Symmetry An Introduction to Group Theory and

This well-organized volume develops the elementary ideas of both group theory and representation theory in a progressive and thorough fashion. Designed to allow students to focus on any of the main fields of application, it is geared toward advanced undergraduate and graduate physics and chemistry students. 1963 edition. Appendices.

Preface

vii

Groups

Symbols and the group property

1

(5)

Definition of a group

6

(1)

The multiplication table

7

(2)

Powers, products, generators

9

(2)

Subgroups, cosets, classes

11

(2)

Invariant subgroups. The factor group

13

(1)

Homomorphisms and isomorphisms

14

(2)

Elementary concept of a representation

16

(2)

The direct product

18

(1)

The algebra of a group

19

(3)

Lattices and Vector Spaces

Lattices. One dimension

22

(3)

Lattices. Two and three dimensions

25

(2)

Vector spaces

27

(1)

n-Dimensional space. Basis vectors

28

(3)

Components and basis changes

31

(2)

Mappings and similarity transformations

33

(5)

Representations. Equivalence

38

(3)

Length and angle. The metric

41

(6)

Unitary transformations

47

(2)

Matrix elements as scalar products

49

(2)

The eigenvalue problem

51

(3)

Point and Space Groups

Symmetry operations as orthogonal transformations

54

(5)

The axial point groups

59

(10)

The tetrahedral and octahedral point groups

69

(6)

Compatibility of symmetry operations

75

(3)

Symmetry of crystal lattices

78

(7)

Derivation of space groups

85

(6)

Representations of Point and Translation Groups

Matrices for point group operations

91

(4)

Nomenclature. Representations

95

(10)

Translation groups. Representations and reciprocal space

105

(4)

Irreducible Representations

Reducibility. Nature of the problem

109

(1)

Reduction and complete reduction. Basic theorems

110

(6)

The orthogonality relations

116

(5)

Group characters

121

(3)

The regular representation

124

(1)

The number of distinct irreducible representations

125

(1)

Reduction of representations

126

(5)

Idempotents and projection operators

131

(2)

The direct product

133

(7)

Applications Involving Algebraic Forms

Nature of applications

140

(1)

Invariant forms. Symmetry restrictions

141

(6)

Principal axes. The eigenvalue problem

147

(3)

Symmetry considerations

150

(1)

Symmetry classification of molecular vibrations

151

(8)

Symmetry coordinates in vibration theory

159

(7)

Applications Involving Functions and Operators

Transformation of functions

166

(4)

Functions of Cartesian coordinates

170

(4)

Operator equations. Invariance

174

(7)

Symmetry and the eigenvalue problem

181

(6)

Approximation methods. Symmetry functions

187

(3)

Symmetry functions by projection

190

(5)

Symmetry functions and equivalent functions

195

(2)

Determination of equivalent functions

197

(6)

Applications Involving Tensors and Tensor Operators

Scalar, vector and tensor properties

203

(3)

Significance of the metric

206

(2)

Tensor properties. Symmetry restrictions

208

(3)

Symmetric and antisymmetric tensors

211

(7)

Tensor fields. Tensor operators

218

(6)

Matrix elements of tensor operators

224

(7)

Determination of coupling coefficients

231

(4)

Appendix 1 Representations carried by harmonic functions

235

(6)

Appendix 2 Alternative bases for cubic groups

241

(4)

Index

245

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Symmetry An Introduction to Group Theory and Its Applications

Summary

Table of Contents

Supplemental Materials