Applied Group Theory

Applied Group Theory

The Commonwealth and International Library: Selected Readings in Physics

1st Edition - January 1, 1968

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  • Author: Arthur P. Cracknell
  • eBook ISBN: 9781483182476

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Applied Group Theory covers group theory and its applications and is designed to cater undergraduate students. This text is comprised of two parts; the first of which discusses topics such as symmetry; crystallographic groups; vibrations in molecules and solids; and electronic states in atoms, molecules, and solids. This book then explains in the second part topics including the elastic characteristic vibrations of symmetrical systems; theory of Brillouin zones and symmetry properties of wave functions in crystals; and magnetic symmetry of crystals. This text also gives hints to solutions of the exercises provided in each discussion. This selection will be invaluable to undergraduate students of mathematics who are in need of reference materials in group theory that are understandable to them. Mathematics instructors will also find this book helpful.

Table of Contents

  • 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 Solids

    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


    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


Product details

  • No. of pages: 430
  • Language: English
  • Copyright: © Pergamon 2013
  • Published: January 1, 1968
  • Imprint: Pergamon
  • eBook ISBN: 9781483182476

About the Author

Arthur P. Cracknell

About the Editor

D. ter Haar

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