Theories and Techniques of Crystal Structure Determination

by Shmueli, Uri

• ISBN13: 9780199213504
• ISBN10: 019921350X
• Format: Hardcover
• Copyright: 2007-09-07
• Publisher: Oxford University Press

Summary

This concise book for chemists, material scientists, and physicists who deal with description of crystalline matter and the determination of its structure, and would like to gain more understanding of the principles involved. The main purpose of the book is to introduce the reader to principles of crystallographic symmetry, to discuss some traditional, as well as modern, experimental techniques, to formulate the phase problem of crystallographic symmetry, to discus some traditional, as well as modern, experimental techniques, to formulate the phase problem of crystallography, and present in some detail the methods for its indirect and direct solution which are indispensable for further work. The book also contains discussions of structure-factor statistics, or value for resolving space-group ambiguities, and atomic displacement parameters, which form an inseparable part of the structure. A discussion of the refinement of structural parameters, conventional, constrained and restrained, concludes the book. Derivations are as far as possible, self contained and wherever mathematical detail might disrupt the line of reasoning the reader is referred to on of four appendices present in the book. The book is of course valuable for students of crystallography at a graduate and upper undergraduate level. No previous course on crystallography is a prerequisite for graduates in the above fields.

Author Biography

Uri Shmueli
Professor Emeritus
School of Chemistry
Tel Aviv University Tel Aviv University 1966 - present
Editor of Volume B of International Tables for Crystallography, from 1983; appointed by the Excutive Committee of the International Union of Crystallography (IUCr).
Ex officio member of the IUCr Commission on International Tables, from 1983.
Ex officio member of the IUCr Commission on Crystallographic Nomenclature, from 1983.

Table of Contents

Symmetry in crystals: fundamentals	p. 1
Introduction	p. 1
The lattice plane	p. 5
The reciprocal lattice	p. 10
Permissible rotational symmetries of a three-dimensional lattice	p. 13
Every lattice is centrosymmetric	p. 16
Orientation of symmetry axes with respect to the lattice	p. 17
Permissible combinations of axes of rotation	p. 18
Crystal systems	p. 22
Exercises for Chapter 1	p. 26
Point groups and lattice types	p. 29
Introduction	p. 29
Fundamentals of group theory	p. 29
Some important definitions	p. 33
Matrix representation of symmetry operations	p. 33
Isomorphism, coset decomposition, and the Lagrange theorem	p. 34
Crystallographic point groups	p. 37
The axial point groups	p. 38
Completion of the presentation of the point groups	p. 39
Bravais Lattices	p. 45
Exercises for Chapter 2	p. 47
Space-group symmetry	p. 49
Introduction	p. 49
Space-group symmetry operators	p. 49
Nonsymmorphic space-group operators	p. 52
Some general considerations	p. 59
Classification and tabulation of crystallographic space groups	p. 65
Location of symmetry elements	p. 65
Graphical representation of space groups	p. 68
Some computational aspects	p. 76
Description of Table 3.4 - explicit space-group symbols	p. 82
Exercises for Chapter 3	p. 83
X-ray diffraction techniques	p. 85
Introduction	p. 85
Diffraction conditions	p. 86
The Laue and Bragg equations	p. 86
Ewald's sphere of reflection	p. 90
Production of X-rays	p. 91
The X-ray tube	p. 91
Synchrotron radiation	p. 96
Detectors of X-rays	p. 100
X-ray film	p. 100
Imaging plate	p. 101
Charge-coupled device (CCD) detector	p. 102
The rotating-crystal method	p. 102
Moving-crystal-moving-film methods	p. 105
The Weissenberg method	p. 106
The de Jong-Bouman method	p. 106
The Buerger precession method	p. 106
The four-circle diffractometer	p. 107
Geometrical considerations	p. 107
The orientation matrix	p. 110
Coordinates and angles	p. 111
Comments on the experiment	p. 112
The Laue method	p. 113
Principle of the method	p. 113
Calculation of the Laue pattern	p. 114
Exercises for Chapter 4	p. 117
The structure factor and the electron density	p. 119
Introduction	p. 119
Scattering by a free charged particle	p. 119
Scattering by a material unit	p. 123
Scattering by a periodic array of material units	p. 125
The atomic scattering factor	p. 128
Space-group symmetry and the structure factor	p. 130
The Fourier synthesis of the electron density	p. 131
Computational considerations	p. 133
The phase problem	p. 134
Effects of space-group symmetry	p. 135
Effect of the lattice type	p. 138
Effect of screw axes and glide planes	p. 139
Exercises for Chapter 5	p. 142
The Patterson function	p. 143
Introduction	p. 143
The Patterson function	p. 143
Periodicity of the Patterson function	p. 145
The heavy atom method	p. 146
Overlap of interatomic vectors	p. 147
Classification of Patterson vectors	p. 148
Symmetry of the Patterson function	p. 149
Harker peaks	p. 151
Overview of Advanced Applications	p. 152
Superposition methods	p. 152
Rotation functions	p. 153
Translation functions	p. 155
Molecular replacement	p. 157
Exercises for Chapter 6	p. 159
Structure-factor statistics	p. 161
Introduction	p. 161
Wilson's method of scaling the intensities	p. 162
Basic Wilson statistics	p. 164
Cumulative distributions	p. 167
Moments of E	p. 168
Non-ideal structure-factor statistics	p. 170
Exercises for Chapter 7	p. 173
Direct methods	p. 177
Introduction	p. 177
Phase invariants and seminvariants	p. 179
Phase invariants	p. 179
Structure seminvariants	p. 181
Specification of the origin	p. 183
Inequalities	p. 186
The Sayre equation and the tangent formula	p. 190
Conditional probability density of a three-phase invariant	p. 192
Other probabilistic considerations	p. 198
Some practical comments on the solution of the phase problem	p. 201
Atomic displacement parameters	p. 207
Introduction	p. 207
Atomic displacement parameters in the Gaussian approximation	p. 210
Non-Cartesian representations	p. 211
Cartesian representations	p. 213
A rigid-body model	p. 215
Introduction	p. 215
Rotational displacement	p. 215
Rigid-body model parameters	p. 217
Refinement of structural parameters	p. 219
Introduction	p. 219
The linear least-squares method	p. 219
An example of a linear least-squares problem	p. 223
The non-linear least-squares method	p. 224
Conventional refinement	p. 226
Rigid-group constraints	p. 228
Introduction of restraints	p. 231
Appendix
Some geometrical considerations	p. 235
Transformation to a Cartesian system	p. 235
Change of basis and trace invariance	p. 236
The finite rotation operator	p. 237
Fundamentals of tensor notation	p. 241
Introduction	p. 241
Transformations	p. 242
The scalar product and metric tensors	p. 242
Examples	p. 243
Basic notions from theory of probability	p. 247
Introduction	p. 247
Random variables	p. 247
The probability density function and related quantities	p. 248
Joint and conditional pdfs	p. 249
Characteristic fanction and cumulants	p. 250
The discrete Fourier transform	p. 253
Some properties of the discrete Fourier transform	p. 253
The Fast Fourier Transform	p. 257
References	p. 261
Index	p. 267
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