Symmetry : an introduction to group theory and its applications /

Preface p. vii
Chapter 1 Groups
1.1 Symbols and the group property p. 1
1.2 Definition of a group p. 6
1.3 The multiplication table p. 7
1.4 Powers, products, generators p. 9
1.5 Subgroups, cosets, classes p. 11
1.6 Invariant subgroups. The factor group p. 13
1.7 Homomorphisms and isomorphisms p. 14
1.8 Elementary concept of a representation p. 16
1.9 The direct product p. 18
1.10 The algebra of a group p. 19
Chapter 2 Lattices and Vector Spaces
2.1 Lattices. One dimension p. 22
2.2 Lattices. Two and three dimensions p. 25
2.3 Vector spaces p. 27
2.4 n-Dimensional space. Basis vectors p. 28
2.5 Components and basis changes p. 31
2.6 Mappings and similarity transformations p. 33
2.7 Representations. Equivalence p. 38
2.8 Length and angle. The metric p. 41
2.9 Unitary transformations p. 47
2.10 Matrix elements as scalar products p. 49
2.11 The eigenvalue problem p. 51
Chapter 3 Point and Space Groups
3.1 Symmetry operations as orthogonal transformations p. 54
3.2 The axial point groups p. 59
3.3 The tetrahedral and octahedral point groups p. 69
3.4 Compatibility of symmetry operations p. 75
3.5 Symmetry of crystal lattices p. 78
3.6 Derivation of space groups p. 85
Chapter 4 Representations of Point and Translation Groups
4.1 Matrices for point group operations p. 91
4.2 Nomenclature. Representations p. 95
4.3 Translation groups. Representations and reciprocal space p. 105
Chapter 5 Irreducible Representations
5.1 Reducibility. Nature of the problem p. 109
5.2 Reduction and complete reduction. Basic theorems p. 110
5.3 The orthogonality relations p. 116
5.4 Group characters p. 121
5.5 The regular representation p. 124
5.6 The number of distinct irreducible representations p. 125
5.7 Reduction of representations p. 126
5.8 Idempotents and projection operators p. 131
5.9 The direct product p. 133
Chapter 6 Applications Involving Algebraic Forms
6.1 Nature of applications p. 140
6.2 Invariant forms. Symmetry restrictions p. 141
6.3 Principal axes. The eigenvalue problem p. 147
6.4 Symmetry considerations p. 150
6.5 Symmetry classification of molecular vibrations p. 151
6.6 Symmetry coordinates in vibration theory p. 159
Chapter 7 Applications Involving Functions and Operators
7.1 Transformation of functions p. 166
7.2 Functions of Cartesian coordinates p. 170
7.3 Operator equations. Invariance p. 174
7.4 Symmetry and the eigenvalue problem p. 181
7.5 Approximation methods. Symmetry functions p. 187
7.6 Symmetry functions by projection p. 190
7.7 Symmetry functions and equivalent functions p. 195
7.8 Determination of equivalent functions p. 197
Chapter 8 Applications Involving Tensors and Tensor Operators
8.1 Scalar, vector and tensor properties p. 203
8.2 Significance of the metric p. 206
8.3 Tensor properties. Symmetry restrictions p. 208
8.4 Symmetric and antisymmetric tensors p. 211
8.5 Tensor fields. Tensor operators p. 218
8.6 Matrix elements of tensor operators p. 224
8.7 Determination of coupling coefficients p. 231
Appendix 1 Representations carried by harmonic functions p. 235
Appendix 2 Alternative bases for cubic groups p. 241
Index p. 245