Introduction to Group Theory with Applications - 1st Edition - ISBN: 9780121457501, 9781483191492

Introduction to Group Theory with Applications

1st Edition

Materials Science and Technology

Authors: Gerald Burns
Editors: Allen M. Alper A. S. Nowick
eBook ISBN: 9781483191492
Imprint: Academic Press
Published Date: 28th January 1977
Page Count: 446
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Description

Introduction to Group Theory with Applications covers the basic principles, concepts, mathematical proofs, and applications of group theory.

This book is divided into 13 chapters and begins with discussions of the elementary topics related to the subject, including symmetry operations and group concepts. The succeeding chapters deal with the properties of matrix representations of finite groups, the vibrations of molecular and crystals, vibrational wave function, selection rules, and molecular approximations. These topics are followed by reviews of the basic of quantum mechanics, crystal field theory, atomic physics, hybrid functions, and molecular orbital theory. The last chapters describe the symmetry of crystal lattices, the band theory of solids, and the full rotation group.

This book will be of value to undergraduate mathematics and physics students.

Table of Contents


Preface

Acknowledgments

Chapter 1 Symmetry Operations

1-1 Introduction

1-2 Point Symmetry Operations

1-3 The Stereographic Projection

1-4 The 32 Crystallographic Point Groups

1-5 Related Considerations

1-6 Space Group Example

Notes

Problems

Chapter 2 Group Concepts

2-1 Introduction

2-2 Definition of a Group

2-3 Symmetry Operations Form a Group

2-4 Related Group Concepts

2-5 Isomorphism and Homomorphism

2-6 Special Kinds of Groups

2-7 More Involved Group Concepts (including a Factor Group of a Space Group)

Appendix to Chapter 2

Notes

Problems

Chapter 3 Matrix Representations of Finite Groups

3-1 Introduction

3-2 Representations

3-3 Irreducible Representations

3-4 Representations of a Factor Group

Appendix to Chapter 3

Notes

Problems

Chapter 4 Characters of Matrix Representations of Finite Groups

4-1 Properties of Characters of Irreducible Representations

4-2 Character Tables

4-3 Reduction of a Reducible Representation

4-4 Basis Functions

4-5 Examples—Neumann Principle

4-6 Atomic Positions

4-7 The Hamiltonian

Appendix to Chapter 4

Notes

Problems

Chapter 5 Vibrations of Molecules and Crystals

5-1 3N Degrees of Freedom

5-2 General Considerations

5-3 Number and Type of Normal Modes for Molecules

5-4 Internal Coordinates

5-5 Crystals

5-6 Eigenvectors and Symmetry Adapted Vectors

5-7 Projection Operators

5-8 Projection Operators Applied to Normal Coordinates

Notes

Problems

Chapter 6 Normal Modes (Direct Product and Selection Rules)

6-1 Direct Product of Irreducible Representations

6-2 Vibrational Wave Function

6-3 Selection Rules—Infrared and Raman

6-4 Molecular Approximations (Site Symmetry and Davydov Splitting)

Notes

Problems

Chapter 7 Quantum Mechanics

7-1 Atomic Wave Functions

7-2 Transformation of Functions

7-3 Eigenfunctions as Basis Functions

7-4 Proper Rotations and Angular Momentum

7-5 Perturbations

7-6 Matrix Elements (Selection Rules)

7-7 General Secular Equation Problem

Appendix to Chapter 7

Notes

Problems

Chapter 8 Crystal Field Theory (and Atomic Physics)

8-1 Rotations in Terms of Euler Angles

8-2 Representations of the Full Rotation Group

8-3 Reduction of Symmetry

8-4 Energy Level Diagrams (Correlation Diagrams)

8-5 Crystal Double Groups

8-6 Correlation Diagrams including Double Groups

8-7 Other Crystal Field Effects

Appendix to Chapter 8

Notes

Problems

Chapter 9 Hybrid Functions

9-1 Introduction

9-2 Simple Hybrid Functions and Bonding

9-3 Tetrahedral Hybridization

9-4 Other Hybrid Functions

9-5 π-Hybrid Functions

9-6 Comment on Hybrid Orbitals (Slater Determinant)

Notes

Problems

Chapter 10 Molecular Orbital Theory

10-1 Hydrogen Molecular Ion

10-2 Simple MO Theory

10-3 Transition Metal Complexes

10-4 LCAO-MO of π-Electrons in Conjugated Hydrocarbons

10-5 Woodward-Hoffman Rules

Notes

Problems

Chapter 11 Symmetry of Crystal Lattices

11-1 The Real Affine Group

11-2 Space Group

11-3 Translational Lattice

11-4 International Tables for X-Ray Crystallography, International Notation, etc.

11-5 Magnetic Groups (Color Groups)

Notes

Problems

Chapter 12 Band Theory of Solids

12-1 Translational Symmetry

12-2 Symmorphic Space Groups

12-3 Nonsymmorphic Space Groups

12-4 Spin-Orbit Effects on Bands

12-5 Time Reversal Symmetry

Notes

Problems

Chapter 13 The Full Rotation Group

13-1 The Homomorphism between the Special Unitary Group in Two Dimensions, SU(2), and the Three-Dimension Rotation Group

13-2 Irreducible Representations of SU(2)

13-3 Wigner Coefficients

13-4 Irreducible Tensor Operators

13-5 The Wigner-Eckart Theorem

13-6 Survey of 3j and Racah Coefficients

Notes

Problems

Appendix 1 Crystal Systems

Appendix 2 The 32 Point Groups

Appendix 3 Character Tables

Appendix 4 Space Groups

Appendix 5 Matrices, Vector Spaces, and Linear Operators

Appendix 6 Direct Product Tables

Appendix 7 Correlation Tables

Appendix 8 Spherical Harmonics

Appendix 9 Tanable-Sugano Diagrams

Appendix 10 Double Group Character Tables

Bibliography

Index

Details

No. of pages:
446
Language:
English
Copyright:
© Academic Press 1977
Published:
Imprint:
Academic Press
eBook ISBN:
9781483191492

About the Author

Gerald Burns

Affiliations and Expertise

IBM Thomas J. Watson Research Center, Yorktown Heights, New York

About the Editor

Allen M. Alper

A. S. Nowick