Cracknell / Ter Haar Applied Group Theory

Selected Readings in Physics
1. Auflage 2016
ISBN: 978-1-4831-4938-7
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 430 Seiten, Web PDF

Selected Readings in Physics: Applied Group Theory provides information pertinent to the fundamental aspects of applied group theory. This book discusses the properties of symmetry of a system in quantum mechanics. Organized into two parts encompassing nine chapters, this book begins with an overview of the problem of elastic vibrations of a symmetric structure. This text then examines the numbers, degeneracies, and symmetries of the normal modes of vibration. Other chapters consider the conditions under which a polyatomic molecule can have a stable equilibrium configuration when its electronic state has orbital degeneracy. This book discusses as well the effect of an electric field having a given symmetry upon an atom. The final chapter deals with the symmetry of crystals with a magnetic moment. This book is intended to be suitable for final-year students and fresh postgraduate students in physics. Physicists and researcher workers will also find this book extremely useful.

Autoren/Hrsg.

Cracknell, Arthur P.

Ter Haar, D.

Inhaltsverzeichnis

1;Front Cover;1
2;Applied Group Theory;4
3;Copyright Page ;5
4;Table of Contents;6
5;Preface;10
6;PART I;14
6.1;Chapter 1. Symmetry;16
6.1.1;1.1. Symmetry Operations;16
6.1.2;1.2. Definition of a Group;20
6.1.3;1.3. Symmetry in Nature;22
6.2;Chapter 2. Theory of Groups;25
6.2.1;2.1. Further Definitions;25
6.2.2;2.2. Subgroups;27
6.2.3;2.3. Classes;27
6.2.4;2.4. Matrices;31
6.2.5;2.5. Matrices and the Symmetry Operations of the Square;32
6.2.6;2.6. Matrix Representations of a Group;34
6.2.7;2.7. Irreducible (Matrix) Representations of a Group;39
6.2.8;2.8. The Deduction of the Irreducible Representations of a Group;41
6.2.9;2.9. The Direct Product of Two Groups;49
6.2.10;2.10. Group Theory and Quantum Mechanics, Wigner's Theorem;51
6.3;Chapter 3. Crystallographic Groups;55
6.3.1;3.1. Point Groups;55
6.3.2;3.2. The Derivation of the Point Groups and their Character Tables;59
6.3.3;3.3. The Basis of a Representation;65
6.3.4;3.4. Bravais Lattices;69
6.3.5;3.5. Space Groups;72
6.3.6;3.6. Seitz Space-group Symbols;74
6.4;Chapter 4. The Rotation, Symmetric and Lorentz Groups;79
6.4.1;4.1. The Rotation Group;79
6.4.2;4.2. The Representations of the Rotation Group;83
6.4.3;4.3. The Spherical Harmonics and the Rotation Group;88
6.4.4;4.4. The Permutation Group, or Symmetric Group, S(n);92
6.4.5;4.5. Young's Tableaux;94
6.4.6;4.6. Special Relativity and the Lorentz Group;98
6.5;Chapter 5. Vibrations in Molecules and Solids;104
6.5.1;5.1. Vibrations of Molecules, Normal Modes;104
6.5.2;5.2. Example, the Normal Modes of the Methane Molecule;105
6.5.3;5.3. Vibrations of Solids, Phonons;120
6.5.4;5.4. Brillouin Zone Theory;123
6.5.5;5.5. Infrared and Raman Activity of the Normal Modes;126
6.6;Chapter 6. Electronic States in Atoms, Molecules and Solids;132
6.6.1;6.1. Wave Functions of Electrons in Atoms;132
6.6.2;6.2. Wave Functions of Electrons in Molecules;135
6.6.3;6.3. The Jahn–Teller Effect;144
6.6.4;6.4. The Splitting of Atomic Energy Levels in Crystals;145
6.6.5;6.5. Wave Functions of Electrons in Solids;150
6.7;Chapter 7. Atoms, Nuclei and Elementary Particles;156
6.7.1;7.1. The Principle of Antisymmetry and the Pauli Exclusion Principle;156
6.7.2;7.2. The Spherical Harmonics and Angular Momentum;158
6.7.3;7.3. Selection Rules for Electrons in Atoms;161
6.7.4;7.4. Spin, SU(2);166
6.7.5;7.5. Nuclei, Isobaric Spin;169
6.7.6;7.6. "Elementary" Particles, SU(3), etc;174
6.8;Chapter 8. Further Topics;182
6.8.1;8.1. Double Groups;182
6.8.2;8.2. Magnetic Point Groups and Magnetic Space Groups;186
6.8.3;8.3. Time Reversal and the Kramers Degeneracy;192
6.8.4;8.4. Symmetry Properties of Tensors;194
6.8.5;8.5. Waveguide Junctions;198
6.8.6;8.6. Campanological Groups;202
6.9;GUIDED BIBLIOGRAPHY;205
6.10;REFERENCES;206
7;PART II;210
7.1;Chapter 1. The Elastic Characteristic Vibrations of Symmetrical Systems;212
7.2;Chapter 2. The Degeneracy, Selection Rules, and Other Properties of the Normal Vibrations of Certain Polyatomic Molecules;226
7.2.1;Explanation of Tables;228
7.2.2;Discussion of Results;230
7.2.3;Molecules with Three Atoms;232
7.2.4;Molecules with Four Atoms;233
7.2.5;Molecules with Five Atoms;234
7.2.6;Molecules with Six Atoms;237
7.2.7;Molecules with Seven Atoms;238
7.2.8;Molecules with Eight Atoms;243
7.2.9;Molecules with Nine Atoms;245
7.3;Chapter 3. Stability of Polyatomic Molecules in Degenerate Electronic States. I Orbital Degeneracy;246
7.3.1;Introduction;246
7.3.2;1. Two Examples;247
7.3.3;2. General Theorem;250
7.3.4;3. Mathematical Formulation and Group-theoretical Considerations;251
7.3.5;4. Proof of General Theorem;254
7.3.6;5. Conclusion;266
7.3.7;Summary;267
7.3.8;References;267
7.4;Chapter 4. Splitting of Terms in Crystals;269
7.4.1;I. Solution by Theory of Groups;272
7.4.2;II. The Eigenfunctions to Zero-order Approximation in the Crystal;295
7.5;Chapter 5. On the Reduction of Space Groups;310
7.5.1;Introduction;310
7.5.2;1. The Theory of Space Groups;311
7.5.3;2. A Theorem Concerning the Reduction of Finite Solvable Groups;316
7.5.4;3. Additional Simplifying Developments;321
7.5.5;4. Concerning Physical Applications;325
7.6;Chapter 6. Theory of Brillouin Zones and Symmetry Properties of Wave Functions in Crystals;327
7.6.1;I;327
7.6.2;II;330
7.6.3;III;333
7.6.4;IV;337
7.6.5;V;338
7.6.6;VI. Body-centered Cubic Lattice;345
7.6.7;VII. Face-centered Cubic Lattice;348
7.7;Chapter 7. On the Consequences of the Symmetry of the Nuclear Hamiltonian on the Spectroscopy of Nuclei;352
7.7.1;1;352
7.7.2;2;355
7.7.3;3;358
7.7.4;4;369
7.7.5;5;375
7.7.6;6;381
7.8;Chapter 8. "Double" Crystallographic Groups;383
7.9;Chapter 9. Magnetic Symmetry of Crystals;392
7.9.1;1. Introduction;392
7.9.2;2. Some Properties of the Transformations Introduced Above. Statement of the Problem;394
7.9.3;3. Algorithm for Constructing the Point Groups;395
7.9.4;4. List of Groups of Point Transformations for the Symmetry of Crystals with a Mean Current Density;401
7.9.5;5. Possible Applications of the Groups of Point Symmetry Transformations of Crystals;403
8;APPENDIX;405
8.1;The Character Tables of the Thirty-two Point Groups;405
8.2;Hints to Solutions of the Exercises;410
9;INDEX;426