is not a rigid law. 

The examples given hitherto have been particularly simple ones, 
because some of the faces have been at right angles to each other, and 
this has given the clue to the type of unit cell. But many crystals do 
not possess faces parallel to the unit cell faces, and for such crystals 
the type of unit cell, and possible indices for the principal faces, are 
very often not by any means obvious. To approach such problems it is 
necessary to introduce the all-important subject of crystal symmetry. 
The type of unit cell is entirely bound up with the symmetry of the 
atomic arrangement ; it is, in fact, the symmetry of the atomic arrange- 
ment which decides which (if any) of the unit cell angles shall be right 
angles, and how many of its edges shall be equal. Therefore if we can 
recognize the symmetries of any particular crystal, this leads us at once 
to the unit cell type and to the probable directions of unit cell edges. 

And this is not all. Each type of unit cell may arise from a number of 
different types of atomic arrangement, and some of the symmetry 
characteristics of these different types of atomic arrangements are re- 
vealed by shape-symmetries. In classifying crystals we can first of all 
divide them into several systems according to unit cell types, and then 
each system can be divided into several classes according to those sym- 
metry characteristics which are revealed by shape. The consideration 



4458 



34 IDENTIFICATION CHAP, n 

of crystal symmetry may thus take us further than the mere derivation 
of unit cell type. 

Internal symmetry and crystal shape. Consider first one of the 
simplest and most highly symmetrical of atomic arrangements, that 
which is found in crystals of sodium chloride and in many other simple 
binary compounds. The atomic arrangement is shown in Fig. 17 a. The 
unit cell is a cube ; if we take the corner of the unit cell to be the centre 
of a sodium ion, there are also sodium ions at the centre of each face, 
the lattice being a face-centred one; the chlorine ions are half-way 
along the edges and also in the centre of the unit cell. Note first that 
tlie reason why the three mutually perpendicular axes are equal in 
length is that the arrangement of atoms is precisely the same along one 
axis as it is along the other two ; the 100 plane has exactly the same 
arrangement of atoms as the 010 and 001 planes ; secondly, that when 
a sodium chloride crystal grows in a pure solution, it is inevitable that, 
provided the three types of faces have the same chance (in a stirred 
solution, for instance), they grow at the same rate, and the crystal 
becomes a perfect cube. 

If sodium chloride crystals are grown in a solution containing 10 per 
cent, of urea, they grow as regular octahedra ; but although the external 
shape is different from that of crystals grown from a pure solution, the 
internal structure is exactly the same; the same internal lattice is 
bounded by surfaces of a different type in the two sorts of crystals. 
The octahedral faces (111, 111, lTl,Tll, iTT, TlT, TTl, and TTT) are per- 
pendicular to the cube diagonals ; the atomic arrangement on all octa- 
hedral faces is the same, and if we proceed from any point in the crystal 
along any of the eight diagonal directions, we shall come across the 
same atomic distribution (alternate layers of sodium and chlorine 
ions) ; consequently, in uniform growth conditions all the octahedral 
faces grow at the same rate, and the crystals grow as perfectly regular 
octahedra. 

Now although the cube and the regular octahedron are quite different 
solid shapes, yet their symmetries are exactly the same ; and it can be 
seen (in Figs. 17-20) that the symmetries of these solid figures are those 
of the arrangement of atoms in a, sodium chloride crystal. Rotate a cube 
about an axis perpendicular to one of its faces and passing through its 
centre (Fig. 176); after a quarter of a turn it presents exactly the same 
appearance as it did at first ; after half a turn, again the same appear- 
ance, and likewise after three-quarters of a turn ; in fact, it presents the 
same appearance four times during a complete revolution ; the axis is an 



CHAP. II 



SHAPES OF CRYSTALS 



35 



axis of fourfold symmetry. There are three such fourfold axes, all at 
right angles to each other and parallel to the cube edges. A regular 
octahedron likewise has three fourfold axes, passing through its corners 
(Fig. 17 c). These fourfold axes correspond with those of the atomic 



(a) 




FIG. 17. a. The atomic arrangement in sodium chloride, and some of its axes 

of symmetry, b and c. Fourfold axes of cube and octahedron, d and e. Twofold 

axe of cube and octahedron. 

arrangement ; every line which passes through a row of atoms parallel 
to a unit cell edge is an axis of fourfold symmetry, since the atomic 
arrangement (regarded as extending indefinitely in space) presents the 
same appearance four times during a complete revolution round this 
line. Similarly there arc, passing through the edges of both cube and 
octahedron, six axes of twofold symmetry, involving identity of appear- 
ance twice during a complete revolution (Fig. 17 d and e) ; and finally, 
passing through the cube corners and perpendicular to the octahedron 



36 



IDENTIFICATION 



CHAP. II 



faces, four axes of threefold symmetry, involving identity of appearance 
three times during a complete revolution (Fig. 18). All these axes are 
symmetry elements of the atomic arrangement. 






FIG. 18. Centre: atomic arrangement in sodium chloride, seen along a threefold 

axis of symmetry. Left : cube seen along a body diagonal. Right : octahedron 

seen along a fare -normal. 

Sodium chloride crystals also possess another type of symmetry; 
imagine a plane parallel to one pair of cube faces, passing through the 

centre of the crystal (Fig. 19) ; this plane 
divides the crystal into two halves, each 
the mirror image of the other, and is there- 
fore called a plane of symmetry. There are 
two sets of such planes of symmetry: a 
set of three mutually perpendicular planes 
parallel to the three pairs of cube faces, 
and a set of six bisecting the angles be- 
tween the first set. These planes of sym- 

< / \ / / \ % - metry, which are also possessed by the 

(r ~y \\\l regular octahedron, correspond with the 

planes of symmetry of the atomic arrange- 
ment planes passing through sheets of 
atoms. 

There is one other element of symmetry 
possessed by sodium chloride crystals. For 

each face, edge, or corner of the cube or octahedron there is an exactly 
similar face, edge, or corner diametrically opposite ; the centre of the 
cube or octahedron (Fig. 20) is therefore called a centre of symmetry. 
The centre of symmetry possessed by these shapes corresponds with 
the centre of symmetry in the atomic arrangement ; the centre of any 
sodium or chlorine ion is a centre of symmetry, since along any direction 
from the selected ion the arrangement encountered is exactly repeated 





FIG. 19. Planes of symmetry in 
cube and octahedron. 



CHAP. II 



SHAPES OF CRYSTALS 



37 




in the diametrically opposite direction. A centre of symmetry is often 
called a centre of inversion because a particular grouping on one side of 
it is an inverted or mirror-image copy of the 
grouping on the other side, just as a pin-hole 
camera produces an inverted image of the 
original object. 

Turn now to sodium chlorate, NaClO 3 . This 
crystal also has a cubic unit cell, and rapidly 
grown crystals are simple cubes; but slowly 
grown crystals (Fig. 21, left) show subsidiary 
faces on the edges and corners, and if these 
crystals are examined it will be found hat their 
symmetries are different from those of sodium chloride. For instance, 
there are only four 'corner' faces ({11 1} type), not eight ; and the axes 
passing through the centres of the cube faces are in this case not four- 
fold but only twofold. Similarly, when we encourage the growth of 1 1 1 



2 

e- 



FIG. 20. Both cube and 
octahedron possess a centre 
of symmetry, which corre- 
sponds to the centre of 
symmetry in each atom of 
the crystal. 



\] 




FIG. 21. Sodium chlorate crystals with tetrahedron faces. 

faces by the presence of sodium thiosulphate in the solution, we ob- 
tain tetrahedra, not octahedra (Buckley, 1930) ; a regular tetrahedron 
(Fig. 21, right) has three mutually perpendicular twofold axes but no 
fourfold axes. Evidently the rate of growth of four of the faces of 
type 1 1 1 is much less than that of the other four. The known atomic 
arrangement (Fig. 22) shows clearly the reason why there is a difference. 
The chlorate ion (C10 3 ) has the form of a low triangular pyramid with 
the chlorine atom as apex and the oxygen atoms forming an equilateral 
triangular base. The arrangement of these pyramidal ions on faces of 
type 111 is rather complex, for there are four different orientations; 
but for the present purpose we need not consider this in detail; we 
need only note that on four of the planes of type 111 there are pyramidal 
ions with their bases facing outwards (and none with an exactly reversed 
orientation), while on the other four it is the apexes which face outwards ; 



38 



IDENTIFICATION 



CHAP. II 



hence the surface forces on four of the planes are quite different from 
those of the other four, and the rates of growth are therefore different 
so much so that one set never appears on crystals at all. The tetrahedron 
has no centre of symmetry, and each threefold axis is called a polar axis 
since its two ends are not equivalent. 

If sodium chlorate grew always in the form of regular tetrahedra 
we might think the atomic arrangement has planes of symmetry, for the 
regular tetrahedron is a solid figure which has such planes. But crystals 
of this substance grown very slowly in pure solution (Fig. 23) present 
evidence of an internal symmetry even lower than that of a simple 





FIG. 22. Structures of left- and right-handed sodium chlorate crystals. 

tetrahedron. Truncating the cube edges there are not only {1 10} faces 
but also faces of type {210} ; but only twelve out of a possible twenty- 
four of this type are present, one on each edge ; thus, on a particular 
crystal, 210 is present but not 120. The threefold axes (cube diagonals) 
are maintained, as they are in all crystals belonging to the cubic system ; 
and so are the twofold axes characteristic of a tetrahedron ; but in con- 
sequence of the presence of this half -set of {210} faces, the crystal has 
no planes of symmetry. If we look down a threefold axis of the crystal 
shown in Fig. 23, left, we see a 210 type of face always in advance 
(clockwise) of a 110 type of face. The reason can again be seen quite 
directly from the known atomic structure of the crystal ; the C10 3 ions 
are placed so that their chlorine-oxygen bonds do not point to the 
corners of the tetrahedral faces ; the ions are rotated to a 'skew' position. 
It should be noted that in addition to the crystal illustrated on the left 
of Fig. 23, there is an equivalent but not identical type (Fig. 23, right) 
in which the faces of type 210 are on the other side of the 110 faces ; in 
these crystals, evidently, the C10 3 ions are twisted round in the opposite 
direction to those in the first-mentioned crystals.f The two types of 

t It is not known whether the orientations of the chlorate groups in the two types of 
crystal are as shown or the reverse. 



CHAP. II 



SHAPES OF CRYSTALS 



39 



crystal are mirror-images of each other, both as regards their external 
shape and their atomic arrangements; they are, like left- and right- 
handed gloves, equivalent but not identical. 

The external form of a crystal may thus reveal, not only the shape 
of the unit cell, but to some extent the symmetries of the internal 
atomic arrangement. For each different type of unit cell (each different 



/ 


111 / 021 


/ 

\m 


/ 


210 


010 




110- 


/ 


\ 
^ 


1 021 / " 



100 



101 



ifo- 

120 

> 

\ 


f 


JOI 


/ 


V 

/ 


liu 

/ 


W^ 

010 

Jc 

7 


\ 


201 


V w 


i tr\ 


100 


S 


NO 
\ 


CD 

f 

"ftf 


f \ 


201 







100 



FIG. 23. Shapes of left- and right-handed sodium chlorate crystals, and orientation 
of C1O 3 groups 011 111 faces. (Point-group symmetry of sodium chlorate 23.) 

crystal system) there are several types of internal symmetry which may 
be revealed by crystal shape ; in the cubic system, for instance, there are 
five different classes recognizable by external shape-symmetry, that of 
sodium chloride having the highest and that of sodium chlorate thfe 
lowest symmetry. Such information is not always obtainable, however ; 
very often, especially when crystals grow rapidly, they have too few 
faces, and the apparent symmetry of the crystals is higher than the real 
internal symmetry ; but when this information can be obtained, it is of 
value for identification purposes and for structure determination. The 
possibilities of identity for a crystal observed to have the form of a 
regular tetrahedron are in some degree limited by the obvious fact that 
it cannot belong to the most highly symmetrical class of the cubic 



40 IDENTIFICATION CHAP, n 

system ; its internal symmetry is not higher than tetrahedral (though it 
might be lower). And in setting out to determine the atomic arrange- 
ment of a crystal having a tetrahedral habit all arrangements having 
fourfold axes of symmetry are ruled out from the start. 

The above remarks on symmetry of shape apply only to crystals 
grown in uniform external conditions. When external conditions are 
not uniform, crystallographically equivalent faces are often found to be 
very unequal in size ; but, however unequal thtey are in size, the angles 
between them are constant, and the symmetries of the internal atomic 
arrangement, though not shown by the shape as a whole, are exhibited 
by the interfacial angles. The best way of thinking of such cases is to 
imagine lines drawn outwards fropi a point within the crystal, each line 
being perpendicular to a crystal face ; this assemblage of perpendiculars 
or 'poles' (which is best represented on paper by the stereographic 
projection) exhibits the symmetries of the atomic arrangement. There 
is an important possible source of confusion here ; certain faces may be 
missing owing to accidental local variations of growth conditions. 
However, it will usually be qbvious that such absences are accidental, 
as opposed to the systematic absences like those shown by sodium 
chlorate crystals. For instance, when only one of a set of eight faces is 
missing the absence is obviously accidental. It is only when a set of 
faces is halved or quartered, for instance, that the circumstance has any 
significance with regard to internal symmetry. The examination of a 
number of crystals from the same batch will usually resolve such diffi- 
culties ; not all the crystals will have the same accidental absences or 
accidental variations of shape, and examination of a number of crystals 
will usually give a sound idea of shape-symmetry. 

Crystal shapes idealized in this way may be regarded as the result of 
the co-operation of selected elements of symmetry. In crystals belonging 
to the cubic system we find the planes and axes of symmetry occurring 
in sets of three or four or six, in consequence of the identity of atomic 
arrangement along three mutually perpendicular directions; but in 
crystals belonging to some of the other systems we may find them in 
smaller sets or in isolation. Crystals of cassiterite, Sn0 2 , for instance, 
which belong to the tetragonal system, exhibit a single fourfold axis, 
perpendicular to two sets of two twofold axes ; there are also planes of 
symmetry in sets of one or two (Fig. 24). Crystals of sodium meta- 
periodate trihydrate, NaI0 4 .3H 2 0, have one threefold axis (of polar 
character) as their only element of symmetry (Fig. 25). Meta-bromoni- 
trobenzene (orthorhombic system) has one twofold (polar) axis and two 



OHAP. II 



SHAPES OF CRYSTALS 



41 



planes of symmetry parallel to this axis (Fig. 26). Paraquinone (mono- 
clinic system) has one twofold axis and a plane of symmetry perpendi- 
cular to this axis (Fig. 27). 

The number of different types of symmetry elements is very small. 
In addition to the symmetry axes already mentioned, the only other 




FIG. 24. Cassiterite, SnO 2 . Left: general view, showing axes of symmetry 

ar\d equatorial plane of symmetry. Right : view down fourfold axis, showing 

vertical planes of symmetry. Class 4/mmm. 




381 




in 



FIG. 25. NaIO 4 .3H 2 O (class 3). Left: general view. Right: view along 
threefold axis. 

straightforward rotation axis is the sixfold axis, involving identity 
of appearance six times in the course of one complete revolution. 
Crystals of potassium dithionate, K 2 S 2 Q 6 , exhibit this type of sym- 
metry (Fig. 28). 

Axes of fivefold or greater-than-sixfold symmetry do not occur in 
crystals, though it is possible to construct solid figures showing such 
symmetries. The reason is that space-patterns regular repetitions of 
structural units in space cannot have such symmetries. Nor, for that 



IDENTIFICATION 



CHAP. II 



matter, can plane-patterns ; it is easy to confirm this by drawing patterns 
of dots on paper. 

Finally, there is another type of symmetry axis which involves, 
not simple rotation, but combined rotation and inversion through a 
point. Crystals of urea, 0:C(NH 2 ) 2 , are prisms of square cross-section, 
terminated at each end by a pair of sloping faces (Fig. 29) ; all four 





FIG. 26. Symmetries of meta- 
bromonitrobenzene (class mm). 



FIG. 27. Symmetries of para-quinono 
(class 2/m). 




, 








FIG. 28. Potassium dithionate, K 2 S 2 O 6 (class 6/wmm). Left: general view. Right: view 
down sixfold axis. (Note. Atomic arrangement has lower symmetry.) 

sloping faces make the same angles with the prism faces, but if we wish 
to imagine a bottom face say 111 moved into the position of a top 
face, we must rotate through 90 and invert through a point at the 
centre of the crystal, thus arriving at 1 1 1 or 1 1 1 . All four sloping faces 
can be accounted for by repetitions of this compound operation. The 
prism axis of such a crystal is known as a fourfold axis of rotatory in- 
version, or fourfold inversion axis. There are also three- and sixfold 
inversion axes. The threefold inversion axis, which is equivalent to an 
ordinary threefold axis plus a centre of symmetry, is exemplified in 
crystals of dioptase, CuH 2 Si0 4 (Fig. 30). The sixfold inversion axis is 



CHAP. II 



SHAPES OF CRYSTALS 



43 



equivalent to a straightforward threefold axis with a plane of symmetry 
normal to it. A twofold inversion axis is equivalent to a plane of sym- 
metry, and is usually known by the latter name. 

All idealized crystal shapes bounded by plane faces exhibit either no 
symmetry at all or else a combination of some of the elements of sym- 
metry in this very short list. Crystals having no symmetry are very rare. 

It has been said at the beginning of this section that the symmetries 
displayed by the shapes of crystals grown in uniform surroundings are 




III -A I" / 




FIG. 29. The fourfold inversion axis. 
Urea, O:C(NH 2 ) 2 . Class 42m. 



FIG. 30. The threefold inversion axis. 
Dioptase, CuH 2 SiO 4 . Class 3. 



those of the atomic space -pattern (or at any rate are not lower than 
those of the atomic space-pattern). This statement needs amplification. 
In some atomic space-patterns parallel contiguous repetitions of units 
of pattern there can be discerned types of symmetry elements involv- 
ing translation : screw axes involving combined rotation and transla- 
tion, and glide planes involving combined reflection and translation. 
(Examples will be found in Chapter VII.) Such symmetry elements 
involving translation naturally cannot be displayed by crystal shapes, 
which are, to speak formally, assemblies of face-types round a point, 
having no element of translation. Crystal shapes therefore display 
symmetry elements which may be regarded as screw axes and glide 
planes deprived of their elements of translation ; that is to say, an 
atomic space-pattern having screw axes gives rise to a crystal shape 
displaying the corresponding simple rotation axes, and a space-pattern 
having glide planes gives rise to a crystal shape displaying straight- 
forward reflection planes. Thus, several different types of atomic 



44 IDENTIFICATION CHAP, n 

space-patterns (space-group symmetries) give rise to the same crystal 
shape-symmetry (point-group symmetry). The space-group symmetries 
are considered more fully in Chapter VII ; here we are concerned only 
with point-group symmetry. 

Nomenclature of symmetry elements and crystal classes. 
It is convenient to have short, self-explanatory symbols with which to 
refer to the various crystal classes. Some of the names used formerly 
for the crystal classes are rather cumbrous (e.g* monoclinic hemimorphic 
hemihedry), and others (e.g. tetragonal hemihedry of the second type) 
are not self-explanatory. Moreover, different authorities have quite 
different name-systems. The point-group nomenclature recently adopted 
internationally and given in Internationale Tabellen zur Bestimmung von 
Kristallstrwkturen (1935) provides symbols which are not only extremely 
concise, but also self-explanatory in that they present the essential sym- 
metries of the point-groups. 

Two-, three-, four-, or sixfoldrotation axes of symmetry are represented 
by the numbers 2, 3, 4, and 6, while three-, four-, and sixfold inversion 
axes have the symbols 3, 4, and 6; In conformity with this scheme, asym- 
metry is represented by the figure 1 (only one repetition in a complete 
rotation), and a centre of symmetry, or inversion through a point, by 1. 
A plane of symmetry is represented by the letter m ('mirror'). 

In putting together the symbols to denote the symmetries of any 
crystal class the convention is to give the symmetry of the principal 
axis first for instance, 4 or 4 for tetragonal classes. If there is a plane 

of symmetry perpendicular to the principal axis, the two symbols are 

4 
associated thus: ( e four over m'), or, more conveniently for printing, 

m 

4/m. Then follow the symbols for the secondary axes, if any, and then 

4 

any other symmetry planes. (Note that 4/mmm means mm, that is, 

m 

the second and third ra's refer to planes of symmetry parallel to the 
fourfold axis.) 

Secondary axes may be in sets, but there is no need to mention more 
than one. Thus if, to a principal fourfold axis, we add a secondary 
twofold axis (perpendicular to the principal axis), the action of the four- 
fold axis inevitably creates another twofold axis at right angles to the 
first ; and further, we find that there are inevitably two more twofold 
axes bisecting the angles between the first two. This is illustrated in 
Fig. 31, a stereographic projection in which a point represents the pole 
of a general plane ; if A is the secondary twofold axis which is first 



CHAP. II 



SHAPES OF CRY.STALS 



47 



system, obtained by adding planes of symmetry both parallel and per- 
pendicular to the principal rotation axis. The enantiomorphous classes 
are those in the first and sixth rows those possessing rotation axes 
only. 

Note how, very often, the association of two elements of symmetry 
inevitably creates further elements. We have already seen an example 
of this in class 42 (Fig. 31). The class symbols given in Fig. 32 are, first, 
those which conform to the scheme of derivation in this diagram (these 
are sufficient sometimes more than sufficient to characterize the 
classes uniquely), and following these the conventional symbols given 







010 



FIG. 33. Triclinic system, a. Unit cell type. b. CaS 2 O a .6H 2 O. Class L c. CuSO 4 .5H 2 O. 
Class 1. d. 1,4 dinitro 2,5 dibromo -benzene. Class 1. 

in Internationale Tabellen, which in some cases are longer (4mm, 42m), 
and in others shorter (w3m), than the first-mentioned symbols. 

The cubic classes stand somewhat apart from the rest. They have as 
their distinctive feature -four threefold axes lying along cube diagonals ; 
these are secondary axes. The primary axes may be either twofold or 
fourfold. 

Examples of crystals are shown in Figs. 33-8 and in various other 
drawings in this book. Familiarity with crystal symmetry is, however, 
best attained by handling and contemplating idealized models of 
crystals. 

The unit cell types or crystal systems. 

Triclinic (sometimes called anorthic). Crystals lacking symmetry of 
any kind naturally have the most 'general' type of unit cell, the three 
axes of which are all inclined to each other at different angles and 
unequal in length. The addition of a centre of symmetry does not alter 
the situation, for this most general type of unit cell has a centre of 
symmetry and is appropriate for this class also. These two classes, 
1 and T, constitute the triclinic system (Fig. 33). 



48 



IDENTIFICATION 



CHAP. II 



The lattice points of a triclinic crystal may be joined in various ways 
to form differently shaped unit cells (see p. 141). It is usually most con- 
venient to use the cell with the shortest edges, unless there is some 
special feature which recommends some other direction as a unit cell 
edge. Donnay (1943) recommends that the shortest axis shall be called 
c and the longest 6 ; and that the angles a and /J shall be obtuse. 

When axes are chosen on morphological grounds there is a convention, 
not always followed, that the principal zone* axis is called c, and that, 




601 


S 


100 


no: 








\ / 


tbi 





FIG. 34. Monoclinic system. (See also Fig. 27.) a. Unit cell type. 6. Left- and right- 
handed tartaric acid. Class 2. c. 2,4,6 Tribromobenzonitrile. Class m. d. p-Dinitro- 
benzene. Class 2/ra. e. (CH 3 COO) 2 Pb.3H 2 O. Class 2/w. 

of the other two, the longer is called b ; and the obtuse angles between 
the axes are usually specified, rather than the acute angles. 

Monoclinic. The single twofold axis of class 2 is an obvious direction 
for a unit cell edge, and this is called b. The existence of the twofold axis 
means that neighbouring lattice points lie on a plane normal to the 
twofold axis ; therefore all the lattice points lie in planes normal to b ; 
thus the a and c edges of the unit cell are both normal to 6, but since 
there is no other element of symmetry, they are inclined to each other ; 
and the three axes are unequal in length. 

This same type of unit cell is appropriate for class m, the a and c axes 
lying in the plane of symmetry and the b axis being normal to this plane. 
It is equally appropriate for class 2/w. The three classes, 2, w, and 2/w, 
constitute the monoclinic system (Fig. 34). 

It would have been better if this unique axis were called c, to bring 



CHAP, n 



SHAPES OF CRYSTALS 



49 



the nomenclature into line with that of tetragonal, hexagonal, and the 
polar orthorhombic crystals, which all have their unique axes labelled c ; 
but the 6 convention for monoclinic crystals seems now too well estab- 
lished to be altered. Of the other two axes, the shorter is called c, and 
the obtuse angle /? between a and c is usually specified, rather than the 
acute angle. 

Orthorhombic (sometimes catted rhombic). In class mm ( = 2mm) the 
lattice points lie in planes normal to the twofold axis ; they also lie in 




FIG. 35. Orthorhombic, system. (Soo also Fig. 26.) a. Unit cell type. 6. ( 
Class 222. Left- and right-handed crystals, c. 1-Brom, 2-hydroxy-naphthalene. Class 222. 
d. Picric acid. Class mm. e. Oxalic acid. Class mmm. /. C 2 I3r 6 . Class mmm. 

the mutually perpendicular planes of symmetry m which are parallel to 
the twofold axis. The lattice is thus entirely rectangular, and the unit 
cell is a rectangular box with unequal edges. The twofold axis is usually 
called c ; of the other two, the longer is called b. 

The same type of unit cell is appropriate for classes 222 and mmm 
( = 2/m 2/m 2/m) ; the cell edges lie along the twofold axes.- Donnay 
(1943) recommends that the longest shall be called b and the shortest c. 
When axes are chosen on morphological grounds the axis of the principal 
prism zone is labelled c, while b is the longer of the other two. The three 
classes mm, 222, and mmm constitute the orthorhombic system (Fig. 35). 

Hexagonal and trigonal. In many crystals having a single three- or 
sixfold rotation axis or inversion axis the unique axis is taken as one of 

4458 v 



50 



IDENTIFICATION 



CHAP. II 



the unit cell edges, and this axis is called c. In all these crystals there 
are, in a plane normal to the principal axis, three equivalent directions 
which are at 120 to each other (see p. 26). Any two of these may be 
called a and 6. The unit cell thus has a diamond-shaped base, with a 
and b edges at 120 to each other and equal in length ; c is perpendicular 
to a and b and different in length. 

The twelve classes which may be referred to such a unit cell are : 3, 3m, 
32 ; 3, 3m ; 6, 6m2 ; 6, 6/m, 6mm, 62, 6/mmm. *Eor examples, see Fig. 36. 



0001 






d 



1 



FIG. 36. Hexagonal and trigonal systems. (Seo also Figs. 25, 28, and 30.) a. Hexa- 
gonal-type unit cell. 6. Apatite, 3Ca 3 (PO 4 ) 2 .CaF 2 . Class 6/m. c. Hydrocinchonine 
sulphate hydrate, (C 19 H2 4 ON 2 ) 2 . H 2 SO 4 . 11 H 2 O. Class Gm. d. Rhombohedral-type unit 
cell. e. A habit of calcite, CaCO 3 . Class 3m. /. KBrO 3 . Class 3. 

It is often more convenient to refer some trigonal crystals to a rhombo- 
hedral cell which has three equal axes making equal angles not 90 with 
each other. The three equal rhombohedral axes are equally inclined to 
the c axis of the hexagonal-type cell. 

Tetragonal. In all crystals having a single fourfold rotation axis or 
inversion axis there are, normal to this unique direction, two equivalent 
directions perpendicular to each other. The unit cell is thus entirely 
rectangular, with two edges (a and b) equal in length, and the remaining 
edge (the fourfold axis) different in length. The seven classes of the 
tetragonal system are: 4, 42m; 4, 4/m, 4mm, 42, and 4/mmm (Fig. 37). 

Cubic (sometimes called isometric, or tesseral). All crystals having four 
secondary threefold axes have three mutually perpendicular directions 



CHAP. II 



SHAPES OF CRYSTALS 



51 



SO" 






III 




Fin. 37. Tetragonal system. (See also Fig. 29.) a. Unit cell type. 6. Thloroglucinol 

diethyl ether. Class 4/wi. c. Wulfenite, PbMoO 4 . Class 4. d. Anatase, TiO 2 . Class 

4/mmm. e. Zircon, ZrSi() 4 . Class 4/mram. 



102 





i 






J ---* > tjP 


"r- -y* 









c-a 











A 










210 




\00 


W 


010 


a*? 


90" b-a 


^J 




A 


- 


- 


^ 


SO 








in 


y 




'021 







FIG. 38. Cubic system. (See also Figs. 17-23.) a. Unit cell typo. 6 andc. Two habits 

of pyrites, FoS 2 . Class w3. d. Tetrahedrito, Cu 3 SbS 3 . Class 43m. c. Spinel, MgAl,O 4 . 

Class w3?w. /. Almandine (Garnet), Fe 3 Al 3 (SiO 4 ) 3 . Class m3m. 

all equivalent to each other. The unit cell is thus a cube, the secondary 
threefold axes being the cube diagonals. The five classes of the cubic 
system are: 23, w3 ( == 2/w3), 43m, 43 ( = 432), and w3w ( = 4/m 3 2/m). 
Examples are shown in Fig. 38. 



52 



IDENTIFICATION 



CHAP. II 



The various names used formerly for the crystal classes are to be 
found, collected in a table of concordances, in Internationale Tabellen 
zur Bestimmung von Kristallstrukturen (1935). 

The essential symmetries and unit cell types for the different crystal 
systems are summarized in Table. I. 

TABLE I 



System 



Triclinic. 

Monoclinic. 

Orthorhombic. 

Trigonal and 
Hexagonal. 

Tetragonal. 
Cubic. 



Essential symmetry 
No planes, no axes. 

One twofold axis, or one plane. 

* 

Three mutually perpendicular 
twofold axes, or two planes inter- 
secting in a twofold axis. 

One threefold axis, or one sixfold 
axis. 

One fourfold axis or fourfold in- 
version axis. 

Four threefold axes. 



Unit cell 



Angles a, ft, and y unequal and not 
90. Edges a, 6, and c unequal. 

a = y = 90*. not 90. a, 6, and 
c unequal. 

a = j3 = y 90. a, 6, and c un- 
equal. 

(1) a - |3 - 90. y - 120. a =- b. 
c different from a and 6. 

(2) a = ft -= y, not 90. a =~ 6 = c. 

a = p ^ y =^ 90. a 6. c differ- 

ent from a and 6. 
a = j8 = y ^ 90. a b c. 



Deduction of a possible unit cell shape and point-group sym- 
metry from interfacial angles. When all the interfacial angles of a 
crystal have been measured on the goniometer, and the symmetries 
deduced by the contemplation of stereographic projections, the proce- 
dure in deducing the relative lengths of the unit cell edges and the angles 
between them follows from the contents of the foregoing notes. In most 
classes the directions of the edges are prescribed by the symmetry 
elements ; when they are not, the principle of simplest indices is called 
in to indicate the probable directions. In some of the tetragonal and 
hexagonal classes there are two sets of secondary axes or symmetry 
planes, providing alternative positions for the secondary (a and 6) edges 
of the unit cell ; the principle of simplest indices is again called in, but 
its verdict will not necessarily be correct ; X-ray diffraction photographs 
are necessary to settle such questions. In certain other tetragonal and 
hexagonal classes there is a single set of secondary twofold axes which 
are naturally chosen as probable unit cell edges. But this again is not 
necessarily correct : in some crystals the unit cell edges are parallel to 
twofold axes, while in others they bisect the twofold axes. The morpho- 
logical axes are, however, entirely adequate for morphological purposes ; 
and the morphological axial ratio c/a is related in a simple way to the 



CHAP. II 



SHAPES OF CRYSTALS 



53 



axial ratio of the unit cell usually by a factor of V2 in the tetragonal 
system and A/3 in the hexagonal system. 

Attention has already been drawn to the fact that the idealized shape 
of a crystal may exhibit a symmetry higher than that of the arrangement 
of atoms. Sodium chlorate crystals when grown rapidly in pure solution 
are cubes, the symmetry of which is holohedral (m3m) ; when sodium 
thiosulphate is present ip the solution the crystals grow as tetrahedra 
(symmetry ?3w); only when grown slowly in pure solution do the 
crystals exhibit the symmetry of the atomic arrangement that of the 
enantiomorphous class 23. In this case, and in many others, the true 



r 



\\\ 






c 



FIG. 39. Loft, bisphenoid of PbCJ 2 . Centre and right, arrangements of atoms 

on 111 and 111. The atoms depicted are those which lie on, or not far bolow, 

the plane of the corner atoms. 

point-group symmetry was known before the atomic arrangement was 
discovered by X-ray methods ; but in the case of sodium nitrite, NaN0 2 , 
which is orthorhombic, the habit of the crystals gives no evidence that 
the symmetry is other than holohedral (mmm), yet the X-ray diffraction 
pattern leaves no doubt that the atomic arrangement has the point- 
group symmetry mm the polar class of the orthorhombic system. 
(See Chapter IX.) 

The opposite may occur if crystal growth takes place in a solu- 
tion containing particular impurities. Miles (1931) showed that when 
lead chloride crystals, whose internal structure has the orthorhombic 
holohedral symmetry mmm, grow in a solution containing dextrine, 
they form bisphenoids, the symmetry of which is 222. It seems curious 
that a holohedral crystal should in any circumstances assume a hemi- 
hedral (holoaxial) shape. The reason is that the substance in solution 
which modifies the shape of the lead chloride crystals is itself asym- 
metric, and only left-handed molecules are present. Consider the 
arrangement of atoms at a particular level on the 111 plane of the lead 
chloride crystal (Fig. 39). This plane-pattern has no symmetry, and if 
we call the arrangement on 111 left-handed, the arrangement on Til is 



54 IDENTIFICATION CHAP, n 

right-handed. Now, modification of crystal habit by dissolved impuri- 
ties is due to adsorption of the impurity molecules on specific crystal 
faces, this adsorption reducing the rate of growth of these faces. If 
asymmetric left-handed molecules are present in the solution, and these 
are adsorbed on the 111 face, they are not likely to fit well on the 111 
face; consequently the rate of growth of 111 (as well as that of the 
equivalent faces TTl, TlT, and iTT) is reduced, while that of Til (and 
TTT, 111, and ill) is not, and the resulting crystal is entirely bounded 
by the first-mentioned set of planes and thus has a hemihedral form. 
To produce an effect of this sort, molecules of the dissolved impurity 
need not be entirely without symmetry, but they must lack planes of 
symmetry, inversion axes, and a' centre of symmetry. 

Such effects are probably rare, and when crystals are grown from 
solutions of high purity there is little danger of the occurrence of shapes 
which are misleading in this way. Nevertheless, the knowledge that 
such phenomena can occur prompts caution in accepting morphological 
evidence on internal symmetry when the conditions of growth are 
incompletely known (see p. 247). 

The shapes and orientations of the etching pits formed in crystal 
faces by appropriate solvents are also used as clues to internal symmetry 
(Miers, 1929). Here again, solvent molecules having only axial sym- 
metry must be avoided, as they may produce misleading effects, for 
reasons similar to those given in the case of crystal shape (Herzfeld and 
Hettich, 1926, 1927). 

The use of shape-symmetry and other morphological features in the 
study of the internal structure of crystals will be considered further in 
Part 2 of this book (Chapters VII and VIII). Here we are concerned 
with crystal shapes in so far as they afford evidence useful for the pur- 
pose of identification. 

Identification by shape. When a substance which it is desired to 
identify consists of well-formed single (that is, not aggregated) crystals 
of sufficient size to be handled, the interfacial angles may be measured 
on the goniometer; it is then possible to look up the morphological 
information on likely substances either in Groth's Chemische Krystallo- 
graphie or in papers scattered through the literature (chiefly chemical 
and mineralogical journals). An indirect method of this sort is, however, 
not always entirely satisfactory: possible substances may be overlooked. 
The desire for a direct method has led to attempts to devise a system in 
which morphological characteristics are measured and the results re- 
ferred to a classified index. Barker (1930) has devised a system in which 



CHAP, ii SHAPES OF CRYSTALS 55 

certain 'key' angles of the measured unknown crystal are looked up in 
an index in which substances are arranged in order of the magnitudes 
of these key angles. The selection of the key angles for an unknown 
crystal involves the indexing of all the faces on the crystal, and thus 
implies the deduction of a possible unit cell shape. Barker does not use 
the term 'unit cell', and does not claim for his system anything more than 
that it is a consistent scheme for the morphological description and 
identification of crystals t but the term 'unit cell' will be retained here, 
since the treatment in this book is entirely based on this conception. 

For the purpose of identification the fact that the 'morphological unit 
cell' does not always coincide with the true unit cell does not matter, 
provided that all crystals of the sam species give the same morpho- 
logical cell in the hands of different investigators. The problem is to 
devise rules which ensure this, even in the triclinic system, where none 
of the axial directions are fixed by symmetry. The rules devised by 
Barker, together with some additions by Porter and Spiller (1939), 
constitute a sound system, and a card index for the method is in exist- 
ence, though it is not yet (1944) published. The rules will not be 
described here in detail ; but we may observe that the system is based 
on a thorough -going acceptance of the principle of simplicity of indices, 
and that a definition of simplicity is given all indices composed only 
of O's and 1's being regarded as equally simple, and all others complex. 
Another point is that class-symmetry within a particular system is 
ignored ; this is necessary in view of the frequency with which crystals 
display in their shapes too high a point-group symmetry (this being in 
some cases variable with growth conditions). 

One limitation of morphological methods has already been mentioned : 
some crystals, especially those grown rapidly, are entirely bounded by 
faces parallel to the unit cell sides, and measurements of the interfacial 
angles of such crystals can only give the angles between the axes, not 
their relative lengths (except where symmetry indicates that two or 
more axes are equal in length, as in the cubic, tetragonal, hexagonal, 
and trigonal systems). Another limitation arises from the fact that all 
crystals belonging to the cubic system have the same shape of unit cell, 
and therefore cannot be identified by purely morphological methods. 
It is true that different crystals belonging to the cubic system often 
have different bounding faces, some growing normally as octahedra, 
others as tetrahedra, and so on ; but there are many different crystals 
of octahedral habit, and many others of tetrahedral habit. In addition, 
it must be remembered that the shape may be completely changed by 




56 IDENTIFICATION CHAP, u 

the presence of certain impurities in the solution. Thus, shape is of very 
little use for identification in the case of crystals belonging to the cubic 
system. 

It is not intended to describe purely morphological methods of 
identification in any more detail in this book, for we are concerned with 
the crystals found in the average experimental or industrial product, 
and for these crystals the practical limitations imposed by small size 
or irregular shape <are often sufficient to rule eut goniometric methods. 
With regard to size, it should be realized that crystals as small as otie- 
qr two-tenths of a millimetre in each direction can be handled and 

measured on the goniometer. 
Generally speaking, however, 
crystals suitable for goniometric 
measurements are either speci- 
ally selected mineral specimens 
or crystals specially grown for 
the purpose. Not all crystals can 
be grown under laboratory con- 
ditions to a size suitable for 

FIG. 40. Orthorhombic crystal lying on 

(001) on microscope slide. handling ; very sparingly soluble 

substances, for instance, might 

require a geological age for growth to such a size. Moreover, it may 
be desired to identify the products of chemical reactions in which it is 
not possible to prescribe suitable crystallization conditions. 

For well-formed microscopic crystals the scope of purely morpholo- 
gical methods is usually limited to qualitative observations which may 
enable us to deduce the type of unit cell. Sometimes it may be possible 
to measure interfacial angles approximately, but only when the crystals 
lie in such a way that the two faces in question are both parallel to the 
line of vision ; for instance, if an orthogonal crystal is lying on the slide 
on its (100) face (Fig. 40), the angle between (110) and (010) faces could 
be measured by bringing them successively parallel to the eyepiece 
cross-wire, and reading off the angle through which the slide or the 
eyepiece has been turned. This would give us an approximate value for 
the axial ratio. For very many crystals, however, interfacial angles 
cannot be measured ; we may be able to conclude that a crystal probably 
has a tetragonal or a monoclinic unit cell, but we cannot deduce the 
relative dimensions of the cell. 

Further, it may often be desired to identify poorly formed crystals 
such as needle-like crystals without definite faces, or skeletal growths > 



CHAP. II 



SHAPES OF CRYSTALS 



67 



or even completely irregular fragments. This can only be achieved by 
measuring some properties of the crystal material itself, properties 
which are independent of the shape of the crystals. Of such properties, 
by far the most important and the most convenient for measurement 
are the optical properties, especially the refractive indices. An ele- 
mentary account of the optical properties of crystals will be found in 
Chapter III. 

Before leaving the subject of crystal shape there are a few other 
morphological features which are sometimes encountered and must be 
mentioned briefly. 





FIG. 41. Twinning, a. Gypsum, CaSO 4 . 2H 2 O. Two individuals joined at a well-marked 
plane (100). b. 'Interpenetration' twin of fluorspar, CaF 2 . One individual is rotated 60 
with respect to the other. The junction surface in such twins is often very irregular, 
c. 'Mimetic' twin of ammonium sulphate, (NH 4 ) 2 SO 4 . Six individuals, with three 
different orientations (numbered). 

Twinning. Two or more crystals of the same species are sometimes 
found joined together at a definite mutual orientation, this orientation 
of the individual crystals being constant in different examples of any 
one species. Such crystals are said to be twinned. Certain species show 
this phenomenon frequently, and some species invariably. The most 
frequent type of twinning is that of calcium sulphate dihydrate (gyp- 
sum), which is often found in the form shown in Fig. 41 a. The two 
crystals appear to be joined at the 100 plane. At the junction there is 
presumably a sheet of atoms common to the two individuals ; when the 
crystal nucleus was formed, two lattices were probably built by deposi- 
tion on opposite sides of this common sheet of atoms. 

Sometimes twinned crystals appear to be interpenetrating, as in the 
calcium fluoride twin illustrated in Fig. 41 6. Here we may imagine 
(in the crystal nucleus) a common 111 sheet of atoms, the symmetry of 
which is trigonal ; the crystal on one side of it is rotated 60 with respect 
to the one on the other side. The twin plane is not always respected 
during subsequent growth ; one individual may encroach on the domain 
of the other, so that the junction surface in the final crystal is irregular. 



68 IDENTIFICATION CHAP, n 

There are many other types of composite shape which arise as the result 
of twinning ; for further examples, see the text-books of Miers and Dana. 

Twinning always involves the addition of a plane or an axis of sym- 
metry, and the symmetry of the composite shape may thus be higher 
than that of an individual crystal of the same species. When the com- 
posite shape has no re-entrant angles it may appear deceptively like 
that of a single crystal of higher symmetry ; thus, ammonium sulphate 
crystals grown in solutions containing ferric iflns form hexagonal prisms 
(Fig. 41 c). The atomic arrangement in ammonium sulphate crystals has 
orthorhombic symmetry, but the conjunction of six sectors with three 
different orientations (opposite sectors having the same lattice orienta- 
tion) gives rise to apparent hexagonal symmetry. The same thing occurs 
in aragonite, the orthorhombic form of calcium carbonate. (For the 
atomic structure on the twin plane see Bragg, 1924 a.) The occurrence 
of such mimetic twinning may cause confusion if its existence is not 
realized. The study of such phenomena is greatly assisted by the use 
of the polarizing microscope; this is dealt with in the next chapter. 

In some crystals the energy of addition of material to a crystal face in 
such a way as to start a new twinned individual may be almost the same 
as that of carrying on a single-crystal structure ; frequent changes may 
thus occur, giving rise to a fine lamellar 'repeated-twinning' structure. 
Here again the polarizing microscope may reveal at once the composite 
character of the structure. 

Cleavage. The cohesion of crystals is not the same in all directions. 
It may be very strong in some directions and very weak in others ; so 
much so that many crystals, on crushing or grinding, break almost 
exclusively along certain planes. The most striking of familiar examples 
is mica, a potassium aluminium silicate mineral which readily cleaves 
into thin sheets. Similarly crystals of calcite, the rhombohedral variety 
of calcium carbonate, break into small rhombohedra ; sodium chloride 
crystals tend to break along planes parallel to the cube faces ; calcium 
fluoride (fluorspar) crystals cleave along the octahedral planes. Minerals 
like chrysotile ('asbestos') have more than one cleavage parallel to the 
same crystal direction and very readily split into fibres. 

Cleavage planes are always planes of high reticular density of atomic 
or molecular packing and large interplanar spacing, the cohesion being 
strong in the plane and weak at right angles to the plane. Cleavage 
planes thus have simple indices, and in fact are often parallel to the 
principal faces of the crystal ; thus calcite, when precipitated in the 
laboratory, often grows in the form of simple rhombohedra whose faces 



CHAP, ii SHAPES OF CRYSTALS 69 

are parallel to the cleavage planes. But this statement, like the principle 
of simplest indices for the faces of growing crystals, is only a broad 
generalization, not a rigid rule. An exception, for instance, is shown 
by calcium fluoride, which usually grows as cubes but cleaves along 
octahedral faces (Wooster, N., 1932). Another is penta-erythritol, 
C(CH 2 OH) 4 , which grows as tetragonal bipyramids but has basal 
cleavage (001). 

Polymorphism. Some substances form, under different conditions, 
crystals of quite different internal structure ; they are then said to be 
polymorphic. The different structures are different packings of the same 
building units. Sometimes one particular structure can only exist within 
a definite temperature range, and if tte temperature goes outside this 
range there is a rapid reorganization of the building units (atoms, mole- 
cules, or ions) to form a different arrangement. Sulphur, for instance, 
forms an orthorhombic arrangement at room temperature and a mono- 
clinic arrangement above 95 C. An extreme example is ammonium 
nitrate, which exists in five different crystalline forms, each of which 
changes to another at a definite temperature. Other substances exist 
in two or more forms which are apparently equally stable at the same 
temperature. Calcium carbonate, for instance, occurs in a rhombohedral 
form, calcite, and an orthorhombic form, aragonite, both of which have 
existed in the earth's crust for geological ages. Actually calcite is prob- 
ably slightly more stable than aragonite at all temperatures, but the 
atomic motions in aragonite crystals at ordinary temperatures are so 
small that no reorganization is possible. There is also a much less stable 
form, jLt-CaC0 3 or vaterite, which is apparently hexagonal. 

Isomorphism and mixed crystal formation. The atomic arrange- 
ment in crystals of ammonium sulphate, (NH 4 ) 2 S0 4 , is entirely analogous 
to that found in potassium sulphate (K 2 S0 4 ) crystals, the ammonium 
ion playing the same role in the structure as the potassium ion. 
The unit cell dimensions of the two crystals are very nearly the same, 
and the shapes of crystals grown under similar conditions are almost 
the same. Accurate gonibmetric measurements would be necessary to 
distinguish between the two crystals by morphological methods. Such 
crystals are said to be isomorphous. The reason for this close resem- 
blance is that ammonium and potassium ions are very similar in size and 
chemical character ; they can therefore fit into the same arrangement 
with sulphate ions. When the ionic sizes are closely similar, they can 
replace each other indiscriminately in the lattice; a mixed solution 
of ammonium and potassium sulphates deposits crystals which may 



60 



IDENTIFICATION 



CHAP. II 



contain any proportions of the two substances, and which have unit cell 
dimensions intermediate between those of the pure components. Such 
crystals are called 'mixed crystals' or 'crystalline solid solutions'. 

Not all isomorphous substances form mixed crystals. Calcite (CaC0 8 ) 
and sodium nitrate (NaN0 3 ) form similar atomic arrangements, their 
unit cells are both rhombohedra of very similar dimensions, and also 
the corresponding ions are closely similar in size; but they do not 
form mixed crystals : the reason presumably'is that their solubilities in 
water are extremely different. 




FIG. 42. Oriented overgrowths of urea on ammonium chloride. 

Oriented overgrowth. Isomorphous substances which do not form 
mixed crystals may do the next best thing ; one crystal may grow on the 
other in parallel orientation. ^Sodium nitrate grows on calcite in this 
way. Isomorphism is not, however, a necessary condition for oriented 
overgrowth ; it is sufficient if the arrangement of atoms on a particular 
plane of one crystal is similar, both in type, dimensions, and distri- 
bution of electrostatic charges, to the arrangement on one of the planes 
of the other crystal ; the two structures may be in other respects com- 
pletely different from each other (Royer, 1926, 1933). Thus, tetragonal 

urea, 0=C<^ 2 , grows with its 001 plane precisely oriented in 
JN xlo 

contact with the cube faces of ammonium chloride, NH 4 C1, Fig. 42 ; 
the two structures are completely different except for a formal and 
dimensional similarity on the planes in question (Bunn, 1933). 



CHAP, n SHAPES OF CRYSTALS 61 

Of the morphological phenomena mentioned in the last few para- 
graphs, that of twinning is likely to be of most frequent value in 
identification problems. But all the phenomena are significant from the 
point of view of structure determination. The subject of crystal 
morphology in relation to internal structure will not, however, be 
pursued further at present ; it will be taken up again in Chapters VII 
and VIII. For the present, we shall oontinue our consideration of the 
problem of the identification of microscopic crystals ; we pass on to 
discuss crystal optics, the relation between optical properties and 
crystal shape and symmetry, and the determination of refractive indices 
and other optical characteristics under the microscope. 



Ill 

THE OPTICAL PROPERTIES OF CRYSTALS 

THE physical properties of crystals, such as refractive index, absorption 
of light, and conduction of heat and electricity, are in general not the 
same for all crystal directions; in other words, a three-dimensional 
graph of any characteristic showing its magnitude for all directions is 
not, except in certain special cases, a sphere, but a less symmetrical 
figure, owing to the fact that on passing through a crystal the sequence 
of atoms encountered depends on the direction taken. The type of shape 
of the three-dimensional grapK is not the same for all characteristics 
and naturally varies with crystal symmetry, but one generalization 
that can be made is that the figure must necessarily exhibit a symmetry 
at least as high as that of the atomic pattern in the crystal. The 
symmetry of the figure may be higher than that of the atomic pattern 
(just as the shape of a crystal may have a higher symmetry than that 
of the atomic pattern), but it cannot be lower. If there is a plane of 
symmetry in the atomic pattern, then there must be a corresponding 
plane of symmetry in the figure ; if there is an axis of symmetry or a 
centre of symmetry in the atomic pattern, then these also are necessarily 
exhibited by the figure. 

In this chapter we are concerned chiefly with the refractive indices 
of crystals and other phenomena depending on the refractive indices. 
The absorption of light and the rotation of the plane of polarization 
are also considered briefly. The treatment of crystal optics followed in 
this book is restricted to those aspects which are most generally useful 
for purposes of identification or structure determination. The finer 
points of crystal optics, and aspects which are of physical rather than 
chemical interest, may be pursued in more comprehensive text-books, 
such as Miers's Mineralogy, Tutton's Crystallography and Practical 
Crystal Measurement, Hartshorne and Stuart's Crystals and the Polariz- 
ing Microscope, Wooster's Crystal Physics, and Preston's Theory of 
Light. 

The refractive index of a solid is usually defined in terms of SnelTs 
law, which states that when a ray of light changes its direction on pass- 
ing from one medium to another the ratio of the sine of the angle of 
incidence to that of the angle of refraction is a constant ; this constant 
is the refractive index of the second medium with respect to the first. 
For the consideration of the optical properties of crystals, however, it 



CHAP, ra OPTICAL PROPERTIES 63 

is better to think of the refractive index, not as a measure of the bending 
of a ray of light when it passes from air into the solid, but as a measure 
of the velocity of light in the solid : the refractive index of a solid with 
respect to air is the ratio of the velocity of light in air to the velocity 
in the solid. By thinking in this way we are focusing our attention on 
a particular direction in the crystal. 

The first point to be made is that in a crystal the refractive index 
depends not on the direction in which the electromagnetic waves are 
travelling but on the direction of the electrical disturbances transverse 
to the line of travel the Vibration direction'. We have to consider 
the shape of the graph connecting refractive index with vibration 
direction for each crystal system, and the methods available for 
measuring the refractive indices of crystals in different vibration 
directions. 

Cubic crystals. Crystals with cubic unit cells have the same atomic 
arrangement along all three axial directions; consequently all the 
properties of the crystal are identical along these three directions. 
The optical properties are found to be the same, not only along these 
three directions, but also for all other directions. An attempt at an 
explanation of this would take us too deeply into the electromagnetic 
theory of light ; we shall therefore simply accept the fact that a cubic 
crystal is optically isotropic it behaves towards light just like a piece 
of glass ; its refractive index is the same for all vibration directions of 
the light. To identify a cubic crystal it is usually sufficient to measure 
its one refractive index. 

Measurement of refractive index under the microscope. The 
measurement of the refractive index of an isotropic transparent solid 
under the microscope is extremely simple. The principle is to keep a 
set of liquids of known refractive indices, and to find which liquid has 
the same (or nearly the same) refractive index as the solid in question. 
When the solid particles are immersed in this liquid they become 
invisible ; the light, in passing from liquid to solid and from solid to 
liquid, is not refracted, and consequently the edges of the particles 
cannot be seen ; as far as the light is concerned, the whole complex is 
a homogeneous medium. 

The procedure is to immerse particles of the solid in a drop of liquid 
of known refractive index on a microscope slide, cover the drop with a 
thin cover-glass, and observe the particles, using a low or moderate 
magnification ( J-inch objective and 4 or 10 times eyepiece, for instance) 
and parallel or nearly parallel light. If the particles show up plainly, 



64 



IDENTIFICATION 



CHAP, in 



their refractive index must differ considerably from that of the liquid ; 
other liquids of different refractive indices are then tried, until a liquid 
is found in which the particles are invisible or very nearly so. The 
search for the right liquid is not as laborious as one might suppose, 
because it is possible, by observing certain optical effects, to tell whether 
the refractive index of the liquid is higher or lower than that of the 
crystal; and with experience, one can estimate roughly how much 
higher or lower. These optical effects are illustrated in Fig 43, Plate II, 
in which a shows cubic crystals of sodium chlorate (refractive index 
n = 1-515) immersed in a liquid of n = 1-480. If the crystals are first 
of all focused sharply, and if then the objective is raised slightly (by 
means of the fine adjustment of the microscope), a line of light (the 



'SOUP 



(a) 




(b) 



FIG. 44. The 'Becke line' effect. 

'Becke line') is seen inside the edges of each crystal ; as the objective is 
raised more and more, the line contracts farther and farther within the 
boundaries of the crystal. This is what happens when the refractive 
index of the liquid is less than that of the crystal ; but if the reverse is 
true, as in Fig. 436, Plate II, the Becke line appears round the outside 
of the crystal when the objective is raised and expands as the objective is 
raised farther. The shape of the particles does not matter ; the Becke 
line always follows the outline of the particle ; the determination of the 
refractive index of irregular fragments of crystals, or of particles of 
glass, is just as easy as that of well-formed crystals. 

The simplest way of regarding the Becke line effect, as well as the best 
way of remembering which way the line moves, is to think of a particle 
as a crude lens which, if it has a refractive index higher than that of 
the medium surrounding it, tends to focus the light at some point 
above it (Fig. 44) ; when the objective is raised it is focused on a plane 
PP above the particle, and in this plane the refracted light waves 
occupy a smaller area than they do in a plane nearer the particle, and 
thus the boundary line of light moves inwards as the objective is raised. 
If the refractive index of the particle is lower than that of the sur- 
rounding liquid, it will have the opposite effect and act as a negative 







FIG. 43. a. Crystals of Hodium chlorate, NaGlO,, in liquid of refractive index 148; 
objective raised. 6. The same substance in liquid of refractive index 1-55; objective 
raised, c. Crystal of monammoriium phosphate, NH 4 H 2 PO 4 , in liquid of refractive index 
1-500; polarized light, vibration dirertion vertical; objective raised, d. Tho same, 
vibration direction horizontal; objective raised, e. Mixture NaBr.2H 2 O and NaBrO, 
in liquid of refractive index 1-54. The NaBrO 3 crystals show up in relief. 



CHAP, m OPTICAL PROPERTIES 65 

lens (Fig. 44) ; consequently on raising the objective the boundary line 
of light expands. 

By observing this effect and trying various liquids in turn, it is 
possible to find in a few minutes a liquid in which the particles are nearly 
invisible. In practice, it is convenient to keep a set of liquids with 
refractive indices differing by 0-005. Refractive index values are nearly 
always given for sodium D light, and the liquids are therefore standar- 
dized for this wavelength. (For suitable liquids, see Appendix 1.) 
Usually, of course, the refractive index of the particles is found to lie 
between those of two of these liquids ; its value can be estimated from 
the magnitude of the Becke line effects in the two liquids. In this way 
the refractive index of isotropic particles can be found within limits of 
0-002. It must be mentioned that solid particles are seldom quite 
invisible in liquids, because the dispersion of the liquid (variation of 
refractive index with colour of light) is usually different from that of the 
solid ; consequently, if the refractive indices of solid and liquid are 
equal for yellow light, they are not equal for red or blue light, and there- 
fore in white light, coloured Becke line fringes will be seen round the 
edges of the crystals. For this reason it is sometimes suggested that 
monochromatic light should be used for refractive index determinations ; 
in practice, however, sufficient accuracy for identification purposes is 
usually obtainable by the use of white light, which is also more pleasant 
in use. 

Tetragonal, hexagonal, and trigonal crystals. Preliminary. 
The simple method just described is applicable as it stands only to 
isotropic solids, that is, to glasses and amorphous solids in general, and 
to crystals belonging to the cubic system. In all other crystals the 
refractive index varies with the direction of vibration of the light in 
the crystal ; the optical phenomena are more complex, and it is necessary 
to disentangle them. 

If tetragonal crystals of monammonium phosphate, NH 4 H 2 P0 4 
(Fig. 45), lying on the microscope slide on their prism faces, are examined 
in the way already described with ordinary unpolarized light, it is not 
possible to find any liquid in which they are nearly invisible. In liquids 
with refractive indices below 1*479 it is clear that the crystals have a 
higher index than the liquids; in liquids with indices above 1-525 it is 
equally clear that the crystals have lower indices than the liquids; 
but in liquids with indices between 1-479 and 1-525 confusing effects 
are seen Becke lines can be seen both inside and outside the crystal 
edges. This is because the crystal resolves light into two components 

445S v 



66 



IDENTIFICATION . 



CHAP. Ill 



(a) 
A 




A 




(c) 


t 
1-479 


1-52S 


X 



\7 



vibrating in different planes,| and the refractive indices of the two 
components are unequal ; the crystal thus 'shows' two different indices 
at the same time. 

In order to observe one refractive index at a time, we must evidently 
use polarized light light vibrating in one plane only and adjust its 
plane of vibration to coincide with one of the planes of vibration in 
the crystal itself. In the polarizing microscope, plane polarized light 
is obtained by means of a Nicol prism placed between the light source 
and the microscope slide ; it is usually located immediately below the 
condenser which concentrates light on the slide. The plane of vibration 

can be adjusted with respect to the 
crystal either by rotating the Nicol 
prism or (on other types of micro- 
scope) by rotating the microscope 
slide; the two cross-wires in the 
eyepiece indicate planes parallel 
and perpendicular to the plane of 
vibration of the light transmitted 
by the polarizer. 

If a crystal of monammonium 
phosphate is immersed in a liquid of 

refractive index 1-500, and observed in light vibrating along the four- 
fold axis, the Becke line effect (Fig. 43 d, Plate II) shows that the index 
of the crystal is lower than that of the liquid ; if the polarizer is turned 
through 90, the index of the crystal is seen to be higher than that of the 
liquid. 

If now we immerse the crystals in various liquids, and observe each 
crystal in light vibrating parallel to its fourfold axis, we observe con- 
sistent effects as in the case of isotropic solids in ordinary light, and we 
find the refractive index is 1-479. If we use light vibrating perpendicular 
to the fourfold axis of the crystal, we again observe consistent effects 
and this time find the refractive index to be 1-525. (Fig. 45 a and b). 

That tetragonal crystals should have one refractive index for light 
vibrating along the fourfold axis and a different index for vibration 
directions perpendicular to this axis is only to be expected, since the 
arrangement of atoms along the c axis (the fourfold axis) is different 
from that along the a and b axes. The same is true for hexagonal and 
trigonal crystals; the refractive index for light vibrating along the 

t The plane of vibration is the plane containing the direction of propagation and the 
direction of the electrical disturbances associated with the waves. 



FIG. 45. Refractive indices of monammo- 
nium phosphate, NH 4 H 2 PO 4 . Arrows 
indicate vibration directions. 



CHAP, ra OPTICAL PROPERTIES 67 

unique sixfold or threefold axis is different from the index for light 
vibrating in directions perpendicular to this axis. The only new and 
perhaps unexpected phenomenon to be grasped is that the crystal 
actually resolves the light into two components vibrating at right angles 
to each other, and that the crystal therefore 'shows' two different 
refractive indices simultaneously except when the incident light is 
polarized and vibrates along one of the crystal's vibration directions. 

Use of crossed Nicols. Extinction directions. Interference 
colours. To set the polarizing NicoFs vibration plane parallel to one of 
th6 crystal's vibration planes is simple for crystals such as those already 
considered. But suppose we have crystals which are irregular fragments, 
so that there are no edges to guide us*? The vibration planes of such 
crystals are found by making use of a second Nicol prism, the 'analyser', 
which is'jplaced somewhere between the crystal and the observer's eye ; 
in the polarizing microscope it is located either in the tube of the micro- 
scope or above the eyepiece. The vibration plane of the analyser is set 
perpendicular to that of the polarizer, so that the light passed by the 
polarizer, as long as it continues to vibrate in the plane imposed on it 
by the polarizer, will be completely stopped by the analyser, f If we 
look through the microscope with the Nicols 'crossed' in this way we 
shall see a dark background. If the particles we are observing happen 
to be isotropic we shall see nothing at all ; but if, like monammonium 
phosphate crystals, they are birefringent that is, have two different 
refractive indices we shall see that most of the crystals are illuminated, 
often with beautiful colours. Moreover, if we rotate the Nicols together 
(keeping them exactly crossed all the time), or alternatively rotate the 
microscope slide, we shall see that each crystal is 'extinguished' at a 
certain position, only to reappear as the Nicols or the slide are rotated 
further. It will be found that the extinction positions for any one 
crystal are 90 apart ; extinction occurs four times during a complete 
revolution. 

The explanation of these phenomena is as follows. Suppose the 
polarizer transmits light vibrating in the plane P (perpendicular to the 
page), Fig. 46 a ; when it gets to a crystal of monammonium phosphate 
which happens to be lying in such a position that its vibration directions 
are not parallel to either of the cross-wires (vibration directions of the 
Nicols), it is resolved by the crystal into two components, vibrating in 

t When light is resolved into a vibration plane which makes an angle with its original 
vibration piano, the resolved part has an intensity equal (apart from absorption effects) 
to a fraction cos a of the original intensity. 



68 



IDENTIFICATION 



CHAP. Ill 



the crystal's own vibration directions X and Z. When this light, which 
now consists of the two components X and Z, passes through the 
analyser, each component is again resolved by the analyser into its 
own vibration direction A, so that the light emerges from the analyser 
as a single component but now vibrating in plane A. In this position, 
therefore, the crystal transmits light. But now consider Fig. 46 b in 
which the crystal's vibration directions coincide with the vibration 
directions of the Nicols. Light from the polarizer, vibrating in plane P, 
on arriving at the crystal continues vibrating in plane P since this is 
also the crystal's own vibration plane X ; the resolved part in plane Z is 
zero. On arriving at the analyser, all the light is necessarily stopped, 

(c) 



Z 
P 
FIG. 46. Crystals of monammoniiirn phosphate between crossed Nicols. 



since it is still vibrating in plane P and the analyser cannot transmit it, 
the resolved part in the analyser's vibration plane A being zero. In this 
position, therefore, the crystal is extinguished. The same thing occurs 
when the fourfold axis of the crystal is parallel to A (Fig. 46 c), and this 
position, 90 from the first-mentioned position, is thus also an extinction 
position. At all intermediate positions the crystal will be illuminated, 
the intensity of illumination being greatest at the 45 position. 

Thus, extinction occurs when the vibration directions of the Nicols 
coincide with those of the crystal. 

This explains illumination and extinction ; but what of the colours ? 
To understand the production of colours we must consider the relative 
velocities of the two components X and Z in the crystal. We have 
already seen that in a crystal of monammonium phosphate the refractive 
index for component Z is greater than for component X ; this means 
that light vibrating along Z travels through the crystal more slowly than 
light vibrating along X, the ratio of the velocities being inversely 
proportional to the ratio of the refractive indices. The frequency v of 
any monochromatic component of the white light naturally remains 
constant ; therefore, since vA = velocity, the wavelength A is smaller 
for component Z than for component X. The two components start in 



CHAP. Ill 



OPTICAL PROPERTIES 



69 



phase with each other at the bottom of the crystal (Fig. 47), but when 
they reach the top of the crystal it is likely that they are no longer 
exactly in phase with each other. When they reach the analyser they 
are resolved into the same plane of vibration and are able to interfere 
with each other. Whether or not they entirely cancel each other out 
depends on the difference of phase. Now, for a given thickness of a 



INTENSITY 
ZERO 



INTENSITY 
STRONG 



TWO RAYS POLARIZED 
AT RIGHT ANGLES 




FIG. 47. Birefringent crystal between crossed Nicols, in 45 position. 

particular crystal, only one particular wavelength of light will be 
completely cut out by interference; for other wavelengths there will 
be only a diminution of intensity. If the thickness of the crystal and 
the values of the two refractive indices are such that blue light is entirely 
cut out by interference, the colour we shall see will consist of the rest of 
the spectrum a yellowish polour ; if red light is cut out, we shall see 
a greenish colour, and so on. For increasing thicknesses of crystal the 
colours given are in the order known as 'Newton's scale' ; it is the same 
order as that of the interference colours given by very thin films, such 
as oil films on a wet road. The order can be studied on any birefringent 
crystal of varying thickness, such as the pyramidal ends of the crystals 



70 IDENTIFICATION . CHAP, in 

of monainmonium phosphate ; the colours appear as bands like contour 
lines on the crystals. The colour produced is determined by the bire- 
fringence of the crystal (the difference between the two refractive 
indices) and its thickness. 

To return to the extinction phenomenon. We now know how to set 
the polarizer so that its vibration direction coincides with one of the 
vibration directions of the crystal: we make use of the extinction 
phenomenon in the following way. Keeping tlie polarizer always in the 
illuminating beam, focus a particular crystal ; introduce the analyser 
(crossed with respect to the polarizer) and rotate either the crystal or 
the coupled pair of Nicols until extinction occurs ; then remove the 
analyses and observe the Becked line effect. Reintroduce the analyser, 
and turn either Nicols or crystal through 90 to the other extinction 
position; after removing the analyser once again, observe the Becke 
line effect for the second time. These observations reveal the relations 
between the refractive indices of the crystal and that of the liquid. 
Suppose one index of the crystal is lower and the other higher than that 
of the liquid (Fig. 43 c, d, Plate II). Try liquids of lower index until 
one is found whose index is equal to the lower of the two indices of the 
crystals ; and subsequently, seek the higher of the two indices of the 
crystals in a similar way. 

Crystals which are all lying in the same position, such as monam- 
monium phosphate crystals lying on their prism faces, give consistent 
results when examined in this way ; but if these crystals are crushed to 
provide irregular fragments capable of lying on the microscope slide in 
any orientation, and these fragments are examined in the same way, it 
will be found that although the upper index of each fragment is con- 
stant and equal to 1-525, the lower index is different for each fragment, 
and may have any value between 1'479 and 1-525. This brings us to a 
general consideration of the refractive indices for all possible orienta- 
tions with respect to the transmitted light. 

The Indicatrix. Imagine a point within a crystal, and from this 
point lines drawn outwards in all directions, the length of each line 
being proportional to the refractive index for light vibrating along the 
line. It is found that for all crystals the ends of these lines fall on 
the surface of an ellipsoid, a solid figure all sections passing through 
the centre of which are ellipses. This ellipsoidal three-dimensional 
graph of refractive indices is called the 'indicatrix'. For monammonium 
phosphate crystals and for all tetragonal, hexagonal, and trigonal 
crystals the indicatrix is a special type of ellipsoid (Fig. 48) in which 



OPTICAL PROPERTIES 



71 



OPTIC 




FIG. 48. Left: positive uniaxial indicatrix. Right: negative 
uniaxial indicatrix. 



LINE 

OF 

VISION 



OPTIC 
AXIS 



two of the principal axes are equal to each other and the third different 
in length (it may be longer or shorter) ; it is an 'ellipsoid of revolution* 
obtained by rotating an ellipse round one of its principal axes in 
the case of monammonium phosphate, 
round the minor axis. The ellipsoid 
thus has one circular section perpen- 
dicular to the unique axis. The unique 
axis of this ellipsoid of revolution 
necessarily coincides with the unique 
(fourfold, sixfold, or threefold) sym- 
metry axis of the crystal. 

The vibration directions and refrac- 
tive indices of crystal fragments of 
monammonium phosphate lying on a 
microscope slide in any orientation are 
given by the indicatrix in the following 
way. A crystal fragment, oriented with 
its unique axis at any angle 6 to the line 
of vision, is mentally replaced by the 
indicatrix (Fig. 49). Perpendicular to the 
line of vision, imagine a section PQ passing through the centre of the 
ellipsoid ; this section is an ellipse, and its principal axes (the maximum 
and minimum radii of the ellipse) represent the vibration directions 
and refractive indices of the crystal fragment. Now the maximum 
radius OD of every such ellipse is also a radius o> of the one circular 




FIG. 49. Uniaxial indicatrix- 
general orientation. 



72 IDENTIFICATION CHAP, ui 

section of the indicatrix.f The minimum radius c' of the ellipse, how- 
ever, varies with the angle 0. In general, the length e' lies between c 
and a) ; when 8 = 90, as it is for well-formed monammonium phosphate 
crystals lying on their prism faces as in Fig. 45 a, c' is equal to c, the 
unique axis of the indicatrix ; when = 0, as it is for crystals of this 
substance standing on end as in Fig. 45 c, e is equal to to, the radius of 
the one circular section. The observed refractive indices of crystal 
fragments of monammonium phosphate are in line with this: every 
fragment has an upper index equal to 1-525, but the lower index varies 
in different fragments between 1-479 and 1-525. 

The method of finding the principal refractive indices of such crystals 
even when quite irregular is therefore simple : numerous fragments are 
examined, each fragment being observed in its two extinction positions ; 
the two principal refractive indices are the extreme upper and lower 
values observed. The upper principal index is (for this particular sub- 
stance) the easier to find because every fragment, however oriented, 
gives this value as its upper index. The lower principal index is the 
lowest of the lower values of all fragments. 

When we are looking along the unique axis, both indices of the 
crystal or fragment are equal to 1-525; the crystal will therefore not 
show any interference colours when examined between crossed Nicols ; 
it will appear to be isotropic. This direction of apparent isotropy is 
called the optic axis ; there is only one such direction in the crystals we 
have hitherto dealt with tetragonal, hexagonal, and trigonal crystals 
and such crystals are therefore described as optically uniaxial. The 
optic axis necessarily coincides with the principal symmetry axis. 

The principal refractive indices of uniaxial crystals are usually symbol- 
ized a) or n w for the more important of the two, the one which is con- 
stant for all orientations, and e or n for the other one. When e is less 
than co (as in monammonium phosphate) the crystal is described as 
uniaxial negative ; when e is greater than o>, as in quartz, SiO 2 (o> = 
1-544, c = 1-553), the crystal is described as uniaxial positive. 

The method for the determination of the principal refractive indices 
of irregular fragments has been described, not only because such 
material may often be encountered in chemical work, but also for 
another reason. Well-formed crystals of many uniaxial crystals are 
of such a shape that, when lying on any one of their faces on a micro- 
scope slide, they do not show both the principal indices. Rhombohedra 

t Wire models will make this and other features of the optical indicatrix clearer than 
plane diagrams can possibly do. 



CHAP, in 



OPTICAL PROPERTIES 



73 



or bipyramids, for instance, do not show c ; they necessarily show w as 
one of their indices, but the other index lies between eo and c. In such 
circumstances it is advisable to break the crystals so as to provide 
irregular fragments, and to seek in the way already described. 

Orthorhombic crystals. The symmetries of the orthorhombic 
classes either three mutually perpendicular planes of symmetry, or 
three mutually perpendicular twofold axes, or two perpendicular planes 
intersecting in a twofold axis demand that the indicatrix, which is of 
the most general type with all three principal axes of unequal length, 
has these three axes parallel to the crystallographic axes. The inequality 

c \1-S06 



*/ m 

1-526 





(a) 



(b) 



MM 





f-506 




(c) 

FIG. 50. Refractive indices of sodium carbonate monohydrate, 
Na 2 C0 3 .H 2 0. 

of the refractive indices for light vibrating along the three crystallo- 
graphic axes is a consequence of the fact that the arrangements of 
atoms encountered along these axes are all different from each other. 
Any one indicatrix axis may coincide with any crystallographic axis. 

For well-formed crystals of suitable shape the three principal refrac- 
tive indices can be found quite easily. Crystals of sodium carbonate 
monohydrate (Fig. 50), which can be made by evaporating a solution 
of sodium carbonate above 40 C., are suitable for demonstration 
because they lie on a microscope slide either on their 001 faces or their 
100 faces (Fig. 50 b and c). It will be found that the extinction directions 
that is, the vibration directions are parallel and perpendicular to 
the long edges of the crystals for both orientations ; crystals lying on 
001 give refractive indices of 1-420 for light vibrating along the crystal 
and 1-526 for light vibrating across the crystal ; those lying on 100 give 
1-420 for the vibration direction along the crystal and 1*506 for the 
other vibration direction. If crystals standing on end can be found on 
the microscope slide, they will give indices of 1-506 and 1-526 for the 
vibration directions shown in Fig. 50. These three values 1-420, 1-506, 



74 



IDENTIFICATION 



OPTIC 



OPTIC 
AXIS 



-CIRCULAR 
SECTIONS 



and 1*526 are the principal refractive indices, and are symbolized a or 
n a for the lowest, j3 or np for the next, and y or n y for the highest. 

Many orthorhombic crystals are of such a shape that, when lying on 
one of their faces on a microscope slide, they do not show any of the 
principal refractive indices. Such crystals may be broken to provide 
fragments which lie in a variety of orientations. If fragments of crystals 
of sodium carbonate monohydrate oriented in all possible ways are 
examined, both indices of each fragment bfeing observed, it will be 
found that the lower of the two indices may have any value between 

a and j8, while the upper index lies 
between J3 and y. The determina- 
tion of a and y for identification 
purposes is in principle quite 
simple: a is the lowest index for 
any vibration direction, and y is 
the highest index for any vibration 
direction. 

To find the intermediate prin- 
cipal index /J is less simple. One 
method of finding it makes use of 
the fact that in any ellipsoid having 
three unequal axes there are two 
circular sections. Thus, referring 
to Fig. 51, there is, somewhere be- 
tween a and y on the surface of the ellipsoid, a point j8' such that 
0j8' = Of}, and the section passing through this point and the centre of 
the ellipsoid is evidently a circle. Further, there is another point /J" 
for which 0/J" is equal to 0$, and /?/?" is therefore another circular 
section. This means that crystals seen along either of the two directions 
OP and OQ which are perpendicular to these circular sections (directions 
known as 'optic axes') have one refractive index only and will appear 
isotropic. Moreover, this one refractive index is equal to j8. Therefore, 
to find )3, search for fragments which appear isotropic or nearly so 
(giving very low order interference colours) ; these fragments give /J or 
values very near it. 

It is not always easy to find crystals or fragments oriented so that 
one is looking along an optic axis; hence it is necessary to mention 
another method of finding j8. This method depends on the fact that no 
crystal, whatever its orientation, can give two refractive indices above 
j8 or two refractive indices below /J. One index must be between a and j8 




FIG. 51. Biaxial indicatrix. 



CHAP, m OPTICAL PROPERTIES 75 

(for particular orientations it may be equal to a or /?), and the other 
must be between /? and y (for particular orientations it may be equal to 
/? or y). Therefore, to find /3 we observe upper and lower values for 
numerous fragments, and ft is the highest of the lower values or the 
lowest of the higher values. 

To sum up, the method of determining the three refractive indices of 
an orthorhombic crystal is to observe the upper and lower indices (for 
the two extinction directions) of numerous fragments, a is the lowest 
of the lower values, y the highest of the higher values, while jS is the 
highest of the lower values or the lowest of the higher values. If we 
find fragments oriented so that we are looking along an optic axis, 
upper and lower values are both equal to /?. 

Since orthorhombic crystals have two optic axes (that is, two direc- 
tions of apparent isotropy), they are termed optically biaxial. The 
angle between the optic axes is known as the optic axial angle. The 
three principal axes of the indicatrix are known as the acute bisectrix 
(of the optic axes), the obtuse bisectrix, and the third mean line. The 
last-mentioned the third mean line is in all cases the vibration 
direction of p. The acute bisectrix is either the vibration direction of 
y in which case the costal is known as biaxial positive or else it is 
the vibration direction of a, in which case the crystal is known as bi- 
axial negative. Note that this nomenclature conforms with that of 
uniaxial crystals. If we regard a uniaxial crystal as having an optic 
axial angle of 0, we may say that both optic axes coincide with the 
acute bisectrix. This unique direction is the vibration direction of e, 
and when this is the highest index (corresponding to y for a biaxial 
crystal), the crystal is known as a uniaxial positive crystal. For weakly 
or moderately birefringent biaxial crystals it is nearly correct to say 
that a positive crystal has /? nearer to a than to y, while a negative 
crystal has /? nearer to y than to a. But for strongly birefringent 
crystals (y a > 0-1) the dividing line between positive and negative 
crystals (where the optic axial angle is 90) occurs when j3 is appreciably 
different from J(a+y). 

Monoclinic and triclinic crystals. The indicatrix for monoclinic 
and triclinic crystals is of the same type as that for orthorhombic 
crystals an ellipsoid with all its three principal axes unequal in length. 
(This is the least symmetrical type of ellipsoid, so that any diminution 
of crystal symmetry below orthorhombic cannot alter the form of the 
ellipsoid.) The measurement of the three principal refractive indices of 
a monoclinic or triclinic crystal is therefore carried out in the manner 



76 



IDENTIFICATION 



CHAP. Ill 



described for orthorhombic crystals, random orientation being assured 
by crushing the crystals if necessary. 

The orientation of the indicatrix with respect to the unit cell axes, 
however, obviously cannot be the same as for orthorhombic crystals, 
since the unit cell axes in monoclinic and triclinic crystals are .not all 
at right angles to each other. 

In monoclinic crystals the 6 axis is either an axis of twofold symmetry 
or is normal to a plane of symmetry (or both) ; {herefore, since the orienta- 
tion of the indicatrix must conform 
to the crystal symmetry, one axis 
of the indicatrix (it may be either 
a, j8, or y) must coincide with the 6 
axis of the unit cell. This is the only 
restriction on indicatrix orienta- 
tion; its other two axes must ob- 
viously lie in the plane normal to 
b the ac plane, but they may be 
in any position in this plane, though 
of course remaining at right angles 
to each other. This is illustrated in 
Fig. 52, which shows a gypsum 
crystal lying on its 010 face, the b 
axis being normal to the paper. The 
vibration direction of /? happens to 
be the one which coincides with the 
b axis, hence a and y lie in the ac plane, and it is found that the vibration 
direction of a makes an angle of 37 with the c axis. If gypsum crystals 
are examined under the microscope, it will be found that the extinction 
directions are inclined to the crystal edges, and refractive index a (1-520) 
is shown when the vibration direction of the light from the polarizer 
makes an angle of 37 with the long edge of the crystal. If these 
crystals can be observed edgewise (in a crowd of crystals, especially 
when immersed in a viscous medium, some may be found suitably 
oriented) it can be seen (Fig. 52, right) that the extinction directions 
are parallel and perpendicular to the long edges of the crystal, and that 
refractive index /? (1-523) is shown when the vibration direction of the 
light is perpendicular to the long edges, that is, along the 6 axis ; for 
the vibration direction parallel to the long edges the index lies between 
a and y, its value being given by the length OZ. 

One consequence of the freedom of position of the indicatrix in the 




FIG. 52. Orientation of indicatrix in 
gypsum crystal. (The differences between 
the refractive indices for example, the 
lengths Ooc and Oy are greatly ex- 
aggerated.) 



CHAP, in OPTICAL PROPERTIES 77 

ac plane is that the extinction position need not be the same for all 
wavelengths of light ; its position for red light may be, and often is, 
appreciably different from that for blue light ; consequently some mono- 
clinic crystals, when lying on the microscope slide on their 010 faces, 
do not show complete extinction at any position of the crossed Nicols ; 
the illumination passes through a minimum on rotation of the Nicols, 
and in the region of the minimum, abnormal interference colours may 
be seen, reddish for one setting of the Nicols (where blue light is 
extinguished) and bluish when the Niools are turned a degree or two 
(when red light is extinguished). This occurs in crystals of sodium 
thiosulphate pentahydrate, Na 2 S 2 O 3 .5H 2 O, and sodium carbonate 
decahydrate, Na 2 C0 3 .10H 2 0. This phenomenon does not occur in 
orthorhombic crystals lying on faces parallel to crystallographic axes,f 
since the indicatrix axes are fixed by symmetry along the crystal axes 
and are therefore unable to vary in position with the wavelength of 
light. Nor does it occur for monoclinic crystals lying on any face 
parallel to 6, since one ellipsoid axis is fixed by symmetry along 6. 

In triclinic crystals there are no restrictions at all on the position 
of the indicatrix with respect to the crystal axes. No axis of the ellipsoid 
need coincide with any one of the crystal axes. Consequently the 
position of the ellipsoid may vary with the light wavelength for all 
crystal orientations ; incomplete extinction with abnormal interference 
colours at the position of minimum illumination may therefore be seen 
for any crystal orientation. 

Use of convergent light. The phenomena so far described are those 
which are seen when approximately parallel light is used. For any 
particular crystal orientation they give information about the proper- 
ties of the crystal for one particular direction of propagation of light 
(the line of vision). If strongly convergent light (given by a high power 
condenser) is used, phenomena can be seen which give information 
about a wide range of directions of propagation of light : in fact, in 
some circumstances, the phenomena show at a glance whether a crystal 
is uniaxial or biaxial, and if it is biaxial, they indicate the magnitude 
of the optic axial angle. 

A bundle of parallel rays which all take the same direction through 
the crystal and then pass through the objective lens of the microscope 
are necessarily brought to a focus at a point a little above the objective 
(in the focal plane of the objective, the plane in which the image of a 

t Orthorhombic crystals lying on hkl faces such as (111) may, however, show this 
phenomenon. 



78 IDENTIFICATION CHAP, in 

distant object would be produced) ; all the rays taking another direction 
through the crystal are focused at a different point in the same plane. 
Consequently, if we look at the optical effects in this plane we shall see 
a pattern which represents the variation of optical properties over the 
range of directions taken by the objective lens. When the crystal is 
between crossed Nicols the pattern of colours indicates the variation of 
birefringence with direction in the crystal. 

This pattern, known as the 'interference figure', 'directions image', or 
'optic picture', may be seen by removing the eyepiece of the micro- 
scope and looking straight down the tube ; it appears to be just above 
the objective lens. If the microscope is fitted with a Bertrand lens a 
special auxiliary lens which can be inserted in the tube it is not 
necessary to remove the eyepiece. For the observation of small crystals 
a Bertrand lens with a small diaphragm, located just below the eye- 
piece, is most suitable, as it picks out the directions image produced by 
a small crystal which occupies only a small fraction of the field of view. 
The objective lens used should have a high numerical aperture, so that 
it takes in a wide angular range of directions ; a 6- or 4-mru. lens of 
numerical aperture 0-7-0-8 is suitable. The crystals are preferably 
immersed in a liquid whose refractive index is not far from /? or to. 

A uniaxial crystal with its optic axis along the line of vision gives a 
directions image consisting of a black cross with concentric coloured rings 
(Fig. 53 a, left). The centre of the figure is dark because it represents 
the direction of the optic axis a direction of apparent isotropy. The 
arms of the black cross represent the vibration directions of the crossed 
Nicol prisms, while the rings show interference colours whose order 
(see p. 81) increases with their radius, owing to the rising birefringence 
of the crystal for directions increasingly inclined to the optic axis. 
Suitable crystals for demonstrating this type of figure are the hexagonal 
plates of cadmium iodide, CdI 2 , which lie correctly oriented on the 
microscope slide. Crystals lying so that the optic axis is a little inclined 
to the line of vision give a directions image displaced from the centre 
of the eyepiece field. 

Biaxial crystals under similar optical conditions produce directions 
images like that shown in Fig. 53 6, when the acute bisectrix of the 
optic axes lies along the line of vision and the vibration directions of 
the crossed Nicols are at 45 to the extinction directions. There are 
black hyperbolae and coloured lemniscate rings. A sheet of muscovite 
mica is a suitable specimen for demonstration. The distance between 
the black hyperbolae is a measure of the optic axial angle. If various 



OHAP. in 



OPTICAL PROPERTIES 



79 



crystals of known optic axial angle are observed, and the distances 
between the black hyperbolae are measured by means of a micrometer 
eyepiece, a calibration can be made so that the optic axial angle of any 





FIG. 63. 'Directions images 1 or 'optic pictures*, a. Uniaxial crystal with optic axis 

parallel (loft) and slightly inclined (right) to line of vision. 6. Biaxial crystal with acute 

bisectrix parallel (left) and inclined (right) to line of vision. 



AIR 




IMMERSION \ / 
MEDIUM 



CRYSTAL* 2V 



l 



//t\\ 



LIGHT 
Fia. 54. Optic axial angle in crystal (2V) and in air (2J57). 

crystal can subsequently be determined. The angle thus measured is 
the angle the optic axial directions make with each other on emerging 
from the crystal into air (Fig. 54) ; this angle 2E is related to the true 
optic axial angle 2V by the expression sinjE? = jSsinF. 



80 IDENTIFICATION . CHAP, in 

The optic axial angle 2 t V is related to the three principal refractive 
indices by the expression 

tanF = 



If three of these quantities are known, the fourth can *be calculated. 
Thus, when it is possible to measure all three principal refractive 
indices, the measurement of the optic axial angle is, strictly speaking, 
superfluous. But in some cases it may be possible to measure only two 
of the principal refractive indices. For instance, some organic crystals 
have y higher than any available immersion liquid. In such circum- 
stances a measurement of the optic axial angle gives the necessary in- 
formation for calculating the third index ; this measurement of the optic 
axial angle must include the determination not only of its magnitude but 
also of its sign. The distinction between positive and negative crystals 
can be made by the use of the quartz wedge ; this forms the subject of 
the next section. 

Use of the quartz wedge. When needle crystals of a uniaxial sab- 
stance such as urea (tetragonal uniaxial positive) are being examined 
between crossed Nicols, it may be seen that when one crystal lies across 
another of similar thickness, and at right angles to it, the apparent 
birefringence (as shown by the interference colour) at the point where 
they cross is very low or actually zero. The effect is seen perhaps most 
conveniently by examining thin threads of fibres such as rayon or 
nylon which behave optically like uniaxial crystals ; in a yarn of such 
materials the threads are of uniform diameter, and where they cross 
each other at right angles, the apparent birefringence is zero. But if 
one thread lies on another parallel to it, the interference colour is of 
much higher order than that given by a single thread. The interference 
effects are thus subtractive when the threads or crystals are at right 
angles to each other, and additive when they are parallel. This is 
because crystal 1 (Fig. 55 a) retards waves vibrating along A relative 
to those along J2; but subsequently, when the waves go through 
crystal 2, the waves vibrating along B are retarded relative to those 
along A, thus neutralizing or compensating the effect of crystal 1, so 
that no interference colours are shown for the crossed position. Con- 
versely, if the crystals are parallel, the retardation effects are additive 
and a higher order interference colour is produced. 

This effect can be used for finding which vibration direction gives 
the higher index for any birefringent crystal. It is most convenient to 
use the quartz wedge, a thin slice of quartz with its length parallel to 



CFAP. Ill 



OPTICAL PROPERTIES 



81 



the hexagonal axis of the crystal (the vibration direction which has 
the higher index) and uniformly tapering in thickness. f If it is pushed 
into the polarizing microscope at 45 to the vibration directions of the 
Nicols (a slot is provided for the purpose), the interference colours of 
Newton's scale can be seen grey near the thinnest part of the wedge, 
and passing through near- white, brownish-yellow, red, and violet of the 
first order, then peacock blue, yellowish-green, yellow, magenta, and 
violet of the second order, then emerald green, yellowish, and pink of 
the third order, and thence through alternating, progressively paler 




Fio. 55. a. Two urea crystals of the same tliickness, crossed at right angles. At the 

centre overlapping portion the combination appears isotropic. 6. Effect of quartz 

wedge on crystal of NH 4 H 2 T?O 4 . As the wedge advances, the colour contours move 

towards the thicker part of the crystal. 

shades of green and pink of the higher orders. If a crystal of mpnam- 
monium phosphate is examined, and the quartz wedge pushed in parallel 
to the fourfold axis (Fig 55 6), it can be seen that the interference 
colours decrease in order as an increasing thickness of quartz overlaps 
the crystal. This shows at once that the optical characlsr of this crystal 
is opposite to that of quartz the waves vibrating along the crystal 
have the lower refractive index. Perhaps the best method of observa- 
tion is to watch the colour contours on the pyramidal ends of the 
crystal ; these contours retreat towards the thicker part of the crystal 
as the quartz wedge advances and neutralizes the retardation. The 
effect may be checked by pushing the wedge in at right angles to the 
fourfold axis of the crystal; the birefringence effects are now additive, 

t This is the commonest type. But quartz wedges having the opposite orientation 
(with the vibration direction for the lower index parallel to the length of the wedge) 
are also made. The phenomena they give are naturally opposite to those described. 

4468 



82 IDENTIFICATION . CHAP, in 

and the colour contours move towards the pointed ends of the crystal 
as the quartz wedge advances. 

The distinction between the vibration directions of higher and lower 
refractive indices can always be made in this way for crystals having 
inclined extinction no less than for those with parallel extinction. When 
refractive indices are measured by the methods already given, the use 
of the quartz wedge is hardly necessary (unless for confirmation of 




FIG. 56. Effect of quartz wodge on 'directions images', a. Uniaxial positive 
6. Uniaxial negative, c. Biaxial positive, d. Biaxial negative. 



conclusions already reached) ; but in other circumstances (for instance, 
when crystals are being examined in their mother liquor), quartz wedge 
observations are useful clues to optical character. 

The quartz wedge may be used in a quantitative manner for finding 
the magnitude of birefringence of a crystal, that is, the difference 
between the two refractive indices the crystal is showing. For this 
purpose it is necessary to know the thickness of the crystal. The quartz 
wedge is pushed in until the birefringence of the crystal is just neutral- 
ized ; the interference colour given by the wedge alone at this point is 
noted, and the corresponding retardation can then be read off on a chart 
like that given by Winchell (1931). The relation between the retardation 
of one wave behind the other (R), the birefringence ()>' a'), and the 
thickness t is (y'~~ ') = -R- Such measurements can be done more 



CHAP, in OPTICAL PROPERTIES 83 

conveniently by means of the Babinet compensator, in which two 
quartz wedges slide over each other in response to the turn of a screw 
(Tutton, 1922). Such methods are useful when it is difficult or impos- 
sible to measure individual refractive indices by the immersion method ; 
for instance, the birefringence of stretched sheets of rubber has been 
measured in this way (Treloar, 1941), 

The optic sign of a crystal can be discovered by observing the effect 
of the quartz wedge on the interference figure. For a uniaxial positive 
crystal the vibration directions of higher index lie along the radii of the 
coloured circles (Fig. 56 a). Consequently, when the quartz wedge 
moves across the figure, additive effects occur along the radii parallel 
to the wedge (since the direction of higher index for the wedge is along 
its length), and subtractive effects along the radii perpendicular to the 
wedge ; the coloured circles therefore move inwards along radii parallel 
to the wedge and outwards along radii perpendicular to the wedge. 
The converse is true for uniaxia] negative crystals. Similar effects for 
biaxial crystals are illustrated in Fig. 56 6. 

Such observations may be useful in those cases when complete 
refractive index measurements by the methods already described are 
not possible ; for instance, when the maximum refractive index of an 
organic crystal is too high to be matched by any available liquid. 

Dispersion. The principal refractive indices of a crystal vary in 
magnitude with the frequency of light ; and in crystals of monoclinic or 
triclinic symmetry, the vibration directions of the principal indices may 
vary with frequency. Such variation is known as dispersion. 

The indicatrix for a cubic crystal is a sphere; the only variation 
which can occur is a change in the size of the sphere with the frequency 
of light. The colour fringes often seen round the edges of a crystal when 
it is immersed in a liquid of nearly the same refractive index are due 
to a difference between the dispersion of the crystal and that of the 
liquid. 

For uniaxial crystals (those of tetragonal, trigonal, and hexagonal 
symmetry) the indicatrix is an ellipsoid of revolution, the orientation 
of which is fixed by symmetry (see earlier section). But the magnitudes 
of a) and . may vary with frequency in different degrees, so that the 
birefringence varies with wave-length. This is not likely to give rise to 
noticeable phenomena under the microscope unless the birefringence is 
very low, when abnormal interference colours may be seen when the 
crystals are observed between crossed Nicols in parallel light. For 
instance, the mineral rinneite, FeCl 2 . SKCl.NaCl, is practically isotropic 



84 



IDENTIFICATION 



CHAP. Ill 



for yellow light, but appreciably birefringent for blue light ; fragments 
of suitable thickness do not show first order yellow, but a bluish 
tinge. In benzil, C 6 H 5 .CO.CO.C 6 H 5 , the changes of co and e with 
frequency are such that it is positive for most of the visible spectrum, 
isotropic in the violet, and negative for the far violet end of the spectrum 
(Bryant, 1943). 

In orthorhombic crystals the vibration directions of the three 
principal indices are fixed by symmetry, but their magnitudes may 
vary independently, and this may lead to appreciable variation of 



A. 

BLUE 


i 




-7 

._* 


IIIF 


2. 




i 




~7 


/ 


/ 


A 


RED 
A 



.XI 






S r \ x" 




s 


. 






^^ 


) 


7 




^ 


GREEN 
A 


BLUE 
A 




FIG. 57. Dispersion of optic axes in orthorhombic crystals, a. p > v. 
b-d. Crossed axial plane dispersion. 

optic axial angle with frequency. This effect modifies the appearance 
of the directions image produced in convergent light. The acute 
bisectrix of the optic axes (to which the centre of the directions image 
corresponds) is fixed by symmetry along one of the crystal axes, and 
the plane of the optic axes (the ay plane of the indicatrix) is one of the 
faces of the unit cell. Consequently the two planes of symmetry of 
the directions image are fixed in the same positions for all frequencies. 
Therefore the hyperbolae which indicate the positions of the optic axes 
may move towards or away from each other with change of frequency, 
but always symmetrically with respect to the fixed lines AB and CD in 
Fig. 57 a. For small dispersions the directions image produced by white 
light will show a red fringe on one side of the hyperbola (where blue 
light is missing) and a blue fringe on the other side (where red light is 
missing), both hyperbolae being the same. When the red fringes are 
on the side nearer to the acute bisectrix, as in the diagram, the optic 
axial angle for blue light is evidently smaller than for red light. The 



CHAP, m OPTICAL PROPERTIES 85 

usual symbol for recording this condition is p > v. In extreme cases, 
as in brookite (the orthorhombic form of titanium dioxide, Ti0 2 ), the 
optic axial angle narrows to zero and then opens out again in a plane 
at right angles to the first, as the frequency of the light is changed 
(see Fig. 57 b-d). This means that the refractive index for vibration 
direction CD, which for red light is , approaches that for the vibration 
direction AB (a for a positive crystal), becomes equal to it for green light 
(so that the crystal is fortuitously uniaxial), and falls below it for blue 
light, so that the index for vibration direction CD is now called a, while 
that for AB is called j8. The white light directions image in such 
circumstances is very abnormal ; to elucidate the relation between optic 
axial angle and frequency, it is necessary to make observations in 
monochromatic light of variable frequency. (See Bryant, 1941.) 

In monocliiiic crystals the indicatrix may not only change its dimen- 
sions, but may rotate round whichever axis coincides with the 6 crystallo- 
graphic axis ; and in triclinic crystals it may rotate in any direction 
whatever, with change of frequency. These movements may give rise 
to less symmetrical types of dispersion of the optic axes, though it is 
only rarely that the magnitude of the effect is great enough to render 
the phenomenon a useful criterion for identification. These types of 
dispersion will therefore not be described in detail ; it will merely be 
observed that the type of dispersion is conditioned by the symmetry 
of the crystal, and that when appreciable dispersion occurs, the 
symmetry of the polychromatic directions image, or the movement of 
the monochromatic figure as the frequency is changed, is a reliable 
indication of maximum crystal symmetry. A polychromatic figure 
which has only a centre of symmetry, or is symmetrical about only 
one line, can only be produced by a crystal having monoclinic or 
triclinic symmetry ; a figure having no symmetry can only be produced 
by a triclinic crystal. For further information, see Miers (1929) and 
Hartshorne and Stuart (1934). 

Pleochroism. When crystals absorb light the positions of the 
absorption bands and their intensities are likely to vary with the 
vibration direction of the light, and therefore, when the absorption 
bands are in the visible region, the colour shown is likely to depend on 
the vibration direction of the light. All coloured anisotropic crystals 
that is, all coloured crystals except those belonging to the cubic system 
are likely to show, in polarized light, colours which vary as the 
polarizer is rotated. This will be noticed when the refractive indices of 
coloured crystals are being measured. Thus, when crystals of potassium 



86 IDENTIFICATION CHAP, in 

ferricyanide K 3 Fe(CN) 6 are examined in polarized light, rotation of the 
polarizer causes the colour of some of the crystals to change from 
yellow to orange-red; crystals 'showing* the refractive index a are 
yellow, while those 'showing' y are orange-red. Such crystals are 
said to be 'pleochroic'. These absorption effects, which are shown 
when only the polarizer of the microscope is in use, should not be 
confused with the interference colours produced when crossed Nicols 
are in use. 

The three-dimensional graph showing the variation in the absorption 
of any frequency with crystal direction is, like that of the refractive 
indices, an ellipsoid. Cubic crystals necessarily have the same absorp- 
tion for all vibration directions, just as they have a constant refractive 
index. For optically uniaxial crystals (those belonging to the tetragonal, 
hexagonal, and trigonal systems) the absorption for the co vibration 
direction may be different from the absorption for the e vibration 
direction different in respect of both the proportion of light absorbed 
and the wave-length ranges of the absorption bands ; and when an index 
between CD and c is shown, the absorption is intermediate. (Strictly 
speaking, the phenomenon in uniaxial crystals should be termed 
dichroism, since there are only two different absorptions.) For biaxial 
crystals, a, /?, and y may all show different colours. Thus in crystals of 
Fe 3 (P0 4 ) 2 . 8H 2 (the mineral vivianite) a is cobalt blue, j3 is nearly 
colourless, while y shows a pale olive-green colour (Larsen and Berman, 
1934). 

The observation of the colour and the degree of absorption associated 
with each index is of obvious value for identification purposes ; the 
larger the number of characteristics observed, the more certain the 
identification. Observations of pleochroism may also be useful as indica- 
tions of certain features of molecular structure. (See Chapter VIII.) 

Very strongly pleochroic crystals, which absorb almost completely 
for one vibration direction and hardly at all for another, can be used as 
polarizers. Tourmaline, a complex aluminosilicate mineral of trigonal 
symmetry, has a very low absorption for light of all colours vibrating 
along the trigonal axis, and a very high absorption for vibration direc- 
tions perpendicular to this axis ; when unpolarized light passes through 
the crystal, it is resolved in the usual way into two components vibrat- 
ing parallel and perpendicular to the threefold axis ; but the component 
vibrating perpendicular to this axis is almost completely absorbed by 
even very thin crystals, while the other component is transmitted with 
little loss of intensity ; consequently the light which emerges from the 



CHAP, in OPTICAL PROPERTIES 87 

crystal is practically completely plane polarized. The polarizing sheets 
known as 'Polaroid* have similar characteristics ; each sheet consists, not 
of a single crystal, but of a large number of submicroscopic crystals, all 
oriented parallel to each other and embedded in a suitable medium. 
The first crystal used for this purpose was 'herapathite', strychnine 
sulphate periodide ; some other substances of this type (periodides) have 
similar properties. 

Rotation of the plane of polarization. When plane polarized light 
passes through crystals belonging to certain classes, the plane of polari- 
zation may be rotated. The phenomenon is readily observed only in 
cubic crystals and in birefringent crystals seen along an optic axis; 
these, when examined between crossed Jficols, using parallel white light, 
do not appear dark (as they would if no rotation occurred), but coloured ; 
and when the crossed Nicols are rotated, no extinction occurs, the 
intensity and colour of the light remaining constant. Light is trans- 
mitted because the plane of vibration of the light from the polarizer is 
rotated by the crystal, so that it is no longer extinguished by the 
analyser ; and the reason for the colour is that the amount of rotation 
usually varies considerably with the wave-length of the light, and con- 
sequently the proportion of light passed by the analyser (resolved into 
its own plane of vibration), is different for each wave-length, the net 
transmitted light being therefore coloured. The rotation is usually 
greatest for the blue end of the spectrum ; consequently for thin crystals 
in which the amount of rotation is much less than 90 for all wave- 
lengths, the light which passes the analyser is predominantly blue. 
Thus, for microscopic crystals, rotation of the plane of polarization is 
indicated by the appearance of a bluish light which does not extinguish 
as the crossed Nicols are rotated, but remains of constant colour and 
intensity. As a check, the analyser should be rotated so that it is no 
longer exactly crossed with the polarizer; the colour should change, 
and the sequence of changes shows the sense of rotation of the plane 
of polarization ; if the analyser is rotated clockwise, a change of colour 
in the order blue, violet, yellow shows that the crystal is rotating to 
the right (clockwise). 

If monochromatic light is used, Nicols exactly crossed will trans- 
mit some if it, but by rotating the analyser extinction can be 
achieved. 

Rotation of the plane of polarization naturally modifies directions 
images. When rotation occurs in a uniaxial crystal the black arms of 
the directions image fade towards the centre, and the centre itself is 



88 IDENTIFICATION CHAP, in 

coloured, not black ; and if rotation occurs along the optic axial direc- 
tions of a biaxial crystal the image will show coloured 'eyes', the black 
hyperbolae being interrupted at these points. 

Such evidence of rotation of the plane of polarization is not likely to be 
detected in microscopic crystals unless the specific rotation is exception- 
ally large. The phenomena mentioned above are usually exhibited only 
by crystals at least several millimetres thick. Suitable subjects for 
observation are sodium chlorate (cubic), quartz (trigonal, uniaxial), and 
cane sugar (monoclinic, biaxial). 

The crystal classes which may rotate the plane of polarization of light 
are, first of all, the enantiomorphous classes those which lack planes 
of symmetry, inversion axes, and a centre of symmetry. But in addition 
to these, one crystal belonging to class m (that is, having a plane of 
symmetry, but no centre of symmetry) is known to exhibit the pheno- 
menon (Sommerfeldt, 1908) ; and therefore presumably some others 
possessing planes but no centre of symmetry may do the same. To a 
chemist, familiar with the conditions necessary for rotation of the plane 
of polarization by dissolved molecules (that is, absence of both planes 
and a centre of symmetry in the molecular geometry), this may appear 
surprising ; but the surprise disappears when it is realized that the two 
situations on the one hand, a mass of randomly oriented molecules, 
and on the other, a single crystal composed of precisely oriented 
molecules are not comparable. Reconciliation of ideas is effected by 
the following considerations. A crystal or a single molecule having a 
plane of symmetry but no centre of symmetry can rotate the plane of 
polarization, but the rotation varies with the direction in which the 
light travels, arid if there is left-handed rotation along any selected 
direction on one side of the plane of symmetry, there must be, along 
the mirror-image direction on the other side of the plane of symmetry, 
right-handed rotation of the same magnitude. Therefore in a mass of 
randomly oriented molecules (or crystals) some will rotate in one 
direction and others (differently oriented) in the opposite direction, the 
net rotation being exactly zero. Thus it is not true to say that a single 
molecule or a single crystal having a plane of symmetry cannot rotate 
the plane of polarization of light ; provided it has no centre of symmetry, 
it can and does cause rotation for light travelling in any direction 
except those parallel and perpendicular to the plane of symmetry ; it is 
the mass of randomly oriented molecules in a liquid or solution which 
fails to show any net rotation. 

For a fuller discussion of the phenomenon, and a list of the crystal 



CHAP. Ill 



OPTICAL PROPERTIES 



89 



classes which (according to current theories) may exhibit it, see 
Wooster, 1938. 

Optical properties of twinned crystals. Each individual in a twin 
exhibits its own optical characteristics. If a gypsuin twin is seen along 
its b axis and examined between crossed Nicols, it can be seen that each 
individual extinguishes independently. The twin plane (100) is a plane 
of symmetry of the composite whole, and the vibration directions of the 
two individuals, like all the other properties, are related to each other 
by this plane of symmetry (see Fig. 58 a). 





FIG. 58. Optical properties of twinned crystals, a. Gypsum, b. Calcium 
sulphate subhydrate. Note orientations of 'directions images' in the three 

sectors. 

The relations between the optical properties of the two individuals 
are clear in the case of gypsum because the crystals lie on the microscope 
slide on their (010) faces, so that the (100) twin planes are parallel to 
the line of vision. In some crystals the twin planes are inclined to the 
line of vision when the crystals are lying on their principal faces so that 
one is looking through two crystals in which the vibration directions are 
not parallel to each other. In these circumstances, in the overlapping 
regions extinction does not occur when the Nicols are rotated. When 
observations are being made for refractive index determination it is 
necessary to confine the observations to those portions of crystals which 
are not overlapped by other individuals. 

Observations of crystals between crossed Nicols are particularly 
valuable in the case of some of those twin combinations which in their 
external shape simulate a single crystal having a symmetry higher than 
that of one of the individuals. The observation of different extinction 
directions in different regions demonstrates at once that the crystal is 
not a single individual but a twinned combination. The hexagonal 
prisms of ammonium sulphate mentioned on p. 58 are in this way shown 



90 IDENTIFICATION CHAP, m 

to be mimetic triplets, since adjacent sectors extinguish at 60 to each 
other. Similarly crystals of calcium sulphate subhydrate grown in nitric 
acid solution are hexagonal<plates, which, however, are not single crystals 
but triplets (Fig. 58 b): three sectors have extinction directions at 120 
to each other, and, moreover, biaxial directions images at 120 to each 
other can be seen by examining each sector in turn. 



IV 

IDENTIFICATION OF TRANSPARENT CRYSTALS UNDER 
THE MICROSCOPE 

IN this chapter the sequence of observations followed in the microscopic 
method of identification is outlined. The immersion method for the 
identification of small separate crystals forms the main subject of this 
chapter, though some remarks on methods for large aggregates will be 
found at the end. When the immersion method is to be used aggregates 
may be crushed or ground carefully. 

A preliminary observation is made in ordinary transmitted light to 
see whether the solid is transparent or not. It must be remembered 
that the amount of light transmitted is greatest when the solid is 
immersed in a medium of similar refractive index ; transparent solids of 
very high refractive index, in air or in a liquid of low index, may appear 
opaque, especially if they are aggregates of small particles, on account 
of the total internal reflection of light at inclined surfaces. Therefore, 
if the particles appear opaque when immersed in a liquid of refractive 
index 1-4-1 -5, a liquid of much higher index say 1-71 '8 should be 
tried. (The polarizer of the microscope, though not necessary for this 
observation, may be left in position ; in fact, it is hardly ever necessary 
to remove it.) 

In general chemical work the great maj ori ty of substances encountered , 
when in the form of small microscopic particles, are likely to be in koine 
degree transparent, and can therefore be studied by methods employ- 
ing transmitted light. For completely opaque particles it must be 
admitted that the chances of identification by any microscopic method 
are rather small, unless well-formed crystals large enough to be handled 
individually are available: such crystals may be mounted on a micro- 
scope stage goniometer, and if sufficient angular measurements can be 
obtained it may be possible to use Barker's morphological method of 
identification (1930). 

For opaque crystals too small to be handled individually, only general 
observations of shape can be made, and for this purpose it is best to use 
diffused light illuminating the crystals from above on one side of the 
microscope. Such observations will not carry us very far we may be 
able to recognize cubes or octahedra or hexagonal prisms or other shapes, 
but in the absence of angular measurements or indeed measurements of 
any characteristics at all, identification in the strict sense of the word is 



92 IDENTIFICATION CHAP, iv 

scarcely possible. In case any readers happen to be metallurgists, I 
hasten to add that experience in dealing with a particular system (in 
the phase rule sense of the word) may show that certain characteristic 
shapes or formations recognizable by simple observation are indicative 
of the presence of certain phases. In metallurgy the body of experience 
built up by a large number of observations of polished and subsequently 
etched surfaces of metal specimens is used with great effect in 'spotting* 
particular constituents. Metallurgical text-books, such as Rosenhain's 
Introduction to Physical Metallurgy (1935), should be consulted for 
further information on this highly specialized branch of crystallography. 
Similar methods may be used, and often are used, for non-metallic 
systems, once the necessary experience has been gained. But experience 
obviously has to be built up for every different system individually ; 
if a new constituent is added, the picture may be entirely changed, 
because new phases may be formed or familiar phases may grow in 
unfamiliar shapes and will have to be identified by methods of general 
validity before the necessary experience for specialized inspection- 
methods can be built up. It is with the methods of general validity that 
we are concerned in this book. 

When a solid substance is seen to be transparent the next step is to 
observe whether it is isotropic or not. The analyser is introduced (crossed 
with respect to the polarizer), and the Nicols (or alternatively the parti- 
cles) are rotated. If the particles remain dark for all positions of the 
crossed Nicols they are isotropic, and their refractive index can be 
measured by the method described at the beginning of the previous 
chapter. Note at this point that crystals belonging to the optically 
uniaxial systems which happen to grow as thin plates (of tetragonal, 
hexagonal, or trigonal outline) tend to lie flat on the microscope slide, 
and in this position their optic axes lie along the line of vision and the 
crystals therefore appear isotropic. If, however, the iris diaphragm of 
the substage condenser is opened to give strongly convergent light, such 
crystals will show interference colours, thus betraying their birefringent 
character ; and an observation of the 'directions image* will confirm that 
they are uniaxial. In any case, it is unlikely that all the crystals will 
be lying flat ; in a crowd of crystals some will almost certainly be tilted 
or even standing on edge, and in parallel light these will show inter- 
ference colours, revealing their birefringent character. In case of doubt 
the crystals may be deliberately tilted. If a 'universal stage* is available 
the microscope slide may be readily tilted in any direction. If not, the 
crystals should be immersed in a viscous liquid such as glycerol or 



CHAP, iv TRANSPARENT CRYSTALS 93 

dibutyl phthalate ; if the microscope is tilted so that its stage is not 
horizontal, or in any case if the cover-glass is disturbed, the liquid will 
flow slowly and the crystals wiU turn over ; observation between crossed 
Nicols while the crystals are moving will show whether they are bire- 
fringent or not. (This is also a useful way of studying the shapes of 
microscopic crystals, the analyser being removed for this purpose.) 

Cubic crystals and amorphous substances. Isotropic solids, 
if they are truly isotropic (not merely aggregates of very small bire- 
fringent crystals too small to shov/ interference colours), are either 
crystals belonging to the cubic system or amorphous substances like 
glasses or gels in which there is no regular arrangement of atoms. 
Crystalline substances are likely to showeome signs of regular structure ; 
if they are well formed and their shape is obvious, isotropic crystals 
should have a shape consistent with cubic symmetry. (See Fig. 38.) 
Even broken fragments of crystals are likely to show occasional edges, 
corners, or cleavage surfaces suggesting the original shape. Substances 
such as ammonium chloride and bromide which grow in skeletal forms 
often have rounded surfaces, but the occurrence of fragments branch- 
ing at right angles does give an indication of an ordered internal 
structure. 

The magnitude of the refractive index of an isotropic crystal usually 
leads to unequivocal identification. In the tables published by Winchell 
(1931) for inorganic laboratory products and Larsen and Berman (1934) 
for minerals, crystals are arranged in order of their principal refractive 
index, and it is therefore a straightforward matter to find which crystal 
has the refractive index which has been measured. It may happen that 
the measured value does not correspond with any in the lists ; in this 
case, there are two possibilities. One is that the substance is a mixed 
crystal or crystalline solid solution, the refractive index of which varies 
continuously with the composition (the tables mentioned indicate the 
known variations) ; a hint of such variation is often given by the sample 
itself some crystals may have a slightly higher index than others. 
The second possibility is that the substance is one whose refractive index 
has not previously been measured, in which case it obviously cannot be 
identified by this method. 

Glasses may reveal their nature by exhibiting conchoidal fractures. 
The composition of a one- or two-component glass may be deduced 
from its refractive index if the system has previously been studied ; 
the indices of a number of glasses are given in WinchelTs tables. For 
three-component glasses the refractive index alone cannot give the 



94 IDENTIFICATION CHAP, iv 

composition ; but if the refractive index and one other property say the 
density can be measured, it may be possible to specify the composition. 

Precipitated amorphous substances usually appear to be irregular 
isotropic masses. They usually tend to hold varying amounts of solvent 
and therefore show variable refractive indices. Usually they cannot be 
identified with certainty. 

Irregular masses which appear isotropic may consist of aggregates of 
anisotropic crystals which are individually too small to show inter- 
ference phenomena between crossed Nicols; each crystal may be of 
submicroscopic size. The single measurable refractive index is an average 
value lying between the principal indices of the crystal in question. 
Weakly birefringent substances are the most likely to appear in this 
form, but any substance may do so provided the individual crystals are 
small enough ; the higher the birefringence, the smaller the individual 
crystals must be in order to appear isotropic. Slaked lime, Ca(OH) 2 , 
which has a moderate birefringence (to = 1-57, e = 1-54), sometimes 
forms apparently isotropic masses ; in such cases it is always advisable to 
increase the intensity of illumination (still using crossed Nicols) by open- 
ing the iris diaphragm of the condensing lens of the microscope, when 
it may happen that vague patches of feeble interference colours (greys 
of the first order) indicate the presence of minute birefringent crystals. 
Strained glass may also show weak birefringence, but the glassy 
character will probably be betrayed by conchoidal fractures. In any such 
case the specimen should be referred to the higher court of inquiry by 
X-ray examination ; this method is dealt with in the next chapter. 

Optically uniaxial crystals . When a crystalline substance is found 
to be birefringent one proceeds with the determination of its principal 
refractive indices by the methods already described. If the crystals are 
flat plates, apparently ifiotropic when lying flat on the slide, they are 
evidently uniaxial ;f the principal index o> is given by the apparently 
isotropic plates, while plates standing on edge give co for light vibrating 
in the plane of the plate and c for light vibrating normal to the plate, 

For crystals which are not plate -like it may not be possible to decide 
from the appearance of the crystals whether they belong to one of the 
uniaxial systems (tetragonal, hexagonal, trigonal) or one of the biaxial 
systems (orthorhombic, monoclinic, triclinic). It should then be assumed 
initially that there are three principal indices a, ft, and y to be measured ; 
evidence of uniaxial or biaxial character is bound to turn up in the course 

f Flat biaxial crystals in which one of the optic axes happens to be precisely 
normal to the plane of the plate will appear Lsotropic; but this situation is rare. 



CHAP, iv TRANSPARENT CRYSTALS 95 

of the observations. Thus, the general procedure is to observe the upper 
and lower indices (for the two extinction positions) of numerous frag- 
ments in a range of liquids, random orientation being assured by crush- 
ing if necessary. The index a is the lowest of the lower values, y is the 
highest of the upper values, while /? is the highest of the lower values 
or the lowest of the upper values. If the crystals happen to be uniaxial 
positive, then ft will be found to be equal to a that is to say, every 
crystal will give a constant lower value: /? = a = co. If the crystals are 
uniaxial negative, ft will be found to be equal to y every crystal will 
give a constant upper value: j8 r= y = CD. 

The uniaxial character may be checked, if possible, by observing (on 
a crystal which appears isotropic or nearly so) the directions image 
produced by strongly convergent light, either by introducing the 
Bertrand lens or by removing the eyepiece. It is useful to do this 
because some biaxial crystals have two indices so close together that it 
is scarcely possible to detect the difference by the immersion method. 
Thus, potassium nitrate has y = 1-335, jB = 1-505 6 , y = 1-506 4 . The 
directions image shows, however, not the black cross of a uniaxial 
crystal, but (for the 45 position of the Nicols) the two black hyperbolae 
of a biaxial crystal ; careful observation is necessary to confirm this, 
because the hyperbolae are very close together (the optic axial angle 
27 being only 7 and 2E 101). 

If a uniaxial directions image is seen the optical sign of the crystal 
may be checked by the use of the quartz wedge in the manner described 
in the previous chapter. This is not necessary (except as confirmation) 
unless for any reason it is not possible to obtain actual measurements 
of both co and e. 

Needle-like crystals naturally lie on the microscope slide with their 
long axes parallel to the slide, and it may not be possible to find tilted 
crystals ; and even crushing may not yield fragments which lie in all 
possible orientations. However, even when the needle axis is invariably 
parallel to the slide, all orientations obtainable by rolling a needle are 
likely to be encountered, and observations of a number of crystals 
should be sufficient to give all the information required. The first thing 
to do is to observe the extinction direction ; if extinction is consistently 
parallel to the length, the crystals may be uniaxial, the direction of 
elongation being necessarily the unique geometrical axis and therefore 
also the optic axis ; but they may also be biaxial either orthorhombic, 
the direction of elongation being any one of the three axes, or mono- 
clinic elongated along the 6 axis (since the 6 axis of a monoclinic crystal 



96 IDENTIFICATION CHAP, iv 

is the only direction which has an axis of the indicatrix coincident with 
it). For all these types the refractive index for light vibrating along 
the needle axis is constant ; but for light vibrating perpendicular to the 
needle axis the refractive index is constant only for uniaxial crystals ; 
for biaxial types it is variable. We return to the biaxial types in the 
next section; meanwhile the position is that needle crystals with 
parallel extinction which give two constant refractive indices are uni- 
axial. It only remains to discover which of these indices is co and 
which e. The latter is the value for light vibrating along the needle 
axis (the optic axis) ; if the vibration direction of the polarizer is known, 
it will be obvious which of the two measured indices is c ; if not, the 
use of the quartz wedge will decide the question. 

Uniaxial bipyramids and rhombohedra (usually recognized by shape 
and symmetrical extinction), when lying on the slide on their faces, will 
not give e but a value lying somewhere between CD and e. Hence the 
need for breaking the crystals to give random orientation. Crystals 
having good rhombohedral or pyramidal cleavages (like calcite) may, 
even when crushed, give many fragments which still lie inconveniently 
on the cleavage faces ; nevertheless, irregular-shaped fragments which 
will lie in random orientation are sure to be produced in sufficient num- 
bers for the determination of e. 

Optically uniaxial crystals may be tetragonal, hexagonal, or trigonal, 
If it is possible to recognize a shape characteristic of a particular system 
this information is useful supplementary evidence; but it must be 
emphasized that the refractive index values by themselves are usually 
sufficient for identification. 

Optically biaxial crystals. The measurement of the three princi- 
pal refractive indices of a biaxial substance presents no difficulties when 
the crystals are large enough to be crushed to provide irregular frag- 
ments which will lie on the slide in random orientation. When the 
crystals are too small for crushing to be desirable or effective, and are 
bounded by a very few plane faces, some caution is necessary ; one must 
make sure of observing not only those crystals which are lying on their 
principal faces but also crystals tilted in various ways (because crystals 
lying on their principal faces may not give their principal indices). If 
a universal stage is available this presents no difficulty ; but even with- 
out the universal stage it is not as difficult as might be supposed to 
find suitably oriented crystals; even thin plates, the worst type of 
crystals in this respect, may be found tilted at various angles or standing 
on edge, especially in a crowd of crystals. 



CHAP, iv TRANSPARENT CRYSTALS 97 

Crushing has been recommended as a primary method because it is 
safe and will lead to the determination of the principal refractive indices 
of any crystalline substance, provided a sufficient number of randomly 
oriented fragments is observed ; it is a beginner's method. But the more 
experienced worker may often dispense with it, when the crystals being 
examined have a well-defined polyhedral shape. If the relation between 
crystal shape and optical properties is properly understood, it is possible 
to determine the principal indices by a limited number of observations 
on crystals selected because they lie in such positions that they neces- 
sarily show their principal indices. 

For instance, crystals which appear to possess three mutually per- 
pendicular planes of symmetry, or two planes intersecting in a twofold 
axis, or three twofold axes, are probably orthorhombic, with rectangular 
unit cells; and if, on looking along the presumed axial directions, 
extinction is parallel to crystal edges or bisects the angles between 
crystal edges, this conclusion is confirmed. Any crystal lying so that 
an axial direction lies along the line of vision necessarily shows two of 
the principal refractive indices; and views down two different axial 
directions yield the three principal refractive indices. Crystals such as 
those of sodium carbonate monohydrate (Fig. 50) are ideal for such 
observations. At the same time, these observations yield a knowledge 
of the orientation of the principal vibration directions (the principal axes 
of the indicatrix) with respect to the crystal axes ; thus, for sodium 
carbonate monohydrate the vibration direction for a is the direction of 
elongation of the crystal, while the vibration direction for j3 is the zone 
axis of the terminal faces. 

Crystals which appear to possess one twofold axis, or one plane of 
symmetry, or both (the twofold axis b being normal to the plane of 
symmetry) are probably monoclinic ; if so, crystals lying with their 
presumed 6 axes parallel to the microscope slide will show extinction 
parallel to this 6 axis, and the refractive index for this vibration direc- 
tion is one of the principal refractive indices. The other two principal 
indices will be shown by crystals lying with their 6 axes along the line 
of vision ; for this aspect of the crystal, extinction is not parallel to a 
principal edge or to the bisector of edge angles.f 

Crystals which appear to possess only a centre of symmetry or no 
symmetry at all a^e probably triclinic, and will probably not show their 
principal refractive indices when lying on their faces; such crystals 
should be crushed. 

t Note that in rare cases extinction angles may be so small as to escape detection. 
4458 



08 IDENTIFICATION CHAP, iv 

Biaxial crystals of inorganic substances can usually be identified by 
their refractive indices alone; it is true that biaxial crystals are far 
more numerous than uniaxial ones, but this is balanced by the fact that 
they have three principal refractive indices three different measurable 
characteristics as against two for the uniaxial types ; it is rare to find 
two substances having their three principal refractive indices equal 
within the limits of experimental error. Nevertheless, it is always 
desirable to discover, if possible, the crystal system and the relation 
between the principal vibration directions and the crystal axes. This 
information will often be simply confirmatory, but for certain mineral 
systems in which considerable variation of composition (and therefore 
of refractive indices) may occur./ the magnitudes of the refractive indices 
alone are not enough for unequivocal identification ; it is necessary to 
discover the crystal symmetry and the orientation of the indicatrix 
with respect to the crystal axes. For orthorhombic crystals the principal 
axes of the indicatrix necessarily coincide with the unit cell axes, and 
it is simply a matter of observing which vibration directions lie along 
characteristic axial directions (such as a direction of elongation, a 
principal prism zone, or a polar axis). For monoclinic crystals it is 
necessary to find which vibration direction lies along the 6 axis, and the 
angles made by the other vibration directions with respect to the a and 
c axes. For triclinic crystals the indicatrix is not fixed in any way by 
symmetry ; it may be possible to determine extinction directions with 
respect to characteristic morphological directions, though not to define 
precisely the orientation of the indicatrix, unless a universal stage is 
available. The necessary information for these purposes is normally 
gathered in the course of the determination of the refractive indices ; 
it is applied to the operation of identification by the use of the tables 
of Larsen and Berman (1934) for minerals and Winchell (1931) for 
laboratory chemicals. 

For organic substances, the available information has been collected 
and arranged by Winchell (1943). For some substances the only re- 
fractive indices which have been recorded are those given by crystals 
lying on their principal faces ; these are of course not always principal 
indices, but they may be equally useful for identification purposes. 
Such information is included in Winchell's tables. 

If the crystals being examined are not well-formed polyhedra, the 
scope of such observations is naturally more limited. Perhaps the 
commonest type of partly defined shape is a rod somewhat rounded so 
that there are no definite faces on it. The only definite morphological 



CHAP, iv TRANSPARENT CRYSTALS 99 

feature here is a single direction the long axis of the rod. If extinction 
is consistently parallel, and it is found that the crystals are biaxial (see 
pp. 95-6), they are almost certainlyf either orthorhombic or else mono- 
clinic with the b axis as the direction of elongation. It is possible to 
determine which vibration direction lies parallel to the rod by the 
methods already given (for instance, by the use of the quartz wedge). 
If extinction is inclined, the crystals are either monoclinic with a or c 
as the direption of elongation, or else triclinic. The extinction angle 
will vary with the orientation of the rod-like crystal on the slide. If 
it is found that crystals which show the maximum extinction angle also 
show two of the principal indices, then the substance is probably mono- 
clinic, and the maximum extinction artgle represents the angle made 
by one of the principal vibration directions with the direction of elonga- 
tion. Otherwise, the crystals are triclinic. 

The observation of directions images in convergent light may often 
provide confirmation of the orientation of the indicatrix. The plane of 
the optic axes is the cxy plane, while the normal to this plane is the /? 
vibration direction. For a positive crystal the acute bisectrix is the y 
vibration direction, while for a negative crystal it is the OL vibration 
direction. 

Even when the crystals being examined are quite irregular fragments, 
it may be possible to obtain some information on their symmetry, if 
certain types of dispersion of the optic axes are observed (see pp. 83-5). 

The optical properties of crystals are usually quite reliable criteria 
for identification ; but occasionally crystals have submicroscopic cracks 
and cavities, and although appearing quite normal, give refractive 
indices lower than those of an entirely solid crystal. This phenomenon, 
which is obviously very misleading, is fortunately very rare, but has been 
observed in anhydrite (calcium sulphate) and calcite (calcium carbonate) 
prepared in the laboratory. In cases of doubt, X-ray powder photo- 
graphs should be taken see Chapter V. 

Mixtures. When the constituents of a mixture differ markedly from 
each other in appearance the refractive indices and other optical 
properties of each can be determined without difficulty. This is very 
frequently the case even when the shapes of the crystals are not very 
well defined; for instance, a mixture may consist quite obviously of 
three constituents, one in the form of comparatively large, rounded, 
roughly equidimensional crystals, another in the form of small rod- 
like crystals, and a third in the form of small rectangular or cubic 

t See footnote to p. 97. 



100 IDENTIFICATION . CHAP, iv 

crystals. There is no difficulty in measuring the properties of each con- 
stituent in such a mixture as this. Even when the differences between 
constituents are much slighter and less easy to specify, they may be 
none the less obvious. 

Even when there are no morphological distinguishing features, how- 
ever, it is very often possible to measure the refractive indices of differ- 
ent constituents. Two or more isotropic substances can be identified, 
provided that their refractive indices are not closer than 0-002. A 
mixture of one anisotropic substance with one, two, or more isotropic 
substances likewise presents no difficulty. Fig 43 e shows a mixture 
of sodium brtanate (cubic, n = 1-616) and sodium bromide dihydrate 
(monoclinic, a = 1-513, /? = 1-(>19, y = 1-525) immersed in a liquid of 
refractive index 1-54. In this case the constituents are distinguishable 
by two features one substance (sodium bromate) is not only isotropic 
but also its refractive index is much higher than those of the other 
substance. Two anisotropic constituents can be identified if the refrac- 
tive indices of one lie wholly above those of the other ; and in fact, any 
number of anisotropic constituents can be identified if their respective 
ranges of refractive indices are quite distinct. Serious difficulties only 
occur if there are present in a mixture two or more anisotropic consti- 
tuents whose refractive index ranges overlap for instance, if the y of 
one constituent is higher than the a of another. It will be evident that 
there are two (or more) constituents in the mixture, since two (or 
more) values of /J are observed ; but, unless there are some distinguishing 
features (such as differences of shape or size, or the presence of striations 
or other marks on one constituent, or differences of dispersion), it will 
not be possible to measure the other indices. Identification may some- 
times be achieved on the basis of the /? values alone or perhaps by /? 
values aided by measurements of optic axial angles ; if not, the mixture 
is one of those which cannot be identified by microscopic methods. 
This situation is most likely to arise when one of the constituents 
is a very strongly birefringent substance such as a carbonate or a 
nitrate. 

Identification when it is not possible to measure refractive 
indices. In some circumstances it may be desired to identify sub- 
stances without removing them from their mother liquor. Direct identifi- 
cation that is, by measuring properties and looking up the measured 
values in tables is not possible, and the evidence obtainable is con- 
fined to shape, vibration directions, optic axial angles, and the like ; but 
if such characteristics of all the substances likely to be formed in the 



CHAP, iv TRANSPARENT CRYSTALS 101 

particular circumstances are known, it may be possible to conclude that 
the crystals can only be one of the likely substances. For instance, it 
may be known that one of the possible substances grows as plate-like 
crystals of a certain shape, which when lying flat on the slide give a 
(convergent-light) directions image showing part of a biaxial figure, 
oriented in a particular way with respect to the crystal edges ; if none 
of the other likely substances has similar characteristics, these obviously 
form a good criterion for identification. Hartshorne and Stuart (1934) 
give numerous examples of the value of such observations. 

The value of measurements of the magnitude of the optic axial 
angle has been urged by Bryant (1932); and where dispersion of the 
optic axes occurs, the variation of the optic axial angle with light 
frequency is a highly characteristic feature which is valuable evidence 
for identity. (See Bryant, 1941, 1943.) 

These methods of 'spotting* constituents of particular systems (in 
the phase rule sense of the word) are akin to those of the metallurgist, 
but have far greater scope on account of the wealth of observable or 
measurable characteristics in transparent crystals. They are most 
closely allied, however, to those of the petrologist, who by observing 
birefringence, extinction directions, optic axial angles, and the like in 
thin slices of rocks, and referring the information to his knowledge of 
the characteristics and occurrence of mineral species, is able to identify 
such species with rapidity and certainty. For information on these 
methods, see Rogers and Kerr's Optical Mineralogy (1942). 

The thin -section methods of the petrologist may be used for artificial 
specimens which are in the form of large aggregates specimens of such 
materials as refractories, bricks, and boiler scales. Instead of powder- 
ing them and using immersion methods, it is possible to grind thin 
sections and examine them. When it is simply a question of distinguish- 
ing between a few possible constituents of known characteristics, this is 
a useful method. But in unfamiliar systems the powder method is likely 
to be more useful for identification purposes ; the principal function of 
the thin-section method in such circumstances is to provide information 
on the distribution, orientation, or size of the crystals of the different 
constituents. 

Substances which are too opaque for the use of transmitted light 
methods are rare, apart from metals; they include chiefly sulphides 
and a few oxides. Aggregates of such materials may be examined by 
the metallurgist's method of grinding and polishing a flat surface. The 
scope of such methods is much greater for non-cubic than for cubic 



102 IDENTIFICATION CHAP, iv 

substances, since by the use of reflected polarized light it is possible to 
measure birefringence. (See Phillips, 1933.) 

This book is concerned with purely physical methods of identification. 
It is, however, relevant to mention in this chapter on microscopic 
methods the use of a combination of physical and chemical methods. 
Chemical reactions may be carried out on a small scale on microscope 
slides, the crystallization of reaction products being watched. Tests for 
particular ions or atom groups have been devised, the criterion of identity 
being, not solubility or colour as in macroscopic qualitative chemical 
analysis, but crystallographic properties. For information on such 
methods, see Handbook of Chemical Microscopy, by Charaot and Mason 
(1931). 



V 
IDENTIFICATION BY X-RAY POWDER PHOTOGRAPHS 

SOLID substances cannot always be identified by measuring crystal 
shapes and optical properties. In the first place, the crystals in a speci- 
men may be too small to be studied as individuals under the micro- 
scope. Secondly, even if the individual crystals are large enough, the 
information obtainable by microscopic methods may not be sufficient 
for unequivocal identification. The measured refractive indices may be 
(Within the limits of error of measurement) equal to those of two different 
substances ; this is rare, but may occur if for any reason only rough 
measurements can be made. Or it may happen that the measured 
refractive indices do not correspond exactly with those of any known 
substance, either because the specimen in question is an unfamiliar 
substance whose optical properties have not previously been recorded, 
or because it is a mixed crystal whose refractive indices lie between 
those of the pure constituents. Finally, if crystals are completely opaque 
(as in metals and alloys), microscopic technique is limited to observa- 
tions by reflected light. In metallurgical specimens, often the only 
evidence available is that provided by the shapes of the intergrown 
constituents in the polycrystalline aggregate ; in familiar systems, such 
evidence may be sufficient for identification, but in unfamiliar systems 
(especially the more complex ones) it is likely to be inadequate. In any 
of these circumstances, examination by X-ray methods may provide an 
answer to the problems involved. 

The production of X-rays. X-rays are electromagnetic waves of 
very high frequency, and are produced when rapidly moving electrons 
collide with atoms ; the electrons at the higher energy levels in the atom 
are disturbed, and the energy liberated in their transitions from higher 
to lower energy levels is given out in the form of X-rays. 

In X-ray tubes the electrons are produced either by ionization of air 
at a moderately low pressure (in 'gas tubes') or by emission from a 
heated filament at a much lower pressure (in 'hot cathode' or Coolidge 
tubes). In most commercially obtainable X-ray tubes, one of which is 
illustrated in Fig. 59, the latter method is used. An electrically heated 
tungsten filament A emits electrons, which are accelerated by a high 
voltage of some tens of thousands of volts maintained between the 
filament and the target B. (Actually, high-voltage A.C. is applied, but 



104 



IDENTIFICATION 



CHAP. V 




only one-half of the cycle is passed 
by the one-way electron stream.) 
In practice the anode, which is 
water-cooled, is kept at earth 
potential, while the filament is at 
a negative high voltage. A metal 
shield C surrounding the filament 
and kept at the same potential 
has the effect of focusing the 
electron stream on a small area 
of the target. The acceleration of 
the electrons by the high voltage 
gives them sufficient energy to 
bring about the emission of 
X-rays on striking the metal 
target B. The part of the tube 
surrounding filament and target 
is made of brass (the rest being 
porcelain) and is cooled by a 
water-jacket. The tube is con- 
tinuously evacuated, through a 
wide tube, by a diffusion pump 
using low-vapour-pressure oil, 
backed by a rotary pump. The 
X-rays are emitted from the 
target in all directions, but only 
a small proportion is used: win- 
dows D of thin aluminium foil 
allow the exit of only those rays 
which make a small angle with 
the target face. This type of 
X-ray tube is demountable: the 
target and windows are readily 
detachable, and the whole of the 
porcelain part of the tube can be 
taken off to fit a new filament. 
All joints are simply flat or 



COOUH6 
WATER 



FIG. 59. A demountable X-ray tube. A, 

filament ; B, target ; C, focusing shield ; 

D, windows ; E, target holder. 



CHAP, v X-RAY POWDER PHOTOGRAPHS 105 

conical surfaces in contact, sealed by 'plasticine' made from low- 
vapour-pressure grease. Targets of different materials can be mounted 
in duplicate holders such as E, so that rapid changes may be made. 
The demountable type of X-ray tube is probably the most convenient 
for research purposes. Sealed glass X-ray tubes can also be obtained ; 
their advantage is that they do not have to be continuously evacuated, 
require no attention during operation, and demand no expenditure of 
time for maintenance ; on the other hand, if more than one type of 
target must be used, extra complete tubes are required; also the 
emission decreases during the useful life of the tube owing to the 
deposition of a solid containing tungsten (from the filament) all over 
the inside of the tube, including the* target and the insides of the 
windows ; and the X-ray beam may be contaminated with undesired 
wave-lengths from this same deposit. 

In the X-ray tube illustrated the filament is a close helix of tungsten 
wire, set horizontally ; the shield surrounding it brings the electrons to 
a focus along a horizontal line. For most crystallographic purposes a 
narrow. X-ray beam is taken at a small angle 5-7 to the plane of the 
target ; at tliis angle the line focus appears foreshortened so that the 
source is effectively a point or, more precisely, a small area, not much 
larger than the collimating systems used in X-ray cameras. Ideally, 
the X-ray source should be as small, as possible, but in practice, if the 
focus is made too sharp, a hole is burnt in the target after a short period 
of use. 

X-ray wave-lengths. The wave-length distribution in the X-ray 
beam depends on the material of the target and on the accelerating 
voltage used. Fig. 60 shows the sort of wave-length distribution given 
by a copper target when bombarded by electrons accelerated by 50,000 
volts. TJiere is a continuous band of wave-lengths, often referred to'as 
'white' radiation, the limit of which on the short wave-length side is 
rigidly determined by the quantum relationship Ve = hv, where V is 
the accelerating potential, e the electronic charge, h Planck's constant, 
and v the frequency of the shortest waves. If the potential is expressed 
in volts (V), the shortest wave-length in A. is given by 1-234 x 10 4 /F'. 
In addition to the continuous band, and superimposed on* it, are very 
narrow peaks of great intensity, the wave-lengths of which are rigidly 
determined by the nature of the target material. This set of sharply 
defined radiations is known as the K series of copper ; by far the strong- 
est peaks are a 1? 2 , and j3 components ; the wave-lengths of o^ and a 2 
are so nearly identical that resolution of them occurs only in special 



106 



IDENTIFICATION 



CHAP. V 



circumstances. The intensity of c^ (1-5374 kX) is twice that of a 2 
(1-5412 kX).f 

For most crystallographic purposes a monochromatic beam, that is, 
a beam consisting of one wave-length only, is desirable. Actually the ex 
components are so strong in comparison with all other wave-lengths 
present that the unfiltered beam may be used for some purposes ; the 
'white* radiation merely increases the background intensity of X-ray 



CO 




WHITE 
RADIATION 



8 1-0 \-2 
/VAVELENGTH IN A 



1-6 1-8 



FIG. 60. Intensity distribution in X-ray beam from copper target ; accelerating: voltage, 
50,000 V. Kfi has about 1/6 the intensity of Kx. A nickel filter 0-021 mm. thick reduces 
this ratio to 1/600 owing to the form of the absorption curve. Kot is actually 

a very close doublet. 

diffraction photographs, and this may not constitute a serious dis- 
advantage ; the /? component produces its own diffraction effects, but 
provided a and jS diffractions are readily distinguished, the presence of 
the latter can be tolerated. For powder photographs it is best always 
to remove the /? component, and this may be done by placing a suitable 
filter in the beam ; the absorption coefficient of any chemical element 
suddenly changes at a particular wave-length corresponding to the 
resonance frequency (see Fig. 60), and by choosing an element whose 
absorption edge lies between the wave-lengths of the a and /} components 



f To convert to Angstrom units (10~ 8 cm.), multiply these figures by 1*00202. (See 
footnote to Table II, p. 107.) Cu K^ = 1-5405 A. Cu Ka 2 = 1-5443 A. 



CHAP. V 



X-RAY POWDER PHOTOGRAPHS 



107 



in the X-ray beam, the intensity of the /? component may be reduced 
to a negligible level. For the very frequently used copper radiation, 
nickel foil accomplishes this end; if it is of thickness 0*021 mm., it 
reduces the intensity of Kfi to 1/600 that of KQL, and at the same time 
reduces the intensity of some of the 'white' radiation in comparison with 

TABLE II 
Targets, Wave-lengths and Filters 





Targe 


t 






P Filter 




Element with 
A torn ic 
dumber 


Line 


Wave-length 
in 
kX units} 


Peak 
KV 


Element and 
absorption edge 
(kK units) 


gm./cm.* 


Thickness in mm. 
necessary to reduce 
KpfKato 1/600 


Mo, 42 


Ej 


0-707831 
0-712105 


80 


Zr, 0-6874 


0-069 


0-108 




Kfi\ 


0-630978 










Cu, 29 


K^ 


1-337395 


50 


Ni. 1-4839 


0-019 


0-021 




K(Xj 


1-541232 












Kfii 


1-38935 










Ni, 28 


K*! 


65450 


50 


Co, 1-6040 


0-015 


0-018 




Ka 2 


65835 












K$i 


49705 










Co, 27 


Kat! 


78529 


45 


Fo, 1-7394 


0-014 


0-018 




Ka a 


78919 












Kfii 


61744 










Fe, "26 


K*i 


932076 


40 


Mri, 1 8916 


0-012 


0-016 




Kct 2 


936012 












Kfii 


753013 










Or, 24 


Xa, 


2-28503 
2-28891 
2-0806 


35 


V, 2 2630 


0-009 


0-016 



t The kX unit is 1/3-02904 of the cleavage spacing of calcite at 18 C. ; it was based 
on a former value for Avogadro's number, according to which the kX unit was 10~ 8 cm. 
or 1 Angstrom unit. The most recent value for Avogadro's number, however, gives the 
kX unit a sliphtlv different value: to convert tho above figures to Angstrom units, 
multiply by 1-00202. The kX unit is retained because the relative values of X-ray 
wave-lengths are known more accurately than the absolute values. (Lipson and Riley, 
1943; Siogbahn. 1943; Wilson, 19436.) 

K a. The filter may, if desired, be made the window of the X-ray tube ; 
or, if the window is aluminium, the filter may be placed in a holder in 
front of the window or on the camera. If it is necessary to remove 
'white' radiation altogether, this is accomplished by reflecting the X-ray 
beam by a particular face of a large crystal set at the correct angle ; 
sodium chloride, pentaerythritol, and urea nitrate crystals have been 
used for this purpose since they giye very strong reflections (Fankuchen, 
1937 ; Lonsdale, 1941). Urea nitrate gives the strongest reflected beam. 



108 IDENTIFICATION CHAP, v 



The reflected beam consists of K^ and Ka 2 , with negligible proportions 
of other wave-lengths ; its intensity is much reduced in comparison with 
the primary beam, so much so that exposures must be increased ten-fold 
or more; this is the price of strict monochromatism. According to 
Pankuchen (1937), the intensity of the reflected beam is increased by 
grinding an artificial surface on the reflecting crystal at a suitable angle. 
Guinier (1937) achieved a focusing effect by using a curved crystal as 
monochromator. 

To reduce exposures, high-power X-ray tubes are being developed ; 
the limiting factor here is the heat generated at the focal spot on the 
target. To avoid melting, the target is rotated so that the heat is spread 
over an increased area (Muller,1929; Astbury and Preston, 1934). 

The wave-length of the KOL radiation is determined by the atomic 
number of the target material; the higher the atomic number, the 
shorter the wave-length. (The wave-length A is given almost exactly by 
the expression A K/(Nl) 2 , where N is the atomic number and K a 
constant.) The wave-lengths of some frequently used radiations, with 
the filters suitable for removing the j3 component, are shown in Table II, 
taken from a paper by Edwards and Lipson (1941); this paper also 
gives similar information for the less frequently used wave-lengths, 
together with useful information on the preparation of filters. 

X-ray powder photographs. When a narrow monochromatic 
beam of X-rays passes through a small specimen of a powdered crystal- 
line solid, or through any poly crystalline specimen in which the 
crystals are oriented at random, numerous cones of diffracted beams 
emerge frorqi the specimen, and they can be recorded either as circles on 
a flat photographic plate or film placed behind the specimen at right 
angles to the X-ray beam, or better still, as arcs on a strip of film en- 
circling the specimen as in Fig. 61. The latter is preferable because the 
angular range of diffracted cones which can be recorded on a circular 
film is much greater than on a flat film. The powder method was first 
used by Debye and Scherrer (1916) and independently by Hull (1917). 

With regard to the origin of the diffraction cones, it is sufficient for 
the present to remark that each cone consists of a large number of small 
diffracted beams, each from a small crystal ; and that all the diffracted 
beams in any one cone are 'reflections' by one particular type of crystal 
plane. Any particular crystal plane can reflect monochromatic X-rays 
only when it is at a particular angle to the primary beam ; all the little 
crystals which happen to lie with this plane at this angle 6 to the primary 
beam give a reflection. The angle of reflection is equal to the angle of 



CHAP. V 



X.RAY POWDER PHOTOGRAPHS 



109 



incidence ; hence the reflected beam makes an angle 26 with the primary 
beam. The reflected beams from all the little crystals which happen to 
be suitably oriented therefore form a cone of semi-vertical angle 20 
having the primary beam as its axis. Each different type of crystal 
plane requires a different angle of incidence, and therefore gives a re- 
flected beam at a different angle to the primary beam ; thus, numerous 
cones of reflected beams are produced at specific angles, each cone 
coming from a different crystal plane. 




ill 



FILM FLATTENED OUT 

FIG. 61. Arrangement for taking powder photographs. The angle ESX is 26, 
where 9 is the angle of incidence on a set of crystal planes. 

For the present we shall not inquire into the reason why reflections 
are produced only at specific angles, nor into the type of crystal plane 
responsible for each cone ; we shall merely accept the fact that each 
crystalline species produces its own characteristic pattern which is 
different from the patterns given by other species. It is possible to 
identify substances by means of their X-ray powder photographs with- 
out any knowledge of the structures of the crystals or of the theory of 
diffraction, just as it is possible to use optical emission spectra for the 
identification of elements without any knowledge of the electron transi- 
tions responsible for the emitted rays. 

Powder cameras. A powder camera consists essentially of an 
aperture system to define the X-ray beam, a holder for the specimen, 
and a framework for holding the photographic film. For most identi- 
fication purposes a camera 9-10 cm. in diameter is found satisfactory; 
an X-ray beam about 0-5 mm. wide is generally used, the powder 
specimen being a little narrower than this of the order of O3 mm. 

X-ray cameras are usually made almost entirely of brass ; this mate- 
rial is not ideal for the aperture system, as its absorption of the commonly 
used wave-lengths is only moderate ; but in practice it is usually found 



110 IDENTIFICATION . CHAP, v 

satisfactory. The best aperture system (see Fig. 62) consists of a circular 
hole O5 mm. in diameter drilled through a brass cylinder A. To prevent 
X-rays scattered by the edgea of the aperture from reaching the film 
there is a guard tube B, I mm. in diameter. Using this aperture system, 
powder photographs like that shown in Fig. 63, Plate III, are produced. 
It is found that not much deterioration of quality of the photograph 
occurs if a slit up to 2 mm. long (and still 0-5 mm. wide) is used instead 
of the circular 0-5 mm. tube. For instance, in Fig. 63, Plate III, the 
photographs of zinc oxide and a alumina were taken with the smaller 
aperture system, while those of sodium sulphite and dickite were taken 
with the slit system 2 mm. long ; in the latter the ends of the arcs are 
a little diffuse, but the centres <io not suffer much, except at very small 
angles. The time of exposure is reduced by using the slit ; the whole 
brass cylinder AB can be withdrawn and replaced by the slit system 
when required. 

Several methods of mounting specimens are used. Ideally, the only 
solid material in the X-ray beam should be the specimen material itself, 
but this is only possible if the specimen is a coherent piece of material 
such as a metal wire ; usually, a powder specimen must be held together 
in some manner, using as little extraneous material as possible. It is 
sometimes mixed with a trace of adhesive and stuck to a hair or fine 
glass fibre ; the hair must be kept taut by hanging a small lead weight on 
it. Another method is to mix the powder with some adhesive to form 
a paste, and extrude a rod of this from a capillary tube. Substances 
which are affected by solvents, or are deliquescent, are packed into 
capillary tubes of lithium borate glass ('Lindemann glass') which can 
then be sealed. Lithium borate is used because it contains only elements 
of low atomic number, and consequently does not absorb X-rays to any 
great extent.")" Powders containing heavy elements may be mixed with 
a light diluent such as powdered gum tragacanth (Rooksby, 1942) to 
reduce the absorption. Metals and alloys are usually examined in the 
form of filings, and the preparation of uncontaminated specimens 
thoroughly representative of the lump from which they were filed 
presents special problems of its own (Hume-Rothery and Raynor, 1941). 

The specimen P (Fig. 62), however it is made, should be not more than 
0-5 mm. wide. It is best to rotate it to ensure random orientation of the 
crystals (otherwise discontinuous spotty arcs may be produced on the 
photograph), and for this purpose it is mounted on a rotating holder. 

t Lithium borate glass capillaries become* devitrified in moist air in a few days, but 
will keep indefinitely if stored over anhydrous calcium chloride. 




FIG. 62. .Essential parts of a powder camera. A, aperture system 
J3 9 guard tube ; CD, trap ; E, knife edges ; JP, specimen. 



112 IDENTIFICATION x CHAP, v 

; 
Centring may be done by hand, or better, by using a holder fitted with 

adjusting screws. 

To prevent fogging of the film by the primary beam a trap is provided. 
Its construction is sufficiently explained by Fig. 62 ; the edges D must 
be so placed that X-rays XE scattered in the trap cannot reach the film. 
At the back of the trap is a screw which can be removed (a) for centring 
the specimen by looking through the hole at a light placed at A, and (b) 
for adjusting the camera in relation to the X-ray tube, for which purpose 
a fluorescent screen is placed at C. (This must be done before the 
photographic film is put in the camera.) It is sometimes useful to have 
the position of the primary beam recorded on the film, but its strength 
must be very much reduced to avoid fogging. This can be done by 
drilling out the screw just mentioned until the thickness of brass re- 
maining to obstruct the beam is 1-5 mm. ; this reduces the primary beam 
to about the same level of intensity as a strong diffraction arc on an 
average powder photograph. 

The film is in contact with the cylindrical brass frame of the camera 
and is held in position by springs S. Sharp edges E terminate the ex- 
posed part of the film abruptly. Light is excluded by a brass cover 
which fits over the whole camera ; the X-ray beam is admitted through 
a hole covered with black paper. Further details of the construction 
and use of powder cameras can be found in a paper by Bradley, Lipson, 
and Fetch (1941). 

Cameras in which powder photographs of substances maintained at 
high temperatures may be taken are much used, especially in metallurgy 
(Jay, 1933; Dorn and Glockler, 1936; Hume-Rothery and Reynolds, 
1938). Other special cameras have been designed for low temperatures 
(Pohland, 1934) and high pressures (Frevel, 1935). 

General characteristics of X-ray powder photographs. A few 
examples of X-ray powder photographs, all taken with copper Koc 
radiation, are shown in Fig. 63, Plate III. They vary greatly in com- 
plexity; chemically simple crystals of high symmetry give strong 
patterns containing few arcs, while crystals of complex chemical consti- 
tution or of low symmetry give patterns consisting of a large number of 
less strong arcs. The time of exposure necessary to produce a photo- 
graph of convenient intensity is related to the complexity. Under 
typical conditions for instance, when a self-rectifying X-ray tube is 
passing 20 milliamperes and the unsmoothed peak voltage is 60 kV 
well-exposed photographs of metals can usually be taken in 10-15 
minutes, using the 2-mm. slit aperture in the 9-cm. camera just described, 



PLATE III 




















''' - " 
FIG. 63. X-ray powder photographs. 



CHAP, v X-RAY POWDER PHOTOGRAPHS 113 

while a complex silicate may require over an hour under the same 
conditions. 

Exposures may be shortened by putting a fluorescent screen behind 
and in contact with the film, so that the optical fluorescence reinforces 
the direct X-ray effect on the photographic emulsion ; or two screens 
might be used, one in front and one behind the film. (Note that the 
intensity relations are much changed by the optical fluorescence : weak 
reflections come out relatively much too weak.) 

There is always a certain background intensity on X-ray powder 
photographs, due partly to the presence of 'white' radiation, which gives 
diffracted rays over a wide angular range from each crystal plane, 
partly to a certain amount of incoherent scattering of the K a radiation 
by the crystals themselves, and partly to X-ray fluorescence of the 
crystals, which absorb the primary rays and re-emit the energy in the 
form of longer waves. The last effect may in some circumstances be so 
strong that serious fogging of the film occurs, and for this reason the 
wave-length of the primary beam used must be chosen with this effect in 
mind. X-ray fluorescence is strongest when the wave-length of the 
absorption edge of the irradiated element (which is almost equal to that 
of Kp for the element) is slightly longer than the wave-length of the 
irradiating beam ; under these circumstances, in addition to the part 
of the primary beam diffracted by the crystals another part is converted 
into the K series of the irradiated element. For instance, the shortest 
wave-length of the K series (and the absorption edge) of iron is not much 
longer than that of copper KOL, and consequently iron-containing crystals 
fluoresce strongly in copper KOL radiation. For substances containing 
iron, therefore, copper KOL radiation is quite unsuitable ; cobalt or iron 
KOL should be used. The elements preceding iron in the periodic table 
also fluoresce in copper KOL radiation, but less strongly than iron, the 
effect diminishing with the atomic number ; for calcium, for instance, it 
is practically negligible. 

Fluorescent radiation, when it is not too serious, may be partly 
absorbed by placing a suitable filter between the specimen and the film. 
For instance, titanium compounds in copper radiation give rather foggy 
photographs owing to X-ray fluorescence ; the fog is reduced by putting 
the nickel filter (necessary in any case for removing copper Kfi) between 
the specimen and the film, instead of in the more usual position in front 
of the camera ; it absorbs titanium radiation much more than it does 
copper KOL, and the background intensity is therefore reduced in com- 
parison with the diffraction arcs. 



114 IDENTIFICATION CHAP, v 

Some substances, such as alkaline-earth sulphides, emit visible light 
when irradiated by X-rays ; for these substances it is essential to have a 
sheet of optically opaque material (such as black paper) between the 
specimen and the film. 

The wave-length used does not, in general, affect the relative inten- 
sities of the various arcs in any pattern, but it does control the scale of 
the pattern ; the longer wavelengths spread out the pattern, while the 
shorter wave-lengths contract it. 

The simplest procedure in identifying a substance by an X-ray method 
is to compare its powder photograph with those of known substances 
taken in the same camera with X-rays of the same wave-length. The 
patterns given by different substances are usually so obviously different 
that visual comparison is sufficient for certainty. Often, however, it 
may not be possible to obtain the reference substances required ; nor is 
this necessary in many cases, for a limited amount of interpretation 
makes it possible to use published results, obtained, maybe, with 
cameras of different radius or X-rays of different wave-length. Interpre- 
tation of powder photographs for this purpose usually need go only as 
far as the calculation of the spacings of the crystal planes responsible 
for the various arcs. This demands no knowledge of the crystal struc- 
ture, but only the use of a simple equation, the derivation of which forms 
the subject of the next section. 

Diffraction of X-rays by a crystal. Diffraction by a three-dimen- 
sional array of atoms might be expected to present a complex geo- 
metrical problem, but in actual fact the fundamental equation, known 
as Bragg's law, turns out to be extremely simple : 

d _ A 
n ~~ 



where A is the X-ray wave-length, d the distance between successive 
identical planes of atoms in the crystal, the angle between the X-ray 
beam and these atomic planes, and n any whole number. (W. L. Bragg, 
1913.) It may seem curious that the arrangement of the atoms in each 
atomic plane does not come into the expression, which contains merely 
d, the distance between the atomic planes. The reason for this can be 
appreciated by considering, first the diffraction of rays by a row of 
points, then by a plane arrangement of points, and finally by a three- 
di#iensional array of points. 

If a train of waves, whose wave-front is perpendicular to the direction 
of propagation, is scattered by an array of points, the scattered rays 



CHAP. V 



X.RAY POWDER PHOTOGRAPHS 



115 



interfere with each other except when they happen to be in phase, that 
is, when the* difference between the path-lengths of rays scattered by 
different points is either zero, or one wave-length, or two wave-lengths, 
or any whole number of wave-lengths. If a single row of equally 
spaced points (spacing = a) is perpendicular to the beam (Fig. 64 a), ray 
II is a cos <f> behind ray I, and thus the rays will be in phase only when 



WAVE 
FRONT 





=/ 



n-0 



d~3 cos $ which must - nX 
(a) 



(b) 




Differencf of path-length = OA-PB 
- a cos - a cos B This must -/?A 

FIG. 64. Diffraction by a row of points. 

n\ = a cos <. For particular values of n and A. <f> is constant, that is, 
the diffracted rays form a cone with the point-line as axis or rather, two 
cones, one on each side of the incident beam. The second-order diffrac- 
tions form a narrower cone than the first-order diffractions. The zero- 
order cone is a plane surface (Fig. 64 b). If the incident beam is not 
perpendicular to the point-line (Fig. 64 c) the diffraction surfaces are 
still cones, but their semi-vertical angles <f> are given by 



and the upward and downward cones for the same value of n have 
different angles. Note that for diffracted rays having the same 



116 



IDENTIFICATION 



path -length (n = 0), <f> will equal 6, and the angle of this (zero-order) 
cone of diffracted beams is independent of the point-spacing a, and 
depends only on 6, which may have any value ; in other words, an X-ray 
beam falling on a line of equally spaced diffracting points at any angle 
gives rise to a zero-order cone of diffracted beams whose semi-vertical 
angle 6 is the same as the angle between the incident beam and the 
point-line in other words, the incident beam forms part of the cone. 
Consider now a regular array of points in a plane, such as that in 
Fig. 65 ; this may be divided into rows of points in many different ways ; 




FIG. 65. 'Reflection' by a regular array of points in a plane. 

each type of point-row would, by itself, produce its own diffraction 
surfaces, but the diffracted rays from different types of point-rows such 
as BAC and DAE will not co-operate except where the different surfaces 
intersect that is, along certain straight lines. We need not consider 
this in detail, except to show that the zero-order diffracted beam 
(n = 0) is a sort of reflection of the incident beam by the point-plane. 
The incident beam OA strikes 1/he surface at an angle 6, Fig. 65. One 
row of points (suppose it is BAC) is bound to lie exactly under the 
incident beam ; that is, LOB A = LOBD 90. This point-row would, 
by itself, give a zero-order diffraction cone of angle 0. Any other row of 
points such as DAE is at a larger angle (f> to the incident beam, and by 
itself would give a zero-order diffraction cone of angle <. The two cones 
cut in a line, and this will be the direction of the diffracted beam pro- 
duced by both point-rows together. We have to find the direction of 
this line AP. Make AP = GA,AC = AB, and AE = AD; join PC, 
PE, and CE. Now the solid figure PCAE has three angles 0, <, and oj 
(meeting at A) equal to corresponding angles of the figure OB AD, and 
the three sides enclosing these angles (AP, AC, and AE) also equal to 



CHAP. V 



X-RAY POWDER PHOTOGRAPHS 



117 



the corresponding sides of the figure OB AD ; hence the two solid figures 
are similar in all respects; therefore the angle PCE = OBD = 90, 
and thus AP lies in the plane OAB. The same could be shown for every 
possible point-row ; hence all the points acting together produce a zero- 
order diffraction along a single direction, the angle of diffraction being 
equal to the angle of incidence, with both incident and diffracted beams 
lying in a plane perpendicular to the point-plane : the zero-order diffrac- 
tion is, in fact, a 'reflection* of the incident beam by the point-plane. 
Such a 'reflection* can be produced for any angle of incidence of the 
primary beam; and it is important to notice that the spacing and 
arrangement of the points in the plane do not affect the process. 




FIG. 66. The condition for 'reflection' by a crystal lattice. Difference 
of path -^ GY+YH = 2cJsm0, which must -= nX. 

If a fferee-dimensional point-array is to produce a diffracted beam, 
the diffracted waves from all the points must be in phase ; some of them 
will have the same path-length, those from other points will be one 
wave-length behind the first set, still others will be two wave-lengths 
behind, and so on. It has just been shown that any plane of points by 
itself would be capable, at any angle of incidence, of producing a dif- 
fracted beam consisting of waves all of the same path-length, and this 
beam would be a reflection of the primary beam by the plane ; but if the 
waves from the next lower plane of points are to be in phase with those 
from the first-mentioned plane, this imposes strict limitations on the 
permissible angles of incidence. Thus in Fig. 66, JP, Q, and R are 
successive planes of points seen edgewise. Plane P, if it were alone, 
would reflect the primary beam AX in XD, the angle of reflection DXP' 
being equal to AXP ( 8) ; this would happen whatever the value of 0, 
and does not depend at all on the spacing and arrangement of the points 



118 IDENTIFICATION CHAP, v 

in the plane. Plane Q likewise, if it were alone, would also reflect the 
beam at this same angle; but since Q is lower than P, the path BYE 
traversed by waves reflected in Q is longer than the path A X D traversed 
by waves reflected in P, and if the two sets of waves are to be in phase, 
then the difference of path-length must be a whole number of wave- 
lengths. The difference in path-length is G Y-\- YH, where XG and XH 
are perpendicular to BY and YE respectively. XY is drawn perpendi- 
cular to PP' and QQ', so that its length is d, the spacing of the planes. 
It follows that GY and YH are each equal to dsin0, hence 

= nX. 



This equation is Bragg's law. ' 

A crystal is not, in actual fact, a simple array of points, each of which 
is a pattern-unit. In the first place, an atom is not a point ; its electrons, 
which scatter the X-rays, occupy a volume commensurable with the 
interatomic distances. Secondly, each pattern-unit in a crystal often 
consists, not of one atom, but a group of atoms. The pattern-unit is 
thus not a point, but has a diffuse and often irregular form. However, 
for the purpose of diffraction theory, as far as it is carried in this chapter, 
the diffuse pattern-unit may be mentally replaced by a point. It will 
be shown in Chapter VII that the form of the pattern-unit affects the 
intensities of the diffracted beams ; but it does not affect their positions, 
which depend only on the space-lattice, the fundamental arrangement 
of identical pattern-units. 

Each of the many different sets of planes in a crystal may produce a 
'reflected' beam, but only if it is at the appropriate angle to the primary 
beam, this angle being determined by d, the spacing of the set of planer 
in question. The angle between the reflected beam and the primary 
beam will be 20. The Bragg equation means that if we turn a crystal 
about at random in an X-ray beam, in general no reflected beam will be 
produced ; but at certain definite positions of the crystal, when the 
condition nX = 2d sin is satisfied for a particular set of planes, a re- 
flected beam flashes out. A set of planes having a large spacing d 
produces a first-order reflection close to the primary beam (that is, is 
small), and higher-order reflections (n = 2, 3, and so on) at larger angles. 
A set of closely spaced planes produces its first-order reflection at a large 
angle. For any particular X-ray wave-length there is a lower limit to the 
possible values ofd/n, set by the fact that sin cannot be greater than 1. 
The lower limit for d/n is equal to A/2. 

In a crystalline powder the crystals are oriented at random. If a 



CHAP. V 



X.RAY POWDER PHOTOGRAPHS 



119 



narrow X-ray beam is sent through the powder, most of the crystals will 
give no diffracted beams, because none of their planes make a suitable 
angle with the beam. Some crystals, however, lie in such positions that 
a particular set of crystal planes say the 100 set is at exactly the 
appropriate angle for giving the first-order reflection (the angle must be 
within a few minutes of arc of the angle specified by the Bragg equation ; 
see p. 203) ; all the little crystals which reflect with their 100 planes give 
a reflected beam at the same angle, this angle depending on the spacing 
of the 100 planes. The locus of all directions making a particular angle 
with the primary beam is a cone having the primary beam for its axis. 
Other crystals in the powder happen to lie in such a way that they can 



FILM 



100 




CRYST/ 



INCIDENT 
X-RAY 
BEAM 

FIG. 67. Each arc on a powder photograph represents 
a 'reflection' by a particular crystal plane. 

reflect with their 110 planes, and all these will produce a cone of re- 
flected rays, but the 110 cone will have a different angle from the 100 
cone, since the spacing of 110 planes is different from that of 100 planes. 
(See Fig. 67.) Each cone of rays cuts the photographic film in an arc. 

This, therefore, is the origin of the arcs on a powder photograph ; 
each arc represents the combined diffracted beams from all the crystals 
which happen to be suitably oriented for reflecting with one particular 
set of planes. 

Measurement of powder photographs. From the measured 
position of each arc on a powder photograph, 6 can be calculated, and 
thence din by the Bragg equation. Since on powder photographs the 
position of the undeviated primary beam is usually not precisely defined, 
it is necessary to measure from an arc on one side of the photograph to 
the corresponding arc on the other side, the distance x between which 
represents 40. If the radius of the film when it was in the camera was r, 



120 IDENTIFICATION . CHAP, v 

the circumference Ztrr represents 360, or, since we are to divide through 
by 4, 27TT represents a Bragg angle of 90 ; for each arc is thus obtained, 
and d/n is calculated from 9, r, and the known X-ray wave-length used 
for the photograph. If many films have to be measured, it is best in the 
long run to make a table giving d/n for all values of x. The effective 
radius r of the camera is best determined by the methods mentioned 
in the section on high-precision methods (p. 180). 

The X-rays used are not strictly monochromatic ; copper KOL radia- 
tion consists not of one wave-length but of two slightly different wave- 
lengths, 1-5374 and 1-5412 kX. These produce reflected beams from any 
particular crystal plane at slightly different angles; but in ordinary 
powder cameras the two reflections are not resolved except at angles 
near 90, as may be seen by inspecting the powder photographs in Fig. 63, 
Plate III ; the last few reflections are plainly doublets, resolution at large 
angles being a consequence of the fact that when is near 90 a small 
difference of sin 6 (produced by a small difference of A) means a com- 
paratively large difference of 8. Therefore, for the arcs at small angles a 
weighted averagef value (1-5387 kX for copper) must be assumed for 
calculations, while for the doublets at large angles calculations can be 
made for both individual wave-lengths. For identification purposes 
measurement with a steel rule graduated in millimetres or half-milli- 
metres is usually sufficiently accurate, but for precision work a travelling 
microscope may be used. For still greater precision photometric measure- 
ment of the distribution of blackening on the film may be made ; each 
reflection appears as a hump on the blackness-distance curve, and the 
position of the peak can be taken as the 'position' of the reflection. In work 
of such precision as this, some account must be taken of several sources 
of error arising from the particular experimental circumstances. 

Spacing errors in powder photographs . Owing to the appreciable 
thickness of the powder specimen and the absorption of X-rays in it, 
the diffraction arcs are produced more by the outer layers of the speci- 
men than by its centre (Fig. 68) ; on this account, corresponding arcs on 
opposite sides of the photograph tend to be slightly too far apart, and 
spacings d/n calculated from the arc angles therefore tend to be low. 
Such errors are greatest for small angles of reflection (large values ofd/n), 
and diminish towards zero for large angles. For specimens containing 
only light atoms, such as carbon, oxygen, aluminium, or sodium, they are 
very small at all angles ; but for specimens containing large proportions 
of heavy elements, such as iodine or lead, they are appreciable, though 

f The oi 1 wave-length is given twice the weight of a , since it is twice as strong. 



CHAP. V 



X-RAY POWDER PHOTOGRAPHS 



121 



not sufficiently large as to be likely to cause confusion in identification. 
The diameter of the specimen should be as small as possible (not more 
than 0-3 mm. f rfor the camera described here) to minimize such errors. 
An effect often produced by strongly absorbing specimens is the splitting 
of small-angle reflections into narrow doublets, owing to the beam 
passing round both sides of the specimen but not through its centre ; 
at larger angles, only the outer component of the doublet is present, as 

CENTRE OF 
REFLECTION > 
,'T*. WEAKLY :'\ 

ABSORBING 
SPECIMEN 



,'/K CENT RE OF 
'I ^REFLECTION 
STRONGLY 
ABSORBING 
SPECIMEN 




X-RAY 
BEAM 

FTG. 68. Apparent displacement of reflections owing to absorption. (Much exaggerated.) 
Reflections at large angles are less affected than those at small angles. 

in Fig. 68. Other errors may occur if the specimen is not strictly in the 
centre of the film or if a long slit is used. Correction terms can be calcu- 
lated if certain factors are known (Claassen, 1930; Bradley, 1935; see 
also Int. Tab. (1935), p. 583). Finally, photographic films shrink on 
development and drying; errors from this cause can be avoided by 
printing fiducial marks on the film. In the type of camera described 
here (Bradley and Jay, 1932) the exposed part of the film (always 
defined by the general background blackening) is terminated by a knife 
edge whose position represents a definite angle which can be accurately 
measured ; assuming uniform shrinkage, the true angle for any arc can be 
calculated by simple proportion. 



122 IDENTIFICATION CHAP, v 

One method of correcting for all possible errors is to mix the substance 
with a standard substance whose spacings are accurately known ; the 
X-ray photograph shows both patterns superimposed. For this purpose 
it is desirable to use a simple substance giving few lines, otherwise 
overlapping of arcs will be frequent ; sodium chloride is often used. 
Measurement of the sodium chloride arcs gives a calibration curve, 
which can then be used for interpolating the precise spacings of the 
substance under investigation. 

Further information on high precision methods will be found on p. 180. 

Identification of single substances, and classification of powder 
photographs. Each crystalline substance has its own set of plane- 
spacings, which is different from those of other crystalline substances. 
The relative intensities of the various reflections are also characteristic. 
Each substance thus gives its own characteristic powder photograph, 
the scale of which, however, depends on the wp,ve-length of the X-rays 
used and the diameter of the camera. 

The indirect method of identification, in which the pattern of the 
unknown is compared with those of likely substances, has been much 
used ; but to eliminate the chances of possible substances being over- 
looked, and to deal with the occurrence of quite unexpected substances, 
a direct method is desirable, in which 'key' spacings of the unknown are 
looked up in an index, in the manner used in the identification of optical 
emission spectra. In the direct method of identification the main 'key' 
is the spacing of the strongest arc: a card index is made, in which all 
substances are arranged in order of the spacing of the strongest arc. 
If two or more arcs appear equally strong to the eye, the innermost the 
one with the greatest spacing should be used as the key. To identify 
an unknown substance, the spacing of its strongest arc is measured ; 
reference to the index may indicate several substances having the 
correct key spacing within the possible limits of error of the photograph. 
Some will be out of the question, in view of the origin of the specimen ; 
for the rest, the remaining arcs will decide. In the card index published 
by the joint committee of the American Society for X-ray and Electron 
Diffraction and the American Society for Testing Materials there are 
three cards for each substance, one for each of the three strongest arcs ; 
consequently, measurement of the second and third strongest arcs, 
followed by reference to the index, may lead to unequivocal identifica- 
tion, which, however, should only be accepted as final when the whole 
pattern is compared with that in the index and found to agree, both in 
spacings and relative intensities. Visual estimates of relative intensities, 



PLATE TV 




Ka 








, Ka 




Ka: 




Ka 




Fro. 69. X-ray powder photograplis. 



CHAP, v X-RAY POWDER PHOTOGRAPHS 123 

classified as 'very strong', 'strong', 'medium', 'weak', and so on, are 
usually sufficient for identification purposes. If the index does not lead 
to identification, the literature may be searched for the patterns of 
likely substances. The 'Strukturbericht' published by the ZeitschriftfUr 
Kristallographie is useful here. A collection of 1,000 patterns was 
published by Hanawalt, Rinn, and Frevel (1938) ; these are included in 
the A.S.T.M. index. For minerals there are the determinative tables 
of Mikheev and Dubinina (1939), and for the ore minerals the compre- 
hensive list of Harcourt (1942). 

If the spacings of the arcs on a powder photograph do not lead to 
identification, the determination of unit cell dimensions from the powder 
photograph may be attempted ; the methods are described in Chapter 
VI. If crystals large enough to be handled individually can be picked 
out of the specimen, single-crystal rotation photographs may be taken 
and used for identification ; this also is dealt with in Chapter VI. 

Identification and analysis of mixtures. A mixture of two or 
more substances gives a pattern consisting of the superimposed patterns 
of the individual components, provided that these components exist as 
separate crystals in the powder specimen (see Fig 69, Plate IV). The 
identification of simple mixtures, therefore, does not differ in principle 
from that of single substances. The principle is to find the spacing of the 
strongest arc; reference to the index may result in identification of 
one probably the main constituent, and this accounts for some of 
the arcs. The spacing of the strongest of the remaining arcs is then used 
in the same way for identifying the second constituent ; and this pro- 
cedure is repeated until every arc on the photograph has been accounted 
for. Overlapping of the arcs of different constituents may sometimes 
cause confusion. 

If a mixture is found to give a very complex pattern which appears 
to consist of several superimposed patterns, many of the arcs of which 
overlap, it may be desirable to attain greater resolution, and this can be 
done either by using X-rays of longer wave-length to spread out the 
diffraction pattern or by modifying the camera or specimen. 

Chromium KOL radiation (A = 2-29 A.) is suitable for the first method ; 
but the absorption of this wave-length by air is appreciable, and it is 
desirable either to evacuate the camera or to fill it with hydrogen. 
Even longer wave-lengths, like the characteristic K radiation of calcium 
or even magnesium, have been used for special purposes, but it is doubt- 
ful whether identification problems would ever call for the use of such 
long waves. Their use entails further experimental difficulties ; owing to 



124 IDENTIFICATION CHAP, v 

the high absorption of these waves by air, the camera should be evacuated 
and built straight on to the X-ray tube, forming part of the same 
evacuated system. (Clark and Corrigan, 1931 ; Hagg, 1933.) 

The second method is preferable. Either the specimen and aperture 
system may be reduced in size or the camera increased in size. The pre- 
paration of very narrow powder specimens is not easy, and the tendency 
now is to use larger cameras (Bradley, Lipson, and Fetch, 1941). 
Exposures are necessarily increased, but this is the unavoidable price 
of greater resolution. 

X-ray powder photographs are now very widely used for identifica- 
tion. Two typical investigations will be mentioned. The first is the 
identification of the crystalline Constituents of Portland cement, one of 
the more important building materials of the present day. It is made 
by heating to a high temperature such raw materials as chalk or lime- 
stone, clay, and sand. Its chemical composition may be expressed in 
terms of lime, silica, alumina, and ferric oxide, but its actual constitu- 
tion cannot be deduced by stoichiometric methods. X-ray powder 
photographs, together with evidence obtained by the determination of 
optical properties under the microscope, have shown that the principal 
crystalline constituents are Ca 3 Si0 5 and j3 Ca 2 Si0 4 , together with smaller 
amounts of Ca 3 (A10 3 ) 2 , 4CaO.Al 2 3 .Fe 2 O 3 , and MgO (Brownmiller and 
Bogue, 1930 ; Insley, 1937 ; Insley and McMurdie, 1938). It may be said 
that the recent great progress in our understanding of the chemistry of the 
setting of cements is largely due to crystallographic investigations of this 
type. (SeeTheChemistryofCementandConcrete,byIjea,a,ndT)esch, 1935.) 

The second example is the investigation of the constitution of 
'bleaching powder', which is made by the action of chlorine gas on 
slaked lime. The constitution of this widely used material had remained 
obscure for many years, since, although it contains calcium chloride as 

well as hypochlorite (2Ca(OH) 2 +2Cl 2 > Ca(OCl) 2 +CaCl 2 -f 2H 2 0), it 

is not deliquescent; moreover, it is difficult to carry chlorination to 
completion. These features had led to many suggestions of the existence 
of double compounds suggestions which could not be tested by older 
methods of investigation, on account of the small size of the crystals. 
X-ray powder photographs showed that bleaching powder consists of 
two substances a crystal of variable composition consisting chiefly of 
Ca(OCl) 2 , and the basic chloride CaCl 2 . Ca(OH) 2 . H 2 . It is the latter, 
a very stable non-deliquescent substance, which is responsible for the 
difficulty of complete chlorination and the non-deliquescent nature of 
the material. (Bunn, Clark, and Clifford, 1935.) 



CHAP, v X-RAY POWDER PHOTOGRAPHS 125 

In both these investigations the X-ray method was not used alone ; 
measurements of the optical properties of crystals under the microscope 
supplied evidence on certain points. The desirability of using micro- 
scopic and X-ray methods in conjunction with each other cannot be too 
strongly emphasized. This applies also in another field where the 
X-ray method of identification has been widely used the determination 
of phase boundaries in metallurgical equilibrium diagrams. (Bradley, 
Bragg, and Sykes, 1940;Hume-RotheryandRaynor, 1941 ;Lipson, 1943.) 

It is possible not only to identify the components of a mixture but 
also to estimate the proportions of the different components from the 
relative intensities of the patterns. No simple mathematical relation- 
ship between the proportions of the* components and the relative 
intensities of particular diffraction arcs can be given, and therefore the 
method of analysis must be empirical ; when the constituents have been 
identified, powder photographs of mixtures containing known propor- 
tions of the constituents must be taken, and that of the unknown mixture 
compared with them. Estimates of proportions to within 5 per cent, can 
be made by visual comparison, but the probable error can be reduced to 
the order of 1 per cent, by measuring the intensities of selected diffraction 
arcs by means of a micro-photometer; the relation between known 
composition and the relative photographic densities of particular arcs is 
found empirically, and the composition of the unknown mixture inter- 
polated from these results. On account of the somewhat variable 
characteristics of X-ray films and the circumstance that the relation 
between photographic density and X-ray exposure is not linear except 
at low densities, it is better for this purpose to print on each film a strip 
giving a series of known X-ray exposures, and from this to calibrate the 
mixture patterns in terms of X-ray exposure rather than photographic 
density. (See Chapter VII.) 

This method of analysis is particularly valuable when chemical 
methods are inadequate or inapplicable. For instance, for complex 
mixtures where the different elements or ions may be associated in many 
different ways, all compatible with the analytical figures ; or for mixtures 
of polymorphous forms of the same substance, such as the three crystal- 
line forms of CaCO 3 (calcite, aragonite, and vaterite) or the three 
crystalline forms of FeO(OH) (goethite, lepidocrocite, and j3 FeO(OH) 
see Bunn, 1941) mixtures for which chemical analytical methods 
are irrelevant. 

One limitation in the use of X-ray powder photographs for the 
identification and analysis of mixtures must be mentioned. It is very 



126 IDENTIFICATION . CHAP, v 

often not possible to detect less than 5 per cent, of a constituent. The 
minimum proportion of a substance which can be detected varies 
enormously; it is usually specific for each crystalline substance, and 
depends on many factors, such as the symmetry of the crystal and the 
diffracting power of the atoms composing it. Highly symmetrical 
crystals of simple substances such as sodium chloride (cubic) and the 
rhombohedral form of calcium carbonate (calcite) can be detected even 
when present to the extent of only 1 per cent, or even less, but less 
symmetrical crystals such as monoclinic CaSO 4 .2H 2 O (gypsum) can be 
detected only if 5 per cent, or more is present. This statement is valid 
under normal conditions, that is, when the X-rays used contain a certain 
proportion of 'white' radiation* in addition to the principal a wave- 
lengths ; but the figures given can be reduced by using strictly mono- 
chromatic radiation, thus diminishing the background intensity of the 
photographs and making it possible to detect weaker arcs. 

'Mixed crystals' or 'crystalline solid solutions' (see p. 59) present 
different problems from those of straightforward mixtures. A mixed 
crystal gives a diffraction pattern which is in general intermediate, in 
respect of both the positions and the intensities of its arcs, between those 
of the pure constituents. Identification of the crystal species can be 
effected if this relation between a given pattern and those of known 
pure constituents is recognized, and quantitative analysis is possible if 
the relation between composition and arc position and intensity is 
known for the system in question. An interesting example is given by 
Rooksby (1941). Preparations of zinc and cadmium sulphides are used 
as luminescent powders, the colour of the emitted light depending on 
the proportions of the two constituents. The X-ray diffraction patterns 
show that the solids are mixed crystals: there is a complete range of 
mixed crystals, as is shown by the fact that the positions of the arcs 
change gradually with composition . (Rooksby shows a range of patterns 
for the whole series.) The composition of the mixed crystal phase can 
be determined to within 1-2 per cent, from the X-ray pattern, and the 
X-ray method has the advantage over chemical analysis that it is not 
affected by the presence of oxide. Mixtures of different mixed crystals 
are also used to give other luminescent colours ; and in these the composi- 
tion of each mixed crystal phase can be determined from the X-ray 
pattern ; this probably could not be done at all by any other method. 

Interpretation of the diffraction patterns of mixed crystals, as far as 
the determination of unit cell dimensions, may be desirable. This is 
described in Chapter VI. 



CHAP, v X-RAY POWDER PHOTOGRAPHS 127 

Non-crystalline substances. Some solid substances, such as sili- 
cate glasses, and certain organic polymers like polystyrene, are not 
crystalline ; the atoms of which they are composed are not arranged 
in a precise way, though there may be some approach to regularity. 
The X-ray diffraction patterns of these 'amorphous' solids, like those 
of liquids and gases, consist of broad diffuse bands with perhaps two 
or three intensity maxima at definite angles. Examples are shown in 
Fig. 69, Plate IV. It is obvious that such diffuse patterns afford less 
scope for identification or interpretation than crystal patterns. Never- 
theless, something may be done ; the difference between the patterns of 
polymethylmethacrylate and polystyrene, for instance, is so great that 
the substances could easily be distinguished from each other in this way. 

It has been pointed out (Randall, Rooksby, and Cooper, 1930; 
Randall and Rooksby, 1931, 1933) that, when a substance is capable of 
existing in both amorphous and crystalline forms, the X-ray pattern 
given by the amorphous form may be regarded as a very diffuse version 
of the crystal pattern. There is, in fact, no sharp distinction between 
'crystalline' and 'amorphous' states ; if, starting with a coarsely crystal- 
line solid, we could reduce the size of the crystals by stages, taking an 
X-ray diffraction photograph at each stage, we should find that when 
the crystal size fell below about 10~ 5 cm. the photographs would become 
diffuse ; the effect is analogous to the imperfect resolution of an optical 
diffraction grating containing only a few lines. With reduction of crystal 
size, the reflections become increasingly diffuse until the limit is reached 
at 10~ 7 to 10"- 8 cm. the region of atomic dimensions, where the word 
'crystal', with its implication of precise pattern-repetition, ceases to 
be appropriate. One cannot speak of a crystal only one unit cell in 
diameter, for the term 'unit cell' implies repetition ; this is the justifica- 
tion for the use of the term 'amorphous' in describing glass-like sub- 
stances. 

The breadth of X-ray reflections may be used to calculate crystal 
size within the range in which broadening occurs; the method is 
mentioned in Chapter XI. The interpretation of amorphous patterns 
in terms of atomic structure is also referred to in the same chapter. 

This brings us to the end of the section of this book concerned primarily 
with identification problems. This does not mean that these problems 
will not reappear later ; they do reappear in Chapter VI. But from this 
point onwards the book is concerned mainly with the determination of 
the arrangements of atoms in crystals. 



SECTION II. STRUCTURE DETERMINATION 

VI 
DETERMINATION OF UNIT CELL DIMENSIONS 

IF it were possible to produce, by means of a supermicroscope, images 
of atomic structures, it would not be necessary to undertake the lengthy 
processes of reasoning and calculation which form the subject-matter 
of this book. Up to the time of writing, however, this very desirable 
objective has not been reached. Enormous magnifications have been 
achieved by means of the electron microscope, but the resolving power 
is still well above atomic dimensions. These attempts to extend our 
range of vision are based on the principles of the optical microscope, 
and X-rays have not been used, because although they have sufficiently 
short wave-lengths to respond to the details of atomic structures, they 
cannot be refractedf and focused as visible light can. Electron beams, 
however, since they consist of streams of charged particles, can be 
refracted and focused by magnetic or electric fields; since they also 
behave as wave -trains having (with a suitable accelerating voltage) 
effective wave-lengths short enough to respond to the details of atomic 
structure, it might be possible, by their use, to produce images of atomic 
structures. The difficulties, especially that of making corrected electron 
'lenses' suitable for the enormous magnifications involved, are serious, 
however; by 1941, the best resolution achieved was about 30-40 A. 
(Marton, McBain, and Void, 1941.) 

At present, therefore, the details of atomic structures must be dis- 
covered indirectly. The experimental material for the purpose is the 
X-ray diffraction pattern. (Electron diffraction patterns are very 
similar and could be used in the same way, see p. 373.) We are con- 
cerned here with the diffraction patterns of crystals, the interpretation 
of which falls into two stages first, the determination of the shape 
and dimensions of the unit cell (see Chapter II), and secondly the dis- 
covery of the positions of the atoms in the unit cell. 

It has been assumed, in the previous chapter, that the positions of 
diffracted beams depend only on the repeat distances in the crystal 
that is, on the unit cell dimensions while the intensities of the diffracted 
beams depend on the positions of the atoms in the unit cell. This can 

f X-rays are refracted when they pass through matter, but to such a slight extent 
that it is not possible to make lenses of short focal length. 





FIG. 70. Diffraction of light by lino gratings. Above: grating of evenly spaced lines, 

with diffraction pattern. Below: grating in which the unit of pattern is a pair of lines 

(repeat distance same as in the first), with diffraction pattern. 



CHAP, vi UNIT CELL DIMENSIONS 129 

be demonstrated in principle by means of a simple one -dimensional 
optical analogy. In Fig. 70, Plate V, is shown, first of all, a grating in 
which the lines are evenly spaced. (It was made by drawing black lines 
on white card, and taking a very small photograph on which the lines 
are 0-2 mm. apart.) A beam of monochromatic light, on passing through 
it, produces a set of diffracted beams,f the intensities of which fade off 
regularly in the successive orders. In the lower half of Fig. 70 is another 
grating having the same repeat distance as the first ; but in this grating 
the unit of pattern is not one Jine but two. It can be seen that in the 
diffraction pattern the diffracted beams have the same spacing as those 
of the first pattern, but the intensities of the successive orders do not 
diminish regularly. The diffraction of X-nlys by crystals is more complex 
than this, but not different in principle. Tims, in the determination of 
unit cell dimensions, only the positions of the diffracted beams need be 
considered ; the intensities may be ignored. 

From a powder photograph all we can obtain (apart from the intensities 
of the arcs, which are irrelevant to the present problem) is a set of values 
of d (= A/(2sin0)). Each arc represents a 'reflection* from a particular 
set of paralle] crystal planes, but there is nothing to tell us which set of 
crystal planes produces which arc; nothing, that is, except the magni- 
tudes and ratios of the spacings themselves. We cannot deduce the 
unit cell of the arrangement of pattern-units directly ; our only course 
is the indirect one of thinking what arrangement of pattern-units has 
spacings of the observed magnitudes. This can only be done for the 
more symmetrical arrangements ; for those of low symmetry, the number 
of variables defining the unit cell is too great for such a method to be 
possible, but for arrangements whose unit cells are defined by not more 
than two variables -that is to say, for cubic, tetragonal, and hexagonal 
(including trigonal) crystals it is readily accomplished. 

Cubic unit cells. In a crystal having a simple cubic unit cell that 
is, a crystal in which identically situated points lie at the corners of 
cubes, as in Fig. 71 the distance between the 100 planes of pattern- 
units is evidently a, the length of the unit cell edge; that between 010 
and 001 planes is also a. For other sets of lattice planes it is a matter 
of simple geometry to show that the spacing d = a/V(/& 2 +P+Z 2 ), 
where //, Ic, and I are the indices of the planes. (For a general derivation 
of the expression for the plane-spacings in all crystals having rectangular 
and hexagonal unit cells, see Appendix 2.) Thus the powder photograph 

t Tho experimental arrangement used for photographing the diffraction pattern is 
described un p. 271. 

4458 K 



130 



STRUCTURE DETERMINATION 



CHAP. VI 



of a substance such as ammonium chloride which has a simple cubic 
unit cell shows a set of arcs whose positions correspond* with plane - 
spacings in the ratios 1 : 1/V2 : 1/V3 : 1/V4 : 1/V5, and so on. The first arc 
on the photograph the one with the smallest angle of reflection is 
produced by the planes having the greatest spacing the 100, 010, and 
001 planes; the others follow in order of diminishing spacing. 

Whenever the arc positions 
in a powder photograph are 
found to correspond with 
spacings in these ratios, it is 
evident that the substance 
producing the photograph has 
a simple cubic unit cell. This 
is indeed usually obvious from 
a mere inspection of the photo- 
graph, which shows regularly 
spaced arcs as in the pattern 
of ammonium chloride in Fig. 
121, Plate X. Note that the 
gaps in this photograph are 
due to the fact that not all 
whole numbers are values of 
h z +k z -\-l 2 ', for instance, no 
combination of the squares of 
three whole numbers is equal 
to 7, and therefore there is a 
gap following the sixth arc. 

Some cubic crystals sodium 
chloride, for instance give 
powder photographs in which there are many more gaps than those in the 
ammonium chloride pattern. It will be shown later, in Chapter VII, 
that such absences are due to the fact that the crystal in question has 
a compound (face-centred or body-centred) unit cell, or to certain 
symmetries in the arrangement of atoms in the unit cell. For the 
determination of cell dimensions these absences need not be considered ; 
we need only note that if a pattern shows spacings in the ratios of the 
various values of l/V(A 2 +& 2 +Z 2 ), even though some of the values are 
missing, the crystals producing the pattern have cubic unit cells. 

The calculations necessary to show that a crystal has a cubic unit cell 
show in addition which crystal plane is responsible for each arc. Thus, 




FIG. 71. Spaoings of some simple planes of 
a cubic lattice. 



CHAP, vi UNIT CELL DIMENSIONS 131 

the 100, 010, and 001 planes are responsible for the first arc in the 
ammonium.chloride pattern, the 110, Oil, and 101 planes for the second 
arc, and so on. 

Note that the fourth arc, which is the second-order 'reflection' from 
100 planes, is labelled '200', the order being included in the index 
description. Similarly the second-order 'reflection' from the 110 planes 
is called 220, and the third order 'reflection' from 312 would be called 
936. This practice of including the order in the index numbers has 
become standard, as it makes for uniformity and avoids confusion. 
Looking at it in another way, we may regard 220 as the first-order 
'reflection' from a set of planes having half the spacing of 110 (see the 
bottom right-hand corner of Fig. 71). "This fits in with the definition 
of the indices of crystal planes given in Chapter II (p. 24) the number 
of planes crossed between one lattice point and the next, along each 
axial direction. Another meaning of the indices of X-ray reflections is 
also important. All 'reflections' are first-order 'reflections' from planes 
defined in the above manner ; this means that there is a phase-difference 
of one wave-length between waves from successive planes. Counting 
the number of planes crossed between one lattice point and the next is 
therefore the same thing as counting the number of wave-lengths phase- 
difference between waves scattered by neighbouring lattice points. The 
indices thus represent the phase -differences between waves diffracted by 
neighbouring lattice points along the three axial directions. Thus, if 
we take any one atom as the reference point, the 936 'reflection' is 
produced when waves diffracted by the next similarly situated atom 
along the a axis are 9 wave-lengths in front of those from the reference 
atom, those from the next similarly situated atom along the b axis are 
3 wave-lengths out, and those from the next along the c axis 6 wave- 
lengths out. The indices are thus the three order numbers which 
characterize diffracted beams produced by a three-dimensional grating. 
For an optical line grating a one -dimensional pattern we speak of 
'first', 'second', and succeeding orders of diffraction, the one order 
number being appropriate to the one-dimensional character of the 
pattern ; but for a three-dimensional pattern, three order numbers are 
necessary to describe diffracted beams. The X-ray beam which we 
have called, rather loosely, the 936 'reflection', or the 'third-order 
reflection from the 312 plane', is, strictly speaking, the diffracted beam 
whose order numbers are 936. 

The length a of the unit cell edge can be calculated from the spacing 
of any arc from the expression a d ftkl ^(h 2 +k z +l 2 ). The results from 



132 STRUCTURE DETERMINATION CHAP, vi 

arcs at l^rge angles are more accurate than those from the first few 
arcs for two reasons : firstly, the errors due to the thickness $nd absorp- 
tion of the specimen diminish with increasing diffraction angle (see 
p. 120), and secondly, on account of the form of the Bragg equation 
n\ = 2dsin0, the resolving power increases with 6 (see p. 120), as is 
obvious from the fact that the ^ 2 doublets are only resolved at large 
values of 0. 

Although the interpretation of patterns from cubic crystals can be 
done by way of calculations as above, it is more convenient to use 
graphical methods as described in the next section. 

Tetragonal unit cells* In crystals of tetragonal symmetry the unit 
cell is a rectangular box withbwo edges equal (a) and the third (c) 
different from the first two. The spacings of hkO planes those parallel 
to c are in the same ratios as those of the hkQ planes of cubic crystals, 
that is, in the ratios 1/V1 2 :1/V(1 2 +1 2 ): 1/V2 2 : 1/V(2 2 +1 2 ), and so on. 
But the 001 spacing is not related in any simple way to a ; the ratio c/a 
may have any value and is different for every tetragonal crystal ; and 

/ 7/^2 i 2 j[2\ 

Tiki spacings in general are given by d hkl ~ If /( ~ h~ol- 

/ V \ a 2 c 2 / 

diffraction patterns of tetragonal crystals are thus less simple than 
those of cubic crystals, and there is no regular spacing of the arcs, as 
may be seen in the pattern of urea, Fig. 72 ; the relative spacings are 
different for every different tetragonal crystal, except for the hkQ 
spacings. 

It would be possible to find the unit cell of a tetragonal crystal by 
first picking out those arcs whose spacings are in the ratios 1 : 1/V2 : 1/V4, 
etc. (these being the hkO reflections), and then assigning likely indices 
to the remaining reflections by trial. But this would be a laborious 
process, and there is no need to proceed in this way, since the problem 
can be solved graphically. The relative spacings of the different planes 
are determined by the axial ratio c/d ; if two crystals happened to have 
the same axial ratio but different actual cell dimensions, their patterns 
would show the same relative spacings, though one pattern would be 
more spread out than the other if the same X-ray wave-length were used. 
Graphs connecting the relative values of d and c/a can be constructed, 
and the whole set of arcs in a powder pattern identified by finding where 
their relative spacings fit the chart. In order to deal only with relative 
spacings so that only the shape (not the actual size) of the cell enters 
into the problem, the chart is made logarithmic with respect to d. The 
first such charts were published by Hull and Davey (1921), who plotted 



CHAP. VI 



UNIT CELL DIMENSIONS 



133 



log d for each crystal plane against c/a. These charts are rather small, 
and for small values of c/a do not extend far enough for some purposes. 
A new method of constructing such a chart entirely without calculation 




002 



3V 25 20 



t-6 



12 JO $ -8 7 -6 
AXIAL RATIO % 



FIG. 72. Graphical method for indexing the powder pattern of a tetragonal crystal. 
The pattern shown is that of urea. 

is given in an appendix. The method of using these charts is to pl6t 
on a strip of paper the values of log d (or 2 log d for the new form of 
chart) for all the arcs on the photograph, and move the strip about on 
the chart, keeping it always parallel to the logd axis, until a good 
match between strip points and chart lines is found ; this is illustrated 
in Fig. 72. (Some reflections may.be absent ; this feature may be ignored.) 



134 STRUCTURE DETERMINATION CI*AP. vi 

More than one matching position will be found ; the position giving the 
simplest indices will naturally refer to the simplest unit cell. When 
the correct position is found, the indices of all arcs can be read off on 
the chart. The axial ratio can also be read off approximately, but it is 
better to calculate a and c from the spacings of selected arcs. The length 
a can be obtained from the spacing of any hkO arc, and c from any of 
the OOZ arcs, the most accurate results being obtained from the arcs at 
the largest reflection angles. Arcs representing two or more different 
crystal planes with about the same spacing should naturally be avoided. 
If unambiguous hkO and OOZ arcs are not available, both a and c can 
be calculated from the spacings of any two arcs having different hk and 
I values from the equations * 



the most accurate results being obtained from a pair of arcs (such as 21 1 
and 102 in the urea photograph, Fig. 72), one of which comes from a 
plane with high hk and low I, and the other from a plane with low hk 
and high I ; they should be fairly near together on the photograph so 
that absorption and other errors are about the same for both. Calcula- 
tions should be made from several pairs of arcs, and the results averaged. 
Reflections at large angles give more accurate results than those at 
small angles see Fig. 68. (For high precision methods, see p. 180.) 

Note that a tetragonal cell with an axial ratio of 1 is cubic, and the 
chart at this position can therefore be used for cubic crystals. 

Hexagonal, trigonal, and rhombohedral unit cells. In many 
crystals of hexagonal and trigonal symmetry the unit cell has a diamond- 
shaped base, a and b being equal in length and at 120 to each other; 
c is perpendicular to the base and different in length from a and b. 
The axial ratio c/a is different for each crystal. The spacings d of the 
planes are given by 

/ 4 (h*+hk+k*) , Z*\ 



The indices for the arcs on a powder photograph can be found graphi- 
cally on a suitable chart (see Appendix 3) in a way similar to that 
described for tetragonal crystals, and the axial lengths calculated from 
the spacings of suitable pairs of arcs by the above equation. 

For some trigonal crystals, the unit cell is a rhombohedron a figure 
which has three equal axes which make equal angles not 90 with each 
other; the cell is, so to speak, a cube either compressed or elongated 



CHAP. VI 



UNIT CELL DIMENSIONS 



135 



along a body diagonal. The spacing d of any set of atomic planes hkl 
is given, in terms of the unit cell edge a and the interaxial angle a, by 
the expression 



__ / /( 
hkl ~~ / A/ ((i' 2 ^ 2 



cos 3 a 3 cos 2 a: 






I 
a))' 



It is not easy to determine directly a and a from a powder photograph 
by the use of this rather unwieldy expression ; fortunately, however, 
the atomic arrangement in rhombohedral crystals can always be referred 




FIG. 73. Rhombohedral cell (bold linos) with corresponding hexagonal 
cell (narrow hues) and hexagonal prism (dotted). 

to a larger hexagonal cell (Fig. 73) whose dimensions a n and C H are 
related to those of the rhombohedral cell, a R and a, by the relations 



,2 "// 



Cjf 
"O"' 



. 

sm- = 



3 



Hexagonal indices h H k ff 1 H are related to rhombohedral indices h R k R 1 R 

by the relations , _ 7 7,7 

6ri -= tiff /C//-H, 



31 R = 2h H k n -\-l H . 

The procedure is to find the simplest hexagonal indices on the chart 
already mentioned, to calculate the dimensions of the hexagonal cell, 
and finally to find the dimensions of the true rhombohedral cell by the 
above expressions. 

If it is not known whether a crystal has rhombohedral symmetry or 
not, this question may be settled by assigning hexagonal indices to the 
reflections and then surveying these indices to see whether all of them 
are such that AJJ^+^HJ ^//+2A? jtf +Z J y, and 2h H k H ~^-l H are 
divisible by 3 ; if they are, the true unit cell is rhombohedral. 



136 STRUCTURE DETERMINATION CHAP, vi 

Other types of unit cells. The dimensions of orthorhombic, mono- 
clinic, and triclinic unit cells cannot usually be determined from powder 
photographs. The number of variable parameters is too great to permit 
the use of charts for finding the indices for the arcs. Even for ortho- 
rhombic crystals a three-dimensional figure of a very complex type 
would be necessary, and this is impracticable. The only hope of finding 
the unit cell dimensions from a powder photograph of one of these less 
symmetrical crystals is by trial: that is, by postulating simple indices 
for the first few arcs, calculating the unit cell dimensions on these 
assumptions, and then finding whether the spacings of the remaining 
arcs fit this cell. Unless external evidence is available, such a process of 
trial is likely to be, at the very % least, extremely lengthy, and more often 
than not, quite hopeless. If, however, external evidence is available, 
such as the axial ratios and angles deduced from goniometric or micro- 
scopic measurements, there is more hope, since, as pointed out in Chapter 
II, the shape of the 'morphological' unit cell is either the true shape or 
is closely related to the true shape, for instance by the halving or doubling 
of one of the axes with respect to the others. Such a clue may lead to 
the postulation of correct indices for the first few arcs and hence to the 
indexing of the whole powder photograph. The cell first chosen may 
be too large ; if, for instance, all the h indices are found to be even, then 
the length of the true a axis is half that first chosen: the change to this 
true a axis will halve all the h indices. 

The spacings of the various planes for these crystals are given by 
the following expressions: 

Orthorhombic d hkl ~ 



Monoclinic d 




hkl 



For triclinic crystals the expression is so unwieldy that it is not 
worth while attempting to use it ; a graphical method based on the con- 
ception of the reciprocal lattice should be used (see pp. 156-8). The 
reciprocal lattice method is also more rapid than calculation for mono- 
clinic crystals. 

The difficulties in the interpretation of powder photographs of crystals 
of low symmetry lie in the fact that in a powder photograph all the 



CHAP. VI 



UNIT CELL DIMENSIONS 



137 



information is crowded along one line, and this information consists 
only of the; spacings of the planes, without any geometrical indication 
of the relative orientations of the crystal planes producing the various 
arcs. This is inevitable in a powder photograph on account of the random 
orientation of the crystals in the specimen. Only by departing from the 
randomness of orientation can we obtain X-ray diffraction photographs 
which give geometrical indications of the orientation of the crystal 
planes producing the various reflections. Obviously it is best to use a 

FILM 



CRYSTAL 





(a) 

FIG. 74, Arrangements for taking single -crystal rotation photographs 
(a) on flat films, (6) on cylindrical films. 

single crystal set in some definite way with regard to the X-ray beam, 
so that the reflections from differently oriented planes shall fall on 
different parts of the recording film. 

Single-crystal rotation photographs. The method found most 
convenient for finding the unit cell dimensions of crystals of low sym- 
metry is to send a narrow monochromatic X-ray beam through a single 
crystal at right angles to one of its axes, to rotate the crystal round 
this axis during exposure in order to bring a number of different crystal 
planes successively into reflecting positions, and to record the reflec- 
tions either on a flat plate or film perpendicular to the primary beam 
(Fig. 74 a), or better still (because more reflections are registered) on a 
cylindrical film surrounding the crystal, the cylinder axis coinciding 
with the crystal's axis of rotation (Fig. 746). A single-crystal camera 
differs from a powder camera only in the necessity of having arrange- 



138 STKUCTUBE DETERMINATION CHAP, vi 

ments for the accurate adjustment of the crystal and the use of a much 
longer cylinder of photographic film. The crystal is mounted on the 
stem of a goniometer head ; whenever possible, accurate adjustment is 
effected by making use of the reflection of light by the faces, in the 
manner described in Chapter II. (Ill-formed crystals may be set 
accurately by X-ray methods ; the procedure cannot be given at this 
stage it will be found on p. 173.) Descriptions of 'universal' X-ray 
goniometers suitable for taking (among other things) rotation photo- 
graphs have been given by Bernal (1927, 1928, 1929), Sauter (1933 a), 
and Hull and Hicks (1936) ; Bernal's papers are particularly valuable, as 
they give much useful information on procedure. Aperture systems for 
defining the X-ray beam are tUe same as in powder cameras ; a 0*5 mm. 
channel in a brass tube, with the usual guard tube, is suitable for 
most purposes. Fogging by the primary beam is avoided either (as in 
the powder camera, Fig. 62) by the provision of a trap or by making 
a small hole in the film through which the primary beam passes. 
Goniometer cameras are not usually made light-tight ; instead, the film 
is contained in an envelope of black paper or other material which stops 
light but not X-rays. X-ray goniometers are usually fitted, not only 
for complete rotation of the crystal, but also for oscillations over limited 
angular ranges ; this is usually effected by heart-shaped cams controlling 
the angular movement. 

The crystal is mounted by sticking it to a glass hair (preferably lithium 
borate glass) by a trace of plasticine, shellac, or wax ; the glass hair in 
turn is stuck to the goniometer stem. Crystals which are deliquescent, 
efflorescent, or rather volatile must be sealed inside lithium borate tubes 
(Robertson, 1935 a). It is in some cases necessary to take X-ray photo- 
graphs of crystals immersed in their own mother liquor ; here again, 
thin-walled capillary tubes must be used (Bernal and Crowfoot, 1934 a). 

The type of X-ray photograph given by a crystal rotated round a 
principal axis is illustrated in Fig. 75, Plate VI, which shows the 
diffraction pattern of the orthorhombic crystal potassium nitrate 
rotated round its c axis. The most obvious feature of this photograph 
is the arrangement of the diffraction spots on a series of straight 
horizontal lines. The reason for this will be apparent when it is remem- 
bered that along the c axis there are identical diffracting units (groups 
of atoms) spaced a distance of c apart. It has already been shown (p. 1 1 5) 
that a row of identical, equally spaced diffracting units perpendicular 
to an X-ray beam produces cones of diffracted rays at angles given by 
n\ = c cos <f> 9 where A is the X-ray wave-length, c the distance between 



PLATE VI 






FIG. 75. Single-crystal rotation photographs. Above : potassium nitrate (orthorhombio ; 
rotation axis, c). Centre: gypsum (monoclinic; rotation axis, c). Below: benzil (hexa- 
gonal ; rotation axis, c). 



CHAP, vi UNIT CELL DIMENSIONS 139 

the diffracting units, ^ the semi-vertical angle of the cone of diffracted 
rays, and n a whole number. On a cylindrical film having the point- 
row for its axis, these cones of rays would be registered as a series of 
straight lines. A crystal is not a single row of diffracting units, but 
consists of many identical rows of such units, all parallel to each other 
and packed side by side in a precise way, arid on account of the three- 
dimensional character of the assemblage of diffracting units, diffracted 
beams are produced, not all along each cone, but only along specific 
directions lying on the cone, the directions being such that the Bragg 
equation A = 2d sin 8 is satisfied ; thus we get on the cylindrical film, 
not continuous straight lines, but spots lying on straight lines. The 
lines of spots are usually called 'layer lines'. 

The length of c can be obtained very simply from this photograph 
by measuring the distance y of any layer line from the equator ; if the 
camera radius is r, then r/y is tan< ; c is then given by wA/cos <, n being 
the number of the layer line selected (the equator having n = Q). For 
the potassium nitrate crystal, c is 6-45 A. If a flat film is used instead 
of a cylindrical film, the layer lines are shown, not as straight lines 
but as hyperbolae. Tan< is given by r/y 1 ', where y r is the shortest dis- 
tance from the hyperbola to the equator the distance at the meridian, 

Unit cell dimensions from rotation photographs. The simplest 
way of measuring the lengths of the three edges of the unit cell of an 
orthorhombic crystal is evidently to take three rotation photographs, 
the crystal being rotated round a different axis for each photograph ; 
the axial directions are chosen on the basis of morphological measure- 
ments, and these directions are necessarily, by symmetry, parallel to 
the true unit cell edges. For the potassium nitrate crystal the lengths 
of the edges of the unit cell were found by D. A. Edwards (1931) to be : 
a = 5-42 A ; 6 = 9-17 A ; c = 6-45 A. 

The same method can be used for all crystals, irrespective of sym- 
metry ; axial directions are chosen, and interaxial angles determined, 
by measurements of interfacial angles, while the lengths of the axes 
are determined from X-ray rotation photographs. There are pitfalls 
here, however; the directions selected as crystal axes on the basis of 
morphological measurements may not always be parallel to the edges 
of the simplest unit cell, which will be referred to here as the 'true unit 
cell' the smallest cell which has the correct symmetry and accounts 
for all the X-ray reflections. Consider first the most highly symmetrical 
crystals those belonging to the cubic, tetragonal, and hexagonal 
(including trigonal and rhombohedral) systems. (Although, as we have 



140 



STRUCTURE DETERMINATION 



seen, the unit cell dimensions of such crystals can usually be determined 
from powder photographs, nevertheless it may happen that faint 
reflections not seen on powder photographs are registered on single- 
crystal photographs, and these may necessitate revision of cell dimen- 
sions; hence, single-crystal photographs should be taken whenever 
possible.) 

Axial directions in cubic crystals are fixed by symmetry, as in the 
orthorhombic crystals already considered. But in tetragonal crystals, 



TRUE 110 



MORPHJ'MO" 

'-rmim 




MORPH.'W 
-TRUE 100 



** 




/.A ROTATION ""<*' ROTATION 

1 ' AXIS1 AXIS 1 

FIG. 76. Determination of unit cell dimensions by rotation photographs, 
(a) for tetragonal, (b) for hexagonal crystals. 

although there is no doubt about the direction of the unique c axis, on 
the other hand the morphologically chosen a axis may be at 45 to the 
true a axis : the prism face selected as 100 may really be 1 10 (see Fig.76 a), 
and thus an X-ray photograph with direction 1 as rotation axis would 
give (from the layer-line spacing) the repeat distance a l9 which is V2 
times the true unit cell edge a 2 . Therefore, to find the true a for a 
tetragonal crystal by this method, it would be necessary to take two 
rotation photographs, one with the morphological '[100]' direction 1 
and the other with the morphological '[1 10] ' direction 2 as rotation axis ; 
one repeat distance will be found to be V2 times the other, and the 
smaller of these lengths is evidently the true a. Similarly, for hexagonal 
crystals (Fig. 76 b) it is necessary to take two photographs with 
directions 1 and 2 respectively as rotation axes; one repeat distance 
(a) will be found to be V3 times the other ( 2 ), and the latter is evidently 
the correct a. 



CHAP. VI 



UNIT CELL DIMENSIONS 



141 




FKJ. 77. Alternative monoclinic 
cells (6 projection). 



Monoclinic crystals may present more serious difficulties of a similar 
type : the b axis is fixed by the symmetry (it coincides with the single 
twofold axis or is perpendicular to the single plane of symmetry) ; but 
the a and c axes are not fixed in any such way. We may encounter the 
state of affairs illustrated in Fig. 77, where the morphological '001' plane 
is really 101 of the true unit cell, and the 
morphological '100' is really 10T; and in 
addition, the true angle /? would be differ- 
ent from the morphologically determined 
j8'. Here we should evidently have to 
take X-ray photographs with the crystal 
rotating round directions parallel to OP 
arid OQ in order to obtain the dimensions 
of the simplest unit cell. Note that the 
alternative cell defined by a", c, and j8" 
has the same size as that defined by a, c, 
and jS, and has an equal claim to be 
regarded as the true unit cell, but may 
be less convenient because its j3 is greater. 
It is possible that the relations between 
the morphologically chosen axial direc- 
tions and the edges of the simplest unit 
cell might be more remote, in which case 
it would be difficult to find the latter by 
the simple method hitherto described. In 
the triclinic system, still more difficulties 
may be encountered. In crystals of 
rhombohedral symmetry the simplest 
unit rhombohedron may be one with quite 
different values of a and a from those of the 
morphologically selected rhombohedron ; 
this is so for calcite, for instance (Fig. 78). 

The straightforward way out of these difficulties is to accept provision- 
ally the cell edges selected on morphological evidence, and to find the 
indices of all the reflections on this basis ; then to survey the indices to 
see whether any smaller cell will account for all the reflections, thus 
simplifying the indices. The smallest cell which has a shape appropriate 
to the crystal system and will account for all the reflections is the true 
unit cell. This procedure may appear very laborious, but the graphical 
methods now to be described greatly simplify and shorten the work. 




Fio. 78. Large (32-molecule) unit 
rhombohedron of calcite based on 
cleavage rhomb. The true unit 
cell is the small (2 -molecule) stoop 
rhombohedron shown inside. 



142 STRUCTURE DETERMINATION OHAP. vi 

For the determination of unit cell dimensions, detailed indexing of 
the reflections on single-crystal rotation photographs is only necessary 
in certain cases, as indicated in the foregoing discussion ; the complete 
indexing of rotation photographs of all types of crystals is, however, 
necessary whenever an investigation is to be carried beyond the 
determination of unit cell dimensions (to the discovery of the symmetry 
of the arrangement of atoms in the crystal, or to the elucidation of the 
arrangement in detail), and it will be appropriate to deal with the whole 
subject at this stage. 

One further remark must be made before taking up this subject. 
Morphological features are useful in suggesting possible unit cell edges, 
but it is possible to proceed* with very meagre evidence of this sort 
(such as a single direction, as in rod-shaped crystals lacking well-defined 
faces), or even with none at all. There are initial difficulties in setting 
such crystals in a suitable orientation on the goniometer, but these can 
be solved by X-ray methods ; see p. 173. As soon as a single crystal has 
been set sufficiently well to give an X-ray rotation photograph showing 
recognizable layer lines, the unit cell dimensions and the indices of all 
the reflections can be found by the methods now to be described. 

Indexing rotation photographs. Preliminary consideration. 
The spots on the equator of a rotation photograph are obviously reflec- 
tions from atomic planes which were vertical during the exposure. In 
Fig. 75, Plate VI, the equatorial spots are reflections from planes parallel 
to the c axis, that is, hkQ planes : the third or I index for these reflections 
is by inspection. The other two indices, h and &, of all the equatorial 
reflections may be found from the spacings of the planes, which are 
worked out from the reflection angles 8 by the Bragg equation. The 
spacing d of any hkQ plane of an orthorhombic crystal is given by 

/ //A 2 k 2 \ 

d = 1 / /I--]-); the simplest way of finding h and k for all the reflec- 
/ V \a* 6 2 / 

tions is to plot log d for each spot on a strip of paper and fit this on a 
chart (Fig. 79) similar to, but simpler than, the charts used for indexing 
powder photographs. The construction of such a chart (which shows 
logd for all values of h and k and a wide range of axial ratios a:b) is 
described in Appendix 3. The h and k indices for each reflection are 
read off on the chart when the match position is found. Note that 
in some cases, owing to the absence of many equatorial reflections 
(see pp. 217-40), the simplest match position is not correct. As a guide 
to the correct match position, log rf 100 and log d 010 (already known from 
the other two rotation photographs) sliould be marked on the strip. 



CHAP. VI 



UNIT CELL DIMENSIONS 



143 



For the spots on layer lines above and below the equator, one 
index (I) is .given by inspection. It should be remembered that the 
indices of reflections represent 
phase-differences between waves 
diffracted by neighbouring units 
along the three axial directions 
(see p. 131). The spots on the 
first layer line above the equator 
lie on a cone for which n in the 
equation nX = ccos< (see p. 138) 
is 1 ; this means that waves coming 
from any one diffracting unit are 
one wave-length behind those from 
the next diffracting unit above it ; 
in fact, n in the cone equation is 
Z, the third index number. Thus, 
all spots on the fourth layer line 
(fourth cone) above the equator 
are from hk4: planes (those on the 
fourth layer line below the equator 
are from M4 planes), and so on. 

The I index of every spot is thus 
obvious by mere inspection. The 
other two indices are best obtained 
by a graphical method. Just as all 
spots with the same I indices (in 
the present example) lie on definite 
lines, so all spots with the same hk 
values lie on definite curves. But 
these 'hk curves' have a form less 
simple than the 'I curves'. The 
form of these curves is most 
readily determined by introducing 
a piece of mental scaffolding known i-o -3 -8 -7 -6 -s -4 

r & AXIAL RATIO a /b 




as the 'reciprocal lattice' a con- 
ception which has proved to be a 
tool of the greatest value for the 
solution of all geometrical prob- 
lems concerned with the directions of X-ray reflections from crystals. 
It was introduced by Ewald (1921). 



Fid. 79. Graphical method for indexing 
equatorial reflections on rotation photo- 
graphs. The pattern shown is that of 
polyehloroprene (hkO reflections). 



144 



STRUCTURE DETERMINATION 



CHAP. VI 



The 'reciprocal lattice*. From a point within a crystal imagine 
lines drawn outwards perpendicular to the lattice planes ; c along these 
lines points are marked at distances inversely proportional to the 
spacings of the lattice planes. The points thus obtained form a lattice 
that is, they fall on sets of parallel planes. (For a simple proof that 
they do form a lattice, see Appendix 4.) This imaginary lattice is known 
as the 'reciprocal lattice'. An example is shown in Fig. 80 ; all points 

having the same I index fall on a 
plane, and the plane containing all 
hkl points is parallel to that con- 
taining the hk2 points, and so on. 
By thinking of this imaginary 
lattice, in which the planes of the 
real lattice are symbolized by 
points, we are obviously brought 
nearer to the single-crystal X-ray 
diffraction pattern with its array 
of spots, especially as the layers 
of points in the reciprocal lattice 
correspond with the layers of spots 



on the diffraction pattern. In fact, 
a rotation photograph such as one 
of those in Fig. 75, Plate VJ, is, as 
we shall see, strongly similar to the 
pattern we should get by rotating 
the reciprocal lattice round the c 
axis of the crystal and marking off 
the positions where the reciprocal lattice points pass through a plane 
through the c axis (Fig. 81). The resemblance between this 'reciprocal 
lattice rotation diagram' and the X-ray photograph is closest near the 
centre of the photograph ; elsewhere the X-ray photograph is a some- 
what distorted version of the reciprocal lattice rotation diagram. 

The process of reflection by the real lattice cannot be visualized in 
terms of the reciprocal lattice ; but the condition for reflection by the 
real lattice (the Bragg equation) naturally has its precise geometrical 
equivalent in terms of the reciprocal lattice. This is illustrated in Fig. 82, 
in which XY represents the orientation of a set of crystal planes which 
we will suppose is in a reflecting position. Along the normal to this set 
of planes is the corresponding reciprocal lattice point P, the distance of 
which from the reciprocal lattice origin X is inversely proportional to 




FIG. 80. Reciprocal lattice of an 
orthorhomhicj crystal. 



CHAP. VI 



UNIT CELL DIMENSIONS 



145 



d, the spacing of the planes in question (defined as on p. 131 so as to 
include the 'order* of reflection) ; the unit of length is chosen so that XP 
is equal to \/d rather than 1/d for a reason which will presently appear 
(A is the characteristic X-ray wave-length, which in any particular ex- 
periment is constant). The X-ray beam QX is reflected by the plane 
at an angle 6, the reflected beam XR making an angle 20 with the 
primary beam. If T lies in the plane QXR, the angle QXY = 6. 

At P draw a line perpendicular to XP to meet the primary beam at 
Q. It must also lie on this line, since primary beam, reflected beam, and 




FIG. 81. Formation of reciprocal lattice rotation diagram. 

the normal to the reflecting plane all lie in a common plane. QE and 
X T are parallel to each other (since both are at right angles to XP and 
are in the same plane), hence the angle PQX = QXY 0. Since the 
angle QPX is a right angle, PX/QX = sin 6 ; therefore 

\ld A 



__ _ _ 

sin# sin 9 dsm9* 

But the Bragg equation states that when a set of crystal planes reflects 
X-rays, 



Hence QX = 2. Thus, for every different set of crystal planes when 
in a reflecting position, the above construction brings us the line QX 
of constant length 2 units. For every possible position of P the angle 
QPX is a right angle, hence P always lies on a circle which has QX (=2) 
as its diameter. To plot the positions occupied by all reciprocal lattice 
points when the planes they symbolize are in reflecting positions, rotate 



146 



STRUCTURE DETERMINATION 




RECIPROCAL 
LATTICE 
ROTATES 
HERE 



the circle QPX about its diameter QX ; the sphere QSXP is obtained. 
(The reason why the reciprocal lattice is made on the scale XP = X/d 
is now apparent ; it is to give the sphere QXSP unit radius.) 

In other words, the condition for reflection, in terms of the reciprocal 
lattice, is this : construct a sphere of unit radius having the primary 
beam along its diameter. Place the origin of the reciprocal lattice at 
the point where the primary beam emerges frpm the sphere. As the 

crystal turns, the reciprocal lattice, in 
turning about its origin, passes through 
the sphere (Fig. 83), and whenever a 
reciprocal lattice point (distant X/d from 
the origin) just touches the surface of 
the sphere (the 'sphere of reflection') a 
reflected beam flashes out, being re- 
flected by the crystal plane correspond- 
ing to the reciprocal lattice point. 

Note that if in Fig. 82 we join O, the 
centre of the sphere, to P, the angle 
OQP = OPQ = 0, and thus the angle 
XOP = 29 \ OP is therefore parallel to 
XB and, equally with XR, represents 
the direction of the reflected ray. The 
problem of finding the position of any 
reflected spot on an X-ray photograph 
therefore resolves itself into (1) finding 
where the reciprocal lattice point for 
the plane in question touches the sur- 
face of the sphere of reflection, and then (2) finding where the (produced) 
radius through this point strikes the film. This procedure is valid for 
all types of single-crystal X-ray photographs. In the particular case 
of a crystal rotating round a principal axis (say c) which is perpendicular 
to the X-ray beam, the reciprocal lattice points are in layers parallel 
to the equatorial section of the sphere of reflection (Fig. 83), and remain 
on the same level as the reciprocal lattice rotates round X C ; consequently 
all the points on any one layer that is, all points having the same I 
index pass through the surface of the sphere at various points lying 
on the circle PNM (Fig. 82) which is parallel to QSX. If we joined the 
centre of the sphere to each of these points we should get a set of Unes 
lying on the surface of a cone. We have thus arrived, by way of the 
conception of the reciprocal lattice, at the same conclusion as that 




FIG. 82. The condition for reflection 
in terms of the reciprocal lattice. 
Reflection occurs when a reciprocal 
lattice point P touches the surface 
of the sphere of reflection. 



CHAP. VI 



UNIT CELL DIMENSIONS 



147 



already drawn from a consideration of diffraction by a row of scatter- 
ing points, namely, that when a single crystal is rotated round its c 
axis and an X-ray beam passes through it perpendicular to its c axis, 
all reflected rays from planes having the same I index lie on a cone. 
The semi -vertical angle of this cone, <, we have already seen is given by 
IX = ccos< ; this is also obvious from Fig. 84 (OU = IX/c = cos<). 

When the reflections are recorded on a cylindrical film the height y 
of each layer of spots above the equator is r cot <, where r is the radius 
of the cylinder. 

RECIPROCAL 
LATTICE 
'ROTATES HERE 
C 




FIG. 83. Reciprocal lattice passing through sphere of reflection 
as it rotates. 

Since the directions of reflected rays are obtained by joining the 
centre of the sphere to points on its surface, the crystal itself may be 
regarded as rotating in the centre of the sphere of reflection, while the 
reciprocal lattice of this same crystal rotates about a different point 
the point where the beam emerges from the sphere. If this seems odd, 
it must be remembered that the reciprocal lattice is a geometrical 
fiction and should not be expected to behave other than oddly; the 
fact is, the reciprocal lattice is concerned with directions ; its magnitude 
and the location of its origin are immaterial. 

As for the precise position of each reflected beam, and the point at 
which it strikes the film, this evidently depends (for any one layer of 
reciprocal lattice points, any one cone of reflections) on the distance of 
the reciprocal lattice point from the axis of rotation. A point whose 
distance from the axis of rotation is equal to the shortest distance from 



148 



STRUCTURE DETERMINATION 



CHAP. VI 



this axis to the circle PNM would just touch the sphere at T (Fig. 84 a 
and 6), which lies on the line UV, parallel to the primary beam; the 
reflected beam for this plane would travel along OT, striking a film of 

^ on ^ e meridian 



W 




RECIPROCAL LATTICE 
ROTATES HERE of the film, directly above the 

central spot X. Points nearer 
the axis of rotation than T 
would never touch the sphere 
at all, and the planes they 
represent would never reflect. 
Other points whose distance 
from the axis of rotation lies 
between TV and N V touch the 
sphere at points such as P ; 
what we want to know is the 
angle PUV (or </r), since this 
angle determines the distance x 
of the reflected spot from the 
meridian of the film. To find 
we have to solve the triangle 
PUV. Now UV is 1 (the radius 
of the sphere). UP, the radius 
of the circle of contact, is 



Therefore, if we know , all three 
sides of the triangle are known 
and the angle ifj can be found. 
is x/r radians. 

In practice, we want to find 
the coordinates of a reciprocal 
lattice point from the measured 
position of a spot on the film. 
This is most simply done by a graphical method, as described below. 
If, however, it is desired to do it by calculation, for the sake of greater 
accuracy, the following expressions are required. If the coordinates of 
a spot on a cylindrical film are x (along the equator) and y (along the 
meridian), the distance of any reciprocal lattice point from the equa- 
torial plane (the circle QSX) is cos^ -- cos(cot"" 1 2//r). The distance f 



JNW 



FIG. 84. Sphere of reflection surrounded by 

cylindrical film of unit radius, a. Elevation. 

6. Plan. 






\ \ \ \ \ \ \ \ \ 



if/// / / / 



\X\\WM 



I \\\\\\\\\\\ 



/ / i III I III ./."/---../-.,/.'... 
/ //////////// 
/ //////////// 
////I I I I I I I I I 



\\\\\\\\\\\\\\\\\ N 
\\\\\\\\\\\\\ \ \\\ 






/ 



\\v\\\\\\\\v 



, f ,,,, 
////// 1 1 1 ! Ill 



v\\\\\\\\\\\ 



//// /nirm 




Fia. 85. Berrial chart giving g and for positions on cylindrical film. (Bernal, 1926) 



CHAP, vi UNIT CELL DIMENSIONS 149 

of the point from the axis of rotation is (solving the triangle PUV 
having two sides of length 1 and sin <, and the included angle iff) given by 



sin 2 {cot~%/r)} 2 sin{cot-%/r)}cos(o;/r)]. 

Bernal (1926) worked out and for all positions on a cylindrical 
film, and gives a chart showing contours of equal and equal , suitable 
for a camera of diameter 10 cm. ; it is only necessary to place a 
rotation photograph on the chart, and read off the and coordinates 
for every spot on the film. For other camera sizes this chart (illus- 
trated on a smaller scale in Fig. 85) may be photographed and re- 
produced on the correct scale. A similar chart for photographs on flat 
films is also given in the same paper. Greater accuracy is attained 
by measuring the positions of spots on the photograph, using a 
millimetre rule or a travelling microscope, and then plotting these 
positions on a special large-scale Bernal chart. 

For the purpose of visualizing the geometry of the reciprocal lattice 
in terms of the actual camera dimensions, it is perhaps useful to multiply 
the dimensions of the reciprocal lattice and of the sphere of reflection 
by r, the radius of the cylindrical film, since in this case the origin of 
the reciprocal lattice is the point where the primary X-ray beam strikes 
the film, and the axis of rotation of the reciprocal lattice is the vertical 
line through this point. Let us recapitulate the geometrical construc- 
tion on this scale. 

We have, first of all (Fig. 86 a), the primary beam passing through 
the crystal at right angles to its axis of rotation and striking the cylindri- 
cal film (radius r) at the point X. Erect the axis XC parallel to the 
cylinder axis; the reciprocal lattice will rotate round XG (its origin 
being at X) while the crystal itself rotates round OS. In the cylinder, 
describe a sphere having the same radius r (Fig. 866) ; this will be the 
sphere of reflection. Any reciprocal lattice point P (distance from the 
origin = rX/d) rotates round XC ; as soon as it touches the surface of 
the sphere (Fig. 86 c), a reflected beam flashes oat, and strikes the film 
at Y!. On further rotation (Fig. 86 d), the point passes again through 
the surface of the sphere at P 2 , and a reflection again flashes out, 
striking the film at Y 2 . 

A Bernal chart for a cylindrical can\era of any radius may be con- 
structed graphically by drawing the plan and elevation of this model. 
Thus, if the height of any reciprocal lattice point above the origin is r 



150 



STRUCTURE DETERMINATION 



CHAP. VI 



and its distance from the axis of rotation is rg, the position of the 
reflection on the film is obtained in the following way. Draw a circle 
of radius r (Fig. 86 e), and then a chord NUT at a distance r from the 
centre (this is the circle of contact seen edgewise) ; UT is the radius of 



BEAM 




FIG. 86. a-d. See text, e and / illustrate giaphical construction 
of a Bernal chart. 



the circle of contact for this reciprocal lattice point. Join OT and pro- 
duce to W on the line XC which is parallel to OU. WX is then the 
ordinate y of the spot on the film. Now draw the plan, that is, draw 
another circle of radius r (Fig. 86 /) and in it describe a circle of radius 
UT. On this circle NT mark off the points P x , P 2 which are at a distance 
rg from X, and produce UP l to Y l and UP 2 to 7 2 . The arcs XY l and 
X Y 2 are the abscissae x of the two reflections on the film produced by 
this plane. By doing this for a number of different values of r and r, 
the complete chart is obtained. 



OHAP. VI 



UNIT CELL DIMENSIONS 



151 



Indexing rotation photographs by reciprocal lattice methods. 
Orthorhombic crystals. First of all, the coordinates and for each 
reflection on the photograph (Fig. 87) are found in one of the ways 
just described; these coordinates may be plotted as in Fig. 88 a to 
form the reciprocal lattice rotation diagram. The problem now is to 
decide which point of the reciprocal lattice itself corresponds to each 
spot on the rotation diagram. 




I -2 -3 4 -5 '6 7 '8 '9 10 M 
FIG. 87. Coordinates of spots on rotation photograph of orthorhombic crystal. 



Consider first the equatorial reflections. For a crystal rotated round 
its c axis, the equatorial reflections are those of hkO planes. To assign 
correct indices it is only necessary to make a diagram (Fig. 886) of 
the zero level of the reciprocal lattice (the dimensions being already 
known from layer-line spacings on other photographs), and to measure 
with a ruler the distance f of each point from the origin ; it is then 
obvious which reciprocal lattice point corresponds to each spot on the 
rotation diagram. 

As for the upper and lower layer lines of the rotation diagram, it is 
immediately obvious that the spots on them lie exactly above or below 
equatorial spots the values for spots on all layer lines are the same 
(except where certain spots are missing). The reason is that the 101 
point of the reciprocal lattice is at the same distance from the axis of 
rotation as 100 (Fig. 80), and in general a point hkl is at the same distance 
from the axis of rotation as the corresponding MO point. Therefore, 



152 



STRUCTURE DETERMINATION 



CHAP. VI 



knowledge of the indices of the equatorial spots immediately leads to 
the correct indices for all the remaining spots. The vertical lines of 




400 040 





















s 


(00 
) 6 * 














oio v 

a* 


Js ^ 


PIO 


020 


030 

..*HO 


040 


050 






100 \ \ 


\ 






140 








200 \ 
^JJO\ 


'-. 














300 


310 


tP\ 

^\ 












400 






450 










500 













FIG. 88. a. Reciprocal lattice rotation diagram corresponding to Fig. 87. 
b. Graphical determination of f values for an orthorhombic crystal. 

spots having the same hk indices are known as 'row lines*. The row lines 
are often obvious on the photograph itself, though they are not straight 
lines see the photograph of benzil in Fig. 75, Plate VI. 



CilAP. VI 



UNIT CELL DIMENSIONS 



153 



It should be noted that some reflections may be missing from the 
photograph on account of certain symmetries in the atomic arrangement 
(see Chapter VII), others because they are so weak that they do not 
produce a perceptible blackening on the film. Still others (such as 001, 
002) are absent because the crystal planes have not been in reflecting 
positions ; the reciprocal lattice points which do pass through reflecting 
positions are contained within a circular area of radius 1, corresponding 
to the boundary of the sphere of reflection (see Fig. 88 a). It is useful 




FIG. 89. Reciprocal lattice (hQl plane) of moiio- 

rlinic crystal. Tho 6 projection of the real coll is 

also shown (a, r, /?). 



FIG. 90. Monoclinic reciprocal 
lattice rotated round 6. 



to remember that the distance from the origin to each point on the 
rotation diagram is X/d for the corresponding crystal plane. 

Monoclinic crystals. The procedure already described is followed 
as far as the determination of and for each point and the construction 
of the reciprocal lattice rotation diagram. But, on account of the lower 
symmetry of the monoclinic cell, the rotation diagram is less simple than 
that of an orthorhombic crystal. 

A monoclinic unit cell has its a and c axes at an angle /J not 90, and 
its b axis normal to the ac plane. The reciprocal lattice has a similar 
form, but it should be noted that, whereas in the orthorhombic system 
all three reciprocal axes are parallel to the real axes, in the monoclinic 
system only the 6* axis of the reciprocal lattice is parallel to the real 
6 axis. The a* and c* reciprocal axes are not parallel to the a and c 
axes of the real cell : a* (length = A/d 100 ) is perpendicular to c, and c* 
(length = A/d 001 ) is perpendicular to a (Fig. 89) ; and the angle /?* of 
the reciprocal cell is the supplement of the angle )3 of the real cell. 

If the crystal is rotated round its b axis (Fig. 90) the equatorial spots 



154 



STRUCTURE DETERMINATION 



CHAP. VI 



are reflections from hOl planes. The | values for these spots are found 
as before by measuring the distance from the origin to each point of 
the (non-rectangular) hOl net plane (Fig. 89). Note that the indexing of 
equatorial reflections in this case cannot be done by a log d chart, since 
there are three variables, a, c, and /? ; the reciprocal lattice method is 
essential. Once the indices for the equatorial reflections have been 










\ \ \ 


















121 


120 


w 


021 


020 


to 


i2i 


120 


m 


m 


220 


221 




II 


10 


111 


on 


0\0 


on 


in 


110 


HI 


21! 


210 


211 




10! 


100 


ioi 


00] 


000 


001 


IOI 


100 


toi 


201 


20d 


201 


FT 


III pllp 


Oil |II7|IJO 


j/i pffpo 


2/1 



FIG. 91. Monodinic reciprocal lattice rotated round normal to a*6* piano 

(c axis of real cell). Above : general view. Right : real roll, same orientation. 

Below: view (on smaller scale) looking straight down c axis. 

found, those of the reflections on upper and lower layer lijies follow at 
once, since all reciprocal lattice points having the same h and I indices 
(such a set as 201, 211, 221, 231, and so on) are at the same distance 
from the axis of rotation and thus form row lines. 

Rotation round the a or c axis of a monoclinic crystal (Fig. 91) results 
in a different type of photograph ; the spots fall on layer lines, as always 
when a crystal is rotated about a principal axis, but not on row lines. 
Consider first the equatorial spots on a photograph obtained by rotating 
the crystal round the c axis ; these are from hkO planes. The zero (hkQ) 



CHAP. VI 



UNIT CELL DIMENSIONS 



155 



level of the reciprocal lattice is a rectangular array of points, from which 
values are obtained as before by measurements from the origin. The 
other levels are also rectangular networks, but they do not lie directly 
above or below the zero level, being displaced in the direction of a* by 

















230 
220 


130 


030 


a 


230 




\g224 


120 

fiu 


020 
> ..* 


120 ..,.- 

-"$224 


220 




210 \ 
\ 

K 


*>;*' 

l^-O-^O^ 

3 2/ V 
X- <^\ 


xim" 

TTT 


no 


210 




200 4 


4 


200 




210 


110 


oro 


IfO 


2io 



YIG. 92. Graphical method for determining f values for non-equatorial 
reflections of monoclinic crystal rotated round c. 



A //I? 

''... 



hk3 


""Wfe ""TTirTn ? *""? 




\ 


\ \ / . ; \ ;ll /; / 




\ ' 


\ ^ / \ /? 


,; / 




* t .' 


Ms/ \/! 




hk2 


.' 






wzjm v 


2 MHW2 H 


/ 




/ \ 


/,|\ /K 


/ 


hkl 


-w^{ 


t^-i'if^- 


z 


5 


' "*"vC^ * * ' * ' i ,""' '*' 




LKO 






f 0/0 100 W) m /2020c(o3of220 


210 J30 



FIG. 93. Part of reciprocal lattice rotation diagram for monoclinic crystal 
rotated round c axis, constructed by measurement of Fig. 92. 

distances which are multiples of c*cosj3*. The 101 point, for instance, 
is not the same distance from the axis of rotation as the 100 point, and 
hence 10Z reflections on the photograph do not lie on row lines but on 
curves whose form depends on j3*. The distances of the non-equatorial 
points from the axis of rotation might be obtained by drawing the pro- 
jections of the various levels on the equatorial plane, as in Fig. 91 ; but 



156 STRUCTURE DETERMINATION CHAP, vi 

it is simpler to use the same network the already drawn zero level 
for all layers, marking off along a* a set of new origins, one for each 
level (Fig. 92). It is important to note that the origins for the upper 
layers (positive values of I) lie along the negative direction of a*. The 
f values for all hkl points are found by measuring the distance from 
the origin for Z = + 1 to the appropriate hkO point ; and so on for other 
layers. Note that for 10T (or T01) is smaller than f for 101. The 
rotation diagram produced in this way is shown in Fig. 93. Only OOZ 
points, and hOl points having h constant, lie on straight lines (inclined 
to the itxis at the angle j8*), all others lying on curves. 

The rotation diagrams of monoclinic crystals can also be used for 
graphical determination of the spacings of the planes ; this is done (as 
in Fig. 93) by measuring the distance of each point to the origin. This 
graphical method is much more rapid than calculation. 

Triclinic crystals. None of the angles of a triclinic cell are right 
angles ; in consequence, none of the axes of the reciprocal lattice are 
parallel to those of the real lattice, and the angles a*, )3*, and y* of 
the reciprocal lattice are all different from those (a, /?, and y) of the real 
lattice. The relations between these quantities are as follows : 

cos 6 cosy cos a 

', 



_ _ , 

- ; ' ; , 

sin p sin y 

cosfi* = cos r cos "- CQS ff 

sin y sin a ' 

A cos a cos fi cos y 
cosy* = -- r - . y 
sin a sin p 

a* = -fccsina, 

6* = casinjS, 

* A A 
c* = - aosmy, 

where D = a&cV(l + 2cosacos/?cosy cos 2 a cos 2 /? cos 2 y). 

These formulae are so unwieldy that it is better to derive the reciprocal 
lattice elements directly from the spacings and angles of the planes : 

... A r * A j. A 

0* = -; b *==^ ; C *-=T-' 

a !00 "'010 a 001 

and 
a* = ^(010): (001), 0* = L (100): (001), y* =-- L (100): (010). 




FIG. 94. Triclinic reciprocal lattice. Above, left: general view. Right: real cell, 
Below: view looking straight down r, showing zero and first layers only. 




FIG. 95. Graphical determination of f values for non -equatorial 
reflections of triclinic crystal rotated round c. 



158 



STRUCTURE DETERMINATION 



CHAP. VI 



If a triclinic crystal is rotated round any axis of the real cell (Fig. 94), 
the photograph exhibits layer lines (since the various levels of the 
reciprocal lattice are normal to the axis of rotation), but not row 
lines, since none of the points on upper or lower levels are at the same 

distance from the axis of rotation as 
corresponding points on the zero level. 
The indices for points on the zercr level 
are found in the same way as for photo- 
graphs of monoclinic crystals rotated 
round the b axis: for the zero level of 
a triclinic crystal rotated round c, a net 
with elements a*, &*, and y* is con- 
structed (Fig. 95), and distances f of 
points from the origin are measured. 
The other levels, projected on to the 
equator, are displaced with regard to the 
zero level in a direction which does not 
lie along an equatorial reciprocal axis; 
the simplest way of measuring values 
is, as before, to use the zero level net- 
work, marking off a set of alternative 
origins, one for each level, along a line 
OL in Fig. 95. The angle S this line makes with a* is given by 

-, cos a* COS fi* cosy* 

tanS = ^_L___ 

cosp*smy* 

and the distances of the alternative origins are multiples of 




I20\fl0 
WO 020 



220 120 



FIG. 96. Reciprocal lattice rota- 

tion diagram for triclinic crystal, 

constructed by measurements on 

Fig. 95. 



The rotation diagram has the appearance of Fig. 96 ; the only points 
lying on a straight line (apart from the layer lines) are the OOZ set. Note 
that the values for hkl, hkl, hkl, and fikl are all different. 

The spacings of the planes of a triclinic crystal are best determined 
from the rotation diagram, by measuring the distance of each point 
from the origin. 

Oscillation photographs. It often happens that on rotation photo- 
graphs the positions of two or more possible reflections are so close 
together that it is impossible to decide whether a particular spot is 
produced by one of the crystal planes in question, or the other, or indeed 
both together. For the purpose of determining unit cell dimensions 



CHAP. VI 



UNIT CELL DIMENSIONS 



159 



this usually does not matter, but if the investigation is to be carried 
further, to the determination of atomic positions (see Chapter VII), it 
is important 'to identify every reflection unequivocally and to measure 
its intensity. 

One method of separating reflections is to take photographs while 
the crystal is, not rotating completely, but oscillating through a limited 
angular range. A set of several photographs is required to cover all 
reflections ; on each photograph only certain spots appear, because only 




FIG. 97. Reciprocal lattice diagram for oscillation photograph. 
Orthorhombic crystal, equatorial level. 

certain sets of crystal planes pass through their reflecting positions in 
the course of the oscillation of the crystal through the selected angular 
range. Thus, it is usually possible to decide that because a particular 
spot appears on one photograph and not on others it must have been 
produced by one crystal plane and not another. 

The orientation of the crystal necessary for the production of each 
reflection is determined graphically by a method (Bernal, 1926) which 
follows naturally from the reciprocal lattice methods already described. 
Consider first the equatorial reflections given by an orthorhombic 
crystal oscillated about its c axis. These reflections are produced as the 
zero level of the reciprocal lattice (containing the hkO points) passes 
through the sphere of reflection. In Fig. 97 the axis of rotation is 
normal to the plane of the paper. Suppose that the crystal is oscillated 
through 15, one extreme position being with the X-ray beam jB x normal 



160 



STRUCTURE DETERMINATION 



CHAP. VI 



to the 100 plane. At this position the circle of contact, which for the 
zero level is the equator of the sphere (radius =1), has its diameter 
along a* (position I). When the crystal rotates, the reciprocal lattice 
rotates about 0, but it is simpler for graphical purposes to keep the 
reciprocal lattice still and rotate the beam (in the opposite direction), 
and with it the circle of contact, which for the other extreme position 
of the 15 oscillation reaches position II. During this movement the 
only reciprocal lattice points which pass through the circumference of 




FIG. 98. Reciprocal lattice diagram for oscillation photograph. 
Orthorhomhic crystal, third layer line. 

the circle are those marked with spots ; therefore the only equatorial 
reflections which appear on this photograph are those from planes having 
the indices of these points, and if we look at the photograph as if looking 
along the beam, reflections OlO, 230, 330, 430, and 530 appear on the left 
of the film, while reflections 120, 220, 520, 620, 710, and 700 appear on 
the right. 

Consider now the reflections on an upper layer line say the third. 
These are produced when the points on the third (upper) level of the 
reciprocal lattice pass through the sphere of reflection. The circle of 
contact (MNP in Figs. 82-4) has a radius less than that of the sphere ; 
the radius is actually </{! (3A/c) 2 }. (For the nth layer the radius would 
be ^{1 (n\/c) 2 }.) During the oscillation of 15 (see Fig. 98) this circle 
of contact moves from position I to position II ; the only crystal planes 
which reflect during this movement are those whose reciprocal lattice 



CHAP. VI 



UNIT CELL DIMENSIONS 



161 



points lie in the areas L and jR that is, 123, 333, 433, and 623, giving 
spots on the^left of the photograph, and 113, 223, 323, 423, 523, and 
613, giving spots on the right. 

All reciprocal lattice levels and all angles of oscillation can be dealt 
with in this way, care being taken always to use the correct radius for 
the circle of contact. In the same way, if for any purpose it is desired 
to know at what angle any plane reflects, it is only necessary to draw 




"IT -> B, 



FIG. 99. Reciprocal lattice diagram for oscillation photograph. 
Triclinic crystal, first layer line. 

the circle of contact on tracing-paper, and rotate it until the appro- 
priate reciprocal lattice point touches the circumference ; the position 
of the diameter BO then gives the necessary orientation of the beam 
with respect to the reciprocal lattice net and thus to any chosen reference 
direction in the crystal. Bernal, in the paper already mentioned (1926), 
gives a transparent chart showing the circles of contact for various 
reciprocal lattice levels. 

The same procedure is followed for all crystals. In dealing with photo- 
graphs of monoclinic crystals oscillated round a or c, or triclinic crystals 
oscillated round any axis, care should be taken to use the appropriate 
origin for each reciprocal lattice level. (See Figs* 92 and 95.) As an 
example, the procedure for the first (Tiki) level of a triclinic crystal is 
illustrated in Fig. 99. 

The oscillation method just described is essentially a method of 
checking indices which have already been assigned on the basis of f 



J62 



STRUCTURE DETERMINATION 



CHAP. VI 




W 



f *> 



FIG. 101. Rotation of a tilted 
crystal. 




values, and of separating those reflections which on complete rotation 
photographs are found to overlap. It defines the positions of reciprocal 
lattice points only within the 15 or 10 angle used for the oscillation 
photograph. It would be possible to define the angular positions of 
these points more closely by oscillating through smaller angles, or by 
taking photographs covering slightly overlapping angular ranges ; but 
this would be tedious. It is better to use one of the methods which 
have been devised to define the precise positions of reciprocal lattice 
points in other words, methods whereby the reciprocal lattice may be 

plotted directly from the coordinates of 
reflections on the photographs. The best 
* methods of doing this are those (to be 
described later) in which the film is 
moved while the crystal is rotating, so 
that one coordinate of a spot on the film 
is related to the position occupied by 
the crystal when that reflection was 
produced. If, however, a moving-film 
goniometer is not available, it is often 
possible to achieve the same result by 
using the ordinary rotation-and-oscilla- 
tion goniometer in a special way: the 

crystal, instead of being rotated round a principal axis, is rotated round 
a direction inclined at an angle of a few degrees to a principal axis. 

The tilted crystal method. Crystals rotated round a direction 
inclined at a few degrees to a principal axis give X-ray diffraction photo- 
graphs in which the spots are displaced from the layer lines. But the 
amount of displacement is different for each reflection ; on the equator 
of Fig. 100, Plate VII (upper photograph), it can be seen that some 
reflections are doublets, one above and one below the equator, the 
separation being different for each pair of reflections ; a few lie actually 
on or very near the equator and are therefore not resolved; others 
are quadruplets, the separation being again variable. The reason is 
illustrated in Fig. 101, which shows a crystal tilted in a direction lying 
in the 010 plane; this particular plane is still vertical (see Fig. 101 b) 
and therefore gives reflections lying on the equator. But the reflections 
from plane 100 will not lie on the equator ; a reflection to the right when 
the crystal is in position a will lie above the equator, while on turning 
through 180 (position c) the reflection to the right will appear below 
the equator. The reflections from other planes in the hkO zone will be 



PLATE VII 





Fio. 100. X-ray diffraction photographs of a gypsum crystal rotated round a direction 
inclined 8 to the c axis in an arbitrary direction. Above, complete rotation; below, 

90 oscillation* 



CHAP. VI 



UNIT CELL DIMENSIONS 



163 



displaced from the equator by an amount which depends' on the orienta- 
tion of the reflecting plane with respect to the plane of tilt, as well as 
on the angle of reflection. 

The problem is best treated by reciprocal lattice methods. Fig. 102 
gives a general view of the zero layer of the reciprocal lattice of a 
crystal tilted in an arbitrary direction. Fig. 103 is a plan; the axis of 
rotation is normal to the plane of the paper ; the normal to the zero 
layer, as it comes out above the paper, lies a little to the right in the 
plane OT. All the reciprocal lattice spots to the right of A A' lie a little 
below the equatorial level, while those to the left of A A' lie a little 




A 

5 6 * Zero layer of 

reciprocal lattice ' 

FIG. 102. Reciprocal lattice of a tilted crystal. Zero layer (general view). 

above the equatorial level. Now since all the points on this net lie in a 
plane, the distance of any point from the equatorial level is propor- 
tional to the distance x from the line AA' \ if <f> is the angle of tilt, 
x - cosec <t> (see Fig. 102). Hence, if <f> and are known, x can be calcu- 
lated. The angle of tilt < can be fixed experimentally by first setting the 
crystal with a principal axis accurately parallel to the axis of rotation, 
and then tilting it <f> by one of the goniometer arcs. The coordinate of 
each spot can be determined either by using a Bernal chart or more 
accurately by calculation (p. 149). It is also possible to determine the 
distance (Fig. 102) from the origin of the reciprocal lattice: the co- 
ordinate of each spot is determined, either on the Bernal chart or by 
calculation, and from this is given by V( 2 + 2 )* 

The two coordinates x and fix the position of each reciprocal lattice 
point in its own net plane, except in one particular: the sign of the y 
coordinate (Fig. 103) is not determined ; in other words, any reciprocal 
lattice point P may be on either side of the tilt plane OT. Points P 
and Q, for instance, in Fig. 103 are on opposite sides of the tilt plane OT, 



164 



STRUCTURE DETERMINATION 



CHAP. VI 



but there is nothing in the treatment so far to tell us which side each 
is on. This ambiguity can be avoided by taking, not a complete rotation 
photograph, but an oscillation photograph in such a way that all reflec- 
tions on one side of the photograph correspond with reciprocal lattice 
points all lying on the same side of OT. For instance, the crystal is 
oscillated through 90 so that the tilt plane OT moves anti-clockwise 
from a position normal to the X-ray beam to a position parallel to the 
beam, and back again. On Fig. 103, in which the reciprocal lattice is 




FIG. 103. Reciprocal lattice of a tilted crystal. Zero layer (plan). 

stationary, this is equivalent to a rotation of the X-ray beam clockwise 
from OB to OC and back again. Reflections on the right-hand side of 
this photograph correspond to reciprocal lattice points through which 
the semicircle! ODB (radius = 1) passes as it rotates to OFC and back 
again ; all these points (lying within the heavily outlined area ODBECF) 
are on the same side of the tilt plane OT. (The left-hand side of the 
same photograph is not free from ambiguity.) The oscillation may, if 
desired, be through a smaller angle within the 90 range mentioned; 
but the X-ray beam may oscillate only between OB and OG if ambiguity 
is to be avoided and it is only avoided on the right-hand side of the 
photograph. 

t Strictly speaking, owing to the tilt of the net plane with respect to the plane of the 
paper all 'circles' in Fig. 103 should be slightly elliptical, with tho,major axis parallel to 
A A' (eccentricity = sin^). 



CHAP. VI 



UNIT CELL DIMENSIONS 



165 



In this way the coordinates of all reciprocal lattice points on the 
zero layer lying within the area ODBECF are directly determined. 
Fig. 104 shows the results obtained from a 90 oscillation photograph 
(Pig. 100, Plate VII) of a gypsum crystal set with its c axis inclined 8| 




Plane of till 



FIG. 104. MO plane of reciprocal lattice of gypsum crystal, determined from the 
photographs in Fig. 100. The length of each arc represents the possible error. 

to the axis of rotation ; in spite of the limited precision in the determina- 
tion of x, there is no doubt about where to draw the net. If the remain- 
ing points are required, the simplest plan is to restore the crystal to 
the untilted position, and then tilt it in a direction at right angles to 
the first by using the second of the arc movements of the goniometer 
head. A second 90 oscillation photograph is then taken, the plane of 
tilt being oscillated as before in relation to the X-ray beam. 

The coordinates of reciprocal lattice points corresponding to spots on 
the upper and lower layers of the same photograph can also be deter- 
mined directly. For these layers it can be shown (Bunn, Peiser, and 



166 STRUCTURE DETERMINATION CHAP, vi 

Turner-Jones, 1944) that if z is the distance of a layer from the equatorial 
layer ' x = z cot <- cosec <f>. 

The other coordinate necessary for the determination of the position of 
a reciprocal lattice point is , the distance of the point from the normal 
to the net plane (i.e. the real axis of the crystal) ; this can be obtained 
either from a photograph of the untilted crystal (in which circum- 
stance = ), or alternatively from the tilt photograph, using the 
expression = V( 2 + 2 z 2 )- It is thus possible to determine the whole 
reciprocal lattice directly from one or two tilt photographs. 

The tilted crystal method can only be used if the layers of reflections, 
though somewhat dispersed, &re distinct from each other: it must be 
possible to recognize at a glance that a particular reflection belongs to 
a particular level of the reciprocal lattice. For this reason, the method 
is most suitable for crystals having at least one short axis. Rotation 
about a direction inclined by a few degrees to the short axis gives a 
photograph in which the layer lines are well separated ; the shorter the 
axis, the larger the angle of tilt which can be used, and therefore the 
greater the displacement of the spots and the more accurate the deter- 
mination of x. This condition is fulfilled by many crystals of aromatic 
substances, since flat molecules often pack parallel to each other ; one 
crystal axis is approximately normal to the plane of the molecules and 
may be as short as 4-5 A. Moreover, the crystals of such substances 
are often needle-like, the short axis lying along the needle axis ; these 
crystals can be conveniently set up on the goniometer with the needle 
axis inclined by a few degrees to the axis of rotation. 

Moving -film goniometers. The advantage of moving the photo- 
graphic film during its exposure to the diffracted X-ray beams from a 
rotating crystal (the movement of the film being synchronized with 
that of the crystal) has already been mentioned : it is that one coordinate 
of a spot on the film is related to the position occupied by the crystal 
when that reflection was produced, and in practice this means that the 
coordinates of reciprocal lattice points can be derived directly from the 
coordinates of the spots on the film. It is true that this can be done by 
means of the ordinary rotation-and-oscillation goniometer if the tilted 
crystal method is used ; but the scope of this method is limited by the 
necessity of keeping the layer lines separate from each other, and even 
in the most favourable circumstances the displacements of the spots 
from the average layer-line levels are small. In moving-film gonio- 
meters a crystal axis is set accurately parallel to the axis of rotation, 



CHAP. VI 



UNIT CELL DIMENSIONS 



167 



and one cone of reflections only is allowed to reach the film, which is 
moved through a comparatively large distance during the rotation of 
the crystal/ 

The earliest of the moving-film goniometers, the one which up to 



FILM 



c=(= 



SCREEN 




/A/SLOT 



SCREEN 





P' 



P F 



W 





oL 


/B 


J29\ 


f90-e\ 


8' 


\ 




V^ 





FIG. 105. a. Principle of the Weissenberg moving-film goniometer, arranged for recording 

equatorial layer by normal beam method. 6. Determination of reciprocal lattice 

coordinates for spots on equatorial layer. 

the present has been most widely used, is that of Weissenberg (1924), 
in which (see Fig. 105 a), while the crystal is rotated, a cylindrical 
film is moved bodily along the axis of rotation, a complete to-and-fro 
cycle taking place during the rotation of the crystal through 180 and 
back again. A slotted screen is adjusted to permit the passage of any 



168 



STRUCTURE DETERMINATION 



CHAP. VI 



selected cone of reflections. Details of the design of this type of gonio- 
meter are to be found in papers by Robertson (1934 6), Buerger (1936), 
and Wooster and Martin (1940). 

The interpretation of Weissenberg photographs is quite simple. 
Consider first the zero layer of reflections, the X-ray beam being perpen- 
dicular to the axis of rotation of the crystal in other words, the reflec- 
tions which would lie on the equator of a fixed-film normal-beam rotation 
photograph, but which in a Weissenberg photograph are spread out as 

in the example in Fig. 107, Plate VIII. 
Imagine the film at one extreme end 
of its range of travel, the crystal being 
in a corresponding position, and in 
Fig. 105 6, let XA (perpendicular to 
the X-ray beam) and XB (along the 
beam) be the axes of reference of the 
reciprocal lattice. P is any reciprocal 
lattice point, whose position with 
respect to XA is given in polar co- 
ordinates by (the distance PX) and 
the angle y (Z.PXA). When the crystal 
rotates anticlockwise, reflection occurs 
when P reaches P' on the surface of 
the sphere of reflection, the direction 
of the reflected ray being OP'. To 
reach this position, the reciprocal 

lattice has rotated through an angle o> (PXP')\ and the film has 
simultaneously moved a distance d which is related to the total travel 

D by the relation = J!j- . 

We wish to find and y for the spot corresponding to the reciprocal 
lattice point P. is obtained from the distance x of the spot from 
the centre line of the film (corresponding to the distance along the 
equator of a fixed-film rotation photograph): if the radius of the 




U 
2F 



FIG. 1^6. Scheme for equi-inclination 
method. When cos ^ = , the Ith cone 

includes the direction of the primary 
beam. 



v J i *i 
cylindrical film 1S ^ -_. _ 



, where 6 is the Bragg angle ; is then 



given by f = 2 sin 0. The angle y is given very simply by the fact that 
/.P'XA (== aj+y) is equal to (since P'XQ is == 90 0): thus 

y = Bw = ~ X 180. 
The whole zero layer of the reciprocal lattice can thus be plotted 



PLATE VITT 





CHAP. VI 



UNIT CELL DIMENSIONS 



169 



directly, using the polar coordinates f and y. Cartesian coordinates 
e and / are usually more convenient, however; these are given by 
e = f cosy dnd / = siny. It is a simple matter to construct a chart 
giving Cartesian coordinates for all positions on the film ; such a chart 
is illustrated in Fig. 107, Plate VIII. 

For other cones of reflections it is best to use the 'equi-inclination* 
method (Fig. 106), in which the X-ray beam is inclined to the axis of 
rotation of the crystal at such an angle that it actually lies on the cone 



SCREEN 




FIG. 108. Moving-film goniometers which record a limited angular range 
of reflections. Left: Robinson, Cox. Right: Schiebold, Sauter. 

of reflections being studied. This occurs when cos (ft = ZA/2c (for rota- 
tion round the c axis). The advantage of the equi-inclination method 
(see Buerger, 1934) is that the chart for the zero layer can be used for 
the other layers ; it is only necessary to remember that to obtain reci- 
procal lattice coordinates on the same scale as those of the zero layer, 
the figures on the chart must be Multiplied by the factor ^{1 (/2) 2 }.f 
Moving-film goniometers intended to record only a limited angular 
range of reflections the range which would appear on a flat stationary 
film are simpler to construct, since reciprocating motion may be 
avoided. B. W. Robinson (1933 a) describes one (Fig. 108) in which 
the equatorial zone of reflections, passing through a slot in a metal 
screen, falls on a cylindrical film rotating round an axis at right angles 
to the axis of rotation of the crystal. A similar type is used by E. G. 
Cox,J who employs synchronous motors to avoid mechanical gearing. 
The position of the film axis is not fixed ; for reflections at small angles 
it may be set at right angles to the beam, while for reflections at large 
angles it can be moved round to the side of the beam. Schiebold (1933) 



t See also p. 189. 



Private communication. 



170 



STRUCTURE DETERMINATION 



CHAP. VI 



and Sauter (19336) use a flat film rotating in its own plane (Fig. 108). 
A design of this form of goniometer employing synchronous motors is 
described by Thomas (1940). This method has the disadvantage that 
the tangential velocity of the film increases with distance from the 
centre, and therefore the intensity of the spots fades off rather rapidly 
with increasing distance from the centre. The previously described 
method is preferable. 

The most interesting of the moving-film cameras is that of De Jong 
and Bounian (1938). (See also De Jong, Bouman, and De Lange, 1938.) 




FIG. 109. De Jong and Bouman's goniometer for undistorted photography of reciprocal 
lattice net planes. Left: camera arrangement. Right: reciprocal lattice equivalent. 

The X-ray beam is inclined to the axis of rotation of the crystal, and a 
flat film is rotated at the same speed in its own plane about an axis 
parallel to, but not coincident with, the axis of rotation of the crystal 
(Fig. 109). One cone of reflections is selected by means of a screen with 
an annular slot. Reflections corresponding to the zero level of the 
reciprocal lattice lie on the cone containing the direction of the X-ray 
beam, and for photography of this cone the beam must pass through 
the centre of the rotating film. When these conditions are fulfilled the 
spots on the film are found to be arranged in a network exactly as. in 
the reciprocal lattice ; in fact, it may be said that the film shows an 
undifitorted photograph of the zero level of the reciprocal lattice. The 
reason is demonstrated in Fig. 109. The scale of the reciprocal lattice 
and its attendant sphere of reflection may be made whatever we choose. 
Suppose we make the radius of the latter equal to Q, the distance 
from the crystal to the centre of the film. We have seen that the origin 
of the reciprocal lattice lies at the point where the beam emerges from 
the sphere of reflection ; evidently, then, the centre of the film is the 



CHAP, vr UNIT CELL DIMENSIONS 171 

origin of the reciprocal lattice, and, in fact, since the film is normal to 
the axis of rotation of the crystal, the plane of the film is the plane of 
the zero level of the reciprocal lattice. Moreover, we have seen that 
when the crystal rotates on its axis, the reciprocal lattice rotates about 
its own origin ; hence, when crystal and film rotate together at the same 
speed, the film keeps pace exactly with the reciprocal lattice in fact, 
the film is the zero level of the reciprocal lattice. Reflections are pro- 
duced when reciprocal lattice points touch the surface of the sphere, 
which they do at various positions in the circle of contact. The circle 
of contact for the zero level of the reciprocal lattice is, in this camera, 
defined by the annular slot in the screen. The directions of reflected 
beams are lines joining the crystal (the cetitre of the sphere) to reciprocal 
lattice points when the latter touch the circle of contact ; hence the 
reflected beams make spots on the film at positions corresponding 
exactly to reciprocal lattice points. We may imagine the reciprocal 
lattice points as already existing in the film, only waiting to be printed 
(as latent images) when reflected beams flash out from the crystal. 

The foregoing description refers to the photography of the zero level 
of the reciprocal lattice. But De Jong and Bouman show that in a 
camera in which both the inclination of the beam and the position of 
the axis of rotation of the film are variable, the various levels of the 
reciprocal lattice may be recorded successively, all on the same scale. 
The advantages of such photographs are obvious : no charts or graphical 
constructions are needed for indexing the spots, the indices being obvious 
by inspection. The only disadvantage of this camera is that the angular 
range of reflections which can be registered on any one film is limited ; 
in this respect, De Jong and Bouman's arrangement is better than that 
of Schiebold and Sauter, but not so good as that of Weissenberg, which, 
for the zero level at any rate, permits the recording of reflections at 
Bragg angles from near to near 90. 

A method not using a moving film has been suggested by Orowan 
(1942): a grid of fine wires, placed between the crystal and the (flat) 
film, rotates at the same speed as the crystal. On the photograph each 
spot is crossed by the shadows of one or more wires, and the orientation 
of these shadows defines the position occupied by the crystal when the 
reflection was produced. This method was devised for the determination 
of the orientation of metal crystals (in wires, for instance) : it is mentioned 
here because in principle it is also applicable to the indexing of single 
crystal photographs. In practice there are two limitations : first, both 
crystal and X-ray beam would have to be rather broad, unless a very 



172 STRUCTURE DETERMINATION CHAP, vi 

fine-mesh grid were used ; secondly, the angular range of reflections 
recorded on a flat film is limited. 

The simplest unit cell. When the indices of all reflections on the 
X-ray photographs of a crystal have been obtained by any of the 
methods described indices based, it will be remembered, on morpho- 
logically chosen axes the whole set of indices can be surveyed to see 
whether any simpler cell would account for all the reflections. The best 
way of doing this is to look at reciprocal lattice diagrams or models. 
For instance, in Fig. 110 it is obvious that the larger, heavily outlined 




110 



FIG. 110. The systematic absences in this reciprocal lattice indicate that a larger reciprocal 

cell (that is, a smaller real cell) can be chosen. The new reciprocal cell is heavily outlined. 

The new indices (underlined) are simpler than the old. 

reciprocal cell extended to form a network accounts for all the 
reflections, and therefore should be accepted in preference to the 
original network based on morphologically chosen axes. The larger 
reciprocal cell represents a smaller real cell, and jgives smaller indices 
for the reflections than the old cell for instance, the former 1 10 becomes 
010, the former iTO becomes 100, and the former 200 becomes 110. 

Bhombohedral crystals are best treated as if they were hexagonal. 
When hexagonal indices have been assigned to all reflections, and the 
simplest hexagonal cell has been chosen, hexagonal indices may be 
transformed to rhombohedral indices by the formulae given on p. 135, 
If, however, a rhombohedral crystal is rotated round an axis of the 
(morphologically chosen) cell, the photographs must be indexed by the 
methods given for triclinic crystals. Examination of a sketch or model 
of the reciprocal lattice deduced in this way will show whether or not 



CHAP, vi UNIT CELL DIMENSIONS 173 

a smaller, differently shaped rhombohedral cell would account for all 
the reflections. 

4 

Where there is more than one cell . of the same volume (and shape 
appropriate to the crystal system) which will account for all the reflec- 
tions, as in monoclinic and triclinic crystals (see Fig. 77), the most 
nearly rectangular cell will usually prove the most convenient to accept. 

When an investigation is to be carried only as far as the determina- 
tion of unit cell dimensions, it is usually not necessary to index the whole 
of the reflections ; it will often be sufficient to index reflections up to a 
Bragg angle of 30-40, or even less for crystals having large unit cells. 
Moreover, photographs taken for only one setting of the crystal are 
usually sufficient ; it is not necessary to* take rotation photographs for 
three settings as in the method mentioned on p. 1 39. For a single setting 
a straightforward rotation photograph gives the cell dimension along 
the axis of rotation (from the layer-line spacing) ; the other cell dimen- 
sions and angles are obtained from the positions of individual spots, 
either on moving-film photographs or, if a moving-film goniometer is 
not available, on tilted-crystal photographs. 

So far it has been assumed that well-formed crystals with plane faces, 
suitable for accurate setting by the optical method, are available. 
Such crystals form the ideal experimental material for any detailed 
crystallographic investigation ; but it is possible, even when the crystal 
symmetry is low, to proceed with far less promising material with 
ill-formed crystals, or with irregular crystal fragments, or even with 
polyerystalline specimens. The additional problems presented by such 
specimens will now be considered. 

The accurate setting of ill-formed crystals. Some crystals have 
imperfect faces which give diffuse optical reflections ; or it may happen 
that the only crystals available have partially defined shapes^ such as 
those considered in the chapter on microscopic methods of identifica- 
tion ^ey may be rod-like or plate-lifee, with fairly well-defined edges 
but too few well-formed faces to permit precise setting by the optical 
method. In such circumstances it is possible to set the crystal by 
preliminary X-ray photographs. The chosen direction is first set 
approximately parallel to the axis of rotation by the optical method, 
and a small-angle (10-15) oscillation photograph is taken, one of the 
arcs of the goniometer head being parallel to the beam for the mean 
position of the crystal. The zero-layer reflections are found to lie, not 
exactly on the equator, but on a curve, and from the form of the curve 
it is possible to deduce in what direction, and by how much, the chosen 



174 



STRUCTURE DETERMINATION 



CHAJP, VI 



axis is mis-set. This, like all such problems, is best appreciated in terms 
of the reciprocal lattice. 

If, as in Fig. Ill a, the c axis of the crystal is displaced from the axis 
of rotation in the plane normal to the beam (for the mean position of 
the crystal), the zero layer (hkO) of the reciprocal lattice is tilted in this 
same direction, and its plane cuts the sphere of reflection in the circle 
AD. During the 15 oscillation a number of hkO points pass through 




FIG. 111. Oscillation photographs for setting ill -formed crystals. 

the surface of the sphere, and thus X-rays reflected by these hkQ planes 
of the crystal strike the film at corresponding points ; on the flattened- 
out film (Fig. 1116) the spots fall on a curve BAD, whose distance from 
the equator is a maximum at a Bragg angle 6 ~ 45 and zero at 9 = 90. 
If, on the other hand, the displacement of the c-axis is in the plane 
containing the beam (Fig. Ill c), the spots on the film fall on a curve 
whose maximum distance from the equator is at 9 == 90 (Fig. Ill d). 
When the displacement of the c axis has components in both directions, 
an intermediate form of curve is obtained (Fig. Ill e). Note that the 
angle < gives the component of displacement in the plane perpendicular 
to the X-ray beam that is, the component for one setting arc; < is 



CHAP, vi UNIT CELL DIMENSIONS 176 

unaffected by the other component. From the curve e it is theoretically 
possible (Kratky and Krebs, 1936; Hendershot, 19376) to calculate 
both compohents, at any rate for crystals having large unit cell dimen- 
sions in the equatorial plane (so that there are sufficient spots on the 
photograph to define the curve of the equatorial layer line). In practice 
it is usually better to take two small-angle oscillation photographs; 
for one of them one of the setting arcs is, at the mean position of the 
oscillation, perpendicular to the X-ray beam; the angle </) of the equa- 
torial layer (Fig. 1116) gives the correction to be applied to this setting 
arc. For the other photograph, the second setting arc is, at the mean 
position of the oscillation, perpendicular to the X-ray beam ; the angle 
</' on this photograph gives, as before, Uhe correction to be applied to 
this arc. This simple method has the advantage that only short 
exposures need be given, since only the strong reflections at small 
angles are used. Another point worth remembering is that unfiltered 
radiation may be used ; the 'white' streak on the equatorial layer helps 
to define the angle </>. 

If the only crystal available is quite irregular in shape, it may be 
set up on the X-ray goniometer in any position, and trial oscillation 
photographs may be taken ; if recognizable layer lines are produced, 
accurate setting may be achieved by the method just given ; if not, the 
setting may be altered at random until recognizable layer lines are 
produced. Examination under the microscope between crossed Nicols 
may be useful: an extinction direction may coincide with, or lie near 
to, a possible crystal axis. 

Oriented polycrystalline specimens. Not every substance occurs 
naturally or can be induced in the laboratory to grow in the form of 
single crystals which can be dealt with by the methods already described. 
In certain fibrous minerals for instance, chrysotile, 3MgO - 2SiO 2 . 2H 2 
('asbestos') even very t^in fibres are found to be, not single crystals, 
but bundles of crystals all having one axis parallel to the fibre axis but 
randomly oriented in other respects. (Warren and Bragg, 1930.) And 
among organic substances the long-chain polymers to which class 
many biologically important substances as well as many useful synthetic 
substances belong usually cannot be obtained in the form of single 
crystals. Fortunately, however, these substances can usually be obtained 
in the form of fibres in which all the little crystals have one axis parallel 
or nearly parallel to the fibre axis (as in chrysotile). Substances such 
as cellulQse, keratin (the protein of hair), and fibroin (the protein of silk) 
occur naturally in this form (Polanyi, 1921 ; Astbury and Street, 1931 ; 



176 STRUCTURE DETERMINATION CHAP, vi 

Astbury and Woods, 1933; Kratky and Kuriyama, 1931). Synthetic 
polymers such as polyethylene and the polyesters and polyamides can 
be drawn out to form fibres in which the same type of orientation 
occurs. (Fuller, 1940) ; and some of the rubber-like substances, although 
amorphous when unstretched, crystallize on stretching, the crystals so 
formed all having one axis parallel to the direction of stretching (Katz, 
1925; Sauter, 1937; Fuller, Frosch, and Pape, 1940). 

When an X-ray beam passes through such a fibre perpendicular to 
its length, the pattern produced is of the same type as that given by a 
single crystal rotated about a principal axis. All orientations perpen- 
dicular to the fibre axis are already present in the -specimen, so that the 
effect of rotation is produced. Examples are shown in Fig. 112, Plate IX. 
The reflections are less sharp than those produced by single crystals, 
for two reasons : firstly, the orientation of the crystals in the fibre is 
not perfect, so that each spot is drawn out to the form of a short arc, 
and secondly, in most polymer fibres the crystals are so small that the 
reflections are inevitably more diffuse than those of large crystals (see 
p. 363). 

The unit cell dimensions can often be deduced from such a pattern 
by the methods already described; it fe true that only one rotation 
photograph is available, but this may be sufficient for the purpose. 
The length of the unit cell edge which is parallel to the fibre axis is 
given directly by the spacing of the layer lines. The other dimensions, 
and the angles, are less easy to discover ; the degree of difficulty depends 
on the symmetry of the crystals. The procedure is first to discover, 
from the spacings of the equatorial planes, the shape and size of the 
projected cell-area seen along the fibre axis. Naturally the simplest 
possibility a rectangular projection is considered first; this is best 
done by calculating logd for each spot, plotting the values on a strip 
of paper, and attempting to find a match point on the logd chart 
(see Fig. 79, p. 143). 

Assuming that the equatorial reflections have been shown to fit a 
rectangular reciprocal lattice net, attention may be turned to the upper 
and lower layer lines. The values for all the spots are read off on 
BernaPs chart, and the reciprocal lattice rotation diagram is constructed 
from these values ; if the values for the upper and lower layer lines 
correspond with those of the equator that is, row lines as well as layer 
lines are exhibited as in Fig. 87 then the unit cell must be ortho- 
rhombic. It should be noted that some spots may be missing from the 
equator, and it may be necessary to halve one or both of the reciprocal 



CHAP. VI 



UNIT CELL DIMENSIONS 



177 




FICJ. 113. Determination of non-rectangular 
equatorial net-plane by trial. 



axes previously found to satisfy the equatorial reflections. The dimen- 
sions of the unit cell, and the indices of all the spots, follow immediately 
from the reciprocal lattice diagrams. 

Suppose, however, that although the equatorial reflections fit a 
rectangular projected cell-base that is, a rectangular zero-level reci- 
procal lattice net the rest of the spots do not fall on row lines. 
This must mean that the remaining axis of the reciprocal lattice is (as 
in Fig. 91) not normal to the zero level; in other words, the unit cell 
is monoclinic, the fibre axis being the c or a axis of the cell. (It is 
customary to call the fibre axis c rather than a.) Indexing is done by 
trial that is, by postulating 
simple indices for the innermost 
spots and then testing the rest 
to see whether they fit the re- 
ciprocal lattice defined in this 
way. The solution is often indi- 
cated very simply by the fact 
that in the rotation diagram 
several orders of OOZ (or JW)0) 
are seen to lie obviously on 
a straight line starting at the 

origin ; the slope of this line at once gives /J*. (This line may be obvious 
in the fibre photograph itself, though the line is not straight ; see Fuller 
and Erickson, 1937 ; Fuller and Frosch, 1939.) This line gives the slope 
of c*, but not its orientation witli respect to the zero-level net. To 
discover this, and to test the remaining values, mark off, on the 
zero-level net, values of nc*cosj8* along one axis, the points to serve 
as origins for their respective layer lines ; measure values as in Fig. 9*2. 
If the measured values do not correspond with those on the rotation 
diagram, these alternative origins should be marked off along the other 
axis of the zero-level net to test the alternative orientation of c*. It 
may be necessary to halve one of the zero-level reciprocal axes to 
account for all spots on other levels. 

If the equatorial reflections do not fit a rectangular net, the crystals 
must be either monoclinic (with 6 parallel to the fibre axis) or triclinic. 
The shape and size of the projected cell-base must be found by trial. 
Simple indices such as 100, 001, 101, Toi are postulated for the first 
few reciprocal lattice points ; thus, mark off along a line the value of 
A/rf for the first spot and call it 100 (Fig. 113); from the origin draw 
arcs of circles with radii equal to X/d for the second and third spots, 



178 STRUCTURE DETERMINATION CHAP, vi 

which will be called 001 and T01 ; then find the position where a line LL 
(parallel to OA) cuts these two arcs at points whose distance apart is 
equal to A/d 100 . This defines a possible net, which can then be extended 
to mark a sufficient range of additional points, the distances of which 
from the origin are compared with the values of X/d for the remaining 
spots. If this does not account for all the spots, it is necessary to try 
halving first one, then the other, and finally (if necessary) both axes of 
the net to account for the whole of the equatorial spots. 

This done, consider the other layer lines on the photograph. A 
reciprocal lattice rotation diagram is prepared as before from the f and 
values of all the spots. If row lines are exhibited, then the remaining 
axis of the reciprocal lattice is normal to the zero-level net, as in Fig. 90 ; 
in other words, the crystals are monoclinic with their 6 axes parallel to 
the fibre axis. It is again necessary to remember that one or both 
reciprocal axes of the zero-level net may have to be halved to account 
for all the points on other levels. 

If row lines are not shown, then the crystals are triclinic. The 
inclination of c* to the vertical may be shown by a row of spots in line 
with the origin ; but the orientation of c* with regard to the zero-level 
net must be found by trial. Distances equal to the f values for OOZ 
planes are marked off along a line on a strip of paper pivoted at the 
origin of the zero-level net, and this line is swung round until the distances 
measured, as in Fig. 95, correspond with the values on the rotation 
diagram. This process is not so difficult or lengthy as it may seem. 

When sheets of certain crystalline polymers are thinned by being 
passed through rollers, or when sheets of certain rubber-like substances 
are stretched, a double orientation of the crystals is effected ; not only 
does one crystal axis become approximately parallel to the direction of 
rolling or stretching, but also a particular crystal plane tends to lie in 
the plane of the sheet. For rubber the best double orientation is obtained 
by stretching a sheet which is short (in the direction of stretching) in 
comparison with its width (Gehman and Field, 1939). Double orienta- 
tion in keratin has been achieved by compressing horn in steam, in a 
direction at right angles to the fibre axis (Astbury and Sisson, 1935). 
If the structure is orthorhombic and the favoured plane happens to be 
a face of the unit cell, then the whole specimen may simulate a single 
crystal; in other circumstances (Fig. 114) there may be two or more 
different orientations of the unit cell in the specimen (for instance, for 
a triclinic unit cell there are four different orientations of the unit cell 
in a doubly oriented specimen). Such doubly oriented specimens are 



OHAP. VI 



UNIT CELL DIMENSIONS 



179 



more useful than singly oriented fibres ; they give photographs which 
are like the oscillation photographs of single crystals or twinned crys- 
tals, and although the limits of crystallite orientation are naturally 






(d) 





! %^- 



1W 



FIG. 114. Orientations of unit cell in doubly oriented polycrystalline specimens. 

a. Orthorhombic, fibre axis c, favoured plane 110 (or any hkO) 

b. Monoclinic, 6, , hQl 



c. 

d. 

e. Triclinic, 

/. Monoclinic, 



c, 
c, 



010 
100 
MO 
hkO 



somewhat indefinite, the photographs can be used like oscillation 
photographs to provide clues to the orientation of crystal planes giv- 
ing particular spots, or to confirm or disprove indices already selected. 
Photographs of such specimens may also be taken, one layer at a time, 



180 STRUCTURE DETERMINATION CHAP, vi 

in moving-film cameras ; instead of the spots given by single crystals, 
streaks are produced, and the position of maximum intensity on any 
streak may be taken as an indication of the position of the reciprocal 
lattice point for the plane in question. 

In drawn metal wires the fibre axis is usually not a crystal axis. The 
problem of the determination of crystal orientation in such specimens 
(and in rolled metal sheets), though closely related to those dealt with 
here, is outside the scope of this book. (The unit cell dimensions, and 
indeed the complete structures of such crystals, are usually known, and 
the problems that arise are questions of correlation of physical proper- 
ties with orientation.) See Schmid and Boas, 1935; Orowan, 1942. 

Determination of unit cell dimensions with the highest accu- 
racy. The greatest precision in the determination of the spacings of 
crystal planes is attained when the angle of reflection (6) is near 90. 
This is in the first place a consequence of the form of the Bragg equation 

d = ; near 90 a very small change of sin 8 (corresponding to a 

2sin# 

very small change in d) means a large change in 0. Hence a certain 
error in the measurement of means a much smaller error in the 
determination of d. In addition to this circumstance, the possible error 
due to the absorption of X-rays in the specimen (see Fig. 68) tends 
towards zero as 9 approaches 90. 

In most high-precision determinations of unit cell dimensions hitherto 
published, powder photographs have been used. Any possible errors 
due to shrinkage of the film on development and drying are avoided by 
printing fiducial marks on the film ; in the type of camera designed by 
Bradley and Jay, which is well adapted for this type of work, the sharp 
termination of the exposed part of the film by a knife-edge in the camera 
serves this purpose. The reflection angles of several arcs (they are all 
! 2 doublets in the large-angle region) are determined from their 
positions in relation to the fiducial marks ; values of d are calculated 
from the angles, and from these values the axial dimensions are worked 
out for each arc. By plotting the several different calculated values of 
the axial dimensions against cos 2 and extrapolating to cos 2 = (that 
is, B = 90), the most probable values are obtained (Bradley and Jay, 
1932). An analytical least-squares method of finding the most probable 
value has also been suggested (Cohen, 1935). Determination of the 
camera constant (the angle which the fiducial marks represent) can be 
done either by an optical method or by taking a powder photograph 
of quartz, the lattice dimensions of which are accurately known (Wilson 



CHAP, vi UNIT CELL DIMENSIONS 181 

and Lipson, 1941). In favourable cases axial dimensions may be deter- 
mined in this way with a possible error of no more than 1 part in 
50,000. (The small deviations from the Bragg equation caused by 
refraction of X-rays are usually negligible Hagg and Phragmen, 1933 ; 
but see also Wilson, 1940, and Lipson and Wilson, 1941.) 

For cubic, tetragonal, hexagonal, and trigonal crystals, powder photo- 
graphs alone may be used. The same method may be used for crystals 
of orthorhombic or even lower symmetry, provided the indices of the 
reflections at large angles can be found ; for this purpose it is usually 
necessary to make use of information derived 
by single-crystal methods. 

Single-crystal rotation photographs fnay 
also be used for very accurate determinations 
of lattice dimensions ; the positions of the 
doublets at large angles of reflection are 
measured. A Weissenberg camera suitable for 
this purpose is described by Buerger (1937). 

Powder cameras of a type different from 
that hitherto described have been used for 
high-precision determinations of unit cell 
dimensions. In this type of camera (first ^ . . , f 

* FIG. 115. Principle of powder 

used by Seeman (1919) and Bohlin (1920)) a camera of focusing type, 
divergent X-ray beam is used, and a focusing 

effect is obtained by making the powdered specimen and the recording 
film parts of the same circle, which passes through the point of diver- 
gence of the X-ray beam (Fig. 1 15). The focusing effect depends on the 
fact that all angles subtended by the same arc of a circle are equal. 
Thus, if the angle of reflection for a particular crystal plane is 20, the 
reflections from crystals at two points A and B on the arc-shaped 
specimen reach the film at the same point R. Sharp reflections are 
thus produced, and the accuracy attainable is as high as in the Debye- 
Scherrer camera, as Owen and his collaborators have shown (Owen and 
Iball, 1932 ; Owen, Pickup, and Roberts, 1935). The specimen AB need 
not be diametrically opposite to the point of entry of the X-ray beam ; 
it may be anywhere on the circumference of the circle. Focusing cameras 
are, however, less suitable for general purposes than the Debye-Scherrer 
type, and are not widely used. 

Applications of knowledge of unit cell dimensions. 1. Identi- 
fication. The use of powder photographs for identification has been 
described in Chapter V ; the simplest method is to calculate the spacings 




182 STRUCTURE DETERMINATION CHAP, vi 

of the crystal planes from the positions of the reflections, and to use 
these spacings together with the relative intensities as determinants. 
If this information does not lead to identification, it may be worth while 
to attempt to discover the unit cell dimensions, since for many sub- 
stances unit cell dimensions have been determined and published, but 
the details of the X-ray diffraction photographs have not been recorded. 
If only a powder photograph is available, this will usually be possible 
only for cubic, tetragonal, and hexagonal (including trigonal) crystals. 
For cubic crystals a list is available in which substances are arranged in 
order of the length of the unit cell edge ; it includes results published 
up to 1931 (Knaggs, Karlik, and Elam, 1932). For the other types it 
is necessary to think of likely substances and look up their unit cell 
dimensions. 

If the spacings given by the powder photograph do not fit any of 
these types of unit cell, the crystal (assuming that there is only one 
crystalline species in the specimen) is probably orthorhombic, mono- 
clinic, or triclinic. Single-crystal photographs are usually necessary for 
the determination of the dimensions of the unit cell in these types of 
crystal ; if it is possible to pick out from the specimen single crystals 
large enough to be handled and set up on the X-ray goniometer, or if 
the specimen can be recrystallized to give sufficiently large crystals, 
the unit cell dimensions may be determined by the methods described 
earlier in this chapter. 

Single-crystal photographs are extremely sensitive criteria for identi- 
fication, much more sensitive than powder photographs. The latter yield 
a set of spacings and relative intensities, which are quite sufficient for 
the unequivocal identification of the great majority of substances. 
However, complex organic substances which are closely related to each 
other (for instance, large molecules differing only in the position of a 
single constituent atom) may give powder photographs which are very 
similar to each other ; but the single-crystal photographs of such sub- 
stances are sure to display some differences. Many reflections which 
would overlap on powder photographs are separated on single-crystal 
photographs, which show not only the spacings of the crystal planes 
and the intensities of the reflections, but also the relative orientations 
of the planes. Even if the unit cell dimensions of two substances 
are closely similar, the relative intensities of some of the reflections are 
likely to be different, since, as we shall see in the next chapter, the 
intensities of reflections change rapidly with small changes in atomic 
positions. 



CHAP, vi UNIT CELL DIMENSIONS 183 

As an example of the use of single-crystal photographs for identifica- 
tion, vitamin B4 was shown by Bernal and Crowfoot (1933 6) to be 
identical with adenine hydrochloride. The extensive survey of the 
crystallography of substances of the sterol group by Bernal, Crowfoot, 
and Fankuchen (1940) gives a vast amount of information on these 
crystals, including unit cell dimensions; this paper also contains a 
discussion of the identification problems in this group. 

If single crystals cannot be obtained, and the only available X-ray 
photograph is a powder photograph which cannot be interpreted directly, 
the possibilities of identification are not exhausted. Indirect methods 
may perhaps be used. For instance, it may be suspected on chemical 
grounds that a substance is one which is known to form monoclinic 
crystals whose single-crystal photographs or unit cell dimensions have 
been published ; from the published information it may be possible to 
calculate the spacings and intensities which would be shown on powder 
photographs. To do this for a large number of reflections would be a task 
of considerable magnitude, for which the knowledge in the next chapter 
is required ; but calculation of the spacings of a few of the strongest 
reflections would soon show whether it is worth while to proceed further. 

Another possibility is that the suspected substance has not previously 
been studied by X-ray methods, but morphological axial ratios have 
been published. The axial ratios of the simplest unit cell are either the 
same, or are closely related to, those selected by morphological methods. 
Consequently, a determination of unit cell dimensions by whatever 
X-ray method is practicable may give the clue to identification. 

For chain compounds, such as the paraffin wax hydrocarbons, fatty 
acids, and other derivatives, it is possible to use a single spacing, or 
the various orders of diffraction from one particular crystal plane, for 
purposes of identification. A thin layer of the substance is crystallized 
on a glass plate; the leaf-like crystals grow with their basal planes 
parallel to the glass surface, so that if an X-ray beam grazes the surface 
while it is oscillated through a small angle, the various orders of diffrac- 
tion from the basal plane are recorded. The long molecules are either 
normal to the basal plane or somewhat inclined to the normal; the 
spacings are thug related to the lengths of the molecules, and can thus 
be used for identifying particular members of a series of homologues. 
The relative intensities of the different orders may be used to locate 
substituent atoms, as in the ketones. (See Piper, 1937.) 

2. Determination of composition in mixed crystal series. The unit cell 
dimensions of mixed crystals crystals in which equivalent positions in 



184 STRUCTURE DETERMINATION CHA*. vi 

the lattice are occupied indiscriminately by two or more different types 
of atom or molecule are intermediate between those of the separate 
constituents. If the relation between composition and unit 'cell dimen- 
sions has previously been established, then in practice the unit cell 
dimensions may be used to determine the composition of the mixed 
crystal. For instance, certain metals form mixed crystals, often over 
a wide range of composition. The crystals are usually highly symmetri- 
cal cubic or hexagonal and therefore powder photographs may yield 
very precise unit cell dimensions, which lead to an accurate determination 
of composition. Of course, for a simple two-component system the 
composition could be determined chemically. But if other constituents 
are present, it may not be known which of them crystallize together ; 
there might be a double compound of A and B and mixed crystals of 
A and C, and in such a case the X-ray method would be very valuable. 
Even in a simple two-component system the results may not be as simple 
as might be expected. For instance, mixed crystals of copper and nickel 
were prepared by co-precipitation of the hydroxides, conversion to 
oxides by heat, followed by low-temperature reduction. But in one 
experiment the resulting solid was shown by its X-ray powder photo- 
graph to consist of two different compositions of mixed-crystal, one 
rich in copper and the other rich in nickel ; the two patterns of cubic 
type appeared together on the film, and the lattice dimensions gave 
the compositions of the two phases. 

Whenever the composition of a crystal lattice varies, and with it the 
lattice dimensions, this method may be used. Certain 'interstitial' com- 
pounds, such as iron nitrides and carbides, come under this heading 
(though they are not usually called 'mixed crystals') ; in these crystals 
varying numbers of carbon or nitrogen atoms fit into the holes between 
metal atoms (Hagg, 1931). Zeolitic crystals, in which the water content 
may vary without essential change of crystal structure, are also of this 
type (Taylor, 1930, 1934). A simpler substance of the same type is 
calcium sulphate subhydrate, CaSO 4 .0-|H 2 0; the water content may 
be determined from the lattice dimensions (Bunn, 1941). 

In certain circumstances it may be possible to use accurate values 
of lattice dimensions as criteria of the purity of a substance. If the 
impurities likely to be present in small quantities are such as form 
mixed crystals with the main substance, then the lattice dimensions 
determined by a high-precision method form sensitive criteria of purity. 
No generalizations can be made on the sensitivity of the test, which is 
entirely specific to each substance and each possible impurity. 



CHAP. VI 



UNIT CELL DIMENSIONS 



185 



3. Determination of molecular weight. From the unit cell dimensions, 
the volume V of the unit cell may be calculated. Multiplying the volume 
V by the dfensity p, we get the weight W of matter in the unit cell: 
W = Vp. The weight of a molecule M is either equal to W (if there is 
only one molecule in the unit cell), or is a simple submultiple of W (if 
there is more than one molecule in the unit cell). Therefore, if we know 
the approximate weight of a molecule and this knowledge is usually 
known from chemical evidence we can find the number of molecules 
n constituting the unit of crystal pattern. Having found n, we can then 
use W to get an accurate value for M , which == Wjn = Vpjn. 




FIG. 116. Volume of unit cell -= r x area of c projection. 
Left: monoclinic. Right: triclinic. 

The volume of a rectangular cell (cubic, tetragonal, or orthorhombic) 
is, of course, the product of the three edge-lengths. For non-rectangular 
cells the following expressions give the volume : 

Hexagonal : V = abc sin 60 ; 

Rhombohedral : 

Monoclinic : V = abc sin /} ; 

Triclinic : V = abc sin /J sin y sin 8 , where 



IT o . o * f i &' sin a/2 

V = a 3 sm 2 asmS , where sin- = : ; 

2 sin a 



i i=y{ si < 



It is worth remembering that to determine the volume of the unit 
cell of a monoclinic or triclinic crystal it is not necessary to find all the 
edge-lengths and angles of the unit cell. The volume of such a unit cell 
is, for instance, the area of the c projection multiplied by the length of 
the c axis (Fig. 116). Consequently, if the crystal is set up with c as the 
axis of rotation, determination of the dimensions of the projected cell- 
base (from the equatorial reflections) and the length of the c axis (from 



186 STRUCTURE DETERMINATION CHAP, vi 

the layer-line spacing) gives all the information required to calculate F. 
It is not necessary to find /? for the monoclinic cell,f or any of the real 
angles of the triclinic cell. 

If the unit cell dimensions have been expressed in cm., using the 
latest values for X-ray wave-lengths in cm. (Lipson and Riley, 1943) 
and the density in gm./c.cm., the molecular weight obtained is in 
grams. To put it on the chemical scale against = 16, the figure must 
be divided by 1-6604 x 10~ 24 , the weight in grams of a hypothetical atom 
of atomic weight 1-0000. (Birge, 1941.) Thus, 



" 1-6604 xlO-* 4 "" ~~ 1-6604 " 

t 

If X-ray wave-lengths in kX units have been used, so that unit cell 
dimensions are also in kX units, these figures may be multiplied by 
10~ 8 , and the molecular weight result divided by the older value for 
the weight of the unit atom, that is, 1-6502 x 10~ 24 gm. Thus, 

M ^ F(inkX 3 )xp 
1-6502 ' 

The most convenient method of measuring the density of crystals is to 
suspend them in a liquid mixture, the composition of which is adjusted 
by adding one of the constituents until the crystals neither float nor 
sink. The use of the centrifuge increases the sensitivity of this method 
(Bernal and Crowfoot, 1934 6). The density of the liquid is then deter- 
mined by the standard pycnometer method. 

The X-ray method is often the most accurate way of finding the 
molecular weight of a substance. Usually the chief error is likely to be 
in the value for the density. 

It should be noted that in crystals of long-chain polymers, the unit 
cell shown by the X-ray photographs contains only sections of molecules. 
A small group of atoms, often only one or two monomer units, is 
repeated many times along a chain molecule, and the precise side-by- 
side packing of these chains gives rise to the crystalline pattern of atoms. 
By a 'crystal' of a long-chain polymer is meant a repeating pattern of 
monomer units, not of whole molecules (see Fig. 143). Chain molecules 
thread their way through the unit cell. A calculation of the foregoing 
type leads, for such substances, to a knowledge of the number of 

t For a monoclinic cell centred on the 100 face (symbol A BOO p. 223) the equatorial 
reflections alone yield an apparent projected cell-base having a b axia half the true length. 
For molecular weight determination this does not matter : the method gives tho weight 
of matter associated with each lattice point, which is either the molecular weight itself or 
a multiple of it. 



CHAP. VI 



UNIT CELL DIMENSIONS 



187 



monomer units in the unit cell. It should also be noted that the 
measured density of specimens of such substances is lower than the 
true density of the crystalline regions, on account of the presence of a 
certain amount of less dense amorphous material (Mark, 1940; Bunn, 
1942 c) ; therefore the calculated value of n (assuming M ) is always a 
little low; but if it comes to 3-8, for instance, it is obvious that there 
are really four monomer units in the cell. In such circumstances, the true 
density of the crystalline regions can be calculated from M, n, and V. 





7 ~/ 7 7 7" / 

FIG. 1 1 7. Molecules of very different shapes, packed in identical unit cells. 

4. Shapes of molecules, and orientation in the unit cell. A knowledge 
of the dimensions of the unit cell does not, by itself, lead to a knowledge 
of molecular shape, even when there is only one molecule in the cell 
and all the molecules in the crystal are therefore oriented in the same 
way. For instance, Fig. 117 shows how a projected cell of given dimen- 
sions can accommodate molecules of very different shapes. In conjunc- 
tion with other evidence, however, unit cell dimensions may lead to 
valuable conclusions on molecular structure and orientation. For 
instance, alternative formulae for a particular substance may be 
suggested on chemical grounds ; models can be made, using the known 
interatomic distances and bond angles, and these may be packed 
together to see which will fit the known unit cell. 

The evidence from optical and other physical properties (see Chapter 
VIII) is likely to be very useful in conjunction with unit cell data. It 



188 



STRUCTURE DETERMINATION 



CHAP. VI 



is often possible to form a general idea of the shape and orientation of 
molecules from such evidence ; after which the actual overall dimensions 
follow from the unit cell dimensions. 

One-molecule unit cells are, however, not common. Usually there are 
two, four, or more molecules in the unit cell, and in such circumstances 
it is necessary to discover the manner of packing ('space-group sym- 
metry' see pp. 224-52) before considering molecular dimensions. 

5. Chain-type in crystals of linear polymers. In drawn fibres of 



Z-S3A 




FIG. 118. Left: structure of molecule of polyvinyl alcohol. 
Right: structure of molecule of polyvinyl chloride. 

these substances the molecules are (in all cases so far known) approxi- 
mately parallel to the fibre axis,| and the unit cell dimension along 
the fibre axis is also the identity -period of the molecule itself. The fact 
that it is possible so simply to determine intra-molecular distances has 
far-reaching consequences. The magnitude of this identity-period may 
lead directly to a knowledge of the geometry of the chain, and some- 
times to a knowledge of the geometry of the whole molecule, including 
side-substituents. For instance, it is known that the fully extended 
zigzag form of the saturated carbon chain has an identity-period of 
2-53 A. (Bunn, 1939.) Polyvinyl alcohol ( CH 2 CHOH ) tt also has 
this same period, and therefore its chain also has the fully extended 

t In a recent paper, Fuller, Frosch, and Pape (1942) suggest that in fibres of certain 
polyesters the chains may be inclined to the fibre axis. There are, however, other possible 
interpretations of the X-ray photographs, which are suggested in the same paper; a 
meandering chain-configuration, with the general direction of the chain parallel to the 
fibre axis, is likely. 



CHAP, vi UNIT CELL DIMENSIONS 189 

zigzag form ; moreover, all the OH groups must occupy corresponding 
stereo-positions (Halle and Hofmann, 1935). The geometry of this 
molecule (Fig. 118) is therefore settled by this one measurement, 
together with the assumption that the carbon valencies are tetrahedrally 
disposed. Polyvinyl chloride has twice this period, 5-1 A., and accord- 
ingly it has a fully extended zigzag chain, but, unlike polyvinyl alcohol, 
has its chlorines in alternating stereo-positions, as in Fig. 118 (Fuller, 
1940). In a similar way, the identity-period of poly-hexamethylene 
adipamide (nylon 66) is 17-3 A. Assuming the usual interatomic 
distances (C C 1-53 A., C N 1-47 A.) and tetrahedral bond-angles, 
the chain must be a fully extended (or very nearly fully extended) 
zigzag, and the geometry of the whole' molecule is therefore approxi- 
mately settled (Fuller, 1940). 

The chains of some polymer molecules are not fully extended ; the 
identity -periods leave no doubt of that. By rotation round the single 
bonds the chains are crumpled, shortened. The magnitude of the 
identity -period may by itself indicate the geometry of the chain, but 
more probably it will not be possible to draw unambiguous conclusions 
without the aid of further stereochemical considerations. Examples of 
the use of identity -periods in conjunction with stereochemical con- 
siderations to deduce possible molecular structures for chain polymers 
are (1) the prediction (subsequently confirmed by detailed structure 
determination Bunn and Garner, 1942) of the chain-form of rubber 
hydrochloride (see p. 323), and (2) Astbury's suggestion of a possible 
structure for a-keratin on the basis of a knowledge of the repeat 
distance along the molecule and a consideration of the packing of side- 
chains. (Astbury, 1941.) It must be emphasized that such concep- 
tions, in the case of chains with long periods, are suggestions only; 
they cannot be accepted as proved unless and until detailed structure 
determination is achieved. 



Additional note to p. 169. 

t For further information on the interpretation of Weissenberg photographs, see 
Buerger (1935, 1942) and Crowfoot (1935). 



VII 

DETERMINATION OF THE POSITIONS OF THE ATOMS IN 
THE UNIT CELL BY THE METHOD OF TRIAL AND ERROR 

HITHERTO only the positions of the X-ray beams diffracted by crystals 
have been considered; unit cell dimensions are determined from the 
positions of diffracted beams without reference to their intensities. To 
discover the arrangement and positions of the atoms in the unit cell it 
is necessary to consider the intensities of the diffracted beams. 

The ideal method would be to measure these intensities, arid then 
combine them, either by calculation or by some experimental procedure, 
to form an image of the structure. Unfortunately it is usually not 
possible to proceed in this direct manner. To appreciate the reason 
for this, and to approach the whole subject in a simple way, it is useful 
to refer once more to the one-dimensional optical analogy already 
introduced at the beginning of Chapter VI (Fig. 70, Plate V). This 
experiment demonstrates the fact that the relative intensities of the 
successive orders of diffraction depend on the details of the grating 
pattern ; the problem now is how to recombine the diffracted beams to 
give an image of the original grating. The possibility of doing this is 
suggested by the fact that, if this grating were put on the microscope 
stage and illuminated by monochromatic light, diffracted beams would 
be produced ; it is these diffracted beams which are collected by the 
objective lens of the microscope. The formation of the magnified image 
in the microscope is obviously the recombination of the diffracted beams ; 
so, one would suppose, if the diffracted waves do, as an experimental 
fact, recombine, it ought to be possible to combine them by calculation. 
The difficulty here, however, is to know the phase relations between the 
various diffracted waves: to combine waves by calculation we must 
obviously know their phase relations as well as their intensities. The 
best way of thinking of the situation is to trace back all the diffracted 
waves to some one particular point in the pattern, taking this particular 
point as the origin of the 'unit cell' of the one-dimensional pattern. 
This is illustrated in Fig. 119, the chosen origin being O. Monochromatic 
light passing through the patterned grating at right angles is scattered 
at each line; interference occurs, except where the path-difference 
between waves from successive similar points in the pattern (P and Q, 
for instance) is a whole number of wave-lengths. In the upper diagram 
the path -difference is one wave-length, while in the lower diagram it is 



CHAP. VII 



POSITIONS OF THE ATOMS 



191 



three wave-lengths. When waves from P and Q are in phase (and so on 
all along the grating), then R and S are also in phase with each other ; 
but what decides the intensity of the diffracted beam js the phase- 
relationship between R and P (or 8 and Q). In both cases chosen the 
resultant diffracted beam is strong ; but the point of interest at the 
moment is that, if we choose a moment when a crest of the incident 
waves strikes the pattern, and trace 
back the resultant diffracted beams 
to the point of reference 0, we find 
that for the first-order diffraction 
there is a trough at this point, while 
for the third order there is a crest. 
In recombining the diffracted waves 
by calculation this would have to be 
taken into account; if the wrong 
phase relations were assumed, the 
wrong picture would be obtained. 
(Some other pattern would give first 
and third orders having the same 
phase at the origin.) 

Turning back to X-ray diffraction 
patterns, the problem is quite ana- 
logous ; it is more complex, because a 
three-dimensional diffraction grating 
is involved, but exactly the same in 
principle; and the difficulty is that 
we cannot determine experimentally 
the phases of the diffracted X-ray 
beams, and usually have no means 
of knowing anything about them. There are rare cases in which the 
phase relations can be deduced directly, using crystallographic evi- 
dence, and when this is so an image of the atomic structure can be 
calculated directly ; or alternatively, by substituting light waves for 
X-rays, an image can be formed experimentally; the methods are 
described in Chapter X. For the great majority of crystals, however, 
the phases are not known. It. is therefore necessary to use indirect 
methods. The method of postulating a likely structure, calculating the 
intensities of diffracted beams which this structure would give, and 
comparing these with the observed X-ray intensities, was used for all 
the earlier structure determinations, and must still be used for many 




FIG. 119. Diffraction of light by the 
patterned lino -grating of Fig. 70. First 
order (above) and third order (below). 
The resultant diffracted beam is in each 
case traced back to the point O (a centre 
of symmetry); the phases (referred to 
this point) of the first and third orders 
are opposite. 



192 STRUCTURE DETERMINATION CHAP, vn 

crystals. But although, in absence of a knowledge of the phases, atomic 
positions cannot be obtained, nevertheless an appropriate synthesis in 
which all the, beams have the same phase can give atomic vectors, 
that is, interatomic distances coupled with directions. From vector 
diagrams it may be possible to deduce some at least of the atomic 
positions ; or, in rare cases, all of them. The methods are also described 
in Chapter X. 

The present chapter deals first with all the preliminary steps which 
must be taken to obtain suitable data for structure determination 
(whether by direct or indirect methods) the measurement of the 
intensities of diffracted beams, and the application of the corrections 
necessary to isolate the factois due solely to the crystal structure from 
those associated with camera conditions. It then goes on to deal with 
the effect of atomic arrangement on the intensities of diffracted beams, 
the procedure in deducing the general arrangement, and finally the 
methods of determining actual atomic coordinates by trial. It follows 
from what has been said that, as soon as atomic positions have been 
found to a sufficient degree of approximation to settle the phases of the 
diffracted beams, then the direct method can be used ; this, in fact, is 
the normal procedure in the determination of crystal structures. 

Measurement of X-ray intensities. The method first used by 
W. H. and W. L. Bragg (1913) for the measurement of the intensities 
of X-ray reflections makes use of the fact that X-rays ionize gases, and 
the resulting conductivity is a measure of the intensity of the X-ray 
beam. The gas the inert gas argon is most suitable (Wooster and 
Martin, 1936) is contained in a chamber, to which the narrow X-ray 
beam is admitted through a fine aperture. A voltage is applied between 
an internal electrode and the wall of the chamber; the current is 
measured, and this is proportional to the intensity of the beam entering 
the chamber. The chamber is mounted on the rotating arm of a spectro- 
meter, the central table of which is occupied by the single crystal or 
block of crystal powder (Wyckoff, 1930) under investigation. By 
suitable movements of the specimen and ionization chamber, reflections 
at all angles may be explored, for one crystal zone at a time ; and the 
primary beam itself may be measured in the same way. The record 
produced a curve relating X-ray intensity to angle of reflection 
shows peaks, e$ch representing the reflection of X-rays by a different 
set of internal planes. The 'intensity* of each reflection is the integrated 
intensity, which is proportional to the area under the peak (W. H. Bragg, 
1914). This is the most direct method for the measurement of 'absolute' 



CHAP, vn POSITIONS OF THE ATOMS 193 

intensities the intensities of crystal reflections in relation to that of 
the primary beam. 

The great convenience and rapidity of photographic methods, how- 
ever, has led to their development and widespread use. Many crystal 
structure determinations have been based on the relative intensities of 
the reflections among themselves, measured photographically; it is, 
however, sometimes desirable to put the whole set of intensities on an 
absolute basis, and this too may be done photographically by com- 
paring a few of the strongest reflections with some of the reflections of 
sodium chloride (James and Firth, 1927) or anthracene (Robertson, 
1933 a). If the intensity of the X-ray beam can be kept constant, 
known exposures may be given, first to one specimen and then to 
another in the same camera, using two pieces of the same film which 
are subsequently developed together. If the intensity of the X-ray 
beam cannot be kept constant over a long period, the safest method is 
to use a special camera in which the two crystals are admitted alternately 
into the beam. Wooster and Martin (1040) have designed a two- 
crystal Wcissenberg goniometer for this purpose. 

The procedure in the photometry of X-ray photographs depends on 
the type of photograph and the type of photometric equipment avail- 
able. It is simplest for powder photographs. On the margin of the film 
(which was shielded from X-rays during the taking of the difiraotion 
photograph) is printed a calibration strip consisting of a row of patches 
exposed to X-rays for known relative times. By measuring tho light 
transmission of each patch on a microphotometer the relation between 
X-ray exposure and light transmission is established; arid since tho 
calibration strip is printed on the same film as the photograph, and thus 
passes through the same development process, any possible errors dno 
to variation of film characteristics or development conditions are a voided . 
The series of patches may be obtained conveniently by means of a brass 
sector wheel rotating in front of a slit in a brass piate (Fig. 120 ft). For 
visible light such a method cannot be used, but for X-rays it is sound 
(Bouwers, 1923). Alternatively, a strip showing a continuously varying 
opacity may be printed by using an appropriately shaped cam in place 
of the sector wheel (Fig. 120 6) ; in this case the distance alon<r the strip 
indicates the X-ray exposure. In taking a photometer record of a 
powder photograph each arc is traversed. The intensity of each reflec- 
tion is proportional to the area under the peak, which can be rapidly 
measured by the use of a planimeter. In th* simplest typo of photo- 
electric photometer (Jay, 1941) the light transmission tit any point on 

445S 



194 



STRUCTURE DETERMINATION 



CHAP, vn 



the film is proportional to a galvanometer deflexion. In the better 
types of photometer the light transmission through the film is balanced 
against a known light intensity adjusted by an optical wedge. 

For single-crystal photographs the requirements are different. It is 
necessary to obtain the integrated intensity over the whole of each spot. 
In B. W. Robinson's photometer (1933 b ; also Dawton, 1937) the cali- 
bration strip itself (of the continuously varying type) is used as an 
optical wedge, against which the light transmission through any point 



SCREEN 




FIG. 120. Arrangements for printing intensity calibration strips on X-ray photographs. 
a. Sector wheel to give a stepped wodgo. 6. Cam to give a continuous wedge. 

on a spot is balanced by means of a pair of opposed photocells coupled 
to an amplifier and galvanometer. Each spot is traversed at several 
levels and the readings at a number of points all over the spot are added 
mechanically. The X-ray intensities measured by this instrument com- 
pare very well with those given by the ionization spectrometer. The 
measurement of the intensities of several hundred spots is, however, a 
lengthy task ; to accelerate such work Dawton has devised a photometer 
in which each spot is scanned in the manner used in television, the 
integrated intensity being given as a single galvanometer reading 
(Robertson and Dawton, 19*41). 

Another rapid method, also developed by Dawton (J938), is to print 
a positive from the X-ray 'negative'. When suitable photographic 
materials are used, the light transmission through a whole spot on the 
positive film is proportional to the integrated X-ray intensity. The 
merits of the different photographic methods have been summed up 
by Robertson and Dawton (1941). For structure determination, great 



|,)|-;;^pj||f;i 



; '\ .' : 

(-' 




100-110 200 211 



NH 4 CL 



221 311 320 400 411 420 332 
300 330 



101 111 211 002 301 321 222 312 




110 200210 220310202 400330 



112 301 







Fia. 121. Powder photographs of NH 4 C1 (room temperature form), TiOj (rutile), and 
urea O=C(NH 2 ) 2 . (All taken with copper Kot radiation.) 



CHA*. vii POSITIONS OF THE ATOMS 195 

accuracy in the measurement of intensities is usually not required, and 
it is probable that the positive film method, which does not require 
specialized apparatus, is adequate for most purposes. 

Many of the simpler structures have been solved by consideration of 
the relative intensities estimated visually. The intensities of X-ray 
reflections are very sensitive to small changes of atomic positions; 
comparatively small movements of atoms mean large changes in the 
relative intensities of the various reflections. Consequently, by adjust- 
ing postulated atomic positions until mere qualitative agreement 
between calculated and observed intensities is attained that is, the 
arrangement of the calculated intensities in order of strength is the 
same as that observed on the photographs a surprisingly good approxi- 
mation to the truth can be achieved. Visual estimates may even be 
used for moderately complex structures ; comparison of reflections with 
calibration spots of known relative intensity is a method capable of 
yielding a set of reflection intensities suitable for all but the most 
precise investigations (Hughes, 1935). 

Whatever photographic method is used for estimating intensities, it 
will be found that the range of intensities is far too great to be recorded 
satisfactorily on a single film : an exposure suitable for recording weak 
reflections at convenient strength would show the strongest reflections 
so opaque that measurement would scarcely be possible. A suitable set 
of films may be obtained in a single X-ray exposure by placing several 
films one behind the other in the camera ; successive photographs are 
related by a constant exposure ratio of about 2:1. Much additional 
information on the measurement of intensities by photographic methods 
can be found in a paper by Robertson (1943). 

Calculation of intensities. Preliminary. Each spot or arc on an 
X-ray diffraction photograph may be regarded as the 'reflection* of 
X-rays by a particular set of parallel crystal planes. The intensity of 
this reflection is controlled by several factors the diffracting powers 
of the atoms, the arrangement of the atoms with regard to the crystal 
planes, the Bragg angle at which reflection occurs, the number of 
crystallographically equivalent sets of planes contributing towards the 
total intensity of the spot or arc, and the amplitude of the thermal 
vibrations of the atoms. In any powder photograph for instance, that 
of ammonium chloride (Fig. 121, Plate X) two features immediately 
strike the eye ; firstly, there is a general diminution of intensities with 
increasing reflection angle, and secondly, the intensities vary from one arc 
to the next in an apparently irregular manner. The general diminution 



196 STRUCTURE DETERMINATION CHAP, vn 

of intensities with increasing reflection angle is due to a decrease in 
the diffracting powers of atoms with increasing angle 0, to the polari- 
zation of the X-rays on reflection (to a degree depending 1 again on 0), 
to a geometrical factor, and to the thermal vibrations of the atoms. 
The apparently irregular variation of intensity from one arc to the next 
is due to the effect of the relative positions of the atoms in space the 
'structure factor' and to the variation in the number of equivalent sets 
of planes contributing to the spot or arc a number which depends ,on 
the type of plane. It is the structure factor in which we are chiefly 
interested, but in order to isolate it we must allow for all the other 
factors. Each factor will now be considered. 

The diffracting powers 'of atoms. X-rays are diffracted, not by 
the positively charged core of an atom, but by the cloud of electrons 
forming the outer parts of the atom. The diffracting power of an atom 
is determined, in the first place, by the number of electrons surrounding 
the central nucleus, that is, by the atomic number of the element its 
place in the periodic table. Atoms such as iodine and lead, which have 
high atomic numbers, have much higher diffracting powers than those 
like sodium and oxygen, which have low atomic numbers; in fact, at 
small angles the diffracting power of an atom is proportional to the 
number of electrons in the atom. In the ammonium chloride crystal 
the ammonium ion NH^" may be treated as a single entity ; the hydrogen 
nuclei (protons) are embedded in the electron cloud which includes the 
electrons from both nitrogen and hydrogen atoms, minus one which is 
given up to the chlorine, f The number of electrons in NH^f" is thus 
7+41 = 10. Since the Cl- ion has 17+1 = 18 electrons, the diffract- 
ing power of Cl~ is nearly twice that of NH^ ; if the ions behaved as 
scattering points, or if the variation in diffracting power with angle (see 
later section on 'angle factors') were the same for both ions, the ratio 
would be exactly 18/10. Actually, the diffracting powers of the two ions 
vary with the angle of diffraction in slightly different degrees ; but the 
difference is not great, and in fact the ratio of diffracting powers is never 
far from 18/10. 

The structure amplitude, F. In an ammonium chloride crystal 
the unit 'Cell is a cube containing one NH^" and one Cl- ion. If the 
centre of a chlorine ion is taken as the corner of the unit cell, then the 
ammonium ion lies in the centre of the cell (Fig. 122). 

Consider the reflection of X-rays by the 001 plane of the crystal ; it 

f Also, at room temperature, the whole ion is rotating and in effect has spherical 
symmetry. 




CHAP, vn POSITIONS OF THE ATOMS 197 

is this reflection, together with the exactly similar reflections by the 
100 and 010 planes, which gives rise to the first arc on the powder photo- 
graph (Fig. 12 1 , Plate X) , an arc of moderate intensity. The 00 1 reflection 
is produced when X-r&ys from one plane of chlorine ions (M in Fig. 123) 
are exactly one wave-length behind those from the next plane of chlorine 
ions N. But when this happens, waves from ammonium ions in plane 
P must be exactly half a wave-length behind those from the chlorine 
ions N, since the ammonium plane P is exactly half-way between the 
chlorine planes M and N. Waves from 
ammonium ions are thus exactly opposite 
in phase to those from chlorine ions, and 
this is true throughout the crystal. Inter-* 
ference will occur, but the intensity of 
the resultant diffracted beam will not be 
zero, because the diffracting power of the 
ammonium ion/ NH< is little more than half 
that of the chlorine ion / C1 , and the am- 
plitude of the resultant wave (= /CI~/NH.) 
is thus reduced to slightly less than half 
what it would be if chlorine ions alone FIG. 122. Structure of ammonium 
were present. The intensity of a beam is, chloride - ( The sizes of the spheres 

f < r ,. , i i are arbitrary ; they do not repre- 

for an imperfect f crystal such as ammo- 80nt effective packing sizes.) 
nium chloride, proportional to the square 

of the amplitude of the waves, hence the intensity for 001 is less than 
one-quarter what it would be for chlorine ions alone. 

But now consider reflections by planes of type 101 (such as 10T, 110, 
Oil, and so on), which give rise to the second arc a very strong one. 
Here, the ammonium ions lie on the same planes as the chlorine ions 
(Fig. 123), the 101 reflection being produced when waves from plane L 
are one wave-length behind waves from plane K ; the waves from all 
the ions in the crystal are therefore in phase with each other, and the 
resultant amplitude (/ C i+/NH 4 ) is about l\ times what it would be for 
chlorine ions alone ; the intensity is therefore about twice what it would 
be for chlorine ions alone. The intensity of a 101 reflection is thus 
something like eight times as great as that of a 100 reflection. This, 
then, is the principal reason why the second arc on the photograph is 
so very much stronger than the first. (There is another reason, but this 
will be considered later.) 

The third arc is composed of reflections from 111 planes. The sheets 

t See p. 207. 



198 



STRUCTURE DETERMINATION 



CHAP. VII 



of chlorine ions which define these planes are interleaved by sheets of 
ammonium ions (Fig. 124) ; hence, the situation is the same as for 001 : 
waves from the ammonium ions oppose those from the chlorine ions, 
and a weak reflection is the result. 




FIG. 123. Reflection of X-rays by 001 and 101 planes of ammonium chloride. 




FIG. 124. Ill planes of ammonium chloride. 

The fourth arc consists of 200+020+002. These reflections occur 
when waves from chlorine plane M (Fig. 125) are two wave-lengths 
behind those from the next chlorine plane N ; but, since the ammonium 
ions in plane P are half-way between planes M and N 9 waves from P 
j&re one wave-length behind those from N and one wave-length in front 



CHAP, vii POSITIONS OF THE ATOMS 199 

of those from M ; therefore they are in phase with those from M and N, 
and a strong reflection is the result. 

A rough idea of the relative intensities of all the reflections may be 
gained in this way. On account of the position of the ammonium ions 
in the centres of the unit cells defined by the chlorine ions, it is found 
that, for all planes in which the sum of the indices A+i-f-Z is even, waves 
from ammonium and chlorine ions are in phase with, and therefore 
co-operate with, each other, giving a strong reflection, while for all 
planes having A+&+Z odd, waves from ammonium ions oppose those 
from chlorine ions, and the reflection is therefore relatively weak. 




FIG. 125. Reflection of X-rays by 002 planes of ammonium chloride. 

The ammonium chloride crystal forms a particularly simple example 
of the effect of atomic arrangement on the intensities of the various 
reflections. The structure amplitude will be treated more generally in 
a later section. 

The number of equivalent reflections, p. The statement just 
made is true in a general way, but it may be noted (in Fig. 121, Plate X) 
that, for instance, 211 (the sixth arc) is stronger than 002 (the fourth 
arc), though h+k+l is even for both. This is because there are more 
planes of type 211 than there are of type 002. There are only three 
different planes of type 002 namely, 002, 020, and 200 ; a crystal turned 
to all possible orientations would give reflections from this type of plane 
for six different orientations with respect to the X-ray beam (002 and 
002 being reflections in opposite directions from the same plane). But 
there are twelve different planes of type 211 namely 211, 121, 112, 
2lT, 2Tl, 2TT, T21, 121, T2T, Tl2, lT2, TT2, and therefore twenty-four 
reflections. Stated generally, there are six reflections of type AOO, twelve 
of type hhQ, twenty -four of type hkQ, eight of type hhh, twenty-four of 
type hhl, and forty-eight of type hkl. In a powder there are crystals 
oriented in all possible ways ; the number of them which happen to lie 
in a position suitable for reflection by a particular type of plane is 



200 



STRUCTURE DETERMINATION 



CHAP. VII 



evidently proportional to the number of differently oriented, crystallo- 
graphically equivalent planes. 

In Table III, the numbers of equivalent reflections in crystals of all 
possible symmetries are set down. These members are applicable to 
powder photographs, in which all planes having the same spacing give 
reflections on the same arc. For single-crystal normal-beam rotation 
photographs the situation is different: planes parallel to the axis of 
rotation give reflections in two places on the equator of the photograph, 
one each side of the meridian, while the reflections from other planes 
are distributed among four positions, one in each quadrant. The multi- 
plicities for single-crystal rotation photographs are therefore given in 
another Table (IV). These multiplicities depend on the symmetry of the 
axis round which rotation occurs ; they are obvious by inspection in 
tilted crystal photographs. It should be noted that in some cases there 
are two or more sets of reflections having the same spacing but different 
intensities ; this depends on the 'Laue-symrnetry' of the crystal see 
p. 241. These features also are obvious in tilted crystal photographs. 

TABLE III 
Numbers of equivalent reflections (multiplicity) 



Point yron[) symmetry '-Fypv f reflection, and multiplicity^ 


Tricliilic: 1,1 All types: 2 


j 


Monoclinio: OA-0 hdl \ hkl 






nt t 2, 2/w 2 2 ; 4 






Orthorhombic: /tOO, OA*0, OOJ 


MO, OJM, hOl 


hkl 


mm, 222, minm 2 - 


4 


8 


Tetragonal: GO/ MO, MO 


MO 


OA-0 


hid 


hkl 


4, 4, 4/w* 2 i 4 


4+4 


8 


8 


8 + 8 


42m, 4//1M, 42, 4///zmm 2 4 


8 


8 


8 


16 


Trigonal and hexagonal : , 001 /eOO, MO 


MO 


OM 


hhl 


hkl 


3, 3 26 


6 + 6 


6 + 6 


6 + 6 


6+6+6+6 


3m, 32, 3w 2 


12 


6 + 6 


12 


12 + 12 


6, 6, 6/m , 2 6 


6+6 


12 


12 


12+12 


62m, 6mm, 62, Gjnnnm 2 j 6 


12 


12 


12 


24 


Cubic: ! AOO MO i MO 


Mth 


hhl 


hid 




23, m3 6 12 1 12 + 12 


8 


24 


24+24 




43m, 43, m3m 6 12 } 24 


8 


24 


48 





| Where the multiplicity is given as, for example, 6+6, this signifies that there are 
two sot& of reflections at the same angle but having different intensities. 

Angle factors. The two factors already mentioned the crystal 
structure amplitude and the factor for the number of similar planes 
give a general idea of the reason for the variation of intensity from one 



CHAP. VII 



POSITIONS OF THE ATOMS 



201 



arc to another in the powder photograph of ammonium chloride. We 
have now to consider the general diminution with increasing angle. 

TABLE IV 



Laue 
symmetry 


Axis 
vertical 


Maximum 
equatorial 
multiplicity^ 


Maximum 
layer -line 
multiplicity^ 


I 




2 


1 


2/m 


Twofold 


2 


2 


mmm 


Twofold 


4 


4 


3 


Threefold 


6 + 6 


3+3+3+3 


3m 


Threefold 


12 


6+6 


4/m 


Fourfold 


4 + 4 


4+4 


4/ww 


Fourfold 


8 


8 


6/m 


Sixfold 


6+6 


6+6 


6/m?nm 


Sixfold 


12 


12 


m3 


Twofold (c) 


4+4 


4+4 








(8 for hkk'a 








and hhVa) 


w3w 


Fourfold 


8 


8 


i 







f" Whore the multiplicity is given as, for example, 4+4, 
this signifies that there are two sets of spots on the same 
layer line and at the same Bragg angle, but having different 
intensities. 

Atoms in crystals cannot be regarded as scattering points; the 
'diameter* of the electron cloud of an atom is of the same order of size 
as the distance between the centres of adjacent atoms in fact, to a 




Fio. 126. Weakening of reflection by out-of-phase waves 
diffracted by outer regions of atoms. 

first approximation, the atoms in many crystals may be regarded as 
spheres of definite radius in contact with each other ; the electron clouds 
of adjacent atoms may be regarded as just touching; each other. The 
consequences of this are illustrated in Fig. 126, in which (in the forma- 
tion of a particular reflection) waves diffracted by electrons on the 
equatorial plane BB of one atom are one wave-length behind those 
diffracted by the equatorial electrons AA of the next atom ; in these 
circumstances, waves diffracted by outer electrons CC and DD of these 
atoms oppose, to some extent, those from A A and BB, and therefore 



202 



STRUCTURE DETERMINATION 



CHAP. VII 



reduce the intensity of the resultant diffracted beam. The reduction of 
intensity is not considerable in the circumstances mentioned, because 
the electron density in the outer regions of an atom is law compared 
with the density near the centre, the electron cloud being in effect a 
sort of diffuse atmosphere having maximum density in the inner regions 
and a low density in the outer regions ; but for the higher order diffrac- 




FIG. 127. Diffracting powers of a few common atoms. 

tions (that is, 'reflections' from closely spaced planes), when the phase 
difference for waves from the inner regions of adjacent atoms is several 
wave-lengths, waves from regions of not very dissimilar electron densities 
interfere with each other, and the intensity of the resultant diffracted 
beam is therefore much reduced. The apparent diffracting power of an 
atom is evidently dependent on the spacing of the reflecting planes ; it 

is usually given as a function of -r ( = ^ 1, and the diffracting powers 

A \ 2d) 

of all atoms (symbolized /) for a wide range of ^- are to be found in 

A 

Int. Tab. A few are shown in Fig. 127. For many atoms, diffracting 



CHAP, vn POSITIONS OF THE ATOMS 203 

powers have been deduced from the measured intensities of reflections 
from crystals whose structures are firmly established (James and 
Brindley, 1931), but it is also possible (Hartree, 1928) to calculate the 
values from the electronic structures of atoms, and these calculated 
values agree well with the experimental values (James, Waller, and 



Hartree, 1928). The units used for / are such that for = 0, / is 

A 

equal to the number of electrons in the atom. For ionized atoms the 
number is increased or decreased by the magnitude of the charge. In 
polyatomic ions such as NO^" the constituent atoms are charged ; the 
appropriate figures to be used in calculations, for the nitrite ion taken 
as example, are those for O~ 2 and N+* 1 . It should be noted that the 
figures given in the tables are not valid when appreciable X-ray fluor- 
escence occurs ; fluorescence, due to the absorption of X-rays followed 
by re-emission as longer waves characteristic of the atoms, is intense 
when the wave-length of the incident X-rays is slightly shorter than Kf$ 
for the atoms, and in these circumstances the intensity of the diffracted 
beam is naturally reduced. 

Another cause of diminution of the intensities of X-rays with increas- 
ing angle of reflection is the polarization which occurs on reflection. 
The intensity of any reflection is reduced by this effect to the fraction 
1+ cos 2 20 
2 * 

Yet another angle factor is what is known as the Lorentz factor L, 
which expresses, for rotating crystal photographs, the relative time any 
crystal plane spends within the narrow angular range over which reflec- 
tion occurs. When a perfect crystal turns slowly through the reflecting 
position, reflection occurs only over a range of a few seconds of arc. 
Most crystals are not perfect, and reflect over a range of some minutes 
or even as much as half a degree (Bragg, James, and Bosanquet, 1921), 
but the reason for this is that different portions of the lattice are not 
quite parallel to each other ; this spread may be regarded as part of the 
rotation of the crystal, and is irrelevant to the present point, which 
concerns only the angular range within which the small perfect sections 
of the lattice reflect. In terms of the conception of the reciprocal lattice, 
each 'point' of the reciprocal lattice has a finite size, and as the reciprocal 
lattice rotates through the sphere of reflection, each 'point' spends a 
finite time passing through the surface of this sphere. This factor varies 
with the distance of the reciprocal lattice point from the origin, which 
is of course related to the angle of reflection. The Lorentz factor varies 



204 STRUCTUKE DETERMINATION CHAP, vn 

with the type of photograph. For the equatorial reflections on a normal 

rotation photograph, it is . - (Darwin, 1922). For other layers on 

sin 2i\j * 

normal rotation photographs, it is . nf . x ~ r/ z~r ~-^ where A 

sm20 ^/(cos 2 < sin 2 0) 

is the angle between the reflecting plane and the axis of rotation ; it 
increases the strength of spots on upper and lower layers in comparison 
with those on the equator, the increase being greatest near the meridian 
of the photograph; Cox and Shaw (1930) give a chart showing this 
function for all positions on a normal rotation photograph. f (The same 
paper gives corrections for the obliquity of incidence of X-rays on non- 
equatorial positions on a cylindrical film.) Expressions for rotation 
photographs taken with the beam not normal to the axis of rotation of 
the crystal are given by Tunell (1939) (for equi-inclination Weissenberg 
photographs) and Bouman and De Jong (1938) and Buerger (1940) (for 
the general inclination used in the De Jong and Bouman camera). 

For powder photographs it is also necessary to take into account the 
fact that all the reflected beams from all the little crystals are spread 
over a cone which is narrow for reflections at small angles but much 
wider for reflections at larger angles, when 26 is near 90. The fraction 
of intensity per unit length of arc (which decides the degree of blacken- 
ing of the film) is thus less at the larger angles than at the smaller ones. 
The cones are smaller again for "back reflections' at Bragg angles 
approaching 90 (that is, angles of reflection 29 approaching 180), so 
that here again there is a greater fraction of intensity per unit length of 
arc. The effect of this factor at large angles can often be seen on powder 
photographs as a tendency towards increasing intensity at the very ends 
of the film where the Bragg angle is approaching 90. (See, for instance, 
Fig. 63, Plate III.) To account for this eflFect, the expression for the 

intensity must be multiplied by the factor -^ ^. 

sm 2 0cos0 

Thermal vibrations . Atoms in crystals vibrate at ordinary tempera- 
tures with frequencies very much lower than those of X-rays ; at any 
one instant, some atoms are displaced from their mean positions in one 
direction while those in another part of the crystal are displaced in 

t This expression and tho chart mentioned imply that, for a spot on the meridian 
(tf> = 90 8), the intensity is infinite which is absurd. This arises from the fact that an 
integration occurring in the derivation of the expression implies that the angular range 
over which reflection occurs is vanishingly small an assumption which is not quite true. 
The point is of no practical importance, for the chance of a spot occurring sufficiently 
near the meridian to render the expression inaccurate is very small. 



CHAP, vii POSITIONS OF THE ATOMS 205 

another direction ; consequently diffracted X-rays which would be 
exactly in phase if the atoms were at rest are actually not exactly in 
phase, and the intensity of the diffracted beam is thus lower than it 
would be if all the atoms were at rest. For crystal planes of large spacing 
(those giving reflections at small angles), the thermal displacements of 
the atoms are small fractions of the plane-spacing, and therefore do 
not affect the intensities much ; but for the more closely spaced planes 
(those giving reflections at the larger angles) the atomic displacements 
may be comparable with the plane-spacing, and therefore the intensities 
of these reflections may be much reduced. The effect is thus greater, 
the larger the angle of reflection ; and it naturally increases with rising 
temperature X-ray diffraction patterns taken at high tempera- 
tures are always weaker than those of the same substance at low 
temperatures. 

Note that it is largely in consequence of considerable thermal vibra- 
tions that the diffraction patterns of crystals of many organic substances 
(taken at room temperature) fade away to nothing at Bragg angles of 
50-60 (using copper Kot radiation, A = 1-54 A), while those of inorganic 
salts and metals which are far below their melting-points show strong 
reflections over the whole range to 90, because the thermal vibrations 
are very small compared with the interatomic distances. 

The ratio T between the actual intensity of a diffracted beam and the 
intensity which it would have if there were no thermal vibrations is 

f ~2#( ) > where B is a constant for a particular crystal. B is related 



to the amplitude of vibration of the atoms by the expression B = 
where u 2 is the mean square amplitude; it can usually be estimated 
only approximately (see Int. Tab., p. 569), but this need not deter us 
from quantitative study of diffraction patterns, since an inaccurate 
estimation of B would only mean that the intensities of the reflections 
fall away with increasing Bragg angle rather more slowly or more 
rapidly than was expected. In practice it is found that uncertainty in 
the value of B does not lead to appreciable doubt about the interpreta- 
tion of X-ray diffraction patterns. A typical value of B for an ionic 
crystal is 1-43 X 10- 16 for Nad (Int. Tab., p. 570). 

The use of the above expression implies that all the atoms vibrate 
with equal amplitudes. This is not strictly true : for instance, in sodium 
chloride the amplitude of vibration of the lighter sodium ion is greater 
than that of the chlorine ion ; and in general, thermal vibrations must 
be different for every crystallographically different atom in a unit cell, 



206 STRUCTURE DETERMINATION CHAP, vii 

since they depend on the surroundings of the atom as well as on its 
inertia. 

Instead of making a separate correction for B it is often possible to use 
experimentally determined values for the diffracting powers of atoms, 
which include the temperature effect as an additional diminution of the 
apparent diffracting power with increasing angle of reflection. Empirical 
diffracting powers for elements occurring in silicate crystals are given by 
Bragg and West (1929) ; and for hydrocarbons by Robertson (1935 a). 

Another assumption implied in the use of the expression given above 
(or the empirical diffracting powers referred to in the last paragraph) 
is that the thermal vibrations of the atoms have the same magnitude in 
all directions in the crystal, lliis is not strictly true ; it is a sufficient 
approximation to the truth for many crystals, but there are cases in 
which the vibrations are markedly anisotropic. For instance, the vibra- 
tions of long-chain molecules are almost entirely perpendicular to their 
long axes, while flat molecules vibrate chiefly in a direction normal to 
the plane of the molecule. f Furthermore, it has been found that some 
crystals give diffuse 'extra' reflections which are undoubtedly due to 
the thermal vibrations of the atoms. In terms of the conception of the 
reciprocal lattice, the reflecting power, which hitherto has been assumed 
to be confined to the points of the reciprocal lattice, is actually to some 
extent spread in varying degrees along the principal lines of the 
reciprocal lattice. It is as if the diminution of reflecting power referred 
to at the beginning of this section is not lost, but reappears along the 
lines of the reciprocal lattice, giving rise to extra spots and streaks on 
single-crystal photographs. A valuable summary of work on this sub- 
ject up to 1942 is given in a paper by Lonsdale (1942). 

Absorption . The effect of absorption of X-rays in a powder specimen 
is to diminish the intensities of reflections at small angles much more 
than those of the 'back reflections' (see Fig. 68). Corrections can be 
calculated for cylindrical specimens of known diameter (Bradley, 1935), 
these corrections being valid also for cylindrically shaped single crystals. 
For crystals of natural shape completely bathed in the X-ray beam, it 
is possible to calculate absorption corrections (Hendershot, 1937 a; 
Albrecht, 1939), but the calculations are laborious, and it is much better, 
if possible, to reduce single crystals to cylindrical shape. For rod- 
shaped crystals this may sometimes be done by rolling them on a 
ground-glass plate ; for soluble substances it may be possible to adjust 

t For the effect of anisotropic thermal vibrations on the relative intensities of 
X-ray reflections, see Helmholz (1936) and Hughes (1941). 



CHAP, vii POSITIONS OF THE ATOMS 207 

the crystal on the goniometer, and then rotate it while holding against 
it a fine paint brush charged with solvent. 

To keep absorption corrections as low as possible it is best, when 
working with strongly absorbing substances, to use smaller crystals 
than when working with transparent substances. But a limit is set by 
the increase in exposure times as well as by difficulties of manipulation: 
a suitable size for a strongly absorbing crystal is l/20th to l/10th mm. 

Bradley (1935) points out that, since the effect of the absorption 
factor is opposite to that of thermal vibrations, the two may in some 
cases cancel each other approximately ; consequently it may be justifiable 
to ignore both factors. This naturally applies only to crystals of moderate 
or high absorption ; it does not apply to most organic substances, for 
which absorption is small and thermal vibrations large, so that the 
effect of *the latter far outweighs the absorption effect. 

Complete expression for intensity of reflection. Perfect and 
imperfect crystals. If relative intensities are being calculated, it is 
sufficient to multiply the structure amplitude (which is treated generally 
in the next section) by all the correction factors mentioned. Thus, for 
a powder photograph, the intensity of each arc is proportional to 

14- cos 2 26 
F 2 p -v-gTj - * X temperature factor (T) X absorption factor (A), 

while for ar normal-beam single-crystal rotation photograph, the in- 
tensity of each spot is proportional to 



/l+cos 2 2/A/ cos6 \ 
*\ sin 26 /\V(cos^-sin 2 0)/ 



The advantage of using of absolute intensities, especially for complex 
structures, has been urged by Bragg and West (1929). If absolute 
intensities are being calculated, the following expressions must be used : 
(1) For powder photograph on cylindrical film, radius r. 

If diffracted energy in a length of arc I = P' and energy of primary 
beam per second per unit area = / , 

P' __N*eWlV p2 l+cos 2 20 
/ 327rm 2 cV P sin 2 cos 6 ' 

where JV = number of unit cells per unit volume, 
e, m = electronic charge and mass, 
A = X-ray wave-length, 
V = volume of powder in the beam, 
c = velocity of light. 



208 STRUCTUBE DETERMINATION CHAP, vn 

(2) For normal-beam single-crystal rotation photograph, using crystal 
of volume V completely bathed in X-rays : 



p -= 



cos0 \ ' 

2 ^~-siii^)/ ' 



sin 2(9 /y 
where p is the integrated reflection, defined thus : 

p = Eaj/I , where E is the total energy in a given reflected beam 
when the crystal has been rotating for time r at constant angular 
velocity o>. Some crystal structure determinations (notably those of 
complex silicates worked out by W. L. Bragg and his school) have 
been based on measurements of reflections from the faces of crystals 
much larger than the primary beam, or reflections transmitted through 
crystal sections. Formulae appropriate to these experimental con- 
ditions will be found in Int. Tab. 

These expressions containing F 2 are valid only for 'ideally imperfect* 
crystals, to which class most known crystals belong. It is a curious 
fact that really perfect crystals like certain diamonds, in which all 
portions of the lattice are parallel to a high degree of precision, give an 
integrated intensity which is proportional directly to .F (not to its 
square), and is thus smaller than that given by an imperfect crystal of 
the same substance. 

If a perfect crystal is turned slowly through the reflecting position, 
using an extremely narrow X-ray beam, reflection occurs only over a 
range of a few seconds of arc (Allison, 1932). Reflection, when it occurs, 
is total the whole beam is reflected but this happens over such a 
small angular range that the integrated reflection (which is always 
measured in crystal structure determination) is less than that given by 
an imperfect crystal which reflects less strongly, but over a much wider 
angular range. 

Most actual crystals are imperfect ; different portions of the lattice 
are not quite parallel, and the crystal behaves as if it consisted of a 
number of blocks (of the order of 10~ 5 cm. in diameter) whose orientation 
varies over several minutes or even in some cases up to half a degree. 
This imperfection is perhaps connected with the manner of growth in 
thin layers (see Chapter II and Fig. 8, Plate I) ; each layer might be 
slightly wavy, and there may be cracks or impurities between the layers. 
Most crystals are imperfect in this way, and in structure determination 
it is usually safe to assume that the intensity of any reflection is propor- 
tional to the square of the structure amplitude. 

The intensities of crystal reflections are in some circumstances 



CHAP, vii POSITIONS OV THE ATOMS 209 

reduced by effects known as primary and secondary extinction. If the 
crystal is not 'ideally imperfect' but consists of rather large lattice 
blocks, the Intensities of the reflections are proportional to a power of 
F between 1 and 2 ; this is 'primary extinction'. 'Secondary extinction' 
affects only the strongest reflections and is due to the fact that the 
top layer of a crystal (the part nearest the primary beam) reflects away 
an appreciable proportion of the primary beam, thus in effect partially 
shielding the lower layers of the crystal ; the strongest reflections are 
therefore experimentally less strong than they should be in comparison 
with the weaker reflections. The relation between the actual intensity 
p and the intensity p which would be obtained if there were no secondary 
extinction is , 

P = r4?- 

where g is a constant the 'secondary extinction coefficient' (Bragg 
(W. L.), James, and Bosanquet, 1021, 1922; Bragg (W. L.) and West-, 
1929). 

Both primary and secondary extinction effects may usually be avoided 
by powdering a crystal. For this and other reasons the intensities of the 
arcs on powder photographs arc likely to be more reliable than those 
of other types of photograph ; but in practice, in structure determina- 
tion it is only possible to use 'powder intensities' alone for very simple 
structures ; for complex crystals reflections from different planes over- 
lap seriously. 

In most structure determinations small crystals 0-1-0-5 mm. across 
are now used. Primary extinction is rare and not likely to be en- 
countered, while secondary extinction for crystals of this size is usually 
not serious. 

It should be remembered that the strongest reflections which arc most 
seriously affected by secondary extinction occur at small angles and 
are less likely to overlap on powder photographs ; therefore it may often 
be best to measure the intensities of the small-angle reflections on a 
powder photograph and the rest on single-crystal photographs. Another 
useful procedure is to measure the strongest reflections on both powder 
and single-crystal photographs, and by comparing them (assuming 
the powder results to be free from extinction effects) to estimate the 
secondary extinction coefficient which can then be applied to the 
single-crystal results (Wyckoff, 1932; Wyckoff and Corey, 1934). 

General expression for the structure amplitude. We are inter- 
ested primarily in the arrangement of the atoms in crystals and the 



210 STRUCTURE DETERMINATION CHAP, vn 

effect of the arrangement on the intensities of diffracted X-ray beams. 
A general idea of the effect of atomic arrangement on the intensities of 
various reflections of a very simple crystal has been presented in an 
earlier section. For this crystal, ammonium chloride, the phase relation- 
ships between the waves from the two types of ions are very simple ; 
the waves from ammonium ions are, for every type of crystal plane, 
either exactly in phase with those from chlorine ions or exactly opposite 
in phase, owing to the position of the ammonium ion in the exact 
centre of the cube defined by the chlorine ions. 

In most crystals, however, the coordinates of some or all the atoms 
are not simple fractions of the unit cell edges, and the phase relation- 
ships between waves from different atoms are therefore not simple. 
Consider, for instance, the atomic arrangement in rutile, one of the 
crystal forms of Ti0 2 (Vegard, 1916). The unit ceU (Pig. 128 a) is 
tetragonal (a = 4-58 A, c = 2*98 A) and contains two titanium and 
four oxygen atoms. Titanium atoms are at the corners and centres of 
the cells (coordinates 000 and |H respectively) ; oxygen atoms lie on 
the base diagonals and at similar positions half-way up the cells, the 
coordinates being (1) 0-31a, 0-316, 0*0c, (2) 0-31a, 0-316, 0-Oc, 
(3) ~_0-19a, +0-196, 0-50c, (4) + 0-19a, 0-196, 0-50c. A powder 
photograph of this substance, taken with copper KQL radiation, is shown 
in Fig. 121, Plate X. Consider the intensities of some of the reflections. 

First of all, no reflection from 001 appears on the photograph; the 
reason is that when waves from plane P (Fig. 128 a) are one wave-length 
behind those from plane M, then waves from plane N are half a wave- 
length behind those from plane M, and are thus exactly opposite in 
phase ; and since there is exactly the same combination of . atoms on 
N as on M (one Ti and two per unit cell), the waves from N exactly 
cancel out those from M ; and so on throughout the crystal. Reflection 
002, however, does appear on the photograph, because a phase-difference 
of two wave-lengths between waves from M and P means a phase- 
difference of one wave-length between waves from M and N, and thus 
all the waves co-operate. The intensity of 002 is not very great because 
there are only two reflections of this type (002, 002). The 100 reflection 
is absent because the 100 planes of Ti atoms are interleaved (at exactly 
half-way) by exactly similar planes of Ti atoms (see Fig. 128 6), which 
produce waves of exactly opposite phase and the same is true for the 
oxygen atoms. 

For 110 (see Fig. 128 c), observe first that waves from all Ti atoms 
are in phase with each other, and that those from two of the oxygens 



CHAP. VII 



POSITIONS OF THE ATOMS 



211 



(3 and 4) are also in phase, but that the waves from the other two (1 
and 2) are not in phase. The last-mentioned waves are not exactly 
opposite in phase to the rest, however; the phase-difference is 



c=2-98A 



liQ) 



4 [a ^W) J f 



f /00 fa 



7>r>; 




/OO ABSENT 




Ti(1)+Ji(2) 
0/3) 




'Ti(1) ~ Vy ^00 JVfyA/C 

FIG. 128. Intensities of some &&0 reflections of rutile, TiO 2 

( 5 /d no ) X 360, which for 0(1) is 0-62 x 360 and for 0(2) is 0-62 X 360. 
If the waves are represented graphically as in Fig. 128 c, the resultant 
amplitude (shown by the thicker line) is obtained by adding the 
ordinates. It is evident that for this particular reflection the partly- 



212 STRUCTURE DETERMINATION CHAP, vn 

out-of-phase waves do not diminish the intensity much, because there 
are only two oxygens producing them, as against 2(Ti)+2(0) pro- 
ducing the rest, and the net result is a strong reflection. 

For 200, however (Fig. 128 d), waves from all the oxygens oppose, 
to some extent, those from the titanium atoms, and since the diffracting 
power of four oxygens is a substantial fraction of that of two titanium 
atoms, a weak reflection is expected and found (see the powder photo- 
graph Fig. 121) ; the reflection is, however, not as weak as if the oxygen 
atoms had been on planes exactly half-way between the titanium planes, 
which would have meant a phase-difference of 180; the actual phase- 
difference is 2 x 0-31 X 2?r ( = 223), the waves from two oxygens being 
223 in front of, and those from the other two 223 behind, those from 
the titanium atoms. Graphical compounding of the waves as in Fig. 
l2Sd shows the result. 

In practice the compounding of waves from the different atbms is 
done by calculation. The expression for compounding waves from 
different atoms (diffracting power /) situated at different points in the 
unit cell (coordinates x, y, z in fractions of the unit cell edges) is, quite 
generally, for any reflecting plane hkl, 



Where A = 2/cos 2n(hx+ky+h), 

B = ^fam27r(hx+ky+lz). 

This is valid for all crystals, whatever their symmetry ; but whenever 
there is, as in rutile, a centre of symmetry at the origin, there is no need to 
calculate the sine terms, since in the aggregate they are bound to add up 
to zero. (For every atom giving a wave of phase angle there is also an 
identical atom giving a wave of phase angle 6, and sin(--0) = sin 0.) 
Apply this to the 200 and 110 reflections of rutile. The spacing d of 

200 is 2-30 A; . ai .. 

sm0 / 1\ 

A \-23J 

is thus 0'217x 10 8 , and, looking up in Int. Tab. the diffracting powers 
of the atoms concerned, we find / Ti = 14-0 and / = 5-2. The phase 
angles for the two Ti atoms are obviously zero (Fig. 128 d) and the net 
contribution for these atoms is 2/ Ti cosO = 2/ T1 = +28-0. But for 
oxygen No. 1, x = 0-31 and / o cos 2?r(2x 0-31) = / cos223 = 5-2 x 
(0-732) = 3-80. For oxygen No. 2 the phase angle is 223, and its 
contribution is thus the same as that of the first (since cos( 0) = cos 0). 
The other two oxygens are at x = 0-19 and +0-19 respectively and 



CHAP, vn 



POSITIONS OF THE ATOMS 



213 



their phase angles are thus 137 and +137, the cosines of which are 
the same as those of 223 and 223. The net contribution of the four 
oxygens is tKus 4/o cos 223 = 4( 3-80) = 15-2. Adding this to the con- 
tribution of the titanium atoms, we get A (== F) = +28-015-2 = 12-8 ; 
the intensity is proportional to the square of F. 



For 110, d = 3-25 A, = 0-154X 10 8 , and/ Ti and/ are 15-7 and 

A 

6-5 respectively. The phase angles for Ti atoms are again zero, and 
the Ti contribution is thus 2/ Ti = +31-4. For oxygens 1 and 2 the 
phase angles are 27r(0-31+0-31) and 27r( 0-310-31) or +223 and 
223 ; their contributions are therefore 

2/ cos 223 = 13( 0-732) = -9-5. 

For oxygens 3 and 4 the phase angle is zero, and their contribution is 
2/ = +13-0. The total is +31-4-9-5+13-0 = +34-9. The actual 
relative intensities of the 200 and 110 arcs on the powder photograph 
are obtained by multiplying the JP's by the appropriate angle factor 

l+-cos 2 20 

-r~ - - n and the number of similar reflections p (4 for 200, 4 for 110). 

sin 2 0cos0 ^ v ' 

In this way it is found that / 200 = K X 9-7 x 10 4 , 7 no = K X 156 X 10*. 
(The absorption and temperature factors are neglected.) 
As a final example the intensity for the general plane 213 will be 



calculated. Spacing of 213 = 0-888 A; 
/o = 1-9- 



sin0 



= 0-566 X 



^ 8-0, 



Atom 


hz+ky+lz 


phase angle 

CO 


COS CO 


/COS CO 


Ti(2) 
0(1) 
0(2) 
0(3) 
0(4) 


2(0-0) +1(0-0) +3(0-0) = 
2(0-50) +1(0-50) +3(0-50) = 
2(0-31) +1(0-31) +3(0-0) = 
2(-0-31)+l(-0-31)+3(0-0) = 
2(~0-19)+ 1(0-19) +3(0-50) = 
2(0-19) + 1(-0-19)+3(0-60) = 


0-00 
+ 3-00 
+ 0-93 
-0-93 
+ 1-31 
+ 1-69 




+ 335 
-335 
+ 112 
+ 248 


+ 1-0 
+ 1-0 
+ 0-91 
+ 0-91 
-0-375 
-0-375 


+8-0 
+ 8-0 
+ 1-73 
+ 1-73 

-0-71 
-0-71 



^ = 2/eoso>:= +18-04, 
F 2 = 326. 

Angle of reflection = 59 34'. 

l + G f 2 * for 59 34' is 1-443. 
sin 2 cos 8 

Number of equivalent reflections (p) of type 213 = 16 
Hence intensity = J 72 X angle factor xp X K 
== 326 X 1-443 X 16^ = 7-5 X 



212 STRUCTURE DETERMINATION CHAP, vn 

out-of-phase waves do not diminish the intensity much, because there 
are only two oxygens producing them, as against 2(Ti)+2(0) pro- 
ducing the rest, and the net result is a strong reflection. 

For 200, however (Fig. 128 d), waves from all the oxygens oppose, 
to some extent, those from the titanium atoms, and since the diffracting 
power of four oxygens is a substantial fraction of that of two titanium 
atoms, a weak reflection is expected and found (see the powder photo- 
graph Fig. 121) ; the reflection is, however, not as weak as if the oxygen 
atoms had been on planes exactly half-way between the titanium planes, 
which would have meant a phase-difference of 180; the actual phase- 
difference is 2 x 0-31 X 2rr ( = 223), the waves from two oxygens being 
223 in front of, and those from the other two 223 behind, those from 
the titanium atoms. Graphical compounding of the waves as in Fig. 
128d shows the result. 

In practice the compounding of waves from the different atbms is 
done by calculation. The expression for compounding waves from 
different atoms (diffracting power/) situated at different points in the 
unit cell (coordinates x, y, z in fractions of the unit cell edges) is, quite 
generally, for any reflecting plane hkl, 

where A = 

B = 

This is valid for all crystals, whatever their symmetry ; but whenever 
there is, as in rutile, a centre of symmetry at the origin, there is no need to 
calculate the sine terms, since in the aggregate they are bound to add up 
to zero. (For every atom giving a wave of phase angle there is also an 
identical atom giving a wave of phase angle 0, and sin( 0) = sin 8.) 
Apply this to the 200 and 110 reflections of rutile. The spacing d of 

200 is 2-30 A; . , / i x 

sm0/ l\ 

A p2rf) 

is thus 0*217 x 10 8 , and, looking up in Int. Tab. the diffracting powers 
of the atoms concerned, we find / Ti = 14-0 and / o = 5-2. The phase 
angles for the two Ti atoms are obviously zero (Fig. 128 d) and the net 
contribution for these atoms is 2/ Ti cosO = 2/ T1 == +28-0. But for 
oxygen No. 1, x = 0-31 and / cos 27r(2x 0-31) =/ o cos223 = 5-2 x 
(0-732) = 3-80. For oxygen No. 2 the phase angle is 223, and its 
contribution is thus the same as that of the first (since cos( 0) = cos 0). 
The other two oxygens are at x = 0-19 and +0'19 respectively and 



CHAP. VH 



POSITIONS OF THE ATOMS 



213 



their phase angles are thus 137 and +137, the cosines of which are 
the same as those of 223 and 223. The net contribution of the four 
oxygens is th*us 4/o cos 223 = 4( 3-80) = 15-2. Adding this to the con- 
tribution of the titanium atoms, we get A (= F) = +28-0 15-2 = 12-8; 
the intensity is proportional to the square of F. 



For 110, d = 3-25 A, = 0-154X 10 8 , and/ Ti and/ o are 15-7 and 

A 

6*5 respectively. The phase angles for Ti atoms are again zero, and 
the Ti contribution is thus 2/ Ti = +31-4. For oxygens 1 and 2 the 
phase angles are 277(0-31+0-31) and 27r( 0-31 0-31) or +223 and 
223; their contributions are therefore 

2/ cos 223 = 13( 0-732) = -9-5. 

For oxygens 3 and 4 the phase angle is zero, and their contribution is 
2/ = +13-0. The total is +31-4 9-5+13-0 = +34-9. The actual 
relative intensities of the 200 and 110 arcs on the powder photograph 
are obtained by multiplying the F*'s by the appropriate angle factor 

_IL- - an d the number of similar reflections p (4 for 200, 4 for 110). 
sin 2 0cos0 ^ v ' 

In this way it is found that J 200 = ^Tx9-7x 10 4 , J 110 = Kx 156x 10 4 . 
(The absorption and temperature factors are neglected.) 
As a final example the intensity for the general plane 213 will be 

calculated. Spacing of 213 == 0-888 A ; 55* = 0-566 x 10 8 ; / Ti = 8-0, 
/o = 1-9. 



Atom 


Aaj-f&y-ffe 


phase angle 
& 


coscu 


/COS (0 


Ti(l) 
Ti(2) 
0(1) 
0(2) 
0(3) 
0(4) 


2(0-0) + 1(0-0) +3(0-0) = 0-00 
2(0-50) +1(0-50) +3(0-50) = +3-00 
2(0-31) +1(0-31) +3(0-0) = +0-93 
2(-0-31)+l(-0-31)+3(0-0) = -0-93 
2(-0-19)+ 1(0-19) +3(0-50) = +1-31 
2(0-19) +1(-0-19) + 3(0-50) = +1-69 




+ 335 
-335 
+ 112 
+ 248 


+ 1-0 
+ 1-0 
+0-91 
+0-91 
-0-375 
-0-375 


+ 8-0 
+ 8-0 
+ 1-73 
+ 1-73 

-0-71 
-0-71 



+18-04, 
F 2 = 326. 

Angle of reflection 6 = 59 34'. 

l + G f 2 i ^ 59 34' is 1-448. 
sin 2 cos 

Number of equivalent reflections (p) of type 213 = 16 
Hence intensity = F 2 x angle factor xp X K 
= 326xl'443xl6#= 7-5 



214 



STRUCTURE DETERMINATION 



CHAP. VII 



The expressions used are valid for crystals of all types, from cubic to 
triclinic: the structure amplitude depends on atomic coordinates (as 
fractions of the unit cell edges), irrespective of the shape ctf the cell. 

Variation of intensities of reflections with atomic parameters. 
In the foregoing calculations the general arrangement of Fig. 128 a was 
accepted, and the known parameter of the oxygen atoms (0-31) was 




FIG. 129. Determination of the single parameter in the rutile structure. 

used ; the calculated intensities agree with those actually observed. The 
accepted value of 0*31 for this parameter was determined by Vegard 
(1916) by calculating the intensities of a number of reflections for a 
range of parameters. To demonstrate how sensitively the intensities 
are related to the parameter x, Fig. 129 shows how the calculated 
intensities for several powder reflections vary with x. The curves are 
shown only for values of a: up to 0-5, since they are symmetrical about 
x = O5. (The ordinates are not the intensities themselves, but their 
square roots, which correspond better with visual impressions of in- 
tensities on X-ray photographs.) It is evident that, taking into account 
mere visual impressions of the intensities of the first few powder reflec- 



CHAP, vn POSITIONS OF THE ATOMS 215 

tions, the oxygen parameter must be about 0-3, since only for this value 
are the various intensities in the correct order 110 very strong, 211 
strong, 101 fairly strong, 111 and 301 medium, 310 and 002 medium 
weak, 200 and 210 weak. The intensities of the higher-order reflections 
vary more rapidly with the value of the parameter, and therefore a 
more accurate value may be obtained by comparing the intensities of 
the various high-order reflections. 

Fig. 129 thus not only demonstrates how rapidly the relative in- 
tensities of different reflections vary with the atomic coordinates, but 
also indicates the straightforward way in which a single parameter can 
be determined : when the general arrangement is known, the intensities 
of a number of reflections are calculated for a range of values of the 
single parameter. In practice in the example given it would not be 
necessary to carry out calculations for the complete range of values of 
x, since it is obvious from the start that a weak 200 reflection could only 
result if the oxygens are about midway between the 200 planes of 
titanium atoms that is, if x is not far from 1/4. Calculations would 
therefore be carried out only for a restricted range round x = 1/4. 

But how to discover the general arrangement ? Calculations such as 
those just described cannot be made until the general arrangement is 
known. This problem of the deduction of the general arrangement of 
atoms in any crystal forms the subject of the next few sections of this 
chapter. Briefly, the consideration of the relative intensities of the 
reflections is begun by observing which reflections have zero intensity. 

Atomic arrangements. Preliminary. For simple structures such 
as those of many elements and binary compounds the determination 
of atomic positions by trial presents little difficulty; there are few 
possible arrangements, and the task of examining them in turn and 
calculating the intensities of the reflections for a range of coordinates 
in each arrangement is not a very complex or lengthy one. Some 
examples are given in Chapter IX. But for complex crystals such as 
silicate minerals or crystals of most organic substances the problem 
may seem to be of bewildering complexity. When there are many atoms 
in the unit cell the number of possible arrangements may be very large, 
and for each arrangement there may be a number of variable parameters 
to be considered. 

There are two lines of approach to such complex problems. In the 
first place, a systematic consideration of the possible types of arrange- 
ment in crystals in general and the influence of the type of arrangement 
on the X-ray diffraction pattern as a whole leads to general principles 



216 STRUCTURE DETERMINATION CHAP, vn 

which render the complexities less formidable and save much time and 
effort. 

Atoms, molecules, or ions tend at low temperatures to 'form that 
arrangement which has the lowest energy; and the arrangement of 
lowest energy is a regular repetition of a pattern in space. The pattern 
usually exhibits symmetries of one sort or another; some types of 
symmetry, as we have already seen, are displayed in the external shapes 
of crystals ; other types of symmetry, as we shall see in this chapter, 
affect the X-ray diffraction pattern they cause certain types of reflec- 
tion to be absent ; systematic absences of certain types of reflection 
therefore give a straightforward clue to the type of arrangement in the 
crystal ; they may limit the possibilities to two or three arrangements, 
or even to a single type of arrangement. 

The second line of approach to the complexities of crystal structures 
is by way of the body of existing knowledge of structure types and 
interatomic distances. The prospects of success in the attempt to find 
the correct arrangement and parameters in a complex crystal depend 
to a considerable extent on the amount of knowledge and experience 
available with regard to related structures previously determined. 

When the general arrangement is known it is then necessary to 
determine precise atomic coordinates. Sometimes the positions of 
certain atoms are invariant they arc fixed by symmetry considera- 
tions but in complex crystals most of the atoms are in 'general' 
positions not restricted in any way by symmetry. The variable para- 
meters must be determined by successive approximations; here the 
work of calculating structure amplitudes for postulated atomic positions 
can be much shortened by the use of graphical methods, to be described 
later in this chapter. It cannot be denied, however, that the complete 
determination of a complex structure is a task not to be undertaken 
lightly; the time taken must usually be reckoned in months. 

The classification of atomic arrangements into types, together with 
the consideration of the effect of the type of arrangement on the diffrac- 
tion pattern, will be considered in two stages. First of all, unit cells are 
either simple, with only one pattern-unit in the cell, or compound, with 
two or more pattern-units in the cell (see Chapter II). Crystals with 
compound unit cells give patterns from which many reflections are 
absent, and the recognition of a compound cell is a very simple matter. 
The type of arrangement of pattern-units is called the 'space lattice'. 
Secondly, the group of atoms forming a pattern-unit the group of 
atoms associated with each lattice point may have certain symmetries, 



CHAP, vii POSITIONS OF THE ATOMS 217 

and some of these symmetries cause further systematic absences of 
certain types of reflections from the diffraction pattern. The complex 
of symmetry elements displayed by the complete arrangement is known 
as the 'space group'. 

Simple and compound unit cells. The 'space lattices 9 . In 
Chapter II it has been mentioned that for some crystals it is most 
appropriate to consider as the unit cell, not the smallest parallelepiped 
from which the whole crystal could be built up by parallel contiguous 
repetitions, but a larger parallelepiped containing two or more pattern- 
units. The example given (in Figs. 5 c and 6) were the metals iron (in its 
room-temperature or a form) and copper ; the accepted unit cell of a-iron 
contains two pattern-units of one atom eal3h, and that of copper contains 
four pattern-units of one atom each. In the first place, it is obviously 
far more convenient to use these compound unit cells which are cubic 
in shape than the true structure-units which are not even rectangular. 
But convenience is not the only basis for accepting the compound unit 
cells ; a more fundamental reason is that the symmetries of these crystals 
naturally lead to their classification with crystals which have simple 
cubic cells. For instance, pure a-iron crystals are rhombic dodecahedra 
and copper crystals octahedra (Groth, 1906-19), and both these shapes 
are typical of the cubic system ; and if the atomic arrangements in these 
two crystals are examined in the way described on pp. 34r-6, it will be 
found that they possess the same elements of symmetry as caesium 
bromide which has a simple cubic unit cell. All three arrangements 
possess the highest symmetry possible in the cubic system. 

The compound two-atom unit cell of a-iron is termed 'body-centred 5 . 
The arrangement is similar to that in ammonium chloride (Fig. 122), 
with the important difference that in a-iron the atoms in the centres of 
the cells are the same as those at the corners, whereas in ammonium 
chloride (which is not called 'body-centred') there are weakly diffracting 
ammonium ions at the cell centres and more strongly diffracting chlorine 
ions at the cell corners. On account of this difference, all those reflec- 
tions which are weak in the ammonium chloride pattern (owing to 
opposition of the waves from corner and centre atoms) are necessarily 
completely absent from the a-iron pattern (see Fig. 130, Plate XI). 
Thus, for the 010 reflection (Fig. 131), waves from all corner atoms are 
in phase with each other, while waves from centre atoms are exactly 
opposite in phase because the planes of centre atoms are exactly half-way 
between planes of corner atoms. For ammonium chloride this situation 
merely produces a weak reflection, but for a-iron the diffracting powers 



218 



STRUCTURE DETERMINATION 



CHAP. VII 



of centre atoms are the same as those of corner atoms, and so waves 
from corner atoms are exactly cancelled by those from centre atoms. 
(Remember that there are just as many centre atoms as comer atoms.) 
All planes for which this is true (010, 111, and 210 are examples) have 
h+k+l odd. (Since the coordinates of the centre atom are JJt, the 
phase angle 27r(hx-\-ky+lz) for this atom is 180 when h+k+l is odd.) 
Therefore, reflections from planes having h+k+l odd are not found in 
the diffraction pattern of a-iron. On the other hand, all planes such as 
110, 200, 310, and 211 which have h+k+l even give strong reflections, 
because all the atoms lie on these planes. (In other words, the phase 



- 020' 020^ 

/LA A 


A 


1 (/' 


V 




1 \r 


t 


y-~ 


"7- 


v / 


/ 




J 



\ 



old ' 

FIG . 131. Planes of body -centred lattice . 

angle for the centre atom is when h+k+l is even.) It is important 
to note that this is true not only for body-centred cubic crystals but 
for all other body-centred crystals, whatever their symmetry. 

The compound four-atom unit cell of copper is termed 'face-centred' ; 
the cubic unit cell has atoms not only at the corners but also at the 
centre of each face. If the various planes are examined in the same 
way as for a-iron, it will be seen that in the first place 010 is absent, 
because the 010 planes (Fig. 132) comprising one corner atom and one 
of the face-centring atoms of each cell (atoms 1 and 2) are interleaved 
by other planes comprising the other two face-centring atoms of each 
cell (atoms 3 and 4). For the same reason, 110 is absent. But 111 is 
strong, and so is 020, because all the atoms lie on these planes. It will 
be found that all reflections having h+k or k+l or l+h odd are absent, 
while all planes having h+k, k+l, and l+h even are present (see Fig. 130, 
Plate XI). It is easier to remember that the only reflections present 
are those whose indices are either all even or all odd for example 111, 
200, 220, 311. This again is true for all cells which are centred on all 
three faces, irrespective of symmetry. 

In the examples just given one atom is associated with each lattice 



PLATE XI 



110 100' 211 ' 220 




511 600 
Bp ' 442 




200 311 331 422 


531 




55 


It 220 1 222 400 

J 1 


420 ' 

I 




4f-0 


6 


20 


' 1 




1 




i 







620 -822 444 551 64C 




too 



FIQ. 130. X-ray powder photographs of a iron, copper, sodium chloride, and potassium 

chloride, The photograph of a iron was taken with cobalt Kot radiation, the others 

with copper Kot radiation. 



CHAP, vn 



POSITIONS OF THE ATOMS 



219 



point, and the lattice points have been chosen for convenience at the 
centres of these atoms. But the rules are equally true for crystals in 
which thef e are several atoms associated with each lattice point ; this 
will be evident when it is remembered that lattice points are defined as 
those points which have identical surroundings. In such circumstances 
we must think (in the case of the body-centred lattice) of the combined 
diffracted wave from the group of atoms associated with the corner 
lattice point being cancelled by the combined diffracted wave from the 
identical group of atoms associated with the centre lattice point, for 
reflections having h-\~k-{-l odd. 






FIG. 132. Planes of face-centred lattice. 

The recognition of simple or body-centred or face-centred lattices is 
thus quite straightforward. Indeed, for many cubic crystals of elements 
and binary compounds it is obvious by mere inspection of the powder 
photograph, provided enough reflections are registered. The grouping 
of reflections in each of the three types of diffraction pattern is illustrated 
in Fig. 133 : a crystal with a simple cubic lattice gives an X-ray powder 
pattern like that of ammonium chloride, in which the arcs are regularly 
spaced up to the sixth, after which there is one gap (because 7 is not a 
possible value for A 2 +fc 2 +J 2 ) ; body-centred crystals give patterns in 
which the regular spacing is maintained beyond the sixth arc; face- 
centred crystals give the grouping shown in the copper pattern (Fig. 130, 
Plate XI) two arcs fairly close together (111 and 200), then a gap to 
the third (220), a similar gap to the fourth (311), and the fifth (222) 
close to the fourth. 

The face-centred cubic lattice is very common. Many metallic elements 
crystallize in this form; so also do many binary compounds such as 
alkali halides and the oxides of divalent metals. Thus the powder photo- 
graph of sodium chloride (Fig. 130, Plate XI) shows the same grouping 



220 STRUCTURE DETERMINATION CHAP, vn 

of arcs as that of copper. Note, however (this is something of a digres- 
sion, but a useful one at this stage), that while the arcs of copper are 
all strong, some of those of sodium chloride are much wg&ker than 
others. The reason is, of course, to be found in the existence of two 
types of ion of different diffracting powers, placed in a particular 
position in relation to each other: in addition to the face-centred lattice 
formed by the chlorine ions (Fig. 134), there is an equal number of 
sodium ions which lie along the edges and in the centre of the cell, always 
half-way between chlorine ions (forming, by themselves, another face- 
to! 
[300] 311 320 

wo no /// 200 210211 m m msu 



SIMPLE 



BODY- CENTRED 



FACE -CENTRED I I I II 



FIG. 133. Powder patterns of simple, body-centred, and face-centred cubic crystals, 

centred lattice) ; and these naturally affect the intensities of the reflec- 
tions. Thus, for the 111 reflection, waves from all the chlorine ions are 
in phase with each other, but since there are planes of sodium ions mid- 
way between the planes of chlorine ions, the waves from the sodium ions 
oppose those from the chlorine ions and thus weaken the reflection (the 
diffracting power of sodium being half that of chlorine). On the other 
hand, all the ions lie on the 200 planes, and all the waves therefore 
co-operate to give a strong reflection. For such reasons, all reflections 
with odd indices are weak, while those with even indices are strong ; 
that this is so may be seen on Fig. 130, Plate XI. Such effects are still 
more marked in the powder photograph of potassium chloride which is 
also shown in Fig. 130, Plate XI so much so that the odd type reflec- 
tions are absent altogether, and the pattern looks like that of a simple 
cubic cell with an edge half the true length. The reason for the complete 
absence of the odd type reflections is that the diffracting power of the 
potassium ion is almost exactly equal to that of the chlorine ion, since 



CHAP. VII 



POSITIONS OF THE ATOMS 



221 



both ions contain the same number of electrons (K + 21 1, Cl~ 19+1). 
The X-ray photograph is in a way misleading, since it appears to indicate 
a simple cubic cell containing only one atom ; but we know, of course, 
from chemical evidence that there are two sorts of ions, and in view of 






200 

FIG . 1 34. Planes of sodium chloride crystal. 

this the absence of the odd type reflections is really a striking demonstra- 
tion of the equality of the diffracting powers of K + and Cl~ and of the 
effect of ionic positions on the intensities of X-ray reflections. 

The three types of lattice which have been mentioned simple (or 
primitive), body-centred, and face-centred 
are the only ones possible in the cubic system. 
Note that a lattice centred on one pair of 
opposite faces, or two pairs, would not have 
cubic symmetry (the essential elements of 
which are the four threefold axes running 
diagonally through the cell). If the possi- 
bilities in the other crystal systems are 
examined, it will be found that there are 
fourteen kinds of lattice in all. This was 
first recognized by Bravais in 1848, and the 
different types are therefore often referred 
to as 'the fourteen Bravais space-lattices'. 
Certain kinds of lattice which at first 
thought might be expected to exist will on 

examination be found either to have the wrong symmetry or to be 
equivalent to other kinds. For instance, in the tetragonal system there 
is no face-centred lattice : if to a simple tetragonal lattice we add extra 
lattice points at the centres of the faces (Fig. 135), the lattice so formed 



B 

FIG. 135. To a primitive tetra- 
gonal lattice ABCDEFOH add 
extra lattice points at the face 
centres. The new lattice is 
equivalent to the body -centred 
lattice BJCIFLGK, 



STRUCTURE DETERMINATION 



CHAP. VII 



will be found to have a body-centred unit cell with a square base whose 
edges are 1/V2 times the length of those of the original cell. All the 
fourteen types of lattice are illustrated in Fig. 136. 




12 



13 



FIG. 136. The fourteen Bravais space-lattices. 1, Triclinic (P). 2-3, Monoclinic (Pand 
C). 4-7, Orthorhombic (P, C, J, and F). 8, Hexagonal (C see text for explanation). 
9, Rhombohedral (R). 10-11, Tetragonal (P and /). 12-14, Cubic (P, /, and F). 

All body-centred crystals, whatever their symmetry, can give only 
reflections having h-{-k+l even, and all face-centred crystals, whatever 
their symmetry, can give only reflections having either all odd or all 
even indices. The only additional type of lattice encountered in non- 




CHAP.VH POSITIONS OF THE ATOMS 223 

cubic crystals is the lattice centred on one face only. If it is the 001 face 
which is centred, it can easily be seen (by reasoning similar to that used 
in the foregoing pages) that all reflections having h+k odd must be 
absent, but that reflections having k+l or l+h odd may be present, 
provided that h+k is even. Thus 210, 012, 121, and 122 are all absent, 
because h+k is odd in each case, but 110, 112, and 312 may be present 
because h+k is even ; the values of k+l and l+h do not matter as far 
as the compound nature of the lattice is concerned. 

In referring to the symmetries of atomic arrangements, concise 
symbols (similar to those of the point-groups see Chapter II) are used. 
In a set of symbols characterizing a 
space-group, the first is always a 
capital letter which indicates whether * 

the lattice is simple (P for primitive), 
body-centred (/ for inner), side- 
centred (A, B, or C), or centred on 
all faces (F). For the rhombohedral 
lattice the special letter B is used. 
For the hexagonal lattice the letter 
C is used, because it can be regarded 

as an orthorhombic lattice centred 

on the C face (see Fig. 137). In Some FlG ' 137 ' Alternative cell-bases for 
ri_ i .LI hexagonal crystals. 

descriptions of hexagonal crystals 

the letter H will be found ; this refers to a larger cell containing three 
lattice points (Fig. 137). It is not a new lattice type, but merely a 
different description of the simple hexagonal lattice ; the H cell is used 
only where it is desired to adhere to the a and b axes selected on mor- 
phological grounds, even when these are not the edges of the smallest 
unit cell. For similar reasons some tetragonal crystals are described 
by a face-centred cell ; this again is not a new lattice type, for a C 
face-centred tetragonal lattice with base-edge a is equivalent to a 
primitive lattice of base-edge a/V2. In structure determination it is 
really simplest to use the smallest unit cell, transforming the indices of 
the reflections if necessary, as described on p. 172. 

The types of absent reflections so far mentioned are those arising 
from the compound nature of certain lattices ; it is important to note 
that a compound lattice causes systematic absences throughout the 
whole range of reflections. These are not the only types of systematic 
absences ; certain other types, which may occur in addition to, or instead 
of, those already mentioned, are due to certain types of symmetry in 



524 STRUCTURE DETERMINATION CHAP, vn 

atomic arrangements. These will be described in the following sections ; 
for the moment all we need note is that these symmetry elements cause 
absences only throughout particular principal zones of reflections, or 
among the various orders of reflection from a particular principal plane ; 
thus they do not affect the determination of the lattice type from the 
X-ray diffraction pattern, which is done by examining the list of reflec- 
tions for any systematic absences throughout the whole range of reflec- 
tions. 

The first stage in the determination of the structure of a crystal 
the discovery of the lattice type is thus quite simple and straight- 
forward. In this chapter we will proceed with the story of the further 
stages in the elucidation of crystal structures in general ; but some of 
the simpler structures can be solved completely by a determination of 
the lattice type, with perhaps a very limited consideration of the 
intensities of a few reflections ; examples of such structures are given 
at the beginning of Chapter IX. 

The symmetries of atomic arrangements. Point groups and 
space groups. Crystals consist of groups of atoms repeated regularly 
in space. A crystal structure may be imagined as being built up by 
assembling a particular group of atoms, and then repeating the same 
grouping in exactly the same orientation at regular intervals in space. 
The smallest group from which the whole crystal may be constructed 
in this way is the unit of pattern. Each such group may be regarded as 
associated with a lattice point ; in other words, we mentally replace a 
group of atoms by a symbolic point. In the previous section the 
various possible arrangements (both simple and compound) of such 
lattice points have been mentioned. We now have to consider 
the arrangement of atoms round each lattice point and the effect of 
the arrangement on the X-ray diffraction pattern. In speaking of the 
symmetries of the arrangement of atoms round a lattice point it is 
customary to use the term 'point-group', and for the symmetries of the 
complete arrangement in the crystal the term 'space-group'. 

Consider first a few of the possible ways of arranging atoms round a 
point, ignoring crystal structure for the moment and thinking of groups 
of atoms in isolation. For this purpose we cannot do better than to 
recall the structures of a few simple molecules and ions, Fig. 1 38 is a 
gallery of simple types. (It should be noted that in these drawings the 
spheres mark the positions of atomic centres ; the effective external radii 
of the atoms are much larger.) 

The trans form of 1,2 dichlorethylene C1HC=CHC1 (Brockway, Beach, 



CHAP. VII 



POSITIONS OF THE ATOMS 



225 



and Pauling, 1935) has a single plane of symmetry (m) passing through 
all the atoms, a twofold axis (symbol 2) normal to this plane and, arising 
out of this combination, a centre of symmetry ; the symmetry of this 




HO 



d tori?) (CH 3 )BrHC CHBHCH 3 ) meso(CH 3 )BrHC CHBrfCHs) 
Symmetry, 2 Symmetry,! , , 

Benzene o/mmm 

FIG. 138. The symmetries of some simple molecules and ions. 

molecule (2/ra) is the same as that of a crystal belonging to the holohedral 
class of the monoclinic system. 

In the carbonate ion C0^~ (W. L. Bragg, 1914, 1924 a) the three 
oxygen atoms lie at the corners of an equilateral triangle, at the centre 
of which the carbon atom is situated. The ion has a plane of symmetry 
(m) passing through all the atoms, and a threefold axis (3) normal to 

4458 Q 



226 STRUCTURE DETERMINATION CHAP, vn 

this plane and passing through the carbon atom ; this combination is 
more concisely described as a hexagonal inversion axis (6). There are 
also three planes of symmetry intersecting in the threefold axis, and 
thres twofold axes of symmetry, each passing through the carbon atom 
and one oxygen atom. The conventional point-group symbol is 62m 
(though 6m would be a sufficient description). 

In the chlorate ion CIQ$ the three oxygen atoms form an equilateral 
triangle, but the chlorine atom is not in the plane of this triangle 
(Dickinson and Goodhue, 1921 ; Zachariasen, 1929) ; the whole ion has 
a threefold axis with three planes of symmetry intersecting in this axis, 
like the carbonate ion, but lacks the three twofold axes and the plane 
of symmetry perpendicular to the threefold axis ; its point-group symbol 
is 3m. 

The urea molecule 0=C(NH 2 ) 2 has two planes of symmetry inter- 
secting in a single twofold axis the symmetry found in crystals belong- 
ing to the polar class of the orthorhombic system (Hendricks 1928 a) ; 
the point group symbol is mm (== 2mm). 

Molecules of the meso form of 2,3 dibromobutane Br(CH 3 )HC 
CH(CH 3 )Br have a centre of symmetry (1) as their only element of 
symmetry! (Stevenson and Schomaker, 1939), resembling in this respect 
crystals belonging to the holohedral class of the triclinic system. It is 
worth noting that the d and / isomers are not asymmetric ; they possess 
one twofold axis.f 

The benzene molecule (Pauling and Brockway, 1934) has a sixfold axis 
of rotation, together with all the additional symmetry elements possessed 
by a crystal belonging to the holohedral hexagonal class 6/mmm. 

Finally the carbon tetrachloride molecule CC1 4 has its chlorine atoms 
arranged tetrahedrally round the carbon atom (Pauling and Brockway, 
1934) ; it has four threefold axes, each passing through the carbon atom 
and one chlorine atom, three mutually perpendicular fourfold inversion 
axes which bisect the Cl C Cl angles, and six planes of symmetry, 
each passing through one carbon atom and two chlorine atoms ; but it 
has no fourfold rotation axes or centre of symmetry . The point-group 
symmetry is that of the tetrahedral class of the cubic system 43m. 

Before considering the placing of point-groups in space lattices it 
must be observed that the pattern-unit of which a crystal is built up 
the group of atoms associated with each lattice point is by no means 

f The two halves of these molecules rotate, with respect to each other, round the 
C C bond as axis; the remarks on symmetry refer to the most stable configuration, 
that in which the molecule spends most of its time. 



CHAP. VII 



POSITIONS OP THE ATOMS 



227 



always a single molecule or a pair of ions ; more often than not, two or 
three or four, or perhaps even more, molecules form the pattern-unit. 
These mofecules are often 
arranged in such a way that 
the group exhibits some of 
the symmetry elements al- 
ready mentioned. There are 
many ways of grouping mole- 
cules round a point, and the 
general problem confronting 
those who wish to catalogue 
all the possible point-groups 
is to think of all the possible 
ways of attaining symmetry 
of one type or another by 
arranging asymmetric ob- 
jects round a point. Thus 
two identical asymmetric 
molecules may be related 
(Fig. 139) by an axis of 
symmetry ; and two enantio- 
morphous molecules may be 
related by a plane of sym- 
metry or a centre of sym- 
metry. Larger numbers of 
molecules may be arranged 
to attain higher symmetries. 
The number of possible sym- 
metries of isolated groups of 
atoms is unlimited; but we 
are concerned here only with 
those symmetries which can 
also exist in repeating space- 
patterns; and, as we have 
already seen, this restricts us 

, ji . i FIG. 139. Arrangements of two asymmetric 

to those arrangements having molecules, 

two-, three-, four-, or sixfold 

axes. There are only thirty-two different symmetry combinations (point- 
groups) which fulfil these conditions thirty-two, including the asym- 
metric case in which only one object is used. This number is the same 





228 STKUCTUBE DETERMINATION CHAP, vn 

as the number of crystal classes ; in fact, the various possible symmetries 
of molecules or groups of molecules correspond with the various types 
of crystal shape which are catalogued in Chapter II ; the' problem of 
arranging diiferent types of atoms round a point is formally the same 
as that of arranging different types of crystal faces round a point. 

We have now to think of the possible ways of placing the various 
types of atomic arrangement in the various types of lattices. Suppose a 
molecule having certain symmetries is to be associated with each point of 
a particular space lattice. The first thing to realize is that the molecule 
can only be placed in a lattice having the appropriate symmetry if 
both molecule and lattice are to retain their original symmetries. Thus 
a molecule of tetrahedral symmetry such as tin tetra-iodide (Dickin- 
son, 1923) fits appropriately into a cubic lattice, the threefold axes of 
the molecule lying along the threefold axes of the cubic lattice. But to 
put a molecule of hexagonal symmetry in a triclinic space lattice would 
seem like sheer waste of good symmetry ; and since the forces between 
neighbouring molecules would not be hexagonally disposed, there would 
be a tendency for all the molecules to distort each other. How much 
effect this would have would depend on the rigidity of the molecule in 
relation to the forces around it tending to distort it. Structures of this 
kind, in which molecules are apparently inappropriately placed, are, 
however, not uncommon. In fact, the state of affairs j ust mentioned a 
molecule of hexagonal symmetry in a triclinic lattice actually occurs 
in the crystal of hexamethylbenzene. In this crystal the rigidity of th0 
molecule is such that no distortion has been detected, but the tendency 
to distortion must be there. The point is that, from the formal point 
of view, the molecules in the crystal do not possess hexagonal symmetry ; 
the only symmetry element they possess is the only one possible in a 
triclinic lattice a centre of symmetry. The reason why hexamethyl- 
benzene molecules arrange themselves to form a triclinic crystal is, no 
doubt, that intermolecular forces and the requirements of good packing 
are satisfied better by a triclinic arrangement than they would be by 
a hexagonal or any other arrangement ; though it is difficult to see just 
why this is so. (See Mack, 1932.) It very frequently happens that some 
of the symmetry elements possessed by free molecules are not utilized 
in the formation of crystalline arrangements,! though the neglect is not 
often so striking as in hexamethylbenzene. 

t Note also that non-crystallographic symmetries in molecules are inevitably 
ignored ; thus, in gaseous cyclopentane the molecule may have a fivefold axis (Pauling 
and Broekway, 1937); this could not be retained in a crystal in other words, there 
would be a tendency to distortion. 



CHAP, vn POSITIONS OF THE ATOMS 229 

Conversely, a molecule of low symmetry cannot by itself form the 
pattern-unit of a crystal of high symmetry. If identical asymmetric 
molecules were placed singly at the corners of a ceD of orthorhombic 
shape, the result would necessarily be that the asymmetrically disposed 
forces between the molecules would distort the cell and make it tri- 
clinic. The only way of making an orthorhombic crystal out of asym- 
metric molecules is to group at least four of them together to form a 
pattern-unit having the symmetry appropriate to the orthorhombic 
lattice. For instance (confining our attention for the moment to the 
symmetry elements so far mentioned), two left-handed and two right- 
handed molecules might be arranged so that the group exhibits two 
planes of symmetry at right angles to erfch other (with, arising out of 
this, a twofold axis at their intersection) that is, point-group symmetry 
mm ; or four left-handed molecules might be arranged so that the group 
exhibits three mutually perpendicular twofold axes that is, point- 
group symmetry 222. Such groups could be placed at the points of 
an orthorhombic lattice without changing the symmetry of either the 
groups or the lattice ; they must of course be oriented correctly, with 
twofold axes parallel to cell edges and planes of symmetry parallel to 
cell faces. These remarks apply to asymmetric molecules ; naturally, if 
molecules themselves possess some symmetry, fewer of them may be 
required to form an arrangement of particular point-group symmetry, 
provided that the natural symmetries of the molecules are utilized. 

Thus, each of the thirty -two point-groups must be placed, correctly 
oriented, in a lattice having appropriate symmetry. Bearing in mind 
the existence of compound lattices having the same symmetries as 
simple ones, we realize that the number of arrangements possible on 
this basis is considerably greater than thirty- two. 

But this does not end the tale of possible arrangements. Hitherto we 
have considered only those symmetry operations which carry us from 
one atom in the crystal to another associated with the same lattice 
point the symmetry operations (rotation, reflection, or inversion 
through a point) which by continued repetition always bring us back 
to the atom from which we started. These are the point-group sym- 
metries which were already familiar to us in crystal shapes. Now in 
many space patterns two additional types of symmetry operations can 
be discerned types which involve translation and therefore do not 
occur in point-groups or crystal shapes. 

Symmetry elements involving translation. These elements are 
the glide plane, which involves simultaneous reflection and translation, 



230 



STRUCTURE DETERMINATION 



CHAP, vn 



and the screw axis, which involves simultaneous rotation and transla- 
tion. By continued repetition of these symmetry operations we do not 
arrive back at the atom from which we started; we arrive at the 
corresponding atom associated with the next lattice point, and then the 
next, and so on throughout the crystal. These operations will be illus- 
trated first by an isolated molecule that of polyethylene (Fig. 140), a 
chain polymer molecule so long that it may be regarded for the present 
purpose as indefinitely long. This molecule may be constructed in 




FIG. 140. Left: twofold screw axis. Right: glide plane. 

imagination by repeating a group of two carbon and four hydrogen 
atoms the groups marked M and N over and over again along a 
straight line ; the distance from any atom to the next similarly situated 
atom along the axis the repeat distance of the molecule will be 
called c . Some of the symmetries of this molecule are those already 
familiar planes of symmetry, twofold axes, and a centre of symmetry ; 
these are not marked on the pictures. But now consider how it is 
possible to move group M into the position of N 9 and N into the position 
of P, and so on. One way (Fig. 140, left) is to imagine group M rotated 
round the c axis for half a revolution and at the same time moved along 
this axis for a distance of |c ; in this way we arrive at the next CH 2 
group N ; if we repeat the process, we arrive at the next CH 2 group P, 
and so on. Thus, two repetitions of the operation bring us, not to the 



OHAP. vn POSITIONS OP THE ATOMS 231 

original group M, but to the next corresponding group along the axis 
of the molecule. The term 'screw axis' aptly describes the process, and 
the accepted symbol for this symmetry element is 2 1 . 

Another way of going in imagination from group M to groups N and 
P is (Fig. 140, right) to imagine group M reflected in the plane marked 
c (at right angles to the plane Containing the carbon atoms) and then 
moved along the molecular axis a distance of c ; this brings us to group 
N 9 and another repetition of the process brings us to group P. This 
symmetry element is appropriately called the glide plane, and the 
accepted symbol for it is an 
italic letter in this case c indi- 
cating the direction along which 
translation occurs. 

Screw axes and glide planes are 
of great importance. In mole- 
cular crystals the molecules are 
usually related to each other by 
these symmetry elements rather 
than by rotation axes or reflec- 
tion planes; the reason is pre- 2 

sumably that, since atoms are 

. , . , . i Fio. 141. The effective shape of the molecule 

more or less spherical in shape, O f giycine, NH,. CH,.COOH. 

and when linked by covalent 

bonds are partly merged in each other, a molecule is a rather knobbly 
object (Fig. 141 ) and there is a tendency for the knobs of one molecule to 
fit into the hollows of its neighbours an arrangement which is likely to 
give rise to screw axes, glide planes, or centres of symmetry rather than 
to rotation axes or reflection planes. (The latter would bring knobs in 
opposition to each other see Fig. 139.) An example of a crystal struc- 
ture exhibiting twofold screw axes and glide planes the structure of 
benzoquinone (Robertson, 1935 a) is shown in Fig. 142. Molecule M can 
be moved into the position of P either by rotation round the screw axis 
S 1 and translation half-way along 6, or by reflection in the glide plane G l 
and translation half-way along a. (Note that these symmetry elements 
do not occur singly; the existence of the screw axes S l automatically 
gives rise to $ 2 , S 3 , and S 4 , half-way between them ; and the glide plane 
<?! is inevitably accompanied by another, (? 2 , at a distance 6 from it.) 
Other examples will be found later in this book notably the structure of 
durene, Fig. 149. The arrangement of polyethylene molecules in crystals 
of that substance (Fig. 143) is also worth studying. Note that not only 




STRUCTURE DETERMINATION 



CHAP, vn 



is the twofold screw axis possessed by a single molecule retained in the 
crystalline arrangement, but also there are twofold screw axes relating 
the molecules to each other ; there are three sets of such twofold screw 
axes, one set parallel to each unit cell axis. Perpendicular to the b axis 
are glide planes, the translation being |a (symbol a) In addition to 




FIG. 142. Crystal structure of benzoquinone. 

these there is another set of glide planes perpendicular to the a axis, 
but the translation is not simply along an axis but along the be diagonal ; 
a special symbol n is used for such glide planes involving a translation 
half-way along the diagonal of a cell-face. There are also planes of 
symmetry, but not all the planes of symmetry possessed by a single 
molecule are retained in the crystal; those perpendicular to c (the long 
axis of the molecule) are retained, but the plane of symmetry parallel 
to the long axis of a single molecule is ignored in the crystal arrange- 



CHAP. VII 



POSITIONS OF THE ATOMS 



233 



ment in fact, formally speaking, the molecule in the crystal no longer 
has a plane of symmetry parallel to its length. 

A twofold screw axis has no left- or right-handed sense of helical 
movement, since rotation through 180 to the left brings us to the 
same place as rotation through 180 to the right. But some threefold, 
fourfold, and sixfold screw axes may be either left- or right-handed. 




^ /Carbon O Hydrogen 



FIG. f43. Crystal structure of polyethylene. 



This is illustrated in Figs. 144-6, which also show in a formal way the 
other types of symmetry axes which may be found in trigonal, tetragonal, 
and hexagonal crystals. The symbols such as 6 X and 4 3 have this signifi- 
cance : the main figure gives the amount of rotation and the subscript 
the translation. Thus, 6 X means a rotation of one-sixth of a turn com- 
bined with a translation of one-sixth the length of the c axis, the spiral 
motion being such as to give rise to a right-handed screw ; 6 6 means a 
rotation of one -sixth of a turn in the same direction combined with a 
translation of 5c/6 which amounts to the same thing as c/6 in the opposite 



234 



STRUCTURE DETERMINATION 



CHAP. VII 



direction, the result being that 6 5 is a left-handed screw, the mirror 
image of 6 r Similarly 4 3 is the mirror image of 4j, and 3 2 the mirror 
image of 3 r 

The inversion axes 3, 4, and 6 (also shown in Figs. 144r-6) have the 
same significance as in morphology ; thus 3 means rotation through one- 
third of a turn combined with inversion through a point. 



-iO 



A- 

3, 



A 

3 



FIG. 144. Types of threefold axes. 



FIG. 145. Types of fourfold axes. 

The only other symmetry element involving translation which remains 
to be mentioned is the glide plane having a translation of one-quarter 
of a cell-face diagonal ; this type of glide, symbolized d, is found only 
in a few space-groups. 

Effects of screw axes and glide planes on X-ray diffraction 
patterns. The existence in a crystal of screw axes or glide planes is 
necessarily not revealed by the shape of the crystal, since the shape of 
a polyhedron cannot exhibit symmetry elements possessing translation. 
Shape-symmetry may tell us that a particular crystal has a fourfold 



CHAP, vn 



POSITIONS OF THE ATOMS 



235 



axis, but it cannot tell us whether-this axis is a simple rotation axis or 
a screw axis. Nor is it possible by examining the shape of a crystal to 
distinguish between a reflection plane and a glide plane. But X-ray 
diffraction patterns do make such distinctions, and in a very straight- 
forward manner: just as it is possible to detect compound ('centred') 
lattices by noticing the absence of certain types of reflections (p. 217), 
so also it is possible to detect screw axes and glide planes, for the 
presence of atoms or groups of atoms related by translations which 
are simple submultiples of a unit cell edge (one-half, one-third, one- 



H 4 



H 



FIG. 146. Types of sixfold axes. 

quarter, or one-sixth) necessarily causes the absence of particular types 
of reflections. 

Consider, for instance, first of all the twofold screw axis and its effect 
on X-ray beams reflected by the crystal plane perpendicular to the screw 
axis. The first-order reflection 001 would be produced when waves from 
atomic plane M M ' (Fig. 147) are one wave-length ahead of waves from 
atomic plane PP' ; but, exactly half-way between these planes is an 
exactly similar sheet of atoms NN', waves from which would be half a 
wave-length ahead of those from PP' and thijs of exactly opposite phase 
(and, of course, equal amplitude). Waves from atomic planes NN' 9 
QQ', and so on evidently cancel out those from M M' , PP', etc.^ and the 
resultant intensity of the 001 reflection is zero. There may be other 
atoms in the crystal, formally independent of those just mentioned; 
but since any other atoms in the crystal are also related to each other 
by the screw axes, the same conclusion is valid. The second-order 
reflection 002, however, is produced when waves from M M ' are two 
wave4engths ahead of those from PP', and when this occurs, waves 
from NN' are one wave-length ahead of those from PP' and therefore 



236 



STRUCTURE DETERMINATION 



CHAP, vn 



in phase with them; the reflection 002 is therefore not cancelled. 
Similarly, all odd-order reflections from this plane are bound to be 
absent, while all even-order reflections may be present; ki fact, the 
spacing along the c axis appears to be halved, since if we observed 
only the different orders of OO/, we should call the first reflection 001 and 
thus be led to suppose that the c axis has a length half the true value 
which is obtained when the first reflection is given its true indices 002. 




Co 




Fio. 147. Twofold screw axes. Effect on X-ray reflections. 

For all other crystal planes there are no simple phase relations between 
waves from M and those from N, and therefore no further systematic 
absences. Thus, the distance x of the atoms from the screw axis in the 
direction of the a axis is not, except by accident, a submultiple of a , 
and therefore there are no systematic absences of AOO reflections. One 
or two of these may not appear on the photograph because the structure 
amplitudes happen to be very small ; but the point is that there are no 
systematic absences. The same is true for all other planes 101 for 
instance (Fig. 147 6), since the distance s between such a plane of atoms 
as NQ' and the plane through P is not, except by accident, a simple 
submultiple of the spacing d wl . 

Thus the only systematic absences caused by a twofold screw axis 
are the odd orders of reflection from the plane perpendicular to the 
screw axis. 



CHAP. VII 



POSITIONS OF THE ATOMS 



237 



In a similar way, in a crystal exhibiting a threefold screw axis ^ or 
3 2 , identical atoms are repeated on planes spaced one-third the length 
of the axis ;<jfcherefore the reflections from the plane normal to the screw 
axis would, by themselves, appear to indicate a repeat distance only 
one-third the true axial length. The first of these reflections would be 
the third-order reflection in reference to the true repeat distance, while 
the second would be actually the sixth-order reflection. In other words, 
a threefold screw axis 3 l or 3 2 causes the absence of the first and second 
orders of reflection from the plane perpendicular to the screw axis, as 
well as the fourth and fifth and 
indeed all orders not divisible by 
3. When a fourfold screw axis 4 X 
or 4 3 is present, all orders not 
divisible by 4 are absent, while a 
sixfold screw axis 6 X or 6 5 cancels 
all but the sixth, twelfth, and other 
orders divisible by 6. Throughout, 
the only crystal plane whose re- 
flections are affected in this way 
is the plane perpendicular to the 
screw axis* 

No such absences occur when 
the axes of symmetry are simple 
rotation axes; thus in Fig. 148 there are no subdivisions of the c axis 
and therefore no absences of OOZ reflections. 

Glide planes are more devastating in their effects on X-ray reflections ; 
they cause absences among a whole zone of reflections. Consider the 
structure of durene, 1, 2, 4, 5 tetramethylbenzene (Robertson, 19336). 




Fief. 148. Ordinary twofold axes. No 
systematic absences of reflections. 




The unit cell is monoclinic and contains two molecules, and the space 
group is P2 1 /a. The two molecules are related to each other by a glide 
plane perpendicular to the 6 axis and having a translation of a/2. Con- 
sider first Fig. 149 a, which is a view looking straight down the b axis. In 
this representation it can be seen that molecule B is differently oriented 
from molecule A ; in order to move B into the position of A it would be 
necessary to reflect it in the plane of the paper and, ipove it half-way 



238 



STRUCTURE DETERMINATION 



CHAP, vn 



along the a axis. But now consider Fig. 149 6, in which only the positions 
of atomic centres are shown ; molecules A and B now look exactly the 
same ; in fact, it looks as if the unit cell has an a axis only Lalf the true 





FIG. 149. Crystal structure of durene, 1, 2, 4, 5 tetramethylbenzene. (a) As seen when 
looking straight down the b axis. (b) The same ; but only the positions of atomic centres 
are marked, (c) As seen when looking straight down the c axis. 

length. It is the atomic coordinates in this projection which settle the 
intensities of the hOl reflections ; hence the only hOl reflections present 
are those which would be given by the half-sized apparent unit cell 
reflections for which the true h is even. (Reflections having h odd are 
those for which the phase-difference of waves from molecules A and C 



OHAP.VH POSITIONS OF THE ATOMS 230 

is an odd number of wave-lengths ; but in these circumstances the phase- 
difference of waves from A and B is an odd number of half wave-lengths ; 
hence wa^WjErom B cancel waves from A. All reflections having h odd 
are therefore absent.) From all other viewpoints, such as the c projec- 
tion, Fig. 149 c, the a axis does not appear to be halved, and therefore 
there are no systematic absences in any zone other than hOl. The only 
other systematic absences are the odd orders of 0&0, owing to the two- 
fold screw axis along 6. 

In some crystals there is a glide plane n having a translation, not along 
an axis, but half-way along a face-diagonal of the cell. In fact, the dame 




FIG. 150. Crystal structure of durene, showing alternative 
unit cell a'c having symmetry P2JH, 

crystal that we have already described as having the space-group 
symmetry P2 1 /a could alternatively be described by a different cell a'c 
(Fig. 150) having the symmetry P2JH, one molecule being derived from 
the other by reflecting in the glide plane and translating half-way along 
the diagonal of the a'c plane. If, as before, we look along the 6 axis, we 
see molecule B looking exactly the same as A but translated half-way 
along the a'c diagonal ; in other words, the projected cell seen from this 
view-point appears to be centred, and therefore all reflections from hQl 
planes having h+l odd (for example, 100, 001, 102 X 201, 302, 203, and 
so on) must be absent. These are the only systematic zone absences, 
since from all other view-points there is no apparent centring. These 
absent reflections are of course the same as in the first description of 
the cell; they are merely denoted by different indices, 101 of P2Ja 
being 20T of P2JH, and T01 of P2Ja being 001 of P2Jn; and so on. 
In a similar way a glide plane normal to the c axis of a crystal 



240 



STRICTURE DETERMINATION 



CHAP. VII 



having a translation of one-quarter of the ab diagonal (symbolized d) 
nullifies all MO reflections except those for which h+k is divisible by 4. 

In some hexagonal, tetragonal, and cubic space groups tj^#e are glide 
planes which are not normal to cell edges; they are normal to the 
diagonals of cell faces. Glide planes of this type, normal to the diagonals 
of the 001 cell face and having a translation of c/2, cause all hhl reflec- 
tions with I odd (for example 223) to be absent. 

Ordinary reflection planes m cannot be detected in this way because 
they cause no systematic absences ; thus when two molecules related by 
a reflection plane are seen from a direction normal to the reflection plane, 
one molecule is exactly eclipsed by the other; there is no apparent 
halving of an axis or a diagonal, and therefore there are no systematic 
absences due to a plane of symmetry. 

Thus, while screw axes and glide planes can be detected and distin- 
guished from each other by X-ray diffraction phenomena alone, ordi- 
nary rotation axes and reflection planes cannot be detected in this way, 
since neither type leads to any systematic absences of reflections. 

The presence of symmetry elements having translation, together 
with the lattice type, can always be deduced, as in the above discussion, 
from first principles. The types of absences and the elements of trans- 
lation causing them are summarized in Table V. The absences for all 
the space-groups are tabulated by Astbury and Yardley (1924), and 
also in Int. Tab. 

TABLE V 



Element of translation 


Symbol 


Absent reflections 


Body -centred lattice 
Lattice centred on 001 face 
Face-centred lattice (all faces) 


I 
O 
F 


hkl with h + k + 1 odd 
hkl with h 4- k odd 
hkl with ft-f k or &+Z or l+h odd 


Glide plane J_c, translation - 


a 


MO with h odd 


lc, -- 6 


n 


MO with 7t+^ odd 


flt""f~ O 
> . I Ci ft Y n 


d 


hkO w^hen h-\~k not divisible by 4 


1110, \ 


c 


hhl when h odd 


Twofold screw axis || c 
Threefold 
Fourfold 


2, 


001 with Z odd 
OOZ when Z not divisible by 3 
OOZ when I not divisible by 4 
OOZ with Z odd 


Sixfold 


62, 65 

6 2 , 6 4 
63 


OOZ when Z not divisible by 6 
OOZ when Z not divisible by 3 
OOZ with Z odd 



CHAP. VII 



POSITIONS OF THE ATOMS 



241 



Diffraction symmetry in relation to point-group symmetry. 

So far, in our consideration of the intensities of X-ray reflections in 
the proces^if discovering the general arrangement in a crystal, we 
have dealt only with reflections of zero intensity, and we have seen 
that when certain types of reflections have zero intensity the presence 
of elements of translation in the structure may be inferred. We now 
consider in a general way the intensity relations between the reflections 



3\0 




3/0 




FIG. 151. Above: urea (c projection). 310 and 310 planes have the same structure 

amplitude. Below: penta-erythritol (c projection). 310 and 310 planes have different 

structure amplitudes. Note positions of atoms with respect to planes in each case. 

which are recorded on the photographs, for in certain circumstances 
the symmetry of the diffraction effects gives some useful information. 
The nature of the relation which exists between the symmetry of 
diffraction effects and that of the crystal may be gathered by considera- 
tion of the hkO intensities of two simple tetragonal crystals urea 
(O C(NH 2 ) 2 ), whose point-group symmetry is 42m, and penta-erythri- 
tol (C(CH 2 OH) 4 ), belonging to class 4. In the case of urea (see Fig. 151), 
the intensity of 310 is the same as that of 3lO, as is obvious from the 
relation of the molecules to the traces of these planes. But in the case 
of penta-erythritol, the relation of the molecules to the 310 planes is 
quite different from their relation to the 3lO planes, and therefore the 
intensity of 310 is very different from that of 3lO. The same is true 



4498 



242 STRUCTURE DETERMINATION CHAP, vn 

for the general planes : hkl and hkl intensities are the same for urea, but 
different for penta-erythritol. 

The planes in penta-erythritol which have correspondin^andices, yet 
give different intensities of X-ray reflections, are just those planes 
which, as crystal faces, have different rates of growth, since the orienta- 
tion of the molecules with respect to the crystal planes determines both 
these properties. Therefore it might be thought that the symmetry 
of diffraction effects the symmetry of the reciprocal lattice, if we 




002 



FIG. 152. Reflection of X-rays in opposite directions by a non-centro- 
symmetrieal structure. The_phase differences of waves from different atoms 
are the same for 002 as for 002 ; hence the intensities are the same. Hence the 
symmetry of an X-ray diffraction pattern ('Laue-symmetry') is the point- 
group symmetry of the crystal plus a centre of {symmetry. 

think of the points of this lattice as eacli having a 'weight* proportional 
to the structure amplitude of the corresponding set of crystal planes- 
is the same as that of the crystal shape, or in other words the point- 
group symmetry. And so it is, except in one important respect : it is 
not possible, except in special circumstances, to decide from X-ray 
diffraction effects whether a crystal has a centre of symmetry or not 
(Friedel, 1913). In a crystal lacking a centre of symmetry (Fig. 152), 
the intensity of reflection by a set of crystal planes depends simply on 
the phase-differences between the waves from different atoms, and these 
phase-differences are normally just the same for reflection in one 
direction 002 in the diagram as they are for reflection in the opposite 
direction 002. Therefore the diffraction symmetry of a crystal is 
normally the point-group symmetry plus a centre of symmetry. Friedel's 
law has been shown to break down when the wave-length of the X-rays 



PLATE XII 





FIG. 153. Laue photographs of ammonium chloride (above) and 
penta-erythritol (below). X-ray beam along fourfold axis. 



POSITIONS OF THE ATOMS 243 

is near that of an absorption edge for some of the atoms in the crystal : 
there is an anomalous phase-change on diffraction which is not reversed 
when the (l>$ction of reflection is reversed (Coster, Knol, and Prins, 
1930; see also Bragg, The CrysMline State, p. 93). But except in these 
very special circumstances the law holds, and therefore the diffraction 
symmetry corresponds with one of the eleven different point-groups 
which are obtained by adding a centre of symmetry to each of the 
thirty-two true point-groups (see Table VI). 

The diffraction symmetry is strikingly shown in Laue photographs 
diffraction patterns produced by sending a beam of X-rays comprising 
a wide range of wave-lengths ('white' X-rays) along a principal axis of 
a stationary crystal. ('White' X-rays are best obtained from an X-ray 
tube with a tungsten target, run at 60 kv. If a copper target is used, 
the characteristic K wave-lengths should be removed by an iron filter.) 
Each crystal plane reflects only those X-rays which have such a wave- 
length that the Bragg equation is obeyed. Laue photographs of 
ammonium chloride (cubic) and penta-erythritol (tetragonal) are shown 
in Fig. 153, Plate XII. Both these crystals have a fourfold axis of 
symmetry, and the X-ray beam is sent down the fourfold axis in both 
eases. It is immediately obvious that in ammonium chloride there are 
apparent planes of symmetry parallel to the fourfold axis, while in 
penta-erythritol there are not. The conclusions that may be drawn 
from these patterns are that ammonium chloride (cubic) must belong 
to one of the point-groups having diffraction symmetry m3m (classes 
43ra, 432, and m3m), while penta-erythritol must belong to one of 
the tetragonal classes having diffraction symmetry 4/m (classes 4/w, 
4, and 4). 

Since the diffraction symmetry is shown so strikingly in Laue photo- 
graphs, it is often called the 'Laue-symmetry'. The information on 
diffraction symmetry is of course all contained in moving-crystal photo- 
graphs taken by monochromatic X-rays, provided that reflections with 
similar indices are separated, as they are in tilted crystal photographs 
and moving film photographs; but Laue photographs show it much 
more obviously. | 

The information obtainable from the Laue-symmetry is meagre ; it 
consists simply in the distinction between crystal classes, and then only 

f Lauo photographs wore formerly muc.h used in structure determination, especially 
by American workers in the years 1 92030 (for methods, see the books by Wyckoff (1931) 
and Davoy (1934)); but the methods described here, in which monochromatic X-rayt 
are passed through rotating crystals, have important advantages and have superseded 
the Laue method. 



244 STRUCTURE DETERMINATION CHAP, vn 

in the more symmetrical systems cubic, tetragonal, hexagonal, and 
trigonal (see Table VI). But it is useful in cases in which morphological 
features do not give clear evidence on this point. ^ 

TABLE VI 



Crystal classes 



Triclmie: 1, 1 
Monoclinio: m, 2. 2jm 
Orthorhombio : mm, 222, mmm 
Tetragonal: (a) 4, 4, 4/w 

(b) 42m, 4mm, 422, 4/mmm 
Trigonal and hexagonal : 

(a) 3, 3 _ 

(b) 3w, 32 3m 

(c) 6, 6, 6/w 

(d) 62m, 6mm, 622, 6/mmm 
Cubic: (a) 23, m3 

(6) 43m, 432, m3m 



Laue-symmetry 



1 

2/m 
mmm 
4/m 
4/mmm 

3 

3m 

6/m 

6/mmm 

m3 

m3m 



Space- group symbols. All the symmetry elements which can be 
discerned in all possible arrangements of atoms have now been 
mentioned. The number of different symmetry elements is not large ; 
nevertheless, as may be imagined, the number of different ways of 
arranging asymmetric groups of atoms by combining the various 
symmetry elements in different ways is considerable. The total number 
is in fact 230. Three different crystallographers, SchOnflies, Fedorov, 
and Barlow, all working independently, had derived the complete list 
in the years 1890-4 long before the advent of X-ray methods made 
it possible to utilize the knowledge. Diagrams showing the symmetries 
of all the space-groups are to be found in Int. Tab. ; one of them is 
reproduced in Fig. 161 to illustrate the conventional representation (in 
a projection) of the commonest symmetry elements. 

In referring to any particular space-group, the symbols for the 
symmetry elements are put together in a way similar to that used for 
the point-groups. First conies a capital letter indicating whether the 
lattice is simple (P for primitive), body-centred (/ for inner), side- 
centred (A 9 B, or C), or centred on all faces (F). An exception is the 
hexagonal lattice, which is, strictly speaking, primitive but is described 
by the letter C for reasons given previously. For some hexagonal 
structures the H cell containing three lattice points is used (Fig. 137). 
The rhombohedral lattice is also described by a special letter J?. 
Following the capital letter for the lattice type comes the symbol for 
the principal axis, and if there is a plane of symmetry or a glide plane per- 



CHAP, viz POSITIONS OF THE ATOMS 245 

2 2 

pendicular to it, the two symbols are associated thus: P >, P -^ or, 

m c 

more conveniently for printing, P2/m, P^/c. Then follow symbols for 
the symmetries of secondary axes, and planes of symmetry or glide 
planes parallel to the axes. The set of symbols is often abbreviated, 
only such symbols as are necessary for unique characterization of the 
space-group being given. Thus, it is not necessary to write P2/m 2/m 2/m 
since Pmmm implies the existence of twofold axes as well as planes of 
symmetry. Note that P222 is a different space-group having no planes 
of symmetry. 

Procedure in deducing the space-group. The number of possible 
space-groups for a crystal under investigation is, of course, limited by 
the knowledge (usually already possessed at this stage) of the crystal 
system to which it belongs. From this point it is often possible to 
identify the space-group unequivocally from the X-ray diffraction 
pattern. 

In examining a list of X-ray reflections for this purpose, it is best 
to look first for evidence of the lattice type whether it is simple (P) 
or compound; systematic absences throughout the whole range of 
reflections indicate a compound lattice, and the types of absences show 
whether the cell is body-centred (7), side-centred (A, J5, or 6 Y ), or face- 
centred (F). When this is settled, look for further absences ; systematic 
absences throughout a zone of reflections indicate a glide plane normal 
to the zone axis, while systematic absences of reflections from a single 
principal plane indicate a screw axis normal to the plane. The result 
of such a survey, followed by an examination of the list of absences for 
all space-groups (see Int. Tab., or Astbury and Yardley, 1924) may be 
to settle the space-group unequivocally. Thus orthorhombic crystals 
of methyl urea CH 3 .NH.CO.NH 2 give all types of reflections except 
AOO with h odd, O&O with k odd, and OOZ with I odd. The lattice is 
evidently primitive, and there are no glide planes ; but there are screw 
axes parallel to all three edges of the unit cell. The only possible space- 
group is P2 t 2 X 2! (Corey and Wyckoff, 1933). 

As another example, ^-azoxyanisole 



CH 3 N=N OCH 3 

A ^-^ 

is monoclinic, and since it gives all types of general (hkl) reflections, 

its lattice is primitive; but in the hOl zone, reflections for which h+l 
is odd are absent, indicating a glide plane with a translation of 



246 



STRUCTURE DETERMINATION 



CHAP. VII 



GL'Q 

I 



1 



N' 



(symbol n), and also the odd orders of OJfcO are absent, indi- 
cating that there is a screw axis parallel to b. The only possible 
space-group is P2 1 /n (Bernal and Crowfoot, 1933 a). (Tb& could be 
described alternatively, with a change of a and c axes, as P2 l /a or 
P2 1 /c.) 

Our third example will more complex. The X-ray diffraction patterns 
of 1,2 dimethylphenanthrene (Bernal and Crowfoot, 1935) show no hkl 
reflections having k~\-l odd; the lattice is therefore centred on the a 
face (symbol A). In the hOl zone, reflections with / odd must of course 
be absent, since for these k is and k+l is thus odd; but it is found 
that hQl reflections with h odd are also absent. 
Evidently there is, perpendicular to the b axis, a 
glide plane with translation a/2. The only other 
zone showing additional systematic absences is Qkl ; 
not only are reflections having k-\-l odd absent 
(owing to the A face-centred lattice), but also all 
reflections having k odd or h odd are absent; it 
therefore appears that there is, perpendicular to 
the a axis, a glide plane having one translation of 
^/^ an( ^ another translation of c/2. This appears at 
first sight a new sort of glide plane not previously 
mentioned ; but actually, owing to the A face-centred 
lattice, a glide of 6/2 is indistinguishable from a glide 
of c/2. This is illustrated in Fig. 154, which sym- 
bolizes the projection in question. A group of atoms 
M at the corner of the cell is repeated at the face-centre P ; if we imagine 
these groups reflected in the plane of the paper and translated 6/2, M 
reaches N and P reaches Q. Exactly the same result would be obtained 
by translating c/2 ; M would reach Q', which is equivalent to Q, and P 
would reach JV', which is equivalent to N. It therefore does not matter 
whether we call this a glide of \b or- \c ; and the space-group may be 
called Aba or Aca\ convention calls it Aba. Reference to the list of 
space-groups shows that Aba is the only one causing this particular 
combination of absences. (The verdicts on space-groups in these ex- 
amples could have been arrived at mechanically, by simply noting the 
absent reflections and looking up the list of space-groups ; but it is best 
to approach such problems from first principles. Reference should, 
however, always be made to the list of space-groups to avoid missing 
any possibilities.) 

Other examples will be found in Chapter IX ; these include rutile, 



Fro. 154. Lattice 
side-centred on 100 
(symbol A); glide 
plane perpendicular 
to the a axis (that is, 
in the plane of the 
paper). 



CHAP, vii POSITIONS OF THE ATOMS 247 

the structure whose general arrangement was assumed earlier in this 
chapter. 

X-ray diffraction patterns alone do not always settle the space-group 
uniquely; for instance, an orthorhombic crystal whose X-ray diffrac- 
tion pattern exhibits no systematic absences may have either space- 
group symmetry Pmmm, or alternatively P222, or Pmm. In such a 
case morphological features may indicate whether there are three 
reflection planes (holohedral class mmm), three twofold axes and no 
reflection planes (enantiomorphic class 222), or two reflection planes 
intersecting in a single twofold axis (polar class mm). Caution is 
necessary here, because the shape of a crystal may have a symmetry 
higher or lower than that of the atomic? arrangement (see Chapter II). 
If, in crystals grown from a solution or melt of high purity, there is 
definite evidence of enantiomorphic or polar character, the crystal 
class and with it the space-group are settled, but if the crystal shape 
has holohedral symmetry, it is by no means certain that the atomic 
arrangement has holohedral symmetry. 

When the conditions of growth of a crystal are unknown the 
evidence of its shape should be regarded with reserve. There are, indeed, 
cases in which the shape of a crystal is inconsistent with clear X-ray 
evidence on atomic structure for instance, cuprite Cu 2 O (Greenwood, 
1924; Bragg, 1937; Miers, 1929). Possibly this is due to the presence 
of impurities during growth (see p. 53). 

If the combination of X-ray and morphological evidence does not 
determine the space-group uniquely, additional information may be 
sought by tests for piezo -electric and pyro-electric properties, and by 
an optical examination for any evidence of rotation of the plane of 
polarization. (See Chapter VIII.) The results of such tests may settle 
the matter, since only certain crystal classes have these properties. 
Only positive results are decisive ; the apparent absence of piezo- 
electric or pyro-electric effects may be due to feeble phenomena. 

If after such tests the space-group is in doubt, there is no other 
course than to proceed with the next stage in the interpretation of the 
X-ray patterns, trying arrangements in each of the possible space- 
groups in turn. There may be stereochemical reasons for supposing 
that one arrangement is more likely than others, and this arrangement 
will naturally be tried first. Such possibilities cannot be discussed in 
general terms ; they are specific for each crystal. Familiarity with the 
general background of crystal chemistry and molecular stereochemistry 
is desirable. 



248 



STRUCTURE DETERMINATION 



CHAP. VII 



Information given by a knowledge of the space-group. 

1. Molecular or ionic symmetry. If the space-group of a particular 
crystal has been determined unequivocally, this knowledge yiay make 
it possible to draw certain definite conclusions about the 'symmetry of 
the molecules or ions of which the crystal is composed and this 

without any attempt to discover the 
positions of individual atoms. 

Consider the space-group sym- 
metry P2 1 /a. A structure having 
this symmetry can be built by 
placing a group of atoms in any 
general position in the unit cell, and 
repeating this group in accordance 
with the demands of the complex 
of symmetry elements. Thus, in 
Fig. 155, if group A (symbolized by 
a question mark because we do not 
yet know anything about the posi- 
tions of individual atoms) is regarded 
as the reference group, the glide 
plane a t gives rise to group B ; and 
the lower screw axis creates C from 
A, as well as a second glide plane a 2 
from a l9 and thence D from C. Now 
the group of atoms forming the 
element of structure need have no 
symmetry at all. The symmetry 
P2j/a can be attained by arranging 

four asymmetric groups in the manner indicated. (All monoclinic 
crystals of the holohedral class with primitive lattices contain four 
asymmetric groups.) But if we find that a particular crystal has this 
symmetry but contains only two molecules in the cell, then each 
molecule must have twofold symmetry of some kind. If the substance 
is a high polymer, each molecule may possess either a screw axis or 
a glide plane or a centre of symmetry, since all these twofold elements 
of symmetry are present in the group P2j/a ; but if the substance is a 
mpnomer, the molecules cannot have symmetry elements of translation, 
and therefore must each have a centre of symmetry. The asymmetric 
element of structure is half a molecule ; each molecule consists of two 
asymmetric halves related by a centre of symmetry. 




FIG. 155. Space-group P2j/a. 



CHAP. VII 



POSITIONS OF THE ATOMS 



249 




A good example of the value of such evidence is the conclusion that 
the molecule of diphenyl C 6 H 5 . C 6 H 5 has a centre of symmetry. The 
situation ia>. exactly that just described the space-group symmetry is 
P2 1 /a and there are only two molecules in the unit cell ; hence each 
molecule has a centre of symmetry (Hengstenberg and Mark, 1929; 
Clark (G. L.) and Pickett, 1931). Assuming that the benzene ring is 
planar and that the connecting link also lies 
in the nuclear plane, the fact that in the 
crystal the molecule has a centre of symmetry 
leads at once to the important stereochemical 
conclusion that the two rings are coplanar; 
any twist at the single bond would destroy 
the centre of symmetry (Fig. 156). Even if 
no assumptions are made, it is still certain 
that the mean planes of the two rings are 
parallel. 

But suppose a crystal having this same sym- 
metry P2 1 /a is found to contain four molecules ? 
It would appear at first thought that each 
molecule is asymmetric, since it requires four 
asymmetric objects (two left-handed and two 
right-handed) to make up this symmetry. This 

conclusion, however, would not necessarily be correct; it embodies the 
assumption that the asymmetric object is to be identified with the 
molecule an assumption which is not warranted. The asymmetric 
unit in a crystal may be, and often is, a molecule ; but it is not neces- 
sarily one particular molecule it may be half one molecule and half 
another, the two molecules being geometrically different and unrelated 
by symmetry operations. Consider an actual example. The crystal of 
stilbene, C 6 H 5 .CH=CH.C 6 H 5 , has the space-group symmetry P2 2 /a, 
and there are four molecules in the unit cell (Robertson, Prasad* and 
Woodward, 1936; Robertson and Woodward, 1937). If the asymmetric 
object were any one particular molecule, then there would be in the 
unit cell two left-handed and two right-handed molecules, mirror 
images of each other. It turns out, however, that there are two types 
of unrelated molecules, both having a centre of symmetry; the asym- 
metric unit is half one of these molecules and half the other, the halves 
being obtained by mentally bisecting the molecules through their 
centres of symmetry (see Fig. 157). These two types of molecules are 
geometrically slightly different from each other; they are not stereo- 



Fio. 156. Structure of di- 
phenyl molecule (in the 
crystal). The two benzene 
rings must be coplanar (left) ; 
any twist at the connecting 
link (right) would destroy 
the centre of symmetry. 



250 



STRUCTURE DETERMINATION 



CHAP. VII 



isomers in the ordinary sense the difference is due to the distorting 
effects of the different surroundings of the two types. The differences 
are more marked in trans-azobenzene, C 6 H 5 .N=N.C 6 H 5 ,^which has 
a similar structure ; half the molecules are flat, while in the rest the ben- 
zene rings are rotated 15 out of the plane of the central C N=N C 
group (Lange, Robertson, and Woodward, 1939). That chemically 
identical molecules should arrange themselves so that the surroundings 
of half of them are different from those of the rest seems odd. Evidently 




FIG. 157. Structure of stilbene. The asymmetric unit (ringed) consits of half 

one molecule and half another, the two types of molecules being unrelated by 

symmetry elements (Robertson and Woodward, 1936). 

intermolecular forces and the requirements of good packing are satis- 
fied better by this arrangement than they would be by an arrangement 
of crystallographically equivalent molecules ; but the phenomenon does 
not appear to be understood in detail. These examples show that 
caution is necessary in forming conclusions on molecular symmetry: it 
is possible to conclude, from a knowledge of the space-group and the 
number of molecules in the unit cell, only that the molecule in its crystal 
setting has a certain minimum symmetry. In truth the molecular 
symmetry may be higher (if, as in the examples given, there are non- 
equivalent molecules), but it cannot be lower than that indicated by 
the sort of evidence considered here. The minimum symmetries of 
molecules in the various space-groups, taking into account different 
possible numbers of molecules in the unit cell, are given in Astbury 
and Yardley's tables (1924). (These symmetries do not refer to High 
polymer molecules.) 



CHAP, vii POSITIONS OF THE ATOMS 251 

The above remarks refer to the symmetries of molecules in crystals. 
It is very important to remember that the symmetry of a molecule in 
its crystal seating is not necessarily the full symmetry of an isolated 
molecule, since, as we have seen, the full symmetries of molecules are 
not always utilized in forming crystalline arrangements. Suppose, for 
example, that X-ray and other evidence leads to the definite conclusion 
that certain molecules in their crystal setting have no symmetry. It 
does not follow that these molecules in isolation are asymmetric : it may 
be that in isolation they would have axes of symmetry or planes of 
symmetry, but that these are ignored in the formation of the crystalline 
arrangement. One element of symmetry, however, they are not likely 
to possess a centre of symmetry. Centrosymmetrical molecules do not 
usually form non-centrosymmetrical arrangements though it cannot 
be said that this is impossible. (There is one interesting case in con- 
nexion with this point. The space-group of 4,4' dinitrodiphenyl is said to 
be the non-centrosymmetrical PC, though the molecules themselves are 
centrosymmetrical or nearly so (Niekerk, 1943). The coordinates of 
the atoms are, however, not significantly different from those of the 
centrosymmetrical space-group P2 a /c, and the case is therefore not a 
convincing one: it deserves further investigation.) 

When the number of molecules in the unit cell is greater than the 
number of asymmetric units necessary to give rise to the space-group 
symmetry, it is certain that there are in the crystal two or more 
crystallographically non-equivalent types of molecules. This is so in 
ascorbic acid (vitamin C), for instance (Cox, 1932 a ; Cox and Goodwin, 
1936). 

2. Molecular dimensions. In research on substances of unknown or 
partially known constitution, a knowledge of the shape and size of the 
molecules may be of great value in confirming or rejecting suggested 
structural formulae. It has been pointed out (p. 187) that when a unit 
cell contains only one molecule, the dimensions of the cell suggest 
possible molecular dimensions. But clearly, When there is more than 
one molecule in the unit cell, the shape and size of the cell merely 
suggest the possible dimensions of a group of molecules : the dimensions 
of an individual molecule cannot be deduced directly, for the cell might 
be divided into two or more identical volumes in an infinite variety 
of ways. If, however, the space-group is known, this knowledge may 
lead to the conclusion that the molecules are related to each other by 
certain symmetry operations, and this restricts the number of ways of 
subdividing the unit cell. Still further restrictions may be imposed if 



252 STRUCTURE DETERMINATION CHAP, vn 

the constitution of the molecule is partly known. A knowledge of some 
of the physical properties, particularly the birefringence, pleochroism, 
or magnetic anisotropy, may lead to definite conclusion on the 
orientation of whole molecules or of particular atom groups (see 
Chapter VIII), and these conclusions may impose still further re- 
strictions on the mode of subdivision of the unit cell, and in fact may 
determine definitely the overall size and shape of each molecule. This 
subject cannot be discussed in general terms ; each substance presents 
its own specific problems. It is desirable in considering the possi- 
bilities to use models in which the atoms have the correct effective 
external radii : abstract thinking or drawing diagrams on paper is apt 
to be misleading. * 

An outstanding example of the use of such methods is the work of 
Bernal (1932) on substances of the sterol group. X-ray and optical 
evidence led to the conclusion that these molecules have the approxi- 
mate dimensions 5x 7-2x 17-20 A, and this played an important part 
in the abandonment of earlier structural formulae and the elucidation 
of the correct constitution (Rosenheim and King, 1932). 

3. Location of atoms in relation to symmetry elements. Equivalent 
positions and their multiplicities. After gaining as much knowledge as 
possible on the general shape, orientation, and symmetry of the 
molecules or ions in a crystal, the next step is to try to locate particular 
atoms. This it is sometimes possible to do by reasoning of the type 
already used. Thus, if an atom is placed in a space-group having six- 
fold rotation axes, it is inevitably multiplied by 6, unless it is placed 
actually on one of the sixfold axes; therefore, if, in a space-group 
having sixfold rotation axes, it is found that there is only one atom of 
a particular kind in the unit cell, that atom must lie on a sixfold axis, 
for only in this position can sixfold repetition be avoided. This brings 
us to a general consideration of the multiplicity of different types of 
positions in the various space-groups: 

A 'space-group' is a group of symmetry elements. If an atom is 
placed in a quite general position in the unit cell, it is inevitably 
multiplied by the symmetry elements, and thus other atoms, exactly 
equivalent to the first, are found at other positions which are related 
in a precise way to those of the first. Each space-group has its own 
characteristic number of equivalent general positions. Thus all primi- 
tive (that is, not centred) space-groups in the polar and holoaxial classes 
of the orthorhornbic system (classes mm and 222) have four equivalent 
general positions, while primitive holohedral space-groups (class mmm) 



CHAP. VII 



POSITIONS OF THE ATOMS 



253 



have eight equivalent general positions. Thus, if in space-group Pmm, 
an atom is placed in a general position xyz (see Fig. 158, left), one 
plane of 'symmetry creates another at x yz, and the second plane of 
symmetry creates another from each of these, at xyz and xyz respectively. 
General positions in space-group Pmm are thus fourfold positions. 

Suppose, however, we place an atom on one of the planes of sym- 
metry, m 2 , at position xOz. Plane m 2 does not create a second atom ; it 
merely makes one-half of the atom the mirror image of the other half. 
But plane m l does create another atom, at position xOz (Fig. 158, right). 
Similarly if an atom were placed on m 1 at position Qyz, only one more 




^4 



m 7 



FIG. 158. Space-group Pmm. Left: fourfold (general) positions. 
Right : twofold positions. 

atom would be created, at Oyz. Thus positions on planes of symmetry 
are twofold positions. It should be noted that there are also planes of 
symmetry half-way along a and b edges of the cell, and therefore x\z 
is a twofold position (reflected by m[ as xjz), as is also \yz (reflected by 
m 2 as \yz). 

Finally it will be evident that, if an atom is placed at any position 
on a twofold axis where two planes of symmetry intersect for instance, 
at OOz it is not multiplied at all ; OOz is thus a onefold position. Other 
onefold positions are 0z, OJz, and \\z\ in other words, in this space- 
group any position on any one of the twofold axes is a onefold position. 

If, in a crystal having this space-group symmetry Pmm, there is only 
one atom of a particular kind, it must necessarily lie on one of the two- 
fold axes. If there are two chemically identical atoms, some caution 
is necessary; they may occupy one set of twofold positions, but another 
possibility which cannot be excluded is that the atoms may actually 
occupy two of the (crystallographically non-equivalent) onefold 
positions. Similarly when there are four chemically similar atoms, 



254 



STBUCTUBE DETERMINATION 



CHAP. VII 





CO 



they may occupy the fourfold (general) positions, or two independent 
sets of twofold positions, or even four independent onefold positions. 

In general, any position lying on a plane of symmetry, a Dotation axis, 
an inversion axis, or a centre of symmetry (these are the symmetry 
elements which do not involve translation) is a special position, having 
fewer equivalent companions than the general positions. But note care- 
fully that positions lying on screw axes or glide planes are not special 
positions (unless they also lie on non-translatory elements) : an atom 

lying on a screw axis or a glide plane is 
multiplied just as surely as if it were 
lying at a distance from the symmetry 
'element see Fig. 159. And in space- 
groups exhibiting translatory symmetry 
elements only, all positions in the cell 
are general positions; this is so, for 
instance, in the frequently occurring 
space-group P2 1 2 l 2 l , the equivalent 
positions in which are illustrated in 
Fig. 1 60 : every position in the cell has 
fourfold multiplicity. 

In many space-groups some of the 
positions of low multiplicity arc fixed 
positions in the cell : no variable para- 
meters are involved. Thus in the 

monoclinic space-group P2Jc the twofold positions are centres of 
symmetry. There are four different pairs of centres of symmetry, 
the coordinates of which are : 

(a) 000, 

(b) 00, i 

(c) OOJ, 

(d) M, 

There are no onefold positions, and all other positions in the cell are 
fourfold. If in a unit cell having this symmetry there are only two 
atoms of a particular kind, these must lie in one pair of centres of sym- 
metry; and we may choose which pair we like, for all give the same 
arrangement as far as these atoms are concerned. This is the situation 
in platinum phthalocyanine : the two platinum atoms in the cell lie 
without any doubt in a pair of symmetry centres (see p. 342). 

Note, however, that if in this same space-group P2 3 /c there is one 
pair of atoms of one kind and one pair of another kind, each pair of 



FIG. 159. Atoms lying on screw axes 

or glide planes are multiplied, no loss 

than those at a distance. 



CHAP. VIJ 



POSITIONS OF THE ATOMS 



255 



atoms must occupy a pair of symmetry centres ; but the relation between 
the two pairs of atoms is not uniquely defined, for the arrangement 
with one pairjui (a) and the other in (6) is not the same as the arrange- 
ment with one pair in (a) and the otfyer pair in (c) ; and the combination 
(a)+(d) is a third different arrangement. The correct arrangement may 
be found by calculating the intensities of the X-ray reflections for the 
different arrangements; this must be done simultaneously with the 
determination of the parameters of the rest of the atoms in the cell. 
(See next section, and also p. 307.) 




FIG. 100. Equivalent positions in space-group P2 1 2 1 2 1 . 

In all except very simple structures most atoms lie either in partly 
restricted positions involving one or two variable parameters or in 
general positions involving three variable parameters. We shall pass 
to the methods of determining these variable parameters in the next 
section ; but before leaving this section there is one other very useful 
consideration to be mentioned. If a structure is known to have twofold 
axes (not screw axes) and certain atoms do not lie on the twofold axes, 
the distance of the centre of one of these atoms from a twofold axis 
cannot be less than the radius of the atom, for any smaller distance 
would mean that two equivalent atoms on opposite sides of the twofold 
axis overlap each other. Similarly the centre of an atom must lie 
either on a plane of symmetry (a reflection plane, not a glide plane) or 
else at a distance from it not less than the atomic radius. The same is 
true for a centre of symmetry. For three-, four-, and sixfold rotation 
axes we have to imagine a ring of atoms round the axis 4 none of them 
can touch the axis, hence the smallest permissible distance is greater 
than the atomic radius. Such considerations have played a large part 



256 



STRUCTURE DETERMINATION 



CHAP, vn 



in the solution of silicate structures such as that of beryl, Be 3 Al 2 Si fl 18 
(Bragg, W. L., and West, 1926). 

The space-groups illustrated here are simple ones having a maximum 
multiplicity of 4. In many space-groups, however, the multiplicity of 
the general positions is much higher 16, or 32, or even 96 in some of 
the highly symmetrical space-groups. The multiplicities and the co- 
ordinates of equivalent positions (both general and special) are given 
in Int. Tab. Specimen diagrams from Int. Tab. are reproduced in 
Fig. 161 to show the conventional way of representing equivalent 



particle at +z 



-0 
-O 



left-handed 
particle at +z 



Ltft-tanM 
, particle ail-z 

-0 + 



\ \ 



ne (translation 
)atU 

\ \ 



Oi- - 



t- -o 

0* *+O 



-O O* 



i- -O 

+ 



O i 

Ot- -0 I ^ 
i- -O O 



o* 



Oi- 



'c glide \nglidc \ 






I 

2atz=0 



I' I 




FIG. 161. Symmetry elements and equivalent positions in space-group 
Omca (Internationale Tabellen, 1, p. 141). 

positions and the commonest symmetry elements. Some examples of 
the use of the tables of equivalent positions will be found in Chapter IX ; 
attention is drawn first of all to the derivation of the structure of sodium 
nitrite NaN0 2 , since the arrangement has the symmetry I mm, closely 
related to the one (Pmm) used as an example here. 

The application of the theory of space-groups to crystal structure 
determination involves the assumption that equivalent sites in the 
crystal are occupied by identical atoms. Fortunately this is true for 
the majority of crystals, though there are some exceptions. Considera- 
tion of the abnormal types of crystals is deferred to the end of Chapter 
IX. For the present we will continue with the account of the methods 
of solving normal structures. 

Determination of atomic parameters . At this stage in the attempt 
to find the positions of the atoms in the unit cell it is known that the 
complete arrangement has certain symmetries. If the space-group is 
P2 1 2 1 2 1 (Fig. 160), a particular atom M l which may be tal^en as the 
standard of reference is repeated by the symmetry elements in other 



CHAP, vn POSITIONS OF THE ATOMS 257 

definite positions in the cell M 2 , M 3 , and M 4 ; a second type of atom 
N v which is formally independent of the first-mentioned, is also 
repeated in other definite positions JV 2 , N- 3 , and JV 4 by the same scheme 
of symmetry 'elements ; and so on. Each type of reference atom has 
three parameters x, y, and z, which are the coordinates of the atom 
with respect to the unit cell edges expressed as fractions of the lengths 
of thsse edges. If the coordinates of each reference atom can be found, 
the other three of each group are bound to be at x, y, +z, at 
f +z, \y, z, and at x, %+y, \z (if the origin of the cell is, in 
accordance with convention, a point half-way between neighbouring 
screw axes as in Fig. 160). 

The procedure in determining the parameters will naturally vary 
with circumstances, but a few general principles can be given. First, 
as to the most convenient method of calculation. We have seen (p. 212) 
that for any crystal plane hkl the contribution of each atom (coordinates 
xyz) to the expression for the structure amplitude consists of a cosine 
term /cos 2n(hx+ky+lz) and a sine term /sin 2ir(hx+ky+lz). Equiva- 
lent atoms (those related to each other by symmetry elements) have 
coordinates which are related to each other in a simple way, and on 
this account the cosine terms for the whole group may be combined to 
form a single expression, the evaluation of which is usually more rapid 
and convenient than the process of dealing with each atom separately. 
Thus, the cosine terms for four equivalent atoms in space-group P2 1 2 1 2 1 
(coordinates given above), on being expanded and combined, are found 
to be equal to the expression 

A = 



The sine terms combine in a similar way to form the expression 
B = ^4/sin 



If the combined expressions are used, it is only necessary to consider 
each reference atom in turn ; the expressions take care of the rest of 
each group. The cosine term for each independent group is evaluated 
and then all the cosine terms are added together. Sine terms for all the 
independent groups are likewise added together. A 2 +B Z is then = F", 
which for an 'ideally imperfect' crystal is proportional to the intensity 
the hkl reflections would have if the atoms really were in the postulated 
positions. The combined expression for the contribution of a set of 

4458 S 



258 STRUCTURE DETERMINATION CHAP, vn 

equivalent atoms, for each space-group, is to be found in Int. Tab. 
and in a book by Lonsdale (1936), 

When there is only one variable parameter in a crystal stiucture its 
determination is straightforward : calculation of the intensities of various 
reflections for a range of values of this parameter (as was done for rutile 
on p. 214) leads directly to the selection of the value which gives the 
best agreement with the observed intensities. It is usually necessary 
to calculate only for a small range of parameters, for it is likely to be 
obvious, from the strength or weakness of certain reflections, that the 
parameter can only be near a certain value (see again the evidence on 
rutile, p. 215). Usually, reflections at small angles fix it approximately, 
while the high-order reflections at large angles lead to a more accurate 
value.f 

Similar methods can be used whenever a parameter can be isolated. 
This is well illustrated by the determination of the structure of 
ammonium hydrogen fluoride NH 4 HF 2 by Pauling (1933). The crystals 
are orthorhombic, and the unit cell has the dimensions a = 8-33 A, 
6 = 8* 14 A, c 3-68 A and contains four molecules. The general 
arrangement, which has the symmetry Prnan, was suggested by refer- 
ence to the already known structure of KHF 2 (Bozorth, 1923) and a 
consideration of the modifications brought about by the formation of 
hydrogen bonds between nitrogen and fluorine atoms : in this arrange- 
ment the nitrogen atoms are at JJzp |Jz 1 , Jfzj, and ffzp half the 
fluorine atoms at #00, 00, (J+#)iO an( i (| #)iO an d the others at 
\y*K \yz* 0($ y)z 2 , and 0(J+y)z 2 . 

The intensities of the JiQQ reflections depend only on the value of x 
(this is the only parameter along the a axis) ; hence the relative intensi- 
ties of the various orders of hOQ lead to the determination of x. (The 
results are presented in Pauling's paper in the form of curves like those 
for rutile in Fig. 129.) Similarly y was found from the relative intensities 
of the O&O reflections. Along the c axis there are two parameters- z x and 
2 2 , but Zj was isolated from z 2 by considering only those likl reflections 
which have h odd and k odd ; if the expressions for the contributions 
of all the atoms in the cell are combined together, the complete structure 
amplitude for these reflections is found to be 

F = 4/ F (cos 27T&E cos 2irky cos 27rZz 2 ), 

f This ij not true in all circumstances. For instance, the diffracting power of the oxygon 
atom falls off with increasing angle of diffraction much more rapidly than does that 
of iron so much so that in lepidocrocite (^-FeO.OH) the intensities of the high-order 
diffractions are determined almost entirely by the positions of the iron atoms ; the para- 
meters of the oxygen atoms must be fixed by means of some of the low-order diffractions. 



CHAP, vii POSITIONS OF THE ATOMS 259 

an expression which does not contain z v The already known values 
of y and x were used, and thus z 2 , the only variable in the expression, 
could be determined in the same straightforward way. This left only 
one parameter, z 1? to be determined ; this could have been found from 
the intensities of various remaining types of reflections ; actually it was 
found from the relative intensities of hkl reflections having h and k 
both even, the structure amplitude for these being 

/h-4-k \ 
F = 4/ Nn4 cos27r[--^-^ + Z2j+4^^ 



In the more complex structures it is usually necessary to determine 
a number of parameters simultaneously ; in crystals of complex organic 
compounds, for instance, more often than not the atoms are all in general 
positions and there are therefore three parameters for each atom. Such 
problems cannot be solved by the methods described for isolated para- 
meters. It is necessary to postulate likely positions for the atoms, to 
calculate the intensities which these positions would give, and to com- 
pare these calculated intensities with those observed. The prospects 
of success depend on whether the postulated positions are anywhere 
near the correct positions, giving some measure of agreement with 
observed intensities ; if they are, the correct positions can be found by 
judicious small displacements of some or all the atoms from the positions 
first chosen. 

The following discussion relates chiefly to complex crystals in which 
all the atoms have much the same diffracting powers crystals such as 
those of many organic compounds ; for it is in these circumstances that 
the indirect method of trial and error must still be used. For crystals 
containing a minority of heavy atoms together with a larger number 
of lighter atoms in the unit cell, the direct or semi-direct methods 
described in Chapter X arc more appropriate. 

The first step should be to gain a general idea of the distribution of 
the atoms. A general idea of the shape and orientation of the molecules 
may have been gained by reasoning based on a knowledge of the unit 
cell dimensions and the space-group, together with physical properties 
such as optical or magnetic birefringence, and whatever stereochemical 
knowledge is cavailable on the type of substance in question. If so, this 
will lead to the. postulation of approximate atomic coordinates. 

Failing this, a direct attack on the problem of the interpretation 
of the intensities of the X-ray reflections may be made. In any case, 
this is the next stage. The only X-ray intensities so far considered 




260 STRUCTURE DETERMINATION CHAP. VTI 

quantitatively (in discovering the space-group) have been zero intensities ; 
the next to be considered are those of maximum intensity, since these 
may indicate the whereabouts of the greatest concentrations of atoms. 
The strongest reflections are usually those at small angles . If one of these 
is much stronger than any others, it may be that all the atoms lie on or 
near the lattice planes responsible for this outstandingly strong reflec- 
tion. If absolute (not merely relative) intensities are available, this 
idea can be checked directly ; if the intensity is about the maximum 
, possible by co-operation of diffracted 

waves from all the atoms in the cell 
(taking into account all angle factors), 
then all the atoms must lie on or near 
the reflecting planes. Even if absolute 
intensities are not available, it is still 
possible to check the idea, though in a 
less direct way; if all the atoms lie on 
particular crystal planes, then the struc- 
ture amplitudes for all orders of reflection 
FIG. 162. Orientation of molecules fr m these plaiies must be equal and there- 
in the crystal of benzoquinone. fore the actual intensities of the different 

S ZS?S p.JSft orders diminish re g ularl y from one to the 

the great strength of 201 shows next, owing to the effect of the angle 

facto - This ar g* has been ch 
used; for an example, see the structure 

of sodium nitrite, p. 303. If a decline is maintained only for two or 
three orders, the higher ones being apparently erratic, or if the decline 
is greater or smaller than the normal decline due to angle factors, then 
the atoms must be somewhat dispersed from the planes in question. 
As an example, in the determination of the structure of benzoquinone 
(Robertson, 1935 a) the great strength of the 20T reflection and the 
rapid decline of the intensities of the second and third orders (402 and 
603) were used as evidence thut the flat molecules lie nearly, but not 
quite, along the 201 planes (see Fig. 162); and the final result of the 
analysis showed that this is correct. In crystals containing atoms of 
widely different diffracting powers, such evidence may mean that the 
more strongly diffracting atoms lie on the planes in question, with the 
more weakly diffracting ones somewhere in between. 

If two or three reflections in the same principal zone are much stronger 
than all the others, the atoms lie approximately along lines parallel to 
this zone axis. Fibres of crystalline chain polymers, for instance, often 



OHAP. vn 



POSITIONS OF THE ATOMS 



261 



give two or three very strong reflections from planes parallel to the fibre 
axis ; this shows that in the crystalline regions the chain molecules are 
fairly well 'extended (not meandering) and parallel to the fibre axis: if 
we look along the chain axis (usually called c), we see the ab projection, 
and in this projection the chains, seen end-on, appear to be compact 
groups of atioms. Thus, in polyethylene (Bunn, 1939) and some of the 
polyesters (Fuller, 1940) the ab projection of the unit cell is rectangular 
(Fig. 163) and two chain molecules run through it; the, 200 and 110 



I x 



V. 

X f 


T ; r 


^ x 


s ^ 

no b 

V -* 


} ^. : x X 

Y px V Y 


A x 
x o 

y x 


rfo 

2 


Jv^a, ^X 
00 -LA 2 


A V 

;?o 

00 



110 



110 





FIG. 163. Packing of chain molecules 
in polyethylene and some of the 
polyesters. The chains, sden end-on, 
appear to be compact groups of 
atoms at P and Q. 



0/0 

FIG. 164. High polymers whose unit 
cells projected along the fibre axis are 
not rectangular may give three very 
strong reflections. If the fibre axis is c, 
these reflections may be 100, 010, 
and lIO. 



reflections are far stronger than any others, and this shows that the 
atoms appear from this viewpoint to be in compact groups at the 
corners and centres of the projected cells. In polyisobutene the 200 
and 110 planes have the same spacing (Fuller, Frosch, and Pape, 1940), 
and in the fibre photograph (Fig. 112, Plate IX) these reflections are 
superimposed, forming a spot of very great intensity. If the projected 
cell is not rectangular, there may be three very strong reflections, from 
planes such as those in Fig. 164. 

Such considerations, applied to the low-order (small angle) reflections, 
give a general idea of the distribution of the atoms, if these happen to 
be in sheets or in strings. In the case of molecular crystals they indicate 
the general orientations of sheet or chain molecules. The occurrence in 
these crystals of the atomic centres in compact groups is due partly to 
the chain or sheet structure of the molecules, and partly to the fact 
that the effective external radii of atoms (which govern the distances 



262 



STRUCTURE DETERMINATION 



CHAP. VIJ 



between molecules) are much greater than the distances between 
atoms linked by primary bonds. 

Any outstandingly strong reflections at large angles may give similar 
information about the distribution of the atoms within the molecules. 
For instance, the crystal of chrysene (monoclinic, a = 8-34 A,6 = 6-18A, 
c = 25-OA, j8= 115-8, space-group I2/c) gives a very strong 0.3.17 
reflection, and this suggests that the atoms lie on planes of this 
type. But it is not known whether the phase of the reflection (the sign 




(a) k^ (b) 

FIG. 165. Structure of chrysene. 

of the structure amplitude) is positive with respect to the centre of 
symmetry at the origin which would mean that the atoms lie on the 
'positive' planes in Fig. 165 b or negative, as it would be if the atoms 
lay on the planes in Fig. 165 a. However, the chemical structure and 
probable dimensions of the molecule are known, and if such a molecule 
is placed in the cell oriented as in Fig. 165 b (an orientation suggested 
by the cell dimensions), the atoms do lie on the 'negative 7 0.3.17 planes ; 
and this postulated orientation is confirmed by the fact that the 060 
reflection is also strong Fig. 165 b shows that the atoms lie on 'nega- 
tive' 060 planes. (Iball, 1934.) 

Another example of the use of this type of evidence is to be found in 
the determination of the structure of crystalline rubber (Bunn, 1942 a). 
First of all it must be mentioned that the chain molecules, seen end -on, 
lie as in Fig. 166. (The great strength of the 200 and 120 reflections, 
coupled with the fact that there are four molecules passing through the 



POSITIONS OF THE ATOMS 



263 




10.0.0 + 



unit cell, gives this information.) A strong hint of the distribution of 
the atoms is given by the fact that the 10.0.0 reflection is recorded; 
it is wea, bjit the fact that it is there at all (considering the large 
angle of reflection) must mean that the structure amplitude for this 
reflection is large. Again, the atoms may lie on 'positive' planes as in 
Fig. 166 a or 'negative* planes as in Fig. 166 6 ; but the probable con- 
figuration of the chain mole- 
cules, derived from reasoning 
based on the repeat distance of 
the molecules (the length of the 
c axis of the cell), is such that 
the first of these alternatives is fa) 
evidently the correct one. 

This is as far as it is possible 
to go by simple inspection ; the 
discovery of the positions of 
individual atoms, and the de- 
termination of the parameters 
with as much accuracy as the 
experimental evidence allows, 
must rest on detailed calcula- 
tions. When atomic positions 
have been postulated as the 
result of considerations such as _ J __ J _ J __ 

those just given, it is usually 10.0.0 - 

best to concentrate on one prin- 
cipal projection of the structure ; 
the projection chosen should be 
the one expected to give the 
clearest resolution of the differ- 
ent atoms, and calculations 
should first of all be carried out for the zone of reflections relevant to 
this projection. (The hkQ reflections yield the x and y coordinates of 
the atoms that is, they give a picture of the structure projected along 
the c axis; and so on.) Following this (assuming that the intensities 
of this first zone of reflections are successfully accounted for and a 
reasonable projected structure is obtained), the other principal pro- 
jections may be considered ; and finally the whole structure is checked 
by calculations of all the hkl intensities. 

In the attempt to obtain correct calculated intensities the following 




FIG. 166. Structure of rubber. The great 
strength of 200 and 120 shows that the four 
chain molecules, seen end-on, lie in the posi- 
tions shown. The large structure amplitude of 
10.0.0 shows that there are concentrations of 
atoms either on the 10.0.0+ planes (above) 
or the 10.0.0 planes (below). 



264 



STRUCTURE DETERMINATION 



CHAP. VII 



PI 



P2 



Pm 



Pb 



Cm 




V 



Pmm 



Pba 



\ 



Pbm 



Cmm 



X 



/Y 



problem arises : suppose a particular set of atomic 
coordinates gives some measure of agreement of 
calculated with observed intensities for' one pro- 
jection, but nevertheless there are still considerable 
discrepancies, especially among the higher order 
reflections a situation which probably means that 
the postulated atomic coordinates are roughly cor- 
rect, but need adjustment. Which atoms should be 
moved, and by how much ? 

It should be remembered that when a reference 
atom is moved, all the atoms related to it by sym- 
metry elements move also in a manner determined 
by the symmetry elements ; and the problem is to 
know, for any particular reflection, the direction in 
which to move the reference atom so that the con- 
tribution of the whole group of related atoms either 
increases or decreases. This problem is best solved 
by the use of charts which show at a glance the 
magnitude of the structure amplitude for such a 
group of atoms for all coordinates of the reference 
atom. 

Graphical methods and machines for evalu- 
ating structure amplitudes. The evaluation of 
the structure amplitudes for a large number of re- 
flections (it may run into hundreds) is a task of 
such magnitude that methods of shortening it are 
very welcome. For any principal zone (MO, hQl, or 
O&Z) only two coordinates of each atom are concerned 
in the structure amplitude, and therefore a graphical 
method may be used : charts showing the values of 
such functions as cos 2ir(hx+ ky) or cos Znhx cos 2nky 
may be constructed (Bragg and Lipson, 1936), and 
on them the contribution of a group of related atoms 
may be read. Great accuracy is not required in 
structure amplitude calculations, and the 1 P er 
cent, or so obtainable by such graphical methods is 
adequate. Such charts have, in addition, the very 
great advantage that they show at a glance what 



FIG. 167. Some of the plane groups. 



CHAP, vn 



POSITIONS OF THE ATOMS 



265 



movements of atoms are required to increase or decrease the calculated 
intensity for any reflection. 

Projected Arrangements may be described as 'plane-groups', for 
which a nomenclature conforming to that of the space-groups is used ; 
the most frequently encountered plane-groups are illustrated in Fig. 167. 
A pair of atoms having coordinates xy and xy (that is, related by an 




FIG. 168. Chart for the estimation of cos 2ir(2x+3y) for all values of x and y. 

apparent centre of symmetry at the origin) form plane-group PI ; the 
contribution of this pair to the structure amplitude is 2/cos 27r(hx-}- ky). 
For the 230 reflection a chart showing cos27r(2o;-{-3y) for all values of 
x and y is required ; such a chart is shown in Fig. 168. It is made square 
for convenience ; the real shape of the projected cell does not matter, 
since structure amplitudes do not depend on the shape of the cell but 
only on atomic coordinates expressed as fractions of the unit cell edges. 
A complete chart would show contours at intervals of 0-1 filling all 
the spaces between the nodes ; but it is unnecessary to draw anything 
but the nodes, since the value at any point can be readily found by 



266 



STRUCTURE DETERMINATION 



CHAP. VII 



using a 'master key' which is fitted between the nodes in the way shown 
in the diagram : the structure amplitude at the point P is obviously 0-73. 
If the chart is made on transparent material (tracing clpth'is durable 
material for this purpose), atomic positions can be plotted, as fractions 
of unit cell edges, on white paper ; for crystal plane 230 the chart show- 
ing cos 2?r(2a:+%) is laid on the plot of atomic coordina,tes and the con- 
tribution of each atom is read off. Note that this same chart can be 



Jin Cos + Cos Cos 
[Sin sm {as Sin 




FIG. 169. ('hart for the estimation of cos '27r2x cos 2ir%y (and other 
similar functions) for all values of a; arid y. 

used not only for plane 230, but also for 230, 320, and 320 by turning 
it over or reversing the axes. But a different chart is required for each 
different pair of numerical values of h and k. In order to serve for sine 
as well as cosine terms each chart is extended in one direction, the 
origins for sine and cosine terms being 7r/2 apart. 

Four atoms having coordinates xy, xy, (l+x)(^ y) and (i x)(l+y) 
form plane-group Pba (see Fig. 167). The combined structure ampli- 
tude for such a group of atoms is 4 cos 27rhx cos 2nky when Ji-\~ k is even, 
and 4 sin "Zirhx sin Z-rrky when h+k is odd. 

Charts showing these functions are divided by the nodes into rect- 
angular sections filled by curved contours (Fig. 169). Here it is even 
more desirable to construct skeleton charts with a set of 'master keys' 
to fit the differently shaped rectangular sections. The same chart 
can be used either for the two functions already mentioned or for 



CHAP, vii POSITIONS OF THE ATOMS 267 

sin 2-n-hx cos 2rrh/ or cos 27rAa: sin 277% (which are required for some 
plane-groups) by appropriate placing of the origin. 

A set of charts for the functions already mentioned suffices for 
graphical evaluation of the structure amplitudes for the principal zones 
(hkQ, hQl, Qkl) of all triclinic, monoclinic, and orthorhombic crystals ; 
since, in Barker's words, this group is the centre of gravity of the crystal 
kingdom (a large proportion of known substances including nearly all 
organic substances crystallize in these systems), such a set of charts 
has a very wide range of usefulness. The same charts can be used for 
hQl zones of hexagonal and tetragonal crystals. It should be noted that 
for certain projections of space-groups containing glide planes the pro- 
jected plane unit cell area appears to have one or both axes subdivided ; 
in other words, the unit cell dimensions of the 'plane-group' are sub- 
multiples of those of the corresponding 'space-group' ; in such circum- 
stances it is necessary to remember to multiply the atomic coordinates 
and divide the indices of the reflections to make them appropriate to 
the plane-group in question. 

For basal plane projections (hkQ) of hexagonal and tetragonal crystals, 
and for projections of cubic crystals (for which MO, hQl, and OH are all 
equivalent), the charts showing the combined contributions from a set 
of crystallographically equivalent atoms have a more complex form. 
Charts for these and all possible 'plane-groups' are illustrated in the 
paper by Bragg and Lipson. 

Figures for hexagonal plane-groups have been evaluated by Beevers 
and Lipson (1938). It should be noted that if charts for the hexagonal 
and square plane-groups are not available (the labour of constructing 
them is considerable), sets of equivalent atoms may be divided into 
subgroups whose contributions may be separately read off on the charts 
for the less symmetrical plane-groups. Thus, for any pair of atoms related 
by a (projected) centre of symmetry (plane-group Pi), the contribution 
is given by 2 cos 2rr(hx-\-ky) ; any four atoms related in the projection by 
two apparent planes of symmetry at right angles (plane-group Pmm) 
contribute 4 cos 2irhx cos 2irky to the structure amplitude ; and so on. 

The value of these charts, in showing what adjustments of atomic 
parameters will increase or decrease the structure amplitude, has already 
been mentioned. To this little need be added except a suggestion on 
procedure in dealing with several independent reference atoms simul- 
taneously. This is best explained by an actual example. In attempting 
to find the x and y coordinates of the atoms in the hkO projection of 
polychloroprene (Bunn 1942 a) a particular set of postulated coordinates 



268 STRUCTURE DETERMINATION CHAP, vii 

gave roughly correct intensities for the hkO reflections, but there were 
still some discrepancies, the most serious of which were that the calcu- 
lated structure amplitude of 240 was too small and that of 420 too great. 
By using the charts it was readily seen that the movements of atoms 
which could rectify these discrepancies are as follows : 

Ci C 2 C 3 4 Cl Ib \ 

To increase 240 . . . / f \ / -> \^ a ] 
To decrease 420 . . . -> f / \ - 

These verdicts are fairly consistent; in particular, it appears that a 
movement of C 2 upwards or Cl to the right would have the desired effect. 
Since the diffracting power of Cl is much greater than that of C, a 
movement of Cl would be much more effective than a similar movement 
of C. But if Cl is moved, it is found that the intensities of some of the 
other reflections are adversely affected; in particular, 410 becomes too 
strong. The movements now required to weaken 410 are as follows: 

\J\ v^2 >-^3 v^ Ol 

To weaken 410 . . . f -> f \ -*- 
It is not now desired to shift Cl ; and of the other atoms it is not 
desirable to move C 4 downwards, as this would weaken 240 again; 
therefore it seems that the correct thing to do is to move C x and C 3 
upwards ; in this way, not only is 410 weakened, but 240 is strengthened 
further. The magnitudes of the movements required were found by trial, f 

It will be evident that these methods can only be used after approxi- 
mately correct atomic positions have been postulated. If the positions 
postulated are not approximately correct, the calculated intensities are 
nothing like those observed, and it is not possible to decide what move- 
ments are necessary to put matters right. Everything therefore depends 
on whether the preliminary reasoning leads to an approximately correct 
structure. This is the great limitation of the method of trial. 

Three-dimensional charts are not practicable, but for many space- 
groups the plane charts just described can be utilized for general (hkl) 
reflections in the following way. For space-group P2 1 2 1 2 1 the structure 
amplitude is J(A*+B 2 ), where 



A ^ At O k h- k \ O 1 7 *-A O /7 l ~-' h \ 

A = > 4/cos27T [hx -- \co82TT\ky -- cos27ruz -- 
4* \ 4 / \ 4 / \ 4 / 

and 



D \? At o i 

B = > 4/sin27r(Aa: 



\ 4 / \ 4 / \ 

f A numerical *least squares' method of adjusting parameters is described by Hughes 
(1941). 



CHAP, vii POSITIONS OF THE ATOMS 260 

The xy positions of the atoms are plotted on a transparent square chart 
in the manner already described ; the z positions are plotted on a separ- 
ate strip. 'To^find A, the product of the first two cosine terms is read off 
on a chart giving cos %-nhx cos Znky by suitably displacing the origin ; for 

instance, for plane 231, 27r(- ] is 90 and 27r( -J is +180 ; using 

the 23 chart, the origin of the atomic position chart is shifted 90 back- 
wards along the x axis and 180 forwards along the y axis. The last cosine 

term is read off on the strip, using a one-dimensional chart giving 

(I _ M 
- 1 is 90, and the origin of the strip must 

therefore be displaced backwards 90. The only calculation is then the 
multiplication of the two graphically estimated figures and/, the diffract- 
ing power of the atom. Sine terms are obtained in a similar way from 
the same charts. 

The procedure is of course longer than for planes of type MO, etc., 
but it is very much shorter than straightforward calculation of hkl 
structure amplitudes. 

As another example, the structure amplitude for space-group P2 1 /a 
is given by 



A = 4coa27rhx+lz+cos2ky~, (B = 0). 
\ 4 / \ 4 / 

The x and z positions of the atoms are plotted on a square chart, and 
the first cosine terms read off on the Bragg and Lipson chart for plane- 

group P2, the origin being displaced by 2nl -I ; the y positions are 

plotted on a strip as in the previous example, and the second term read 
off, again after the appropriate displacement of the origin ; the two terms 
are then multiplied. 

For the general planes of triclinic crystals the structure amplitude is 
given by 



The Bragg and Lipson charts cannot be used for these expressions. The 
following method uses slide-rule technique and has been found to save 
much time and effort. A strip A is prepared (Fig. 170) on which phase 
angles and the corresponding cosine and sine values are marked. For a 
particular atom in the structure the postulated coordinates are marked 
on a separate strip of paper B. The phase angle for any crystal plane 
hkl is found by placing the two strips together, first of all with their 



270 



STRUCTURE DETERMINATION 



CHAP. VII 



origins opposite to each other (Fig. 170 a) and then displacing strip B in 
the following way. For crystal plane 1 1 1 , for instance, a pointer (a needle 
point or a sharp pencil point) is placed at x, and strip B is f moved along 
so that its origin comes opposite the pointer (Fig. 170 6) ; the pointer is 
now moved to y (Fig. 170c) 3 the origin shifted to the pointer, and so on. 
For 121 two y displacements would be made. For 231 two x displace- 
ments, three negative y displacements, and one z displacement would 
be made. The value of the cosine (or sine) for the final phase angle is 



[ 


POINTER 


+1-0 
\sm , cos V 

Iiii ih 111 i i i A 


-5 
iiiiiiiiti 1 1 1 


-10 

, | 


-5 -f-5 +/-0 

1 1 1 llllllllllllll 1 1 1 


1 I 

B * 2 


y 


1 


A 







(fa) 



r 




A 


. +1-0 { 

ifin R.A 


-5 
Hiiliiiii 1 1 1 


-1-0 


-5 +t 
1 1 1 1 iiiiln t it i 


+w 


u 


x 


z 


y 


M 



(c, 



r 




0. 


./ft.riLf,,,"! ^ 


-J +'5 +10 

In.L.lM,, 1 


A 


jgl i z y 1 


1 



FIG. 170. Slide-rule method for determination of cos (or sin) 2Tr(hx-\-ky-}-lz). 

read off, and only needs multiplying by / to give tho contribution of 
the atom in question. A separate paper strip should be prepared for 
each independent atom in the structure. If the phase angle becomes so 
large that it is outside the range of strip A, it must be transferred back 
by shifting the pointer 2?7. This disadvantage of a slide rule could be 
remedied by making A and B circles or disks. 

Machines to carry out calculations of structure amplitudes have been 
devised. Evans and Peiser (1942) have a machine (made largely of 
'Meccano' parts) which evaluates/cos (orsin) 2ir(kx+ky) that is to say, 
for an atom having coordinates x and y along the a and b axes, it 
reckons up the phase angle, and then multiplies its cosine (or sine) by 
the diffracting power /; this can be done for all the MO planes by 
suitable numbers of turns of two handles (representing h and k). For 
other atoms, x, y, and / are reset, and the process repeated. The 



CHAP, vii POSITIONS OF THE ATOMS 271 

machine does not add up the contributions of all the independent atoms 
to the structure amplitude of any one plane ; this must be done separ- 
ately. The machine can also be used for general (hkl) planes by re- 
setting for each value of /. 

Cox has a machine which is undoubtedly more rapid than the one 
already mentioned. As an electrical analogy to the process of multiply- 
ing a diffracting power by the cosine (or sine) of a phase angle, a con- 
denser whose capacity is proportional to the diffracting power / of an 
atom is charged by a voltage proportional to the cosine (or sine) of the 
phase angle 6, the resulting charge on the condenser (== capacity x 
voltage) being therefore proportional to /cos 6. (See discussion of paper 
by Beevers, 1939.) 

The optical diffraction method. W. L. Bragg (1944) has suggested 
a method of avoiding the calculation of structure amplitudes altogether, 
at any rate in the early stages of structure determination. This method 
consists in making, on a scale small enough to give optical interference 
effects, a plane pattern corresponding to the postulated crystal structure 
as seen from one particular direction (usually a unit cell axis) ; the 
diffraction pattern produced when monochromatic light passes through 
this imitation crystal corresponds to the X-ray diffraction pattern 
produced by the real crystal that is to say, the relative intensities of 
the optical diffraction spots correspond with those of the X-ray diffrac- 
tion spots for the particular zone selected. Thus, suppose the c projec- 
tion of a crystal is being considered ; x and y coordinates for all the 
atoms are postulated and a picture is made, on a very small scale, 
showing many repetitions of the projected unit cell ; this is done by a 
photographic reduction method (described below). The diffraction 
pattern may be observed by looking through a telescope at a point- 
source of monochromatic light several feet away ; when the pattern 
which represents the crystal structure is placed between the source 
and the telescope, many images of the source are seen ; the relative 
intensities of these images should correspond with those of the hlcO 
X-ray reflections, if the postulated structure is correct. Another simple 
method of observing the diffracted beams is to set up a microscopic 
objective lens (of, say, 2 inches focal length) to produce an image of a 
monochromatic point-source several feet away, and to examine this 
image by means of a microscope ; when the pattern representing the 
crystal structure is placed between the source and the 2-inch lens, many 
images are seen; the 2-inch lens may conveniently be screwed into 
the substage of the microscope. That this method does give correct 



272 STRUCTURE DETERMINATION CHAP, vn 

intensities is shown by Fig. 172, Plate XIII, in which the upper photo- 
graph shows part of a pattern representing the b projection of the 
phthalocyanine crystal (actually 676 repetitions were madu), and the 
lower photograph shows the diffraction pattern given by it. The actual 
structure amplitudes obtained from the hOl X-ray reflections (Robert- 
son, 1935 c, 1936 a) are given in Table VII, and it can be seen that there 
is a close correspondence between the optical and X-ray intensities; 
there are some discrepancies among the weaker spots, but the agree- 
ment is on the whole very good quite good enough to show that the 
method can be used for the purpose of finding approximate atomic 
positions. 

TABLE VII 



I 


















7 


+ 24 





-13 


+ 26 





-25 








6 


-50 


-72 


-67 


-33 


+ 21 





+ 16 





5 


-60 


+ 10 





-76 





+ 12 


+ 13 





4 


+ 17 


-13 





-37 





+ 62 





-30 


3 


-39 


+ 6 


-13 


+ 38 


+ 20 


+ 20 


-13 


-10 


2 


-85 


-92 


-46 


-40 





+ 28 


+ 52 


+ 11 


1 


+ 78 





+ 80 


-55 


+ 15 


+ 13 


+ 12 


+ 16 







+ 97 


-85 


-31 





-14 


-41 


+ 9 


T 


+ 78 


+ 17 


-74 





-13 


+ 27 


-31 


-22 


2 


-85 


+ 68 


-48 


-16 








-42 


-37 


3 


-39 


-61 


+ 38 


+ 31 


-15 


-46 


-13 





4 


+ 17 


+ 36 





+ 31 


+ 42 


+ 15 


+ 10 





5 


-60 





+ 43 


+ 17 


+ 44 











6 


-50 


+ 22 


+ 59 


-22 


-15 





+ 9 





7 


+ 24 





+ 13 





-38 


-13 













1 


2 


3 


4 


5 


6 


7 h 



The multiple pattern is made in the following way (see Fig. 171). 
The atoms in one projected unit cell are represented by illuminated 
holes in an opaque screen. (A square coordinate system is used for 
convenience, just as square charts are. used in the graphical methods.) 
A multiple photograph of this one unit cell is taken by means of a 
multiple pinhole camera (Fig. 171 a) which consists of 676 pinholes in 
an area 5 mm. square (made by drawing a large pattern of black dots 
on white card and taking a small photograph on a fine-grained plate). 
The distances between the illuminated holes and the pinholes, and 
between pinholes and photographic plate, must be such that the images 
are properly contiguous, to preserve correct coordinates in the unit cell. 
Atoms of different diffracting power may be represented either by holes 
of different sizes, or else, if the holes are all the same size, by covering 
those holes representing the lighter atoms during part of the exposure. 




PLATE XIII 



FlQ. 172. Above: pattern representing the b projection of tho phthaloeyanine structure. 
Below: optical diffraction pattern given by it. 



CHA*.VH 



POSITIONS-OF THE ATOMS 



273 



Another method of making patterns by means of the multiple pin- 
hole camera is to use a single small light-source (such as a car headlamp 
bulb) which is photographed in various positions each of which repre- 
sents the position of an atom in the projected unit cell. Atoms of 
different diffracting powers are represented by exposures of different 
lengths. 

Concluding remarks on the method of trial and error. The 
general procedure in structure determination by trial has been described. 

PHOTOGRAPHIC 
*^ PLATE 

V'\ r .MULTIPLE 
/J ^ PIN HOLE 
t \ \ CAMERA 




' / 



\ \ 



/ / \ ^ 



/ 

/ / 

/ / 



\ \ 

\ \ 
\ 



\ 



/ / 

t 1 
f t 




/^< '.- : 

*~2 * 


\ \!XUNIT 


CbLL 



w 



Fia. 171. a. Part of multiple pinhole camera. 6. Arrangement for making 

repeating patterns by the multiple pinhole camera. Atoms in the unit cell 

are represented by points of light. 

It remains to mention certain experimental and theoretical devices 
which are of less general application but which are valuable in special 
circumstances. 

a. In some diffraction patterns certain types of reflections- the types 
which if absent altogether would indicate the presence of glide planes 
or screw axes arc consistently very weak, suggesting the existence of 
pseudo glide planes or pseudo screw axes. For instance, if all kOl 
reflections with Jt odd are very weak, this means that in the 6 projection 
there are, about half-way along the a edges of the cell, groups of atoms 
(molecules, perhaps) looking much the same as, but not identical with, 
groups of atoms at the cell corners. (See the structure of ascorbic acid, 
p. 315.) Similarly, it is not uncommon to see in a rotation photograph 
that the odd layer lines are much weaker than the even ones ; this 
means that there is a pseudo unit cell with one edge (the edge parallel 

4458 r 



274 STRUCTURE DETERMINATION CHAP, vii 

to the axis of rotation) half the true length: one half the unit cell is 
similar to, but not identical with, the other. 

6. Two isomorphous crystals in which atoms of different diffracting 
powers occupy corresponding sites give diffraction patterns in which 
corresponding reflections have different intensities. The differences of 
structure amplitudes may be used to locate these atoms : the differences 
may be regarded as the structure amplitudes which would be given by 
a hypothetical crystal consisting only of hypothetical atoms having a 
diffracting power equal to the difference between the diffracting powers 
of the replaceable atoms in the real crystals. By solving the structure of 
the hypothetical crystal, using the differences of structure amplitudes, 
the positions of the replaceable atoms can be found independently of the 
remaining atoms in the real cell. (The assumption on which this method 
rests is true only for the stronger reflections : for the weaker reflections, 
the replacement of one atom by another may cause a change of sign 
of the structure amplitude, in which case the difference in magnitude of 
the structure amplitudes does not correspond with the required arith- 
metical difference. The stronger reflections may, however, provide 
sufficient evidence for successful location of the atoms in question.) 

c. In crystals containing atoms of very similar atomic numbers 
(copper and manganese, for instance) it may not be possible by normal 
methods to distinguish between the two species of atom because their 
diffracting powers are very similar for most X-ray wave-lengths. But if 
an X-ray wave-length lying between the absorption edges of the atoms 
concerned is used, the difference between the diffracting powers are 
enhanced, so that distinction is possible. (Bradley and Rodgers, 1934.) 

One final remark. Adjustment of postulated atomic positions by 
trial need be carried only so far as to settle the phases of the majority 
of the reflections ; from that point the direct method described in 
Chapter X can be used. 

The background of crystal chemistry. Ideally, crystal structures 
should be deduced from the X-ray diffraction patterns of crystals 
(together with such physical properties as are rigorously determined 
by internal symmetry) without making any stereochemical assump- 
tions. Most of the simple structures, and some of the more complex 
ones, have been determined in this way. In the early days of the use 
of X-ray methods for the determination of crystal structures it was 
necessary that structures should be deduced by rigorous reasoning from 
physical data, so that the foundations of crystal chemistry should be 
well and truly laid. In some of the more complex structures, however, 



CHAP, vii POSITIONS OF THE ATOMS 275 

it would be difficult to determine all the atomic positions by such 
methods alone; and in these circumstances the obvious course is to 
make use of the wealth of information contained in the large number 
of crystal structures already established, as well as stereochemical 
information obtained by other methods, and those physical properties 
which have been shown by experience to give reliable structural informa- 
tion. There is no reason why the fullest possible use should hot be made 
of the generalizations resulting from previous studies, providing one 
retains an open mind with regard to the possibilities of deviations from 
or exceptions to these generalizations. After all, such considerations 
are only used to indicate approximate atomic positions which, it is 
hoped, will give approximately correct X-ray intensities ; the atoms are 
then moved about independently until the best possible agreement 
between the calculated and observed intensities of X-ray reflections 
from a wide range of planes is obtained. The proof of the correctness 
of the structure is this agreement, and it does not matter how it is 
attained -whether by rigid deduction from the X-ray diffraction pattern 
alone or by reasonable induction from general principles arising from 
a survey of previously determined structures. 

The danger of using the non-rigorous methods is that the possibility 
of there being more than one arrangement of atoms satisfying the X-ray 
intensities may be overlooked ; there is a danger that the arrangement 
first selected, if it gives good agreement between observed and calculated 
intensities, may be accepted without further question. The only remedy 
here is ruthless self-criticism on the part of the investigator. The chance 
that two or more arrangements of atoms are equally compatible with 
the X-ray results is small. It is only in crystals containing both heavy 
and light atoms that there is an appreciable chance of ambiguity ; the 
heavy atoms may usually be placed with certainty, but lighter atoms, 
since they contribute comparatively little to the X-ray intensities, may 
be moved appreciably from selected sites or even to quite different sites 
without altering radically the calculated intensities. Wyckoff (The 
Structure, of Crystals) draws attention to some cases in which alternative 
structures have been proposed; potassium dithionate (Helwig, 1932; 
Huggins, 1933) may be cited as an example. Such ambiguities have 
usually been resolved by subsequent investigations more precise in 
technique and critical in approach. 

In crystals in which different atoms have much the same diffracting 
power (many organic crystals, for instance) the interchange of different 
atoms such as nitrogen, oxygen, and carbon will have little effect on 



276 STRUCTURE DETERMINATION CHAP, vn 

the calculated intensities ; but usuaDy in such circumstances chemical 
evidence indicates that all combinations but one are definitely ruled out 
or wildly improbable. Granted that there is only one possible chemical 
grouping of atoms in the molecule, the fact that the diffracting powers 
of all the atoms are similar is actually an advantage, for the chance that 
two different crystal structures give approximately the same calculated 
intensities is in these circumstances very small. 

It is not within the scope of this book to describe the principles of 
crystal chemistry and molecular stereochemistry which have so far 
emerged. The reader is referred for an account of the former subject 
to Evans's Crystal Chemistry (1939) and Pauling's Nature of the 
Chemical Bond (1940), and for the latter to text-books of organic 
chemistry such as Freudenberg's Stereochemie (J 932-4). It must suffice 
to observe here (in broadest outline) that the mode of packing of atoms, 
ions, or molecules in crystals may be regarded as controlled by two 
principles the principle of close packing (the closest packing obtainable 
in view of the shapes and sizes of the building units), and, where ions 
are concerned, the tendency for an electrically charged unit to surround 
itself with units of opposite charge. Some of the silicate mineral struc- 
tures may be regarded as close -packed arrangements of the compara- 
tively large spherical oxygen ions, with the positive ions fitting into the 
spaces between them; and the use of this generalization played an 
important part in the solution of some of these structures (Bragg 1930). 
Pauling (1929) went farther and formulated a set of rules based on the 
principle of local satisfaction of electrostatic forces. In molecules and 
complex ions atoms are joined by the comparatively short, strong, and 
precisely directed covalent bonds; the covalent radii in such units 
are only about half the external radii of the atoms, and a molecule or 
complex ion may thus be regarded as an assemblage of partly merged 
spheres ; Fig. 141 illustrates this. The configurations of aliphatic organic 
molecules are determined first of all by the tetrahedral disposition of 
the four bonds of a carbon atom ; in double- and triple-bonded group- 
ings the joining of atoms by pairs or triplets of bonds results in a 
planar configuration of atomic centres in the group 

a \ A 



b 

and a linear configuration of a CE=C b. In chain molecules com- 
paratively free rotation round single bonds as axes leads to molecular 
flexibility; nevertheless certain configurations those in which the 



CHAP. VH POSITIONS OF THE ATOMS 277 

single bonds of covalently linked atoms are staggered (see Fig. 204, 
p. 324) are more stable than others, and these configurations are 
found in crystals (Bunn, 19426). Aromatic ring molecules including 
fused-ring structures like anthracene and chrysene are flat. 

As for the physical properties of crystals, some account of crystal 
morphology and optics has been given in Chapters II and III, where, 
however, these subjects were developed only as far as was necessary for 
identification purposes. For structure determination further considera- 
tion of both these subjects, as well as others such as the magnetic, pyro- 
electric, and piezo-electric properties of crystals, is desirable ; this will 
be found in Chapter VIII. 

Examples of the use of stereochemical generalizations and physical 
properties in structure determination will be found in Chapter IX. 

The present chapter on the methods of structure determination would 
not be complete without some mention of the fact that two structural 
principles which have so far been tacitly assumed are not always obeyed. 
The first of these principles is that atoms in crystals occupy precise 
positions, about which they merely vibrate to a degree depending on 
the temperature ; when the atoms are the constituents of molecules or 
polyatomic ions this means that the molecules or ions have precisely 
defined orientations as well as precise mean positions. The second 
principle is implied in the application of the theory of space-groups to 
structure determination: it is assumed that the members of a set of 
crystallographically equivalent positions are all occupied, and that 
they are occupied by identical atoms. In the majority of crystals these 
principles are obeyed, but there are some in which one or the other of 
them is violated: in some crystals, at certain temperatures, whole 
molecules or ions rotate ; in others equivalent positions are occupied 
indiscriminately by two or more different kinds of atoms ; in still others 
some members of a set of equivalent positions are empty, the gaps 
being randomly distributed. This subject will not be pursued farther 
at present ; some account of it will be found towards the end of Chapter 
IX, where examples of different types of abnormal structures are given. 
Here, in this chapter on the general principles of structure determina- 
tion, it is only necessary to point out that, in setting out to determine 
the structure of any crystal, it is obviously necessary to bear in mind 
the possibility of abnormalities of this sort. 



VIII 

EVIDENCE ON CRYSTAL STRUCTURE 
FROM PHYSICAL PROPERTIES 

A STUDY of the physical properties of a crystal its shape and cleavages, 
its optical and magnetic characteristics, or piezo- and pyro-electric 
behaviour cannot lead to a detailed knowledge of its structure, but it 
can give valuable information on the general features of the structure ; 
it may lead to a partial knowledge of the internal symmetry, to definite 
conclusions on the general shape and orientation of molecules or poly- 
atomic ions in the crystal, or to a general idea of the arrangement of 
the molecules or the distribution of forces. 

Shape and cleavage. The general shape of a crystal gives an indi- 
cation of the relative rates of growth of the structure in different direc- 
tions. Crystals which are roughly equi-dimensional have much the same 
rate of growth on all the faces which have developed, but those which 
are markedly plate-like or rod-like have very unequal rates of growth 
in different directions, and this anisotropy of rate of growth is due either 
to the effect of the shape and arrangement of the molecules, or to the 
particular distribution of forces in the crystal, or to both these factors. 

In molecular crystals held together by weak undirected van der 
Waals forces the shape and arrangement of the molecules appear to 
decide the relative rates of growth. Long molecules tend to pack parallel 
to each other, forming plate-like crystals in which the long molecules are 
perpendicular or nearly perpendicular to the plane of the plate (Fig. 
173 a). It is apparently easier to add a molecule to an existing layer 
than to start a new one. Flat molecules sometimes form needle-like 
crystals in which the planes of the molecules are approximately per- 
pendicular to the needle axis (Fig. 1736); it is easier to add a flat 
molecule to an existing pile than to start a new pile alongside the first. 
On the other hand, either long or flat molecules may form arrangements 
in which the'molecules are not all parallel, giving approximately equi- 
dimensional crystals. It is evident that when the general shape of a 
molecule is known through chemical evidence, the shape of the crystals 
may indicate the general arrangement. Molecules which are roughly 
spherical, such as hexamethylene tetramine, form roughly equi- 
dimensional crystals. 

In many crystals it is not possible to distinguish individual molecules ; 
in silicates, for instance, there may be continuous one-, two-, or three- 



CRAP, vm 



PHYSICAL PROPERTIES 



279 



dimensional networks. In plate-like crystals such as mica and the clay 
minerals, as well as the simpler 'layer lattices' such as CdI 2 and MoS 2 , 
there are sheets of atoms extending through the whole crystal ; in any 
one sheet the atoms are held together by strong ionic forces, but between 
neighbouring sheets the forces are much weaker. The cleavage, as well 
as the anisotropy of rate of growth, is due to this distribution of forces. 
In other crystals, such as chrysotile, 3MgO . 2SiO 2 . 2H 2 O ('asbestos'), 
there are continuous strings of atoms held together by strong ionic 
forces, the strings being held together by weaker forces ; the result is a 
needle-like or fibrous habit and easy cleavages parallel to the fibre axis. 





FICJ. 173. (a) Long molecules, packed parallel, give plate-like crystals, while 
(6) flat molecules, packed parallel, give needle-like crystals. 

In molecular crystals held together by ionic forces (for instance, salts 
of organic acids) or polar forces such as 'hydrogen bonds' (for instance, 
alchohols and amides), the two influences, shape and distribution of 
forces, may not co-operate, and it is difficult to form any definite con- 
clusions on the structure from crystal shape and cleavage, though it is 
well to keep these properties in mind during structure determination, 
for any suggested structure should account for them. 

The above remarks refer only to the relative dimensions of crystals. 
A consideration of the indices of the principal bounding faces may lead 
to further conclusions, at any rate for molecular crystals. The bounding 
faces on crystals are apparently those planes having the greatest reticu- 
lar density of atoms or molecules ; the indices of the bounding faces 
may therefore indicate the general arrangement of the molecules. For 
instance, when a crystal is found to be bounded entirely or mainly by 
faces of 110 type (110, Oil, 101, etc.) it is likely that there are molecules 
at the corners and centres of the unit cells, since this is the arrangement 



280 STRUCTURE DETERMINATION CHAP, vm 

that gives greatest reticular density on these planes. For instance, 
crystals of hexamethylene tetiamine (CH 2 ) 6 N 4 are rhombic dodeca- 
hedra, all the bounding faces of which are of type 110, an(J the molecular 
arrangement is body-centred (Dickinson and Raymond, 1923). For 
similar reasons, in crystals bounded entirely by 111 faces the molecules 
are likely to be arranged in a face-centred manner, and in prismatic 
crystals bounded by faces of 110 type a base-centred arrangement is 
probable. It should be noted, however, that the arrangements need 
not be centred in the strict (space-lattice) sense ; the molecules present 
at cell-centres or face-centres need not be oriented in the same way as 
those at the corners of the cell. Benzene crystals, for instance, grow as 
orthorhombic bipyramids bounded by {111} faces, but the molecular 
arrangement is not strictly face-centred; there are molecules at the 
corners and face-centres of the unit cell, but they are not all oriented 
in the same way. (Cox, 1928, 1932 b.) 

When ionic or polar forces play an important part in binding atoms 
or molecules together in a crystal, matters are more complex, since 
the rates of growth of crystal faces appear to be influenced by the 
distribution of electric charges as well as the reticular density (Kossel, 
1927). The subject has not so far received much attention, and it is 
unwise to attempt to formulate generalizations. 

A more detailed consideration of the types of faces present on a 
crystal may lead to definite conclusions on the point-group symmetry 
of the atomic or molecular arrangement. This subject has been dis- 
cussed in Chapter II ; the necessity for caution in accepting morpho- 
logical evidence on internal symmetry should be remembered. Accord- 
ing to Donnay (1939), it is possible to go farther, and to deduce (from 
the relative 'importance' of the different faces) the presence or absence 
of glide planes and screw axes and in fact the whole space-group 
symmetry. But although the correct space-group symmetry of some 
crystals has been deduced from morphological evidence, it would be 
unwise to place too much reliance on such considerations, for there are 
some striking exceptions to 'Donnay 's law'. (See Donnay and Harker, 
1937.) In any case, where X-ray methods are used it is unnecessary to 
attempt to use morphological evidence to this extent. 

Optical properties. The relations between the refractive indices of 
crystals and their atomic structures were first pointed out by W. L. 
Bragg (1924 fe), who succeeded not only in correlating birefringence 
with structure in a general way but even in calculating, from the known 
structures of several crystals (first of all calcite and aragonite), the 



CHAP, vm 



PHYSICAL PKOPERTIES 



281 



actual values of the refractive indices and getting them approximately 
correct. Here we are concerned, .for the moment, with the reverse 
process the. use of refractive indices to provide clues to the atomic 
arrangements in crystals. 

Bragg's theory is roughly this : consider (to take the simplest situa- 
tion) the effect of a diatomic molecule or ion on light passing through it, 
first of all when the electric vector of the light waves lies along the line 
joining the atoms or, as we usually say, the vibration direction of the 
light is parallel to the line joining the atoms (Fig. 174 a). Each atom 
becomes polarized that is, positive and negative parts suffer a relative 
displacement in the direction of 
the electric vector, to an extent 
which depends on the strength 
of the electric field and the 
'polarizability' of the atom. But 
the two atoms will also affect 
each other; the presence of 
dipole A increases (by induc- 





ELECTOC 
VECTOR 



(a) W 

FIG. 174. Illustrating Bragg's theory of 
the refractive indices of crystals. 



tion) the polarization of J5, and 
the presence of B increases the 
polarization of A. Each atom 
is thus polarized more than it 
would be if the other were absent. If, however, the electric vector 
of the light waves is perpendicular to the line joining the atoms, as in 
Fig. 174 6, the induction effect of dipole A is to decrease the polariza- 
tion of jB, and similarly the presence of B decreases the polarization of 
A. The effective dielectric constant is therefore much greater for situa- 
tion a than for situation 6, and since the refractive index is proportional 
to the square root of the dielectric constant,! the refractive index is 
much higher when the electric vector lies along the line joining the atoms 
than when it is perpendicular to this direction. 

In a similar way it is easy to show that a flat molecule or polyatomic 
ion (such as COj- ion, in which all the atoms lie in a plane and the 
oxygen atoms form an equilateral triangle round the carbon atom) has 
a higher refractive index when the electric vector lies in the plane of 
the group of atoms than when it is perpendicular to this plane. 

In crystals, the molecules or polyatomic ions are surrounded by others, 
the presence of which complicates matters ; but the distances between 
atoms in neighbouring molecules or polyatomic ions are much greater 

f See for instance Richtmeyer, 1928, p. 111. 



282 STRUCTURE DETERMINATION CHAP, vm 

than those between atoms linked by primary bonds, and since the 
induction effect is very sensitive to distance (inversely proportional to 
the cube of the distance), the induction effects of neighbouring molecules 
or polyatomic ions are small. The main factor controlling the bire- 
fringence of crystals containing strongly birefringent molecules or poly- 
atomic ions is the relative orientation of these units. Where they are 
all parallel, as in most carbonates and some nitrates, the refractivities 
of the crystal are approximately those of the individual polyatomic ions ; 
where the birefringent units are inclined to each other their individual 
effects are partially, or in cubic crystals completely, cancelled out. 

The effect of interatomic distance is strikingly illustrated by the fact 
that the birefringence of nitrates in which the nitrate ions are all 
parallel is much greater than that of carbonates of similar structure, 
though the interatomic distances in the nitrate ion are only slightly 
less than those in the carbonate ion : 

Distances 



KNO 3 . . . 1-335 1-506 1-506) 

NaNO 3 . . . 1-336 1-586 (- |B)/ 

CaCO 3 (calcite) . 1-486 1-658 (= j8)i 



O O 
2-30 



2-18 



N(or C) O 
1-21 



1-26 



(aragonite) . 1-530 1-681 1-686] 

In view of this, it is not surprising that neighbouring ions have only 
minor effects, since the distances between oxygen atoms in neighbour- 
ing ions are 2-7 A. Nor is it surprising that in crystals in organic sub- 
stances, where the distances between linked carbon atoms are 1-3-1-5 A 
and the distances between carbon atoms in neighbouring molecules 
3-5-4-2 A, the refractive indices depend almost entirely on the refrac- 
tivities of individual molecules and the relative orientations of these 
molecules in the crystals. 

The examples in Table VIII (in addition to those already quoted) 
will give some idea of the birefringence to be expected for various 
groups. (See also Wooster, 1931.) 

The very high negative birefringence of aromatic molecules may be 
partly due to the conjugated double-bond systems in these molecules. 
At all events, conjugated double bonds in a chain increase the refractive 
index enormously for the vibration direction along the chain, as is 
shown by the properties of crystals of methyl bixin 

CH 3 0~CO CH={CH C(CH 3 )=CH CH=} 4 CH CO OCH 3 , 
which has a refractive index of 2-6 for the direction along the polyene 
chain, the other principal indices being 1-47 and 1-65. (Waldmann and 
Brandenberger, 1932.) 



CHAP. VIII 



PHYSICAL PROPERTIES 

TABLE VIII 



283 






a. 


ft 


y 


Sign 


y a 


Reference 













Very 




LinearF aN > 








-f 


large 


Wooster, 1938. 


^ mear \Ca(OCl) 2 .3H 2 


(~ P) 


1-535 


1-63 


4- 


0-095 


Bunn, Clark, and Clifford, 1935. 


Obtuse \ 














" N NQ 
V-shaped / 2 


1-340 


1-425 


1-655 


+ 


0-315 


Bunn, unpublished. 


Low 1 \ 














pyramid/ 3 


1-410 


1-517 


1-524 





0-114 


Zachariasen, 1929. 


/PbC0 3 


1-804 


2-076 


2-078 


__ 


0-274 


Larsen and Berrnan, 1934. 


Flat NaHC0 3 


1-380 


1-500 


1-586 


_ 


0-206 


Winchell, 1931. 


with 1 C 6 (CH 3 ) 6 


1-503 


1-748 


1-801 





0-298 


Bhagavantarn, 1930. 


planes \ C 6 H 3 (C 6 H 5 ) 3 


1-524 


1-867 


1-873 





0-349 


Orelkin and Lonsdale, 1934. 


parallel C 10 B 8 


1-442 


1-775 


1-932 


_ 


0-490 


Bhagavantam, 1929. 


^(naphthalene) 















When flat molecules have their planes inclined at a large angle to 
each other but all parallel to a line, the refractive index for light 
vibrating along this line is high, but in all directions perpendicular 
to this line the refractive index is moderately low (corresponding 
to an average value for the other two principal directions in the 
molecule); the birefringence is thus positive, not negative. Thus 
in the tetragonal crystal of urea O=C(NH 2 ) 2 , the planes of the 
Y-shaped molecules are perpendicular to each other but parallel to 
a line (the c axis); consequently this crystal is uniaxial positive 
(co = 1-481, = 1-602). 

It will be evident that the birefringence and the orientation of the 
indicatrix may be used in a semi- quantitative manner as evidence of 
the orientation of strongly birefringent groups of atoms in crystals. 
The birefringence of molecules or polyatomic ions may sometimes be 
calculated theoretically (this has been done for oxalic acid and some of 
its salts, for instance Hendricks and Deming, 1935), and the values so 
obtained used in conjunction with the measured indices of crystals as 
evidence of the orientation of the groups in the crystals. But the princi- 
pal use of the methods is likely to be qualitative and empirical ; when a 
crystal containing flat molecules or ions is found to have strong negative 
birefringence it can be assumed that the flat groups are all roughly 
parallel to each other and perpendicular to the direction of lowest index ; 
and when a crystal containing chain molecules or ions is found to have 
Strong positive birefringence it can be assumed that the chains are all 
roughly parallel to each other and to the direction of highest index. 
Some idea of the birefringence to be expected for a particular group 
can often be obtained from the properties of crystals of already known 



284 



STRUCTUKE DETERMINATION 



CHAP. Yin 



structure. Evidence of this type is, in general, all that is required in 
Structure determination; the details the precise orientation of the 
groups and the atomic positions must be settled by the interpretation 
of X-ray diffraction patterns ; the conclusions from optical properties 
merely provide the starting-point for trial stuctures. 

A few examples will show the value of such considerations. Naphtha- 
lene (monoclinic, a = 8-29 A, 6 = 5-97 A, c = 8-68 A, j8 = 122 42') 
has refractive indices 1-442, 1-775, and 1*932. The birefringence is so 
strong that we should not be far wrong in assuming that the planes of 




FIG. 175. Structure of naphthalene. 

the two molecules in the cell are roughly parallel ; and if we associate 
the three indices with the three vibration directions of the molecule 
itself, we are led to suppose that the longest axis of the molecule lies 
parallel to the vibration direction of y (that is, nearly parallel to c), and 
the intermediate axis along 6 (the j8 vibration direction). 

The complete structure, determined by Robertson (1933 c), shows 
that this is substantially correct ; the two molecules have their longest 
axes almost exactly parallel to the direction of highest index, and their 
intermediate axes tilted (one in one direction and the other in the 
opposite direction) 29 to the b axis. (Fig. 175.) 

In a crystal of hexamethylbenzene (triclinic, with one molecule in 
the unit cell) the vibration direction for the lowest refractive index is 
almost exactly normal to the 001 plane (Bhagavantam, 1930), indicat- 
ing that the plane of the molecule is almost exactly parallel to 001 ; 
and again this is correct (Lonsdale, 1929). 



CHAP, vni PHYSICAL PROPERTIES 285 

Potassium chlorate is monoclinic ; it has strong negative birefringence, 
and the vibration direction for the lowest refractive index lies in the 
ac plane, making an angle of 56 with the c axis. The conclusion that 
the low pyramidal (that is, more or less flat) chlorate ions have their 
oxygen triangles normal to this direction of least refractive index is 
correct within 1 (Zachariasen, 1929). 

Turn now to crystals whose structures are as yet unknown. The 
substance C 6 H 5 CH^CH CO C 6 H 4 CH 3 , the molecules of which 
are expected to be elongated in shape, forms orthorhombic crystals 
having the refractive indices and vibration directions a = 1*607 (|| a), 
j8 = 1-634 (|j c), y =. 1-881 (|| 6). j* The strong positive birefringence shows 
that all the molecules in the crystal have their long axes roughly parallel 
to each other, and to the vibration direction having the highest refrac- 
tive index, that is, the 6 axis. Also, the fact that the two low indices 
are so similar indicates that the planes of the benzene rings in the crystal 
are not all parallel to each other. 

Vaterite, or /x-CaC0 3 , which grows in the form of hexagonal plates, 
is interesting, because it has fairly strong positive birefringence (w = 
1-550, = 1-650), in contrast to calcite andaragonite, which are strongly 
negative. The strong positive birefringence shows that the negative 
carbonate ions cannot be parallel to the plane of the crystal plate ; the 
planes of the flat carbonate ions must be roughly perpendicular to 
the plane of the crystal plate. An arrangement in which the planes of the 
carbonate ions are parallel to the apparent sixfold axis but not parallel 
to each other would give similar to the highest index of calcite or 
aragonite (actually a little lower because the density of ju,-CaC0 3 is 
low), and the indices for vibrations in the plane of the plate definitely 
higher than the low indices of calcite and aragonite. The reported 
indices of /i-CaC0 3 (Winchell, 1931) are indeed of this order. These 
considerations also lead to a further conclusion of structural significance. 
The unit cell is stated to contain two molecules (Olshausen, 1925). 
Now in a carbonate ion the only symmetry axes in the plane of the 
atoms are twofold axes. Trigonal or hexagonal symmetry cannot be 
achieved by any arrangement of two cax'bonate ions oriented as the 
optical properties demand. Hence, if the cell really contains two 
molecules, its symmetry cannot be trigonal or hexagonal ; or alterna- 
tively, if the symmetry is trigonal or hexagonal, the true unit cell must 
be larger than that reported. 

Observation of the absorption of light in different vibration directions 

t Groth, 1906-19. 



286 STRUCTURE DETERMINATION CHAP, vm 

may also be useful. Not very much work has yet been done on this 
subject, but it seems that for molecules containing chromophoric groups 
such as a polyene chain ( CH=CH ) n , or quinonoid 



or azo N=N groups, the absorption is largely confined to the 
vibration direction parallel to the double bonds. Thus, in a crystal of 
methyl bixin, the vibration direction along the polyene chain is character- 
ized not only by a very high refractive index as we have already seen, 
but also by vejy high absorption ; it is practically black for this direc- 
tion, and red or yellow for other directions. For other examples of the 
use of such evidence see Bernal and Crowfoot, 1 933 a (azoxy compounds), 
W. H. Taylor, 1936 (rubrene, etc.), and Perutz, 1039 (parallelism of the 
four chromophoric groups in molecules of methaemoglobin and oxy- 
haemoglobin). In crystals of the complex substances used as dyestuffs 
the colours (that is, the positions of the absorption bands) for the 
principal vibration directions are often very different from each other ; 
a study of these absorptions in relation to the chemical constitution of 
the molecules and their orientation in the crystals should throw much 
light on the problem of the relation of colour to chemical constitution ; 
and this knowledge, in turn, will be useful in the determination of 
crystal structures of new substances. 

The infra-red absorption spectra for different vibration directions in 
a crystal should also give information on the orientation of particular 
groups (Ellis and Bath, 1938). Very little experimental evidence, how- 
ever, is yet available. 

One other optical character which may sometimes contribute informa- 
tion useful in structure determination is the rotation of the plane of 
polarization. In cases where the shape or the X-ray diffraction pattern 
or other properties do not yield unequivocal evidence on point-group 
symmetry, a positive observation of the phenomenon may settle the 
question. (For experimental method, see Chapter III.) 

A question which may sometimes be asked is this : 'If an enantio- 
morphous crystal that is, one possessing neither planes nor a centre 
of symmetry is dissolved in a solvent, does the solution necessarily 
rotate the plane of polarization of light ?' The answer to this question 
is, 'Not necessarily'. If the molecules or ions of which the crystal is 
composed are themselves enantiomorphous, then the solution will be 
optically active. But it must be remembered that enantiomorphous 



CHAP, vni PHYSICAL PROPERTIES 287 

crystals may be built from non-centrosymmetrical molecules which in 
isolation possess planes of symmetry these planes of symmetry being 
ignored in the crystal structure ; such molecules in solution would not 
rotate the plane of polarization of light. (A molecule of this type, in 
isolation, would rotate the plane of polarization of light (see p. 88), 
but the mass of randomly oriented molecules in a solution would show 
no net rotation.) An example is sodium chlorate NaClO 3 ; the crystals 
are enantiomorphous and optically active, but the solution of the salt 
is inactive because the pyramidal chlorate ions (see Fig. 138) possess 
planes of symmetry. 

Centrosymmetrical molecules do not usually form non-centrosym- 
metrical crystals ; therefore, if a molecular crystal is found to be enantio- 
morphous, it is probably safe to conclude that the molecules lack 
centres of symmetry ; but they may have planes of symmetry. 

Magnetic properties. Interest in the magnetic properties of 
crystals has grown rapidly in recent years, and anisotropy of diamag- 
netic susceptibility has been used, in much the same way as optical 
anisotropy, as evidence of molecular orientation in crystals. 

All substances composed of ions, atoms, or molecules having no 
resultant orbital or spin moment (this includes organic substances and 
inorganic salts, except those containing transition elements like iron 
and platinum) are diamagnetic. This means that when placed near a 
magnet they are repelled; or, more precisely, when placed in a non- 
uniform magnetic field they tend to move to a weaker part of the field. 
Evidently a piece of a diamagnetic substance when placed in a magnetic 
field becomes (by induction) a magnet in opposition to the inducing 
field behaviour opposite to that of ferromagnetic and paramagnetic 
substances. The force of repulsion is exceedingly minute, but can be 
measured if a powerful magnet and delicate suspensions are used. The 
ratio of the induced magnetism to the field strength is known as the 
diamagnetic susceptibility. 

Crystals, except those belonging to the cubic system, are anisotropic 
in this respect ; the force of repulsion varies with the orientation of the 
crystal with respect to the direction of the field. The graph representing 
vectorially the diamagnetic susceptibility in all directions in a crystal 
is an ellipsoid, whose orientation with respect to the unit cell is restricted 
by symmetry in exactly the same way as that of the optical indicatrix. 
Thus, for uniaxial crystals the magnetic ellipsoid is an ellipsoid of revolu- 
tion whose unique axis coincides with the threefold, fourfold, or sixfold 
axis of the crystal; for orthorhombic crystals the ellipsoid has three 



288 STRUCTURE DETERMINATION CHAP, vm 

unequal axes which necessarily coincide with the three axes of the 
crystal ; for monoclinic crystals the only restriction is that one of the 
principal axes of the magnetic ellipsoid must coincide with the 6 axis 
of the crystal ; while for triclinic crystals the orientation of the ellipsoid 
is not restricted in any way. 

The available methods for the determination of diamagnetic suscepti- 
bilities in crystals will not be described here. Papers by Rabi (1927), 
Krishnan and his collaborators (1933, 1934, 1935), and the excellent 
review of the whole subject by Lonsdale (1937 a) should be consulted. 

For many aromatic molecules, and for the flat nitrate and carbonate 
ions, the relative dimensions of the magnetic ellipsoid are opposite to 
those of the refractive index ellipsoid : the susceptibility is numerically 
much greater in the direction normal to the plane of the molecule or 
ion than in directions lying in the plane. This is partly a matter of 
relative electron density in the different directions; but for aromatic 
substances, with their large conjugated ring systems, it seems that the 
large orbits of the resonance electrons play an important part. (Pauling, 
1936; London, 1937; Lonsdale, 1937 6.) 

Crystals in which the molecules are all parallel to each other have the 
same characteristics as the individual molecules. But when there are 
two or more differently oriented molecules in the unit cell, the magnetic 
anisotropies of the individual molecules are to some extent neutralized. 
The magnetic properties of a crystal are, very precisely, the vectorial 
sum of those of the constituent molecules. (For equations, see Lonsdale, 
1937 a.) The effects of neighbouring molecules on each other are negli- 
gible, the reason being that induced magnetic effects are exceedingly 
feeble. Magnetic properties therefore have, at any rate theoretically, 
some advantage over optical properties for the determination of mole- 
cular orientation, since molecular interaction does play a small part in 
determining refractive indices. However, in structure determination, 
physical properties are needed only to indicate approximate molecular 
orientations, and for this purpose optical properties are quite satis- 
factory, and usually much easier to measure than magnetic properties. 
The precise details of the structure are settled by X-ray analysis. 
Magnetic properties are likely to be most valuable in circumstances in 
which refractive indices are not easily measured. For instance, crystals 
of many substances used as dyes are so strongly coloured that even 
minute crystals are almost opaque, so that it is scarcely possible to 
measure refractive indices. 

The magnetic ellipsoid of a crystal or a molecule is not always the 



CHAP, vin PHYSICAL PROPERTIES 289 

inverse of the refractive index ellipsoid. This is shown by the properties 
of potassium chlorate, KC10 3 . Optically, the chlorate ion is strongly 
negative like the nitrate and carbonate ions; but magnetically it is 
also negative, in contrast to the nitrate and carbonate ions which are 
positive. (Krishnan, Guba, and Banerjee, 1933.) The reason, no doubt, 
lies in the pyramidal form of the chlorate ion (Fig. 138). The refractive 
indices are determined largely by the triangle of oxygen atoms forming 
the base of the pyramid ; the chlorine atom at the apex has little effect, 
because Cl +8 is less polarizable than O~ 2 . Magnetic properties are 
determined by quite different factors, electron density being important 
and for this reason the comparatively heavy and dense chlorine atom 
and its position outside the plane of the oxygen atoms plays a very 
important part. The effect does not appear to have been quantitatively 
explained, and the facts prompt caution in interpreting magnetic 
properties except for molecules or complex ions of well-established 
characteristics . 

Crystals composed of aliphatic chain molecules provide further 
examples of special effects which give rise to diamagnetic characteristics 
different from those which might have been expected. The susceptibili- 
ties of several such crystals have been shown to be numerically greater 
along the chain molecules than across them; thus the magnetic 
characteristics of these chain molecules (one large susceptibility and 
two smaller ones) are the same as those of flat aromatic molecules, not 
inverse as might have been expected. These magnetic properties of 
chain molecules have been interpreted as an indication that the electron 
clouds of the chain CH 2 groups are flattened in the plane normal to the 

TT 

chain axis that is, the C^ plane. (Lonsdale, 1939.) There is X-ray 

evidence pointing in the same direction. (Bunn, 1939.) 

The magnetic properties of crystals composed of aromatic polynuclear 
molecules may give information on the relative orientations of the 
benzene rings to each other. Thun, Clews and Lonsdale (1937) con- 
cluded from the magnetic anisotropy of crystals of o-diphenylbenzene 
that the planes of the o-phenyl groups are inclined at 50 to the plane of 
the main ring. 

The relations between paramagnetic and ferromagnetic properties 
and structure are less simple than in the case of diamagnetic substances, 
and the subject is in its infancy. It will not be dealt with here ; the 
reader is referred to the review by Lonsdale (1937 a). 

Pyro-electric and piezo-electric tests. When a crystal belonging 

4458 



290 STRUCTURE DETERMINATION CHAP, vni 

to one of the iion-centrosymmetrical classes is heated or cooled, it 

develops electric charges and becomes positive at one end and negative 

at the other end of each polar axis. Therefore, if a crystal is found to 

be pyro-electric, it must belong to one of the classes which" lack a centre 

of symmetry. Various qualitative tests for pyro-electric character have 

been used. The three most suitable for small crystals are the following : 

(a) Crystals are placed on a metal plate or spoon and dipped in liquid 

air. When the grains have cooled, the plate is tilted until it 

becomes vertical ; pyro-electric crystals stick to the metal, others 

fall off. (Martin, 1931.) 

(6) Small crystals are attached to fine silk threads, and two or more 
are dipped in liquid air ; pyro-electric crystals tend to stick to each 
other, others do not. (Robertson, 1935 c.) 

(c) A crystal is heated ; the charges formed on it are then dissipated 
by passing it through a flame. It is then allowed to cool in a bell 
jar full of magnesium oxide smoke (made by burning magnesium 
in it) ; the charges developed on cooling cause fine filaments of 
magnesium oxide to grow out from the crystal along the lines of 
force, forming a pattern like that of iron filings round a magnet. 
(Maurice, 1930.) 

Only positive results are significant: feeble pyro-electricity may 
escape detection by these tests. 

Piezo-electricity is the property, possessed by some crystals, of 
developing electric charges when compressed or extended in particular 
directions. Conversely, when a potential difference is applied to suitable 
points on such a crystal, it expands or contracts. Piezo-electric proper- 
ties can occur in all crystals lacking a centre of symmetry, except those 
belonging to the cubic class 432 (Wooster, 1938). A test for such 
properties, suitable for small crystals or even powders, is the following. 
The crystals are placed between the plates of a condenser which forms 
part of an oscillating circuit. An audio-frequency amplifier, with head- 
phones or speaker, is connected to the oscillator. When the frequency 
of the oscillator is changed continuously by means of a variable con- 
denser in the circuit, clicks (or, for a large number of small crystals, 
rustling noises) are heard. The reason is that whenever the frequency 
of the oscillator happens to coincide with a natural frequency of one of 
the crystals, there is a sudden change of current through the condenser 
and consequently an impulse which is amplified by the audio-frequency 
amplifier. For a suitable circuit see Wooster, 1938. 

Other physical properties. Anisotropy of thermal and electrical 



CHAP, vm PHYSICAL PROPERTIES 291 

conductivity, coefficient of thermal expansion, elasticity, and dielectric 
constant may also provide information on internal structure. These 
properties, however, have so far been little used in structure determina- 
tion, because they are less easily measured than those already con- 
sidered ; consequently not very much experimental evidence is available 
for the purpose of generalizing on the relations between such properties 
and structural features. For further information on these subjects, see 
Wooster (1938). 

A sudden change in average dielectric constant (measured by using 
powdered material) when a substance is heated has been taken as 
evidence of the onset of molecular rotation at the temperature of the 
sudden change. (White and Bishop, 1940 ; White, Biggs, and Morgan, 
1940; Turkevitch and Smyth, 1940.) Specific heat anomalies also 
accompany the onset of molecular rotation. (Fowler, 1935; Eucken, 
1939.) 



IX 

SOME EXAMPLES OF CRYSTAL STRUCTURE 
DETERMINATION BY TRIAL 

THE principles of the methods by which atomic positions are deduced 
from X-ray diffraction patterns have been described in Chapters VI 
and VII ; and examples of the separate stages (determination of unit 
cell dimensions, deduction of space-group, calculation of structure 
amplitudes, and so on) have been given. It is now intended, in this 
chapter, to describe the complete process of structure determination 
in several examples. The structures described are all relatively simple 
ones; they have been chosen on the ground that they .display the 
utilization of the essential principles in relatively simple circumstances. 
In some of the examples the help given by physical properties (the 
subject of Chapter VIII) is an important feature. (The train of reasoning 
by which each structure is deduced does not, in all cases, coincide with 
that followed in the original investigations.) Many structures of far 
greater complexity than these have been worked out completely ; but 
success in such cases has usually been possible through the application 
of stereochemical principles derived from simpler structures; the 
principles of interpretation of the X-ray patterns and the physical 
properties are essentially the same. The use of stereochemical principles 
is brought out in some of the later examples. The chapter ends with a 
section on abnormal structures in which the crystallographic ideals 
embodied in the application of the theory of space-groups are not 
followed. 

In setting out to discover the relative positions of the atoms in a 
crystal, it is best, when the unit cell dimensions have been determined 
and the intensities of the reflections measured, to calculate F for each 
reflection. (See Chapter VII.) Absolute values of F* derived from 
intensities in relation to that of the primary beam, form the ideal experi- 
mental material, though very many structures have been determined 
from a set of relative JP's. The reliability of the set of figures depends 
on the success with which the corrections for thermal vibrations, absorp- 
tion, and extinction effects have been estimated. 

Some of the simplest structures of all are those of many metallic 
elements, in which there is one atom to each lattice point. These need 
not detain us long; for clearly, as soon as the lattice type has been 
deduced (by inspection of the indices of the reflections), the whole 



CHAP. 



EXAMPLES OF DETERMINATION BY TRIAL 



293 



structure is completely determined. Thus aluminium has a cubic unit 
cell containing four atoms and gives only reflections having all even or 
all odd indices ; hence the lattice is face-centred, and there is one atom 
to each lattice point. The structure presents no further problems. 
Similarly, molybdenum has a cubic unit cell containing two atoms and 
gives only reflections having h+k+l even ; hence the lattice is body- 
centred, there is only one atom to each lattice point, and the whole 
structure is settled. 











a, 


CaO 








a ^ 1 




|| 




\ 


iK l?r 


111200 220 3//I 


400 331 \ 422 511 


440 531 BOO 


222 


420 


333 





CuCL 


, 1 1 


1 1 , 1 


Vi". 


I1IZOO 220 311 400 331 


422 511 440 531 
333 


620 533 



FIG. 176. Diagram representing powder photographs of calcium oxido and cuprous 

chloride. Abscissae represent distances of arcs along the film; ordinates represent 

relative intensities (estimated visually). 

Many binary salts, oxides, and sulphides are a little more complex, 
two atoms being associated with each lattice point ; it is necessary to 
discover the relative positions of the two atoms. This can be done by 
mere inspection of the set of structure amplitudes, and confirmed by 
a very moderate amount of calculation. Two examples will be given 
calcium oxide and cuprous chloride. 

Calcium oxide, CaO, is cubic, and the unit cell (a = 4-80A) contains 
four molecules of CaO. The only reflections present (Fig. 176) are those 
having all even or all odd indices ; the lattice is therefore face-centred. 
If the origin of the unit cell is taken as the centre of a calcium atom, 
then there are also calcium atoms at the centres of the cell faces (Fig. 
177 a). It is now necessary to place the oxygen atoms. Note first of all 
that the oxygen atoms by themselves also form a face-centred lattice 
(an oxygen atom might have been chosen as the origin of the cell the 
two sorts of atoms obviously have equal rights in this respect) ; the only 
problem therefore is the relation of the oxygen lattice to the calcium 



294 



STRUCTUBE DETERMINATION 



CHAP. IX 




lattice. Inspection of the powder photograph shows that reflections 
with odd indices, such as 111 and 531 , are weaker than those with even 
indices at about the same angle. (With one exception thb pair 311 
and 222, which are about equally strong; but, since the number of 
equivalent reflections is 24 for 311 and only 8 for 222, it is evident that 

F for 311 is much smaller than that for 
222.) Consider the placing of one oxygen 
atom. To weaken 111 the oxygen must 
be somewhere on or near the plane ABC 
(marked 111 in Fig. 177 a). Since 200 
is strong, it must be somewhere on or 
near the planes marked 200+ ; and since 
220 is strong, it must be on or near the 
planes marked 220+ . Its position is 
evidently somewhere near where these 
three types of planes intersect, that is, 
at A or By or the similar positions 
Z>, Ey F, etc. There is no need to choose 
between these positions, for if we place 
an atom at any one of them, say A, 

^y* -7,0 -^ o identical atoms immediately spring into 

t" T~C- ! ^ -Lrr* being at B, C, D, etc., forming a face- 
centred lattice ; and it should be noted 
that, to preserve the symmetries of the 
cubic system, they must be exactly half- 
way along the edges and in the centre of 
the cell (Fig. 177 b). In confirmation, it 
is found that structure amplitudes calcu- 
lated for this arrangement agree with 
those experimentally determined, and 
the structure, which is analogous to that 
of sodium chloride (see p. 220), is established. Note that in this arrange- 
ment (a very common one among binary compounds) every atom is 
equidistant from six of the other kind of atorA. 

Cuprous chloride, CuCl, is also cubic (a = 5-41 A) with four 
molecules in the unit cell. Since the only reflections present on the 
powder photograph (Fig. 176) are those with all even or all odd indices, 
the lattice is, like that of calcium oxide, face-centred. It is, however, 
immediately obvious from the photograph that the arrangement must 
be different from that of calcium oxide, since the 111 reflection of 




FIG. 1 77. Structure of calcium 
oxide, CaO. 



CHAP, ix EXAMPLES OF DETERMINATION BY TRIAL 



295 




cuprous chloride is much stronger than 200 ; in the calcium oxide pattern 
the opposite is true. It is evident that the face-centred chlorine lattice 
must be placed in such a way with respect to the face-centred copper 
lattice that for 111 the chlorine atoms lie on or near the same planes as 
copper atoms, while for 200, chlorine planes interleave copper planes. 
Note also that 220 is strong, and thus 
all the atoms, both copper and chlorine, 
lie on or near 220 planes. Taking the 
centre of a copper atom as the origin 
of the cell (Fig. 178 a) and focusing 
attention on the placing of one chlorine 
atom (the rest will follow inevitably 
from the first), the position A seems 
a possible site, since at this point the 
planes marked 11 1 + , 200 , and 220+ 
intersect. This will not do, however ; 
if the other reflections on the photo- 
graph are examined, it will be found 
that 222 is absent ; if the chlorine were 
at A, which lies on the plane DCEF, 
then 222 would be strong its F would 
be as great as that of 111. (There are 
other reasons why position A will not 
do, but this one will suffice.) To account 
for the absence of 222, the chlorine 
must be moved out of the 111+ plane 
but not too far, in view of the strength 
of 111. The absence of 222 can be 
accounted for by moving the chlorine 
away from plane DEF by a distarice equal to half the spacing of 222, 
since at this position waves from the chlorine will oppose those from the 
copper atoms (and the same will be true for ail the other chlorine atoms) ; 
actually the intensity should not be zero, but evidently this reflection 
is too weak to show on the photograph. Half the spacing of 222 is one- 
quarter the spacing of 111, and therefore the intensity of 111 will not 
be adversely affected to any serious extent. The chlorine atom, in 
moving away from plane DEF, must keep to the line AG, to preserve 
the correct intensities for 200 and 220. The position necessary to 
account for all the intensities so far considered is thus P, half-way 
between A and /. It is also to be noted that P is equidistant from copper 



o 




FIG. 178. Structure of cuprous 
chloride, CuCl. 



296 STRUCTURE DETERMINATION CHAP, ix 

atoms B, C, D, and H, and, moreover, when the other chlorine atoms 
are placed so that they all form a face-centred arrangement (Fig. 178 6) 
the essential symmetries of the cubic system (the diagonally disposed 
threefold axes) are preserved. P, for instance, lies on the 'diagonal DJ. 
The symmetry is not holosymmetric but tetrahedral ; the crystal class 
(point-group) is 43m and the space -group F 43m. Calculation of the 
remaining structure amplitudes and comparison with those experi- 
mentally determined show that this arrangement is indeed correct. 
This arrangement, in which each chlorine atom is tetrahedrally sur- 
rounded by four copper atoms (in contrast to CaO, with its octahedral 
six-coordination), is found in many of the less polar binary solids. In 
diamond the same arrangement, but with all the atoms identical, is 
found, and reflections which are weak for Cud are absent altogether for 
diamond, t In view of the preferred tetrahedral configuration of carbon 
bonds this arrangement in diamond is not surprising. 

Titanium dioxide, TiO 2 (rutile). In the structures so far con- 
sidered, all the atoms have occupied very special positions in the unit 
cell ; there were no continuously variable parameters to be determined. 
The structure of rutile, now to be considered, is a simple example of a 
structure in which there is one parameter. This structure has been 
described on p. 210, where it was introduced in connexion with the 
calculation of structure amplitudes. The general arrangement of the 
atoms was assumed, and the effect of the variation of the oxygen para- 
meter on the structure amplitudes of the reflections was demonstrated 
(Fig. 129). Here we shall consider the evidence which leads to a know- 
ledge of the general arrangement of the atoms. 

The tetragonal unit cell, the dimensions of which (a 4-58 A, 
c = 2-98 A) can be calculated from either powder or single -crystal 
photographs, contains two titanium and four oxygen atoms. A survey 
of the indices of the reflections (see the powder photograph in Fig. 121) 
shows that there are no systematic absences among those of the general 
(hkl) type ; hence the lattice is primitive. For the principal zones J the 
only systematic absences are hOl reflections having h+l odd (together 
with the equivalent type OK with k-\-l odd) ; hence there are glide planes 

f Except that diamond gives a very weak 222 reflection. This is taken as an indication 
that the electron cloud of the carbon atom is not spherical, but has tetrahedral symmetry. 
(Bragg, W. H., 1921.) 

J It should be remembered that in the cubit;, tetragonal, and hexagonal systems 
there may be glide planes perpendicular to the diagonals of the basal plane ; hence the 
set of hhl reflections constitutes a 'principal zone' ; absence of hhl reflections having I odd 
indicates the existence of such a glide plane. 



CHAP. IX 



EXAMPLES OF DETERMINATION BY TRIAL 



297 



n having diagonal translation perpendicular to both a and b axes. If 
the list of tetragonal space-groups is consulted, it will be found that the 
only possible space-groups are P4nm and P4/rawn. (Remember that 

4 
the latter means Pnm.) 



m 



It should be noted that in both these space-groups the fourfold axes 
are of the 4 2 type ; this is not expressed in the conventional space-group 
symbol because the existence of the glide planes having diagonal trans- 
lation n implies the 4 2 type of fourfold axis. 

b 



^ , 
a 
C 


\j ' 
) C 

i r\ 4 


' V 

a 
) 

f< 


y v 
h 6 


V ' 

a 
b < 


. 

0* 



O 



O 



(c) and ft) (e) (f) (g) 

FIG. 179. Fourfold positions in space-group P4/mnw. 

The shapes of rutile crystals give no hint of polar character, hence 
the holohedral space-group P^jmnm is the more likely to be correct. 
It will be considered first. 

'Consider the positions of the titanium atoms. There are only two of 
these in the unit cell ; if one is placed at the corner of the unit cell, then 
the other can only be at the centre of the cell: the glide planes having 
diagonal translation demand it.f 

The positions of the four oxygen atoms can best be deduced by refer- 
ring to the lists of equivalent positions in Int. Tab. There are five sets 
of fourfold positions in space-group P^/mnm : 

(c) OJO; J00;0il;i0l. 

(d) OJi; OJ; 0||; 0f. 

(e) OOz; OOz; i, i i+z; i i, i-z. 
(/) xxOixxO; 

(g) xxQ: xxQ', 

All except the last two of these sets can be dismissed very simply by 
the following consideration: the x and y coordinates of (c) and (d) are 
iO and OJ, and the c projection looks like Fig. 179. This projected 
arrangement is centred, and if it were correct, all hkO reflections having 
h+k odd would be absent. Actually, such reflections are present for 

t In Int. Tab. a second set of twofold positions is given: 00 J and JJO. This set, how- 
ever, represents the same arrangement as the first set 000, JJ. 



298 



STRUCTUBE DETERMINATION 



example, 210. Hence these arrangements can be dismissed. Similarly 
the (e) arrangement in projection appears centred, and can likewise be 
dismissed. It is therefore certain that the oxygen atoms occupy either 
the (/) or the (g) set of fourfold positions. Since (/) and'(gr) are equiva- 
lent that is, they give rise to exactly the same complete arrangement 
(see Figs. 179. and 180) we can use either. 

The general arrangement is thus settled, and it remains only to deter- 
mine the value of the single variable parameter x ; the weakness of 200 
indicates that it is not far from 0*25 ; its precise magnitude is found by 
calculating the intensities of a number of reflections for a range of 



--0-- 



)-- 




a '-*+ a 

FIG. 180. Structure of ru tile, TiO 2 . Arrangements in P4/mnm (left) 
and P&nm (right). 

positions around this value. The best agreement between calculated 
and observed intensities is obtained for x 0*31. 

Since the whole X-ray diffraction pattern is accounted for by this 
arrangement whose space-group symmetry is P4/mnm, this appears to 
be the correct structure. It is, however, necessary to consider whether 
any arrangement in the other possible space-group P4nm would account 
for the intensities equally well. In this space-group the titanium atoms 
are in the same positions as before, but the oxygen atoms occupy the 
following set of fourfold positions : 

(c) (xxz), (xxz), (\+x, | #,i+z), (\ x, l+x, l+z), 

giving the arrangement illustrated in Fig. 180. It differs from the 
P4:/mnm arrangement in that the oxygen atoms are all shifted along 
the c axis by a distance z. The x parameter must be either 0-31 or 
0-19, to account for the hkO intensities (see Fig. 129). The z parameter 
is given by the intensities of the other reflections ; we need go no farther 
than the consideration of 002, the intensity of which is such that z can 
only be about zero : any value far from zero would give 002 too weak 
in comparison with the hkO intensities. Other intensities involving z 
establish that its value is exactly zero. (At the same time, x is estab- 



CHAP, ix EXAMPLES OF DETERMINATION BY TRIAL 299 

lished as 0-31, not 0-19.) But this arrangement with z zero is none other 
than the P4[mnm arrangement we have already considered ; this struc- 
ture is therefore established as correct beyond doubt. 

Urea, O=^C(NH 2 ) 2 . Crystals of urea are tetragonal, and their 
distinctive habit (Fig. 29) places them without any doubt in class 42m 
(tetragonal scalenohedral in Groth's nomenclature, ditetragonal alternat- 
ing in Miers's). The unit cell dimensions are a = 5-67 A, c = 4-73 A, 
and these figures, together with the known density of 1-335, lead to the 
conclusion that this unit cell comprises two molecules of urea. There 
are no systematic absences among the hkl reflections, hence the lattice 
is primitive (P). There are no systematic absences among hkQ, tiki, or 
hOl reflections, hence there are no glide planes. In fact, the only 
systematic absences arc AGO reflections for which h is odd (and, of course, 
(WfeO reflections for which Jc is odd, since the a and 6 axes are equivalent) ; 
the only symmetry elements involving translation are therefore screw 
axes (2j) parallel to a and b. If the list of tetragonal space-groups and 
their systematic absences is consulted, it will be found that the only 
possible space-group is P42 1 m. 

In considering the positions of the atoms in the unit cell we are 
entitled to assume that the atoms are linked together in molecules in 
the manner established by chemical evidence : 

H 2 N NH 2 

2 \/ 


I! 
o 

This means that we may consider first of all (in order to attain a general 
idea of the atomic positions) the symmetry of the molecule, and the 
relation of the two molecules in the cell to the symmetry elements of 
space-group P42jW, the c projection of which is shown in Fig. 181. 

Consider the axes of symmetry in the crystal. There are fourfold 
inversion axes, twofold axes, and twofold screw axes. Now a molecule 
having the chemical structure O^C(NH 2 ) 2 cannot have a fourfold 
inversion axis; neither can it have a screw axis (since it is a finite 
molecule). Hence the molecules cannot lie on these crystal axes; the 
two molecules must be related to each other by these axes. On the other 
hand, a molecule of this structure may well possess a twofold axis pass- 
ing through the C and O atoms ; consequently the twofold axes (A in 
Fig. 181) are likely sites for molecules. Furthermore, it is to be noted 
that each twofold axis stands at the intersection of two mutually per- 
pendicular planes of symmetry and these also are likely to be possessed 



300 



STRUCTURE DETERMINATION 



CHAP. IX 



by a molecule of urea (see Fig. 138). Further consideration shows that 
all other positions are impossible ; for instance, if we put a molecule at 
B, it is inevitably repeated at B', B", and B"' ; this is out of the question, 
since we know there are only two molecules in the unit' cell, not four. 
It is therefore certain that each molecule lies on a twofold axis, and 
thus the C and O atoms of the molecule lie, one above the other, on this 
twofold axis. Moreover, the nitrogen* atoms must lie on the symmetry 
planes; for, suppose them displaced from the symmetry planes, as 





FIG. 181. Structure of urea. Left: arrangement of_moleeules ; general view. Right: 
symmetry elements of space-group P42jm (c projection). 

at C in Fig. 181 ; multiplication would inevitably occur, and there 
would be four nitrogen atoms to each molecule, which we know is 
incorrect. 

It is thus certain that the molecules are arranged as in Fig. 181, with 
the carbon and oxygen atoms on twofold axes and the nitrogen atoms 
on the diagonally placed planes of symmetry. Hydrogen atoms need 
not be considered, since their positions cannot be found by X-ray 
methods. 

An alternative argument will now be given , which arrives at the same 
conclusion as that already given, but takes a different course ; it starts 
with a consideration of the equivalent positions in the space-group 
P42 1 m, and only introduces the concept of molecular structure at the 
end. Both arguments are included here, because it is often useful to 
think in both ways ; consideration of the placing of molecules of known 
chemical structure is often more appropriate for organic crystals, while 
the argument from equivalent positions is more likely to be useful for 
ionic structures. 

In the unit cell of urea we have to place two carbon, two oxygen, and 



CHAP, ix EXAMPLES OF DETERMINATION BY TRIAL 



301 



four nitrogen atoms. Consider first the carbon and oxygen atoms. The 
twofold positions in space-group P42 A m are 

(a) 000; 

(b) 

(c) 

Suppose we put the carbons at (a) and the oxygens at (b). In this case 
an oxygen atom would be equidistant from 6\ and C 2 (Fig. 182 a) ; in 
other words its distance from the carbon atom in its own molecule would 
be the same as its distance from a carbon atom in a different molecule, 
which is very unlikely. Suppose now we put carbons at (a) and oxygens 
at (c) ; the same situation develops an oxygen would be equidistant 

^L 



) 

< 


A 




^ 

c 


J 

v^ 


-c 


1 
$r 


J 


^ 


/^ 


r 


-/ "~~V. 

' 












w 



FIG. 182. Consideration of possible atomic positions in urea. Left: C atoms 

at (a), O atoms at (b). Centre: C atoms at (a), O atoms at (c). Right: C and 

O atoms at (c), N atoms at (d). 

from C l and (7 3 (Fig. 182 6). Only by putting both carbons and oxygens 
at two sets of (c) positions, with different values of z (Fig. 182 c), can 
we keep the intramolecular and intermolecular carbon-oxygen distances 
different from each other. 

For the nitrogens consider the fourfold positions 

(d) OOz; OOz; Jz; fjz; 

(e) x, %+x, z; , \x,z\ %+x, x, z\ \x, x, z. 

Suppose we put them at (d) positions. Remembering that carbons 
and oxygens are at (c), it is evident that a nitrogen is equidistant from 
two carbon atoms (Fig. 182c); the (d) positions can therefore be 
rejected for the same reason as before. We are left only with the (e) 
positions, and are thus brought to the same conclusions as in the 
previous form of argument that is, that the structure is as shown in 
Fig. 181, a structure in which there are four independently variable 
parameters to be determined, # N , Z N , z c , and z o . 

The best experimental determinations of the structure amplitudes for 
the various reflections are those of Wyckoff (1930, 1932) and Wyckoff 
and Corey (1934), who measured the intensities of the reflections from 
a pressed cake of powder and from a single crystal in the form of a 



302 



STRUCTURE DETERMINATION 



cylindrical rod, using the ionization spectrometer. The powder data 
were used to indicate the corrections for secondary extinction to be 
applied to the single crj r stal data (see p. 209). 

It is best to determine # N first, by considering the hkO intensities ; for 
this projection # N is the only variable; since carbon and oxygen atoms 
are fixed (one underneath the other) as in Fig. 183. It is simply a ques- 
tion of calculating the hkO intensities for a range of positions along the 
diagonal line AB in Fig. 183. This is done most rapidly by Bragg and 
Lipson's graphical method (Fig. 169), the chart for plane group Pba 
being used. It is important to remember to use the 'cos cos' origin for 

6 






CWO 



FIG. 183. Urea. View along 
c axis. 



FIG. 184. Uroa. View along a axis. 
Plane group l y bnt. 



reflections having h~}-k even and the 'sin sin' origin for those having 
h+k odd. It is also important to use the correct scattering powers ; the 
NH 2 group may be regarded as a single scattering unit containing nine 
electrons, consequently scattering powers in the ratios 6:8:9 are 
appropriate for C, 0, and NH 2 respectively. The value of X N giving the 
best overall agreement between calculated and observed F's is 0-145 
(Wyckoff, 1932). 

The three z parameters must all be determined together ; it is simplest 
to consider first the Qkl intensities, which will give positions in the a 
projection, Fig. 184. To use the graphical method, shift the origin to P 
in Fig. 184, and on the chart use the 'cos cos' origin for reflections 
having k even and the 'sin sin' origin for those having k odd, these 
being the appropriate expressions for this plane-group Pbm. It is 
important to remember that there are two NH 2 groups and only one 
carbon and one oxygen in the structure. It would not be profitable to 
describe in detail the procedure in shifting the atoms about in the 
attempt to obtain correct .F's. But two remarks may be made. The 
first is that, in order to limit the possible atomic positions, it is justifiable 



rf\ 

m 
010 



\ 



CHAP, ix EXAMPLES OF DETERMINATION BY TRIAL 303 

to assume, as Hendricks ( 1 928 a) did in the earliest work , that the distance 
C is somewhere between 1-0 and 1-7 A, while C N is 1-0-1-5 A. 
(In working out organic structures nowadays it would be justifiable to 
assume much narrower limits, owing to the accumulation of knowledge 
since that time.) The second is that when a particular set of atomic 
positions gives a set of structure amplitudes some of which are seriously 
wrong, inspection of the charts shows in which direction each atom 
should be moved in order to increase or diminish the structure ampli- 
tude for a given reflection (see p. 264). 

The whole structure should be checked by calcula- 
tions of hkl intensities. The appropriate expression 
will be found in the Int. Tab. It is possible to use 
the Bragg and Lipson charts to shorten such calcula- 
tions (see p. 268). The final parameters were found /f0 
by Wyckoff and Corey to be z c = 0-335, z o = 0-60, //0- 
X N = 0-145, Z N = 0-18. 

Sodium nitrite, NaNO 2 , forms orthorhombic 
crystals of the shape of Fig. 185. This shape has 
holohedral symmetry mmm: the internal symmetry F J G * 1 . 85 ' . . 

J J 9 j j of sodium nitrite, 

might, however, be lower than this (see p. 247), and NaNO 2 . 

therefore atomic arrangements in all three classes 

of the orthorhombic system (mmm, 222, and 2mm) may be considered. 

The unit cell has the dimensions 

a = 3-55 A, 
b .-= 5-56 A, 
c = 5-38 A, 

and contains two molecules of NaN0 2 (Ziegler, 1931). All reflections 
for which h+k+l is odd are absent, hence the lattice is body-centred (/). 
There is evidently only one molecule of NaNO 2 associated with each 
lattice point ; the problem of structure determination is simply to group 
the atoms of one molecule of NaN0 2 at one corner of the cell; the other 
molecule is arranged in exactly the same way at the centre of the cell. 

There are no further systematic absences ; the absences of odd orders 
of AGO, Oi % 0, and OOZ are included in the general statement that reflec- 
tions having h-\-k-{-l odd are absent. This means that, for a body- 
centred lattice, we cannot tell (from the systematic absences) whether 
twofold screw axes are present or not. The possible space-groups are 
therefore /mmm in the holohedral class, 7222 and /2 1 2 1 2 1 in the enantio- 
morphic class, and 7mm in the polar class. Of these, 72 1 2 1 2 1 can be ruled 
out at once because there are no twofold positions in this space-group. 



304 



STRUCTURE DETERMINATION 



CHAP. IX 



Consider now 1222, which has several sets of twofold positions. Put 
a nitrogen atom A at the origin (its companion B will necessarily be at 
the centre of the cell). At the centres of edges and faces (as well as at 
the corners of the celJ) three twofold axes intersect (Fig. 186 a), and a 
sodium atom, to avoid multiplication, must lie at one of these points, 
that is, either exactly half-way along the c axis (at S) as in the diagram, 
or else at one of the other points mentioned. For the two oxygen atoms 
of the reference molecule the positions available are along the edges 
of the cell, either at D or in similar positions such as E. (For, suppose 
we put them off one of the twofold axes as in Fig. 186 b ; F and G would 




H 



o.. 



o 



-C-2 



-6-2 



"o 

(a) (b) 

FIG. 186. Sodium nitrite. Consideration of arrangement Immm. 

be inevitably repeated at H and /.) We may assume that the two 
oxygen atoms of the nitrite ion are closely associated with the nitrogen 
atom, hence if we have our reference nitrogen atom at the origin, the 
two oxygen atoms belonging to it will be found along one of the axes. 
In Fig. 186 a they are shown on 6, but they might equally well be on a or c. 

It is now necessary to note that this sort of arrangement has planes 
as well as axes of symmetry ; in other words, we cannot place 2NaN0 2 
in an orthorhombic cell to give symmetry 7222 ; the attempt to do so 
leads inevitably to symmetry Immm. 

We now consider the structure amplitudes which this highly sym- 
metrical type of arrangement would give, beginning with the various 
orders of feOO, O&O, and OO/. Only the even orders are present (the lattice 
being body-centred). If the NO 2 groups are as in Fig. 186 a, the oxygen 
atoms lie on both 200 and 002 planes ; hence all the orders of 200 and 002 
would have the same F'a, or in other words, the successive orders of 
both 200 and 002 should show a normal decline of actual intensity. 
Similarly, if we put the oxygens along a, the successive orders of 020 



CHAP, ix EXAMPLES OF DETERMINATION BY TRIAL 



305 



and 002 should show a normal decline ; or, if we put them along c, the 
orders of 200 and 020 should decline regularly : in each case the succes- 
sive orders of two principal planes should decline normally. In actual 
fact, the decline of intensities for the orders of both 002 and 020 is not 
regular (002 vw, 004 w\ 020 vs, 040 w). Hence the actual arrangement 
in the sodium nitrite crystal is not one of those so far considered : the 
correct space-group cannot be Immm. 



< 

o. 


X^JO (XNO 


^cT i cx_^a^ 








o^ 


^ 






c 












c\ 


> 


<^ 


~cs; 


i 


^ 





N 1 


c 


rJ 


b 
a 

b 


P 
b 

b 


l/\ 


s~ 








o b 




b w " 




(b) 



(c; 



N w P /^ a Q ( 
Ck-^O ^ CX^Lo P^fr P>| N 

->gr iTcx.^y^ <rV > ^Ix, 





FIG, 187. Sodiiim nitrite. Arrangements having symmetry Imm. 

(There is also another quite different reason for dismissing such 
arrangements : they would be unstable. For instance, the forces between 
the ions in Fig. 186 a (assuming the orientations of the ions were main- 
tained) would make the a and c axes equal and the symmetry tetragonal 
with b as the fourfold axis.) 

We are thus driven to try arrangements having the lower symmetry 
Imm. This space-group has twofold axes parallel to only one (we do not 
know which) of the cell edges, with planes of symmetry intersecting on 
each twofold axis. This means that, if we put a reference nitrogen atom 
at the origin as before, the oxygen atoms must lie on one face of the cell, 
but need not form a straight line with the nitrogen atom ; the N0 2 ion 
may be V-shaped as in Fig. 187 a. Further, the sodium atoms must lie 
on the twofold axes, but need not be exactly half-way along the cell 
edges ; this point is also illustrated in the diagrams. 



306 STRUCT CJHE DETERMINATION CHAP, ix 

The plane of the nitrite ion can be defined : it must lie in the only 
principal plane showing a normal decline of intensities that is, 200. 
The nitrite ions must therefore lie as in Fig. 187 a (with c as the polar 
twofold axis) or as in Fig. 187 b (with b as the polar twofold axis). 

Before trying to decide which orientation of nitrite ions is correct, 
consider the positions of the sodium ions. The outstanding fact bearing 
on this is that 101 is very strong. With nitrite ions at the corners of 
the cell, the only way of ensuring tins is to put sodium atoms on or 
near the b edges of the cell ; others will fall near the centres of the ac 
faces as in Fig. 187 c. Putting in the nitrite ions in the two possible 
orientations a and 6, we get the two complete arrangements d and e. 
In trying to choose the more likely of these, consider the fact that 020 
is very strong while 002 is very weak. For arrangement d, in which 
sodium atoms are exactly half-way along 6, the only way of ensuring 
that 020 shall be strong and 002 weak is to put the oxygens fairly near 
the c axis and well away from the b axis (about a quarter of the way up c). 
This would give an acute-angled nitrite ion as in /, where y < z. This 
seems improbable. In arrangement e, on the other hand, the intensi- 
ties mentioned can be satisfied by an obtuse -angled ion ; this therefore 
appears to be the more probable arrangement. Calculations of structure 
amplitudes, in the first instance for Okl planesf and finally for all planes, 
confirm that this is correct, and give the precise positions of the atoms. 
The parameters were found by Ziegler (1931) to be (taking the nitrogen 
atorti as the origin) y N& = 0-50, y o = 0-083, z o 0-194. 

The optical properties of the sodium nitrite crystal are fully consis- 
tent with this arrangement. The birefringence (largely due to the nitrite 
ion) is very strong and positive, as would be expected for a crystal con- 
taining roughly linear ions packed parallel: a. 1-340 (|| a), j8 1-425 (|| fc), 
y 1-655 (|| c).J The plane of the V-shaped ions is normal to a, the vibra- 
tion direction of lowest index, while the longest dimension of the ion 
lies along c, the vibration direction of highest index. These facts might 
indeed have been used in deducing the orientation of the nitrite ions 
in the crystal. The derivation from X-ray intensities alone has been 
given, however, as it forms a,good example of the use of such evidence. 

Sodium bicarbonate, NaHCO 3 . The structure of sodium bicar- 
bonate is more complex than that of sodium nitrite, and it would be 
very difficult or impossible to solve it by the use of X-ray data alone. 
The optical properties, however, provide valuable evidence, and the 

f Best done graphically, using Bragg and Lipson's charts. The plane-group for this 
projection is Cm. { Measurements by the author. 



CHAP, ix EXAMPLES OF DETERMINATION BY TRIAL 307 

solution of the complete structure by Zachariasen (1933) forms a very 
good example of the combined use of optical and X-ray evidence, 

The dimensions of the monoclinic unit cell (a = 751 A, b = 9-70 A, 
c 3-53 A, /? = 93 19') were found by using the rotation and oscilla- 
tion photographs of a single crystal. These dimensions, together with 
the known density of 2-20, lead to the conclusion that there are four 
molecules of NaHC0 3 in the unit cell. Absent reflections are those in 
the li()l zone for which h+l is odd indicating a glide plane n perpen- 
dicular to 6 and also the odd O&O reflections, indicating that there are 
twofold screw axes parallel to b. The space-group is evidently P^ifn. 
(This is equivalent to PZJa, with a change of a and c axes see p. 239. 
P2ja is the normal set-up given in Int. Tab.) 

In this space-group the general positions have fourfold multiplicity. 
(Coordinates! xyz, xijz, (i+a)(J-f/)(i+), (i~*)(l+y)(i-).) The 
only special positions have twofold multiplicity; these are pairs of 
symmetry centres. There are four such pairs :f (#) 000, Mi 5 (b) 00, 0|| ; 
(c) OOi, UO] (d) |0|, OJ-O. We have to assign 4Na, 4H, and 4CO 3 to 
appropriate sets of equivalent positions. Hydrogen atoms will be 
ignored for the present: their positions cannot be found directly by 
X-ray methods. 

We are entitled to assume that in this crystal there are carbonate 
groups having the same shape and dimensions as in other carbonate 
crystals that is, equilateral triangles of oxygen atoms with carbon 
atoms at the centres (see Fig. 138). Now the carbonate ion does not 
possess a centre of symmetry ; therefore neither the carbon atoms nor 
any of the oxygen atoms lie at the centres of symmetry ; they lie in 
general positions. 

No such argument applies to the sodium atoms, whicli can be 
assumed confidently to be independent ions ; they may well lie at centres 
of symmetry. It is of course not certain that they do : they may either 
occup}' two pairs of symmetry centres or alternatively one set of 
general positions. 

Assume first of all that the sodium ions do occupy two sets of sym- 
metry centres, and consider the c projection only. Although in space 
there are four different combinations of two pairs of symmetry centres, 
there are only two different projected arrangements, which are illus- 
trated in Fig. 188. Other combinations are equivalent to these: thus 
(a) + (b) is equivalent to (a)+(d), when seen from this viewpoint. 



t Usually expressed more briefly as #i/z, (I + )(} 2 

J The symmetry cent-res are incorrectly paired in the original paper. 



308 



STRUCTURE DETERMINATION 



CHAP. IX 



The orientation of the carbonate groups may be inferred from the 
optical properties of the crystals. The birefringence is very strong 
(a 1-378, ft 1*500, y 1-580), and a is so low that it may be assumed that 
the planes of the carbonate ions are all perpendicular to the vibration 
direction for this refractive index ; this direction lies in the ac plane, 
making an angle of 27^ with the c axis. Zachariasen accepted this, 
and also the dimensions of the carbonate ion as found in other crystals ; 
and attempted to find, by trial, positions in the c projection which 
would satisfy the observed hkQ intensities. This involved moving one 
reference carbonate ion about, and also rotating it in its own plane, 



f 




^ 


> < 


CO 






I ^ 4 



2Na 



2Na 



Km. ] 88. Sodium bicarbonate, c projection. Sodium atoms 
in pairs of symmetry centres. 

for each of the sodium arrangements shown in Fig. 188. The other 
three carbonate ions are of course related to the reference ion by the 
symmetries of P2Jn. 

Positions giving correct relative intensities for the hkQ refections 
could not be found, and Zachariasen therefore concluded that the 
sodium ions do not lie in symmetry centres but in general positions. 
He then moved the sodium ions (or rather, in practice, one reference 
ion) about in this same projection. (Carbonate contributions for various 
positions and orientations of the carbonate ions were already known as 
a result of the first set of calculations.) A set of positions satisfying the 
hkQ intensities was found; the arrangement is shown in Fig. 189. 

The only coordinates remaining to be found were then those along 
the c axis; these involve only two variable parameters, one for the 
sodium and one for the carbonate ion. The values of these parameters 
were found without much difficulty from the relative intensities of 
some of the other reflections ; the principle was to compare structure 
amplitudes for reflections at similar angles. Finally the structure was 
checked by calculations of the intensities of all reflections within a wide 
angular range. 



CHAP. IX 



EXAMPLES OF DETERMINATION BY TRIAL 



309 



Although only visual estimates of intensities were used, the number 
of reflections for which calculations were made is so large that the 
parameters* may be accepted with considerable confidence. A view of 
the structure seen along the 6 axis is shown in Fig, 189. 




FIG, 189. Structure of sodium bicarbonate, NaHCO a . 
Above, c projection ; below, b projection. 

In the calculations hydrogen atoms were ignored ; but their positions 
are indicated by the fact that one oxygen-oxygen distance is abnor- 
mally low (2-55 A) a fact which is taken as evidence for the existence 
of a 'hydrogen bond' between these oxygen atoms. (See Pauling, 1940.) 



310 



STRUCTURE DETERMINATION 



The hydrogen atoms are assumed to be midway between the two 
oxygen atoms concerned. This was the principal result of chemical 
interest which came from this investigation the proof of a striking 
example of hydrogen bond formation, at a time when the existence of 
this type of bond had not long been realized and was exciting con- 
siderable interest. 

. Zachariasen rounded off this work by calculating the three principal 
refractive indices of the crystal on the basis of his structure, accepting 
Bragg's theory (1924). The calculated values are close to the known 
indices of the crystal. 






FIG. 190. p-Diphenylberi/ew. Controsyiniiietricul configurations. 

^-Diphenylbenzene, C 6 H 5 .C 6 H 4 .C 6 H 5 , and dibenzyl, C 6 H 5 .CH 2 . 
CH 2 . C 6 H 5 . The crystal structures of these two substances present very 
similar problems and will be considered together. In both crystals the 
unit cell is rnonoclinic and contains two molecules, and in both crystals 
the space-group symmetry is PSJa. (Absent reflections : hQl with h odd, 
indicating the existence of glide planes perpendicular to the b axis with 
a translation of a/2, and O&O with k odd, indicating the existence of 
twofold screw axes parallel to b.) 

These facts lead at once to a valuable stereochemical conclusion, as 
in the case of diphenyl discussed on p. 249. It takes four asymmetric 
units to give the symmetry P2 JL /a, and therefore, since there are only 
two molecules in the unit cell, each molecule must possess twofold 
symmetiy; and since finite molecules cannot possess either a screw 
axis or a glide plane, they must possess the only other symmetry 
element in the cell a centre of symmetry. 

From this point we shall consider the substances separately. The 
centre of symmetry in jo-diphenylbenzene obviously lies in the middle 



CRAP. IX 



EXAMPLES OF DETERMINATION BY TRIAL 



311 



of the central benzene ring ; and the existence of it means that the 
planes of the terminal benzene rings are parallel to each other. They 
may be at any angle to the plane of the central benzene ring, but they 
must be parallel to each other (Fig. 190). 

Consider now the approximate orientation of the molecules in the unit 
cell. The dimensions of the cell (a = 8-08 A, b = 5-60 A, c = 13-59 A, 
= 91 55') suggest that the long molecules lie very roughly parallel 
to the long c axis ; and the fact that the 201 reflection is very strong 
suggests that the long molecules lie along the traces of these planes, 
as in Fig. 191 a. If this is true, and we look along the c axis, we 




FIG. 191. p-Diphenylbenzene. Approximate orientation of 
molecules in unit cell. 



should see the long molecules more or less end-on ; the strength of 110 
confirms this (see Fig. 191 b). 

It remains to define the orientation of the molecules more precisely 
and to fix the positions of all the carbon atoms. There is no further 
help to be gained from symmetry considerations all the atoms are in 
general positions, as in most organic crystals; atomic positions are 
found by the laborious process of trial. The carbon atoms in the 
asymmetric unit 

xCJuL CJi\ xCH 

CHy xC CC 

\CH CH/ \CH 

must be moved about until the calculated intensities agree with those 
actually observed. Pickett, who worked out the structure in 1933, 
assumed that the benzene ring is a flat regular hexagon, that the C C 
distance is 1*42 A as in hexamethylbenzene, and that the C C link 



312 



STRUCTURE DETERMINATION 



CHAP. IX 



lies in the planes of both rings. The problem therefore was to rotate the 
terminal ring with respect to the central ring, and to alter the orienta- 
tion of the whole molecule to find which position (if any) satisfies the 




FIG. 192. Structure of jo-diphenylbenzene. (Strukturbericht, 1933-5, p. 681.) 

intensities of the reflections. The procedure followed was to attempt 
first to satisfy the intensities of the small-angle reflections, and then 
to work outwards, the atomic coordinates being defined more and 



CHAP. IX 



EXAMPLES OF DETERMINATION BY TRIAL 



313 



(a) 





more closely as this went on. It was found that molecules having all 
three rings coplanar, oriented as in Fig. 192, give correct intensities 
for all ih& reflections. Note that the long axes of the molecules lie 
almost precisely along the 201 planes, in accordance with our prelimi- 
nary expectation ; and, moreover, they are parallel to the ac face of the 
cell, which makes the structure easy to visualize. 

The structure of dibenzyl C 6 H 5 .CH 2 .CH 2 .C 6 H 5 is formally similar 
to tnat of >-diphenylbenzene, but its elucidation (accomplished by 
J. M. Robertson, 1934 a) was a rather more complex problem. We 
may note first of all that a gre'ater variety of molecular configurations 
might be assumed by rotation 
round the three single bonds. 
However, thanks to the existence 
of the centre of symmetry in the 
molecule (which must be half- 
way between the CH 2 groups), 
some of these can be immediately 
rejected: it is certain that the 
three single bonds form a plane 
zigzag as in the paraffin hydip- 
carbons, since this is the only 
configuration of the three bonds 
which has a centre of symmetry. 
It is also certain that the planes 
of the benzene rings are parallel 
to each other ; they may be twisted 
at any angle to the central zigzag (Fig. 193) but they must be parallel 
to each other. 

A rough idea of the orientation of the molecules in the unit cell 
(dimensions a = 12-77 A, b = 6-12 A, c == 7-70 A, j3 = 116) is given 
by the fact that the highest structure amplitude is that of 202 ; 
the long molecules therefore lie approximately along these planes 
(Fig. 194). The atoms must, however, be somewhat dispersed from 
these planes, since the absolute value of the structure amplitude 
falls considerably short of the maximum possible (70 out of 113). 

In attempting to find the atomic coordinates Robertson considered 
first the more symmetrical molecular configurations a and c of Fig. 193, 
and calculated structure amplitudes for various orientations of both 
models. Starting with the inner (small-angle) reflections and working 
outwards as usual, he found that agreement between calculated and 



H-f- 



V 



-4-H 



FIG. 193. Dibenzyl. Centitf symmetrical 
configurations . 



314 



STRUCTURE DETERMINATION 



CHAP. IX 



observed intensities over the whole range of reflections could not be 
achieved by type a, whatever the orientation of the molecules, but 
that good agreement could be obtained for a particular oriehtation of 




FIG. 11)4. Dibenzyl. Approximate orientation of molecules in unit cell. 



Molecule at corner 



Molecule 
atzb 




FIG. 195. Structure of dibenzyl, C 6 H 6 .CH 2 .CH 2 .C 6 H 5 . 

molecules of type c. The latter therefore appears to be correct : it was 
not necessary to consider intermediate configurations such as b. The 
complete structure (b projection) is illustrated in Fig. 195. (It was 
subsequently confirmed by calculations of the distribution of electron 



CHAP, ix EXAMPLES OF DETERMINATION BY TRIAL 315 

density in the crystal in the way described in the next chapter 
Robertson, 19356.) 

Ascorbic acid ( 'Vitamin C ') . The crystal structure of this substance 
cannot be said to be established with the certainty and precision we 
associate with those already described ; nevertheless, there is no reason 
to doubt that the structure suggested by Cox and Goodwin (1936) on 
the basis of a limited study of the X-ray reflections is essentially correct. 
The work is described here because this crystal structure presents some 
very interesting and instructive features. It is also historically inter- 
esting because a preliminary study by optical and X-ray methods 
played a part in the elucidation of the chemical structure of this bio- 
logically important substance. (Cox, 1932 a ; Cox, Hirst, and Reynolds, 
1932; Cox and Hirst, 1933.) 

The crystals have monoclinic sphenoidal symmetry (class 2) and 
grow as almost square tablets having 100 as the principal face. They 
have very strong negative birefringence (a 1-476, /? 1-594, y 1-750), the 
vibration direction of a being parallel to the b axis. This suggests that 
the molecules are flat, with the plane perpendicular to the b axis ; indeed, 
the birefringence is so strong that the molecule may have a flat ring 
structure containing double bonds. 

The X-ray results lead to the same general conclusion. The unit cell 
dimensions are found to be a = 16-95 A, b = 6-32 A, c 6-38 A, 
/? = 102 J, and the unit cell contains four molecules. The 020 reflection 
is very strong, and the regular decline of the ixitensities of subsequent 
orders (040, 060) indicates that most of the atoms lie on or near the 
020 planes ; in other words, the molecules are flat and lie with their 
planes perpendicular to b. 

The X-ray results also lead to a knowledge of the approximate 
molecular dimensions. It is first necessary to note that the only 
systematic absences are the odd orders of OfcO ; hence the space-group is 
either P\ or P2 x /m. The shape of the crystals indicates that P2 X is 
correct. In addition, it is worth noting that P2 l /m is ruled out by the 
fact that the substance in solution rotates the plane of polarized light ; 
hence the molecules are asymmetric, and there cannot be equal numbers 
of d and I molecules in the crystal, as would be required for P2 l /m ; 
therefore the space-group must be P2 X . This space-group, however, 
requires only two asymmetric units ; the unit cell" actually contains 
four molecules, hence a group of two molecules constitutes the asym- 
metric unit of structure ; these two molecules may be grouped in any 
manner whatsoever. 



316 



STRUCTURE DETERMINATION 



CHAP. IX 



This seems to complicate the problem hopelessly. But the situation 
is not as bad as it seems ; for it is found that all hkO reflections for which 
h is odd are extremely weak, and this suggests very strongly that if we 
look along the c axis (Fig. 196 b) there is a molecule, B almost exactly 
half-way between molecules A and C and oriented in almost exactly 
the same way, so that from this viewpoint B looks almost exactly the 



(a) 




1 2 ' 

--. 

IB 



a Sin 



FIG. 196. Ascorbic acid. Arrangement of asymmetric molecules 
(represented by formal shapes) consistent with X-ray data. 

same as A and C and halves the apparent length of the a axis. The 
packing of the nearly flat molecules is of the form shown in Fig. 196 a, 
in which A and B are differently related to the screw axes ; there is no 
question of halving for the b projection, since all types of hOl reflections 
are present. 

The dimensions' of the molecules are therefore likely to be about 

L xcx-, that is, 8*5 X 6-4x3-1 A. These dimensions ruled out some 
suggested constitutions, and played a part in suggesting the following 



CHAP, ix EXAMPLES OF DETERMINATION BY TRIAL 317 

constitution which was eventually established chemically by Herbert, 
Hirst, Percival, Reynolds, and Smith (1933): 

HO OH 



/ Vo 



OH HI 

Y 

CH 2 OH 

The relative positions of the molecules in the unit cell were after- 
wards found by calculating some of the structure amplitudes for various 




FIG. 197. Structure of ascorbic acid (6 projection). Only one 
sheet of molecules is shown. (Cox and Goodwin, 1936.) 

positions. The arrangement in one sheet of molecules is shown in Fig. 
197. (In the published account (Cox and Goodwin, 1936) use is made 
of the conception of a pseudo plane of symmetry perpendicular to the 
plane of the molecule and perpendicular to the c axis ; actually the 
apparent halving of a in the c projection does not demand any such 
pseudo plane of symmetry in the molecule, but can be produced with 
completely asymmetric molecules, as Fig. 196 shows. However, as it 
turns out, the configuration and arrangement of the molecules which 
accounts for the intensities of the principal reflections does show a 
pseudo plane of symmetry in the position mentioned.) 

Long-chain polymers. To conclude this series of examples of 
structure determination by trial, accounts will be given of the elucida- 
tion of the structures of two long-chain polymers. Substances of this 
type are of increasing practical importance, and moreover their 



318 STRUCTURE DETERMINATION CHAP, ix 

molecules are very interesting stereochemically. The experimental data 
available for the study of their crystal structures is more scanty than 
in the case of crystals composed of small molecules: there is no morpho- 
logical evidence on crystal symmetry, and only limited optical evidence 
on molecular arrangement, while on account of the imperfect orienta- 
tion and often small crystal size in fibre specimens, the X-ray reflec- 
tions are less sharp than those of single crystals (see Fig. 112), with the 
result that the weakest reflections tend to be lost in the general back- 
ground of the photographs. Another limitation is that, owing to the 
overlapping of different reflections, only their combined intensities are 
known ; and there is often doubt about systematic absences. Neverthe- 
less, to offset these disadvantages there are some compensating features 
which make the study of chain-polymer structures less difficult and 
uncertain than might be supposed. The principal advantage is that in 
fibre specimens the molecules run parallel to the fibre axis (molecular 
orientation being therefore partially defined from the start), and more- 
over the length of the unit cell edge which lies parallel to the fibre axis 
is a distance iviihin the molecule a feature which has far-reaching 
consequences, as we have seen in Chapter VI. There are also other 
advantages ; for on account of the special character of the molecules, 
special arguments can sometimes be used to limit the possible arrange- 
ments in the crystal. The two examples to be described (f$ gutta-percha 
( CH 2 C(CH 3 )=CH CH 2 ) n a naturally occurring polymer of iso- 
prene and rubber hydrochloride (-H 2 C(CH 3 )C1 CH 2 CH 2 )J 
are also instructive for another reason : they exhibit pseudo-symmetries, 
which may cause confusion if the possibility of their occurrence is not 
realized. 

1. f$ Gutta-percha. Interpretation of fibre photographs shows that the 
unit cell of /? gutta-percha is rectangular (and therefore probably ortho- 
rhombic in symmetry), with the dimensions a = 7-78 A, b = 11-78 A, 
c = 4-72 A (c being the fibre axis). Cold-rolled sheets provide con- 
firmatory evidence, for in them the crystals tend to be oriented, not 
only with their c axes along the direction of rolling, but also with 
their 010 planes in the plane of the sheet; photographs of such speci- 
mens set at particular angles to the X-ray beam can be treated as crude 
oscillation photographs of single crystals, the indices of the various 
reflections being thus checked. Four chain molecules run through 
the cell. 

In interpreting chain-polymer photographs it is best to consider 
first the length of the unit cell edge which is parallel to the fibre axis, 



CHAP, ix EXAMPLES OF DETERMINATION BY TRIAL 



319 



for this length is also the repeat distance of the molecules themselves ; 
the magnitude of this repeat distance often gives valuable information 
on molecular configuration. Gutta-percha forms a striking example 
here. Its repeat distance (4-72 A) is so short that it is probable from 
the start that there is only one chemical unit ( CH 2 C(CH 3 )= 
CH CH 2 ) in this length. With regard to cis and trans positions of 
chain-bonds with regard to the double bond, it is evident from Fig. 198 
that only the trans form of chain 
is likely to have one chemical 
unit in the repeat distance (the 
cis form having two). Now a 
trans chain with all its carbon 
atom centres in a plane would 
be expected to have a repeat dis- 
tance of 5-04 A, if bond lengths 
and angles are normal; this 
figure is considerably in excess 
of the observed repeat distance. 
The only way of shortening 
such a ohain without serious 
and improbable alterations of 
bond lengths and angles is to 
make it non-planar: in other 
words, starting from a planar 
chain, we make rotations round 
the bonds. Rotations round the 
double bond are unlikely : all the 
atoms attached to the double-bonded pair of carbon atoms are likely to 
be in a plane. We must therefore rotate round single bonds. The only 
possibility is to move bond 4 la (Fig. 198) out of plane 12345 by 
rotation round bond 3 4; at the same time, in order to keep unit 
Ia2a3a4a5a strictly parallel to unit 12345 so that the two remain 
crystallographically equivalent, it is necessary to rotate bond la 4 
round bond la 2a. Thus one chemical unit has been moved towards 
the other along the chain axis while maintaining the correct distance 
between atoms 4 and la and maintaining the angles 341a and 41a2a at 
109. In this shortening movement bond 4 la can be rotated either 
clockwise or anti-clockwise, giving the two types of asymmetric mole- 
cule shown in Fig. 199 ; they are mirror images of each other. 

The next step in the interpretation of chain-polymer photographs is 




FIG, 198. Planar poly-isoprene molecules. 

(a) Cis chain, (b) End view of (a), (c) Trans 

chain, (d) End view of (c). 



320 



STRUCTURE DETERMINATION 



CHAP. IX 



usually the consideration of the projection of the structure along the 
fibre axis, to deduce the side-by-side arrangement of the molecules. 
The intensities of the equatorial reflections on the normal fibre photo- 
graph are the experimental material for this purpose. A survey of the 



left 



righfc 



4-7A 






FIG. 199. Molecular models assuming planar isoprene units (trans) and 
repeat distance of 4-7 A. Left- and right-handed molecules. 

indices of these reflections on the /J gutta-percha pattern shows that all 
reflections having k odd are absent. It looks as if there is a glide plane 
normal to c with a translation of 6/2 ; but some caution is necessary in 
this particular case, for it will be observed in Fig. 199 that left- and 
right-handed molecules, seen along the fibre axis from either end, look 
almost identical as far as the positions of atomic centres are concerned ; 
pseudo symmetries are obviously possible. Whatever the truth on this 



OHAP. IX 



EXAMPLES OF DETERMINATION BY TRIAL 



321 



point, however, the cell as seen along c apparently has its b axis halved, 
and we can certainly work on the half-size projected cell, which has only 
two chain "molecules passing through it. 

Calculations' of the intensities of the hkQ reflections for various 
positions and orientations of two molecules lead to the conclusion that 
they are disposed approximately as in Fig. 200. It can also be shown 
that no arrangement of planar molecules can possibly satisfy these 
intensities. (In the original investigation the interpretation of these 
intensities was considered before the question of chain-shortening; 
and an unprejudiced consideration 
of what atomic positions could 
possibly satisfy these intensities 
led to the arrangement of Fig. 200, 
implying non-planar chains ; con- 
sequently, when the question of 
chain-shortening (to satisfy the 







FIG. 200. Approximate positions of carbon 
atoms in. the c projection of )3 gutta-percha, 
deduced from intensities of hkQ reflections. 



repeat distance of the molecule) 
was then considered, the attain- 
ment of a non-planar molecular 
configuration having almost ex- 
actly the same end-view was most 
striking and encouraging.) 

It is now possible to consider 
the space-group symmetry of the 
structure as a whole. The evidence of absent reflections will not be 
considered yet, for reasons already given. Instead, the known approxi- 
mate arrangement in the c projection will be the starting-point for a 
consideration of the possible complete arrangements. First, a further 
limitation can be imposed by the following reasoning. )8 gutta-percha is 
made by cooling amorphous ('melted') material rapidly. 'Melted' gutta- 
percha is not a liquid but a rubber-like 'solid'. In such material the 
molecules probably do not move about relative to each other to any 
great extent ; if they did, the material would be fluid. Neither can the 
enormously long molecules turn round to reverse their ends. Hence, 
on crystallization, the molecules settle down in an orderly manner 
while remaining more or less where they happen to be. Crystals form 
where sections of molecules happen to lie in favourable positions. Since 
in any such group there are likely to be equal numbers of molecules 
pointing both ways, we expect to find in a crystal equal numbers of 
molecules pointing both ways ; we may admit the possibility of minor 



322 



STRUCTURE DETERMINATION 



CHAP. IX 



movements of sections of molecules sufficient to convert a random 
arrangement into an ordered crystalline arrangement, without admit- 
ting the wholesale migrations which would mean fluidity. Thus, of 
the four molecules passing through the unit cell, two are likely to be 
upside-down with respect to the others an 'up' molecule being defined 
as one with its methyl groups above the double bonds and a 'down* 
molecule the reverse. 



KoSoe/ 

f o 



a , , a , , 

>OC/ ROOU r *<OC/ 

_ I Rdfru I 

I R<&o b Rdfcd* Wocf ^ lopoaf* 

o 

M I J /gogo^ 

^ODW L flopoi/ 

^ f^,%W ^ ^ fZM(B) * & 



- Rcft) tf 

o - 







Pea 




/?C&(/ 
J *~ 



^ 



Rc&ou 

* o 



J> 







FIG. 201. Possible arrangements of molecules in j3 gutta-porcha. Note that for Pea the 
axial nomenclature is changed; this is done to attain the conventional orientation of 
symmetry elements in this space-group. (One symmetry element in this arrangement is 
not shown : there is a glide plane parallel to the paper, having translation a/2.) R right, 

L ~ left, u = up, d down. 

When all possible arrangements allowed by these limitations are 
considered, it is found that there are only five with orthorhombic 
symmetry (Fig. 201). The indices of the unambiguous reflections 
definitely present on the photographs do not allow us to rule any of 
these out, though the symmetry P2 1 2 1 2 l seems more likely than P2 X 2 X 2 
in view of the absence of 001 and 003 from photographs taken with the 
fibre axis oscillating with respect to the X-ray beam. 

The B forms of P2 1 2 1 2 1 and P2!2 X 2 can be ruled out by a simple 
consideration: the Oil reflection is fairly strong. Since the chain atoms 
can contribute little to its intensity, the side methyl groups must be as 
in Fig. 202. The choice between the remaining arrangements can only 



CHAP. IX 



EXAMPLES OF DETERMINATION BY TRIAL 



323 



be made by detailed calculations of intensities for a range of molecular 
positions % in each case. It would not be profitable to describe this 
rather laborious process here. It must suffice to observe that correct 
relative intensities were only obtained for the P2 1 2 1 2 1 arrangement. 
The best carbon positions are those shown in Fig. 203. f Hydrogen 
atoms are usually ignored in work on organic substances; but it is 
noteworthy that in the work on /? gutta-percha it was found necessary 
(in order to improve the agreement between observed and calculated 
intensities) to assume that the diffracting powers of C, CH, CH 2 , and 
CH 3 groups are in the ratios 6:7:8: 9 these being the numbers of 



CH 3 



FTO. 202. Approximate positions of CH 3 groups in the a projection of j3 gutta-percha. 

electrons in the groups. More recently, Levi and Corey (1941) in theii 
work on alanine have found a distinct improvement when the contribu- 
tions of hydrogen atoms at definite positions in the structure were taker 
into account. 

The results of stereochemical interest which came out of this wort 
may be indicated (Bunn, 1942 a-c). It paved the way to a solution ol 
the crystal structure of rubber itself (the cis isomer of poly-isoprene 
and of the synthetic rubber-like substance polychloroprene ( CH 2 CC 
~CH CH 2 ) n ; it led to suggestions with regard to the moleculai 
basis of rubber-like properties and to stereochemical explanations o 
the differences between the physical properties of the substances 
mentioned ; and finally it led to a general consideration of the stereo 
chemistry of chain polymers and to a new generalization on the con 
figurations of aliphatic molecules. (See next section.) 

2. Rubber hydrochloride ( CH 2 - -C(CH 3 )C1 CH 2 CH 2 ) w . This 
cellophane-like substance is made by the action of hydrochloric acic 
on rubber. X-ray diffraction j)attcrns show that it is crystalline 
(Gehinan, Field, and Dinsmore, 1938), and interpretation of the pattern! 

t See note on p. 334. 



324 



STRUCTURE DETERMINATION 



CHAP. IX 



given by drawn fibres and sheets (Bunn and Garner, 1942) yields the 
information that the unit cell is rectangular, with the dimensions 



... +. 



down 



up 



up 



down 



4-- 



down 



up 



down 



down 




FIG. 203. Structure of right-handed ft gutta-percha crystal, 
seen (A) along the c axis, (B) along the a axis. 




c b a 

FIG. 204. Bond positions in saturated carbon compounds. 

a = 5-83 A, 6 = 10-38 A, c = 8-95 A, the last being the fibre axis. 
There are four chemical units in this cell. 

The length of the repeat distance along the molecule suggests that it 



CHAP, ix EXAMPLES OF DETERMINATION BY TRIAL 



325 



comprises two chemical units. The question of the chain configuration 
is obviously more complex than in the case of gutta-percha : by rotation 
round single bonds to various degrees, all sorts of configurations, all 
having the correct repeat distance, could be obtained. But although 
rotation round single bonds occurs in liquids and gases, certain con- 
figurations are more stable than others, 
and when crystallization occurs, molecules 
settle down in these preferred configura- 
tions. In all the well-established crystal 
structures containing such molecules the 
bonds of singly linked carbon atoms are 
found to be staggered (Fig. 204). Various 
types of chain may be constructed by 
using various sequences of the three 
possible bond -configurations a'da, a!db, 
and a'dc, which will be referred to as A, B, 
and C respectively. Now the only chain 
with, an 8-atom period which has a repeat 
distance of about 8-95 A is the chain 
AAABAAAC, shown in Fig. 205. This is 
therefore likely to be the configuration of 
the rubber hydrochloride molecule. This 
chain, moreover, seems probable from 
other points of view; one would expect 
the CH 2 CH 2 CH 2 portions to be 
plane zigzags as in polyethylene (A 
sequences), while a different bond-con- 
figuration at every fourth carbon atom 
(those which carry substituents) is an 
obvious possibility. The chlorine and 
methyl substituents may be either as in 
Fig. 205 or reversed; their positions, and the arrangement of the 
molecules, must be discovered from the intensities of the X-ray 
reflections. 

There are only two molecules passing through the unit cell ; if the 
symmetry is orthorhombic (as one would suppose, from the rectangular 
character of the cell), four asymmetric units are required ; therefore 
each molecule must have twofold symmetry of some kind. Now the 
molecule in Fig. 205 has a glide plane as its only element of symmetry ; 
therefore this must be used in the crystal structure. A survey of the 




GLIDE PLANE, 

TRANSLATION . 

c/2(SEEN 

EDGEW/5E 




a 



FIG. 205. Rubber hydrochloride. 
Molecular configuration sug- 
gested by the repeat distance and 
the principle of staggered bonds. 
(Hydrogen atoms omitted.) 



326 



STRUCTURE DETERMINATION 



CHAP. IX 



reflections shows that hOl reflections having / odd are absent, indicating 
that there is, normal to 6, a glide plane having a translation of c/2. 
Our expectations are thus confirmed, so far ; and the orientation of the 
molecule in the cell is settled : it has its glide plane normal to b. 

Assuming still that the symmetry is orthorhombic, the two molecules 
must be related to each other by a symmetry element ; and this can 
only be either a plane of symmetry or a glide plane, perpendicular to 
the glide plane already mentioned, giving an arrangement in the polar 
class mm. (Arrangements in the holosymmetric class mmm would 




Pia. 206. Rubber hydrochloride. Projection along a. Consideration 
of arrangement with n glide plane parallel to the paper. 

require eight asymmetric units ; arrangements using axial symmetry 
where this does not imply an extra plane or glide plane have mono- 
clinic symmetry.) Molecules are not usually related by planes of sym- 
metry (see p. 231); moreover, the a axis (normal to which the plane 
would be) is so short (5-83 A) that there is not room for two molecules 
along it. Therefore we look for a glide plane normal to either a or c. 
In the hkO zone there is certainly no glide plane. In the Okl zone the 
presence of a fairly strong 013 reflection rules out glide planes with 
b or c translation. (This reflection, from its position, might be 003 or 
013 or both ; but in view of the structure of the molecules, 003 is bound 
to be absent.) There remains the possibility of an n glide. The great 
strength of the spot indexed as 021 + 111 (one of the strongest on the 
photograph) suggests that 021 is present, which would rule out an n 
glide ; but we cannot be quite certain of this. However, the arrange- 
ment with an n glide (Fig. 206) can be ruled out, because it would give 
a strong Oil reflection; Oil is actually absent. 

It appears, therefore, that the symmetry cannot be orthorhombic. 



CHAP. IX 



EXAMPLES OF DETERMINATION BY TRIAL 



327 



It may be monoclinic with the angle /J equal to 90. In this case the c 
glide plane possessed by the molecules themselves need not necessarily 
be used in the crystal structure ; however, the existence of a c glide 
in the crystal structure suggests that it is used. In looking for other 




>10-364. 
Cl Cl 

FIG. 207. Structure of rubber hydrochlorido. c projection. 




b 10-38 A. 
FIG. 208. Structure of rubber hydrochloride. a projection. 

symmetry elements (for the relation of the two molecules to each 
other), we find evidence (in the absence of odd O&O reflections) of a screw 
axis along 6, pointing to the space-group P2 1 /c, the arrangement being 
as in Fig. 207. 

The test of this structure is begun, as usual, by considering the c 
projection. It can be shown that the arrangement of Fig. 2.07 gives 



328 



STRUCTURE DETERMINATION 



CHAP. IX 



approximately correct intensities for the hkO reflections, if the chlorine 
atoms are placed on p bonds and the methyl groups on q bonds. The 
positions of the molecules along the c axis are found by calculating 
intensities, first for hOl and Okl reflections and finally for the general 
(hkl) reflections. It is found that satisfactory agreement between 
calculated and observed intensities is obtained for the positions shown 
in Figs. 207-9 ; this is evidently the structure of rubber hydrochloride. 
The mode of packing of the molecules shows clearly the reason why 
the angle j8 is 90. The chlorine atom of one molecule (Cl' in Fig. 209) 



Cl 




Fid. 209. Structure of rubber hydrochloride. b projection. 

fits into the hollow formed by groups CH 3 , CH 2 (2), and CH 2 (3) attached 
to carbon atom C of the molecule in front (that is, the next molecule 
along the a axis). This packing ensures that the z coordinate of Cl' is 
about the same as that of C, and since the carbon-chlorine bond is at 
right angles to the chain axis, the angle /} must necessarily be approxi- 
mately 90. 

The coordinates of atoms in chain-polymer crystals cannot be deter- 
mined with the precision attained in single-crystal work, on account of 
the smaller number of reflections available and the overlapping of some 
of them. But the amount of evidence available is sufficient (for polymers 
of the degree of complexity of those considered here) to leave little 
doubt of the general arrangement, as well as the approximate coordi- 
nates of the atoms. 



CHAP, ix EXAMPLES OF DETERMINATION BY TRIAL 329 

The principal interest of the rubber hydrochloride structure (apart 
from its bearing on the theory of the relation between the physical 
properties and the molecular structure of polymers) is that it formed the 
first test of validity and usefulness of the principle of staggered bonds. 

Abnormal structures. In all the structures considered so far two 
structural principles have been obeyed: firstly, atoms have occupied 
precisely defined positions, and secondly, positions which are equivalent 
according to the theory of space-groups have been occupied by identical 
atoms. It has already been mentioned (at the end of Chapter VII) that 
there are some crystals in which one or the other of these principles is 
violated ; and it is now intended to pursue this subject by giving a few 
examples. 

Crystals which are abnormal in this way are in a minority ; but they 
are important in their bearing on our understanding of the physical 
properties of crystals and the relations between the crystalline and 
liquid states ; and, moreover, if in attempting to determine the structure 
of any crystal it is found impossible to account for the intensities of 
the X-ray reflections by any structure based on the acceptance of the 
two principles mentioned, it must be considered whether any structure 
in which these principles are ignored can account for the X-ray pattern. 

Molecular rotation. In a normal crystal every atom occupies a 
precise mean position, about which it vibrates to a degree depending 
on the temperature; molecules or polyatomic ions have precisely 
defined orientations as well as precise mean positions. When such a 
crystal is heated, the amplitude of the thermal vibrations of the atoms 
increases with the temperature until a point is reached at which the 
regular structure breaks down, that is, the crystal melts. But in a few 
types of crystal it appears that rotation of molecules or polyatomic 
ions sets in below the melting-point ; in other words, rotation does not 
disturb the arrangement sufficiently to disorganize it entirely. Molecules 
which behave in this way are either roughly cylindrical, so that they 
may rotate about a particular axis without unduly disturbing their 
neighbours, or else roughly spherical and rotate about more than one 
axis. (White and Bishop, 1940; White, Biggs, and Morgan, 1940.) 
Evidently, when the molecules start rotating, the forces between them 
are still sufficient to ensure three-dimensional regularity until at a 
higher temperature the links between the molecules are broken by 
additional thermal vibrations, and the crystal melts. The onset of 
molecular rotation is often, but not always, accompanied by a change 
of symmetry. 



330 



STRUCTURE DETERMINATION 



CHAP. IX 



J-54A 



In some crystals such rotation occurs at room temperature. One of 
the simplest examples is potassium cyanide, KCN ; the structure is of 
the sodium chloride type (Fig. 134), and this can only mean that the 
CN~ ion is rotating ; it does not necessarily mean that aH orientations 
are equally probable, but it does mean that frequent changes of orienta- 
tion occur, such that the effective symmetry of the ion is the highest 
possible in the cubic system ; neither carbon nor nitrogen atoms occupy 
specific positions in the structure but are in effect 'spread over' a 

number of positions. 

Long molecules sometimes rotate about their 
long axes, and disk -shaped molecules or ions 
like NO^~ may spin in the plane of the disk. 
The history of the study of the long-chain 
primary alkyl-ammonium halides such as 
C 5 H 11 NH 3 C1 is interesting and instructive. 
These substances form tetragonal crystals 
with two molecules in the unit cell. It ap- 
peared at first (Hendricks, 1928 b) that the 
carbon chain in these substances is linear, with 
a C C distance of 1*25 A. Yet in other long- 
chain molecules the carbon chain is a zigzag, 
with a C C distance of 1-54 A and bond 
angles of about 110; this form of chain is 
quite incompatible with the tetragonal sym- 
metry of the alkyl-ammonium halide crystals. 
The dilemma was resolved by the suggestion that rotation of the chain 
about its long axis occurs, since in this way the zigzag chain may 
attain, in effect, tetragonal symmetry (Fig. 210) ; the spacing of 1-25 A 
is the projection on the chain axis of a bond 1-54 A in length inclined 
at 35 to the axis. It was then found that at low temperatures the 
structure changes (Hendricks, 1930) ; probably in this low- temperature 
form, the molecules are not rotating. 

Crystals in which molecules rotate still have three-dimensional regu- 
larity ; they must not be confused with 'liquid crystals', in which there 
is only two-dimensional or one-dimensional regularity (see JBernal and 
Wooster, 1932; Randall, 1934; G. and E. Friedel and others, 1931; 
Oseen and others, 1933). 

Evidence of molecular rotation may be given by non-crystallographic 
evidence; the transition from a rotating to a non-rotating state is 
accompanied by sudden changes in specific heat and in dielectric con- 




FIG. 210. The zigzag hydro- 
carbon chain in C 6 H 11 NH 3 C1 
attains, by rotation, tetra- 
gonal symmetry. 



CHAP, ix EXAMPLES OF DETERMINATION BY TRIAL 331 

starit (see Chapter VIII). Molecular rotation in crystals also leads to 
an abnormally high melting-point and a small temperature interval 
between melting- and boiling-points (Baker and Smyth, 1939). 

Since molecular rotation does occur in certain crystals, it is necessary, 
when attempting to determine the structure of any crystal, to consider 
this possibility. If there appears to be a conflict between the symmetry 
of a molecule in the crystal and the expectation based on stereochemical 
principles, or if it is found impossible to obtain correct calculated 
intensities on the assumption that the molecules are fixed, it should 
be considered whether the hypothesis of molecular rotation provides 
an explanation. 

Mixed crystals and 'defect' structures. Certain substances 
which, by themselves, form crystals of the same structural type 
are able to crystallize together in the form of a 'mixed crystal', in which 
equivalent sites are occupied indiscriminately by different atoms. The 
example of K 2 S0 4 and (NH 4 ) 2 S0 4 has already been mentioned in Chap- 
ter II ; there is nothing surprising in the formation of mixed crystals of 
these substances, since the structures of the pure substances are entirely 
analogous and the potassium and ammonium ions are similar both in 
chemical character and size. (Radii: K+, 1-33 A; NH+, 1-43 A.) Still 
simpler examples are found in some alloy systems ; for instance, copper 
and gold, which by themselves form face-centred cubic crystals, are able 
to form mixed crystals containing any proportions of the two elements. 
Here again, similarity of chemical character and size of the two types 
of atoms is the underlying cause. (Radii: Cu 1-28 A, Au 1-44 A.) The 
proof that in these alloys crystallographically equivalent sites are 
occupied indiscriminately by the two different types of atoms is very 
simple. The alloy crystals are face-centred cubic, with four atoms to 
the unit cell, that is, one atom to each lattice point ; it therefore appears, 
from the X-ray evidence, that the atoms in the crystal are all identical ; 
it is, however, known that two different types of atom are present. 
The only way out of this dilemma is the conclusion that the equivalent 
lattice points are occupied indiscriminately by the two types of atom. 
(A large number of unit cells is concerned in the formation of an X-ray 
reflection; local irregular variations of composition are not detected.) 
In agreement with this conclusion is the fact that the length of the unit- 
cell edge of such an alloy lies between those of the pure components 
indeed, the relation between unit-cell size and composition is almost 
exactly linear. 

The difference between a mixed crystal and a compound is well 



332 



STRUCTURE DETERMINATION 



CHAP. IX 




brought out by other phenomena which occur in the copper-gold 
system. The mixed-crystal alloys mentioned in the last paragraph are 
only obtained by quenching from high temperatures. If these mixed 
crystals are cooled slowly, or annealed at a suitable temperature, the 
atoms sort themselves out and form a more regular arrangement. The 
type of arrangement depends on the composition. Thus, an alloy of 
composition Cu 3 Au, when annealed, gives an X-ray diffraction pattern 
containing many more reflections than that of the quenched specimen. 
The reflections fit a cubic unit cell of about the dame size as that of the 
quenched specimen, but the pattern exhibits no systematic absences ; 

in fact the lattice is primitive, not face-centred. 
Detailed analysis shows that the arrangement 
of the atoms is that of Fig. 211, a properly 
ordered arrangement in which equivalent sites 
are occupied by identical atoms. This is a 
very simple and clear example of the effect of 
ordered and disordered arrangements on X-ray 
diffraction patterns; ordered arrangements, 
which are stable at low temperatures, give 
patterns containing more reflections than the 
disordered arrangements which are stable at 
higher temperatures. (In alloy systems the ordered structures obtained 
from mixed crystals by annealing are called 'superlattices'.) 

Mixed crystals are mentioned hero chiefly as an introduction to the 
idea that sites equivalent according to space-group theory may in some 
circumstances be occupied by different atoms. As far as structure 
determination is concerned, we need not be detained by further con- 
sideration of mixed crystals ; nobody is likely to attempt to determine 
the structure of a mixed crystal without first knowing the structures of 
the pure constituents. The function of X-ray analysis here is to deter- 
mine, as in the example of Cu 3 Au, whether a substance thought to be 
a mixed crystal is really a true mixed crystal or a 'compound' character- 
ized by a superlattice ; the latter is amenable to the normal methods of 
structure analysis based on space-group theory, and likewise need not 
detain us further. 

More surprising than the formation of mixed crystals is the occurrence 
of substances which are apparently compounds of fixed composition, 
yet in which different atoms are scattered indiscriminately among 
crystallographically equivalent sites ('defect structures'). Crystals of 
lithium ferrite LiFe0 2 , for instance, give an X-ray diffraction pattern 



FIG. 211. Structure of 
Cu,Au. 



CHAP, ix EXAMPLES OF DETERMINATION BY TRIAL 333 

which indicates the sodium chloride type of structure, with an oxygen 
in place of each chlorine atom and, to all appearances, |Li+-}-|Fe 3+ 
in place of each sodium a result which can only mean that lithium 
and ferric ions (which have the same radius, 0-67 A) are scattered 
indiscriminately over the positive sites (Posnjak and Barth, 1931). 
The constancy of composition is due to the fact that the interchange- 
able ions have different charges; any variation of the proportions 
of Li+ and Fe 3+ would mean that the whole crystal would not be 
electrically neutral. Li 2 TiO 3 has a similar structure (Kordes, 1935 b). 
More complex examples of the same sort of thing occur among substances 
having the spinel type of structure, an arrangement common to many 
mixed oxides having the formula AB 2 4 . In a normal spinel like 
ZnAl 2 4 the cubic unit cell corttains eight 'molecules' ; the space-group 
is FdSih, oxygen ions occupy a 32-fol4 set of positions, zinc ions an 8-fold 
set of positions in which each is surrounded tetrahedrally by four 
oxygens, and aluminium ions a 16-fold set of positions in which each is 
surrounded octahedrally by six oxygens. But in some spinels such as 
MgFe 2 O 4 , the positive ions are distributed differently over the same 
pattern of sites. In the example given, half the ferric ions occupy the 
8-fold positions, while^the other half, together with all the magnesium 
ions, are distributed at random over the 16-fold positions. (Barth and 
Posnjak, 1932.) The evidence for this arrangement is of course provided 
by the intensities of the X-ray reflections. This example serves to 
remind us that, if satisfactory agreement between observed and calcu- 
lated intensities cannot be achieved on the basis of the assumption 
that equivalent sites are occupied by identical atoms, then arrangements 
ignoring this principle should be tried. 

Still more surprising are certain crystals in which a set of equivalent 
positions is only partially occupied, some sites here and there at random 
being empty. The spinel group also provides examples of this type of 
structure. The cubic (y) form of Fe 2 3 , for instance, gives an X-ray 
diffraction pattern very similar to that of Fe 3 4 , which is a normal spinel 
Fe 2 +Fef H 4 . In fact it appears that in the unit cell of y-Fe 2 3 there 
are 32 oxygen ions arranged in the same way as in Fe 3 4 ; this leads to 
the surprising conclusion that there are, on the average, 21 \ iron atoms 
in the unit cell, these being scattered indiscriminately over the positive 
ion sites and the intensities of the reflections confirm this (Verwey, 
1935; Hagg, 1935). The structure of y~Al 2 3 is of the same type 
(Kordes, 1935 a ; Hagg and Soderholm, 1935). 

A simpler example is the iron sulphide pj^rrhotite, the composition of 



334 STRUCTURE DETERMINATION CHAP, ix 

which is roughly FeS but which always contains rather too little iron. 
The X-ray pattern indicates the sodium chloride type of structure, and 
it appears that while the negative ion positions are fully occupied by 
sulphur, there is a deficiency of iron atoms in the positive ion sites. 
(Laves, 1930; Hagg and Sucksdorff, 1933.) 

The zeolite group of minerals provides further examples of defect 
structures. These are complex aluminosilicates, the crystals of which 
have a rigid framework of Al, Si, and O atoms in which there are 
continuous channels ; water molecules may enter or leave the crystals 
by way of these channels, the amount of water in the crystals being 
variable (W. H. Taylor, 1930, 1934). (In normal hydrates the structure 
collapses when water is removed, a new structure being formed.) A 
simple substance in which the same thing occurs is calcium sulphate 
subhydrate CaSO 4 .0-fH 2 O (Bunn, 1941). 

An extreme type of defect structure is the a form of Agl, which is 
stable above 146 C. In this crystal the iodine atoms form a cubic body- 
centred arrangement, but the silver atoms apparently have no fixed 
positions at all; they wander freely through the iodine lattice (Strock, 
1934, 1935). 

The evidence for the various types of defect structures is (it is hardly 
necessary to repeat) provided by X-ray diffraction patterns. The unit- 
cell dimensions, the chemical analysis, arid the density settle the composi- 
tion of the unit cell, and the intensities of the reflections settle the 
positions of the atoms. Those who studied these structures were forced 
to the rather surprising conclusions by this evidence. The moral of this 
tale is that the implications of X-ray diffraction patterns (in conjunc- 
tion with reliable chemical analyses and densities) should be accepted 
boldly, even if they conflict with geometrical ideals (the application of 
the theory of space-groups) or with stereochemical preconceptions. Only 
in this way is new knowledge and a deeper comprehension of the crystal- 
line state attained. 

Additional note to p. 323. 

Slightly different atomic coordinates were subsequently suggested by Jeffrey (1944). 
The two sets of coordinates gix r o about equally good agreement between observed and 
calculated intensities. Comparison of the two sets of coordinates will give some idea 
of the degree of precision to bo expected in this work. 



ELECTRON DENSITY MAPS AND VECTOR MAPS 

IN the method of trial, crystal structures are determined by considering 
what atomic positions will account for the intensities of the diffracted 
X-ray beams. This method is not only very laborious (except for very 
simple structures) but also has all the disadvantages of an indirect 
method: so much depends on the chances of postulating an approxi- 
mately correct structure. The opposite method is to record and measure 
the diffraction pattern, and then combine the results by suitable mathe- 
matical or experimental operations to give a picture of the crystal 
structure. 

The reason why it is not usually possible to employ this direct method 
for the solution of crystal structures has already been indicated at the 
beginning of Chapter VII : it is that we do not usually know, and cannot 
determine experimentally, the phases of the various diffracted beams 
with respect to a chosen point in the unit of pattern. However, for 
certain crystals we can from the start be reasonably certain of the phase 
relations of the diffracted beams, or can deduce them from crystallo- 
graphic evidence, and in these circumstances we can proceed at once 
to combine the information, either mathematically or by experimental 
methods in which light waves are used in place of X-rays. Otherwise, 
it is necessary to find approximate positions by trial, the approximation 
being taken as far as is necessary to be certain of the phases of a 
considerable number of reflections ; as soon as the phases are known, 
the direct method can be used. 

The subject may be approached most simply by considering the 
process of image formation in the microscope, and in particular the 
formation of an image of the patterned line-grating shown in Fig. 70, 
Plate V. The simple geometrical representation of the formation of 
the image of a large object by a simple lens, in which light waves from 
different points on the object travel independently through the lens and 
are brought to a focus at different points, is not adequate for objects 
bearing fine detail commensurate with the wave-length of light, since 
in these circumstances, waves from neighbouring points interfere with 
each other. For the small patterned line-grating shown in the lower 
half of Fig. 70, Plate V, it cannot be said that an image of each line 
is produced independently of the images of its neighbours ; owing to 
interference between waves from neighbouring lines, a set of diffracted 



336 



STRUCTURE DETERMINATION 



CHAP. X 



beams is produced, each of them coming from the pattern as a whole. The 
formation of two of them the first and third orders is illustrated in 
Fig. 119. Thus, from the pattern of the grating, a very different pattern 
of diffracted beams is produced. Yet an image of the original grating 
is formed by the lens, and this image must evidently be built up by the 
interaction of the diffracted beams after passing through the lens. An 
idea of the part played by each diffracted beam may be gained in the 
following way. 

Suppose first that, in addition to the direct (zero-order) beam, only 
the first-order diffracted beams (one on each side of the primary beam) 




Spectra Lens 



Object Light 
waves 



FIG. 212. Formation of the image of a diffraction grating. Paths of light 
rays (Bragg, 1929/0- 

pass through the lens. The paths of the light rays are shown in Fig. 212, 
for the special case of a small distant monochromatic light-source. The 
parallel direct rays give an image of the source at $ ; the two sets of 
first-order diffracted rays give additional images on either side of the 
central image. There is thus in this plane a diffraction pattern in the 
form of a set of images of the source. Continuing on their way, the first- 
order rays reach the image plane, where they interact, producing an 
ordinary set of interference fringes in which there is a sinusoidal 
distribution of light intensity. (See top of Fig. 213.) In other words 
the image given by the first-order diffracted beams alone is an extremely 
diffuse one ; it merely shows diffuse lines (the spacing of which corre- 
sponds to the repeat distance of the pattern) without any of the details. 
The second-order diffracted beams by themselves would produce a 
set of interference fringes having half the spacing of those of the first 
order, and an intensity proportional to that of the diffracted beams 
concerned. If this set is added to the first-order set, as in the second 
stage of Fig. 213, the image is modified, the resultant distribution of 



OHAP. X 



ELECTRON DENSITY MAPS 



337 



light intensity being that shown by the full line, obtained by adding 
the ordinates of the constituent (dotted) curves. In this particular case 
the second orders make little difference because their intensity is small ; 
but with the Addition of the third orders (also shown in Fig. 213) the 
details of the pattern (the pair of lines constituting the pattern-unit) 
begin to appear, and become sharper when the fourth orders are added. 





FIG. 213. Formation of image of patterned line-grating of Fig. 70 by 

superposition of different sots of interference fringes, each set being 

produced by a pair of diffracted beams. 

It is by the co-operation of all the orders passing through the lens that 
the image is built up ; one may imagine all the sets of interference 
fringes, each with its own spacing and intensity, superposed. The larger 
the number of orders of diffraction taking part, the more faithful the 
image. This is the reason why the resolving power of a microscopic 
objective lens combination depends on the numerical aperture, which 
is a measure of the angular range of diffracted beams collected by 
the lens. 

If it were not possible to obtain an image of the pattern experimentally, 
it would be possible to obtain it mathematically ; but in order to do so 



338 STRUCTURE DETERMINATION CHAP, x 

it would be necessary to know not only the positions and intensities of 
the diffracted beams but also their phase angles, for it is these phase 
angles which place the various sets of interference fringes in correct 
register: for the patterned line-grating which we are using as example, 
the first, second, and fourth orders have negative phase with respect to 
the origin 0, while the third order is positive. Thus, given the position, 
intensity, and phase angle of every diffracted beam produced by a 
unidimensional pattern, we can calculate the actual density distribution 
along that pattern. We effect by numerical computation what is shown 
graphically in Fig. 213: the resultant amplitude of light vibrations at 
any point whose distance from the origin is x (expressed as a fraction 
of the repeat distance) is given by the Fourier series 



cos 7 



where each term represents the contribution of one order, the co- 
efficients A ly AZ, etc., being the amplitudes of the waves (the square 
roots of the intensities of the diffraction spots), and ]_, <x 2 , etc., the 
phase displacements. (^4 is the amplitude of the zero -order dif- 
fraction.) For a centrosymmetrical pattern, such as the one we are 
considering, the phase displacements arc all either or i (in degrees, 
or 180) with respect to the centre of symmetry, and therefore 
the expression A Q ~\- ^A n coB2nnx may be used, each coefficient A 1 , 
A 2 , etc., being given the appropriate sign, positive for a phase angle 
of 0, negative for a phase angle of 180. 

This is a simple example of the synthesis of an image from a diffrac- 
tion pattern by calculation. The synthesis of an image of a crystal 
structure from its X-ray diffraction pattern is more complex (because 
a three-dimensional diffraction grating is involved), but similar in 
principle, because the X-ray diffraction spots produced by an atomic 
pattern are absolutely analogous to the diffracted light beams formed 
by a pattern whose repeat distance is comparable with the wave-length 
of light. 

We have seen that a diffracted X-ray beam may be regarded as a 
reflection from a set of parallel planes of lattice points, and that the 
intensities of the different orders of reflection from this set of planes 
depend on the distribution of atoms between one plane and the next, 
Consequently a synthesis of all the orders of reflection from one set of 
planes leads to a knowledge of the distribution of scattering matter 
between one plane and the next, just as the synthesis of all the orders 



CHAP, x ELECTRON DENSITY MAPS 339 

of optical diffraction from a line grating yields a curve showing the 
distribution of scattering points along the grating. The scattering 
matter in a crystal consists of the electron atmospheres of the atoms, 
hence a syntHesis on the lines indicated yields a curve showing the 
distribution of electron density between one lattice plane and the next. 
The orders of 001 reflections, for instance, yield the distri bution of electron 
density between one 001 plane and the next. The expression used for 
the synthesis is entirely analogous to the one for the line grating : 
Diffracting power at any level z 

cos 27r(z-h ooi)-r- ^002 oos 2 



= ^000+ 2 ^owc 

^001 > ^002' an d so on are the structure amplitudes for these reflections, 
calculated from the intensities in the way described earlier in this book. 
If the projection has a real or apparent centre of symmetry, the phase 
angles with respect to this centre of symmetry are all either or 180. 
The scattering material in a crystal consists of electrons, and if we 
wish to calculate the absolute electron density at any level, we must 
use absolute structure amplitudes in the expression 

/^mn ^'000+ 2^ow cos2Tr (fcH <%)/) 

in which p^ is the absolute electron density at a level 2, rf 001 is the spacing 
of the 001 planes, and J^ ftoo (the structure amplitude for the zero-order 
diffraction) is the number of electrons in the unit cell. If only relative 
jP\s arc available, the constant term 7^ 000 is not known in relation to 
the set of relative F's ; nevertheless, electron densities in relation to an 
arbitrary level can be calculated. 

Turn now to the calculation of an image of a crystal structure as seen 
along a zone axis. This is obtained by a synthesis of all the reflections 
from planes parallel to this zone axis. Thus, the hOl reflections give an 
image of the structure as seen along the 6 axis, an image in which the 
high lights are the points of maximum projected electron density the 
atomic centres. 

One way of approach to this is to consider a two-dimensional pattern 
on the optical scale and the formation of an image of it by a microscope 
lens. A two-dimensional pattern such as that in Fig. 172. Plate XIII, 
when illuminated by parallel monochromatic light, gives a two- 
dimensional diffraction pattern (shown in the lower half of this figure) 
in which each diffracted beam is characterized by two order numbers. 
The diffracted beams, after passing through the objective lens and 



340 STRUCTURE DETERMINATION CHAP, x 

reaching the image plane, form the image by interference; we may 
imagine many sets of interference fringes, one set from each pair of 
diffracted beaitis hQl and toZ, crossing each other in all directions, and 
by their superposition building up the image. , Returning to the X-ray 
reflections from a crystal, we may treat the hOl set of reflections as 
if they were diffracted beams from a two-dimensional pattern which 
is the 6 projection of the crystal structure, and combine them by 
calculation to form an image of this pattern. The two order numbers, 
h and I, for each reflection, enter into the expression in this way: 
Pxz (projected electron density at point 'xz) X A (area of projected unit 
cell) = F {m +^^ f F m Gos27T(hx+ky+a m ). (Bragg, 1929 a.) To obtain 
the complete image it is necessary to calculate the projected electron 
density at a large number of points xz all over the projected cell ; for 
each point a large number of terms (one for each hOl reflection) must 
be added to the constant term F OQ0 . 

Finally, synthesis of all the reflections gives the electron density at 
any point xyz in the unit cell. The expression for this three-dimensional 
synthesis is p xyz (electron density at a point xyz) x V (volume of unit 
cell) = ^000+ 222F m cos2iT(hx+ky+lz+a hkl ). 

The labour involved in a three-dimensional synthesis is very great, 
except for the simplest structures ; for this reason (and others which will 
appear later), the two-dimensional synthesis, giving a view of a projec- 
tion of the structure, is most often used in crystal structure determina- 
tion. No synthesis can be carried out, however, unless the phase angles 
of the reflections with respect to a reference point in the structure are 
known. We now consider under what circumstances it is possible to be 
sure of the phase anglfes from the start, or to deduce them from crystallo- 
graphic evidence. 

For a much more detailed discussion of Fourier series methods, see 
Robertson, 1937. 

Direct structure determination. In some crystal projections the 
phase angles of the reflections relevant to the projection are all the same 
(with respect to a reference point). In such circumstances the image of 
the structure can be calculated directly from the intensities of the 
reflections. The circumstances in which we can be certain that this is so 
are unfortunately rare. Three conditions must be fulfilled. The first is 
that the projection must possess apparent centres of symmetry ; not 
necessarily true centres of symmetry in the crystal, but apparent 
centres of symmetry such as occur when there are twofold axes of sym- 
metry parallel to the zone axis of the projection. With respect to any 



CHAP. X 



ELECTRON DENSITY MAPS 



341 



apparent centre of symmetry, the phase angles of all reflections are 
necessarily either or 180. 

The second condition is that there must be, at the centre of symmetry, 
an atom whos'e diffracting power is very much greater than that of any 
of the other atoms in the cell. The third is that there must be only one 
such atom in the projected unit cell ;t this atom is conveniently taken 
as the origin of the projected cell. In these circumstances, it is certain 
that the phase angles of all the reflections with respect to the origin are 
0, since the wave from the heavy atom at the origin has a phase angle 
of and is so strong that it overrides the effects of all the waves from 






L^-o _ IP-O 

0-</ 5 ^f 



FIG. 214. For this projected structure, in which the black circles Represent 

heavy atoms, all reflections having h~}-k even have a phase angle of 0, but 

those having h-\-k odd may have phase angles of or 180. Therefore direct 

structure determination is not possible. 

the remaining atoms ; the latter, if in opposition to those from the heavy 
atom (that is, have, in combination, a phase angle of 180), reduce its 
intensity but cannot reverse its phase sign. 

The necessity for the third condition may be appreciated by con- 
sideration of Fig. 214, in which there are two molecules or atom groups 
in the projected unit cell. Each molecule contains a heavy atom, and 
both heavy atoms are at centres of symmetry, but the orientation of 
the molecule in the centre of the projected cell is different from that of 
the molecule at the corner. As far as the heavy atoms alone are con- 
cerned, the projected cell is centred ; this means that for all reflections 
having h+k odd, the contribution of the heavy atoms is zero, since 
the wave from the centre atom opposes the wave from the corner atom. 
But with regard to the rest of the molecule, the projection is not centred, 

f Note that for certain views of some space-groups the unit cell of the projection con- 
tains fewer molecules than the real cell ; see p. 238. Also, that where two heavy atoms 
have the same coordinates in a projection, they may be regarded as a single strong 
scattering centre for the present purposes. 



342 STRUCTURE DETERMINATION CHAP, x 

and therefore reflections having h+k odd are produced; they are due 
entirely to the parts of the molecules other than the heavy atoms, and 
their phases may be either or 180. Thus, while it is true that reflec- 
tions having h+Jc even all have a phase angle of owing to the over- 
riding effect of the heavy atoms, reflections having h+k odd may have 
phase angles of either or 180, and there is nothing to tell us which is 
correct for each reflection. 

The three necessary conditions are fulfilled in the b projection of the 
platinum phthalocyanine crystal. (Robertson and Woodward, 1940.) 
The unit cell is monoclinic (space-group P2j/a), and contains two mole- 
cules of PtC 32 H 1( jN s . Since there are only two platinum atoms in the 
cell, and the only twofold positions in this space -group he at centres of 
symmetry, each platinum atom lies at a centre of symmetry. In the 
6 projection the a axis appears to be halved owing to the existence of 
the glide plane ; in other words, this projection has a one-molecule cell 
with the heavy platinum atom at a centre of symmetry. The other 
atoms in the molecule of PtC 32 H 16 N 8 all have diffracting powers very 
much smaller than that of platinum, and therefore the phase angles will 
be those of the platinum atom, that is, for every reflection. This is 
not true for the a and c projections. The direct method can therefore 
only be used for the b projection ; fortunately this is by far the most 
informative on the details of the molecular structure. 

When electron densities are calculated for this projection by the 
expression already given, the map shown in Fig. 215 is obtained. Con- 
tours are drawn at intervals of one electron, except round the platinum 
atom, where the interval is 20 electrons. The detail shown in this 
striking view of the structure is remarkable; every atom is clearly 
resolved, and the positions of atomic centres can be read off with con- 
siderable precision. It should be noted that the flat molecule is inclined 
at 26-5 to the plane of the paper, and is therefore somewhat fore- 
shortened in one direction. 

The swamping effect of the heavy atom is of extreme value in that it 
enables us to assume the phases of all the reflections ; but it should be 
noted that the intensities of the reflections must be determined with as 
much precision as possible, since the swamping effect tends to make 
the intensities more nearly equal than they would be in absence of the 
heavy atom. 

It follows from the foregoing discussion that three-dimensional 
determinations by this direct method (giving the electron densities at 
points in the unit cell) are only possible when the unit cell contains only 



CHAP. X 



ELECTRON DENSITY MAPS 



343 



one heavy atom, which must lie at a centre of symmetry. This situation 
is rare, and, so far, three-dimensional determinations by this method 
have not 'been published. 





4 * 



FIG. 215. Electron densities in the 6 projection of platinum phthalocyanine. 
(Robertson and Woodward, 1940.) 

Phase angles from experimental evidence. There is another 
set of circumstances in which the signs of the terms can be deduced 
from experimental evidence. This has been done for phthalocyanine 
itself. It so happens that the unit cell dimensions and space-group of 
the parent substance and the nickel derivative are identical, and it can 
be assumed that the orientations of the molecules are the same in both 



344 STRUCTURE DETERMINATION CHAP, x 

crystals. The centre of symmetry of the cell is occupied by hydrogen 
in the former and nickel in the latter ; the contribution of hydrogen is 
quite negligible, hence it can be assumed that the centre of symmetry 
is effectively empty in the parent substance. Now the 'phase for the 
parent substance is positive for some planes and negative for others ; 
and it is reasonably assumed that for any particular crystal plane the 
contribution of the organic part of the nickel derivative has the same 
sign as for the corresponding plane of the parent substance. The nickel 
atom, however, being at the centre of symmetry, gives a positive con- 
tribution for every reflection. Therefore if, in the nickel derivative, the 
organic part of the molecule gives, for a particular plane, a negative 
contribution, this will be in opposition to that of the nickel atom, and 
the reflection will therefore be weaker than the corresponding reflection 
of the parent substance. But for some planes the organic part of the 
molecule will give a positive contribution, and this, co-operating with 
that of the nickel atom, will result in reflections stronger than those 
from corresponding planes of the parent substance. The procedure is 
therefore to measure the intensities of the reflections from both parent 
substance and nickel derivative, and compare the absolute intensities 
for corresponding planes. When the intensity for the nickel derivative 
is higher than that for the parent substance, it is assumed that the 
structure amplitude for the latter is positive ; when the reverse is true, 
it is assumed that the structure amplitude for the parent substance is 
negative. In this way the signs of all the structure amplitudes for the 
parent substance are found. The way is thus opened for a direct 
Fourier synthesis. This was done by Robertson (1936tf) ; this indeed was 
the first structure to be determined by the direct method. The electron 
density map so produced is very similar to that of the platinum deriva- 
tive, except that the molecule is more tilted with regard to the piano of 
projection, and therefore appears more foreshortened. The angle of 
tilt is 44. It should be noted that no assumptions of a stcreochemical 
nature were made; the only assumptions were those based on the 
observed isomorphism of the crystals. (Likewise, in the case of the 
platinum derivative, the only assumption was that the phases are all 
the same.) 

Refinement of approximate structures by image synthesis. 
Crystals which fulfil the conditions necessary for the exclusive use of 
direct methods are rare, and most crystal structures must still be solved, 
or partially solved, by indirect methods. But the indirect methods 
the method of trial, or the vector methods mentioned later in this 



CHAP, x ELECTRON DENSITY MAPS 345 

chapter need be carried only as far as the correct placing of such atoms 
as constitute the greater part of the scattering material. As soon as the 
approximate positions of these atoms have been found by trial, or by 
vector methocifc, the phases of some of the reflections are known ; the 
phases for some of the weakest reflections may be wrong, but those of 
the strong reflections are bound to be correct, and with this information 
a preliminary Fourier synthesis can be carried out. The electron 
density map so obtained indicates atomic positions with a little more 
precision than at first, and the new positions can be used to check the 
phases of the weaker reflections ; there will be a few changes, and a 
second synthesis can then be carried out. An alternative procedure is 
to include in the first synthesis only those terms whose signs are bound 
to be correct that is, those reflections which are strong in the photo- 
graphs and also have large calculated structure amplitudes. In the next 
synthesis more terms are included ; and so on. This process constitutes 
a direct method of adjusting atomic coordinates towards more probable 
values ; naturally it is most successful for projections in which the atoms 
are seen clearly resolved from each other. Good examples of structures 
determined in this way are, among inorganic substances, diopside 
CaMg(8i() 3 ) 2 determined by W. L. Bragg (1929 6), and among organic 
substances stilbene C 6 H 6 CH=CH C 6 H 5 determined by Robertson 
and Woodward (1937). They are both illustrated, in Figs. 216 and 217 
respectively. 

Organic iodine derivatives have been used for the sake of the over- 
riding effect of the heavy iodine atom. An example is picryl iodide, 
the structure of which has been determined by Huse and Powell (1940) 
In projections of this crystal structure the iodine atoms are not at 
centres of symmetry, hence the entirely direct method cannot be used 
But it can be assumed that the phases of the reflections are those of 
the combined waves from the iodine atoms alone ; this may not be true 
for a few reflections, because when the net contribution by the iodine 
atoms is small, the phase may be reversed by the rest of the molecule ; 
nevertheless, most of the assumed phases are likely to be right. Con- 
sequently, as soon as the positions of the iodine atoms are known, 
direct image synthesis can begin. This, in fact, is how the structure was 
worked out. (The preliminary location of the iodine atoms was actually 
accomplished, not by trial, but by the Patterson vector method described 
later in this chapter.) 

In most crystal structure determinations in which Fourier series 
methods have been used, two-dimensional syntheses have been made 








FiG.217. Structure of stilbene, C 6 H 8 .CH:CH.C G H & . (Robertson and Woodward, 1937.) 



CHAP, x ELECTRON DENSITY MAPS 347 

for one, two, or three principal zones. Three-dimensional syntheses 
are very laborious and are usually quite unnecessary ; but for special 
purposes it may be desirable to carry out a three-dimensional synthesis, 
not necessarily for points throughout the cell, but in a particular region 
or on a particular plane. This has been done, for instance, for particular 
planes in the crystal of penta-erythritol tetra-acetate (Goodwin and 
Hardy, 1938), and also in the poJy- ethylene structure (Bunn, 1939). 

One-dimensional synthesis is sometimes useful, as in the case of 
quaterphenyl, C 6 H 5 . C G H 4 . C 6 H 4 . C 6 H 6 (Pickett, 1936). 

For crystals not possessing a centre of symmetry (or apparent centre 
of symmetry in a projection) the difficulties of the Fourier synthesis 
method are increased by the necessity of estimating the phase angle 
for each reflection, or in other words the term ot ttkl in the expression 
F f)kl eos27r(hx-\~ky-\-lz J t-a m ). This is done by finding approximate 
positions by trial: atomic coordinates are adjusted until approximately 
correct intensities are obtained. The calculations made for this 
purpose provide the necessary data for determining a fl / d : A (see 
p. 212) = Fco&27rot ; B F sin 2?ra. This has been done for 
resorcinol C 6 H 4 (OH) 2 by Robertson (1936 b), and for the rhombohedral 
crystal form of acetamide CH 3 .CO.NH 2 by Seiiti and Barker (1940). 

Methods of computation. The number of reflections in a single 
zone (MO, h()l, or Okl) is not likely to be so large as to make the labour 
of computation prohibitive. It is, however, a task of considerable 
magnitude, and methods of reducing the time and effort are very 
desirable. Beevers and Lipsoii (1936) convert cos27r(JiX'\-ky) into 
cos 27rhx cos 27T/7/ sin 27rhx sin 2rrky and for the addition of these terms 
for a large number of points xy, use a large set of prepared card strips, 
each of which gives F cos 2nhx (or F sin Zrrhx) for given values of F and 
h and a range of values of x ; thus, a typical strip for F = +23, h = 3 
gives values of 23cos(27rX3xj|j), 23cos(27rx3Xa 1 5 ), 23cos(27rX 3x|>)> 
and so on up to 23cos(27rx3xjg), that is, for points separated by fo 
of the cell edge up to one-quarter of the cell edge. The various reflec- 
tions are divided into groups ; for instance, all those with h 3 are 
grouped together, since it is necessary to add up -F 310 cos ZirZx cos 2ny, 
F 320 cQs27r3xcos27T2y, -F 330 cos 27r3o; cos 27r3?/, and so on or in other 
words cos 27T^x(F^ w cos 2Try+F yio cos 27r2y+F^ cos 2n3y-\- ...). The 
terms in the brackets are added up (for the various values of y) by 
taking out the appropriate strips and adding up the numbers on them. 
This provides, for each value of y, a new coefficient F' ; so we have 
F' cos 27r3o:. Similar operations for other groups of reflections give 

ft *3 



348 STBUCTUKE DETERMINATION CHAP, x 

(for this same value of y) F'coaZnx, JT cos 27r2#, and so on. Strips 

h=l A=2 

for these expressions are now taken out, and the numbers -on them 
are added up, giving the electron densities at the range of points x. 
For positions of x and y beyond one-quarter of the cell edge the anti- 
symmetry of cos 6 about 9 = 90, and the symmetry of sin about the 
same point, are used. For further details of the method, the original 
paper should be consulted. A somewhat different method, using three- 
figure coefficients, is described by Robertson (1936 c). 

A machine has been designed by Beevers (1939) which carries out 
these operations with great rapidity. (See also MacEwan and Beevers, 
1942.) Finally, it is possible to effect a one- or two-dimensional syn- 
thesis by an optical method due to W. L. Bragg ; this is described in 
the next section. Three-dimensional syntheses are very laborious, 
except for very simple structures. It is probably best to use expanded 
expressions, which reduce to different forms for the various space- 
groups. Such expressions, in the forms most convenient for calculation, 
are given by Lonsdale (1936). 

Optical synthesis. We have seen that the formation, by a lens, of 
the image of a microscopic pattern may be regarded as occurring in two 
stages: first, diffracted beams are formed by interference; secondly, 
when the diffracted beams are reunited in the image plane, interference 
again occurs, and the formation of the image may be regarded as the 
result of the superposition of many sets of interference fringes, one set 
from each pair of diffracted beams. The calculation of an image of a 
crystal structure from an X-ray diffraction pattern is simply the 
mathematical equivalent of the second stage : in summing the Fourier 
series we are simply adding up the contributions of the various sets of 
interference fringes. 

The use of optical methods in place of calculations was suggested by 
W. L. Bragg. His first method consists in photographically printing 
sets of imitation interference fringes. For each pair of reflections hOl 
and K,Ql, a set of light and dark bands, having the same distribution of 
intensity as a set of interference fringes and a spacing and orientation 
appropriate to the reflections in question, is printed, the exposure being 
proportional to the structure amplitude. The superposed bands, cross- 
ing in many directions, build an image of the projected crystal struc- 
ture. For examples, see The Crystalline State, by W. L. Bragg, pp. 231-4 ; 
W. L. Bragg, 19296; Huggins, 1941. 

Bragg's second method is much more elegant ; real optical interfer- 
ence effects are produced. Beams of light, one for each X-ray reflection 



CHAP, x ELECTRON DENSITY MAPS 349 

in a chosen zone, are arranged so as to produce, by interference, an 
image of the projected crystal structure. The apparatus used consists 
essentially of two lenses, each of about 6 feet focal length, placed a few 
inches apart. * At the principal focus of one lens is a pinhole source of 
monochromatic light. Between the lenses, therefore, the light is parallel, 
and if there were no obstruction in the path of the light, an image of 
the pinhole source of light would be formed at the principal focus of the 
second lens. If, however, an opaque plate drilled with a pattern of 
holes is put between the lenses, multiple images of the point-source are 
formed at the principal focus Of the second lens : the diffraction pattern 
of the original pattern of holes is produced. If the holes are arranged 
like the points in the reciprocal lattice of a crystal zone, and the area of 
each hole is proportional to the structure amplitude for the reciprocal 
lattice point, the diffraction pattern is a representation of the 
arrangement of atoms in the crystal as seen along the zone axis, 
provided that the phases of all the X-ray reflections concerned are the 
same. The image is of course very small, and must be viewed or photo- 
graphed by means of a microscope. We may regard the formation of 
this image in the following way. The first stage in image formation 
the production of diffracted beams is accomplished by X-rays, since 
these have an appropriate wave-length for the purpose ; then, for each 
diffracted X-ray beam, a beam of visible light is substituted, so that 
the second stage of image formation the recombination of the 
diffracted beams is accomplished in the medium of visible light. This 
view of the process must not be taken too literally : it must be remem- 
bered that, on account of the three-dimensional character of the atomic 
arrangement, the crystal must be moved in relation to the X-ray beam 
in order to give diffracted beams. It is perhaps truer to say that one 
zone of three-dimensional X-ray diffractions is treated as a set of two- 
dimensional diffractions produced by a flat pattern which is the projec- 
tion of the actual crystal structure ; the recombination of these diffracted 
beams yields an image of this flat pattern. Bragg (1939) shows that a 
substantially correct image of the b projection of the diopside crystal 
is formed by this method. 

As described, the method applies to projections for which all phases 
are the same. But it has been suggested that adjustment of the phases 
of the beams of light may be accomplished by the use of small quarter- 
wave sheets of mica placed (in correct orientation) over the holes in the 
plate. To avoid having to drill holes of various sizes in a metal plate 
(as in the earlier experiments), the necessary pattern of holes can be 



350 STRUCTURE DETERMINATION CHAP, x 

produced on a photographic plate provided that the plate is optically 
flat (to a small fraction of a wave), and that there is no gelatin in the 
holes. (Bragg, 1942 a.) 

Resolving power; and other general matters. Just as the resolv- 
ing power of a microscopic objective lens depends on the angular range 
of diffracted beams collected by it, so the resolution in a calculated 
image of a crystal structure depends on the number of diffracted beams 
used in the synthesis. It can be shown (Bragg, W. L., and West, 1930) 
that peaks cannot be distinguished if the distance between them is less 
than O61rf , where rf is the lower limit' of the spacings of the crystal 
planes whose F 7 s are used in the synthesis. If reflections are recorded 
up to a Bragg angle of nearly 90, d Q ( A/2 sin 6) is about A/2, which, for 
the much-used copper Ku radiation, is 0-75 A; therefore peaks are 
resolved only if they are more than about 0-5 A apart. Since atoms in 
crystals are always 1 A or more apart, resolution could always be 
achieved by three-dimensional synthesis; but for the more usual 
(because more practicable) two-dimensional synthesis, projected dis- 
tances are often less than 0-5 A, and therefore the desirability of using 
short waves, giving a lower limit of d 0? a larger number of reflections, and 
thus better convergence of the Fourier series, is indicated (Cox, 1938). 

There is another important point. If a Fourier series is cut off 
sharply when the terms are still appreciable, false detail will appear 
in the electron density map. To avoid this, for crystals giving strong 
reflections at large angles, an artificial temperature factor may be 
applied to the intensities, to make the JF's fade off gradually instead 

_/f/fi!l0\2 

of stopping abruptly. (The intensities are multiplied by e \ A / 9 
where E is a constant.) For examples, see Wooster (W. A.), 19,'W, and 
Wells, 1938. 

Even when this is done, electron density maps usually show, in the 
regions of low density, irregularities which do not appear to have any 
significance ; they are probably due to inaccuracies in the measurement 
of the intensities of the reflections, or to approximations in calculation. 
The positions of the atomic centres, however, are not in doubt. 

Atomic positions fixed by Fourier synthesis are not necessarily more 
accurate than those obtained solely by trial, but the Fourier synthesis 
method is more straightforward ; when, in the trial method, it is desired 
to make final small changes of atomic coordinates to obtain a closer 
approximation between calculated and observed intensities, it may be 
difficult to decide which atoms should be slightly moved, arid by how 
much ; these difficulties are avoided by using the Fourier series method. 



CHAP, x ELECTRON DENSITY MAPS 351 

The final test of any proposed structure, however, is the comparison of 
calculated with observed intensities, and it is usual, when atomic 
positions have been found by Fourier series methods, to calculate all 
the intensities for these atomic positions, for comparison with the 
photographs. This is especially necessary when some of the atoms are 
not clearly resolved on the projections ; final adjustments may have to 
be made by trial. 

If the absolute intensities of the X-ray reflections are not available 
but only relative intensities the value of the constant term (the 
equivalent of jP 000 in the equations given previously) in relation to the 
other terms of the Fourier series (which are in this case in arbitrary 
units) is not known; the figures for the electron density obtained by 
calculation, omitting the constant term, will all be wrong by this 
amount ; but for the purpose of locating atomic centres, this is of no 
consequence: the image formed by the electron density contours is of 
precisely the same form. 

It is worth noting that although, in using Fourier series methods, it 
is desirable to measure the intensities of X-ray reflections as accurately 
as possible, nevertheless surprisingly good approximations to correct 
atomic, positions can be obtained by using mere visual estimates of 
intensities. This is well illustrated by two independent determinations 
of the structure of cyanuric triazide, one (Hughes, 1935) based on 
visual estimates of intensities, and the other (Knaggs, 1935) on accurate 
measurements. There is not a great deal of difference in the results, 
though the latter are naturally to be preferred. This comparison is a 
good illustration of the statement already made (Chapter VII), that 
the intensities of X-ray reflections are very sensitively related to atomic 
positions : small changes of atomic positions mean large changes of the 
relative intensities of different reflections. 

Interatomic vectors. Although, in absence of knowledge of the signs 
of the Fourier terms, it is not possible to deduce directly the actual posi- 
tions of the atoms in the cell, it is theoretically possible to deduce inter- 
atomic vectors, that is, the lengths and directions of lines joining atomic 
centres. Patterson (1934, 1935a)showed that a Fourier synthesis employ- 
ing values of F 2 (which are of course all positive) yields this information. 
The Patterson function P xyz =2221 F IM 1 2 cos Zir(hx+ty+lz) exhibits 
peaks at vector distances from the origin equal to vector distances 
between pairs of maxima in the electron density. Thus, if (Fig. 218 
left) there are atoms at positions A, J5, and (' in the unit cell, 
the function P xyz obtained by the above three-dimensional synthesis 



352 



STRUCTURE DETERMINATION 



would show maxima at A', B', and C' in Fig. 218 right, where 
OA' = AB, OB' = EG, and OC' = AC. A two-dimensional synthesis 
of all the MO intensities (P xy = 22 I^Mol 2cos27r (^+%)) would 
give these peaks projected on to the ab face of the vector cell that 
is, at D, E, and F in Fig. 218. The height of each peak is proportional 
to the product of the scattering powers of the two atoms concerned. 

Calculations of the Patterson function may be carried out in exactly 
the same way as those of electron densities. Bragg's optical method 
may also be used ; indeed, in general it may be applied more readily to 
the formation of vector maps, since (the signs of the F 2 coefficients 





a a 

FIG. 218. Left: atoms in unit cell. Right: corresponding vectors. 

being all positive) the question of phase adjustment does not arise. 
The optical method has been shown to give a correct, vector map for 
the b projection of haemoglobin. 

A straightforward F 2 synthesis gives a large peak at the origin, which 
expresses the fact that any atom is at zero distance from itself. If 
necessary, this origin peak can be removed ; the method for doing this, 
as well as a procedure for sharpening other peaks, are given in Patter- 
son's 1935 paper. 

The usefulness of the F 2 synthesis is subject to the inherent limitation 
of a vector diagram : vectors are all erected from a single point. The 
vector diagram, when obtained, must be interpreted in terms of actual 
atomic coordinates. (For the relations between peak positions on 
vector maps and the equivalent points in the 17 plane groups, see 
Patterson, 1935 b.) For simple structures this presents little difficulty, 
but for more complex structures it may be almost as difficult to interpret 
the vector diagram as it would be to solve the structure by trial. This 
is due, not only to the nature of vector diagrams, but also to the fact 
that, for unit cells containing many atoms (especially those of organic 



CHAP, x ELECTRON DENSITY MAPS 353 

substances, where interatomic distances do not differ much from each 
other), some of the vectors are very likely to lie close together so that 
the individual peaks are not resolved. This is so even in a three-dimen- 
sional vector model ; for instance, in Fig. 218 (right), A' and B' would 
be rather close together, since the vectors AB and BC in the cell are 
similar in length and orientation, and might not be resolved. In a 
projection the chances of overlapping are obviously greater still. 
Further, the number of peaks in a vector map rises rapidly with the 
number of atoms in the cell : if there are N atoms in the cell (N peaks in 
an electron density map), there are N(N 1) peaks in a vector map. 

This is a pity, for it is precisely for the more complex structures that 
some help by a direct method is most needed. However, if some of the 
atoms in the cell have much greater diffracting powers than the others, 
the vectors between the heavy atoms will stand out, and the informa- 
tion thus gained may lead to a knowledge of the coordinates of these 
atoms. The positions of the iodine atoms in picryl iodide were found 
in this way, and the knowledge paved the way to a subsequent F 
synthesis which gave an image of the projected structure (Huse and 
Powell, 1940) ; and in the determination of the structure of NiSO 4 . 7H 2 0, 
an F 2 synthesis threw considerable light on the positions of the Ni and 
S atoms (Beevers and Schwartz, 1935). 

The results for picryl iodide are particularly clear and simple. Crystals 
of this substance are tetragonal, with unit cell dimensions a = 7-03 A 
and c = 1 9* 8 A ; and the absent reflections indicate the enantiomorphous 
pair of space-groups P^i and P^\. (In agreement with this con- 
clusion, the crystals rotate the plane of polarization of light.) There 
are four molecules in the unit cell, and therefore, since the general 
position is eightfold, each molecule must possess twofold symmetry; 
this can only be the twofold axis lying perpendicular to the c axis and 
along the ab diagonal of the cell. These twofold axes necessarily pass 
through the iodine atoms, which lie in the only fourfold positions in the 
ceU: 

x,x t 0; x,x, J; J a, J+s, J; \+x, x, f. 

From the fact that 110 and 220 are both strong, it appears that the 
iodine atoms are not far from the corners and centres of the cells that 
is, x is small. For the determination of a; by the F 2 synthesis the 110 
projection appears to be the most suitable; from this viewpoint the 
iodine atoms appear as in Fig. 219 a ; the vectors V t and F 4 are exactly 
equivalent and will amalgamate to a single peak at Jc (Fig. 219 6) ; so 
will F 2 and F 8 ; and F 5 and F 6 will give a single peak at \c. For the 100 

4458 A a 



354 



STRUCTURE DETERMINATION 



CHAP, x 



projection there would not be this amalgamation, and the larger number 
of peaks might lead to confusion, at any rate for some values ofx. The 
result of Huse and Powell's synthesis of the hhl reflections is shown in 
Fig. 220. The vector marked with an arrow is F 2 + F 3 , arid the distance 
of the peak from the cell edge is 0*045 of the ab diagonal which means 
that x is also 0-046. The 001 vector map (Fig. 221), obtained by a 




/'v 



Ovl 



('C 



\ 







FIG. 219. Fourfold positions in space-group P4 1 2 1 . 
a. View along 1 1 0. b. Corresponding vector diagram. 



FIG. 220. Picryl iodide. 
Vector map, 110 pro- 
jection. ((_!. Huse and 
H. M. Powell.) 



synthesis of the hkQ reflections, confirms this ; the peaks near the origin 
are not resolved, and are thus useless for the purpose, but the two peaks 
near | give the information required. (Corresponding vectors are 
marked in the diagrams.) 

With this information, the signs of the F's for iodine atoms alone 
were calculated, and an F synthesis performed for the 110 projection. 
The resulting electron density map indicated approximate positions for 
the lighter atoms, and doubtful signs were then recalculated, using 
this information. (It is interesting that there were only two changes of 
sign.) Using the altered signs, a second F synthesis was performed, 



CHAP. X 



ELECTRON DENSITY MAPS 



355 



giving the electron density map shown in Fig. 222 ; the vectors in this 
diagram a^re marked to correspond with the peaks in the vector map, 
Fig. 220. Owing to the fact that the lighter atoms are not well resolved 
(this is a consequence of the swamping effect of the iodine atoms), 
final adjustments were made by trial, to get the best possible fit of 
calculated to observed intensities. 

The presence of comparatively heavy atoms may in some circum- 
stances confer a further advantage. Suppose there is one heavy atom 
in a projected cell, and this atom is at a centre of symmetry. We have 





FIG. 221. Picryl iodide, a. Fourfold positions, 001 projection. 
6. Vector map. (G. Huse and H. M. Powell.) 

already seen that if the atom is heavy enough, the signs of all the F's 
will be positive with respect to the heavy atom, and projected electron 
densities may be calculated directly. If, however, the atom is not heavy 
enough to determine all the signs of the jP's, this method cannot be used. 
(For instance, a copper atom in a large organic molecule would not 
determine all the phases as platinum does in platinum phthalocyanine.) 
The next best thing is to carry out an F 2 synthesis, producing a vector 
map. The strongest peaks will be at the corners of the projected vector 
cell, and they naturally present vectors between heavy atoms. The 
next strongest peaks will represent vectors between a heavy atom and 
the lighter atoms ; since the heavy atom is at the origin, these peaks 
represent the actual coordinates of the lighter- atoms. It is true that 
theoretically there should be subsidiary peaks representing vectors 
between light atoms, and these might be expected to confuse the 
picture ; but in practice (since the height of a peak is proportional to 



356 



STRUCTURE DETERMINATION 



CHAP. X 



the product of the atomic numbers of the atoms concerned) confusion 
from this source is not likely to be appreciable. 

Atomic coordinates in relation to symmetry elements. A three- 
dimensional F z synthesis over the whole unit cell would involve a 
prohibitive amount of labour, but a three-dimensional F* synthesis of 




- I. C. *N. 0=0 
Fio. 222. Picryl iodide. Electron density map, 1 1 projection. (Huse arid Powell, 1940.) 

limited scope, giving the Patterson function over a particular plane of 
along a particular line, is practicable, since the labour of computation 
involved is that of a two-dimensional or one-dimensional Fourier 
synthesis. Moreover, as Harker pointed out (1936), provided the 
crystal has planes or axes of symmetry (or screw axes or glide planes), 
it is easy to specify on which plane or along which line of the vector 
cell the most useful information will be found ; the synthesis can there- 
fore be restricted to the appropriate plane or line. If the crystal has 
axes or screw axes of symmetry, vectorial distances of atoms from 
these axes can be obtained, the labour involved being that of a two- 



CHAP. X 



ELECTRON DENSITY MAPS 



357 



dimensional synthesis although the whole of the reflections are used. 
If the crystal has planes or glide planes of symmetry, the labour of a 
one-dimensional synthesis (but again using the whole of the reflections) 
yields the distances of the atoms from these planes. This procedure 
can be followed for all but triclinic crystals, the only ones which have 
no axes or planes of symmetry. 

To specify which plane or line of the vector cell contains the desired 
information, consider the equivalent positions of the atoms in relation 
to the symmetry element. Suppose the crystal in question has a two- 




(a) (b) 

FIG. 223. a. Twofold axis, real coll. b. Corresponding vector cell. 

fold axis parallel to 6. If there is an atom at x, y, z, there is an equiva- 
lent atom at x , y, z (Fig. 223 a). The vector between these (Fig. 223 6) 
lies on the ac face of the vector cell (y = 0), and there will be a maxi- 
mum in the Patterson function P an/z at the point 2x y 0, 2z. On this ac 
face of the cell there will be a maximum for each crystallographically 
different kind of atom ; by halving the coordinates the distances of the 
atoms from the twofold axis are obtained. It is therefore necessary to 
evaluate the Patterson function only on the ac face of the cell, that is, 
for points x, 0, z. 

In working out the values of 

h= + oo k 4- oo Z + oo 

211 IF^GOs^^hx+ky+k) 

/J,as 00 fc 00 Z= 00 

for all points x, 0, z, note that for all planes having the same h and Z for 
example 201, 211, 221, 231, 2Tl, 221, etc. the cosine term is the same, 
since ky = 0. Hence, before starting the calculations, add up the F 2 '& 



358 



STRUCTURE DETERMINATION 



CHAP. X 



for all those planes ; and likewise for other sets of planes having the 
same h and Z. Thus the expression simplifies to 



Ji~= 4. oo Zn 4- o 



k -foo 



For all crystals with axes of symmetry, whether two-, three-, four-, or 
sixfold, the vectors between equivalent atoms are parallel to a face of 
the real unit cell ; the maxima in P xvs therefore lie on a face of the vector 



cell, and their positions can be found by evaluating P xQs 
P xyQ , as the case may be). 



(orP ( 



Oyss 



or 




A 




(a) (6) 

FIG. 224. a. Twofold screw axis, real cell. 6. Corresponding vector cell. 

When a crystal has a twofold screw axis parallel to 6, there are 
equivalent atoms at x, y, z, and x y y+l, z (Fig. 224 a). The vector 
between these has components 2x, , 2z, and there will therefore be a 
maximum in the Patterson function at the point 2x, J, 2z (Fig. 224 6). 
For every pair of atoms related by the screw axis there will be a maxi- 
mum somewhere on the plane y = J, and it is therefore only necessary 
to evaluate P all over this plane, that is, P X ^ Z9 to determine the vector 
distances of the atoms from the screw axis. 

In a similar way, it can be shown that if a crystal has a plane of 
symmetry perpendicular to its 6 axis, the Patterson function has 
maxima along the b axis of the cell (the line 0, y, 0, in Fig. 225) which 
indicate the distance of atoms from the plane of symmetry. For a glide 
plane perpendicular to 6, with a translation c/2, the distance of atoms 
from this plane are indicated by maxima along the line 0, y, |. 

There are other circumstances in which some of the atomic coordi- 
nates in a crystal can be discovered by evaluation of the Patterson 



CHAP. X 



ELECTRON DENSITY MAPS 



359 



function over a particular plane or along a particular line. For instance, 
it may be known, from a consideration of the space-group and the 
equivalent positions in the unit cell, that there is one particular atom 
at the origin of the cell and others somewhere on the plane y = . The 
Patterson function will show maxima on this plane at positions which 
give immediately the actual coordinates of these atoms, Similar con- 
siderations were used in the determination of the structure of potassium 
sulphamate NH 2 S0 3 K (Brown and Cox, 1940) ; it was known that the y 
coordinates of the potassium ions are and |, while those of the sulphur 
atoms are J and ; consequently the Patterson function on the plane 





FIG. 225. Left: When a crystal has a plane of symmetry normal to 6, the 
distances of atoms from this piano are given by maxima along the line OyO 
of tho vector coll. liight: When there is a glide plane perpendicular to 6, with 
translation c/2, the distances of atoms from this plane are given by maxima 
along the line Qy\ of the vector cell. 

y = J shows maxima at positions corresponding to K-S vectors. 
Atomic positions are not given directly, but can be derived from the 
positions of Patterson peaks by a consideration of the equivalent 
positions in the space-group. 

The place of vector methods in structure analysis. The Patter- 
son F 2 synthesis has proved extremely useful; indeed, in studying 
complex structures, the normal procedure nowadays is to carry out an 
F 2 synthesis (either a two-dimensional or else a partial three-dimensional 
one, according to circumstances) to see if it yields any clear information 
on the positions of some of the atoms or the general form or orientation 
of the molecules ; if it does yield such information, this may be sufficient 
to settle the signs of some of the F's, which are then used for a two- 
dimensional F synthesis leading to the first approximate electron 
density map; this is then refined in the way previbusly described. 
From what has already been said it will be evident that interpre- 
tation of vector diagrams of a complex structure is likely to be 
difficult or impossible unless the structure contains a minority of 
heavy atoms. On the relation between the vector method and the 



360 STRUCTURE DETERMINATION CHAP, x 

trial-and-error method, the following opinion by R. C. Evans (1935) 
is worth noting : 

'In more complex structures, the number of interatomic distances will be so 
large that only the most prominent between heaviest atoms will be expected 
to stand out in a Patterson synthesis, and to this extent the method has its 
limitations and perhaps gives little more information than would be deduced 
from general considerations by an experienced worker in structure analysis. 
At the same time, the Patterson synthesis does afford the only means of 
giving unprejudiced presentation of all the information which may be 
derived directly from the experimental material/ 

It is perhaps fair to say that the information contained in X-ray 
photographs is transformed by the Patterson synthesis and presented 
to the investigator in quite a different form. Whether he chooses to 
consider the interpretation of the X-ray intensities themselves, or the 
interpretation of the vector diagram, is partly a matter of taste and 
experience and partly a matter depending on the particular features 
(or suspected features) of the structure he is investigating. 

Special circumstances may weight the scales heavily in favour of the 
vector method, as we have already seen ; therefore it should always be 
considered whether the peculiarities of any substance imply ^ny 
simplification of the problem of interpreting its vector diagrams. 
Perhaps the most generally applicable of the procedures so far devised 
is the use of substances containing a minority of heavy atoms ; an 
example has already been given (picryl iodide). There is no doubt that 
this method will be much used in the study of organic substances. 
Other devices which may be useful in special circumstances are the 
following : 

(a) Vector diagrams from two photographs, taken with two different 
X-ray wave-lengths, one on either side of an absorption edge for 
one type of atom in a crystal, may be compared. The relative 
intensities of some of the peaks are different ; those peaks which 
are affected are due to the atoms whose diffracting power changes 
with the X-ray wave-length. 

(6) If a crystal is known to have identical atoms or groups of atoms 
at the corners and centre of the projected unit cell (the c projec- 
tion, say) though the projection as a whole is not centred then 
an F 2 synthesis employing only the reflections having h+k odd 
will exhibit peaks due only to atoms other than the centre and 
corner atoms. 



CHAP, x ELECTRON DENSITY MAPS 361 

(c) The vector diagrams of isomorphous substances may be com- 
pared ; the differences may indicate which peaks are due to the 
replaceable atoms. 

These are merely examples of possible applications of special pro- 
cedures ; no doubt many others will be devised in future. Even among 
crystals of extreme complexity, such as the proteins, the problem of 
the interpretation of the vector diagrams is not hopeless, in virtue of 
the peculiarities of at any rate some of them. For instance, it seems 
possible that by making use of the shrinkage and change of shape of 
the unit cell of crystalline haemoglobin on drying, and a comparison 
of the vector diagrams, it may be possible to draw some conclusions 
on the internal structure of the molecule, or even to solve the crystal 
structure in detail. (Bernal, Fankuchen, and Perutz, 1938; Perutz, 
1942.) 



XI 

BROADENED X-RAY REFLECTIONS ANJ> 
THEIR INTERPRETATION 

HITHERTO, in this book, we have been concerned chiefty with sharp 
X-ray reflections which occur at the Bragg angle and over a very 
narrow angular range near it the reflections given by crystals which 
are comparatively large (> 10~ 5 cm. in diameter) and are perfectly 
regular in internal structure. Specimens are, however, sometimes 
encountered which give broadened X-ray reflections: for instance, 
powder photographs may be obtained which show rather diffuse lines. 
This may mean that some crystals have slightly different unit-cell 
dimensions from others ; or that the unit-cell dimensions vary in different 
regions of the same crystal, owing to variations of composition or to 
strains. In these circumstances certain crystals or parts of crystals 
give X-ray reflections at slightly different angles from others, and a 
broad line is the result. Alternatively, the broadening may mean that 
the crystals are extremely small. Just as an optical diffraction grating 
with comparatively few lines gives diffuse diffracted beams, so a crystal 
of very small dimensions gives reflections over a wider angular range 
than a large crystal. More complex broadening effects may be caused 
by structural irregularities on a very small scale and by the thermal 
movements of the atoms in crystals. 

The interpretation of the broadening is of obvious importance in 
relation to the physical properties of materials, and much work is being 
done from this viewpoint. In addition, there may be, for certain sub- 
stances, some correlation between the broadening and particular chemi- 
cal properties. The rate of a chemical reaction may, for instance, 
depend on the size of the crystals of a solid reactant (small crystals 
reacting more rapidly than large ones) ; thus, if the size involved lies 
in the range below 10~ 5 cm., it may be possible to correlate reactivity 
with the broadening of the X-ray reflections. The activity of solid 
catalysts is also likely to depend first of all on the size of the crystals : 
the adsorption which is believed to be a prerequisite for many catalytic 
reactions is more extensive the greater the surface area (which is a 
function of crystal size). Size, however, is not the only factor involved 
in catalytic activity. The 'active spots' which are believed to exist on 
catalyst surfaces may be associated with local strains, due perhaps to 
the presence of foreign atoms inserted in the main crystals. ('Promoters' , 



CHAP, xi BROADENED X RAY REFLECTIONS 363 

the secondary substances* often added to increase the activity of cata- 
lysts, may have the function of providing these foreign atoms which 
strain the crystals and thus give rise to active spots.) Little is known 
of these matters at present, but it appears to be a field well worth the 
attention of X-ray workers in chemical laboratories. A convincing 
correlation between catalytic activity and the line-broadening on X-ray 
powder photographs of the catalysts concerned, with an interpretation 
extended to the distinction between the different possible causes of 
line-broadening, would represent a notable advance in our knowledge 
of the mechanism of catalysis: 

The correlation of line-broadening with the chemical conditions of 
preparation of materials also falls within our scope. Consider, for 
instance, pigment materials ; we are not concerned with the colour or 
other physical properties as such ; but the colour of a powder may 
depend not only on the crystal structure of the particles, but also on 
their size; and the size may depend on the chemical conditions of 
preparation. 

In all such circumstances the problem which presents itself is, in the 
first place, that of distinguishing between the different possible causes 
of line-broadening ; and then, if a definite verdict on this point can be 
given, to attempt quantitative interpretation in terms of this factor, be 
it crystal size, or the extent of the variation of lattice dimensions, or the 
periodicity of structural irregularities or thermal movements. 

Relation between crystal size and breadth of X-ray reflections. 
If an extremely narrow X-ray beam is diffracted by a large perfect 
crystal which is rotated, each diffracted beam (making an angle of 20 
with the primary beam, where is the Bragg angle) is extremely narrow 
it has a width of only a few seconds of arc on either side of the 
theoretical angle 20. Most crystals are imperfect in the sense that 
different regions of the crystal are not exactly parallel to each other ; 
but this does not affect the angular range of the diffracted beam : the 
crystal may continue to give a diffracted beam when it is rotated through 
several minutes of arc, or even as much as half a degree, but the 
diffracted beam is not broadened by such imperfections, provided that 
the perfect sections of the crystal are larger than 10~ 5 cm. ; the imperfec- 
tions of the crystal may be regarded as part of its rotation. (On Weis- 
senberg photographs the reflections are often short streaks owing to 
such imperfections.) 

If such a crystal were ground to fragments, and a powder photograph 
were taken, using an extremely narrow X-ray beam and a very small 



364 



STKUCTUBE DETEKMINATION 



CHAP. XI 



specimen, the angular width of the arc would still be only a few seconds, 
provided that the crystal fragments were larger than lO^ 5 cm. in 
diameter. In practice, somewhat divergent X-ray beams, and powder 
specimens of very appreciable width, are used (otherwise exposures 
would be inconveniently long), and it is these circumstances which are 
responsible for the widths of the arcs on typical powder photographs 
such as those in Fig. 63, Plate III. The arcs on the photograph of 
quartz, for instance, would be described as perfectly sharp ; in other 




BACKGROUND 

INTENSITY 

SUBTRACTED 



BACKGROUND 



DISTANCE ALONG FILM 
FIG. 226. Intensity distribution in an arc of a powder photograph. 

words, the breadth of the arcs is almost entirely due to camera condi- 
tions, not to the diffracting properties of the specimen. 

A crystal smaller than about 10~ 5 cm. in diameter gives a broadened 
diffracted beam. For quantitative treatment it is first necessary to 
define 'breadth*. A line on a powder photograph (Fig. 226) shows 
maximum intensity at the centre, and fades away on either side ; a 
simple definition of breadth is the angular width between the points 
at which the intensity falls to half its maximum value. In an 'ideal' 
powder photograph of an extremely narrow specimen taken with a very 
narrow beam, the relation between this 'breadth at half height' /? and 
the size of the crystals would be 

/? (in radians) = sec 6, 
t 

where t is the thickness of the crystal, C is a constant, A the X-ray 
wave-length, and 6 the Bragg angle. (Scherrer, 1920.) Strictly speaking, 



CHAP, xi BROADENED X-RAY REFLECTIONS 365 

this is true only when the particles are spherical, belong to the cubic 
system, and are uniform in size, but it is usually agreed that the equa- 
tion can be used quite generally without risk of serious error. The 
constant C is*, according to Scherrer, 0*94. Somewhat different treat- 
ments by Seljakow (1925) and Bragg (1933) gave the same equation, 
with a value for the constant of 0-92 and 0'89 respectively. 

Laue (1926) dealt with the general case of a parallelepiped belonging 
to any crystal system, and used a different definition of Krie breadth 
the breadth of an imaginary line having uniform intensity equal to the 
maximum and a total intensity equal to that of the actual line (in other 
words, the area of the curve divided by its height). For crystals of 
cubic shape belonging to the cubic system, Laue's expressions reduce 
to the same equation as Scherrer's, with C = 1-42 (Laue gave 0-9, but 
Jones (1938) pointed out and corrected arithmetical errors in Laue's 
calculations). 

The pure diffraction broadening discussed in the last two paragraphs 
increases the breadths of lines on powder photographs when the crystals 
are very small. In order to deduce the size of the crystals it is necessary 
to find the pure diffraction broadening /J, and for this purpose it is 
necessary to know the breadth of lines (b) given by crystals greater 
than 10~ 6 cm. under the same camera conditions. This breadth b is 
determined not only by the divergence of the X-ray beam and the size 
of the specimen, but also by the absorption. (See Chapter V.) Numerous 
attempts have been made to correct for such factors by calculation 
(for summary, see Cameron and Patterson, 1937) ; but the safest method 
is that of Jones (1938), who mixes the solid under investigation with 
another substance consisting of crystals larger than 10~ 6 cm. The 
diffraction arcs of both substances are affected in exactly the same way 
by the camera conditions and absorption in the specimen ; consequently 
the breadths of the sharp lines of the reference substance give 6, while 
those of the substance under investigation give J5, the total breadth 
including diffraction broadening. Both these vary with 0, and the 
procedure is therefore to measure on the microphotometer the intensity 
distribution in various lines of both substances at different Bragg angles 
(converting from photographic opacity to X-ray intensity by the use 
of a calibration wedge, as described in Chapter VII), and to plot the 
breadths 6 of the sharp lines against 6 ; by interpolation, the value of 
6 at the Bragg angle of each of the broadened lines is read off. To obtain 
the pure diffraction broadening /? from B and 6, Scherrer assumed that 
j} = J3 6. Jones (1938) pointed out that it depends on the intensity 



366 STRUCTURE DETERMINATION CHAP, xi 

distribution in a line, and gives, for particular line-shapes, correction 
curves giving the relation between b/B and j8/J5. He also gives a method 
for correcting for the effect of the cx x a 2 doublets. 

In the Scherrer formula ft is proportional to sec 0. For other causes 
of line-broadening, the relation is different ; therefore, in studying a 
particular substance, if /? is found to be proportional to sec 0, it is 
probably justifiable to assume that the broadening is due to the small 
size of the crystals. 

No great accuracy can be expected in work of this sort. A further 
source of doubt is in the effect of the wide range of crystal sizes present 
in most specimens ; Jones states, as a general conclusion, that on this 
account the observed mean size will be greater than the true mean size, 
but that the difference is unlikely to be greater than 30 per cent, of the 
observed mean size. 

The above discussion relates, strictly speaking, only to spherical 
crystals. For other shapes the breadths of different reflections depend 
on the dimensions of the crystals in different directions and on the 
indices. The breadths of different lines thus do not vary regularly with 
the Bragg angle. The subject will not be considered in detail ; but, in 
a general way, it may be observed that the breadth of a line depends on 
the thickness of a crystal in a direction at right angles to the reflecting 
planes. Plate-like crystals, for instance, will give broader reflections 
from planes parallel to the plane of the plate than from those perpen- 
dicular to this plane. For other planes it is necessary to consider the 
volume averagef of the thickness of the crystal in the direction at right 
angles to the reflecting planes. (Stokes and Wilson, 1942.) For further 
information see also papers by Laue (1926), Brill (1930), Jones (1938), 
and Patterson (1939). 

There are other possible causes of differential broadening, as we shall 
see later; and a distinction between the different possible causes is 
much more difficult than in the case of broadening which is some 
function of the Bragg angle only. But if there are external reasons 
for believing that small crystal size is the only likely cause for 
broadening in a particular specimen, then a general idea of the size 
and shape of the crystals may be gained in the way indicated in the 
last paragraph. 

This section will be concluded by an emphasis on three points. First, 

f T m d V 
f The 'volume average' is , where V is the volume of the crystal and T^i 

is the thickness of the crystal in the direction normal to the hkl plane. 



CHAP, xi BROADENED X-RAY REFLECTIONS 367 

that for powders the X-ray method gives the size of crystals, not 
particles.(which may be aggregates of crystals). Second, that the range 
of sizes covered is the range below 10~ 5 cm., and that estimates of 
crystal size Between 10~ 6 and 10~ 5 cm. are very rough, since the line- 
broadening is slight. It is only below 1C"* 6 cm. that broadening is really 
sufficient for reliable measurements. For larger crystals the range 
above 10~ 4 cm. (1 p) is usually dealt with by measuring and counting 
particles or crystals under the ordinary microscope. The range between 
10~ 6 and 10~ 4 cm. is now catered for by the electron microscope; there 
are many recent papers on its Applications see, for instance, Zworykin 
(1943) . The third point is that the line-broadening gives no information 
on the size distribution among thje crystals responsible for the effect ; 
it indicates only an average size. 

Other causes of broadening of X-ray reflections. When a mixed 
crystal phase is formed the composition is likely to vary from one crystal 
to another, and unless a long time is allowed for the attainment of 
equilibrium, this difference of composition between crystals, or even 
different parts of the same crystal, is likely to persist. The lattice 
dimensions will vary with the composition, and therefore a powder 
photograph will show broadened lines. In distorted crystals also the 
lattice dimensions vary, with a similar effect on the powder photograph ; 
this may happen in crystals of fixed composition as well as in mixed 
crystals. If the variation of lattice dimensions is uniform in all direc- 
tions, the breadth of the X-ray reflections is proportional to tan 0. We 
have already seen that broadening due to small crystal size is propor- 
tional to sec 8 ; hence, if data are available for a sufficient angular range, 
it is possible to distinguish between these two possible causes of line- 
broadening. (Stokes, Pascoe, and Lipson, 1943; Lipson and Stokes, 
1943; Smith and Stickley, 1943.) It is interesting to appreciate the 
difference between the two situations in terms of the reciprocal lattice. 
A reduction in the size of crystalsf causes all the points of the reciprocal 
lattice to expand to the same extent (see Fig. 227 a) ; but a variation of 
lattice dimensions means a variation of reciprocal lattice dimensions, 
with the result that the outer points are drawn out more than the inner 
ones (Fig. 227 b). The breadths of reflections are given by the relative 
times taken by different reciprocal 'points' (they are really small 
volumes) to pass through the surface of the sphere of reflection on 

f For non-spherical shapes the reciprocal points expand anisotropically, being greatest 
in tho direction of the smallest real dimension ; all the reciprocal points become the same 
shape and size. 



368 



STRUCTURE DETERMINATION 



CHAP. XI 



rotation; it is thus evident that the relations between breadth of 
reflection and 6 are different in the two cases. 

Another cause of the broadening of X-ray reflections is to be found 
in the thermal motions in crystals. It has already been mentioned in 
Chapter VII that, on account of the thermal motions, the reflecting 
power is not concentrated entirely in the points of the reciprocal lattice, 
but is to some extent spread along the lines joining the points. The 
'points' are really small volumes having a three-dimensional star-shape. 
On single-crystal photographs the result is the formation of extra spots 



Toio 



10 1700 



ft 



W 



W 1100 



MO Tf0 T200 2/0 220 



10 



10 



10 



10 



20- 



20 



20 

















\ 


220 


\ 


2fO 


200 f 


tfto 


/ 
















^^ 


120 


\ 


iio 


TOO 4 


'fro 


^/p 


















020 




010 




010 




^^ 


do 


/ 


1JO 


no ' 


*110 


>^ 
















/ 


220 


/ 


210 


200 \ 


\210 


s 

















FIG. 227. a. One level of the reciprocal lattice of a very small crystal. 
6. Effect of variation of reciprocal lattice dimensions. 

and streaks, much weaker than the normal 'Bragg' reflections, and only 
appreciable in intensity in the case of certain substances and under 
certain experimental conditions. (See Lonsdale, 1942.) On powder 
photographs the effect would be to extend the 'foot' of the curve (Fig. 
226) representing the distribution of intensity in an arc ; this may make 
the background level uncertain; but this effect is likely to be serious 
only in the case of crystals near their melting-points. 

One more cause of broadening will be mentioned. This is the existence 
of structural irregularities in crystals. The nature of such irregularities 
will be illustrated by one of the simplest cases that of the metal cobalt. 
This element may crystallize either with a hexagonal structure or a 
face-centred cubic structure. These structures are different types of 
close-packing of spheres, arising in the following way. If a single layer 
of spheres is arranged in close-packed formation (positions 1 in Fig. 228) 
and a second laver is put on top of the first, resting in the hollows 




CHAP, xi BROADENED X-RAY REFLECTIONS 369 

marked 2, the third layer put over the first, and the fourth over the 
second, the arrangement is hexagonal close-packed. If the third layer, 
instead of being put over the first, is put in the alternative set of posi- 
tions marked 3, and then this 1, 2, 3 succession is repeated, the arrange- 
ment is face-centred cubic (the layers being 111 planes). Some speci- 
mens of cobalt give X-ray photographs which indicate that these two 
possible arrangements occur indiscriminately throughout the crystals. 
The evidence is that while some X-ray reflections are sharp, others are 
broadened in different degrees. The phenomena will not be explained 
in detail, but it may be observed that the building plane of the above 
description (001 of the hexagonal 
form, 111 of the cubic form) is 
perfect, and gives a sharp reflection, 
but that planes inclined to it exhibit 
random faults, which give rise to 
reciprocal lattice points extended 
in particular directions and thus 
to X-ray reflections broadened in 
different degrees. (Edwards and 
Li yson, 1942; Wilson, 1942.) More ^ nn T11 

' , / i_ i_ FIG. 228. Illustrating alternative 

complicated cases which have re- close-packed structures. 

cently been studied are those of 

AuCu 3 (Jones and Sykes, 1938; Wilson, 1943 a), and Cu 4 FeNi 3 (Daniel 
and Lipson, 1943). The streaks which occur on fibre photographs of 
chrysotile (Warren and Bragg, 1930) and certain chain polymers (Fuller, 
Baker, and Pape, 1940) probably have a similar origin. The distribution 
of the faults may give rise to extension of the reciprocal lattice points 
in particular directions and in different degrees, or to the existence of 
satellites round certain reciprocal points. 

It will be evident that when differential broadening is encountered 
in powder photographs, it may often be difficult to decide which is the 
most likely cause. If single-crystal photographs can be obtained, more 
detailed and more certain interpretation may be attempted. But, 
beyond the observation that thermal effects will obviously vary in 
intensity with temperature (Preston, 1939, 1941 ; Lonsdale, 1942), no 
general rules can be given ; each case must be considered individually. 
Mathematical treatment is likely to be difficult, and even qualitative 
consideration far from straightforward; here, the optical diffraction 
methods introduced by Bragg are likely to be very useful (see Bragg 
and Lipson, 1943) : patterns can be made exhibiting particular types of 



370 STRUCTURE DETERMINATION CHAP, xi 

faults, and the optical diffraction effects compared with the X-ray 
diffraction effects of actual crystals. For simple structures, suggestions 
of possible types of faults may be obtained by Bragg's device of using 
an array of surface-packed bubbles as an analogy of a t\tfo-dimensional 
atomic arrangement. (Bragg, 19426.) 

Interpretation of diffraction effects of non- crystalline sub- 
stances. It has been pointed out in Chapter V that there is no sharp 
dividing line between crystalline and 'amorphous' substances : with de- 
crease of crystal size, X-ray diffraction patterns become more and more 
diffuse until finally, any attempt to calculate crystal size by the method 
given earlier in this chapter gives a figure of only a few Angstrom units 
that is, about one unit cell; in these circumstances the word 'crystal', 
with its implication of pattern -repetition, is inappropriate. The alterna- 
tive word 'amorphous' is not entirely satisfactory either: on account of 
the sizes of atoms and their preference for particular environments, the 
distribution of atomic centres cannot be entirely random. The word 
'non-crystalline' is really preferable. 

With increasing diffuseness of the diffraction pattern the possibilities 
of interpretation obviously become more restricted ; but even in the 
extreme cases of glass-like substances and liquids it is possible to draw 
definite conclusions on the manner of association of the atoms, at any 
rate in the simpler cases. As in crystal analysis, there are two ways of 
proceeding. Particular arrangements of atoms may be postulated, and 
the intensity of diffraction at different angles calculated, for comparison 
witL the actual diffraction pattern of the specimen. The opposite 
method is to convert the experimental diffraction pattern, by the 
Fourier series method, into a vector diagram. Provided that there is 
only one type of atom in the specimen, this diagram will represent the 
radial distribution of atoms round any atom in the specimen. In the 
more general case, where there is more than one type of atom, we may 
say that all the vectors in the specimen are superposed in all directions ; 
it is then necessary to consider the interpretation of the vector diagram 
in terms of atomic arrangements. 

In either case, the first step is to record the diffraction pattern of the 
substance ; it must be a true record of the absolute intensity of diffrac- 
tion of monochromatic X-rays over a wide angular range. 'White' 
radiation effects, superimposed on the monochromatic pattern, would 
complicate the situation : hence it is necessary to use strictly mono- 
chromatic X-rays reflected from a crystal (see Chapter V) ; and it is 
also advisable to evacuate the X-ray camera to avoid scattering by air. 



OHAP. xi BROADENED X-RAY REFLECTIONS 371 

(Warren, Krutter, and Morningstar, 1936.) The photographic intensities 
are measured on the microphotometer, and converted to X-ray inten- 
sities in the usual way. It is also desirable to correct for 'Compton 
scattering 5 the incoherent scattering of X-rays with concomitant 
change of wave-length. (See Warren, 1934; Wollan. 1932.) 

In the method of trial the intensity / of scattered radiation at any 
angle is given (for a solid containing two different types of atom) by 
the Debye equation 

T ___ 



where f m and/ n are the scattering powers of atoms m and n, S ~ -~^~r , 

A 

and r mn is the distance from atom m to atom n (Debye, 1915). 

In the Fourier series method the weighted radial distribution func- 
tion, which represents the number of atoms at a distance r from any 
atom, weighted by the products of the diffracting powers, is given by 
the expression 

oo 

2 K m 4irr a /> + f iS sin Sr dS, 

77 J 
* o 

which is evaluated for a range of values of r. In the first term of this 
expression K m is the effective number of electrons in atom m, and p the 
average number of electrons per unit volume ; the summation is over 

a unit of composition. In the second term i = (IVTTJJ"""" M J ^ being 

absolute intensity (in electron units) at an angle 8, N the number of 
atoms in the sample, and / their absolute diffracting power. 

Warren (1937, 1940) calculates iS from the experimental intensity 
curve and plots it as a function of 8 ; he then carries out the integration 
for about forty different values of r ranging from to 8 A, either 
graphically or with a harmonic analyser. 

For an example of the results obtained in this way, see the paper by 
Warren, Krutter, and Morningstar (1936) on silica glass. It appears that 
the immediate surroundings of any one atom are much the same as in 
cristobalite (one of the crystalline forms of silica) ; this is the reason 
why the diffraction pattern of silica glass is like a very diffuse version 
of that of cristobalite. On the larger-scale aspects of glass structure, 
the X-ray data do not give definite information ; they are equally con- 
sistent with Zachariasen's model (1932) of a continuous random net- 
work (Fig. 229 b) and with the ideas of Hfigg (1935) and Preston (1942), 



372 STRUCTURE DETERMINATION CHAP, xi 

who suggest that large discrete groups of atoms may be present 
in other words, that the bond-scheme of Fig. 229 b is not maintained 
continuously throughout the specimen. 

Warren's method may be applied also to powder photographs of 
crystalline substances. Warren and Gingrich (1934), applying it to 
orthorhombic sulphur, found the results consistent with the assumption 
that rings of eight sulphur atoms are present. Medlin (1935, 1936) has 
also used it for several substances ; but the results are of course much 
less detailed than those obtained by single-crystal methods. 




FIG. 229. Two-dimensional representation of the difference between 
a crystal (left) and a glass (right), according to Zachariasen (1932). 

Similar methods may be used in the study of the structures of liquids. 
The continual movements which occur in liquids do not affect the 
determination of the principal interatomic distances. For earlier work 
on the subject see Randall's book (1934). Among more recent papers, 
those of Harvey (1939) on ethanol and Bray and Gingrich (1943) on 
carbon tetrachloride are typical. 

The interpretation of the diffraction effects of gases of simple mole- 
cular structure is simplified by th fact that the molecules are very 
far apart, and therefore it is necessary to consider only the distances 
between the atoms in one molecule. Much information on the structures 
of molecules has been gained in this way by Debye and others. 

The ring patterns formed when electrons are scattered by gases have 
been used in a similar way, and much valuable information on the 



CHAP, xi BROADENED X-RAY REFLECTIONS 373 

geometry of gas molecules has been gained by Pauling, Brockway, and 
others. (See Brockway, 1936.) The subject of electron diffraction will 
not be considered in this book, but it is worth remarking that electron 
diffraction patterns, both of gases and crystals, are strongly similar to 
the X-ray diffraction patterns of the same substances. Electrons are 
scattered chiefly by the nuclei of atoms, while X-rays are scattered by 
the electron clouds of the atoms ; but the positions and intensities of 
diffracted beams are controlled by atomic arrangement in the same way. 
Electron diffraction powder photographs may be used for identification 
in exactly the same way as X-ray powder photographs. Exceedingly 
thin films must be used, in a very high vacuum. The electron diffraction 
patterns of very thin films of high, polymers are also strikingly similar 
to the corresponding X-ray patterns. (Storks, 1938.) The chief chemi- 
cal application of electron diffraction methods (apart from the study of 
the molecular structure of gases, mentioned above) appears to be the 
identification of crystalline substances in very thin films, and the study 
of crystal orientation in similar circumstances. Electron diffraction 
patterns have not so far been much used for crystal structure determina- 
tion ; one reason appears to be that when electrons are diffracted at the 
surfaces of large crystals, the phenomena are more complex than those 
of X-ray diffraction, owing to the fact that the diffraction of electrons 
is much more efficient than that of X-rays. (See Bragg, 1933 ; Thomson 
and Cochrane, 1939.) 



APPENDIX 1 
IMMERSION LIQUIDS AND THEIR STANDARDIZATION 

BY mixing pure liquids any refractive index between those of the pure 
constituents can be attained. If two liquids having widely different refractive 
indices are chosen, a considerable range can be covered by mixing them 
in different proportions. Liquids for mixing in this way should preferably 
have similar vapour pressures at room temperature, so that evaporation does 
not lead to appreciable change of refractive index. Sets of liquids suitable 
for the identification of minerals have been suggested by several writers. 
(Larsen and Berman, 1934.) These are mostly very oily substances, which 
for inorganic chemical work on certain substances have the disadvantage 
that they are intolerant of water: crystals which are slightly damp, when 
immersed in such oily liquids, do not make optical contact with the liquids, 
so that genuine Becke line effects are not seen. It is an advantage to use 
liquids which can dissolve a small proportion of water ; the change of refrac- 
tive index caused by the dissolution of a surface film of water is usually 
negligible. The following liquids have been found suitable : 

1-373 1-396 Ethyl acetate and amyl acetate. 

1-396 1-490 Amyl acetate and xyleiie. 

1-490 1-559 Para-cymene and monobromobeiizene. 

1-559 1-598 Monobromobenzerie and bromoform. 

1-598 1-658 Bromoform and a-bromonaphthalene. 

1-658 1-740 a-Bromonaphthalene and methylene iodide. 

1-74 1-78 Solutions of sulphur in methylene iodide. 

1-78 1-88 Solutions of sulphur, SnI 4 , AsI 3 , SbI 3 , and iodoform in 
methylene iodide. (See Larsen and Berman, 1934.) 

The most important range for inorganic chemicals is between 1-40 and 
1-70. But there are some substances, such as certain oxides and sulphides, 
whose indices lie well above this range, or even well above 2-0. Media which 
are liquid at room temperature and have such high refractive indices are not 
available, but certain mixtures of substances which solidify to glasses may 
be used. A little of the medium in melted on a microscope slide, the substance 
under examination is dusted into the melt, a cover-glass is pressed on, and 
the slide is then allowed to cool. Substances which have been used in this 
way are mixtures of piperine with arsenic and antimony tri-iodides (for 
indices 1-7-2-1), mixtures of sulphur and selenium (2-0-2-7) for details, 
see Larsen and Berman, 1934 and mixtures of the halides of thallium 
(Barth, 1929). 

Crystals of many organic substances are soluble in the liquids given in 
the above list, and for these it is necessary to use aqueous solutions. Potassium 
mercuric iodide is a suitable substance, solutions of which have refractive 
indices up to 1-72. Owing to evaporation of the water, the refractive index 



APPENDIXES 



375 



of a solution may change ; therefore, if stock solutions are used, their refrac- 
tive indices should be checked frequently; or, alternatively, the refractive 
index of *a crystal should be matched by adjusting the composition of 
a solution, which is then immediately checked on the refractometer. To 
lessen the evaporation of water, solutions of potassium mercuric iodide in 
mixtures of glycerol and water may be used. (Bryant, 1932.) 

Stock solutions may be kept in 30-c.c. bottles with ground stoppers carry- 
ing dropping rods. For refractive indices up to 1 -7 it is best to keep a set of 
liquids having refractive indices differing by 0-005 ; from 1-7 to 1-9, intervals 
of 0*01 are sufficient. The refractive indices of liquids are best measured by 
means of the Abb6 refractometev. 

If no refractometer is available, the refractive indices of a set of liquids 
made by mixing known volumes of two pure liquids can be obtained in the 
following way. The refractive indices of the two pure liquids can be found 
in International Critical Tables. Assuming that the relation between volume- 
concentration and refractive index is Linear, a regular series of mixtures, 
0-005 apart in refractive index, is made up. The actual refractive indices, 
which will differ slightly from the expected values, are found by examining 
a known crystalline substance in two adjacent liquids, one above and one 
below, and estimating the relative differences between the refractive indices 
of the liquid and the crystal by the Becke line effect ; since the two liquids 
differ by 0-005, the actual refractive indices are then known. This procedure 
is repeated for other crystals, and a curve is then drawn for the whole range 
of mixtures, the refractive indices of the untested liquids being read off from 
the curve. Cubic and uniaxial crystals are most suitable for this purpose. A 
list of suitable substances is appended. 

List of crystals suitable for checking refractive indices of immersion liquids 



Refractive 
index 


Substance 


Refractive 
index 


Substance 


1-326 


NaF 


1-585| 


NaNO 3 


1-352 


KF 


1-616 


NaBrO 3 


1-392 


LiF 


1-642 


NH 4 C1 


1-410 


KCN 


l-658f 


CaCO 3 (caloite) 


1-434 


CaF 2 (fluorspar) 


1-667 


KI 


1-456 


K 2 S0 4 . A1 2 (SO 4 ) 3 . 24H 2 O 


1-703 


NH 4 I 


1-482 


K 2 SO 4 .Fe 2 (S() 4 ) 3 .24H 2 


1-711 


NH 4 Br 


1-490 


KC1 


l-768f 


A1 2 O 3 (corundum) 


1-515 


NaClO 3 


1-784 


Pb(N0 3 ) 2 


l-525f 


NH 4 H a PO 4 


1-83 


CaO 


1-544 


NaCl 


1-93 


CuCl 


1-559 


KBr 


l-97f 


HgCl 


1-572 


Ba(N0 3 ) 2 







These are co values of uniaxial substances. All the other substances are cubic. 



APPENDIX 2 
THE SPACINGS OF CRYSTAL PLANES 

Rectangular unit cells. In Fig. 230, OA, OB, and OC are orthogonal 
axes. Consider any set of crystal planes; one plane passes through the 
origin, the next (X YZ) makes intercepts of ajh, b/k, and c/l on the axes. Drop 




FIG. 230. The spacings of planes of orthogonal crystals. 

a perpendicular ON from the origin to the plane XYZ; we have to find t'ne 
length of ON (= d) in terms of a, 6, c, h, k, and Z. 



Now 
therefore 



d = cos /.NOK = cos LNOY = cos 






~, and 
6 



~. 
c 



The law of direction cosines states that 

cos 2 SLNOX+CO&* NOY+cos 2 NOZ = 1, 



1, 



therefore 



whence 



and 



For tetragonal crystals (a = b) this expression reduces to 



APPENDIXES 
and for cubic crystals (a = b = c) to 



377 



d = 



J 



Hexagonal unit cells. The c axis is at right angles to a and 6 which 
are equal in length and are at 120 to each other. To use the above formula, 
it is required to find the intercepts made by any plane on orthogonal axes, 
two of which are chosen as OC and OB (Fig. 231), the third being OQ in the 

plane AOB. The intercepts made by plane hkl on OB and OC are ? and y 
respectively. It is required to find the length OW (== r). 




FIG. 231. The spacings of planes of hexagonal crystals. Above: general 

view of plane X YZ, indices hkl. Below : normal view of basal plane ( AOB) 

of this diagram. 



Produce TO to P, and join XP, where tXPO = 90. 

The right-angled triangles YOW and TPX are similar, hence 



TO 
OW 



YP 
PX 



YO+OP 
PX ' 



a/k 
r 



T = 



V3.E. 

2 M 

1 . 1 



V3 

2M k' 



378 APPENDIXES 

From the formula for direction cosines, 



therefore 



+ 

OM-- 

(h* + hk + k*) 

a* "*" 

For rhombohedral unit cells, it is best to transform indices to hexagonal 
Indices, and to use the above formula for the spacings of planes. 

For monoclinic and triclinic cells, the formulae for the spacings are very 
unwieldy. Graphical methods based on the conception of the reciprocal 
lattice are recommended. (Chapter VI.) 



APPENDIX *3 

CHARTS SHOWING THE RELATION BETWEEN PLANE 
SPACINGS AND AXIAL RATIOS, CONSTRUCTED WITHOUT 

CALCULATION 

Tetragonal (including cubic) crystals. Bjurstrom (1931) showed that 
if, on a rectangular framework AMNC (Fig. 232 a), values of h?+k 2 are 




FIG. 232. Bjurstrom's chart. Tetragonal crystals. 

plotted along M A and values of I 2 along NO, and all the points on M A are 
joined to all the points on NO by straight lines, as in the diagram, then for 
a crystal whose axial ratio corresponds to position X along M N, the ordinates 
of all the lines represent the relative values of l/d 2 for all values of hkl. The 



380 



APPENDIXES 



reason for this can be seen by writing the equation for the plane-spacings in 
a tetragonal crystal thus : 



If we make (l/a 2 +l/c 2 ) a constant (this actually means using different units 
of length for every different crystal), the above expression is, for given values 
of A, k, and Z, of the form y K l x+K 2 , that is to say, the graph of y against 




oo 3-0 20 1528 1225 W 816 -654-5 333 
AXIAL RAW cla 

FIG. 233. Logarithmic form of Bjurstrom's chart. Tetragonal crystals. 

a: is a straight line. In Bjurstrom's chart the constant (l/a 2 +l/c 2 ) is the 
length M N of the base of the chart. 

The value of l/d 2 for each arc on a powder photograph is calculated from 
the Bragg equation. Since, however, we do not know on what scale to plot 
these values, it is necessary to represent them with a wide range of scales ; 
that is, to plot them, on transparent paper, along a line DE as in Fig. 232 b, 
and join them all to a point F on the perpendicular to DE. This 'fan' diagram 
is moved about on the chart, keeping FD coincident with MN, until 'fan' 
lines cross chart lines consistently along a line parallel to AM and NC ; along 
this line the indices for each 'fan* line are given by the chart line crossing it. The 



axial ratio c/a is given by the relation 



- = 



/ gji /i> 
1 1 a *-p~ i ic 



but 



* s 



APPENDIXES 381 

to calculate it from suitable pairs of spacings. In practice, this method 
is found, to be very unsatisfactory; the two sets of lines are so confusing 
that it is difficult to find the match position. The difficulty can be removed 
by making the chart logarithmic along the direction MA. Thus, values 
of Iog(/fc 2 -f & 2 ) are plotted along MA, and values of log/ 2 along NC, and 
all these points are joined by logarithmic curves, giving a chart of the 
type shown in Fig. 233, which can be used in the same way as the Hull and 
Davey charts (Chapter VI). The construction of such a chart is naturally 
not quite so simple as that of the straight-line Bjurstrom chart, but, inasmuch 
as no calculations are required, it is very much simpler than that of a chart 
of the Hull and Davey type. Afc an example of the method of construction, 
consider the plotting of the 213 curve, which joins Iog5 (i.e. log(^ 2 +fc 2 )) on 
one side to log 9 (i.e. logZ 2 ) on the other. Nine other points, at intervals of 
MN/10, are enough for the construction of the curve. Since on the straight- 
line Bjurstrom chart the ordinates at these points would be 5*4, 5*8, 6-2, 6*6, 
7-0, 7-4, 7-8, 8-2, and 8-6, we simply plot on the new type of chart log 5-4, 
log 5-8, and so on ; this can be done directly on large-scale logarithmic graph 
paper ; or on ordinary graph paper by reading off the values in a table of 
logarithms. 

This type of chart has one undesirable feature ; at the two sides (that is, 
for very small or very large values of c/a) some of the lines are nearly parallel 
to MA and NC, and appear very crowded. It is better, therefore, to spread 
out the diagram in these regions by plotting against log c/a instead of against 

1 /c 2 
- , . Plotting the curves is no more difficult than in the previous 

l/GT+l/C 2 

type of chart, if it is remembered that (to use the same example as before) 
log 5-4, log 5*8, and so on are plotted against values of log c/a given by 

1/c 2 
i/ 2 I ]/"a 1/W, 2/10, and so on ; straight lines at these values of log c/a are 

first drawn temporarily, and used as a scaffold for the plotting of the log I/a* 2 
curves. This, the best type of chart, is illustrated in principle in Fig. 72. 

Hexagonal and trigonal crystals. A chart for these crystals can be 
constructed by plotting values of log(& 2 -f hk+k z ) along MA and values of 
log I 2 along NC, and joining them by logarithmic curves. The axial ratio 

/ i ii i f A u i ^ MX 4 /3a 2 . ... 

c/a can be obtained from the relation -^-^ = In ' but again it is 

' MN 4/3a 2 +l/c 2 6 

better to calculate it from suitable pairs of spacings. A better form of chart 
is obtained by plotting against log c/a; no calculations are needed if it is 
remembered that in the original Bjurstrom chart the graph of I/a* 2 against 
MX (defined above) is a straight line. Thus, for the 213 curve, h 2 +hk-\-k* = 7, 
while Z 2 = 9; on the Bjurstrom chart, the straight line for these indices 
joins the point at 7 along M A with that at 9 along NC ; for the logarithmic 
chart we plot log 7-2, log 7*4, log 7-6, etc., at points along the log c/a axis 



382 APPENDIXES 

For rhombohedral crystals, which can always be referred to a large 
hexagonal cell containing more than one pattern-unit, this same qhart may 
be used, the indices h a k ff l H relating to the hexagonal cell being subsequently 
transformed to those relating to the rhombohedral cell h R k R l R by the 
relations 



3k R = h H +2k H +l H , 
31 R = -2h a - 



Many reflections which would be given by a simple hexagonal crystal are 
necessarily absent for a rhombohedral crystal. Only those reflections h a k H 1 H 
for which h a k H +l H , h ff +2k H ~{-l Ht and 2h H k H +l H are all divisible 
by three occur when the true unit cell is rhombohedral. In view of this, a 
special chart from which the unwanted* hexagonal lines are omitted may be 
constructed in the way already described. 
Zero layer on single -crystal rotation photographs. The spacings 

/ ;/^2 m 
d of the hOl planes of an orthorhombic crystal are given by d = 1 / / 1 -5 + -5 1 

The spacings for all axial ratios c/a can be represented on a chart which is 
similar to those already described but has values of logh 2 on one side and 
logZ 2 on the other. The tetragonal chart could be used, curves other than hOl 
being ignored ; but the many unwanted curves are confusing, and it is therefore 
better to construct a special chart from which the unwanted lines are omitted. 
Since this chart is symmetrical about the centre line, it is necessary to con- 
struct only half of it. The use of this chart is illustrated in Fig. 79. 

This chart may be used for single- crystal photographs of orthorhombic 
crystals rotated about any axis, for tetragonal crystals rotated about a or c 
(in the latter case the cell base is square and the axial ratio 1). It may also 
be used for hexagonal crystals rotated about a or c ; in the latter case true 
hexagonal indices will not be given, but indices in reference to an ortho- 
rhombic cell having an axial ratio of V3. The orthorhombic indices h k Q are 
converted to hexagonal indices Ji H k H by the relations 



It may also be used for monoclinic crystals rotated about a or c, since the 
projection of a monoclinic cell along either of these axes is rectangular. 

All these charts should be drawn on a large scale ; it is found that if a 
difference of 0-1 in log I/d 2 is made 2 inches, all reflections including those at 
Bragg angles near 90 can be indexed, except in the case of very large unit 
cells. In using the charts, values of 2 log d are plotted on a strip which is 
moved about on the chart in the manner described in Chapter VI. 



APPENDIX 4 
PROOF THAT RECIPROCAL POINTS FORM A LATTICE 



CONSIDEB a crystal with its c axis (OZ in Fig. 234) vertical. All reciprocal 
points corresponding with the vertical hkQ (real) lattice planes lie in the 
horizontal plane x*y*. 

Consider now any set of real lattice planes having indices hkl. If one plane 
passes through the origin 0, the next plane RST makes an intercept of c/l on 
the OZ axis. Draw a perpendicular to this plane, meeting it at N (ON = d t 
the spacing of the planes), and produce ON to P, where OP = X/d. P is the 
reciprocal lattice point corresponding to the set of real lattice planes hkl. 




M 



FIG. 234. Proof that reciprocal points form a lattice. 

Now since the angle RNO is 90, ON (= d) = OR cos = (c/Z)cos< ; 

cos< = dl/c. 

Drop a perpendicular PM on to the horizontal plane x*y*. We have to 
find the length of PM, the height of the reciprocal lattice point above the 
horizontal plane. 

PM = OPGO&Z.MPO. Since PM is parallel to OZ,MPO ==^LNOR = </>. 
Thus PM - OP cos <. But OP = A/d, 

.*. Pif = (A/d)cose 

= (}(/d)(dl/c) AZ/c, which is constant for a given value of Z. 

Thus all points having the same I index lie at the same distance from the 
horizontal plane #**/*, and thus lie on a plane parallel to the plane x*y* ; 
moreover, the distance of each such plane of points from the plane x*y* is 
proportional to Z. Therefore successive sets of points for the successive values 
of the index Z lie on a set of equidistant planes, spaced A/c apart. 

Similarly, it can be shown that the reciprocal hkl points lie on planes 
perpendicular to the a axis (spaced A/a apart), and on planes perpendicular 
to the b axis (spaced A/6 apart). Thus the whole assemblage of points forms 
a lattice. 



APPENDIX 6 
LIST OF SPACE-GROUPS 





Hermann- Mauguin symbols 






Point 


Smallest unit cell 




Schoenflies 




group 


Normal 


Other orientations 


Larger cells 


symbols 


No. 


I 


PI 




Al, Bl, CI, F1,I1 


c\ 


1 


I 


Pi 




Al, Bl, CI, Fl, II 


C}, SJ 


2 


m 


Pm 




Bm " 


CIA, C 1 , 


3 




PC 


Pa, Pn 


Ba, Bd 


f& pa 

1A* * 


4 




Cm 


Am, Im 


Fm 


CIA, CJ 


5 




Cc 


Aa, la 


Fd 


CIA, c i 


6 


2 


P2 




B2 


CJ 


7 




P2, 




B2j 


el 


8 




C2 


A2, 12 


F2 


ci 


9 


2/ra 


P2/m 




B2/m 


C* 


10 




P2 1 /m 




B2 x /m 


c|> 


11 




C2/m 


A2/m, I2/m 


F2/m 


c|* 


12 




P2/e 


P2/a, P2/n 


B2/a, B2/d 


CM 


13 




P2 t /c 


po / -po / 

jr^/a, ir^j/n 


B2 l /&, B2 x /d 


C 2A 


14 




C2/c 


A2/a, I2/a 


F2/d 


ci* 


15 


nun 


Pmm 


. . 




cL 


16 




Pmo 


Pern 




CL 


17 




Pec 






CL 


18 




Pma 


Pbm 




CL 


19 




Pea 


Pbc 




CL 


20 




Pnc 


Pen 




cL 


21 




Pmn 


Pnm 




cL 


22 




Pba 






CL 


23 




Pna 


Pbn 




cL 


24 




Pnn 






cj; 


25 




Cmm 






c al 


26 




Cmc 


Com 




CJJ 


27 




Ccc 






cj; 


28 




Amm 


Bmra 




cj; 


29 




Abm 


Bma 




c aJ 


30 




Ama 


Bbm 




L 5 


31 




Aba 


Bba 






32 




Fmm 






c aJ 


33 




Fdd 




. . 


CaJ 


34 




Imm 






ci! 


35 




Iba 




. . 


c|l 


36 




Ima 


Ibm 




CB 


37 


222 


P222 






V 1 D 1 


38 




P222! 


P2!22, P22^ 


. . 


V 2 D 2 


39 




P2&2 


P22 1 2 1 , P2 1 22 l 




V 3 , DJ 


40 



APPENDIXES 



385 





Hermann-Mauguin symbols 






Point 


Smallest unit cell 




Schoenflies 




group 


Normal 


Other orientations 


Larger cells 


symbols 


JVo. 


222 


P2 1 2 1 2 1 




. . 


V 4 , D 4 , 


41 


cont. 


C222i 


A2 X 22, B22;i2 




V 5 , D| 


42 




C222 


A222, B222 




V 8 , DJ 


43 




F222 






V*,D* 


44 




1222 






V 8 , Df 


45 




I2 1 2 1 2 1 






V 9 , Djj 


46 


mmm 


Pmmm 






VJ, D^ 


47 




Pnnn 




. . 


VJ, D& 


48 




Pccm 


Pbmb, Pmaa 




VJ, D^ 


49 




Pban 


Pcna, Pncb 




VJ, Djk 


60 




Pmma 


Pmmb, Pmam, Pmcm, 


. . 


VJ, D^ 


51 






Pbmm, Pcmm 










Pnna 


Pnnb, Pnan, Pncn, 


. . 


y6 j)6 


62 






Pbnn, Pcnn 










Pinna 


Pnmb, Pman, Pncm, 


. . 


V' D^ 


53 






Pbmn, Pcmn 










Pcca 


Pccb, Pbab, Pbcb, 


. 


V!,D|, 


54 






Pbaa, Pcaa 










Pbam 


Pcma, Pmcb 


. . 


V 9 , D|i 


55 




Pccn 


Pbnb, Pnaa 


. . 


v i, DJJ 


56 




Pbcm 


Pbma, Pcam, Pcmb, 




vi 1 , DB 


67 






Pmab, Pmca 










Pnnm 


Pnmn, Pmnn 




VJ 2 , Dg 


58 




Pmmn 


Pmnm, Pnmm 




V13 T>18 
A ' %h 


59 




Pbcn 


Pbna, Pcan, Pcnb, 


. . 


y"!* J)l* 


60 






Pnab, Pnca 










Pbca 


Pcab 




ylS j)l6 


61 




Pnma 


Pnam, Pbnm, Pcmn, 


. . 


VJ 6 , DJJ 


62 






Pmnb, Pmcn 










Cmcm 


Ccmm, Amma, Amam, 




vj 7 , Dy; 


63 






Bmmb, Bbmm 










Cmca 


Ccma, Abma, Abam, 




ylS^ pl8 


64 






Bmab, Bbam 










Ommm 


Ammm, Bmrnm 




Vl T> 19 
A -^^iA 


65 




Cccm 


Amaa, Bbmb 




ySO T}20 


66 




Cmma 


Abrnm, Bmam 


. . 


y21 j)21 


67 




Ccca 


Abaa, Bbaa 




V 2 2 *n22 
V A ^^ 


68 




Fmmnx 






Vf.DfJ 


69 




Fddd 






V 24 D? 4 


70 




Immm 






V25 T) 25 
V A > -^2* 


71 




Ibam 


Ibma, Imaa 




V26 T\26 
A 2A 


72 




Ibca 






y27 J)27 


73 




Imma 


Imam, Ibmm 




yflS T)28 


74 


4 


P4 




C4 


si 


75 




14 




F4 


SJ 


76 


42m 


P42m 


C4m2 


VJ, D,J rf 


77 



4458 



CC 



386 



APPENDIXES 



Point 
group 


Hermann- Mauguin symbols 


Schoenflies 
symbols 


^0. 


Smallest unit cell 


Larger cells 


Normal 


Other orientations 


42m 


Pl2c 


. . 


C4c2 


V2,Di, 


78 


cont. 


P42 1 m 




Cln^j 


VI Dt 


79 




P42!C 




C4c2! 


vj, r& 


80 




P4m2 




C42m 


vi DI, 


81 




P4c2 




C42c 


VJ. Dfc 


82 




P4b2 




C42b 


VJ,DJ, 


83 




P4n2 




C42n 


VJ, Dfc 


84 




I4m2 




F42m 


VS,DL 


85 




I4c2 




F42c 


vj, DK 


86 




I42m 




F4m2 


vj 1 , Da 


87 




I42d 




F4d2 


VJ 2 , DJJ 


88 


4 


P4 




C4 


cj 


89 




P4i 




C4 t 


CJ 


90 




P4 a 




C4 a 


cj 


91 




P4, 




04, 


C{ 


92 




14 




F4 


cj 


93 




I4i 




F4, 


CJ 


94 


4/m 


P4/m 




C4/m 


ci, 


95 




P4 a /m 




C4 2 /m 


CL 


96 




P4/n 




C4/a 


C 4 S * 


97 




P4 a /n 




C4 a /a 


cu 


"98 




I4/m 




F4/m 


cx* 


99 




I4ja 




F4!/d 


cs, 


100 


4-Ty^rn 


P4mm 




C4mm 


a 


101 




P4bm 




C4mb 


cs. 


102 




P4cm 




C4mc 


CL 


103 




P4nm 




C4ran 


cj v 


104 




P4nc 




C4cn 


05, 


105 




P4cc 


.. 


C4cc 


a 


106 




P4mc 


.. 


C4cm 


cz. 


107 




P4bc 




C4cb 


CL 


108 




I4mm 




F4mm 


a 


109 




I4cm 




F4mc 


01. 


no 




I4md 




F4dm 


cil 


111 




I4cd 




F4dc 


cs; 


112 


42 


P42 




C422 


BJ 


113 




P42 X 




C422 X 


DJ 


114 




P4j2 




C4 X 22 


B! 


115 




P4 1 2 1 




04^2, 


BJ 


116 




P4 a 2 




C4 2 22 


DJ 


117 




P4 8 2 1 




04,22, 


DS 


118 




P4 3 2 




C4 3 22 


i>; 


119 




PVi 




C4 8 22! 


BJ 


120 




142 




F42 


B 4 9 


121 




14,2 





F4 X 2 


DJ 


122 



APPENDIXES 



387 





Hermann-Mauguin symbols 






Point 


Smallest unit cell 




Schoenflies 




group 


Normal 


Other orientations 


Larger cells 


symbols 


^0. 


4/mmm 


P4/mmm 


. . 


C4/mmni 


DJ 


123 




P4/mcc 




04/mcc 


i>2, 


124 




P4/nbm 




C4/amb 


L 


125 




P4/nnc 




C4/acn 


Di 


126 




P4/mbrn 




C4/mmb 


I>JA 


127 




P4/mnc 




C4/mcn 


^ 


128 




P4/nmm 




C4/amm 


DL 


129 




P4/ncc 





C4/acc 


Bf A 


130 




P4/mmc 




C4/mcra 


DJ* 


131 




P4/mcin 




C4/mmc 


DiJ 


132 




P4/nbc 




C4/acb 


I>ii 


133 




-P4/nnm 




C4/amn 


i>a 


134 




P4/mbc 




C4/mcb 


i>2 


135 




P4/mnm 




C4/mmn 


i>u 


136 




P4/nrac 




C4/acm 


i>a 


137 




P4/ncm 




C4/amc 


DS 


138 




14/mmm 




F4/mmm 


Dfi 


139 




14/mcm 




F4/mmc 


DJ! 


140 




14/amd 




F4/ddm 


DJ! 


141 




I4/acd 




F4/ddc 


i>a 


142 


3 


C3 


f t 


H3 


<s 


143 




C3, 




H3 X 


ci . 


144 




C3 2 




H3 2 


c| 


145 




R3 




. . 


CJ 


146 


3 


C3 




H3 


Cfc, SJ 


147 




R3 






ci^s? 


148 


3m 


C3m 




H31m 


a 


149 




C31m 




H3m 


CL 


150 




C3c 




H31c 


CL 


151 




C31c 




H3c 


CJ, 


152 




R3m 






c^ 


153 




K3c 






CL 


154 


32 


C312 




H32 


BJ 


155 




C32 




H312 


D! 


166 




C3J2 




H3 X 2 


3 3 


157 




C3 2 2 




m^ ' 


I>3 4 


158 




C3 a l2 




H3 2 2 


D 


159 




C3 a 2 




H3 2 12 


i>S 


160 




R32 




, . 


i>5 


161 


3m 


C31m 




H3m 


DL 


162 




C31c 




H3c 


r>l, 


163 




C3rti 




H31m 


Dii 


164 




C3c 




H31c 


% 


165 




R3m 






DS 


166 



388 



APPENDIXES 





Hermann-Mauguin symbols 






Point 


Smallest unit cell 





Schoenflies 




group 


Normal 


Other orientations 


Larger cells 


symbols 


No. 


3m con*. 


R3c 






DS, 


167 


6 


C6 




H6 


CJ, 


168 


6m 


C6m 




H62m 


% 


169 




C6c 




H62c 


DL 


170 




C62m 




H6m 


DL 


171 




C62c 




H6c 


^ 


172 


6 


C6 




H6 


Q 


173 




C6! 




H6 X * 


ci 


174 




C6 6 


. . 


H6 5 


c? 


175 




C6 a 




H6 a 


cj 


176 




C6 4 




H6 4 


Q 


177 




C6 8 




H6 8 


CS 


178 


6/m 


C6/m 




H6/m 


cj, 


179 




C6 8 /m 




H6 8 /m 


c| A 


180 


6mm 


C6mm 




H6mm 


CL 


181 




C6cc 




H6cc 


a 


182 




C6cm 




H6mc 


ci. 


183 




C6mc 




H6cm 


a 


184 


62 


C62 




H62 


DJ 


185 




C6 X 2 




H6i2 


B? 


186 




C6 5 2 




H6 5 2 


DJ 


1ST 




C6 8 2 




H6 2 2 


^ 4 


188 




C6 4 2 




H6 4 2 


r> 


189 




C6,2 


. . 


H6 8 2 


S 


190 


6/mmm 


C6/mmm 




H!6/minm 


!> 


191 




C6/mcc 




H6/mcc 


i>l 


192 




C6/mcm 




H6/mmc 


Dt 


193 




C6/mmc 


- 


H6/mcm 


DJ* 


194 


23 


P23 


.." 


.. 


T l 


195 




F23 


. . 


. . 


T 2 


196 




123 




. . 


T 3 


197 




P2 X 3 






T* 


198 




12^ 






T 5 


199 


m3 


Pm3 






Ti 


200 




Pn3 






n 


201 




Fm3 






T 3 


202 




Fd3 




. 


TJ 


203 




Im3 






TJ 


204 




Pa3 






TJ 


205 




Ia3 






TJ 


206 


43m 


P43m 






TJ 


207 




F43m 






TJ 


208 




I43m 


. . 




TJ 


209 




P43n 






Ti 


210 



APPENDIXES 



389 



Point 
group 


Hermann -Mauguin symbols 


Schoenfiies 
symbols 


No. 


Smallest unit cell 


Larger cells 


Normal 


Other orientations 


43m 


F43o 






T 


211 


cant. 


I43d 






TJ 


212 


43 


P43 






O 1 


213 




P4 a 3 






o a 


214 




F43 






s 


215 




F4 X 3 






0* 


216 




143 


' ' 




o e 


217 




P4!3 






O 6 


218 




P4 8 3 






O 7 


219 




I4 X 3 






O 8 


220 


m3m 


Pm3m 






oi 


221 




Pn3n 






oi 


222 




Pm3n 






oi 


223 




Pn3m 






oi 


224 




Fm3m 






oj 


225 




Fm3o 






oj 


226 




Fd3m 






oi 


227 




Fd3c 






of 


228 




Im3m 






oj 


229 




Ia3d 






oi 


230 



REFERENCES AND NAME INDEX 

Abbe ^ . 336, 375 

Albrecht, G., 1939. Rev. Sci. Inst. 10 221 . . . . * . . 206 

Allison, S. K., 1932. Phys. Rev. 41 1 208 

American Society for Testing Materials (ASTM) .... 122, 123 
American Society for X-ray and Electron Diffraction (ASXRED) . . 122 
Angstrom ........... 106-7 

Artem&jv, D. N., 1910. Z. Krist. 48 417 21 

Astbury, W. T., 1941. Chem. and Ind. 60 491 189 

and Preston, R. D., 1934. Nature 133 46,0 108 

and Sisson, W. A., 1935. Proc. Roy. Soc. A 150 533 ... 178 

and Street, A., 1931. Phil. Trans. A 230 75 . . . .175 

and Woods, H. J., 1933. Phil. Trans, A 232 333 . . . .176 

and Yardley, K., 1924. Phil. Trans. A 224 221 . . 240, 245, 251 

Avogadro ........... 107 

Babinet 83 

Baker, W. O., and Smyth, C. P., 1939. J. Amer. Chem. Soc. 61 1695 . 331 

See also Fuller, C. S. 

Banerjee, S. See Krishnan, K. S. 

Barker, T. V., 1922. Book: Graphical and Tabular Methods in Crystallo- 
graphy. London: Murby ........ 30 

1930. Book: Systematic Crystallography. London: Murby 11, 64-5, 91 

Barlow, W 244 

Barth, T. F. W., 1929. Amor. Min. 14 358 374 

and Posnjak, E., 1932. Z. Krist. 82 325 333 

See also Donnay, J. D. H. ; Posnjak, E. 

Bath, J. See Ellis, J. W 

Beach, J. Y. See Brockway, L. O. 

Becke, F. 64-6, 70, 374 

Beevers, C. A., 1939. Proc. Phys. Soc. 51 660 348 

and Lipson, H., 1936. Proc. Phys. Soc. 48 772 . . . 347 

1938. Proc. Phys. Soc. 50 275 267 

and Schwartz, C. M., 1935. Z. Krist. 91 157 353 

Berman, H. See Larsen, E. S. 

Bernal, J. D., 1926. Proc. Roy. Soc. A 113 117 . . 149, 159, 161, 176 

1927. J. Sci. Inst. 4 273 138 

1928. J. Sci. Inst. 5 241, 281 138 

1929. J. Sci. Inst. 6 314, 343 138 

1932. Nature 129 277 252 

and Crowfoot, D. M., 1933a. Trans. Far. Soc. 29 1032 . . 246, 286 

19336. Nature 131 911 183 

1933c. Ann. Reports Chem. Soc. 30 411 . . . . 7 

1934a. Nature 133 794 138 

19346 Nature 134 809 186 

1935. 3. Chem. Soc. 93 246 

Crowfoot, D. M., and Fankuchen, I., 1940. Phil. Trans. A 239 135 . 183 

Fankuchen, I., and Perutz, M. F., 1938. Nature 141 523 . . 361 

and Wooster, W. A., 1932. Ann. Reports Chem. Soc. 28 262 . . 330 

Bertrand 78, 95 



REFERENCES AND NAME INDEX 391 

Bhagavantam, S., 1929. Proc. Roy. Soc. A 124 545 .... 283 

1930. Proc. Roy. Soc. A 126 143 283-4 

Biggs, B.*S. See White, A. H. 

Birge, R. T., 1941. Rev. Mod. Phys. 13 233 186 

Bishop, W. 8.* See White, A. H. 

Bjurstrom, T., 1931. Z. f. Physik 69 346 379-82 

Boas, W. See Schmid, E. 

Bogue, L. H. See Brownmiller, L. T. 

Bohlin, H., 1920. Ann. Physik 61 421 181 

Bosariquet, C. H. See Bragg, W. L. 

Bouman, J., and de Jong, W. F., 1938. Physica 5 817 .... 204 

See also de Jong, W. F. 

Bouwers, A., 1923. Z. Physik. l4 374 193 

Bozorth, R. M., 1923. J. Amer. Chem. Soc. 45 2128 . . , .258 
Bradley, A. J., 1935. Proc. Phys. Soc. 47 879 . . . 121, 206-7 

Bragg, W. L., and Sykes, C., 1940. J. Iron Steel Inst. 141 63 . 125 

and Jay, A. H., 1932. Proc. Phys. Soc. 44 563 ... 121, 180 

Lipson, H., arid Petch, N. J., 1941. J. Sci. Inst. 18 216 . . 112, 124 

and Rodgers, J. W., 1934. Proc. Roy. Soc. A 144 340 . . . 274 

Bragg, W.H., 1914. Phil. Mag. 27 881 192 

1921. Proc. Phys. Soc. 33 304 296 

and Bragg, W. L., 1913. Proc. Roy. Soc. A 88 428 . . . 192 

Bragg, W. L. . . 118-20, 132, 139, 142, 144, 168, 171, 174, 180, 195, 201, 

204-5, 208, 243, 362-6, 368, 380, 383 
1913. Proc. Camb. Phil. Soc. 17 43 114 

- 1914. Proc. Roy. Soc. A 89 4(58 225 

- 1924a. Proc. Roy. Soc. .4 105 16 .... 58, 225, 310 

19246. Proc. Roy. Soc. A 105 370; A 106 346 ... 280, 310 

1929a. Proc. Roy. Soc. A 123 537 340 

19296. Z. Krist. 70 475, 489 345-6,348 

1930. Z. Krist. 74 237 276 

1933. Book: The Crystalline State. London: Bell . 243, 348, 365, 373 

1937. Book : Atomic Structure of Minerals. Cornell, Univ. Press . 247 

1939. Nature 143 678 6, 349, 362 

1942a. Nature 149 470 6, 350, 352 

19426. J. Sci. Inst. 19 148 369 

1944. Nature 154 69 271 

James, R. W., and Bosanquet, C. H., 1921. Phil. Mag. 42 1 . 203, 209 

1922. Phil. Mag. 44 433 209 

and Lipson, H., 1936. Z. Krist. 95 323 . 264, 267, 269, 302-3, 306 

1943. J. Sci. Inst. 20 110 369 



- and West, J., 1926. Proc. Roy. Soc. A 111 691 . . . .266 

1929. Z. Krist. 69 118 206-7,209 

1930. Phil. Mag. 10 823 360 



See also Bradley, A. J. ; Bragg, W. H. ; Warren, B. E. 

Brandenborger, E. See Waldmann, H. 

Brandes, H. and Volmer, M., 1931. Z. Phys. Chem. A 155 466 . . 22 

Bravais, A. ........... 221 

Bray, E. E., and Gingrich, N. S., 1943. J. Chem. Phys. 11 351 . . 372 

Brill, R., 1930. Z. Krist. 75 217 366 

Brindley, G. W. See James, R. W. 

Brockway, L. O., 1936. Rev. Mod. Phys. 8 231 373 



392 REFERENCES AND NAME INDEX 

Brockway, L. O., Beach, J. Y., and Pauling, L., 1935. J. Amer. Chem. 

Soc. 57 2693 224-5 

See also Pauling, L. 

Brown, C. J., and Cox, E. G., 1940. J. Chem. Soc. 1 . . .359 

Brownmiller, L. T., and Bogue, L. H., 1930. Amer. J. Sci. 20 241 . . 124 
Bryant, W. M. D., 1932. J. Amer. Chem. Soc. 54 3758 . . .101, 375 

1941. J. Amer. Chem. Soc. 63 511 85, 101 

1943. J. Amor. Chem. Soc. 65 96 84, 101 

Buckley, H. E., 1930. Z. Krist. 75 15 37 

Buerger, M. J., 1934. Z. Krist. 88 356 . . . . . .169 

1935. Z. Krist. 91 255 169 

1936. Z. Krist. 94 87 168 

1937. Z. Krist. 97 433 . . . ' . . ; .181 

1940. Proc. Nat. Acad. Sci. Washington 26 637 . . . . 204 

1942. Book: X-ray Crystallography. New York: Wiley . . .169 

Bunn, C. W., 1933. Proc. Roy. Soc. A 141 567 .... 22, 60 

1939. Trans. Far. Soc. 35 482 . . . 188, 261, 289, 347 

1941. J. Sci. Inst. 18 70 125, 184, 334 

1942a. Proc. Roy. Soc. A ISO 40 . . . . 262, 268, 323 

19426. Proc. Roy. Soc. A 180 67 277, 323 

1942c. Proc. Roy. Soc. A 180 82 187, 323 

Clark, L. M., and Clifford, I. L., 1935. Proc. Roy. Soc. A 151 141 124, 283 

and Garner, E. V., 1942. J. Chem. Soc. 654 . . . . 189, 324 

Peiser, H. S., and Turner-Jones, A., 1944. J. Sci. Inst. 21 10 . 165-6 

Cameron, G. H., and Patterson, A. L., 1937. Symposium on Radiography , 

and X-ray Diffraction Amer. Soc. for Testing Materials . . . 365 
Chakravorty, N. C. See Krishnan, K. S. 
Chamot, E. M., and Mason, C. D., 1931. Book: Handbook of Chemical 

Microscopy. New York : Wiley . . . . . . .102 

Claassen, A., 1930. Phil. Mag. 9 57 121 

Clark, G. L., and Corrigan, K. E., 1931. Ind. Eng. Chem. 23 815 . . 124 

and Pickett, L. W., 1931. J. Amer. Chem. Soc. 53 167 . . . 249 

Clark, L. M. 23 

See also Bunn, C. W. 

Clews, C. J. B., and Lonsdale, K., 1937. Proc. Roy. Soc. A 161 493 . 289 
Clifford, I. L. See Bunn, C. W. 
Cochrane, W. See Thomson, G. P. 

Cohen, M. TL, 1935. Rev. Sci. Inst. 6 68 180 

Coolidge, W. D 103 

Cooper, B. S. See Randall, J. T. 

Corey, R. B., and Wyckoff, R. W. G., 1933. Z. Krist. 85 132 . . 245 

See also Levi, H. A. ; Wyckoff, R. W. G. 

Corrigan, K. E. See Clark, G. L. 

Coster, D., Knol, K. S., and Prins, J. A., 1930. Z. Physik 63 345 . . 243 

Cox, E. G 169, 271 

1928. Nature 122 401 280 

1932a. Nature 130 205 261, 315 

19326. Proc. Roy. Soc. A 135 491 280 

1938. Ann. Reports Chem. Soc. 35 176 350 

and Goodwin, T. H., 1936. J. Chem. Soc. 769 . . 251, 315-17 

and Hirst, E. L., 1933. Nature 131 402 315 



KEFEKENCBS AND NAME INDEX 393 

Cox, E, G., Hirst, E. L., and Reynolds, R. J. W., 1932. Nature 130 888 315 

and^Shaw, W. F. B., 1930. Proc. Roy. Soc. A 127 71. . . 204 

See also Brown, C. J. 

Crowfoot, D. M., 1935. Z. Krist. 90 215 169 

See also Bernal, J. D. 

Dana, E. S., 1932. Book: Textbook of Mineralogy, 4th edn., revised by 

W. E, Ford* New York: Wiley 58 

Daniel, V., and Lipson, H., 1943. Proc. Roy. Soc. A 181 368 . . 369 

Darwin, C. G., 1922. Phil. Mag. 43 808 204 

Davey, W. P., 1934. Book: A Study of Crystal Structure and its Applica- 
tions. New York: McGraw-Hill 243 

See also Hull, A. W. 

Dawton, R. H. V. M., 1937. J. ScL Inst. 14 198 194 

1938. Proc. Phys. Soc. 50 419 194 

See also Robertson, J. M. 

Debye, P 372 

1915/ Ann. d. Physik. 46 809 371 

and Scherrer, P., 1916. Physik. Zeit. 17 277 .... 108, 181 

Deming, W. E. See Hendricks, S. B. 

Desch, C. H. See Lea, F. M. 

Dickinson, R. G., 1923. J. Amer. Chem. Soc. 45 958 . . . 228 

and Goodhue, E. A., 1921. J. Amer. Chem. Soc. 43 2045 . . 226 

and Raymond, A. L., 1923. J. Amer. Chem. Soc. 45 22 . . . 280 

Dinsmore, R. P. See Gehinan, S. D. 

Donnay, J. D. H., 1939. Amer. Min. 24 184 280 

^ 1943. Amer. Min. 28 313 48-9 

and Barker, D., 1937. Amer. Min. 22 446 280 

Dorn, J. E., and Glockler, G., 1936. Rev. Sci. Inst. 7 389 . . 112 

Dubinina, V. N. See Mikheev, V. I. 

Diirer, A. 11 

Edwards, D. A., 1931. Z. Krist. 80 154 ... 139 

Edwards, O. S., and Lipson, H., 1941. J. Sci. Inst. 7 389 . . 108 

1942. Proc. Roy. Soc. A 180 268 369 

Elam, C. F. See Knaggs, I. E. 

Ellis, J. W. and Bath, J., 1938. J. Chem. Phys. 6 221 . . . . 286 

Erickson, C. L. See Fuller, C. S. 

Eucken, A., 1939. Z. Elektrochemie 45 126 291 

Evans, R. C., 1935. Chem. Soc. Annual Reports 32 193. . . . 360 

1939. Book: Crystal Chemistry. Cambridge Univ. Press . . .276 

and Peisor, H. S., 1942. Proc. Phys. Soc. 54 457 . . . . 270 

Ewald, P. P., 1921. Z. Krist. 56 129 143 

Fankuchen, L, 1937. Nature 139 193 107-8 

See also Bernal, J. D. 

Fedorov, E. S 11, 244 

Field, J. E. See Gehman, S. D. 
>Firth, E. M. See James, R. W. 
Fourier, J. B. J. . 338, 340, 344, 345, 347, 350, 351, 356, 371 

Fowler, R. H., 1935. Proc. Roy. Soc. A 151 1 291 

Freudenberg, K., 1932-4. Book: Stereochemie. Leipzig: Deuticke . .276 

Frevel, L. K., 1935. Rev. Sci. Inst. 6 214 112 



394 BEFEBENCES AND NAME INDEX 

Frevel, L. K. See also Hanawalt, J. D. 

Friedel, G., 1913. C. r. acad. sci. Paris 157 1633 . . . . g . 242 

and Friedel, E. (and others), 1931. Z. Krist. 79 1 . . . ' . 330 

Frosch, C. J. See Fuller, C. S. 

Fuller, C. S., 1940. Chera. Reviews 26 143 . . . . ' 176, 189, 261 

Baker, W. O., and Pape, N. R., 1940. J. Amer. Ohem. Soc. 62 3275 369 

and Erickson, C. L., 1937. J. Amer. Ohem. Soc. 59 344 . . . 177 

and Frosch, C. J., 1939. J. Phys. Chem. 43 323 ; J. Amer. Chom. Soc. 

61 2575 177 

Frosch, C. J., and Pape, N. R., 1940. J. Amer. Chem. Soc. 62 1905 176, 261 

1942. J. Amer. Chem. Soc. 64 154 188 



Garner, E. V. See Bunn, C. W. 

Gaubert, P., 1906. C. r. acad. Sci. Paris 143 936 22 

Gehman, S. D., and Field, J. E., 1939. J. Applied Phys. 10 564 . . 178 

anc l Dinsmore, R. P., 1938. Proc. Rubber Technology Con- 
ference, 961 323 

Gille, F., and Spangenberg, K., 1927. Z. Krist. 65 207 . . . . 13 

Gingrich, N. S. See Bray, E. E. ; Warren, B. E. 

Glockler, G. See Dorn, J. E. 

Goodhue, E. A. See Dickinson, R. G. 

Goodwin, T. H., and Hardy, R., 1938. Proc. Roy. Soc. 164 369 . . 347 

See also Cox, E. G. 

Greenwood, G., 1924. Phil. Mag. 48 654 247 

Groth, P., 1906-19. Book: CliemiscUe Krystallographie. Leipzig: Engol- 

mann 20, 54, 98, 217, 299 

Guha, B. C. See Krishnan, K. S. 

Guinier, A., 1937. C. r. acad. Sci. Paris 204 1115 . . . . 108 

Hagg, G., 1931. Z. phys. Chem. B 12 33 184 

1933. Z. Krist. 86 246 124 

1935. J. Chem. Phys. 3 42 . .371 

1935. Z. phys. Chem. B 29 95 333 

and Phragmen, G., 1933. Z. Krist. 86 306 181 

and Soderholm, G., 1935. Z. phys. Chem. B 29 88 . . . 333 

and Sucksdorff, L, 1933. Z. phys. Chem. B 22 444 . . . 334 

Halle, F., and Hofmann, W., 1935. Naturwiss. 23 770 . . . .189 
Hanawalt, J. D., Rinn, H. W., and Frevel, L. K., 1938. Ind. Eng. Chem. 

(Anal.) 10 457 123 

Harcourt, A., 1942. Amer. Min. 27 63 123 

Hardy, R. See Goodwin, T. H. 

Harker, D., 1936. J. Chem. Phys. 4 381 356 

See also Donnay, J. D. H. ; Senti, F. 

Hartree, D. R., 1928. Proc. Camb. Phil. Soc. 24 89, 111 . . . 203 

See also James, R. W. 

Hartshorne, N. H., and Stuart, A., 1934. Book: Crystals and the Polarizing 

Microscope. London: Arnold . . . . . . 62, 85, 101 

Harvey, G. G., 1939. J. Chem. Phys. 7 878 372 

Helmholz, L., 1936. J. Chem. Phys. 1936 4 316 206 

Helwig, G. V., 1932. Z. Krist. 83 485 275 

Hendershot, O. P., 1937o. Rev. Sci. Inst. 8 324 206 

19376. Rev. Sci. Jnst. 8 436 175 



BEFEBENCES AND NAME INDEX 395 

Hendricks, S. B., 1928a. J. Amer. Chem. Soc. 50 2455 . . . 226, 303 

1928$. Z. Krist. 67 106, 475; 68 189 330 

1930. Nature 126 167 330 

and Deming, W. E., 1935. Z. Krist. 91 290 .... 283 

Herbert, R. W.*, Hirst, E. L., Percival, E. G. V., Reynolds, R. J. W., and 

Smith, F., 1933. J. Chem. Soc. 1270 317 

Hengsteriberg, J., and Mark, H., 1929. Z. Krist. 70 285 ... 249 

Hermann, C., and Mauguin, C. ....... 385-90 

Hertzfeld, K. F., and Hettich, A., 1926. Z. Physik 38 1 . . .54 

1927. Z. Physik 40 327 54 

Hettich, A. See Hertzfeld, K. F. 

Hicks, V. See Hull, R. B. 

Hirst, E. L. See Cox, E. G. ; Herbert, R. W. 

Hofmann, W. See Halle, F. 

Huggins, M. L., 1933. Z. Krist. 86 384 275 

1941. J. Amer. Chem. Soc. 63 66 348 

Hughes, E. W., 1935. J. Chem. Phys. 3 1 ... . 195, 351 

1941.- J. Amer. Chem. Soc. 63 1737 ... . 206, 268 

Hull, A. W., 1917. Phys. Rev. 10 661 108 

and Davey, W. P., 1921, Phys. Rev. 17 549 ... 132, 381 

Hull, R. B., and Hicks, V., 1936. Rev. Sci. Inst. 7 464 . . . . 138 
Hume-Rothery, W., and Raynor, G. V., 1941. J. Sci. Inst. 18 74 . 110, 125 

and Reynolds, P. W., 1938. Proc. Roy. Soc. A 167 25 . . . 112 

Huse, G.. and Powell, H. M., 1940. J. Chem. Soc. 1398 . . 345, 353-6 

Iball, J., 1934. Proc. Roy. Soc. A 146 140 262 

See also Owen, E. A. 

Insley, H., 1937. Bur. Stand. J. Res. 17 353 124 

and McMurdie, H. F., 1938. Bur. Stand. J. Res. 20 173 . . 124 

International Critical Tables, 1926. New York: McGraw-Hill . . 375 

Internationale Tabollen zur Bestimmung von Kristallstrukturen, 1935. 

Berlin: Borntraeger 44, 52, 121, 202, 205, 208, 212, 240, 244, 245, 256, 258, 

297, 303, 307 

James, R. W., and Brindley, G. W., 1931. Z. Krist. 78 470 . . . 203 

and Firth, E. M., 1927. Proc. Roy. Soc. A 117 62 . . . . 193 

Waller, I., and Hartree, D. R., 1928. Proc. Roy. Soc. A 118 334 . 203 

See also Bragg, W. L. 

Jay, A. H., 1933. Proc. Phys. Soc. 45 635; Z. Krist. 86 106 . .112 

~ 1941. J. Sci. Inst. 18 128 193 

See also Bradley, A. J. 

Jeffrey, G. A., 1944. Trans. Far. Soc. 40 517 323 

Jones* F. W., 1938. Proc. Roy. Soc. A 166 16 365-6 

and Sykes, C., 1938. Proc. Roy. Soc. A 166 376 . . . . 369 

Jong, W. F. de, and Bouman, J., 1938. Z. Krist. 98 456; Physica 5 220 

170-1, 204 
- and Lange, J. J. de, 1938. Physica 5 188 . . . .170 



See also Bouman, J. 

Karlik, B. See Knaggs, I. E. 

Katz, J. R., 1925. Naturwiss. 13 411 176 



396 REFERENCES AND NAME INDEX 

Kerr, P. F. See Rogers, A. F. 

King, H. See Rosenheim, O. , 

Knaggs, I. E., 1935. Proc. Roy. Soc. A 150 576 351 

Karlik, B., and Elam, C. F., 1932. Book: Tables of Cubic Crystal 

Structure. London : Adam Hilger . . . . . .182 

Knol, K. S. See Coster, D. 

Kordes, E., 1935o. Z. Krist. 91 193 333 

19356. Z. Krist. 92 139 333 

Kossel, W., 1927. Nachr. Ges. Wiss. Gottingen, 135 ... 22, 280 
Kowarski, L., 1936. J. Chim. Phys. 32 303, 395, 469 . . . .19 
Kratky, O., and Krebs, G., 1936. Z. Krist. 95 253 . . . . 175 

and Kuriyama, S., 1931. Z. phys. Chem. B 11 363 . . . 176 

Krebs, G. See Kratky, O. 

Krishnan, K. S., and Banerjee, S., 1935. Phil. Trans. A 234 265 ; Curr. Sci. 

3 548 288 

Chakravorty, N. C., and Banerjee, 'S., 1934. Phil. Trans. A 232 103 288 

Guha, B. C., and Banerjee, S., 1933. Phil. Trans. A 231 235 . 288-9 

Knitter, H. See Warren, B. E. 
Kuriyama, S. See Kratky, O. 

Lange, J. J. de, Robertson, J. M., and Woodward, L, 1939. Proc. Roy. Soc. 

A 171 398 250 

See also Jong, W. F. de. 

Larsen, E. S., and Berman, H., 1934. Book: Microscopic Determination of 

the Nonopaque Minerals. Washington : U.S. Geol. Survey Bulletin 848. 

2nd edn 86, 93, 98, 283,*374 

Laue, M. von .... ..... 6, 200, 243 

1926. Z. Krist. 64 115 365-6 

Laves, F., 1930. Z. Krist. 73 202 334 

Lea, F. M., and Desch, C. H., 1935. Book: The Chemistry of Cement and 

Concrete. London: Arnold . . . . . . . .124 

Levi, H. A., and Corey, R. B., 1941. J. Amer. Chem. Soc. 63 2095 . 323 

Lindemann, F. A. . . . . . . . . .110 

Lipson, H., 1943. J. Inst. Metals. 69 1 125 

and Riley, D. P., 1943. Nature 151 250, 502 ... 107, 186 

and Stokes, A. R., 1943. Nature 152 20 367 

and Wilson, A. J. C., 1941. J. Sci. Inst. 18 144 . . . . 181 

See also Beevers, C. A. ; Bradley, A. J. ; Bragg, W. L. ; Daniel, V. ; 

Edwards, O. S. ; Stokes, A. R. ; Wilson, A. J. C. 
London, F., 1937. Compt. rend. acad. sci. Paris 205 28; J. Chem. Phys. 

5 837 ; J. Phys. Radium 8 397 288 

Lonsdale, K., 1929. Proc. Roy. Soc. A 123 494 . . . . 19, 284 
1936. Book : Simplified Structure Factor and Electron Density Formulae 

for the 230 Space Groups of Mathematical Crystallography. London: 

Bell 258, 348 

1937a. Reports on Progress in Physics 4 368. . . . 288-9 

19376. Proc. Roy. Soc. A 159 149 288 

1939. Proc. Roy. Soc. A 171 541 289 

1941. Proc. Roy. Soc. A 177 272 107 

1942. Proc. Phys. Soc. 54 314 206, 368-9 

- See also Clews, C. J. B. ; Orelkin, B. 



Lorentz 203 



REFERENCES AND NAME INDEX 397 

MacEwan, D., and Beevers, C. A., 1942. J. Sci. Inst. 19 150 . . . 348 

Mack, E., 1932. J. Amor. Chem. Soc. 54 2142 228 

Marcelin, R., 1918. Ann. do physique 10 185, 189 .... 19 

Mark, H., 1940. J. Phys. Chem. 44 764 187 

See also Hangstenberg, J. 

Martin, A. J. P., 1931. Min. Mag. 22 519 290 

See also Wooster, W. A. 

Marton, L., McBain, J. W., and Void, R. D., 1941. J. Amer. Chem. Soc. 

63 1990 128 

Mason, C. D. See Chamot, E. M. 

Mauguin, C. See Hermann, C. 

Maurice, M. E., 1930. Proc. Camb. Phil. Soc. 24 491 . . . 290 

McBain, J. W. See Marton, L. 

McMurdie, H. F. See Insley, H. 

Mediin, W. V., 1935, J. Amor. Chem. Soc. 57 1026 . . . .372 

1936. J. Amer. Chem. Soc. 5g 1590 372 

Miers, H. A., 1929. Book: Mineralogy. 2nd edn. London: Macmillan 30, 54, 

58, 62, 85, 247, 299 
Mikheev, V. I., and Dubinina, V. N., 1939. Annales de 1'Institut des Mines 

a Leningrad 131 123 

Miles, F. D., 1931. Proc. Roy. Soc. A 132 266 53 

Morgan, S. O. See White, A. H. 

Morningstar, O. See Warren, B. E. 

Muller, A., 1929. Nature 124 128 108 

Newton, 1 69, 81 

Nicol, W 66-8, 70, 72, 77-8, 80-1, 83, 86-7, 89, 92-5 

Niekerk, J. N. van, 1943. Proc. Roy. Soc. A 181 314 . . . . 251 
Niggli, P., 1920. Z. anorg. Chem. 110 55 20, 22 

Olshausen, S. v., 1925. Z. Krist. 61 463 285 

Orelkin, B., and Lonsdale, K., 1934. Proc. Roy. Soc. A 144 630 . . 283 

Orowan, E., 1942. Nature 149 355 171, 180 

Oseen, C. W., and others, 1933. Trans. Far. Soc. 29 883 . . . 330 

Owen, E. A., 1943. 20 190 112 

and Iball, J., 1932. Phil. Mag. 13 1020 181 

Pickup, L., and Roberts, I. O., 1935. Z. Krist. 91 70 . . 181 

Pape, N. R. See Fuller, C. S. 
Pascoe, K. J. See Stokes, A. R. 

Patterson, A. L 345, 356-60 

1934. Phys. Rev. 46 372 361-2 

1935a. Z. Krist. 90 517 351 

19356. Z. Krist. 90 543 352 

1939. Phys. Rev. 56 972, 978 366 

See also Cameron, G. H. 

Pauling, L 372 

1929. J. Amer. Chem. Soc. 51 1010 276 

1933. Z. Krist. 85 380 25S 

1936. J. Chem. Phys. 4 673 28S 

1940. Book : The Nature of the Chemical Bond. 2nd edn. Cornell Univ. 

Press 27f 



398 REFERENCES AND NAME INDEX 

Pauling, L., and Brockway, L. O., 1934. J. Chem. Phys. 2 867 . . 226 
1937. J. Amer. Chem. Soc. 59 1223 228 



- See also Brockway, L. O. 



Poiser, H. S. See Bonn, C. W. ; Evans, R. C. 

Petch, N. J. See Bradley, A. J. 

Percival, E. G. V. See Herbert, R. W. 

Perutz, M. F., 1939. Nature 143 731 286 

1942. Nature 144 491 361 

See also Bernal, J. D. 

Pickup, L. See Owen, E. A. 

Phillips, F. C., 1933. Min. Mag. 23 458 102 

Phragmen, G. See Hagg, G. 

Pickott, L. W., 1933. Proc. Roy. Soc. 142 333 311 

1936. J. Amer. Chom. Soc. 58 2299 347 

See also Clark, G. L. 

Piper, S. H., 1937. J. Soc. Chem. Ind. 56 61 T 183 

Planck, M 105 

Pohland, E., 1934. Z. phys. Chem. B 26 238 112 

Polanyi, M., 1921. Naturwiss. 9 337 175 

Porter, M. W., and Spiller, R. C., 1939. Nature 144 298 ... 55 

Posnjak, E., and Barth, T. F. W., 1931. Phys. Rev. 38 2234 . . 333 

See also Barth, T. F. W. ; Wyckoff, R. W. G. 

Powell, H. M. SeeHuse, G. 
Prasad, M. See Robertson, J. M. 

Preston, E., 1942. J. Soc. Glass Tech. 26 82 371 

Preston, G. D., 1939. Proc. Roy. Soc. A 172 116 369 

1941. Proc. Roy. Soc. A 179 1 .369 

Preston, R. D. See Astbury, W. T. 

Preston, T., 1928. Book: Theory of Light. 5th edn. London: Macmilian . 62 

Prins, J. A. See Coster, D. 

Rabi, 1. 1., 1927. Phys. Rev. 29 174 288 

Randall, J. T., 1934. Book: The Diffraction of X-rays and Electrons by 

Amorphous Solids, Liquids and Gases. London : Chapman and Hall 2, 330, 

372 

and Rooksby, H. P., 1931. Glass 8 234 127 

1933. J. Soc. Glass Tech. 17 287 127 

and Cooper, B. S., 1930. Nature 125 458; J. Soc. Glass Tech. 

14219;Z. Krist. 75 196 127 

Raymond, A. L. See Dickinson, R. G. 

Raynor, G. V. See Hume-Rothery, W. 

Reynolds, P. W. See Hume-Rothery, W. 

Reynolds, R. J. W. See Cox, E. G. ; Herbert, R. W. 

Richtmeyer, F. K., 1928. Book : Introduction to Modern Physics. New York : 

McGraw-Hill 281 

Riley, D. P. See Lipson, H. 

Rinn, H. W. See Hanawalt, J. D. 

Roberts, I. O. See Owen, E. A. 

Robertson, J. M., 1933a. Proc. Roy. Soc. A 140 79 . . . 193 

19336. Proc. Roy. Soc. A 141 594; A 142 659 .... 237 

1933c. Proc. Roy. Soc. A 142 674 284 

1934a. Proc. Roy. Soc. A 146 473 313 



REFERENCES AND NAME INDEX 399 

Robertson, J. M., 19346. Phil. Mag. 18 729 168 

1935a. Proc. Roy. Soc. A 150 106 ... 138, 206, 231, 260 

1935&. Proc. Roy. Soc. A 150 348 315 

1935c. J. Chem. Soc. 615 ....... 272, 290 

1936a. JaChem. Soc. 1195 272, 344 

19366. Proc. Roy. Soc. 157 79 347 

1936c. Phil. Mag. 21 176 348 

1937. Reports on Progress in Physics 4 332 340 

1943. J. Sci. Inst. 20 169 195 

and Dawton, R. H. V. M., 1941. J. Sci. Inst. 18 126 ... 194 

Prasad, M., and Woodward, I., 1936. Proc. Roy. Soc, A 154 187 . 249 

and Woodward, I., 1937. Proc. Roy. Soc. A 162 568 . 249, 345, 346 

1940. J. Chem. Soc. 36* 342-3 



See also Lange, J. J. do. 



Robinson, B. W., 1933a. J. Sci. Inst. 10 165 169 

19336. J. Sci. Inst. 10 233 . 194 

Rodgers, J. W. See Bradley, A. J. 

Rogers, A. F., and Kerr, P. F., 1942. Book: Optical Mineralogy. New York: 

McGraw-Hill 101 

Rooksby, H. P., 1941. J. Sci. Inst. 18 84 126 

1942. J. Roy. Soc. Arts. 90 673 110 

See also Randall, J. T. 

Rosenhain, W., 1935. Book: Introduction to Physical Metallurgy. London: 

Constable 92 

Rosenheini, O., and King, H., 1932. Chem. and Ind. 51 464 . . . 252 

R<jyer, L., 1926. Compt. rend. acad. sci. Paris 182 326 .... 60 

1933. Compt. rend. acad. sci. Paris 196 282 .... 60 

1934. Cornpt. rend. acad. sci. Paris 198 185, 585 .... 22 

Sauter, E., 1933a. Z. phys. Chem. B 23 370 138 

19336. Z. Krist. 84 461 ; 85 156 170, 171 

1937. Z. phys. Chem. B 36 405 176 

Scherrer, P., 1920. 'See Zsigmondy's book Kolloidchemie. 3rdedn. p. 387 364-6 

Schiebold, E., 1933. Z. Krist. 86 370 169, 171 

Schmid, E., and Boas, W., 1935. Book: Kristallplastizitdt. Berlin: Springer 180 
Schomaker, V. See Stevenson, D. P. 

Schonflies, A 244, 385-90 

Schwartz, C. M. See Beevers, C. A. 

Seemann, H., 1919. Ann. d. Physik 59 455 181 

Seljakow, 1ST., 1925. Z. Physik 31 439 365 

Senti, F., and Marker, D., 1940. J. Amer. Chem. Soc. 62 2008 . . 347 
Shaw, W. F. B. See Cox, E. G. 

Siegbahn, M., 1943. Native 151 502 107 

Smith, C. S., and Stickley, E. E., 1943. Phys. Rev. 64 191 . . . 367 

Smith, F. See Herbert, R. W. 

Smyth, C. P. See Baker, W. O. ; Turkevitch, A. 

Sisson, W. A. See Astbury, W. T. 

Soderholm, G. See Hagg, G. 

Sommerfeldt, E., 1908. Neues Jahrb. f. Min. 1 58 . . . .88 

Spangenberg, K., 1928. Neues Jahrb. f. Min. A 57 1197 . . .21 

See also Gille, F. 

Spiller, R. C. See Porter, M. W. 



400 BEFERENCES AND NAME INDEX 

Stevenson, D. P., and Sohomaker, V., 1939. J. Amer. Chem. Soc. 61 3173 226 

Stickley, E. E. 'See Smith, C. S. 

Stokes, A. R., Pascoe, K. J., and Lipson, H., 1943. Nature 151 137 . 367 

and Wilson, A. J. C., 1942. Proc. Camb. Phil. Soc. 38 313 . , 366 

Storks, K. H., 1938. J. Amer. Chem. Soc. 60 1753 . . . .373 

Stranski, I. N., 1928. Z. phys. Chem. 136 259 22 

Street, A. See Astbury, W. T. 

Strock, L. W., 1934. Z. phys. Chem. B 25 441 334 

1935. Z. phys. Chem. B 31 132 334 

Strukturbericht, Zeitschrift fiir Kristallographie .... 123, 312 

Stuart, A. See Hartshorne, N. H. 

Sucksdorff , I. See Hagg, G. 

Sykes, C. See Bradley, A. J. ; Jones, F. W. 

Taylor, W. H., 1930. Z. Krist. 74 1 184, 334 

1934. Proc. Roy. Soc. A 145 80 " . 184, 334 

1936. Z. Krist. 93 151 286 

Thomas, D. E., 1940. J. Sci. Inst. 17 141 170 

Thomson, G. P., and Cochrane, W., 1939. Book: The Theory and Practice 

of Electron Diffraction. London : Macmillan . . . . .373 

Treloar, L. R. G., 1941. Trans. Far. Soc. 37 84 83 

Tunell, G., 1939. Amer. Min. 24 448 204 

See also Donnay, J. D. H. 

Turkevitch, A., and Smyth, C. P., 1940. J. Amer. Chem. Soc. 62 2468 . 291 

Turner -Jones, A. See Bunn, C. W. 

Tutton, A. E. H., 1922. Book: Crystallography and Practical Crystal 

Measurement. London: Macmillan . . . . .30, 62, 83 

Vegard, L., 1916. Phil. Mag. 32 65 210, 214 

Verwey, E. J. W., 1935. Z. Krist. 91 65 333 

Void, R. D. See Marton, L. 

Volmer, M., 1923. Z. phys. Chem. 102 267 19 

See also Brandes, H. 

Waldmann, H., and Brandenberger, E., 1932. Z. Krist. 82 77 . . 282 

Waller, I. See James, R. W. 

Warren, B. E., 1934. Phys. Rev. 45 657 371 

1937. J. Applied Phys. 8 645 371 

1940. Chem. Reviews 26 237 371 

and Bragg, W. L., 1930. Z. Krist. 76 201 .... 175, 369 

and Gingrich, N. S., 1934. Phys. Rev. 46 368 . . . . 372 

Knitter, H., and Morningstar, O., 1936. J. Amer. Ceram. Soc. 19 202 371 

Weissenberg, K., 1924. Z. f. Physik 23 229 . . 167, 168, 171, 181, 363 

Wells, A. F., 1938. Proc. Roy. Soc. 167 169 . * . . . . 350 

West, J. See Bragg, W. L. 
White, A. H., Biggs, B. S., and Morgan, S. O., 1940 . . . 291, 329 

and Bishop, W. S., 1940. J. Amer. Chem. Soc. 62 8 . . 291, 329 

Wilson, A. J. C., 1940. Proc. Camb. Phil. Soc. 36 485 . . . . 181 

1942. Proc. Roy. Soc. A 180 277 369 

1943a. Proc. Roy. Soc. A 181 360 369 

19436. Nature 151 562 107 

and Lipson, H., 1941. Proc. Phys. Soc. 53 245 ... 180-1 



BEFEBENCES AND NAME INDEX 401 

Wilson, A. J. C. See also Lipson, H. ; Stokes, A. R. 

Winchell, A. N., 1931. Book: The Microscopic Characters of Artificial 

Minerals. New York: Wiley 82,93,98,283,285 

1943. Book: The Optical Properties of Organic Compounds. Madison, 

Wis. : Univ. of Wisconsin Press ....... 98 

Wollan, E. O., 1932. Kev. Mod. Phys. 4 233 371 

Woods, H. J. See Astbury, W. T. 

Woodward, I. See Lange, J. J. de ; Robertson, J. M. 

Wooster, N., 1932. Science Progress 26 462 59 

Wooster, W. A., 1931. Z. Krist. 80 495 22 

1936. Z. Krist. 94 375 350 

1938. Book: A Textbook on Costal Physics. Cambridge Univ. Press . 62, 

89, 283, 290-1 

and Martin, A. J. P., 1936. Proc. Roy. Soc. A 155 150 . . . 192 

1940. J. Sci. Inst. 17 83 168, 193 



- See also Bernal, J. D. 



Wyckoff, R. W. G., 1930. Z. Krist. 75 529 192, 301 

1931. Book: The Structure of Crystals. New York: Chemical Catalog 

Co 243, 275 

1932. Z. Krist. 81 102 209, 301-2 

And Corey, R. B., 1934. Z. Krist. 89 102 ... 209, 301, 303 

See also Corey, R. B. 



Yardley, K. (K. Lonsdale). See Astbury, W. T. 

Zachariasen, W. H., 1929. Z. Krist. 71 501, 517 . . . 226, 283, 285 

1932. J. Amer. Chem. Soc. 54 3841 372 

1933. J. Chem. Phys. 1 G34, 640 307-10 

Zeigler, G. E., 1931. Phys. Rev. 38 1040 303, 306 

Zworykin, V. K., 1943. Ind. Kng. Chem. 35 450 367 



SUBJECT INDEX 



Abb6 Refractometer .... 375 

Abnormal Structures . . . 329 ff. 

Absent Reflections 153, 210, 216 ff., 234 ff. 
in Body-centred cells . . 217 ff., 222 
in Face-centred cells . 218 f., 222 f. 

Glide planes 234 ff. 

Lattice 216 ff. 

Screw axes 234 ff. 

Space-group. . 216 ff., 235 ff., 245 ff. 
Tables 240 

Absolute X-ray Intensities 192 f., 207 ff . 

Anthracene 193 

Expressions for . . . 207 f. 
for Fourier syntheses . . . . 361 

Sodium chloride 193 

for Structure of non-crystalline 
substances 370 f. 

Absorption, Infra-Red .... 286 

Absorption of Light 85 f. 

Ellipsoid 86 

Structure determination . . 285 f. 

Absorption of X-rays 

Air 123 

Coefficient 106 

Corrections of intensity . . 206 f. 
Corrections of spacings . 120ff., 132 
Edge . 106 f., 113, 243, 274, 360 

Friedel's law 242 f. 

^-synthesis 360 

Standard substance admixture . 122 

Accurate Measurements 

Cell dimensions , . . 120, 180 f. 
Interplanar spacings . . . . 120 

Aceiamide 

Structure 347 

Acute Bisectrix 75 

Adenine Hydrochloride 

Identification 183 

Air 

Absorption of X-rays . . . . 123 

Alkyl Ammonium Halides 

Molecular rotation 330 

Alloys 

Copper-gold . . . . 331 f., 369 

Copper-ironnickel 369 

Metallurgical identification . . 92 
Solid solution, q.v. . . 274, 331 f. 

Almandiiie 

Crystal symmetry 61 

Alum 

for R.I. liquids standardization . 375 

at Alumina 

Powder photograph ... PI. Ill 
for R.I. liquids standardization . 375 

Y Alumina 

Defect structure 333 

Aluminium 

Structure determination . . . 293 

Ambiguities in Structure Determina- 
tion 275 f. 



Ammonium Bromide 

for R.I. liquids standardization . 375 
Ammonium Chloride 

Laue photograph . . 243, PL XII 

Oriented overgrowths of urea on . 60 

for R.I. liquids standardization . 375 

Skeletal growth 23, PL I 

Structure amplitude . . . 196 ff. 

X-ray powder pattern . 130, PL X 
Ammonium Hydrogen Fluoride 

Structure determination . . 258 f. 
Ammonium Iodide 

for R.I. liquids standardization . 375 
Ammonium Nitrate 

Polymorphism 59 

(mon) Ammonium Phosphate 

Optical properties 65 ff., 71 ff., 81, PL II 

for R.I. liquids standardization . 375 
Ammonium Sulphate 

Crystal habit 26 f., 32 

Isomorphism with K 2 SO 4 . . 59, 331 

Twinning 57 f., 89 . 

Amorphous Substances 

Optical identification . . . . 93 f. 

Optical properties 63 f. 

Structure 370 ff. 

Warren's method 372 

X -ray identification . . . . 127 
Amplitudes of Diffracted Waves 

(see also Intensities of X-ray Re- 
flections) 

Structure, q.v. . 196 ff., 209 ff., 292 ff. 
Amyl Acetate 

as R.T. liquid 374 

Analcite 

Powder photograph ... PL III 
Analysis (Chemical Composition) 

Crystallographic aids . 3 f., 102, 184 

Mixed crystals . . . . 126, 183 f. 

Mixtures. . . 99 ff., 123 ff., PL IV 

Quantitative 125, 184 

Solid solutions 126 

Analysis (Crystal Structure) 

Abnormal structures . . . 329 ff. 

Adjustment of parameters . 264 ff. 

Ambiguities 275 f. 

Atomic parameters, q.v. 256 ff., 344 ff. 

Birefringence 280 ff. 

by Calculation .... 7, 338 ff. 

Crystal chemistry .... 274 ff. 

Defect structures . . . 277, 332 ff. 

Direct 340 ff. 

Electron density .... 335 ff . 

Examples 291 ff. 

Fourier series ...... 335 ff . 

General 6 ff . 

Indicatrix 283 ff. 

Irregularities in structures 362 ff., 368 

Mixed crystals 331 ff. 

One parameter 214f. 



SUBJECT INDEX 



403 



Analysis (Crystal Structure cont.) 
Optical diffraction methods, q.v. 

7, 271 ff., 348 ff., 369 f. 
Physical properties. . . . 278 ff. 

Polymers 317ff. 

Shape of crystals . . 13 ff., 278 ff. 
Simplest structures . 196 ff., 209 ff. 
Space group, q.v. 43 ff ., 217, 224, 244 ff. 
by Trial and error, q.v., . 7 f., 190 ff., 
259 ff., 271 ff., 292 ff., 350 f. 
Unit cell, q,v. ... 128 ff., 217 ff. 

Vector method 351 ff. 

Warren's method 372 

Analysis (Fourier) 

Absolute intensities . . , . t 351 
Beevers-Lipson strips . . . 347 f. 
Electron density .... 339 ff. 

False detail 350 

Image formation 338, 

Methods of computation . . ,'347f. 
Non-crystalline substances . . 370 f. 
Optical synthesis .... 348 ff . 

Patterson 35 J ff. 

Patterson-Harker .... 356 ff. 

Resolving power 350 

Strips 347 f. 

Structure determination . . 335 ff. 

Analyser 67 

Anatase 

Crystal symmetry 51 

Anglo Factors 200 ff . 

Angstrom Units 100 

Anhydrite (see also Calcium Sulphate) 

Crystal habit 31 

Optical anomalies 99 

Anisoie, p-Azoxy- 

Space -group 245 f. 

Anisotropy 

Crystal growth 278 f. 

Diamagnetic susceptibility . 287 ff. 
Optical . 2 ff., 65 ff., 82 f., 86, 94 ff., 

280 ff. 

Other physical properties . . 290 f. 
Anorthic System (see Triclinic System, 

and Biaxial Crystals) 
Anthracene 

Absolute intensities . . . . 193 

Molecular shape 277 

Antimony Iodide . 

as Solute in R.I. liquids . . . 374 
Apatite 

Crystal symmetry 50 

Aperture (X-ray) 

Height correction 121 

Reduced size 124 

Aragonite (see also Calcium Car- 
bonate) 59 

Analysis 125 

Birefringence 282 

Twinning 58 

Arsenic Iodide 

as Solute in R.I. liquids . . . 374 
Artificial Temperature Correction 350 
Asbestos 

Chrysotile, q.v 58, 175. 279 



Ascorbic Acid 

Molecules in unit cell . . . . 251 

Pseudo -symmetry 273 

Structure determination . . 316 ff. 

A.S.T.M. Index 122 f. 

Atomic Number 

Absorption coefficient . . . 106 f. 

Diffracting power .... 202 f. 

X-ray wave-lengths . . . . 108 
Atomic Parameters . . . . 190 ff. 

Adjustment 264 ff. 

Ambiguities 275 f. 

Determination . . . 256 ff., 292 ff. 

Refinement 344 ff. 

Structure amplitude, q.v. . 257 ff. 
Atomic Space Patterns (see Space -Groups) 
Atoms 

Diffracting power, q.v. 126, 196, 200 ff. 

Packing 276 

Polarization 281 

Axes of Symmetry .... 35 ff. 

Effect on ^-synthesis . . . 356 ff. 

Inversion 42 f., 234 f. 

Polar 40 f. 

Screw, q.v. . . 43 ff., 230 ff., 273 f. 
Axial Ratio (see also Unit Cell) 31 ff. 

from Interplanar spacings 132 ff., 376 ff., 

379 ff. 
Azobenzene (trans) 

Crystal structure 250 

Azo -Groups 

Light absorption 286 

p - Azoxyanisole 

Space -group 245 f. 

Babinet Compensator .... 83 
Background Intensity . . . . 113 

Bakelite 1 

Barium Nitrate 

for R.I. liquids standardization 375 
Becke Line Method . . 64 ff., PI. II 
Beevers-Lipson Strips . . . 347 f. 
Benzene 

Habit and structure .... 280 

Molecular symmetry . . . 226 f. 
Benzene, Azo- (trans) 

Crystal Structure 250 

Benzene , Bromo - 

as R.I. liquid 374 

Benzene, m -Bromo -Nitro - 

Crystal symmetry .... 41 
Benzene, 2, 6 Dibromo, 1, 4 Dinitro- 

Crystal symmetry 47 

Benzene, m-Dihydroxy- (see Resorcinol) 
Benzene, ^p-Dinitro- 

Crystal symmetry 48 

Benzene, 1, 4 Dinitro, 2, 5 Dibromo - 

Crystal symmetry 47 

Benzene, o-Diphenyl- 

Magnetic anisotropy .... 289 
Benzene, j>-Diphenyl- 

Struoture determination . . 310 ff. 
Benzene, Hexamethyl- 

Birefringence 283 f. 

Crystal habit 19, 31 



4458 



Dd2 



404 



SUBJECT INDEX 



Benzene, Hexamethyl- (cont.) 

Crystal structure . . . .15, 284 

Symmetry 228 

Benzene, I Hydroxy, 3, 5 Diethoxy- 

Crystal symmetry 51 

Benzene, 1 Hydroxy, 2, 4, 6 Trinitro- 

Crystal symmetry 49 

Benzene, 1, 2, 4, 6 Tetramethyl- 

Symmetry 231 

Systematic absences . . . 237 ff. 
Benzene, 2, 4, 6 Tribromo, 1 Cyano- 

Crystal symmetry 48 

Benzene, 1, 3, 5 Triphenyl- 

Birefringence 283 

Benzil 

Dispersion 84 

Rotation photograph . .152, PL VI 
Benzoquinone 

Structure 231 ff., 260 

Symmetry 231 ff. 

Benzyl, Di- 

Structure determination . . 310 ff. 
Benzylidene-p-Methyl Toluyl Ketono 

Optical properties and structure . 285 
Bernal Chart 149 ff. 

Indexing rotation photographs 151 ff. 

Bertrand Lens 78 

Beryl 

Structure 256 

Biaxial Crystals 73 ff. 

Dispersion 84 f. 

Identification 96 ff. 

Indicatrix 73 ff. 

Optic pictures 78 f. 

Optic sign 75, 83 

Pleochroism 85 ff. 

Birefringence . . . . 65 ff., 94 ff. 

Crystal structure .... 280 ff. 

Identification 94 ff. 

Measurement 82 f. 

Molecular orientation . . . 282 ff . 

Optic sign .... 72, 75, 80 ff. 

Organic substances. . . . 282 ff. 

Polyatomic ions . . . . 281 ff. 

Bisectrix 75 

Bixin, Methyl- 
Birefringence 282 

Pleochroism 286 

Bleaching Powder 

Identification of constituents 124 

Body-Centring ... 217 ff., PL XI 

Absences due to . . . 217 ff., 222 
Boiler Scales 

Identification of constituents . 101 
Bragg's Law 114 ff. 

Reciprocal lattice 145 

Bragg-Lipson Charts . . . 264 ff . 
Bravais Lattices (see also Space - 

Lattices). ... 221 ff., PL XI 

Breadth of X-ray Reflections . 362 ff . 
Bricks 

Identification of constituents . . 101 
Broadened X-ray Reflections . 362 ff. 
1 Brom, 2 Hydroxy Naphthalene 

Crystal symmetry 49 



Bromobenzene 

as R.I. liquid 374 

Bromoform ' 

as R.I. liquid 374 

ex Bromonaphthalene 

as R.I. liquid ...'... 374 
m-Broraonitrobenzene 

Crystal symmetry 41 

Brookite 

Crossed axial dispersion . 84 f. 
Brushes (see Directions Image) 
Butane, 2, 3 Dibromo- 

Molecular symmetry . . . 225 f. 

Cadnjium Iodide 

Layer formation . . . ' . PL I 

Layer lattice 279 

Optic pictures 78 

Cadmium Sulphide 

in liuminescent powders 126 

Caesium Bromide 

Crystal habit 19 f. 

Crystal structure . . . . 15, 217 

'Caking' of Crystals .... 23 

Calcite 59 

Analysis 125 

Birefringence 282 

Cleavage 58 

Crystal symmetry ..... 50 
Detection of small quantities in 

mixture 126 

Isomorphism with NaNO a . . ,60 

Optical anomalies 99 

for R.I. liquids standardization . 376 

Unit cell 141 

Calcium Aluminates 

in Portland cement . . . . 124 
Powder photograph ... PL III 

Calcium Carbonate 

Analysis 125 

Aragonite, q.v. ... 58 f, 282 

Birefringence 282 

Calcite, q.v. . 50, 58 ff., 99, 125 f., 
141, 282, 375 

p 69 

Polymorphism 59 

Vaterite, q.v 59, 285 

Calcium Chloride 

Basic 124 

in Bleaching powder . . . . 124 

Calcium Fluoride 

Cleavage 58 f. 

Crystal habit 59 

for R.I. liquids standardization . 376 

Twinning 67 

Calcium Hydroxide 

Optical properties 94 

Calcium Hypochlorite 

Birefringence 283 

in Bleaching powder . . . . 124 

Calcium Oxide 

for R.I. liquids standardization . 375 

Structure determination . . 293 f. 

Calcium Silicates 

in Portland cement . . . . 124 



SUBJECT INDEX 



405 



Calcium Sulphate 

Anhydrite, q.v 31, 99 

Dihydrate (gypsum, q.v.) 12 f., 31, 
57, 76, 89, 126, 165, Pis. VI & VII 

Subhydrate .... 90, 184, 334 
Calcium Thiosuiphate Hexahydrate 

Crystal symmetry 47 

Calculated Intensities (see also Struc- 
ture Amplitude) 

Relation to observed . . 191 f., 275 
Calculation 

Correction factors, q.v. 132, 199 ff., 350 

Fourier series . . . 335 ff., 347 f. 

Interplanar spacinga . .114, 376 ff. 

Refractive indices .... 80 ft 

Structure amplitude 

196 ff., 209 ff., 264 ff. 

Structures by Fourier series . 335 ff. 
Cameras (X-ray) 

Focusing * 181 

Moving-film, q.v 166 ff. 

Powder, q.v. . . . 109 ff., 123 f. 

Rotation 137 ff. 

Two-crystal 193 

Universal goniometers . . . 138 
Cane-Sugar 

Optical activity 88 

Carbon 

Amorphous 1 

Carbonate Ion 

Polarizability 281 

Shape 307 

Symmetry 225 f. 

Carbonates (see also individual Car- 
bonates) 

Birefringence 282 

Diamagnetism 288 

Carbon Tetrachloride 

Molecular symmetry . . . 225 f. 

Structure 372 

Cassiterite 

Crystal symmetry 40 

Catalysts 

Activity 362 

Cell (see Unit Cell) 
Cellulose 

Orientation 175 

Cement 

Portland 124 

Centre of Inversion (see Inversion Axes) 
Centre of Symmetry . . . . 3C f. 

Friedel's law 242 f. 

Laue symmetry .... 243 f. 

Piezo-electricity 290 

Pyro-electricity .... 289 f. 

X-ray photographs. . . . 242 f. 
Chain Compounds 

Birefringence 282 f. 

Identification 183 

Polymers, q.v. . 127, 175 ff., 186 ff., 
248 f., 261 f., 317 ff., 373 
Characteristic X-radiation . . 105 ff. 
Charts 

Indexing layer lines of rotation 
photographs 142 f., 151 ff., 382 f. 



Indexing powder photographs 

132 ff., 379 ff. 

Lorentz factor correction . . 204 

Structure amplitude . . . 264 ff. 
Chemical Composition 

Analytical methods 3 f., 102, 125 f., 184 

Mixed crystals .... 126, 183 f. 

Mixtures. . 99 ff., 123 ff., PL IV 

Quantitative .... 125, 184 

Solid solutions 126 

Chemical Reactions 

on Microscope Slide . . . . 102 
Chlorate Ion (see also individual 
Chlorates) 

Symmetry 225 f. 

Chromophoric Groups .... 286 
Chrysene 

Molecular shape 277 

Structure 262 

Chrysotile 

Cleavage 58, 279 

Polycrystalline character . . . 175 
Classes of Symmetry 33 ff., 44 ff., 224 

Cubic 50 ff. 

Hexagonal 49 f. 

Monoclinic 48 ff. 

from Morphological data . 52 ff. 

Nomenclature . . . . 44 ff., 384 ff. 

Optical activity 88 f. 

Orthorhombic 49 

Relation to diffraction symmetry 241 ff. 

Relation to molecular symmetry 228 f. 

Relation to space lattices . . 228 f. 

Stereographic projection . 45 ff. 

Tetragonal 50 

Triclinic 47 f. 

Trigonal 49 f. 

Clay minerals 

Layer lattice 279 

Cleavage 58 f. 

Crystal structure 279 f. 

Identification 93 

Cobalt 

Irregularities of structure . . 368 f. 
Compensator 

Babinet 83 

Composition (Chemical) 

Analytical methods 3 f., 102, 125 f., 184 

Mixed crystals .... 126, 183 f. 

Mixtures. . . 99 ft, 123 ff., PL IV 

Quantitative . . . . 125, 184 

Solid solutions 126 

Compton Scattering 371 

Computation 

Correction factors, q.v. 

199 ff., 132, 200 ff., 350 

Fourier series 347 f. 

Interplanar spacings . . . 376 ff. 

Refractive indices .... 280 ff. 

Structure amplitude 196 ff., 209 ff., 

264 ff. 

Structures by Fourier series . 335 ff. 
Conductivity 

Crystal structure .... 290 f. 
Cones of Diffracted X-rays 108 f., 137 ff. 



406 



SUBJECT INDEX 



Conjugated Double-Bonds 

Birefringence 282 

Continuous X-ray Spectrum (see 

White X-Radiation) 
Convergent Light 

Investigations in 77 if . 

Copper 

Crystal habit 20 

Mixed crystals with gold . . .331 f. 

Structure . . 17, 217, PL XI 

Superlattices with gold . . . 332 
Copper (Cuprous) Chloride 

for R.I. liquids standardization . 375 

Structure determination . . 294 ff . 
Copper Di hydrogen Silicate (Dioptase) 

Crystal symmetry 42 f. 

Copper (Cuprous) Oxide 

Misleading crystal shape . . . 247 
Copper Sulphate Pentahydrate 

Crystal symmetry 47 

Correction Factors 

Absorption . . 120 ff., 132, 206 f. 

Angle 200 ff. 

Artificial temperature .... 350 

Intensities 199ff., 350 

Slit 121 

Spacings 120 ff., 132 

Temperature 205 f. 

Corundum 

Powder photograph ... PL III 

for R.I. liquids standardization . 375 
Cox Moving-Film X-ray photo- 
graphs 169 

Cox-Shaw correction charts . . 204 
Crossed Axial Dispersion. . 84 f. 

Crossed Nicols 67 ff. 

Crystal Chemistry .... 274 ff. 
Crystal Growth . . 12 ff., 18 ff., 31 

Anisotropy 278 f. 

Good crystals 23 

Layer formation . . . 19, PL I 

Rate 21, 31, 278 f. 

Crystal Habit 

Crystal structure . . 13 ff., 278 ff. 

Donnay's law 280 

Effect of growth conditions 12 ff., 21, 

31,40 

Identification . . 54 ff., 91, 99 ff. 

Modification . . 13, 34, 37, 53 f. 

Molecular shape . . . . 277 ff. 

Unit cell 30 ff. 

Crystalline Polymers . . 175 ff., 188 f. 

Density 186 f. 

Double orientation . . . 178ff. 

Electron diffraction .... 373 

Orientation 175 ff. 

Strongest X-ray reflections . 261 f. 

Structure determinations . 317 ff. 

Symmetry 248 f. 

X-ray identification .... 127 
Crystalline Solid Solutions . . 59 f. 

Composition 183 f. 

JF-synthesis 361 

Isomorphous crystals, q.v. 59 ff., 126, 

274, 361 



Optical properties 93 

Structure 331 ff. 

X-ray patterns . . . ' 126, 367 

ZnS-CdS * . 126 

Crystal Optics 3 ff., 62 ff., 91 ff., 280 ff. 

Absorption . . . / . 85 f., 285 f. 

Amorphous substances . 63 f., 93 f. 

Anisotropy . . 2, 4 ff., 65 ff., 82 f., 
94 ff., 280 ff. 

Biaxial 73 ff., 96 ff. 

Convergent light .... 77 ff. 

Crystal structure .... 280 ff. 

Cubic 63, 93 f. 

Dispersion, q.v.. 65, 77, 83 ff., 100 f. 

Extinction directions, q.v. 67 ff., 77, 

95 ff., 100 

identification . . . . 3 f., 91 ff. 

Image formation .... 335 ff. 
. Interference colours . . 67 ff., 78 ff. 

Mixed crystals 93 

Mixtures 99 ff. 

Nicol prism 66 ff. 

Opaque crystals . . . 91 f., 101 f. 

Optic axial angle, q.v. 75, 77 ff., 80, 

100 f. 

Parallel light 63 ff. 

Pleochroism 85 ff . ' 

Refractive indices, q.v. . 62 ff. 

Symmetry .... 62 ff., 70 ff . 

Twinned crystals 89 f. 

Uniaxial crystals . . 65 ff., 94 ff. 

Universal stage 1 92 

Crystal Planes 

Interplanar spacing, q.v. 114, 118ff,, 
129 ff., 376 ff ., 379 ff. 

Law of constancy of angles . . 20 

Law of rational indices . 27 f., 31 ff. 

Nomenclature 24 ff. 

Relation to unit cell . . . 31 ff. 

Roticular density .... 20, 58 

Structure amplitude . . . 196 ff. 
Crystal Size 

Effect on X-ray pattern . 127, 362 ff. 

Electron microscope .... 367 

, Microscope count 367 

Crystal Strains 362 ff. 

Crystal Structure Determination 

Abnormal structures . . . 329 ff. 

Adjustment of parameters . 264 ff . 

Ambiguities 275 f. 

Atomic parameters, q.v. . 256 ff., 

344 ff. 

Birefringence 280 ff. 

by Calculation .... 7, 338 ff. 

Crystal chemistry .... 274 ff . 

Defect structures . . . 277, 332 ff. 

Direct 340 ff. 

Electron density .... 335 ff . 

Examples 291 ff. 

Fourier series 335 ff. 

General 6 ff . 

Indicatrix 283 ff. 

Irregularities in structures 362 ff., 368 

Mixed crystals 331 ff. 

One parameter 214 f. 



SUBJECT INDEX 



407 



Crystal Structure Determination (cont.) 

Optical diffraction methods, q.v. 7, 
271 ff., 348 ff., 369 f. 

Physical properties. . . . 278 ff. 

Polymers 317 ff. 

Shape of crystals . . 13 ff., 278 ff. 

Simplest structures . 196 ff., 209 ff. 

Space-group, q.v. . . 43 ff., 217, 224, 

244 ff. 

by Trial and error, q.v. 7 f., 190 ff., 
259 ff., 271 ff., 292ff.,350f. 

Unit cell, q.v. . . . 128 ff ., 21 7 ff. 

Vector method 351 ff. 

Warren's method 372 

Cubic system 

Classes 50 ff. 

Dispersion 83 

Interplanar spacings 129 f., 377, 379 ff. 

List of unit cell dimensions . . 182 

Optical identification . . . . 93 f. 

Optical properties T 63 

Space-lattices 217 ff. 

Unit cell from powder data 129 ff., 377 
Cupric Dihydrogen Silicate (Dioptase) 

Crystal symmetry 42 f. 

Cupric Sulphate Pentahydrate 

Crystal symmetry 47 

Cuprite 

Misleading crystal shape . . . 247 
Cuprous Chloride 

for R.I. liquids standardization 375 

Structure determination . . 294 ff. 
Cuprous Oxide 

Misleading crystal shape . . . 247 
Cyanuric Triazide 

Structure 351 

2?-Cymene 

as R.I. liquid 374 



Debye Equation . 371 

Debye Factor (see Temperature 
Factor) 

Debye-Scherrer X-ray Photographs 

103, 106, 108 ff. 

(see also Powder X-ray Photo- 
graphs) 

Defect Structures . . 277, 332 ff. 

De Jbng-Bouman Moving-Film X- 
ray Photographs .... 170 f. 

Density 

Determination 186 

Molecular packing .... 276 

Polymers 186 f. 

Reticular 20, 58 

Determinative Tables 

Absent reflections 240 

Crystal shapes .... 54 ff., 91 
Refractive indices .... 93, 98 

Space -groups 384ff. 

Unit cell dimensions cubic sub- 
stances 182 

X-ray patterns minerals . . . 123 
X-ray powder patterns . . 122 ff. 

Diagonal Glide Planes .... 239 



Diamagnetic Susceptibility 

Crystal structure .... 287 ff 

Dibenzyl 

Structure determination . . 310 ff 

2, 3 Dibromobutane 

Molecular symmetry . . . 225 f 

2, 5 Dibromo, 1, 4 Dinitro Benzene 
Crystal symmetry 41 

1, 2 Dichloroethylene (trans) 
Molecular symmetry . . . 224 f 

Dichroism 85 ff 

Crystal structure .... 285 f 

Dickite 

Powder photograph ... PL II] 

Dielectric Constant 

Crystal structure .... 290 f 

Diffracting Power of Atoms . . 19( 

Angle factors 200 ff 

Graph (sin0)/A 205 

Intensity of X-ray reflections 126, 19 

Diffraction 

Amorphous substances . . 370 ff 

Bragg's law 114ff 

Cones of .... 108 f., 137 ff 

Electron 372 f 

Geometry .... 108 f., 128 ff 

Line broadening .... 362 ff 

Non-crystalline substances . 370 ff 

Optical, q.v. 129, 190 f., 271 ff., 335 ff., 

348 ff., 369 f., Pis. V, XIII 

Order of, q.v. . . 114ff., 131 

Phase relation . 116 ff., 190 ff., 197 ff., 

210 ff., 340 ff. 

Point array in plane . . . 116f 
Point array in 3 dimensions . 117 f, 

Point row 114ff. 

Symmetry 241 ff, 

Diffuse X-ray Reflections . 206, 362 ff, 

w-Dihydroxybenzene (see Resorcinol) 

1, 2 Dimethyl phenanthrene 

Space-group 246 

p -Dinitrobenzene 

Crystal symmetry 48 

1, 4 Dinitro, 2, 6 Dibromo Benzene 

Crystal symmetry 47 

4, 4' Dinitrodiphenyl 

Structure 251 

Diopside 

Structure 345 f., 349 

Dioptase 

Crystal symmetry .... 42 f, 

Diphenyl 

Symmetry 249 

o -Diphenyl Benzene 

Magnetic anisotropy .... 289 

p -Diphenyl Benzene 

Structure determination . 310 ff. 

Diphenyl, 4, 4' Dinitro- 

Structure 251 

Directions Image 77 ff. 

Bertrand lens 78 

Identification 95, 99 

Optical activity .... 87 f. 
Optic axial angle, q.v. 75, 77 ff., 100 f. 
Optic sign 83 



408 



SUBJECT INDEX 



Direct Structure Determination 340 ff. 

Dispersion 65, 77, 83 ff. 

Biaxial 77, 84 f. 

Crossed axial 84 f. 

Cubic 83 

Identification of mixtures . 100 f. 

Isotropic 83 

Optic axes 84 f. 

Orthorhombic 84 f. 

Uniaxial 83 f. 

Donnay's Law 280 

Double Fourier (Series .... 340 

Double Orientation in Polymers 178 ff. 

Double Refraction 65 f. 

Durene 

Symmetry 231 

Systematic absences . . . 237 ff. 

d-Values . . . 114 ff., 376 ff., 379 E. 
Absorption correction 120 ff., 132 

Bragg's law 114ff. 

Calculation . . . 114, 376 ff. 
Correction factors . . 120 ff., 132 
Cubic ... 129 f., 377, 379 ff. 

Hexagonal . . 134 f., 377 f., 381 

Lower limit 118 

Monoclinic .... 136 f., 156 

Orthorhombic 136 f., 142 f., 376, 382 f. 
Powder photographs . 119 ff., 379 ff. 
Reciprocal lattice, q.v. . . 144 ff. 
Rhornbohedral ... 134 f., 378 
Rotation photographs . . 142 ff. 

Slit correction 121 

Tetragonal . . . 132, 376, 379 ff. 
Triclinic .... 136 f., 156 ff. 
Trigonal . . . 134 f., 378, 381 
Unit cell dimensions . . 376 ff. 

Dyes 

Magnetic properties .... 288 
Optical properties 286 

Elasticity 

Crystal structure . 290 f. 

Electron Density 

Absolute intensities . . . . 351 

Beevers-Lipson strips . . 347 f. 

False detail 350 

Fourier series 339 ff. 

Maps 6f., 335 ff. 

Methods of computation . . 347 f. 

Optical synthesis .... 348 ff. 

Resolving power 350 

Strips 347 f. 

Structure determination . . 335 ff. 
Electron Diffraction .... 372 f. 
Electron Distribution in Atoms 201 f. 
Electron Microscope . . . 128, 367 
Electron Scattering (Diffraction) 372 f. 
Ellipsoids Representing Physical Pro- 
perties (see also Indicatrix) 

Absorption of light 86 

Diamagnetic susceptibility . 287 ff. 

Refractive index . . 70 ff., 283 ff. 
Emission Spectra of X-rays . . 105 ff. 
Enantiomorphism . . . . 38 f ., 286 f . 

Molecules 227 



Equivalent Positions . . . 252 ff. 

General 252 f. 

Special . 253 ff. 

Structure amplitude . . '.. 257 ff. 

Etch-Figures 54 

Ethane, Hexabromo- 

Crystal symmetry . .' . . 45, 49 
Ethanol 

Structure 372 

Ethyl Acetate 

as R.I. liquid' 374 

Ethyl Alcohol (see Ethanol) 
Ethylene, 1, 2 Dichloro- (trans) 

Molecular symmetry . . . 224 f. 
Ettringite 

Powder photograph ... PI. Ill 
Extinction 

Primary and secondary . . 208 f . 
Extinction Directions . . 67 ff., 77 
. Identification by . . 95 ff., 100 f. 
Extra'Spots 206 



Face -centring . 

Absences duo to 
Faces of Crystals 

Crystal structure 

Indices 



218 ff., PI. XI 
218 f., 222 f. 
. . 18 ff. 
. . 279 f. 
24 ff., 31 ff. 



Law of constancy of angles . . 20 
Law of rational indices . 27 f., 31 ff. 

Nomenclature 24 ff. 

Rate of growth . . . . 21,31 
Relation to unit cell shape . 30 ff. 

Fatty Acids f 

Identification 183 

Faults in Crystal Structure (see Ir- 
regularities, and Broadening of 
X-ray Reflections) 

/-Curves 202 f. 

y Ferric Oxide 

Defect structure 333 

Ferroso-ferric oxide 

Structure 333 

Fibre X-ray Photographs 176 ff., 318, 

PL IX. 

Indexing 176 ff. 

Interpretation 317ff. 

Repeat period . . . . 176, 188 f. 

Fibroin 

Orientation 175 

Filters for X-rays . . . 106 f., 113 

Fluorescence (see also Luminescence) 113 f. 
Effect on intensities .... 203 

Fluorite 

Cleavage 58 

Crystal habit 59 

for R.I. liquids standardization . 375 
Twinning 57 

Focusing Cameras 181 

Fogging of X-ray Film . . 113, 138 

Form 27 

Fourier Series 

Absolute intensities . . . . 351 
Beovers Lipson strips . . 347 f. 
Electron density .... 339 ff. 
False detail 350 



SUBJECT INDEX 



409 



Fourier Series (cont.) 

Image formation 338 

Methods of computation . 347 f. 

Non-crystalline substances 370 f. 

Optical synthesis . . . 3483. 

Patterson 351 ff. 

Patterson-Barker . . . 356 ff. 

Resolving power 350 

Strips 347 f. 

Structure determination . . 335 ff. 
.Friedel's Law ...'.. 242 f. 
jF-Synthesis 339 ff. 

False detail 350 

Resolving power 350 

^-Synthesis 

Absorption / 360 

Non-crystalline substances . 370 f. 

Patterson 351 'ff. 

Patterson-Harker .... 356 ff. 
P- Values (see also Structure Ampli- 
tude) 196 ff., 209 ff., 257 ff., 292 ff. 

Garnet 

Crystal symmetry 51 

Gas Molecules 

Geometry 372 f. 

General Positions .... 252 f. 
Structure amplitude . . . 257 ff. 

Glasses 

Lindemann ... 110, PI. IV 
Lithium borate . . 110, PL IV 
Optical identification ... 93 f. 
^Optical properties . . . 63 f., 374 
R.I. measurement .... 374 

Silicate 127 

Specimen mounts 110 

Structure 1, 370 ff. 

Warren's method 372 

X-ray diffraction . . . . 370 f. 
X-ray identification . . . . 127 
X-ray photographs. . . . PI. IV 

Glide Planes 43, 229 ff . 

Absences due to .... 234 ff . 

Diagonal 239 

Effect on ^-synthesis . . 356 ff. 
Effect on X-ray patterns . 234 ff . 

Pseudo 273 f. 

Quarter 234 

Glycine 

Molecular shape 231 

Goethite 

Analysis 125 

Gold 

Mixed crystals with copper . 331 f. 
Superlattices with copper . . 332 

Goniometry . . . . 11 ff., 28 ff. 

X-ray 137 ff., 193 

X-ray moving-film, q.v. . . 166 ff. 

Graphical Methods 

- Indexing fibre photographs . . 176 
Indexing powder photographs 132 ff., 

379 ff. 

Indexing rotation photographs 142 f., 

382 f. 
Structure amplitudes . . . 264 ff . 



Grating 
Image formation of ... 335 ff . 

Growth of Crystals . . 12 ff., 18 ff., 31 

Anisofcropy 278 f. 

Good crystals 23 

Layer formation during ... 19 
Rate 21, 31, 278 f. 

Gum Tragacanth 110 

|3 Gutta-Percha 

Structure determination . . 318 ff. 

Gypsum 

Crystal habit 12 f. 

Detection of small quantities in 

mixture 126 

Optical properties ..... 76 
Rotation photograph ... PL VI 
Tilted crystal photograph 165, PL VII 
Twinning 57, 89 

Habit of Crystals 

Crystal structure . . 13 ff., 278 ff. 

Donnay's law 280 

Effect of growth conditions 12 ff., 21, 

31,40 

Identification ... 54 ff., 91, 99 ff . 

Modification in . . 13, 34, 37, 53 f. 

Molecular shape . . . . 277 ff. 

Unit cell 30 ff. 

Heavy Atoms . 

JF-synthesis 341 ff. 

^-synthesis 353 ff. 

Herapathite 

Light absorption 87 

Hexabromoethane 

Crystal symmetry .... 45, 49 
Hexagonal System (see also Uniaxial 
Crystals) 

Classes 49 f. 

Dispersion 83 

Interplanar spacings . 134 f., 377 f., 

381 

Optical properties .... 65 ff. 

Unit cell from powder data 134 f., 

377 f., 381 

Unit cell from rotation photographs 140 
Hexamethylbenzene 

Birefringence 283 f. 

Crystal habit 19,31 

Crystal structure . . . . 15, 284 

Symmetry 228 

Hexamethylene Tetramine 

Habit and structure .... 280 
Hydrocinchonine Sulphate Hydrate 

Crystal symmetry 50 

Hydrogen Bond 309 f. 

Ice 23 

Ideally Imperfect Crystals . . 208 f. 
Identification of Crystalline Materials 

Cleavage 93 

Crystal shape . 54 ff., 91, 99 f., 103 
Directions Image .... 95, 99 
Extinction directions . 95 ff., 100 f. 

Metallurgical 92 

Microscopic 3, 91 ff. 



410 



SUBJECT INDEX 



Identification of Crystalline Materials 
(cont.) 

Mixed crystals 183 f. 

Mixtures, q.v. . 99 ff., 123 ff., PI. IV 

Morphological . 64 ff., 91, 99 f.,183 

Optic axial angle . . . . 100 f. 

Powder photographs 6, 8, 103 ff., 122 ff. 

Shape . . . 54 ff., 91, 99 f., 103 

Single crystal photographs . 182 f. 

Thin section 101 

Unit cell 123, 182 ff. 

Vibration direction. 95 ff., 100 f. 

Identification of Ions 

Microscope slide reactions . . . 102 
Identification of Non-Crystalline 
Materials 

Optical 93 f. 

X-ray 127 

Identity Period 

Fibre 176, 188 f. 

Layer lines, q.v. 138 f., 142 f., 147, 

382 f. 

Rotation photographs . 138 f., 147 
[mage formation . ... 335 ff. 

[mmersion Liquids . . 63 ff., 374 8. 
Immersion Methods of Refractive 

Index Measurement 63 ff ., 374 ff. 

Identification by . . . . 91 ff. 
Imperfect Crystals 

Defect structures . . . 277, 332 ff. 

Ideally 208 f. 

X-ray reflections . . 363, 367 ff. 
Index (Catalogue) 

A.S.T.M 122f. 

Crystal shapes . . . . 54 ff., 91 

Cubic unit cell dimensions . . 182 

Refractive indices .... 93, 98 

X-ray powder patterns . . 122 ff. 
Indexing of X-ray Photographs 

Bernal chart 149 ff. 

Fibre photographs . . . . 176 ff. 

Layer lines .... 142 f., 382 f. 

Moving-film photographs, q.v. 167 ff. 

Orowan 171 f. 

Oscillation photographs . . 159 ff. 

Powder photographs . 129 ff., 379 ff., 

PL X 

Reciprocal lattice, q.v. 151 ff., 176 ff. 

Rhombohedral . . 135, 172 f. 

Rotation photographs 142 ff., 151 ff., 

382 f. 

Simplest cell 172 f. 

Tilted crystal photographs 162 ff. 

Indicatrix 

Biaxial 73 ff. 

Orientation in monoclinic and tri- 
clinic crystals .... 76 f. 

Structure determination . . 283 ff. 

Uniaxial 70 ff. 

Indices of Crystal Faces . 24 ff., 31 ff. 

Principle df simplest ... 32 f. 
Infra-Red Absorption Spectra 

Crystal structure 286 

Integrated Intensity . . 192, 194 f. 
Intensifying Screens 113 



Intensities of X-ray Reflections 

Absent, q.v. 153, 210, 216 ff., 234 ff., 

f 245 ff . 

Absolute, q.v. 192 f., 207 ff., 351, 370 f. 
Absorption correction . . . 206 f . 
Background . . . f . . . 113 
Calculated and observed . 191 f., 275 
Calculation 195 ff., 209 ff ., 264 ff. 

Characteristic of substance . . 122 
Chemical composition . . . 125 
Correction factors, q.v. 199 ff., 350 

Diffracting power of atoms, q.v. 126, 
196, 200 ff. 

Expressions 207 f. 

for Fourier synthesis .... 351 
Integrated . . . 192, 194 f. 

Measurement 192 ff. 

Relative, q.v. . 122, 125, 192 ff., 207 

Scattered 113 

Structure amplitude . 196 ff., 209 ff., 

257 ff. 

Trial and error method, q.v. 7 f., 

259 ff., 271 ff., 292 ff. 

Interatomic Vectors . . . . 351 ff. 
Maps ... 351 ff., 356 ff., 370 f. 

Interfacial Angles . . . . 11 ff. 
Goniometric measurement . 28 ff. 
Law of constancy of .... 20 
Microscopic measurement . . 56 
Unit cell shape 52 ff. 

Interference Colours .... 67 ff. 
in Convergent light ... 78 ff. 
Quartz wedge .... 80 ff. 

Interference Figure . . . . 77 ff. 

Bertrand lens 78 

Identification 95, 99 

Optical activity .... 87 f. 

Optic axial angle . 75, 77 ff., 100 f. 
Optic sign 83 

Interpenetration Twins .... 57 

Interplanar Spacings . 114ff., 376 ff., 

379 ff. 
Absorption correction . 120 ff., 132 

Bragg's law 114ff. 

Calculation . . . 114, 376 ff. 
Correction factors ... 120 ff., 132 
Cubic . . . 129 f., 377, 379 ff. 

Hexagonal . . 134 f., 377 f., 381 

Lower limit 118 

Monoclinic .... 136 f., 156 

Orthorhombic 136 f., 142 f., 376, 

382 f. 

Powder photographs . 119 ff., 370 ff. 
Reciprocal lattice, q.v. . , 144 ff. 
Rhombohedral . . . 134f., 378 
Rotation photographs . . 142 ff. 

Slit correction 121 

Tetragonal . . . 132, 376, 379 ff. 
Triclinic .... 136 f., 156 ff. 
Trigonal . . 134 f., 378, 381 

Unit cell dimensions . . 376 ff. 

Inversion Axes . . . 42 f., 234 f. 

lodoform 

as R.I. liquid 374 

lonization Chamber 192 



SUBJECT INDEX 



411 



Ions 

Diffracting power 203 

Identification by microscope slide 
reactions 102 

Polyatomic, q.v. . 5 f., 225 f., 276, 

280 ff . 

Rotation 329 ff. 

Scattering power 203 

Symmetry . . . 225 f., 248 ff. 
a Iron 

Crystal structure . 18, 217, PL XJ 
Iron Oxide (y Fe 2 O 8 ) 

Defect structure 333 

Iron Oxide (Fe 3 O 4 ) 

Structure 333 

Iron Oxides (Hydrated) (FeO.OH) * 

Goethite and Lepidocrocite analysis 125 

Lepidocrocite structure . . . 258 
Iron Phosphate Octahydrate 

Pleochroism * 86 

Iron Sulphide 

Pyrites, crystal symmetry . 51 

Pyrrhotite, defect structure 333 f. 

Irregularities in Crystal Structure 362 f., 

368 f. 

Optical diffraction 369 f. 

Isogyres (see Directions Image) 
Isometric System (see also Cubic 

System) 50 ff. 

Isomorphous Crystals . , . 59 ff. 

jF 2 -syntheis 361 

Location of atoms .... 274 

Mixed crystals, q.v. 59 ff., 93, 126, 

183 f., 331 ff., 361, 367 

ZnS-CdS 126 

Jong-Bouman Moving-Film X-ray 

Photographs 170f. 



106 f. 
. 105 ff. 

. . 120 

, . 189 
, . 178 
. 175 

. 183 
, 105 ff . 
105 ff. 
. 120 

. 107 

Lamellar Twinning 58 

Lattice 

Absences due to .... 216 ff. 
Body-centred ... 217 ff., PI. XI 
Bravais (see also Space-) . . 221 ff. 
Face-centred ... 218 ff., PL XI 

Point 18, 24, 118 

Primitive 223 

Reciprocal, q,v. ... 144 ff., 383 

Side-centred 223 

Space-, q.v. . 118, 216 ff., PL .XI 



K-Absorption Edge . 
JST-Emission Spectra 

a -doublet .... 
Keratin 

Chain type .... 

Double orientation 

Orientation .... 
Ketones 

Identification . 
JK-Series of X-rays 

Emission spectra . 

Resolution of a-doublets 
kX Units 

Definition .... 



Laue Symmetry ... 243 f., PL XII 
Laue X-ray Photographs 243, PL XII 
Law of Constancy of Interfacial 

Anjgles 20 

Law of Rational Indices . 27 f., 31 ff. 
Layer Formation 

Crystal growth . . . . . 19, PL I 
Layer Lines 138 f. 

Indexing . . . 142 f., 382 f. 

Reciprocal lattice 147 

Lead Acetate Trihydrate 

Crystal symmetry 48 

Lead Carbonate 

Birefringence 283 

Lead Chloride 

Modification of external crystal 

.symmetry 63 f 

Lead Molybdate 

Crystal symmetry 51 

Lead Nitrate 

for R.I. liquids standardization . 375 

Lemniscate Rings 78 

Lepidocrocite 

Analysis 125 

Structure 258 

Light Diffraction Methods in Struc- 
ture Determination 129, 190 f., 

271 ff., 335 ff., 348 ff., 369 f., Pis. V, XIII 

Faulty pattern 369 f. 

Fourier synthesis .... 348 ff. 

General 7, PL XIIJ 

Geometry 129 

Image formation by lens . . 335 ff. 

Patterson synthesis .... 352 

Phase relation 190 f. 

Structure amplitude , . . 271 ff. 

Trial and error method 271 ff., PL XIII 
Lindemann Glass 110 

Powder photograph ... PL IV 
Line Broadening 362 ff. 

Reciprocal lattice .... 367 f 
Line Grating 

Image formation of . 335 ff., PL V 
Line Spectra 

X-rays 105 ff. 

Liquid Crystals 330 

Liquids 

Diffraction 370 f. 

Immersion .... 63 ff., 374 ff. 

Structure 370 ff. 

Supercooled 1 

X-ray identification . . . . 127 
Lithium Borate 

Glass 110, PL IV 

Lithium Ferrite 

Defect structure .... 332 f. 
Lithium Fluoride 

for R.I. liquids standardization . 375 
Lithium Titanate 

Defect structure 333 

Long-Chain Polymers . . . 188 f. 

Chain type 188 f. 

Density 186 f. 

Double orientation . . . 178 ff. 

Electron diffraction .... 373 



412 



SUBJECT INDEX 



Long-Chain Polymers (cent.) 

Orientation 175 ff. 

Polycrystalline 175 ff. 

Strongest X-ray reflections . 261 f. 

Structure determinations . 317 ff. 

Symmetry 248 f. 

X-ray identification . . . . 127 

X-ray photographs . . Pis. IV,, IX 

Lorentz Factor 203 f. 

Luminescent Powders . . . . 126 

Magnesium Iron Oxide 

Defect structure 333 

Magnesium Oxide 

in Portland cement . . . . 124 

Magnetic Properties 

Crystal structure .... 287 ff. 

Maps (Fourier) 

Absolute intensities . . . . 351 
Beovers-Lipson strips . . . 347 f. 
Electron density .... 339 ff. 

False detail 350 

Image formation 338 

Methods of computation . . 347 f. 
Non-cry stalline substances . 370 f. 
Optical synthesis .... 348 ff. 

Patterson 351 ff. 

Patterson-Harker .... 356 ff. 

Resolving power 350 

Strips 347 f. 

Structure determination . . 33t5 ff. 

Mercurous Chloride 

for R.I. liquids standardization . 375 

Metallurgical Identification . . 92 

Metal Specimens 

Crystal size and strain . . 362 ff. 
Irregularity in crystal structure 362 f., 

368 f. 

Oriented 180 

Orowan's X-ray photographs . 171 
Solid solution, q.v. . . 274, 331 f. 

Methyl Bixin 

Birefringence 282 

Methyleno Iodide 

as R.I. liquid 374 

Methyl Urea 

Space-group 245 

Mica 

Cleavage 58 

Layer lattice 279 

Optic picture 78 f. 

Micro -Photometer 

Accurate spacings 120 

Composition mixture , . . . 125 
Intensities of X-ray reflections 193 ff. 
Structure of non-crystalline sub- 
stances 371 

Microscopic Investigations 

Absorption 85 f., 285 f. 

Biaxial crystals . . 73 ff., 96 ff. 
Chemical reactions . . . . 102 

Cleavage 93 

Convergent light 77 ff. 

Crystal shape ... 56, 93, 100 

Crystal structure .... 280 ff. 



Cubic 63 

Dispersion, q.v. . . 65, 77, 83 ff., 

100 f. 

Extinction directions, q.v. 67 ff., 77, 

95 ff., 100 

Identification . . ... . 3, 91 ff. 

Interference colours ... 67 ff. 

Limitations 103 

Mixed crystals 93 

Mixtures 99 ff. 

Nicol prisms 66 ff. 

Opaque crystals . . 91 f., 101 f. 

Optic axial angle, q.v. 75, 77 ff., 80, 

100 f. 

Parallel light 63 ff. 

Ple6chroism 85 ff. 

Refractive index measurement, 

q.v 62 ff. 

Tilting crystals .... 92 f. 

Twiiming 89 f. 

Uniaxial crystals . . . 65 ff., 94 ff. 

Universal stage 92 

Mimetic Twinning 58 

Minerals 

Table of refractive indices . . 93 

Tables of X-ray patterns . . 123 
Mixed Crystals 59 ff. 

Composition 183 f. 

F z -synthesis 361 

Isomorphous crystals, q.v. 59 ff., 126, 

274, 361 

Optical properties 93 

Structure 331 if. 

X-ray patterns .... 126, 367 

ZnS-CdS 126 

Mixtures 

Identification by optical methods 99 ff. 

Identification by X-ray methods 1 23 ff ., 

PI. IV 

Quantitative composition . . . 125 
Modification of Crystal Habit 13, 34, 37, 

53 f. 

Involving change in symmetry 53 f. 
Molecular Weight Determination 

9, 185 ff. 
Molecules 

Birefringence due to orientation 
of . 282 ff. 

Diamagnetism due to . . 288 ff . 

Dimension 251 f. 

Enantiomorphous 227 

Packing 276 

Rotation 291, 329 ff. 

Shapes . . 4 ff., 187 f., 251 f., 277 ff. 

Structure of simple gas . . 372 f. 

Symmetry .... 224 ff., 248 ff. 
Molybdenum 

Structure determination . . . 293 
Molybdenum Sulphide 

Layer lattice 279 

Monobromobenzene 

as R.I. liquid 374 

Monochromatic X-rays . . . 106 ff. 

Structure of non-crystalline sub- 
stances 370 f. 



SUBJECT INDEX 



413 



Monochromators 107 f. 

Monoclinic System (see also Biaxial 
Crysfals) 

Classed 48 ff. 

Dispersion 85 

Interplanar spacings . . 136 f., 156 
Optical properties .... 75 ff. 
Reciprocal cell shape . . . . 153 
Rotation photographs . . 153 if. 
Unit cell and powder data . 136 f. 
Unit cell from rotation photographs 

141 f. 
Unit coll volume .... 185 f. 

Morphology of Crystals . . . 1 1 ff . 

Classes 52 ff. 

Crystal structure . . 13 ff., 278 ff. 

Donnay's law 280 

Goniometry . . . 11 ff., 28 ff. 
Identification . 54 ff., 91, 99 ff., 183 

Index 54*ff., 91 

Molecular shape .... 277 ff. 
Unit cell . 27 f., 30 ff., 52 ff., 139 f. 

Mosaic Crystals (see Ideally Imper- 
fect Crystals) 

Moving-Film X-ray Cameras . 166 ff. 

Cox 169 

De Jong and Bouman . . 1 70 f . 

Robertson 169 

Schiebold and Sauter . . . 169 f. 

Thomas 170 

Unit cell 166 ff., 181 

Weissenberg 167 ff., 181, 193, PL VIII 

Multiplicity of X-ray Reflections 199 f. 

Naphthalene 

Birefringence 283 f. 

Structure 284 

Naphthalene, 1 -Brom, 2 Hydroxy-, 

Crystal symmetry 49 

Naphthalene, a Bromo- 

as R.I. liquid 374 

Newton's Scale of Colours . . 69, 81 
Nickel Phthalocyanine 

Structure 343 f. 

Nickel Sulphate Heptahydrate 

Structure 353 

Nicol Prisms 66 ff. 

Crossed 67 ff. 

Nitrates (see also individual Nitrates) 

Birefringence 282 

Diamagnetism 28$ 

Ionic rotation 330 

Nomenclature 

Classes 44 ff., 384 ff. 

Crystal planes 24 ff. 

Point-groups ... 44 ff., 384 ff. 

Space-groups ... 244 f., 384 ff. 

Space-lattices 223 

Non-Crystalline Substances 

Glasses, q.v. 1, 63 f., 93 f., 110, 127, 
370 ff., 374 

Identification, q.v. . . 93 f., 127 

Structure 370 ff. 

Warren's method 372 

X-ray diffraction .... 370 ff. 



Notation 

Classes .... 44 ff., 384 ff. 

Crystal planes 24 ff. 

Point-groups ... 44 ff ., 384 ff . 

Space-groups ... 244 f., 384 ff. 

Space-lattices 223 

Nylon 

Chain type 189 



Obtuse Bisectrix . 
Opacity of Crystals . 
Opaque Substances 

Optical study . 
Optical Activity 

Classes 

Directions image . 

Enantiomorphism, q.v. 



75 
23 



91 f., 101 f. 
87 ff. 
88 f. 
87 f. 

38 f., 227, 
286 f. 
Solid and solution .... 286 f. 

Space-groups 286 

Optical Diffraction Methods in Struc- 
ture Determination 

Faulty pattern 369 f. 

Fourier synthesis .... 348 ff . 
General 7, PL XIII 



Image formation by lens . 
Patterson synthesis 
Phase relation .... 
Structure amplitude . 
Trial and error method 271 ff 


335 ff. 
. 352 
190 f. 
271 ff. 
, PL XIII 



Optical Properties 3 ff., 62 ff., 91 ff., 

280 ff. 

Absorption 85 f., 285 f. 

Amorphous substances . 63 f., 93 f. 
Anisotropy . 2, 4ff., 65 ff., 82 f., 94 ff., 

280 ff. 

Biaxial 73 ff., 96 ff. 

Convergent light .... 77 ff. 
Crystal structure .... 280 ff . 

Cubic 63, 93 f . 

Dispersion, q.v. 65, 77, 83 ff., 100 f. 
Extinction directions, q.v. 67 ff., 77, 



95 ff., 100 
3 f., 91 ff. 
, 335 ff . 
67 ff., 78 ff. 



Identification . 

Image formation . 

Interference colours 

Mixed crystals 93 

Mixtures 99 ff. 

Nicol prism 66 ff. 

Opaque crystals . . 91 f., 101 f. 

Optic axial angle, q.v. 75, 77 ff., 80, 

100 f. 

Parallel light 63 ff. 

Pleochroism 85 ff. 

Refractive indices, q.v. . 62 ff. 

Symmetry 62 ff., 70 ff . 

Twinned crystals .... 89 f. 

Unaxial crystals . . 65 ff., 94 ff. 

Universal stage 92 

Optic Axial Angle . . . 75, 77 ff. 

Determination from refractive 

indices 80 

Identification of mixtures . 100 f. 
Optic Axis 72, 74 



414 



SUBJECT INDEX 



Optic Picture 77 ft. 

Bertrand lens 78 

Identification 95, 99 

Optical activity . . , . 87 f. 

Optic axial angle, q.v. 75, 77 ff., 100 f. 

Optic sign 83 

Optic Sign 

Biaxial 75, 83 

Directions image 83 

Quartz wedge 80 ff. 

Uniaxial 72, 81, 83 

Order of Reflection . . . 114ff., 131 

Absent, q.v. 153, 210, 216 ff., 234 ff., 

240 

Zero 339 f., 351 

Oriented Overgrowths .... 60 
Oriented Specimens (Polycrystal- 

line) 175 ff. 

Doubly 178ff. 

Metal 180 

Unit cell determination . . 176 ff. 

X-ray photographs . . . 176 ff. 
Orowan's X-ray Photographs . 171 f. 
Orthorhombic System (see also Biaxial 
Crystals) 

Classes 49 

Dispersion 84 f. 

Indexing rotation photographs 151 ff. 

Interpjariar spacings 

136 f., 142 f., 376, 382 f. 

Optical properties .... 73 ff. 

Unit cell and powder data 136 f., 376 

Unit ceil from rotation photographs 

139, 142 ff., 382 f. 
Oscillation X-ray Photographs 1 58 f. 

Indexing 159 ff. 

Tilted crystal 164 ff. 

Oxalic Acid 

Birefringence 283 

Crystal symmetry 49 

Packing of Atoms 276 

Paraffin Wax 

Identification 183 

Parameters (Atomic) . . . 190 ff. 

Adjustment 264 ff. 

Ambiguities 275 f. 

Determination . . . 256 ff., 292 ff. 

Refinement 344 ff. 

Structure amplitude, q.v. . 257 ff. 

Pattern Unit 118 

Patterson Synthesis . . . . 351 ff. 

by Optical method .... 352 
Patterson-Harker Synthesis . 356 ff. 

Pauling's Rules 276 

Pentaerythritol 

Cleavage 59 

Crystal habit 59 

Diffraction symmetry . . 241 f. 

Laue photograph . . 243, PI. XII 

Monochromators 107 

Pentaerythritol Tetra- Acetate 

Three-dimensional synthesis . . 347 
Perfect Crystal .... 203, 208 f. 
Periclase (see Magnesium Oxide) 



Perspex (see Polymethylmethacrylate) 
Petrological Methods , .... 101 
Phase Angles (see also Structure 
Amplitude A and B terms) 1 

Equal 340 ff. 

Experimental determination . 343 f. 

Non-centro-symmetrical structures 347 

Optical image 338 

X-ray reflections .... 340 ff. 
Phase Relation m X-ray Diffraction 

115 ff., 190 ff., 197 ff., 210 ff., 340 ff. 
Phenanthrene, 1, 2 Dimethyl - 

Space -group 246 

Phloroglucinol Diethyl Ether 

Crystal symmetry 51 

Photographic Film 

Shrinkage 121 

Photometry . . . 120, 193 ff. 

Accurate spacings 120 

Composition mixture . . . . 125 

Intensities of X-ray reflections 193 ff. 

Structure of non-crystalline sub- 
stances 371 

Phthalocyanine 

Nickel 343 f. 

Optical diffraction ... PL XIII 

Phase angles 343 f. 

Platinum, q.v 254, 342 f. 

Structure amplitudes . 272, PL XIII 
Physical Properties 

Crystal structure .... 278 ff. 

Identification . . . 54 ff., 91 ff. 

Optical, q.v. 3 ff., 62 ff., 91 ff., 280 'ff. 
Picric Acid 

Crystal symmetry 49 

Picryl Iodide 

Determination of I positions 353 f. 

Structure 345, 353 ff. 

Piezo-Electricity 

Crystal structure 290 

Pigments 

Colours 363 

Piperine 

for R.I. measurements. . . . 374 

Plane -groups 264 ff. 

Plane of Vibration 66 

Rotation of 87 ff. 

Plane Polarized Light 

Pleochroism . . . 85 ff., 285 f. 

Production 66, 86 f. 

Rotation of Plane .... 87 ff. 
tlanes of Symmetry . . 36, 43 ff. 

Diagonal glide 239 

Effect on ^-synthesis . . 356 ff . 

Glide, q.v. ... 43, 229 ff,, 273 f. 

Quarter glide 234 

Plaster of Paris (see Calcium Sulphate 

Subhydrate) 
Platinum Phthalocyanine 

Direct structure determination 342 f. 

Position of Pt atoms . . 254, 342 f. 
Pleochroism .... 85 ff. 

Crystal structure .... 286 f. 
Point-Groups (see also Classes of Sym- 
metry) 44 ff., 224 



SUBJECT INDEX 



415 



Point-Groups (cont.) 

from Morphological data . 52 ff. 

Nomenclature . . 44 ff., 384 ff. 
Optical activity .... 88 f. 
Relation to diffraction symmetry 

241 ff. 

Relation to molecular symmetry 228 f. 
Relation to space-lattices . . 228 f. 

Symmetry 44 if . 

Polar Axes of Symmetry* . . 40 f. 

Polarizability 281 

Polarization 

Atoms 281 

Light 66, 86 f. 

Rotary 87 ff., 286 

X-rays 203 

Polarizer 67 

Polaroid 87 

Polyamides 

Nylon chain type 189 

Orientation 176 

Polyatomic Ions 

Orientation in crystal structures 

5 f., 280 ff. 

Packing 276 

.Polarizability 281 

Rotation 329 f!. 

Symmetry .... 225 f., 248 ff. 
Polychloroprene 

Structure determination . . . 268 
Polycrystalline Materials 
Crystal size and strain . . 362 ff. 
Double orientation . . . 178 ff. 
Fibre photographs, q.v. 

176 ff., 188 f., 317 ff., PL IX 

Metal 180 

Orientation. . . . 175f., 178 ff. 

Polymers, q.v. . . 127, 175 ff., 248 f., 

261 f., 317 ff., 373. 

Unit cell determination . . 176 ff. 
X-ray photographs . . . 176 ff. 
Polyene Chains 

Birefringence 282 

Light absorption ..... 286 
Polyesters 

Orientation 176 

Strongest reflections . . . . 261 
Polyethylene 

Molecular symmetry . . . 230 f. 

Orientation 176 

Strongest reflections . . . . 261 
Symmetry of structure . . 231 ff. 
Three-dimensional synthesis . . 347 
Polyhexamethylene Adipamide 

Chain type 189 

Polyisobutene 

Fibre photograph . . . . PI. IX 
Polymers 

Chain type 188 f. 

Density 186 f. 

Double orientation . . . 178 ff. 
Electron diffraction .... 373 

Orientation 175 ff. 

Polycrystalline 175 ff. 

Strongest X-ray reflections . 261 f. 



Structure determinations . . 317 ff. 

Symmetry 248 f. 

X-ray identification .... 127 
X-ray photographs. . . Pis, IV, IX 

Polymethylmethacrylate 

X-ray pattern .... 127, PL IV 

Polymorphism 59 

Polystyrene 

X-ray pattern .... 127, PL IV 

Polysynthetic Twinning (see Repeated 
Twinning) 

Polythene (see Polyethylene) 

Polyvinyl Alcohol 

Chain type 188 f. 

Polyvinyl Chloride 

Chain type 189 

Polyvinylidene Chloride 

Fibre photograph .... PL IX 

Portland Cement 

Identification of constituents 124 

Potassium Bromate 

Crystal symmetry 50 

Potassium Bromide 

for R.I. liquids standardization . 375 

Potassium Chlorate 

Birefringence .... 283, 285 

Magnetic ellipsoid 289 

Structure 285 

Potassium Chloride 

Powder photograph . . 220, PL XI 
for R.I. liquids standardization . 375 
Structure .... 220 f., PL XI 

Potassium Cyanide 

Ionic rotation 330 

for R.I. liquids standardization . 376 

Potassium Dithionate 

Crystal symmetry .... 41 f. 
Structure 275 

Potassium Ferricyanide 

Pleochroism 85 f. 

Potassium Fluoride 

for R.I. liquids standardization . 375 

Potassium Iodide 

for R.I. liquids standardization . 375 

Potassium Mercuric Iodide 

for R.I. measurements . . 374 f. 

Potassium Nitrate 

Birefringence 282 

Optical properties 95 

Rotation photograph . .138, PL VI 

Potassium Sulphamate 

Structure 359 

Potassium Sulphate 

Isomorphism with (NH 4 ) 2 SO 4 59, 331 

Powder X-ray Cameras ... 109 ff. 

Focusing 181 

Hydrogen filled 123 

Large radius 124 

Seeman-Bohlin 181 

Vacuum 123 f. 

Powder X-ray Photographs 

103, 106, 108 ff., Pis. Ill, IV, X, XI, 
Absorption, q.v. 

106 f., 120 ff., 132, 206 f. 
A.S.T.M. index .... 122 f. 



416 



SUBJECT INDEX 



Powder X-ray Photographs (con/.) 
Axial ratio . . 132 ff., 376 ff., 379 ff. 

Background 113 

Centred cubic lattices . . 219 f. 
Choice of wave-lengths 113 f., 123 

Crystal size 127, 362 ff. 

Cubic 129 ff. 

Filters 106 f., 113 

Identification . 6, 8, 103 ff., 122 ff. 
Indexing 129 ff., 376 ff., 379 ff., PL X 
Intensities of reflections 192 ff ., 207 ft. 
Interplanar spacings, q.v. 

119ff., 376 ff., 379 ff. 
Limitations in identification . 125 f. 
Line broadening .... 362 ff* 

Measurement 119 ff. 

Mixed crystals . 126, 183 f., 367 

Number of equivalent reflections 199 f. 
Photometry, q.v. 120. 125, 193 ff., 371 
Solid solutions .... 126, 367 
TJnit cell 129 ff., 180 ff., 376 ff., 379 ff. 

Primary Extinction .... 208 f. 

Primitive Cell (see also Simple Cell) 223 

Principal Refractive Indices 72, 74 ff. 

Principle of Simplest Indices . 32 f. 

Principle of Staggered Bonds . . 277 

Projections 

Reciprocal lattice, q.v. 

144 ff., 367 -f. 
Stereographic . . . . 29 ff., 45 ff. 

Pseudo-Symmetry 

Space-group 273 f. 

Twinning 58 

Pyrites 

Crystal symmetry 51 

Pyro-Electricity 

Crystal structure .... 289 f. 

Pyrrhotite 

Defect structure 333 

Quarter Glide Planes .... 234 

Quartz 

Optical activity 88 

Optical properties 72 

Powder photograph . . . PI. Ill 
Wedge 80 ff., 95, 99 

Quaterphenyl 

One-dimensional synthesis . . 347 

p-Quinone 

Crystal symmetry 41 

Quinonoid Groups 

Light absorption 286 

Rational Indices 

Law of . . . . . . 27 f., 31 ff. 

Reactions (Chemical) 

on Microscope slide . . . . 102 

Reciprocal Lattice . . 144 ff., 383 
Doubly oriented polycrystalline 

specimens 179 f. 

Indexing fibre photographs . 176 ff. 
Indexing moving-film photographs 

166 ff. 
Indexing oscillation photographs 

159 ff. 



Indexing rotation photographs 151 ff. 

Indexing tilted crystal photographs 

. 163 ff. 

Line broadening .... 367 f. 

Monoclinic . . . . . . 153 

Point, shape of . . " . 206, 368 

Proof 383 

Rhombohedral .... 172 f. 

Simplest unit cell . . . 172 f. 

Triclinic . * 156 f. 

Reflecting Sphere .... 146 ff. 
Reflection Planes (see Planes of Sym- 
metry) 
Reflections of X-rays . . . 108 ff. 

Absent, q.v.. . . 163, 210, 216 ff. 

Atomic planes 114 ff. 

Bragg'slaw l)4ff. 

Broadened 362 ff. 

. Ideally imperfect crystals . . 208 

Intensity, q.v. ... 122 ff., 192 ff. 

Microphotometry, q.v. 

120, 125, 193 ff., 371 

Number of equivalent . . 199 f. 

Order of 114ff., 131 

Perfect crystals 208 

Spacings, q.v. . . . 114ff., 376 if. 

Structure amplitude, q.v. 

196 ff., 209 ff., 292 ff. 
Refractive Index 62 ff. 

Anisotropy 2 ff ., 65 ff ., 82 f., 94 ff., 280 ff. 

Becke line 64 ff. 

Biaxial crystals . 73 ff., 96 ft. 



Birefringence 
Calculation . 
Crystal structure 
Cubic crystals . 
Glasses . 
Identification . 
Indicatrix, q.v. 
Liquids . 

Measurement 3 f 
Optic axial angle 
Organic substances 
Polyatomic ions 
Principal 
Symmetry 



65 ff., 94 ff. 
, . . 280 ff. 
. . . 280 ff. 
63 ff., 93 f. 
63 f., 93 f., 374 
, . 3 f., 91 ff. 
70 ff., 283 ff. 
63 ff., 374 ff. 
t ff., 93 ff., PL II 
, . . . 80 
. 98, 282 ff. 
. . 281 ff. 
72, 74 ff. 
62 ff., 70 ff. 



Tables 93, 98 

Uniaxial crystals . . . 65 ff., 94 ff. 
X-ray 128 

Refractometer 375 

Refractories 

Identification of constituents . 101 

Relative Intensities 122, 125, 192 ff., 207 
Absent, q.v. 

153, 210, 216 ff., 234 ff., 245 ff. 
Absorption correction . . . 206 f. 

Background 113 

Calculated and observed . 191 f., 273 
Calculation . . 195 ff., 209 ff., 264 ff. 
Characteristic of substance . . 122 
Chemical composition . . . . 126 
Correction factors, q.v. . 199 ff., 350 
Diffracting power of atoms, q.v. 

126, 196, 200 ff. 
Expressions for 207 



SUBJECT INDEX 



417 



Relative Intensities (cant.) 

for Fourier synthesis .... 351 

Measmement 192 ff. 

Scattered 113 

Structure amplitude 

196 ff., 209 ff., 257 ff. 
Trial and error method, q.v. 

7 f., 259 ff., 271 ff., 292 ff. 

Relative Retardation (see Birefringence) 

Repeated Twinning 58 

Repeat Period 

Fibre 176, 188 f. 

Layer lines, q.v. 138 f., 142 f., 147, 382 f. 
Rotation photographs . 138 f., 147 

Resolution 

a-doublet 120 

Atoms in electron density map . 360 
Peaks on vector maps . . 352 f. 
Variation with 6 132 

Resorcinol 

Structure * . . 347 

Reticular Density .... 20, 58 

Rhombic System (see Orthbrhombic 
System and Biaxial Crystals) 

Rhombohedral Cell 134 f., 141, 172 f. 
Indices . . 135, 172 f., 378 

Reciprocal cell .... 172 f. 

Rhombohedral System (see Trigonal 
System and Uniaxial Crystals) 

Rinneite 

Dispersion S3 f. 

Robertson Moving -Film X-ray Photo- 
graphs 169 

Robertson Strips 348 

Rock Slices 

Identification of constituents in thin 
sections 101 

Rotary Inversion Axes . . . . 42 f. 

Rotation Axes. ... 35 ff., 43 ff. 

Rotation of Molecules . "291, 329 ff. 

Rotation of Plane of Polarization 87 ff. 
Space-groups 286 

Rotation X-ray Cameras . , 137 ff. 

Rotation X-ray Photographs 

137 ff., PL VI 

Bernal chart 149 ff. 

Extra spots 206 

Identification 182 f. 

Indexing . , 142 ff., 151 ff., 382 
Intensities of reflections 192 ff ., 207 ff. 
Layer lines, q.v. . 138 f., 142 f., 147 

Monoclinic 153 ff. 

Moving-film, q.v 166 ff. 

Number of equivalent reflections 200 f. 

Row lines 152 

Setting 173 ff. 

Streaks 206 

TUted crystal 162 ff. 

Triclinio 157 f. 

Unit cell ... 139 ff., 181 f., 382 

Row Lines 152 

Rubber 
Double orientation . . . . 178 

Orientation 176 

Structure 262 f. 



Rubber Hydrochloride 

Chain type 189 

Structure determination . . 323 ff. 

Rutile 

Powder photograph ... PI. X 

Structure 210 ff. 

Structure determination . . 296 ff. 



Scattering Power of Atoms (see Dif- 
fracting Power) 
Scattering of X-rays . . . . .113 

Compton 371 

by Electrons 118 

Schiebold-Sauter Moving-Film X-ray 

Photographs 169 f. 

Schoenflies Symbols .... 384 ff. 
Screw Axes .... 43 ff., 230 ff. 

Absences due to . 234 ff. 

Effect on jP 2 -synthesis . 356 ff. 

Effect on X-ray patterns . 234 ff . 

. Pseudo .... .273 f. 

Secondary Extinction * . . 209 

Seoman-Bohlin X-ray Cameras . 181 

Selenium 

for R.I. measurements . . . 374 
Setting X-ray Photographs . 173 ff. 
Shape of Crystals .... 11 ff. 

Classes 52 ff. 

Crystal structure . . 13 ff., 278 ff. 

Donnay'slaw 280 

Effect of growth conditions 

12 ff., 21, 31, 40 

Goniometry . . . 11 ff., 28 ff. 

Identification . . 54 fi., 91, 99 ff. 

Index 64 ff., 91 

Microscopic investigation 56, 93, 99 f. 

Misleading 247 

Modification . . 13, 34, 37, 53 f. 

Molecular shape .... 277 ff. 

Morphology, q.v. 1 1 ff., 52 ff., 91, 278 ff. 

Unit cell 27 f., 30 ff., 62 ff., 139 f. 

Shape of Molecules 

4 ff., 187 f., 251 f., 277 ff. 
Short Wave-Length Limit ... - 105 

Side-Centring 223 

Sign (Optic) 

Biaxial 75,83 

Directions image 83 

Quartz wedge 80 ff. 

Uniaxial 72, 81, 83 

Silica 

Quartz, q.v. 72, 80 ff., 88, 95, 99 t 

PL in 

Silicate Glass . . - 127 

Silk 

Orientation 1*75 

Silver Iodide 

Defect structure 334 

Simple Cell 217 

Simplest Cell 172 f. 

Simplest Indices 32 f. 

Single-Crystal Osculation X-ray Photo- 
graphs 158 ff. 

Indexing 159 ff. 

Tilted crystal 164 f. 



4458 



E e 



418 



SUBJECT INDEX 



Single-Crystal Rotation X-ray Photo- 
graphs .... 137 ff., PL VI 

Bernal chart 149 ff. 

Extra spots 206 

Identification 182 ,f. 

Indexing ... 142 if., 151 ff., 382 
Intensities of reflections . . 192 ff., 

207 ff. 
Layer lines, q.v. . 138 f., 142 f., 147 

Monoclinic 153 ff. 

Moving-film, q.v 166 ff. 

Number of equivalent reflections 200 f. 

Bow Lines 152 

Setting 173 ff. 

Streaks 206 

Tilted crystal 162 ff. 

Triclinic 157 f. 

Unit cell . . . 139 ff., 181 f., 382 

Skeletal Growths ... 23, PI. I 

Slit (X-ray) 

Height correction 121 

Use of reduced size of . . . . 124 

Snell's Law 62 

Sodium Azide 

Birefringence 283 

Sodium Bicarbonate 
Birefringence .... 283, 308 
Structure determination . . 306 ff . 

Sodium Bromate 

Optical properties . . . 100, PL II 
for R.I. liquids standardization . 375 

Sodium Bromide Dihydrate 

Optical properties . . . 100, PL II 

Sodium Carbonate Decahydrate 

Dispersion 77 

Sodium Carbonate Monohydrate 

Optical properties . . . 73 f., 97 

Sodium Chlorate 

Crystal habit .... 12, 37, 53 
Enantiomorphism ... 38 f., 287 

Growth 20 f. 

Modification of habit ... 37, 53 

Optical activity 88 

Refractive index .... PL II 
for R.I. liquids standardization . 375 

Structure 37 ff. 

Symmetry 37 ff. 

Sodium Chloride 

Absolute intensities . . . . 193 

Cleavage 58 

Crystal habit 12 

Detection in mixture . . . . 126 
Layer formation .... PL I 
Modification of habit ... J3, 34 

Monochromators 107 

Powder photograph 130, 219 f., PL XI 
for R.I. liquids standardization . 375 
Standard for 'cf measurements . 122 
Structure . . . 34, 220, PL XI 

Symmetry 35 ff. 

Thermal vibration 205 

Sodium Ferrite 

Powder photograph ... PL IV 

Sodium Fluoride 

for R.I. liquids standardization . 375 



Sodium Hydroxide 

Powder photograph ... PL IV 
Sodium Metaperiodate Trihydrato 

Symmetry 40 

Sodium Nitrate 

for R.I. liquids standardization . 375 

Sodium Nitrite 

Birefringence .... 283, 306 

Crystal symmetry 53 

Space-group symmetry . . . 256 

Structure 260 

Structure determination , . 303 ff. 

Sodium Sulphite 

Powder photograph . .. . PL III 

Sodiufm Thiosulphate Pentahydrate 

Dispersion . * 77 

Solid solutions 59 f. 

Composition 183 f. 

4 .F 2 -cynthesis 361 

Isomorphous crystals, q.v. 

59 ff., 126, 274, 361 

Optical properties 93 

Structure 331 ff. 

X-ray patterns .... 126, 367 
ZnS-CdS 126 

Space-Groups 

43 f., 217, 224, 244 ff., 384 ff. 
Absences due to 216 ff., 235 ff., 245 ff. 
Deduction . . ... 245 ff. 

Donnay'e law 280 

Equivalent positions . . . 252 ff. 
General positions .... 252 f. 

List 384 ff. 

Molecular dimensions . . . 251 f. 
Molecular symmetry . . . 248 ff . 
Multiplicity of atomic positions 252 ff . 

Optical activity 286 

Plane-groups 264 ff. 

Schoenflies symbols . . . 384 ff. 
Special positions .... 253 ff. 
Structure amplitude . . . 257 ff. 
Symbols 244 f. 

Space-Lattices. . 118, 216 ff., PL XI 

List 222 

Nomenclature ...... 223 

Relation to point-groups . 228 f. 

Spacings of Crystal Planes 

114 ff., 376 ff., 379 ff. 
Absorption correction . . 120 ff., 132 

Bragg's law 114 ff. 

Calculation . . . 114, 376 ff. 
Correction factors . . . 120 ff., 132 
Cubic . . . 129 f., 377, 379 ff. 
Hexagonal . . . 134 f., 377 f., 381 

Lower limit 118 

Monoclinic 136 f., 166 

Orthorhombic 136 f., 142 f., 376, 382 
Powder photographs . 119 ff., 379 ff. 
Reciprocal lattice, q.v. .' . 144 ff. 
Rhombohedral ... 134 f., 378 
Rotation photographs . . 142 ff." 

Slit correction 121 

Tetragonal . . . 132, 376, 379 ff. 
Triclinic .... 136 f., 156 ff. 



SUBJECT INDEX 



419 



Spacings of Crystal Planes (con*.) 

Trigonal . , . 134 f., 378, 381 

Unit cell dimensions . . . 376 if. 

Special Positions 253 if. 

Specimen Mountings for X-ray Photo- 
graphs * 110 

Sphere of Reflection . . . . 146 if. 
Spinel 

Crystal symmetry 51 

Defect structures . .* . . . 333 
Staggered Bonds Principle . . . 277 
Stannic Iodide 

as Solute in R.I.' liquids . . . . 374 

Symmetry 228 

Stannic Oxide 

Crystal symmetry * 40 

Stereographic Projections . 29 if . 

Classes , 45 ff . 

Sterols 

Identification .'183 

Molecular dimensions .... 252 
Stilbene 

Asymmetric structure unit . 249 f. 

Structure ..".... 345 f. 

Streaks 206, 368 f. 

Strips (Fourier) 347 f. 

Strontium Formate Dihydrate 

Crystal symmetry 49 

Structure Amplitude 

196 ff., 209 if., 264 if., 292 if. 

A and B terms 212 

Calculation . 19$ if., 209 if. , 264 if. 

'Charts 264 if . 

Equation , 212 

Graphical methods . . . 264 if. 

Machines 270 f. 

Optical method . 271 if., PI. XIII 

Space-group 257 if. 

Triclinic 269 f. 

Structure Factor (see Structure Ampli- 
tude) 
Strychnine Sulphate Periodide 

Light absorption 87 

Sulphur 

Polymorphism 59 

as Solute in R.I. liquids . . . 374 

Structure (orthorhombic) . . . 372 

Superlattices 332 

Symmetry 33 ff. 

Atomic arrangements . . 224 ff . 

Axes, q.v. 35 ff., 42 ff., 230 ff., 256 if. 

Centre, q.v. . 36 f., 242 f., 289 f. 

Classes 33 ff., 44 if., 88 f., 228 f., 241 if. 

Diffraction 241 if. 

Effect on ^-synthesis . . 356 ff . 

Elements 34 if . 

Glide planes, q.v. 43 ff., 229 ff., 273 f. 

Intensity of X-ray reflections. . 126 

Inversion axes . . . . 42 f., 234 f. 

Ions 224 ff ., 248 if. 

Laue 243 f. 

Modification of external . . 53 f. 

Molecules .... 224 ff., 248 ff . 

Optical properties . , . 62 ff., 70 ff . 

Physical properties ; 62 



Polyatomic ions . . 225 f., 248 ff. 

Polymers 248 f. 

Pseudo 58, 273 f. 

Refractive indices . . . 62 ff., 70 ff. 

Screw axes, q.v. 43 ff., 230 if,, 273 f. 

Space-groups, q.v. 43 ff., 217, 224, 244 if. 

Systems, q.v. ... 33 f., 47 ff ., 52 

Translational ... 43 ff., 229 ff. 

Unit cell 28, 217 

Systematic Absences 

153, 210, 216 fl. f 234 ff. 

in Body-centred cells . . 21 7 ff., 222. 

in Face-centred cells . . 18 f., 222 f. 

Glide planes 234 if. 

Lattice 216ff. 

Screw axes 234 ff. 

Space -group . . . 224 ff., 245 ff. 

Tables 240 

Systems of Symmetry 33 f., 47 ff., 52 
'Cubic 50 ff. 

Hexagonal 49 f. 

Monoclinic 48 ff. 

Orthorhombic 49 

Tetragonal 50 

Triclinic 47 f. 

Trigonal 49 f. 



Tables (see also Index) 
Absent reflections . 



Crystal shapes . 
Refractive indices . 



. . . 240 
54 ff ., 91 
93, 98 

Spare-groups 384 ff. 

Unit cell dimensions cubic sub- 
stances ...*.... 182 

X-ray patterns minerals . . . 123 

X-ray powder patterns . . 122 ff. 
Tartaric Acid 

Crystal symmetry 48 

Temperature Factor .... 205 f. 

Artificial 350 

Tesseral System (see also Cubic Sys- 
tem) 50 ff. 

Tetragonal System (set also Uniaxial 
Crystals) 

Classes 50 

Dispersion 83 f. 

Interplanar spacings 132, 376, 379 ff. 

Optical properties .... 65 ff. 

Unit cell from powder data 

132 ff., 376, 380 ff. 

Unit cell from rotation photographs 140 
Tetrahedrite 

Crystal symmetry ..... 51 
1, 2, 4, 5 Tetramethyl Benzene 

Symmetry 231 

Systematic absences . . . 237 ff. 
Thallium Hal ides 

for R.I. measurement . . . 374 
Thermal Expansion 

Crystal structure .... 290 f. 
Thermal Spots .... 206,368 
Thermal Vibrations .... 204 ff. 

Anisotropy 206 

Intensity correction . . . 205 f. 

Reflection broadening . . 362 ff., 368 



EC2 



420 



SUBJECT INDEX 



Thin-Section Methods of Identification 101 

Third Mean Line 75 

Tilted Crystal X-Ray Photographs 

162 ff., PI. VII 
Tilting of Crystals on Microscope Stage 

92 f. 
Tin Dioxide 

Crystal symmetry 40 

Tin Tetra-lodide 

as Solute in B.I. liquids . . . 374 

Symmetry 228 

Titanium Dioxide 

Anatase 51 

Brookite 84 f. 

Rutile, q.v. . . 210 ff., 296 ff., PI. X 
Tourmaline 

Absorption of light 86 

Trans- Azobenzene 

Crystal structure 250 

Trans 1, 2 Dichloro Ethylene 

Molecular symmetry . . . 224 f. 
Translational Elements of Symmetry 

43 ff., 229 ff. 

Trial and Error Method orCrystal 
Structure Determination 

7 f., 190 ff., 259 ff. 

Comparison with Fourier methods 350 f . 

Examples 291 ff. 

Non-cry stalUno substances . . 371 

Optical method . 271 ff., PI. XIII 
2, 4, 6 Tribromobenzonitrile 

Crystal symmetry 48 

Triclinic System 

Classes 47 f. 

Dispersion .... 85 

Interplanar spacings . 136, 158 

Optical properties . , 77 

Reciprocal cell shape . . 157 f. 

Rotation photographs . . 158 

Structure amplitudes . . 269 f. 

Unit cell and powder data . . 136 

Unit cell from rotation photographs 

156 ff. 

Unit cell volume . . . . 185 f. 
Trigonal System (see also Uniaxial 
Crystals) 

Classes 49 f. 

Dispersion 83 f. 

Interplanar spacings 134 f., 378, 381 

Optical properties .... 65 ff. 

Unit cell from powder data 134 f., 381 
1, 3, 5 Triphenyl Benzene 

Birefringence 283 

Twinning 57 f., 89 f. 

Optical properties 89 f. 

Two-Crystal Weissenberg Goniometer 193 



Uniaxial Crystals . . . . 65 ff., 94 ff. 

Dichroism 85 ff. 

Dispersion 83 f. 

Identification 94 ff. 

Indicatrix 70 ff. 

Optic pictures 78 f. 

Optic sign 72, 81, 83 



Unit Cell 

Accurate dimensions 
Axial ratio . 
Body-centred 
Chain type . 
Dimensions . 
Face-centred 
Fibre photographs . 



. . . 180 f, 
31 rf., 379 ff. 
. 18, 217 ff., 222 
. . . 188 f. 
31 ff., 128 ff., 182 
17 f., 218 ff., 222 f, 
176 ff. 



Geometry of diffraction pattern 128 f. 

Hexagonal ., . 134 f., 140, 377 f., 381 

Identification .... 123, 181 ff. 

Interplanar spacings . . . 376 ff. 

Low symmetry . ' . 136 ff., 141 ff. 

Mixed crystals 183 f. 

Molecular weight . . . . 185 ff. 

Monoclinic 141 

Morphology 27 f., 30 ff., 52 ff., 139 f. 

Moving -film photographs, q.v. 

166 ff., 181 

Optfcal properties .... 63 ff. 

Oriented polycrystalline specimens 

176 ff. 

Orthorhombic 

136 f., 139, 142 ff., 376, 382 f. 

Oscillation photographs . . 158 ff. 

Powder photographs 

129 ff., 180 ff., 376 ff., 379 ff. 

Pseudo 273 f. 

Rhombohedral 

134 f., 141, 172 f., 378 

Rotation photographs 

139 ff., 181f., 382 f. 

Shape of molecules . . . 187 f, 

Shapes . 15, 17 f., 30 ff., 47 ff., 52 ff . 

Simple and compound . 17, 216 ff. 

Simplest 172 f. 

Tetragonal . 132 ff., 140, 376, 379 ff. 

Tilted crystal 162 ff. 

Trigonal 134 f., 381 

Types . . 15, 17 f., 47 ff., 52, 216 ff. 
Unit of Pattern .... 118,224 

Universal Stage 92 

Urea 

Birefringence 283 

Diffraction symmetry . . 241 f. 

Molecular symmetry . . . 225 f. 

Optical properties 81 

Oriented overgrowths on NH 4 C1 . 60 

Powder photograph . . . PI. X 

Structure determination . . 299 ff. 

Symmetry 42 f. 

Urea, Methyl- 

Space-group 245 

Urea Nitrate 

Monochromator . . . . , 107 

Vacuum Cameras . . . 123 f., 370 

Vaterite 59 

Analysis 125 

Birefringence 285 

Structure 285 

Vector Maps 
Non-crystalline substances . 370 f. 

Patterson 351 ff. 

Patterson-Harker .... 356 ff. 



SUBJECT INDEX 



421 



Vibration Direction . . . 63, 65ff. 
Extinction directions . . 67 ff., 77 
Identification . , . 95 ff., 100 f. 
Refractive index measurement 

66 f., 73 ff. 

Vitamin B 4 

Identification 183 

Vitamin C 

Constitution 317 

Molecules in unit cell ... .251 

Pseudo -symmetry 273 

Structure determination . . 315 ff. 

Vivianite 

Pleochroism 86 

Warren's Method 

Structure determination . . 371 f. 

Wave-Lengths of X-rays . . 105 ft. 

Atomic number 108 

Bragg's law 114ff. 

Choice 113f., 123 

K-series 105 ff. 

Powder patterns 114 

Resolution KOL doublet . . . 120 
Resolving power atoms . . . 350 

Short limit 105 

TJse of long A's for identification . 123 

Wax (Paraffin) 

Identification 183 

Weissenberg Moving-Film X-ray 

Photographs . . 167 ff., PL VIII 

Equi -inclination 169 

Two-crystal 193 

Unit cell 181 

White X-Radiation .... 105 f. 
Background .... 113, 370 
Laue photographs .... 243 

Removal 107 

Setting photographs . . . . 173 

Width of X-ray Reflections . . 362 ff. 

Wooster-Martin Two -Circle Gonio- 
meter 193 

Wulfenite 

Crystal symmetry 51 

{-Values 148 

Bernal chart 149 ff. 

Moving-film methods . . . 167 ff. 

Rotation photographs . . 148 ff. 

Tilted crystal method ... 163 ff. 
X-ray Emission Spectra . . . 105 ff . 
X-ray Fibre Photographs 176 ff., 318 

Indexing 176 

Interpretation 317 ff. 

Repeat period .... 176, 188 f. 

X-ray Filters 106 f., 113 

X-ray Fluorescence . . . . 113f. 

Effect on intensities .... 203 
X-ray Goniometers 138 

Moving-film, q.v 166 ff. 

X-ray Intensities 

Absent, q.v. 

153, 210, 216 ff., 234 ff., 245 ff. 

Absolute, q.v. 

192 f., 207 ff., 351, 370 f. 



Absorption correction . . . 206 f . 

Background 113 

Calculated and observed . 119 f., 275 
Calculation . . 195 ff., 209 ff., 264 ff. 
Characteristic of substance . . 122 
Chemical composition . . . . 125 
Correction factors, q.v. . 199 ff., 350 
Diffracting power of atoms, q.v. 

126, 196, 200 ff. 

Expressions 207 f. 

for Fourier synthesis .... 351 

Integrated 192, 194 f. 

Measurement 192 ff. 

Relative, q.v. 122, 125, 192 ff., 207 

Scattered 113 

Structure amplitude 

196 ff ., 209 ff., 257 ff. 
Trial and error method, q.v. 

7f., 269 ff., 271 ff., 292 ff. 

X-ray Laue Photographs . . . 243 

X-ray Oscillation Photographs . 158 ff. 

Indexing 159 ff. 

Tilted crystal 164ff. 

X-ray Powder Cameras . . . 109ff. 

Focusing 181 

Hydrogen filled 123 

Large radius 124 

Seeman-Bohlin 181 

Vacuum 123 f. 

X-ray Powder Photographs 

103, 106, 108 ff., Pis. Ill, IV, 
X, XI 
Absorption, q.v. 

106 f., 120 ff., 132, 206 f. 
A.S.T.M. index .... 122 f. 
Axial ratio . . 132 ff., 376 ff., 379 ff. 

Background 113 

Centred cubic lattices . . . 219 f. 
Choice of wave-lengths . 113f., 123 

Crystal size 127, 362 ff . 

Cubic 129 ff. 

Filters 106 f., 113 

Identification . 6, 8, 103 ff., 122 ff. 
Indexing 129 ff., 376 ff., 379 ff., PL X 
Intensities of reflections 192 ff., 207 ff. 
Interplanar spacings, q.v. 

119 ff., 376 ff., 379 ff. 
Limitations in identification . 125 f. 
Line broadening .... 362 ff. 

Measurement 119ff. 

Mixed crystals . . 126, 183 f., 367 
Number of equivalent reflections 199 f. 
Photometry, q.v. 120, 125, 193 ff., 371 
Solid solutions .... 126, 367 
Unit cell 129 if., 180 ff., 376 ff., 379 ff. 

X-ray Rotation Photographs 

137 ff., PL VI 

Bernal chart 149 ff. 

Extra spots 206 

Identification 182 f. 

Indexing . . 142 ff., 151 ff., 382 f. 
Intensities of reflections 192 ff., 207 ff. 
Layer lines, q.v. . 138 f., 142 f., 147 

Monoclinic 153 ff . 

Moving-film, q.v 166 ff. 



422 



SUBJECT INDEX 



X-ray Rotation Photographs (con*.) 

Number of equivalent reflections 200 f. 

Row lines 152 

Setting 173 ff. 

Streaks 206 

Tilted crystal I62ff. 

Triclinic 157 f. 

Unit cell . . 139 ff., 181 f., 382 f. 

X-ray Tubes 103 ff. 

High power ...... 108 

Xylene 

as R.I. liquid 374 

Zeolites 

Defect structures 334 



Zero-Order Diffraction . . 339 f., 351 
{-Values ....<.... 148 

Bernal chart ' 149 ff. 

Rotation photographs . . 148 ff. 

Tilted crystal method . . 163, 166 
Zinc Oxide 

Powder photograph . . . PL III 
Zinc Sulphide 

in Luminescent powders . . 126 
Zircon 

Crystal symmetry 51 

Zone 29 

Axis . . . 29 

Stereographic projection ... 29 



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