Elements of x ray diffraction cullity

Elements of X-RAY DIFFRACTION SECOND EDITION B D CULLITY Department of Metallurgical Engineering and Materials Science University of Nôtre Dame ADDISON-WESLEY PUBLISHING COMPANY INC Reading, Massachusetts - Menlo Park, California London - Amsterdam - Don Mills, Ontario - Sydney This book is in the Addison-Wesley Series in Metallurgy and Materials Morris Cohen Consulting Editor Copyright 1978, 1956 by Addison-Wesley Publishing Company, Inc Philippines copyright 1978 by Addison-Wesley Publishing Company, Inc All rights reserved No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher Printed in the United States of America Published simultaneously in Canada Library of Congress Catalog Card No 77-73950 ISBN 0-201-01174-3 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 determination, 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 experimental 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 examination of nonmetallic materials, inasmuch as the physical principles involved not depend on the material investigated This book should therefore be useful to metallurgists, chemists, physicists, ceramists, mineralogists, etc., namely, to all who use x-ray diffraction 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 metailurgical 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 orientations found in specific metals and alloys are not described, because these topics are adequately covered in existing books In short, x-ray diffraction 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 diffraction pattern X-ray diffraction iii iv Preface phenomena are rather sharply divisible into those effects that are understandable in terms of the Bragg law and those that require a more advanced treatment, based on the reciprocal lattice This book is written entirely in terms of the Bragg l a and can be read without any knowledge of the reciprocal lattice My experience with teaching x-ray diffraction to senior students in metallurgy, for many of whom this book represents a terminal course in the subject, is that there is insufficient time to attain both a real facility for "reciprocal thinking" and a good knowledge of the many applications of diffraction I therefore prefer the Bragg-law approach for a first course Those instructors who wish to introduce the reciprocal lattice at the beginning can interpose Appendix I , which contains the rudiments of the subject, between Chapters and Chapters on chemical analysis by x-ray diffraction and x-ray spectroscopy are included because of the industrial importance of these analytical methods Electron and neutron diffraction are treated in appendices This second edition includes an account of new developments made possible by the semiconductor detector and pulse-height analysis, namely, energy-dispersive spectrometry and diffractometry Applications of position-sensitive detectors are also described A new section is devoted to x-ray topography and other x-ray methods of assessing the quality of single crystals Other additions include a quantitative treatment of the temperature factor and descriptions of the Auger effect, microcameras and Guinier cameras, and microanalysis in the electron microscope References to original papers are now given, and the tables of wavelengths and absorption coefficients have been expanded This edition contains more material on the measurement of preferred orientation and residual stress than the first edition, but the former chapter on chemical analysis by x-ray absorption has been dropped, as being of minor interest to most readers The first edition carried the following acknowledgements: 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 chapters 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 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 Preface v In the preparation of the second edition I have been helped in many ways by Charles W Allen, A W Danko, Ron Jenkins, Paul D Johnson, A R Lang, John W Mihelich, J B Newkirk, Paul S Prevey, B E Warren, Carl Cm Wu, and Leo Zwell To all these, my best thanks Notre Dame, Indiana November 1977 B D Cullity Contents FUNDAMENTALS Chapter Properties of X-rays Introduction Electromagnetic radiation The continuous spectrum The characteristic spectrum Absorption Filters Production of x-rays Detection of x-rays Safety precautions Chapter 2-1 2-2 2-3 2-4 2-5 2-6 2-7 2-8 2-9 2-10 2-1 Chapter 3-1 3 13 19 21 27 29 32 32 34 37 39 41 47 56 58 59 63 81 82 86 88 91 92 99 Geometry of Crystals Introduction Lattices Crystal systems Symmetry Primitive and nonprimitive cells Lattice directions and planes Crystal structure Atom sizes and coordination Crystal shape Twinned crystals The stereographic projection Diffraction I: Directions of Diffracted Beams Introduction Diffraction The Bragg law X-ray spectroscopy Diffraction directions Diffraction methods Diffraction under nonideal conditions vii viii Contents Chapter Diffraction 11: Intensities of Diffracted Beams Introduction Scattering by an electron Scattering by an atom Scattering by a unit cell Some useful relations Structure-factor calculations Application to powder method Multiplicity factor Lorentzfactor Absorptionfactor Temperature factor Intensities of powder pattern lines Examples of intensity calculations Measurement of x-ray intensity EXPERIMENTAL METHODS Chapter 5-1 5-2 5-3 5-4 5-5 Laue Photographs Introduction Cameras Specimens and holders Collimators The shapes of Laue spots 149 150 155 156 158 Chapter Powder Photographs Introduction Debye-Scherrer method Specimen preparation Filmloading Cameras for special conditions Focusing cameras Seemann-Bohlin camera Back-reflection focusing cameras Pinhole photographs Microbeams and microcameras Choice of radiation Background radiation Crystal monochromators Guinier cameras Measurement of line position Measurement of line intensity Chapter 7-1 7-2 Diffractometer and Spectrometer Measurements Introduction 188 General features 189 Contents X-rayoptics Counters (general) Proportional counters Geiger counters Scintillation counters Semiconductor counters Pulse-heightanalysis Special kinds of diffractometry Scalers Ratemeters Monochromatic operation APPLICATIONS Chapter Orientation and Quality of Single Crystals Introduction Crystal Orientation The back-reflection Laue method Transmission Laue method Diffractorneterrnethod Setting a crystal in a required orientation , Crystal Quality Laue methods Topographic and other methods Chapter Structure of Polycrystalline Aggregates Introduction Crystal Size Grain size Particle size Crystal quality Depth of x-ray penetration Crystal Quality Crystal Orientation General The texture of wire (photographic method) The texture of sheet (diffractometer methods) The texture of wire (diflractometer method) Inverse pole figures ~ m o r p h o kSolids 9-12 Amorphous and semi-amorphous solids Summary Chapter 10 Determination of Crystal Structure Introduction : Preliminary treatment of data Indexing patterns of cubic crystals Indexing patterns of noncubic crystals (graphical methods) Indexing patterns of noncubic crystals (analytical methods) The effect of cell distortion on the powder pattern Determination of the number of atoms in a unit cell Determination of atom positions Example of structure determination Chapter 11 Precise Parameter Measurements Introduction Debye-Scherrer cameras Back-reflection focusing cameras Pinhole cameras Diffractometers Method of least squares Cohen's method General Chapter Phase-Diagram Determination curves (disappearing-phase method) curves (parametric method) Chapter 43 Order-Disorder Transformations Chapter 44 Chemical Analysis by X-ray Diffraction Introduction 141 Qualitative Analysis Basic principles Powder diffraction file Procedure Examples of analysis Practical difficulties Chapter references 541 Chapter 16 Measurement of Residual Stress 16.1 John T Norton Norelco Reporter, p 50, April-June, 1968 16.2 John T Norton Materials Evaluation, p 21A, February 1973 (Lester Honor Lecture) 16.3 M E Hilley; J A Larson; C F Jatczak; and R E Ricklefs, editors Residual Stress Measurement by X-Ray Difiction SAE lnformation Report J 784a (1971) Available from the Society of Automotive Engineers, Inc., 400 Commonwealth Drive, Warrendale, Pa 15096 16.4 F Gisen; R Glocker; and E Osswald Tech Physik, 17, 145 (1936) 16.5 A L Christenson and E S Rowland Trans ASM, 45,638 (1953) 16.6 R E Ogilvie M.S thesis, Mass lnst of Technology (1952) Work directed by John T Norton 16.7 D P Koistinen and R E Marburger Trans ASM, 51, 537 (1959) 16.8 M G Moore and W P Evans Trans SAE, 66, 340 (1958) 16.9 H R Letner Trans ASME, 77, 1089 (1 955) 16.10 M R James and J B Cohen Adv in X-Ray Analysis, 19,697 (1976) 16.11 Eric W Weinman; Joseph E Hunter; and Douglas D McCormack Progress, p 88, July 1969 Merai 16.12 Donald A Bolstad and William E Quist Adv in X-Ray Analysis, 8, 26 (1965) 16.13 John T Norton Ado in X-Ray Analysis, 11, 401 (1968) 16.14 J T Norton and D Rosenthal Proc Soc E-up Stress Analysis, 1( ) , 77 (1943) 16.15 Leopold Frommer and E H Lloyd J Inst Metals, 70,91 (1944) 16.16 Donald A Bolstad; Robert A Davis; William E Quist; and Earl C Roberts, Metal Progress, p 88, July 1963 16.17 R J Homicz SAE Paper 670151, SAE Congress, January 1967 16.18 M Raefsky; N Walter; W Lieberman; and J Clark SAE Paper 670154, SAE Congress, January 1967 16.19 Matthew J Donachie, Jr and John T Norton Trans ASM, 55, 51 (1962) 16.20 Paul S Prevey Ado in X-Ray Analysis, 20, 345 (1977) 16.21 Bruce M MacDonald Adv in X-Ray Analysis, 13,487 ( 1970) 16.22 A L Esquivel Adu in X- Ray Analysis, 12, 269 ( 969) 16.23 B N Ranganathan; J J Wert; and W N Clotfelter J of Testin{/ and Eralrration, 4, 218 (1976) 16.24 H R Woehrle; F P Reilly, Ill; W J Barkley, Ill; L A Jackman; and W R Clough Adu in X-Ray Analysis, 8,38 (1965) 16.25 M R James and J B Cohen Adv in X- Ray Analysis, 20,29 ( 1977) 16.26 B D Cullity Adu in X-Ray Analysis, 20,259 (1977) 16.27 D M Vasil'ev and B I Smirnov Societ Physics Uspekhi, 4, 226 (1961) (In English.) 16.28 Eckard Macherauch Exper Mechanics, 6, 140 (1966) 16.29 David N French J Amer Ceramic Soc., 52, 271 (1969) 542 Chapter references 16.30 R E Marburger and D P Koistinen Trans ASM, 53, 743 (1961) 16.31 W.J McG Tegart The Electrolytic and Chemical Polishing of Metals in Research and Industry, 2nd ed (New York: Pergamon, 1959) 16.32 M R James Ph.D thesis, directed by J B Cohen, Northwestern University (1 977) 16.33 F Bollenrath; V Hauk; and W Weidemann Archiv fur das Eisenhuttenwesen, 38, 793 (1967) 16.34 R H Marion and J B Cohen Adv in X-Ray Analysis, 18,466 (1975) Answers to Selected Problems Chapter 1-1 4.23 x 10" sec-', 2.80 x 10-l5 J; 1.95 x 10" sec-', 1.29 x 10-l5 J 1-6 33 cm2/gm 1-8 a) 26.97 cm2/gm, 3.48 x cm-I b) I,/I,, = 0.50 for 20 cm of air 1-11 1.541 A 1-9 8980 volts 1-13 8980 volts; mainly 1.54 A (Cu Ka) and 1.39 A (Cu KB) 1-15 0.80, 26 to 1-17 3.5 to Chapter 2-7 A section on (1210) will show this 211 Shear strain = 0.707 2-14 a) 20°S, 30°W; b) 27"S, 48"E; c) 39"S, 61°E Chapter 3-5 t = 21/(B cos 8) B - - 10" 45 80 0.31" 0.43 1.76 Chapter 4-1 A = A(@ = (1/2p)[1 - exp (-2ptlsin 8)] 4-2 34 percent reduction for copper, 82 percent for lead 4-4 F2 = for mixed indices; F2 = for (h + k + 1) an odd multiple of 2; F2 = 64fz for (h k + I ) an even multiple of 2; FZ = 32fz for (h k I) odd + + + 544 Answers to selected problems + 2p (as 1, 3, 5, ) 8p (as 8, 16, 24 ) 4(2p 1) (as 4, 12, 20, 28 ) 2(2p + 1) (as 2, 6, 10, 14 ) 8p I (as 1, 7, 9, 15, 17 ) 4(2p + I ) It I (as3, 5, 11, 13, 19,21 ) 8~ ( ~+ 1) WP + 1) n and p are any integers, including zero Line hkl Calc Jnt + - - 110 200 21 220 4-9 11 and 200 The ratio is 2400 to 10.0 1.7 3.5 1.1 Chapter 5-1 0.67 cm for (1 11); 0.77 cm for (200) 5-3 a) 3rd to 18th; b) 3rd to 15th Chapter 6-1 41 minutes 64 81" 6-2 A = R tan O/AS 6-6 4.2 cm Chapter 7-1 0.44" 7-2 2.8 percent 7-4 a) 0.39 percent; b) 0.67 percent a) 1.14 (Co) to I (Ni); b) 10.5 Chapter 8-1 8"N, 23"E; 74"S, 90°E; 16"S, 64"W 8-3 26" about beam axis, clockwise, looking from crystal to x-ray source; 3" about EW, clockwise, looking from E to W; 9" about NS, counterclockwise, looking from N to S Chapter 9-2 Diffractometer 9-5 b) 0.1 1,O.l7,0.28, and 0.44, listed in the order in which the incident beam traverses the layers 9-8 b) Decreased by 26 percent Answers to selected problems 545 Chapter 10 10-1 1I1, 200, 220, 311, 222,400, 331,420, 422, and 41 (333); a = 4.05 A 10-4 100,002,101,102,110 10-6 I 11, 220, 311,400, 331,422, 51 (333), 440 Diamond cubic; a = 5.4 a; silicon 10-8 100, 002, 101, 102, 110, 103, 200, 112 Hexagonal close-packed; a = 3.2 5.2 A; magnesium Chapter 11 11-1 + 1.7"C 11-3 4.997 A 11-5 Near = 34" Chapter 12 12-1 + 0.0002 A Chapter 13 13-2 0.0010 Chapter 14 14-1 Bas 14-3 Mixture of Ni and NiO 14-5 13 volume percent Chapter 15 15-1 a) 1.2", 1.9", and 5.9" for 200, 220, and 420 b) 1.1 ",1 So, and 2.4" c) All Chapter 16 161 a) = 1.1702A,d, = 1.1691 A, di = 1.1716A b) -0.1 percent 165 -65 ksi without, and - 63 ksi with, correction by the LPA factor A, c = Index Index Absorption of x-rays, 13 Absorption coefficients, 13 table, 12 Absorption edges, table, 509 Absorption factor, deb ye-Schemer, 132 diffractometer, 133 for reflection from flat plate, 133 for transmission through flat plate, 306 Amorphous solids, 321 Annealing texture, 295 Annealing twins, 59 Anomalous dispersion, 13, 392 Anomalous transmission, 275 Asterism, 261 thermal, 495 ASTM, diffraction data cards, 398 grain size number, 282 Atomic scattering factor, 112 change near absorption edge, 113, 392 table, 520 Atomic weights, table, 567 Atom sizes, 56 table, 506 AuBe structure, 53 AuCu, ordering in, 390 AuCu,, ordering in, 383 Auger effect, 18 Auger electrons, 14 Austenite, 341, 375 Austenite determination, 1 Background radiation, powder method, 179 Back-reflection focusing camera, 172 errors, 358 Back-reflection Laue camera, 152 Back-reflection Laue method, 92 for crystal orientation, 233 Back-reflection pinhole camera, 175 errors, 358 Balanced filters, 227 Basal plane, 45 Baseline, 14 Berg-Barrett method, 270 Body-centered cubic structure, 47 Borrmann method, 275 Bragg law, 84, 86 Bravais lattice, 34 table, 35 Bremsstrahlung, Broad lines, measurement of, 460 Bunn chart, 332 Caesium chloride structure, 51 Cameras (see X-ray cameras) Cell distortion, effect on powder pattern, 340 Cell volumes, equations, 502 Centering translations, 41 Characteristic radiation, wavelength table, 509 550 Index Chemical analysis by diffraction, 397 qualitative, 398 computer searching, 405 quantitative, 407 direct comparison method, 41 external standard method, 409 internal standard method, 15 Chemical analysis by spectrometry, 421 energydispersive, 439 intensity, 439 qualitative, 442 quantitative, 442 resolution, 439 selective excitation, 441 selective filtration, 44 spectrometers, 439, 442 wavelength-dispersive, 425 automatic, 429 counters, 428 crystals, 428 intensity, 432 multichannel, 430 qualitative, 434 quantitative, 434 resolution, 432 single-channel, 425 spectrometers, 422, 425 Chemical analysis by parameter measurement, 407 Choice of radiation, 178 Clustering, 394 Cohen's method, 363 Coherent scattering, 108, 114 Cold work (see Plastic deformation) Collimators, 156, 165 Complex exponential functions, 119 Compton effect, 110 Compton modified radiation, 110, 115 Conservation of diffracted energy, 138 Continuous spectrum, Coordination number, 57 Counters, 28, 199 energy resolution, 202, 210 Geiger, 208 proportional, 204 gas-flow, 207 position-sensitive, 207, 19, 466 scintillation, 208 semiconductor, 210 Counting efficiency, 20 Counting error, 220 Counting losses, 200 Counting-rate meter (see Ratemeter) Crystal monochromators focusing, 181 use with diffractometer, 226 Crystal orientation distribution, 15 Crystal quality, 95, 260, 279 Berg-Barrett method, 270 Borrmann method, 275 Guinier-Tennevin method, 265 Lang method, 27 Laue method, 260 in polycrystals, 285 by rocking curves, 277 Crystal setting, 256 Crystal structure, 47 of compounds, table, 508 of elements, table, 506 Crystal-structure determination, 324 example of, 345 Crystal systems, 34 table, 35 CsCl structure, 51 Cube texture, 298 Cube-on-corner texture, 13 CuZn, ordering in, 389 Debye-Scherrer camera, 162 Debye-Schemer method, 97, 162 errors, 352 film loading, 167 intensity equation, 139 specimen preparation, 166 Decker, Asp, and Harker method, 305 Defect structures, 343, 377 Defocusing error, 10 Deformation texture, 295 Deformation twins, 62 Densities, table, 12 Depth of x-ray penetration, 292 betectors (see Counters) Diamond structure, 52 Diffraction, Index Diffraction and reciprocal lattice, 487 Laue method, 492 powder method, 490 rotating-crystal method, 489 Diffraction methods, 92 Diffmctometer, 98 absorption factor, 133 automatic, 195 double-crystal, 277 errors, 359 general features, 189 intensity equations, 139, 408 optics, 196 single-crystal, 195 specimen preparation, 194 Diffractometry energy-dispersive, 17 time-analysis, 19 wavelength-dispersive, 189 Diffusion studies, by parameter measurement, 407 Disappearing-phase method, 377 Doublet, Electron diffraction, 294, 497 Electron microscope, 444 Empirical-coefficient method, 436 Energies, Kcu line, table, 509 Energy-dispersive diffractometry , 17 Energy-dispersive spectrometry, 439 Enhancement, 435 Escape peak, 215 Ewald construction, 488 Excitation voltage, Extinction, 140, 268, 418 Extinction contrast, 27 Face-cen tered cubic structure, 47 Ferrite, 52, 55 FeSi structure, 54 Fiber axis, 296 Field and Merchant method, 10 Filters, 19 balanced (Ross), 227 table, 21 Fink method, 400 Fluorescence yield, 19 551 Fluorescent analysis (see Chemical analysis by spectrometry) Fluorescent radiation, 15, 14 Fluorescent screens, 27 Focal spot, 24 Focusing cameras, 170 Form, l , Fundamental lines, 385 Fundamental-parameter method, 438 Geiger counter, 208 Ge(Li) detector, 10 Grain growth, 288 Grain size, 281 Greninger chart, 236 Guinier camera, 183 Guinier-Tennevin method, 265 Hanawalt method, 399 Hexagonal close-packed structure, 47 Hexagonal-rhombohedra1 transformation, 504 High-temperature cameras, 169 Hull-Davey chart, 33 Image intensification, 153, 260 Incoherent scattering, 111 Indexing powder patterns cubic crystals, 327 noncubic crystals, analytical, 337 graphical, 330 Indices, of directions, 41 of planes, 42 Integrated intensity, 127, 408 measurement with scaler, 221 Integrating camera, 177 Intensification, image, 153, 260 Intensifying screens, 153 Intensities of powder pattern lines in Debye-Schemer camera, 139 in diffractometer, 139, 408 Intensity calculations, CdTe, 345 copper, 140 ZnS (zinc blende), 142 Intensity measurements, photographic, 185 Internal stress (see Residual stress) 552 Index Interplanar angles cubic system, table, 75 equations, 502 Interstitial solid solutions, 55, 375 Inverse pole figures, 19 Ionization chamber, 204 JCPDS (Joint Committee on Powder Diffraction Standards), 398 kX unit, 4, 90 Lang method, 271 Lattice, 32 Lattice parameters, 33 of some compounds, table, 508 of some elements, table, 506 Lattice-parameter measurements, 350 with back-reflection focusing camera, 358 with Debye-Schemer camera, 352 with diffractometer, 359 with pinhole camera, 358 Laue cameras, back-reflection, 152 specimen holders, 155 transmission, 150 Laue equations, 488 Laue method, 92,492 back-reflection, 92 in color, 149 diffraction spot shape, 158 transmission, 92 Least squares, method of, 360 Leonhardt chart, 249 Limiting sphere, 491 Line broadening, due to fine particle size, 100, 284 nonuniform strain, 286 stacking faults, 287 Long-range order, 383 Long-range order parameter, 386 Lorentz factor, 127 Lorentz-polarization factor, 131 modified, 463 when monochromator is used, 183 table, 524 Low-temperature cameras, 169 Macrostrain, 447 Macrostress, 286 Martensite, 341, 375, 41 Matrix absorption (in fluorescence), 434 MCA (multichannel analyser), 216 microabsorption, 18 Microanalysis, 444 Microbeam, 177 Microcamera, 177 Microdensitometer, 185 Microphotometer, 185 Microprobe, 444 Microstrain, 447 Microstress, 286 Miller-Bravais indices, 44 Miller indices, 42 quadratic forms, table, 516 Monochromators (see Crystal monochromators) Mosaic structure, 103 Moseley's law, 10, 16 Multiple excitation, in fluorescence, 435 Multiplicity factor, 127 table 523 NaCl structure, Neutron diffraction, 394, 498 Nondispersive diffractometry , 17 Nondispersive spectrometry, 439 Nonprimitive cells, 34, 39 Octahedral plane, 45 Octahedral site, 57 Optimum specimen thickness, 176, 307 Order, long-range, 383 parameter, 386 short-range, 394 Order-disorder transformations, 383 in AuCu, 390 in AuCu,, 383 in CuZn, 389 Orientation of single crystals, 233 by back-reflection Laue method, 233 by diffractometer method, 250 by transmission Laue method, 247 Parametric method, 379 Particle size, 284 Index Particle-size broadening, 100, 284 Penetration depth (x-rays), 292 Phase diagrams, determination of, 369 Photoelectrons, 15, 114 Photographic film, 27 Photographic measurement of intensity, 185 Photomultiplier, 208 Physical constants, table, 528 Pinhole method, cameras, 175 conclusions from film inspection, 322 errors, 358 for texture determination, 300 Plane-spacing equations, table, 501 Plastic deformation effect on Laue photographs, 260 effect on powder photographs, 285 Point lattice, 33 Polarization factor, 110 when monochromator is used, 183 Pole figure, 297 Polycrystalline aggregates, 28 crystal orientation, 295 crystal quality, 285 crystal size, 28 1, 284 Polygonization, 264, 288 Polymers, 321 Position-sensitive counter, 207, 19, 466 Powder diffmction file, 398 Powder method, 96, 161, 490 Preferred orientation (see Texture) Primitive cells, 34, 39 Principal stresses, 454, 456 Prismatic plane, 45 Proportional counter, 204 Pseudo-macrostress , 477 Pulse-height analyzer, single channel, 214 multichannel, 216 Pulse-height discriminator, 14 Pyramidal plane, 45 Quadratic forms of Miller indices, table, 516 Quartz, determination in dust, 416 Radiography, Ratemeter, 191, 222 553 calibration, 226 errors, 224 Reciprocal lattice, 480 Recovery, 288 Recrystallization, 288 Recrystallization texture, 295 Residual stress, 286, 448 after grinding, 464 measurement (see Stress measurement) pseudo-macrostress, 477 in weldments, 449, 471 Resolving power for plane spacings, 162, 171, 173 for wavelengths, 173, 432,440 Retained austenite determination, 11 Rhombohedral-hexagonal transformation, 504 Rock-salt structure, Rocking curves, 277 Ross filters, 227 Rotating-crystal method, 95, 340, 489 Safety precautions, 29 Scalers, 191, 219 errors, 219 use in measuring integrated intensity, 22 Scattering (see X-ray scattering) Schemer formula, 102 Schulz reflection method, 308 Schulz transmission method, 308 Scintillation counter, 208 Seemann-Bohlin camera1 170 Semiconductor counter, 10 Setting a crystal in a required orientation, 256 Sheet texture, 297 Short-range order, 394 Short-wavelength limit, Si (Li) detector, 210 (sin @/A values, table, 518 sin2 values, table, 514 Small-angle scattering, 285 of neutrons, 500 Sodium chloride structure, 51 Solid solutions, defect, 343, 377 interstitial, 55, 375 554 Index ordered, 56, 383 substitutional, 54, 376 Soller slits, 197, 427 Space groups, 344 Spectral width, 103 Spectrometer, 88, 422, 425, 439 Sphere of reflection, 488 Standard projections, 72-77 Stereographic projection, 63 Stereographic ruler for back-reflection Laue, 243 for transmission Laue, 250 Straumanis method, 168 Stress measurement, 447 calibration, 472 diffractometer method, 456 automated, 463 line position, 460 special instruments, 466 stress constant, 458, 459, 471 table, 460 variant techniques, 465 Fastress, 467 oscillations in di, 478 photographic method, 469 practical difficulties, 476 precision and accuracy, 474 principal stresses, 454, 456 pseudo-macrostress, 477 subsurface, 464 Structure factor, 19 of BCC element, 122 of FCC element, 122 of HCP element, 125 of NaC1, 123 of ZnS (zinc blende), 142 Subgrain structure, 103 Substitutional solid solutions, 54, 376 Superlattice, 56, 383 Superlattice lines, 385, 391 Surface deposits, identification of, 406 Symmetry elements, 37 table 39 Temperature factor, 135 table, 526 Temperature-diffuse scattering, 138 Tetrahedral site, 57 Texture (preferred orientation), 295 Texture determination diffractometer method (sheet), 303 diffractometer method (wire), I6 photographic method (wire), 300 Thermal asterism, 495 Thickness of specimen, optimum, 176, 307 Thomson equation, 109 Time constant, 223 Time width of slit, 225 Topographic methods, 267 Torsional strain, 266 Transmission Laue camera, 150 Transmission Laue method, 92 for crystal orientation, 247 Twinned crystals, 59, 78 Twins, annealing, 59 deformation, 62 Unit cell, 33 Unit-cell volume, equations, 502 Uranium structure, 50 Vegard's law, 376 Wavelengths of absorption edges, table, 509 of characteristic lines, table, 509 White radiation, Window, 214 Wire texture, 296 Wulff net, 67 X unit, 4, 90 X-ray cameras, 150 back-reflection focusing, 172 Debye-Schemer, 162 Guinier, 183 high-pressure, 169 high-temperature, 169 Laue, 150, 152 low-temperature, 169 microcamera, 177 pinhole, 175 Seeman-Bohlin , 170 stress, 471 Index X-ray density, 88 X-ray images, 446 X-ray microscopy, 267 X-ray scattering, 14 by amorphous solids, 105 by an atom, 111 coherent, 108 Compton modified, 111 by an electron, 107 by gases and liquids, 105 incoherent, 111 by random solid solutions, 387 at small angles, 285 temperature-diffuse, 138 by a unit cell, 115 X-ray spectroscopy, 88 X-ray topography, 267 X-ray tubes, filament, 21 gas, 21 microfocus, 26 miniature, 26 pulsed (flash), 26 rotating-anode, 25 X-rays, absorption of, 13 characterjstic, continuous, depth of penetration, 292 detection of, 27 fluorescent, 15 production of, 21 safety precautions, 28 ZnS (zinc blende) structure, 53 Zone, 45 Zone law, 45, 486 555