Mathematical Techniques in Crystallography and Materials Science

200 Pages·1982·4.746 MB·German

by  Edward Prince Ph.D. (auth.)

Mathematical Techniques
in Crystallography and
Materials Science
Edward Prince
Mathematical Techniques
in Crystallography and
Materials Science
With 28 Illustrations
Springer Science+Business Media, LLC
Edward Prince, Ph.D.
National Bureau of Standards
Washington, D.C. 20234, U.S.A.
Sponsoring Editor: Philip Manor
Production: Abe Krieger
Library of Congress Cataloging in Publication Data
Prince, Edward.
Mathematieal teehniques in crystallography
and materials seience.
Bibliography: p.
Includes index.
1. Mathematieal erystallography. I. Title.
QD911.P75 548'.7 81-18443
AACR2
ISBN 978-3-540-90627-8 ISBN 978-3-662-25351-9 (eBook)
DOI 10.1007/978-3-662-25351-9
© 1982 by Springer Seienee+Business Media New York
Originally published by Springer-Verlag New York, Ine. in 1982.
Softcover reprint ofthe hardcover 1st edition 1982
All rights reserved. No part of this book may be translated or reprodueed in any
form without written permissiun from Springer Science+Business Media, LLC.
The use of general deseriptive names, trade names, trademarks, ete. in this publ i ea
tion, even if the former are not espeeially identified, is not to be taken as a sign that
such names, as understood by the Trade Marks and Merchandise Marks Aet, may
aecordingly be used freely by anyone.
987654321
Preface
In the course of 30 years as a practicing crystallographer I have frequently
been faced with the necessity of finding out a little bit about some general
branch of mathematics with which I was previously unfamiliar. Under
these circumstances I have usually followed the common practice of
seeking out some colleague who would be expected to have a thorough
knowledge of the subject. I would then find mys elf faced either with an
involved lecture in which the colleague would attempt to distill a lifetime of
experience into a form that was comprehensible to a novice with a very
different background, or with a book about the subject, written by a
specialist, that contained far more information than I really wanted to
know. I would have to separate the few kerneis of useful material from a
large volume of what would probably be wheat to someone else, but was
chaff to me. In the course of that time I have acquired a collection of books
to which I frequently refer. Most of these have a small number of
thoroughly dog-eared pages, along with many that have scarcely been
opened.
During the same period I have been privileged to associate and collabo
rate with a number of materials scientists who were not trained as crystal
lographers, but whose interests required them to understand particular
details of some structural problem. In these cases I tried to deal with the
problem at hand without requiring them to learn more than they wanted to
know about crystallography. As a consequence of these experiences I have
used two criteria in choosing the material to be included in this book.
Either they are things that I have had to learn or look up frequently
because I didn't use them enough to retain the details in ready memory, or
they are things that I have frequently explained to other colleagues. It is my
hope that I have covered these topics in sufficient detail to answer the
day-to-day questions, or that, if the detail is not sufficient, I will have
provided enough to get a person star ted on a fruitful search. Most active
vi Preface
researchers will find much that they already know, but I would expect that
there will be very few who will not find something that they do not know as
well as they feel they should.
The level of mathematics I have assumed is that included in college
courses ca lied "Advanced Calculus," or something of that sort. It includes
partial differentiation, infinite series, and improper integrals. I have not
bothered with proving the convergence properties of series and integrals.
The reader is asked to assume that these proofs can be found elsewhere if I
use them. In the discussion of rigid-body thermal motion I have deliber
ately omitted any discussion of Bessel's functions. Although there are
exceptions, it has been my experience that these usually appear because of
a failure to replace a trigonometrie function by its power series expansion
at a suitable point.
My own knowledge of these topics is a result of contact with many other
workers. The discussion of sublattices and superlattices is based on the
work of my colleagues A. D. Mighell and A. Santoro. The discussion of
constrained refinement is based on work in the course of which I have had
a fruitful collaboration with L. W. Finger, and the sections on rigid-body
motion owe at great deal to the work of C. K. Johnson. In the development
of robustjresistant techniques I have been privileged to collaborate with
W. L. Nicholson. If the discussions are correct, they owe much to the
contributions of these workers, as well as to the mentors of my early years,
D. Harker and W. A. Wooster. If any derivations and arguments are not
correct, the blame is entirely mine.
In writing this book I have endeavored to write not a textbook but a
reference book-a vade mecum for active research workers. There have
been many times when I would have found it useful to have such a book
within my reach. I hope others will also find it useful.
Bethesda, Maryland Edward Prince
January, 1982
Contents
Chapter 1 Matrices: Definitions and Fundamental Operations 1
Fundamental Matrix Operations 3
Linear Algebra 5
Eigenvalues 9
Linear Transformations 13
Rotation ofAxes 14
The Metrie Tensor 17
Chapter 2 Sy~metry of Finite Objects 20
Groups 20\
Representations 21
Point Groups 22
Basis Funetions 38
Chapter 3 Symmetry of Infinitely Repeated Patterns 41
Bravais Lattices 41
Space Groups 42
Chapter 4 Vectors 49
Scalar and Vector Products 50
The Reeiprocal Lattice 52
The Orientation Matrix 54
Zones and Forms 56
Sublattiees and Superlattices 56
Chapter 5 Tensors 58
Covarianee and Contravarianee 59
The Multivariate Normal Distribution 60
Anisotropie Temperature Faetors 62
The Equivalent Isotropie Temperature Factor 65
Effeet of Symmetry 65
Contents
VllI
Tensors of Higher Ranks 67
Moments and Cumulants 69
Rigid-Body Motion 72
Chapter 6 Data Fitting 77
Fitting Funetions 77
Finding the Minimum 82
False Minima 90
Chapter 7 Estimation of Uncertainty 93
Estimates 94
The Preeision of Estimates of Preeision 99
Models with More than One Parameter 100
Estimates of Uneertainty When the Algorithm Is Not Least
Squares 103
Chapter 8 Significance and Accuracy 105
The F Distribution 105
Student's t Distribution 109
Correlation 110
The Relation between Preeision and Aeeuraey 113
Uneertainties of Derived Funetions: Propagation of Errors
117
Chapter 9 Constrained Crystal Structure Refinement 119
The Observed Data 119
The Model 120
The General Form for a Constrained Model 122
Shape Constraints 124
Rigid-Body Thermal Motion Constraints 126
Chemieal Constraints 128
Representing non-Gaussian Distributions 129
Appendix A Eigenvalues and Eigenveetors of 3 X 3 Symmetrie
Matriees 134
Appendix B Stereographie Projeetion 136
Appendix C Sublattiees and Superlattiees 140
Appendix D The Probability Integral, the Gamma Funetion, and
Related Topies 142
Appendix E The Harmonie Oseillator in Quantum Meehanies:
Bloeh's Theorem 147
Appendix F Symmetry Restrietions on Seeond-, Third-, and
Fourth-Rank Tensors 155
Appendix G Some Useful Computer Programs 161
Bibliography 188
Index 189
Chapter 1
Matrices: Definitions and
Fundamental Operations
The everyday activities of the crystallographer or materials scientist deal
with the real physical properties of real objects in the real, three
dimensional space in which we live. Most of these properties can be
described, at least approximately, by systems of linear relationships be
tween one set of measurable quantities, "causes," and another set of
measurable quantities, "effects." We shall find that a useful shorthand for
expressing these relationships is the algebra of matrices. To make use of this
algebra we need to start, if only to make sure we agree on notation, with
definitions of matrices and their fundamental operations.
Definition 1: A matrix is a two-dimensional, rectangular array of numbers.
The numbers may be positive or negative, real, imaginary, or complex.
In this text we denote a matrix, when referring to it as a whole, usually
by a capital, boldface Roman letter, e.g., A. We denote the elements of
a matrix by the same capital letter in regular type, and with either
numbers or lowercase italic letters as subscripts, as in A23 or Aij. The
matrix has horizontal "rows" and vertical "columns." The row index is
given first, so that A23, above, indicates the element of A that is a
member of the second row and of the third column. A general matrix
has m rows and n columns. When necessary the matrix is written out,
enclosed in parentheses:
There are certain special types of matrix it is useful to define.
2 Matrices: Definitions and Fundamental Operations
Definition 2: A matrix is square if the number of rows 1S equal to the
number of columns.
Definition 3: A matrix with only one row is a row vector.
Definition 4: A matrix with only one column is a column vector. We shall
use column vectors frequently. They will usually be denoted by lower
case boldface letters, e.g., x, and their elements represented by a single
subscript, as in
Xi'
Definition 5: The elements of a matrix for which i = j are diagonal elements.
The elements for which i =1= j are off-diagonal elements.
Definition 6: A matrix is diagonal if Aij = 0 for all off-diagonal elements.
Definition 7: A matrix is upper triangular if Aij = 0 when i > j.
Definition 8: A matrix is lower triangular if Aij = 0 when i < j. Upper
triangular, lower triangular, and diagonal matrices will usually also be
square. A diagonal matrix is obviously also both upper triangular and
lower triangular.
Definition 9: A matrix B is the transp ose of A, designated by AT, if BJi = Aij'
Neither A nor B need be square, but, obviously, if A has m rows and n
columns, then B has n rows and m columns.
Definition 10: A null matrix is a matrix all of whose elements are equal to O.
It is also useful at this point to define several explicit functions of the
elements of square matrices.
Definition 11: The trace of the matrix A, designated by Tr A, is given by
n
TrA= ~Aii'
i=t
where the ~ denotes "the sum of all such terms as i takes all values
from I to n." The trace is simply the sum of the diagonal elements.