The Mathematical Theory of Symmetry in Solids

This book gives the complete theory of the irreducible representations of the crystallographic point groups and space groups. This is important in the quantum-mechanical study of a particle or quasi-particle in a molecule or crystalline solid because the eigenvalues and eigenfunctions of asystem belong to the irreducible representations of the group of symmetry operations of that system. The theory is applied to give complete tables of these representations for all the 32 point groups and 230 space groups, including the double-valued representations. For the space groups, thegroup of the symmetry operations of the k vector and its irreducible representations are given for all the special points of symmetry, lines of symmetry and planes of symmetry in the Brillouin zone. Applications occur in the electronic band structure, phonon dispersion relations and selectionrules for particle-quasiparticle interactions in solids. The theory is extended to the corepresentations of the Shubnikov (black and white) point groups and space groups.

Arthur P. Cracknell graduated in Physics from the University of Cambridge in 1961, took his DPhil from the University of Oxford in 1964 and worked in the Physics Departments of the University of Singapore and the University of Essex before moving to Dundee as senior lecturer in 1970. He was promoted to Reader in 1974 and to Professor of Theoretical Physics in 1978 and later was transferred to the Carnegie Chair of Physics in the Department of Applied Physics and Electronic & Mechanical Engineering at Dundee University. He retired in 2002. He has won the Society Medal, The Remote Sensing Society, 1989; the Schiwefdsky Medal of the German Photogrammetric Society/ISPRS, 1996, and the President's award, Remote Sensing Society, 1996.
Christopher J. Bradley won the Mayhew Prize for Distinction in Applied Mathematics, Cambridge, in 1959, before becoming Fellow in Applied Mathematics, Jesus College and CUF Lecturer in Mathematics, Oxford 1963-80. He later joined the teaching profession, and was Head of Mathematics, then Director of Studies, at Clifton College, Bristol, 1981-2003. He is a Fellow of the Institute of Mathematics and its Applications.

Symmetry and The Solid State	p. 1
Introduction	p. 1
Group theory	p. 7
Group representations	p. 15
Point groups	p. 24
Space groups	p. 37
Symmetry-Adapted Functions for the Point Groups	p. 51
The matrix elements of the rotation group	p. 51
The generation of symmetry-adapted functions	p. 54
Application to the point groups	p. 56
Symmetry-adapted functions for the crystallographic point groups	p. 57
Active and passive operators	p. 76
Symmetrized and anti-symmetrized products of point-group representations	p. 77
Space Groups	p. 81
Bravais lattices	p. 81
Reciprocal lattices and Brillouin zones	p. 86
The classification of points and lines of symmetry	p. 95
The irreducible representations of the translation groups	p. 119
The classification of the 230 3-dimensional space groups	p. 121
The action of space-group operations on Bloch functions	p. 144
A descriptive account of the representation theory of space groups	p. 146
Examples: cubic close-packed and diamond structures	p. 161
The Representations of A Group in Terms of The Representations of An Invariant Subgroup	p. 171
Induced representations	p. 171
Groups with an invariant subgroup	p. 176
The theory of little groups	p. 181
The small representations of little groups	p. 184
The point groups as semi-direct products	p. 186
The reality of representations induced from little groups	p. 201
Direct products of induced representations	p. 204
Symmetrized and anti-symmetrized squares of induced representations	p. 215
The Single-Valued Representations of The 230 Space Groups	p. 225
Abstract groups	p. 225
The single-valued representations of the 230 space groups	p. 285
The labels of space-group representations	p. 389
Example of the use of the tables of the space-group representations	p. 397
Representation domain and basic domain	p. 407
The Double-Valued Representations of The 32 Point Groups and The 230 Space Groups	p. 418
The double-valued representations of the point groups	p. 418
Symmetry-adapted functions for double point groups	p. 437
The double-valued representations of the space groups	p. 454
An example of the deduction of the representations of a double space group	p. 458
The double-valued representations of the 230 space groups	p. 467
The Magnetic Groups and Their Corepresentations	p. 569
The Shubnikov groups and their derivation	p. 569
The classification of Shubnikov groups	p. 570
Anti-unitary operations and the corepresentations of magnetic groups	p. 605
The Kronecker products of corepresentations	p. 618
The corepresentations of the magnetic point groups	p. 622
The corepresentations of the magnetic space groups	p. 638
Example: the space-group corepresentations of P4′₂/mnm′ and their Kronecker products	p. 652
Spin-space groups	p. 669
Polychromatic point groups and space groups	p. 677
Appendix	p. 682
Bibliography	p. 684
Subject Index	p. 737
Table of Contents provided by Ingram. All Rights Reserved.

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The Mathematical Theory of Symmetry in Solids Representation Theory for Point Groups and Space Groups

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