Applied Group Theory

Preface

Part 1

I. Symmetry

1.1. Symmetry Operations

1.2. Definition of a Group

1.3. Symmetry in Nature

II. Theory of Groups

2.1. Further Definitions

2.2. Subgroups

2.3. Classes

2.4. Matrices

2.5. Matrices and the Symmetry Operations of the Square

2.6. Matrix Representations of a Group

2.7. Irreducible (Matrix) Representations of a Group

2.8. The Deduction of the Irreducible Representations of a Group

2.9. The Direct Product of Two Groups

2.10. Group Theory and Quantum Mechanics, Wigner's Theorem

III. Crystallographic Groups

3.1. Point Groups

3.2. The Derivation of the Point Groups and their Character Tables

3.3. The Basis of a Representation

3.4. Bravais Lattices

3.5. Space Groups

3.6. Seitz Space-group Symbols

IV. The Rotation, Symmetric and Lorentz Groups

4.1. The Rotation Group

4.2. The Representations of the Rotation Group

4.3. The Spherical Harmonics and the Rotation Group

4.4. The Permutation Group, or Symmetric Group, S(n)

4.5. Young's Tableaux

4.6. Special Relativity and the Lorentz Group

V. Vibrations in Molecules and Solids

5.1. Vibrations of Molecules, Normal Modes

5.2. Example, the Normal Modes of the Methane Molecule

5.3. Vibrations of Solids, Phonons

5.4. Brillouin Zone Theory

5.5. Infra-red and Raman Activity of the Normal Modes

VI. Electronic States in Atoms, Molecules and Solids

6.1. Wave Functions of Electrons in Atoms

6.2. Wave Functions of Electrons in Molecules

6.3. The Jahn-Teller Effect

6.4. The Splitting of Atomic Energy Levels in Crystals

6.5. Wave Functions of Electrons in Solid

VII. Atoms, Nuclei and Elementary Particles

7.1. The Principle of Antisymmetry and the Pauli Exclusion Principle

7.2. The Spherical Harmonics and Angular Momentum

7.3. Selection Rules for Electrons in Atoms

7.4. Spin, SU(2)

7.5. Nuclei, Isobaric Spin

7.6. "Elementary" Particles, SU(3), etc.

VIII. Further Topics

8.1. Double Groups

8.2. Magnetic Point Groups and Magnetic Space Groups

8.3. Time Reversal and the Kramers Degeneracy

8.4. Symmetry Properties of Tensors

8.5. Waveguide Junctions

8.6. Campanological Groups

Guided Bibliography

References

Part 2

1. The Elastic Characteristic Vibrations of Symmetrical Systems

2. The Degeneracy, Selection Rules, and Other Properties of the Normal Vibrations of Certain Polyatomic Molecules

3. Stability of Polyatomic Molecules in Degenerate Electronic States. I

4. Splitting of Terms in Crystals

5. On the Reduction of Space Groups

6. Theory of Brillouin Zones and Symmetry Properties of Wave Functions in Crystals

7. On the Consequences of the Symmetry of the Nuclear Hamiltonian on the Spectroscopy of Nuclei

8. "Double" Crystallographic Groups

9. Magnetic Symmetry of Crystals

Appendix. The Character Tables of the Thirty-Two Point Groups

Hints to Solutions of the Exercises

Index