Introduction to Group Theory with Applications

Introduction to Group Theory with Applications

Materials Science and Technology

1st Edition - January 28, 1977

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  • Author: Gerald Burns
  • eBook ISBN: 9781483191492

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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


    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



    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



    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



    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



    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



    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)



    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



    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



    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)



    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



    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)



    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



    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



    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



Product details

  • No. of pages: 446
  • Language: English
  • Copyright: © Academic Press 1977
  • Published: January 28, 1977
  • 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 Editors

Allen M. Alper

A. S. Nowick

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