Elements of X-Ray Diffraction

569 Pages·1978·25.016 MB·English

by  Bernard Dennis Cullity

Elements of
X-RAY
DIFFRACTION
SECOND EDITION
B. D. CULLITY
Department of Metallurgical Engineering and Materials Science
University of N(cid:244)tre 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
0
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 parti-
cle 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 knowl-
edge 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. How-
ever, little or no modification of experimental method is required for the examina-
tion of nonmetallic materials, inasmuch as the physical principles involved do 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 con-
cerned 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 meth-
ods, 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 la6
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 2 and 3.
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, micro-
cameras 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 orien-
tation 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.D unn 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 B. D. Cullity
November 1977

Contents
FUNDAMENTALS
Chapter 1 Properties of X-rays
Introduction . . . . . . . . . . . . . . . . 3
Electromagnetic radiation . . . . . . . . . . . . 3
The continuous spectrum . . 6
.The characteristic spectrum . . . . . . . . . . . . 8
Absorption . . . . . . . . . . . . . . . . . 13
Filters . . . . . . . . . . . . . . . . . . 19
Production of x-rays . . . . . . . . . . . . . . 21
Detection of x-rays . . . . . . . . . . . . . . 27
Safety precautions . . . . . . . . . . . . . . . 29
Chapter 2 Geometry of Crystals
2-1 Introduction . . . . . . . . . . . . . . . . 32
. . . . . . . . . . . . . . . . . .
2-2 Lattices 32
2-3 Crystal systems . . . . . . . . . . . . . . . . 34
2-4 Symmetry . . . . . . . . . . . . . . . . . 37
. . . . . . . . . . .
2-5 Primitive and nonprimitive cells 39
. . . . . . . . . . .
2-6 Lattice directions and planes 41
2-7 Crystal structure . . . . . . . . . . . . . . . 47
. . . . . . . . . . . .
2-8 Atom sizes and coordination 56
. . . . . . . . . . . . . . . .
2-9 Crystal shape 58
2-10 Twinned crystals . . . . . . . . . . . . . . . 59
2-1 1 The stereographic projection . . . . . . . . . . . . 63
Chapter 3 Diffraction I: Directions of Diffracted Beams
. . . . . . . . . . . . . . . .
3-1 Introduction 81
. . . . . . . . . . . . . . . . .
Diffraction 82
. . . . . . . . . . . . . . . .
The Bragg law 86
. . . . . . . . . . . . . .
X-ray spectroscopy 88
. . . . . . . . . . . . . .
Diffraction directions 91
. . . . . . . . . . . . . .
Diffraction methods 92
. . . . . . . . . .
Diffraction under nonideal conditions 99
vii
viii Contents
.
Chapter 4 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 Laue Photographs
5-1 Introduction . . . . . . . . . . . . . 1 49
5-2 Cameras . . . . . . . . . . . . . . . . 1 50
5-3 Specimens and holders . . . . . . . . . . . . 1 55
5-4 Collimators . . . . . . . . . . . . . 15 6
5-5 The shapes of Laue spots . . . . . . . . . . . . . 1 58
Chapter 6 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 Diffractometer and Spectrometer Measurements
7-1 Introduction . . . . . . . . . . . . . . . . 1 88
7-2 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 8 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 9 Structure of Polycrystalline Aggregates
Introduction . . .
Crystal Size
Grain size . . . . . . . . . .
Particle size . . . . . . . . .
Crystal Quality
Crystal quality . . . . .
Depth of x-ray penetration .
Crystal Orientation
General . . . . . . . . . . .
The texture of wire (photographic method)
The texture of sheet (diffractometer methods) .
The texture of wire (diflractometer method) .
Inverse pole figures . . . . . . . .
~morphokS olids
Amorphous and semi-amorphous solids . .
9-12 Summary . . . . . . . . . .
10
Chapter 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 . . . . . . . .
1
Chapter 2 Phase-Diagram Determination
. . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . .
curves (disappearing-phase method) . .
curves (parametric method) . . . . .
Chapter 43 Order-Disorder Transformations
Chapter 44 Chemical Analysis by X-ray Diffraction
141
Introduction . . . . . . . . . . . . . . .
1
Qualitative Analysis
Basic principles . . . . . . . . . . . . . . .
Powder diffraction file . . . . . . . . . . . . . .
Procedure . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . .
Examples of analysis
Practical difficulties . . . . . . . . . . . . . .