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ELEMENTS OF

X-RAY DIFFRACTION

ADDISON-WESLEY METALLURGY SERIES MORRIS COHEN, Consulting Editor

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

Schuhmann— METALLURGICAL ENGINEERING

VOL. I: ENGINEERING PRINCIPLES

Wagner— THERMODYNAMICS OF ALLOYS

ELEMENTS OF

X-RAY DIFFRACTION

by B. D. CULLITY

Associate Professor of Metallurgy University of Notre Dame

ADDISON-WESLEY PUBLISHING COMPANY, INC.

READING, MASSACHUSETTS

Copyright © 1956 ADD1SON-WESLEY PUBLISHING COMPANY, Inc.

Printed ni the United States of America

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

Library of Congress Catalog No 56-10137

PREFACE

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

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

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

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

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

VI PREFACE

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

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

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

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

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

PREFACE Vll

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

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

CONTENTS

FUNDAMENTALS

CHAPTER 1 PROPERTIES OF X-RAYS 1

1-1 Introduction 1

1-2 Electromagnetic radiation 1

1-3 The continuous spectrum . 4

1-4 The characteristic spectrum 6

1-5 Absorption . 10

1-6 Filters 16

1-7 Production of x-rays 17

1 -8 Detection of x-rays 23

1 9 Safety precautions . 25

CHAPTER 2 THE GEOMETRY OF CRYSTALS 29

^2-1 Introduction . 29

J2-2 Lattices . 29

2-3 Crystal systems 30

^2-4 Symmetry 34

2-5 Primitive and nonprimitive cells 36

2-6 Lattice directions and planes °* . 37

2-7 Crystal structure J 42

2-8 Atom sizes and coordination 52

2-9 Crystal shape 54

2-10 Twinned crystals . 55

2-11 The stereographic projection . . 60

CHAPTER 3 DIFFRACTION I: THE DIRECTIONS OF DIFFRACTED BEAMS 78

3-1 Introduction . .78

3-2 Diffraction f. 79

^3-3 The Bragg law * ' . 84

3-4 X-ray spectroscopy 85

3-5 Diffraction directions - 88

3-6 Diffraction methods . 89

3-7 Diffraction under nonideal conditions . 96

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

4-1 Introduction 104

4-2 Scattering by an electrons . . 105

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

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

CONTENTS

4-5 Some useful relations . 118

4-6 Structure-factor calculations ^ 118

4-7 Application to powder method ' 123

4-8 Multiplicity factor 124

4-9 Lorentz factor 124

1-10 Absorption factor 129

4-11 Temperature factor 130

4-12 Intensities of powder pattern lines 132

4-13 Examples of intensity calculations 132

4-14 Measurement of x-ray intensity 136

EXPERIMENTAL METHODS

LPTER 5 LAUE PHOTOGRAPHS 138

5-1 Introduction 138

5-2 Cameras . 138

5-3 Specimen holders 143

5-4 Collimators . .144

5-5 The shapes of Laue spots . 146

kPTER 6 POWDER PHOTOGRAPHS . .149

6-1 Introduction . 149

6-2 Debye-Scherrer method . 149

6-3 Specimen preparation .... 153

6-4 Film loading . . 154

6-5 Cameras for high and low temperatures . 156

6-6 Focusing cameras ... . 156

6-7 Seemann-Bohlin camera . 157

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

6-9 Pinhole photographs . 163

6-10 Choice of radiation . .165

6-11 Background radiation . 166

6-12 Crystal monochromators . 168

6-13 Measurement of line position 173

6-14 Measurement of line intensity . 173

VPTER 7 DlFFRACTOMETER MEASUREMENTS 177

7-1 Introduction . . . • 177

7-2 General features .... 177

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

7-4 Intensity calculations . ... 188

7-5 Proportional counters . . . . 190

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

7-7 Scintillation counters . - 201

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

7-9 Ratemeters . - 206

7-10 Use of monochromators 211

CONTENTS XI

APPLICATIONS

CHAPTER 8 ORIENTATION OF SINGLE CRYSTALS . . . 215

8-1 Introduction . . .... 215

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

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

8-4 Diffractometer method ' . ... 237

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

8-6 Effect of plastic deformation . 242

8-7 Relative orientation of twinned crystals 250

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

CHAPTER 9 THE STRUCTURE OF POLYCRYSTALLINE AGGREGATES . 259

9-1 Introduction . 259

CRYSTAL SIZE

9-2 Grain size 259

9-3 Particle size . 261

CRYSTAL PERFECTION

9-4 Crystal perfection . .... 263

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

CRYSTAL ORIENTATION

9-6 General . .272

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

9-8 Texture of sheet (photographic method) 280

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

9-10 Summary . . 295

CHAPTER 10 THE DETERMINATION OF CRYSTAL STRUCTURE . . . 297

10-1 Introduction . . 297

10-2 Preliminary treatment of data . . . 299

10-3 Indexing patterns of cubic crystals 301

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

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

10-8 Determination of atom positions . 317

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

CHAPTER 11 PRECISE PARAMETER MEASUREMENTS . ... 324

11-1 Introduction .... 324

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

1 1-3 Back-reflection focusing cameras 333

11-4 Pinhole cameras 333

11-5 Diffractometers 334

11-6 Method of least squares .335

Xll CONTENTS

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

11-8 Calibration method . . 342

CHAPTER 12 PHASE-DIAGRAM DETERMINATION . . . 345

12-1 Introduction . 345

12-2 General principles . . 346

12-3 Solid solutions . 351

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

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

12-6 Ternary systems 359

CHAPTER 13 ORDER-DISORDER TRANSFORMATIONS 363

13-1 Introduction . 363

13-2 Long-range order in AuCus 363

13-3 Other examples of long-range order 369

13-4 Detection of superlattice lines 372

13-5 Short-range order and clustering 375

CHAPTER 14 CHEMICAL ANALYSIS BY DIFFRACTION 378

14-1 Introduction 378

QUALITATIVE ANALYSIS

14-2 Basic principles 379

14-3 Hanawait method 379

14-4 Examples of qualitative analysis 383

14-5 Practical difficulties 386

14-6 Identification of surface deposits 387

QUANTITATIVE ANALYSIS (SINGLE PHASE)

14-7 Chemical analysis by parameter measurement 388

QUANTITATIVE ANALYSIS (MULTIPHASE)

14-8 Basic principles . . . 388

14-9 Direct comparison method . . . 391

14-10 Internal standard method . . . 396

14-11 Practical difficulties . . . 398

CHAPTER 15 CHEMICAL ANALYSIS BY FLUORESCENCE 402

15-1 Introduction . ... 402

15-2 General principles . . 404

15-3 Spectrometers ... . 407

15-4 Intensity and resolution . . . 410

15-5 Counters .... . 414

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

15-7 Quantitative analysis ... . . 415

15-8 Automatic spectrometers . . 417

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

15-10 Measurement of coating thickness 421

CONTENTS xiil

CHAPTER 16 CHEMICAL ANALYSIS BY ABSORPTION . . . 423

16-1 Introduction . . . ... 423

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

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

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

16-5 Applications . . 429

CHAPTER 17 STRESS MEASUREMENT . ... 431

17-1 Introduction . 431

17-2 Applied stress and residual stress . . 431

17-3 Uniaxial stress . . 434

17-4 Biaxial stress . 436

17-5 Experimental technique (pinhole camera) 441

17-6 Experimental technique (diffractometer) 444

17-7 Superimposed macrostress and microstress 447

17-8 Calibration 449

1 7-9 Applications 451

CHAPTER 18 SUGGESTIONS FOR FURTHER STUDY . 454

18-1 Introduction 454

18-2 Textbooks . 454

18-3 Reference books . 457

18-4 Periodicals 458

APPENDIXES

APPENDIX 1 LATTICE GEOMETRY . 459

Al-1 Plane spacings 459

Al-2 Cell volumes . . 460

Al-3 Interplanar angles . . . 460

APPENDIX 2 THE RHOMBOHEDRAL-HEXAGONAL TRANSFORMATION 462

APPENDIX 3 WAVELENGTHS (IN ANGSTROMS) OF SOME CHARACTERISTIC

EMISSION LINES AND ABSORPTION EDGES . . . 464

APPENDIX 4 MASS ABSORPTION COEFFICIENTS AND DENSITIES . 466

APPENDIX 5 VALUES OF siN2 8 . 469

APPENDIX 6 QUADRATIC FORMS OF MILLER INDICES . . . 471

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

APPENDIX 8 ATOMIC SCATTERING FACTORS . 474

APPENDIX 9 MULTIPLICITY FACTORS FOR POWDER PHOTOGRAPHS . * . 477

APPENDIX 10 LORENTZ-POLARIZATION FACTOR 478

APPENDIX 11 PHYSICAL CONSTANTS . 480

XIV CONTENTS

APPENDIX 12 INTERNATIONAL ATOMIC WEIGHTS, 1953 481

APPENDIX 13 CRYSTAL STRUCTURE DATA 482

APPENDIX 14 ELECTRON AND NEUTRON DIFFRACTION 486

A14-1 Introduction . ... . . 486

A14r-2 Electron diffraction ... . 486

A14-3 Neutron diffraction .... . 487

APPENDIX 15 THE RECIPROCAL LATTICE . . 490

A15-1 Introduction . .... .490

A15-2 Vector multiplication . ... 490

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

A15-4 Diffraction and the reciprocal lattice . 496

A15-5 The rotating-crystal method . 499

A15-6 The powder method . 500

A15-7 The Laue method . . 502

ANSWERS TO SELECTED PROBLEMS . 506

INDEX ... 509

CHAPTER 1 PROPERTIES OF X-RAYS

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

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

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

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

1

PROPERTIES OF X-RAYS

[CHAP. 1

Frequency Wavelength
(cycles/sec) in millimicrons
1023
1022				_10-5
1021		Gamma-rays <		_io-4 i	X unit
1020	"			_io~3
1019				_10-2
10'8	_J	x-rays	..	__io-] i	angstrom
1017		_1 1	millimicron
10*	J\ Ultraviolet -	*~ "	_io
ID15				_102
1077	M	WHj. Visible 1H	0	_103 1	micron
ion		>• Infrared -		"lo*4
10""	Short radio waves	10
10E		_107 1	centimeter
		_108
iof^		_109 1	meter
	•»	_1010
10^	•uttTiill	~1012 1	kilometer
10f_		1013
iol	Long radio waves	__1014
10 —		1015
iol		_1016

1 megacycle 10£_

1 kilocycle IQl

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

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

bemg lkX= 1.00202A.

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

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

1-2]

ELECTROMAGNETIC RADIATION

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

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

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

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

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

connected by the relation c

X - -. (1-2)

V

where c = velocity of light = 3.00 X 1010 cm/sec.

Electromagnetic radiation, such as a beam of x-rays, carries energy, and the rate of flow of this energy through unit area perpendicular to the direc- tion of motion of the wave is called the intensity I. The average value of the intensity is proportional to the square of the amplitude of the wave, i.e., proportional to A2. In absolute units, intensity is measured in ergs/cm2/sec, but this measurement is a difficult one and is seldom carried out; most x-ray intensity measurements are made on a relative basis in

+E

0 -E

+E i 0

(a) (b)

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

4 PKOPERTIES OF X-RAYS [CHAP. 1

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

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

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

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

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

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

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

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

1-3]

THE CONTINUOUS SPECTRUM

1.0 2.0

WAVELENGTH (angstroms)

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

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

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

PROPERTIES OF X-RAYS [CHAP. 1

c he

12,400

(1-4)

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

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

/cent spectrum = AlZV™, (1-5)

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

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

1-4] THE CHARACTERISTIC SPECTRUM 7

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

0.70926A, Ka2: 0.71354A,

0.63225A.

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

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

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

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

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

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

PROPERTIES OF X-RAYS

[CHAP. 1

60

50

.5 40

1 30

20

10

0

Ka

*-<0.001A

0.2

1.0

0.4 0.6 0.8

WAVELENGTH (angstroms)

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

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

= C(Z - er),

(1-7)

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

1-4]

THE CHARACTERISTIC SPECTRUM

3.0 2.5 2.0

X (angstroms) 1.5 1.0

0.8 0.7

80

70

60

I W

w

50

u

s 40

30

20 -

10

T

I

I

I

T

I

1.0

1.2

1.4

1.6

1.8

2.0 2.2 X 109

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

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

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

M shell

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

FlG ^ Electronic transitions in an at0m (schematic). Emission proc- esses indicated by arrows.

10 PROPERTIES OF X-RAYS [CHAP. 1

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

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

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

of the electrons is given by

ynxr = WK- (1~8)

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

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

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

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

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

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

1-5]

ABSORPTION

11

/* =

Joe-0"""

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

(1-11)

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

It is occasionally necessary to know the mass absorption coefficient of a substance containing more than one element. Whether the substance is a mechanical mixture, a solution, or a chemical compound, and whether it is in the solid, liquid, or gaseous state, its mass absorption coefficient is simply the weighted average of the mass absorption coefficients of its constituent elements. If Wi, w2, etc., are the weight fractions of elements 1, 2, etc., in the substance and (M/P)I, (M/p)2j etc., their mass absorption coefficients, then the mass absorption coefficient of the substance is given by

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

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

M P

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

?-(«n*,gm) ENERGY PER * ^ QUANTUM (erg) * O 5 O § COt— ' tO C*3 C		\	critica for e, electro	1 energ ection u from /	yWK otK nickel
	\
		\	/ ^^
				—
		/'
			K absorption ^edge
		/		/
	y		<^i	V

0 0.5 1.0 1.5 2.0 2. X (angstroms)

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

12 PROPERTIES OF X-RAYS [CHAP. 1

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

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

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

he

where VK and \K are the frequency and wavelength, respectively, of the K absorption edge. Now consider the absorption curve of Fig. 1-8 in light of the above. Suppose that x-rays of wavelength 2.5A are incident on a sheet of nickel and that this wavelength is continuously decreased. At first the absorption coefficient is about 180 cm2/gm, but as the wavelength decreases, the frequency increases and so does the energy per quantum, as shown by the upper curve, thus causing the absorption coefficient to decrease, since the greater the energy of a quantum the more easily it passes through an absorber. When the wavelength is reduced just below

1-5] ABSORPTION 13

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

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

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

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

14

" A	i £	A'«		Ar/3 emission
	"cfl
	1
O
H
O
>* ir
§ L		'
w
w		cs o
		1	La
		0
		X
		0?
HTu			r
				M A/a
Hrv
n				"AT

PROPERTIES OF X-RAYS [CHAP. 1

K state (A' electron removed)

L state (L electron removed)

M state (M electron removed)

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

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

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

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

1-5]

ABSORPTION

15

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

hi> K<*I

(1-15)

1

X/,111

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

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

= WK =

i. he

— ' e\K

12,400

he

— .

*

(1-16)

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

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

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

30 25 920 I- 1, 5 0	\
	\
	\
	\	\
		x^	^	^5^	-__.

0 0.5

1.0 1.5 2.0 X (angstroms)

2.5 3.0

16

PROPERTIES OF X-RAYS

[CHAP. 1

A'a

1.2

1.4 1.6

X (angstroms)

1.8

1.2

1.4 1.6

X (angstroms)

(b) Nickel filter

1.8

(a) No filter

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

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

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

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

1-7]

PRODUCTION OF X-RAYS

17

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

			Filter thickness for
		Incident beam	I(Ka) 500
Target	Filter	I(Ka)	ivm ~ i in trans, beam	I(K<x) trans.
KKfi)	I(Kot] incident
			2 mg/cm	in.
Mo	Zr	3.9	75	0.0045	0.27
Cu	Ni	5.6	19	0.0008	0.40
Co	Fe	5.7	14	0.0007	0.44
Fe	Mn	5.7	13	0.0007	0.43
Cr	V	5.1	11	0.0007	0.44

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

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

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

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

18

PROPERTIES OF X-RAYS

[CHAP. 1

1-7]

PRODUCTION OF X-EAY8

19

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

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

\-ray tube

ri'ISZil

~

high-voltage transformer

M AK Q-0-0-0 Q.ooo Q Q Q Q Q,Q Q .* —

ground autotransformer f 0000001)1)0 "

filament rheostat

000000000

filament transformer

110 volts AC

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

20

PROPERTIES OP X-RAYS

[CHAP. 1

c o

1-8]

DETECTION OF X-RAYS

23

electrons

x-rays

target metal

anode

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

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

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

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

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

24

PROPERTIES OF X-RAYS

[CHAP. 1

(a)

(h)

K edge of silver

(0.48A).

A' edge of bromine (0.92A)

V

1 0 1 5

X (angstroms)

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

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

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

1-9] SAFETY PRECAUTIONS 25

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

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

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

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

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

26 PROPERTIES OF X-RAYS [CHAP. 1

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

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

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

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

PROBLEMS

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

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

PROBLEMS 27

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

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

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

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

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

1-7. (a) Calculate the mass and linear absorption coefficients of air for Cr Ka radiation. Assume that air contains 80 percent nitrogen and 20 percent oxygen by weight, (b) Plot the transmission factor of air for Cr Ka radiation and a path length of 0 to 20 cm.

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

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

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

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

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

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

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

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

1-16. Filters for Co K radiation are usually made of iron oxide (Fe203) powder rather than iron foil. If a filter contains 5 mg Fe203/cm2, what is the transmission factor for the Co Ka line? What is the intensity ratio of Co Ka to Co KQ in the filtered beam? (Density of Fe203 = 5.24 gm/cm3, /i/P of iron for Co Ka radiation = 59.5 cm2/gm, M/P of oxygen for Co Ka radiation = 20.2, pt/P of iron for Co Kfi radiation = 371, JJL/P of oxygen for Co K0 radiation = 15.0.)

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

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

28 PROPERTIES OF X-RAYS [CHAP. 1

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

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

CHAPTER 2 THE GEOMETRY OF CRYSTALS

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

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

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

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

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

29

30

THE GEOMETRY OF CRYSTALS

[CHAP. 2

FIG. 2-1. A point lattice.

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

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

FIG. 2-2. A unit cell.

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

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

2-3]

CRYSTAL SYSTEMS

31

TABLE 2-1 CRYSTAL SYSTEMS AND BRAVAIS LATTICES

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

System	Axials lengths and angles	Bravais lattice	Lattice symbol
r Cubic	Three equal axes at right angles a = /, = r, a = p = J = 90°	Simple Body-centered Face-centered	P 1 F
Tetragonal	Three axes at right angles, two equal a = 6 ^ c , a = p = 7 = 90°	Simple Body-centered	P I
Orthorhombic	Three unequal axes at right angles a ^ b i- c, a = p = 7 = 90°	Simple Body -centered Base-centered Face-centered	.P I C F
Rhombohedral	Three equal axes, equally inclined a = b = c , a = P « 7 * 90°	Simple	P
Hexagonal	Two equal coplanar axes at 120°, third axis at right angles a = b ? c, a = p = 90°, 7 = 120°	Simple	P
Monoclinic	Three unequal axes, one pair not at right angles a * b * c, a - y = 90° * P	Simple Base -centered	P C
Triclinic	Three unequal axes, unequally inclined and none at right angles a * b * c, a ^ p ^ X ^ 90°	Simple	P

* Also called trigonal.

spaced and mutually perpendicular, the unit cell is cubic. In this case the vectors a, b, c are all equal and at right angles to one another, or a = b = c and a = 0 = 7 = 90°. By thus giving special values to the axial lengths and angles, we can produce unit cells of various shapes and therefore various kinds of point lattices, since the points of the lattice are located at the cell corners. It turns out that only seven different kinds of cells are necessary to include all the possible point lattices. These correspond to the seven crystal systems into which all crystals can be classified. These systems are listed in Table 2-1.

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

32

THE GEOMETRY OF CRYSTALS

[CHAP. 2

SIMPLE CUBIC (P)

BODY-CENTERED FACE-C 'ENTERED CUBIC (/) CUBIC1 (F)

SIMPLE BOD Y-( CENTERED SIMPLE BODY-CENTERED

TETRAGONAL TETRAGONAL ORTHORHOMBIC ORTHORHOMBIC

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

BASE-CENTERED FACE-CENTERED RHOMBOHEDRAL ORTHORHOMBIC1 ORTHORHOMBIC (/?)

(O (F)

^^
	c
	120°	a
WJ-**	^^
HEXAGONAL (P)

SIMPLE MONOCLINIC

BASE-CENTERED TRICLINIC (P)

(P) MONOCLINIC1 (C)

FIG. 2-3. The fourteen Bravais lattices.

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

2-3]

CRYSTAL SYSTEMS

33

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

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

N =

Nf

--

2

Nc

— ,

8

(2-1 ;

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

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

/

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

34

THE GEOMETRY OF CRYSTALS

[CHAP. 2

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

2-4 Symmetry, i Both Bravais lattices and the real crystals which are built up on them exhibit various kinds of symmetry. A body or structure is said to be symmetrical when its component parts are arranged in such balance, so to speak, that certain operations can be performed on the body which will bring it into coincidence with itself. These are termed symmetry operations. /For example, if a body is symmetrical with respect to a plane passing through it, then reflection of either half of the body in the plane as in a mirror will produce a body coinciding with the other half. Thus a cub0 has seir-ral planes of symmetry, one of which is shown in Fig. 2-6(a).

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

\ X	c p'1'	>
\		r
/	\	V
	\	X

,	z c.	~7 ?
		r
7	A	/
		/

(b)

(ci)

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

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

2-4] SYMMETRY 35

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

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

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

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

TABLE 2-2 SYMMETRY ELEMENTS

System

Minimum symmetry elements

Cubic

Tetragonal

Orthorhombi c

Rhombohedral

Hexagonal

Monoclinic

Triclinic

Four 3 - fold rotation axes

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

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

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

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

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

None

36

THE GEOMETRY OF CRYSTALS

[CHAP. 2

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

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

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

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

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

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

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

2-6]

LATTICE DIRECTIONS AND PLANES

37

to point Z), being separated from it by the vector c. The coordinates of equivalent points in different unit cells can always be made identical by the addition or subtraction of a set of integral coordinates; in this case, subtraction of 0 0 1 from f ^ 1 (the coordinates of E) gives ^ f 0 (the coordinates of D).

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

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

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

[100]

[233]

[001]

[111]

[210]

HO

[100] '[120]

Indices of directions.

38 THE GEOMETRY OF CRYSTALS [CHAP. 2

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

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

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

1A 2A 3A 4A

(a) (b)

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

2-6]

LATTICE DIRECTIONS AND PLANES

39

Axial lengths Intercept lengths Fractional intercepts

Miller indices

4A 2A

I

I2 16

8A 6A

3 1

4

3A

3A

1

1

3

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

A slightly different system of plane indexing is used in the hexagonal system. The unit cell of a hexagonal lattice is defined by two equal and coplanar vectors ai and a2, at 120° to one another, and a third axis c at right angles [Fig. 2-11 (a)]. The complete lattice is built up, as usual, by

HfeocH

(110)

(110) (111)

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

(102)

40

THE GEOMETRY OF CRYSTALS

[CHAP. 2

[001]

(0001)

(1100)-

[100] '

[Oil]

(1210)

[010]

(1011)

'[210]

(a) (b)

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

repeated translations of the points at the unit cell corners by the vectors EI, a2, c. Some of the points so generated are shown in the figure, at the ends of dashed lines, in order to exhibit the hexagonal symmetry of the lattice, which has a 6-fold rotation axis parallel to c. The third axis a3, lying in the basal plane of the hexagonal prism, is so symmetrically related to EI and a2 that it is often used in conjunction with the other two. Thus the indices of a plane in the hexagonal system, called Miller-Bra vais indices, refer to four axes and are written (hkil). The index i is the recipro- cal of the fractional iiltercept on the a3 axis. Since the intercepts of a plane on ai and a2 determine its intercept on a3, the value of i depends on the values of h and k. The relation is

h + k = -i.

(2-2)

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

Directions in a hexagonal lattice are best expressed in terms of the three basic vectors ai, a2, and c. Figure 2-1 l(b) shows several examples of both plane and direction indices. (Another system, involving four indices, is sometimes used to designate directions. The required direction is broken up into four component vectors, parallel to ai, a2, aa, and c and so chosen that the third index is the negative of the sum of the first two. Thus

2-6]

LATTICE DIRECTIONS AND PLANES

41

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

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

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

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

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

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

[001]

(210) UOO) \

(11°) (210)

,(100)

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

(2-4)

W = /&1/T2 — h?jk\.

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

42

THE GEOMETRY OF CRYSTALS

[CHAP. 2

(13)

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

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

(Cubic) dhki = -^-JL===. (2-5)

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

(Tetragonal) dhki =

(2-0)

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

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

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

2-7]

CRYSTAL STRUCTURE

43

BCC FCC

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

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

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

The next degree of complexity is encountered when two or more atoms of the same kind are "associated with" each point of a Bravais lattice, as exemplified by the hexagonal close-packed (HCP) structure common to many metals. This structure is simple hexagonal and is illustrated in Fig. 2-15. There are two atoms per unit cell, as shown in (a), one at 0 0 0 and the other at § \ | (or at \ f f , which is an equivalent position). Figure 2-15(b) shows the same structure with the origin of the unit cell shifted so that the point 1 0 0 in the new cell is midway between the atoms at 1 0 0 and § \ | in (a), the nine atoms shown in (a) corresponding to the nine atoms marked with an X in (b). The ' 'association" of pairs of atoms with the points of a simple hexagonal Bravais lattice is suggested by the dashed lines in (b). Note, however, that the atoms of a close-packed hexagonal structure do not themselves form a point lattice, the surround- ings of an atom at 0 0 0 being different from those of an atom at § 3 ^. Figure 2-15(c) shows still another representation of the HCP structure: the three atoms in the interior of the hexagonal prism are directly above the centers of alternate triangles in the base and, if repeated through space by the vectors ai and a2, would alsd form a hexagonal array just like the atoms in the layers above and below.

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

44

THE GEOMETRY OF CRYSTALS

(a)

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

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

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

2-7]

CRYSTAL STRUCTURE

i HID

45

[001]

HEXAGONAL CLOSE-PACKED

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

46

THE GEOMETRY OF CRYSTALS

[CHAP. 2

j;

HH

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

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

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

Another example of the "association" of more than one atom with each point of a Bravais lattice is given by uranium. The structure of the form stable at room temperature, a-uranium, is illustrated in Fig. 2-17 by plan and elevation drawings. In such drawings, the height of an atom (ex- pressed as a fraction of the axial length) above the plane of the drawing (which includes the origin of the unit cell and two of the cell axes) is given by the numbers marked on each atom. The Bravais lattice is base-centered orthorhombic, centered on the C face, and Fig. 2-17 shows how the atoms occur in pairs through the structure, each pair associated with a lattice point. There are four atoms per unit cell, located at Or/-}, 0 y f , \ (\ + y} T> and i (2 "~ y) T • Here we have an example of a variable parameter y in the atomic coordinates. Crystals often contain such vari- able parameters, which may have any fractional value without destroying any of the symmetry elements of the structure. A quite different sub- stance might have exactly the same structure as uranium except for slightly different values of a, 6, c, and y. For uranium y is 0.105 ± 0.005.

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

2-7]

CRYSTAL STRUCTURE

47

O CB+

[010]

(a) CsCl

(b) NaCl

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

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

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

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

What is the Bravais lattice of CsCl? Figure 2-1 8 (a) shows that the unit cell contains two atoms, ions really, since this compound is com- pletely ionized even in the solid state: a caesium ion at 0 0 0 and a chlo- rine ion at ^ \ \ . The Bravais lattice is obviously not face-centered, but we note that the body-centering translation \ \ \ connects two atoms. However, these are unlike atoms and the lattice is therefore not body-

48 THE GEOMETRY OF CRYSTALS [CHAP. 2

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

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

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

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

4 Na4" at 0 0 0 + face-centering translations 4 Cl~ at \ \ \ + face-centering translations.

Note also that in these, as in all other structures, the operation of any symmetry element possessed by the lattice must bring similar atoms or ions into coincidence. For example, in Fig. 2-18(b), 90° rotation about the 4-fold [010] rotation axis shown brings the chlorine ion at 0 1 \ into coincidence with the chlorine ion at ^11, the sodium ion at 0 1 1 with the sodium ion at 1 1 1, etc.

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

000 + face-centering translations

1 i I + face-centering translations.

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

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

2-7]

CRYSTAL STRUCTURE

51

O Fe

• C position

<£

(a)

(b)

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

on the lattice of the solvent, while in the latter, solute atoms fit into the interstices of the solvent lattice. The interesting feature of these struc- tures is that the solute atoms are distributed more or less at random. For example, consider a 10 atomic percent solution of molybdenum in chro- mium, which has a BCC structure. The molybdenum atoms can occupy either the corner or body-centered positions of the cube in a random, ir- regular manner, and a small portion of the crystal might have the appear- ance of Fig. 2-21 (a). Five adjoining unit cells are shown there, contain- ing a total of 29 atoms, 3 of which are molybdenum. This section of the crystal therefore contains somewhat more than 10 atomic percent molyb- denum, but the next five cells would probably contain somewhat less. Such a structure does not obey the ordinary rules of crystallography: for example, the right-hand cell of the group shown does not have cubic symmetry, and one finds throughout the structure that the translation given by one of the unit cell vectors may begin on an atom of one kind and end on an atom of another kind. All that can be said of this structure is that it is BCC on the average, and experimentally we find that it displays the x-ray diffraction effects proper to a BCC lattice. This is not surpris- ing since the x-ray beam used to examine the crystal is so large compared to the size of a unit cell that it observes, so to speak, millions of unit cells at the same time and so obtains only an average "picture" of the structure. The above remarks apply equally well to interstitial solid solutions. These form whenever the solute atom is small enough to fit into the sol- vent lattice without causing too much distortion. Ferrite, the solid solu- tion of carbon in a-iron, is a good example. In the unit cell shown in Fig. 2-21 (b), there are two kinds of "holes" in the lattice: one at | 0 £ (marked •) and equivalent positions in the centers of the cube faces and edges, and one at J 0 ^ (marked x) and equivalent positions. All the evidence at hand points to the fact that the carbon atoms in ferrite are located in the holes at f 0 f and equivalent positions. On the average, however, no more than about 1 of these positions in 500 unit cells is occu-

2-8] ATOM SIZES AND COORDINATION 53

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

BCC =

2 '

V2 2 a> (2-7)

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

a2 c2 (between atom in basal plane \ 3 4 and neighbors above or below).

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

Change in coordination Size contraction, percent

12 -» 8 3

12 -> 6 4

12 -> 4 12

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

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

54 THE GEOMETRY OF CRYSTALS [CHAP. 2

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

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

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

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

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

2-10] TWINNED CRYSTALS 55

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

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

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

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

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

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

Occasionally, annealing twins appear under the microscope as in Fig. 2-22 (a), with one part of a grain (E) twinned with respect to the other part (A). The two parts are in contact on the composition plane (111) which makes a straight-line trace on the plane of polish. More common, however, is the kind shown in Fig. 2-22 (b). The grain shown consists of three parts: two parts (Ai and A 2) of identical orientation separated by a third part (B) which is twinned with respect to A\ and A2. B is known as a twin band.

(a)

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

56

THE GEOMETRY OF CRYSTALS

[CHAP. 2

C A B C

PLAN OF CRYSTAL PLAN OF TWIN

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

2-10]

TWINNED CRYSTALS

59

twinning shear

[211]

(1012) twin plane

PLAN OF CRYSTAL

PLAN OF TWIN

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

60

THE GEOMETRY OF CRYSTALS

[CHAP. 2

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

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

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

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

100

010

FIG. 2-25. crystal.

100 {1001 poles of a cubic

M

FIG. 2-26. Angle between two planes.

2-1 1J

THE 8TEREOGRAPHIC PROJECTION

61

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

The angle a between two planes is evidently equal to the angle between their great circles or to the angle between their normals (Fig. 2-26). But this angle, in degrees, can also be measured on the surface of the sphere along the great circle KLMNK connecting the poles PI and P2 of the two planes, if this circle has been divided into 360 equal parts. The measure- ment of an angle has thus been transferred from the planes themselves to the surface of the reference sphere.

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

, projection plane - basic circle

reference sphere

\

point of projection

4

observer

SECTION THROUGH AB AND PC

FIG. 2-27. The stereographic projection.

62 THE GEOMETRY OF CRYSTALS [CHAP. 2

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

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

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

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

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

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

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

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

2-11]

THE STEREOGRAPHIC PROJECTION

63

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

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

64

THE GEOMETRY OP CRYSTALS

[CHAP. 2

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

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

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

2-11]

THE STEREOGRAPHIC PROJECTION

65

PROJECTION

Wulff net

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

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

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

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

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

66

PROJECTION

(a)

FIG. 2-31. (a) Stereo- graphic projection of poles Pi and P2 of Fig. 2-26. (b) Rotation of projection to put poles on same great circle of Wulff net. Angle between poles = 30°.

(b)

2-11]

THE STEREOGRAPHIC PROJECTION

67

sired angle measurement can then be made. Figure 2-31 (a) is a projec- tion of the two poles PI and P2 shown in perspective in Fig. 2-26, and the angle between them is found by the rotation illustrated in Fig. 2-3 l(b). This rotation of the projection is equivalent to rotation of the poles on latitude circles of a sphere whose north-south axis is perpendicular to the projection plane.

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

FIG. 2-32. Method of finding the trace of a pole (the pole P2' in Fig. 2-31).

68

THE GEOMETRY OF CRYSTALS

[CHAP. 2

PROJECTION

FIG. 2-33. Measurement of an angle between two poles (Pi and P2 of Fig. 2-26) by measurement of the angle of intersection of the corresponding traces.

PROJECTION

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

2-11] THE STEREOGRAPHIC PROJECTION 69

We often wish to rotate poles around various axes. We have already seen that rotation about an axis normal to the projection is accomplished simply by rotation of the projection around the center of the Wulff net. Rotation about an axis lying in the plane of the projection is performed by, first, rotating the axis about the center of the Wulff net until it coin- cides with the north-south axis if it does not already do so, and, second, moving the poles involved along their respective latitude circles the re- quired number of degrees. Suppose it is required to rotate the poles A\ and BI shown in Fig. 2-34 by 60° about the NS axis, the direction of mo- tion being from W to E on the projection. Then AI moves to A2 along its latitude circle as shown. #1, however, can rotate only 40° before finding itself at the edge of the projection; we must then imagine it to move 20° in from the edge to the point B[ on the other side of the projection, staying always on its own latitude circle. The final position of this pole on the positive side of the projection is at B2 diametrically opposite B\.

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

For example, we may be required to rotate AI about BI by 40° in a clockwise direction (Fig. 2-35). In (a) the pole to be rotated A} and the rotation axis BI are shown in their initial position. In (b) the projection has been rotated to bring BI to the equator of a Wulff net. A rotation of 48° about the NS axis of the net brings BI to the point B2 at the center of the net; at the same time AI must go to A2 along a parallel of latitude. The rotation axis is now perpendicular to the projection plane, and the required rotation of 40° brings A2 to A 3 along a circular path centered on B2. The operations which brought BI to B2 must now be reversed in order to return B2 to its original position. Accordingly, B2 is brought to JBs and A% to A*, by a 48° reverse rotation about the NS axis of the net. In (c) the projection has been rotated back to its initial position, construc- tion lines have been omitted, and only the initial and final positions of the rotated pole are shown. During its rotation about B^ AI moves along the small circle shown. This circle is centered at C on the projection and not at its projected center BI. To find C we use the fact that all points on the circle must lie at equal angular distances from BI] in this case, measurement on a Wulff net shows that both AI and A± are 76° from B\. Accordingly, we locate any other point, such as D, which is 76° from B\, and knowing three points on the required circle, we can locate its center C.

70

THE GEOMETRY OP CRYSTALS

[CHAP. 2

48°

40°

(b)

(a) (c)

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

2-11]

THE 8TEREOGRAPHIC PROJECTION

71

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

zones.

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

100

100

no

no

110

1)10 Oil

no

111

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

72

THE GEOMETRY OF CRYSTALS

[CHAP. 2

TABLE 2-3

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

1W2I	IWi!
100	110	in	210	211	221	310
100	0
	90
110	45	0
	90	60
		90
111	54.7	35.3	0
		90	70.5
			109.5
210	26.6	18.4	39.2	0
	63.4	50.8	75.0	36.9
	90	71.6		53.1
211	35.3	30	19.5	24.1	0
	65.9	54.7	61.9	43.1	33.6
		73.2	90	56.8	48.2
		90
221	48.2	19.5	15.8	26.6	17.7	0
	70.5	45	54.7	41.8	35.3	27.3
		76.4	78.9	53.4	47.1	39.0
		90
310	18.4	26.6	43.1	8.1	25.4	32.5	0
	71.6	47.9	68.6	58.1	49.8	42.5	25.9
	90	63.4		45	58.9	58.2	36.9
		77.1
311	25.2	31.5	29.5	19.3	10.0	25.2	17.6
	72.5	64.8	58.5	47.6	42.4	45.3	40.3
		90	80.0	66.1	60.5	59.8	55.1
320	33.7	11.3	61.3	7.1	25.2	22.4	15.3
	56.3	54.0	71.3	29.8	37.6	42.3	37.9
	90	66.9		41.9	55.6	49.7	52.1
321	36.7	19.1	22.2	17.0	10.9	11.5	21.6
	57.7	40.9	51.9	33.2	29.2	27.0	32.3
	74.5	55.5	72.0	53.3	40.2	36.7	40.5
			90
331	46.5	13.1	22.0
510	11.4
511	15.6
711	11.3

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

2-11]

THE STEREOGRAPHIC PROJECTION

73

[112] zone

mi]

1110]

[001] zone

[100] // zone

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

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

74

THE GEOMETRY OF CRYSTALS

[CHAP. 2

Jl20

T530,

1321

• 0113.

' " • 14 •l013 5,,4 •"<>'
• 1014 .2203
3 *'QI5 - %>*
0114 •TlO4l°3	•
*OII5 Tl05	23?l
• 0001 . 0
12 F4 ?2l2	T2H
• "05 .0115
104 . *°i4OM3	9

foil

•no. • •ioTs .

53TO

320

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

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

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

cosp

d o/fc'

cos a

d bjk

d

cos r

2-11]

THE STEREOGRAPHIC PROJECTION

75

100

(a) (b)

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

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

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

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

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

100

010

010

(111)

twin plane

100

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

76 THE GEOMETRY OP CRYSTALS [CHAP. 2

PROBLEMS

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

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

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

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

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

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

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

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

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

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

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

(6) Four atoms of the same kind per unit cell located at 0 0 z, 0 J z, 0 f (^ + z),

00(| + 2).

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

(I - y) *, (I -*)(* + y) *•

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

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

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

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

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

(a) Find the angle between the two planes.

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

PROBLEMS 77

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

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

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

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

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

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

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

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

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

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

by

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

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

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

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

CHAPTER 3 DIFFRACTION I: THE DIRECTIONS OF DIFFRACTED BEAMS

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

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

78

3-2]

DIFFRACTION

79

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

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

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

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

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

Two conclusions may be drawn from this illustration :

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

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

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

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

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

3-2| DIFFRACTION 83

We have here regarded a diffracted beam as being built up of rays scat- tered by successive planes of atoms within the crystal. It would be a mistake to assume, however, that a single plane of atoms A would diffract x-rays just as the complete crystal does but less strongly. Actually, the single plane of atoms would produce, not only the beam in the direction 1' as the complete crystal does, but also additional beams in other directions, some of them not confined to the plane of the drawing. These additional beams do not exist in the diffraction from the complete crystal precisely because the atoms in the other planes scatter beams which destructively interfere with those scattered by the atoms in plane A, except in the direc- tion I7.

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

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

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

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

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

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

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

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

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

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

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

As previously stated, diffraction in general occurs only when the wave- length of the wave motion is of the same order of magnitude as the repeat distance between scattering centers. This requirement follows from the Bragg law. Since sin 0 cannot exceed unity, we may write

n\

— = sin0<l. (3-2)

2rf'

Therefore, n\ must be less than 2d'. For diffraction, the smallest value of n is 1. (n = 0 corresponds to the beam diffracted in the same direction as the transmitted beam. It cannot be observed.) Therefore the condi- tion for diffraction at any observable angle 26 is

X < 2d'. (3-3)

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

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

n

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

(3-5)

This form will be used throughout this book.

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

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

3-4]

X-RAY SPECTROSCOPY

85

(100)

(200)

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

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

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

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

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

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

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

weight of atoms in unit cell

Density = - — - — >

volume of unit cell

ZA p = , (3-6)

NV

where p = density (gm/cm3), SA = sum of the atomic weights of the atoms in the unit cell, N = Avogadro's number, and V = volume of unit cell (cm3). NaCl, for example, contains four sodium atoms and four chlo- rine atoms per unit cell, so that

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

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

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

3-4] X-RAY 8PECTRO8COPY 87

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

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

(200) plane spacing of calcite

1 kX = • (3—7)

3.02945 V ;

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

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

(3-8)

1 kX = 1.00202A

This conversion factor was decided on in 1946 by international agreement, and it was recommended that, in the future, x-ray wavelengths and the lattice parameters of crystals be expressed in angstroms. If V in Eq. (3-6) for the density of a crystal is expressed in A3 (not in kX3) and the currently accepted value of Avogadro's number inserted, then the equation becomes

1.66020S4 P = (3-9)

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

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

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

3-5 Diffraction directions. What determines the possible directions, i.e., the possible angles 20, in which a given crystal can diffract a beam of monochromatic x-rays? Referring to Fig. 3-3, we see that various diffrac- tion angles 20i, 202, 203, ... can be obtained from the (100) planes by using a beam incident at the correct angle 0i, 02, 0s, • • • and producing first-, second-, third-, . . . order reflections. But diffraction can also be produced by the (110) planes, the (111) planes, the (213) planes, and so on. We obviously need a general relation which will predict the diffrac- tion angle for any set of planes. This relation is obtained by combining the Bragg law and the plane-spacing equation (Appendix 1) applicable to the particular crystal involved.

For example, if the crystal is cubic, then

X = 2d sin 0 and

1 (ft2 + fc2 + I2}

Combining these equations, we have

X2

sin20 = -— (h2 + k2 + l2). (3-10)

4a2

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

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

4 a2 c2

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

3-6] DIFFRACTION METHODS 89

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

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

Laue method Variable Fixed

Rotating-crystal method Fixed Variable (in part)

Powder method Fixed Variable

The Laue method was the first diffraction method ever used, and it re- produces von Laue's original experiment. A beam of white radiation, the continuous spectrum from an x-ray tube, is allowed to fall on a fixed single crystal. The Bragg angle 6 is therefore fixed for every set of planes in the crystal, and each set picks out and diffracts that particular wavelength which satisfies the Bragg law for the particular values of d and 0 involved. Each diffracted beam thus has a different wavelength.

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

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

(a) (b)

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

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

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

(a)

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

3-6]

DIFFRACTION METHODS

91

Z.A.

(b)

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

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

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

92

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

Z.A.

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

FIG. 3-9. Rotating-crystal method.

The fact that the Laue reflections from planes of a zone lie on the surface of a cone can be nicely demonstrated with the stereographic projection. In Fig. 3-8, the crystal is at the center of the reference sphere, the incident beam 7 enters at the left, and the transmitted beam T leaves at the right. The point representing the zone axis lies on the circumference of the basic circle and the poles of five planes belonging to this zone, PI to P5, lie on the great circle shown. The direction of the beam diffracted by any one of these planes, for example the plane P2, can be found as follows. 7, P2, D2 (the diffraction direction required), and T are all coplanar. Therefore 7>2 lies on the great circle through 7, P2, and T. The angle between 7 and P2 is (90° — 0), and 7)2 must lie at an equal angular distance on the other side of P2, as shown. The diffracted beams so found, D\ to Z>5, are seen to lie on a small circle, the intersection with the reference sphere of a cone whose axis is the zone axis.

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

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

3-6] DIFFRACTION METHODS 93

^m^mm ^'S'lililtt

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

a particular set of lattice planes will, for an instant, make the correct Bragg angle for reflection of the monochromatic incident beam, and at that instant a reflected beam will be formed. The reflected beams are again located on imaginary cones but now the cone axes coincide with the rotation axis. The result is that the spots on the film, when the film is laid out flat, lie on imaginary horizontal lines, as shown in Fig. 3-10. Since the crystal is rotated about only one axis, the Bragg angle does not take on all possible values between 0° and 90° for every set of planes. Not every set, therefore, is able to produce a diffracted beam ; sets perpendicular or almost perpendicular to the rotation axis are obvious examples.

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

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

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

94

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

(a)

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

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

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

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

3-6]

DIFFRACTION METHODS

95

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

(a)

\

26 = 0°

1 t "] o I

1

b )

(b)

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

26 = 180°

26 = 0°

ii

(a)

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

cm.

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

spacing. In the cubic system, for example, d is a maximum when (h2 + k2 + I2) is a minimum, and the minimum v#lue of this term is 1, corresponding to (hkl) equal to (100). The 100 reflection is accordingly the one of lowest 20 value. The next reflection will have indices hkl corre- sponding to the next highest value of (h2 + k2 + /2), namely 2, in which case (hkl) equals (110), and so on.

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

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

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

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

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

3-7J

DIFFRACTION UNDER NONIDEAL CONDITIONS

97

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

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

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

X = 2d sin 6B.

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

FIG. 3-14. diffraction.

Effect of crystal size on

98

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

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

When we consider incident rays that make Bragg angles only slightly different from 0#, we find that destructive interference is not complete. Ray B, for example, makes a slightly larger angle 0i, such that ray L' from the mth plane below the surface is (m + 1) wavelengths out of ph6.se with B', the ray from the surface plane. This means that midway in the crystal there is a plane scattering a ray which is one-half (actually, an integer plus one-half) wavelength out of phase with ray B' from the surface plane. These rays cancel one another, and so do the other rays from sim- ilar pairs of planes throughout the crystal, the net effect being that rays scattered by the top half of the crystal annul those scattered by the bottom half. The intensity of the beam diffracted at an angle 20i is therefore zero. It is also zero at an angle 202 where 02 is such that ray N' from the mth plane below the surface is (m — 1) wavelengths out of phase with ray C' from the surface plane. It follows that the diffracted intensity at angles near 2fe, but not greater than 26 1 or less than 202, is not zero but has a value intermediate between zero and the maximum intensity of the beam diffracted at an angle 20s- The curve of diffracted intensity vs. 28 will thus have the form of Fig. 3-15(a) in contrast to Fig. 3-15(b), which illus- trates the hypothetical case of diffraction occurring only at the exact Bragg angle.

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

202

20i

20

20* 20-

(a) (b)

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

3-7] DIFFRACTION UNDER NONIDEAL CONDITIONS 99

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

B = f (20i - 202) = 0i - 02. The path-difference equations for these two angles are

2t sin 02 = (m - 1)X. By subtraction we find

£(sin 0i — sin 02) = X,

(/> i n \ //) /) \ CM ~"T~ f2 \ i ^1 — ^2 \ 1 sin I ) = X. 2 / \ 2 /

But 0i and 02 are both very nearly equal to 0#, so that

0i + 02 = 200 (approx.) and

sin f — ^J = f — j (approx.).

Therefore

2t[ — -) cos 0B = X,

t = (3-12)

JS cos SB

A more exact treatment of the problem gives

, . _«*_. (3-13)

B cos BR

which is known as the Scherrer formula. It is used to estimate the particle size of very small crystals from the measured width of their diffraction curves. What is the order of magnitude of this effect? Suppose X = 1.5A, d = LOA, and 0 = 49°. Then for a crystal 1 mm in diameter the breadth J5, due to the small crystal effect alone, would be about 2 X 10~7 radian (0.04 sec), or too small to be observable. Such a crystal would contain some 107 parallel lattice planes of the spacing assumed above. However, if the crystal were only 500A thick, it would contain only 500 planes, and the diffraction curve would be relatively broad, namely about 4 X 10~~3 radian (0.2°).

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

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

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

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

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

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

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

3-7]

DIFFRACTION UNDER NONIDEAL CONDITIONS

101

(a)

(1))

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

crystal

liquid or amorphous solid

0 90 180

DIFFRAC TION (SCATTERING) ANGLE 28 (degrees)

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

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

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

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

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

PROBLEMS

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

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

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

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

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

(a) Simple cubic (a = 3.00A)

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

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

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

PROBLEMS 103

3-6. Calculate the breadth B (in degrees of 26), due to the small crystal effect alone, of the powder pattern lines of particles of diameter 1000, 750, 500, and 250A. Assume 6 = 45° and X = 1.5A. For particles 250A in diameter, calculate the breadth B for 0 = 10, 45, and 80°.

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

CHAPTER 4 DIFFRACTION II: THE INTENSITIES OF DIFFRACTED BEAMS

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

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

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

,$ (a) (b)

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

104

4-2]

SCATTERING BY AN ELECTRON

r i

3

105

(a)

(b)

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

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

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

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

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

sin2 a,

(4-1)

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

where /o = intensity of the incident beam, c = velocity of light, and a = angle between the scattering direction and the direction of accelera- tion of the electron. Suppose the incident beam is traveling in the direc- tion Ox (Fig. 4-3) and encounters an electron at 0. We wish to know the scattered intensity at P in the xz plane where OP is inclined at a scattering angle of 26 to the incident beam. An unpolarized incident beam, such as that issuing from an x-ray tube, has its electric vector E in a random direction in the yz plane. This beam may be resolved into two plane- polarized components, having electric vectors Ey and E2 where

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

E,2 = Ez2 = £E2.

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

IQV =