Geared toward students and professionals working on the theory of solids, this high-level volume applies group theory to solid state and molecular physics. It starts with fundamental theorems and the uses of group theory and representations, advancing to the summary of point groups, including the relation between spin and double groups. Additional features include examples in optical absorption selection rules, spin-orbit coupling and crystal field theory, plus a complete demonstration of projection techniques. Short-cut and invariant methods for solving molecular vibration problems and for determining the form of crystal tensors are discussed, along with the translational properties of crystals, including Bravais lattices and space groups. The author explains relevant applications to electron phonon scattering, optical absorption selection rules, electronic energy bands, electron dynamics, and effective Hamiltonians. Over 200 carefully selected problems.
This respected high-level text, aimed at professionals and students working on the theory of solids, is particularly valuable for its application of group theory to solid state and molecular physics. The author -- Distinguished Professor of Physics at City College of New York, widely known for his work in optics, lasers, and solid-state physics -- devotes the first three chapters to a distillation of the most relevant ideas and theorems of group theory (at the group point level) and then leads readers directly into group theory applications.
Professor Lax provides examples in optical absorption selection rules, spin-orbit coupling, and crystal field theory, plus a complete demonstration of projection techniques. He also develops short-cut and invariant methods for solving molecular vibration problems and for determining the form of crystal tensors; develops the translational properties of crystals, including Bravais lattices and space groups; and explains relevant applications to electron phonon scattering, optical absorption selection rules, electronic energy bands, electron dynamics, and effective Hamiltonians. To illustrate several of these topics, and to show the relation between the microscopic and macroscopic views of elasticity, the author presents a complete study of the diamond structure.
Among the unique features of this volume are its use of subgroup techniques, its consideration of the influence of time reversal on selection rules, its use of shell theorems and invariance techniques to construct the form of tensors, and its use of broken symmetry to relate the symmetry of valence and molecular orbitals to the symmetry of electron molecular wave functions. The book also contains more than 200 carefully selected problems.
High-level text applies group theory to solid state and molecular physics. The author develops short-cut and invariant methods for solving molecular vibration problems and for determining the form of crystal tensors; develops the translational properties of crystals; and explains relevant applications. 69 illustrations. 1974 edition.
High-level text applies group theory to physics problems, develops methods for solving molecular vibration problems and for determining the form of crystal tensors, develops translational properties of crystals, more. 1974 edition.
Chapter 1. Relation of Group Theory to Quantum mechanics
1.1 Symmetry Operations
1.2 Abstract Group Theory
1.3 Commuting Observables and Classes
1.4 Representations and Irreducible Representations
1.5 Relation between Representations, Characters, and States
1.6 Continuous Groups
1.7 Summary
Chapter 2. Point Groups
2.1 Generators of the Proper Rotation Group R superscript + (3)
2.2 The Commutator Algebra of R superscript + (3)
2.3 Irreducible Representations of R superscript + (3)
2.4 Characters of the Irreducible Representations of R superscript + (3)
2.5 The Three-dimensional Representation j=1 of R superscript + (3)
2.6 The Spin Representation j=1/2 of R superscript + (3)
2.7 Class Structure of Point Groups
2.8 The Proper Point Groups
2.9 Nature of Improper Rotations in a Finite Group
2.10 Relation between Improper and Proper Groups
2.11 Representations of Groups Containing the Inversion
2.12 Product Groups
2.13 Representations of an Outer Product Group
2.14 Enumeration of the Improper Point Groups
2.15 Crystallographic Point Groups
2.16 Double Point Groups
2.17 Summary
Chapter 3. Point Group Examples
3.1 Electric and Magnetic Dipoles: Irreducible Components of a Reducible Space
3.2 Crystal field Theory without Spin: Compatibility Relations
3.3 Product Representations and Decomposition of Angular Momentum
3.4 Selection rules
3.5 Spin and Spin-Orbit Coupling
3.6 Crystal Field Theory with Spin
3.7 Projection Operators
3.8 Crystal Harmonics
3.9 Summary
Chapter 4. Macroscopic Crystal Tensors
4.1 Macroscopic Point Group Symmetry
4.2 Tensors of the First Rank: Ferroelectrics and Ferromagnetics
4.3 Second-Rank Tensors: Conductivity, Susceptibility
4.4 Direct Inspection Methods for Tensors of Higher Rank: the Hall Effect
4.5 Method of Invariants
4.6 Measures of Infinitesimal and Finite Strain
4.7 The Elasticity Tensor for Group C subscript (3upsilon)
4.8 Summary
Chapter 5. Molecular Vibrations
5.1 Representations contained in NH subscript 3 vibrations
5.2 Determination of the Symmetry Vectors for NH subscript 3
5.3 Symmetry Coordinates, Normal Coordinates, Internal Coordinates, and Invariants
5.4 Potential Energy and Force Constants
5.5 The Number of Force Constants
5.6 Summary
Chapter 6. Translational Properties of Crystals
6.1 Crystal Systems, Bravais Lattices, and Crystal Classes
6.2 Representations of the Translation Group
6.3 Reciprocal Lattices and Brillouin Zones
6.4 Character Orthonormality Theorems
6.5 Conservation of Crystal Momentum
6.6 Laue-Bragg X-ray Diffraction
6.7 Summary
Chapter 7. Electronic Energy Bands
7.1 Relation between the Many-Electron and One-Electron Viewpoints
7.2 Concept of an Energy Band
7.3 The Empty Lattice
7.4 Almost-Free Electrons
7.5 Energy Gaps and Symmetry Considerations
7.6 Points of Zero Slope
7.7 Periodicity in Reciprocal Space
7.8 The k p Method of Analytical Continuation
7.9 Dynamics of Electron Motion in Crystals
7.10 Effective Hamiltonians and Donor States
7.11 Summary
Chapter 8. Space Groups
8.1 Screw Axes and Glide Planes
8.2 Restrictions on space Group Elements
8.3 Equivalence of Space Groups
8.4 Construction of Space Groups
8.5 Factor Groups of Space Groups
8.6 Groups G subscript k of the Wave Vector k
8.7 Space Group Algebra
8.8 Representations of Symmorphic Space Groups
8.9 Representations of Nonsymmorphic Space Groups
8.10 Class Structure and Algebraic Treatment of Multiplier Groups
8.11 Double Space Groups
8.12 Summary
Chapter 9. Space Group Examples
9.1 Vanishing Electric Moment in Diamond
9.2 Induced Quadrupole Moments in Diamond
9.3 Force Constants in Crystals
9.4 Local Electric Moments
9.5 Symmetries of Acoustic and Optical Modes of Vibration
9.6 Hole Scattering by Phonons
9.7 Selection Rules for Direct Optical Absorption
9.8 Summary
Chapter 10. Time reversal
10.1 Nature of Time-Reversal Operators without Spin
10.2 Time Reversal with Spin
10.3 Time Reversal in External Fields
10.4 Antilinear and Antiunitary Operators
10.5 Onsager Relations
10.6 The Time-Reversed Representation
10.7 Time-reversal Degeneracies
10.8 The Herring Criterion for Space Groups
10.9 Selection Rules Due to Time Reversal
10.10 Summary
Chapter 11. Lattice Vibration Spectra
11.1 Inelastic Neutron Scattering
11.2 Transformation to Normal Coordinates
11.3 Quantized Lattice Oscillators: Phonons
11.4 Crystal Momentum
11.5 Infinitesimal Displacement and Rotational Invariance
11.6 Symmetry Properties of the Dynamical Matrix
11.7 Consequences of Time Reversal
11.8 Form and Number of Independent Constants in the Dynamical Matrix for Internal and Zone Boundary Points
11.9 The Method of Long Waves: Primitive Lattices
11.10 Nonprimitive Lattices and Internal Shifts
11.11 Summary
Chapter 12. Vibrations of Lattices with the Diamond Structure
12.1 Force Constants and the Dynamical Matrix
12.2 Symmetry of Vibrations at DELTA = (q, 0, 0)
12.3 R(q) and omega(q) for q = (q, 0, 0)
12.4 Sigma Sum Modes (q, q, 0)
12.5 The Modes LAMBDA = (q, q, q) and L = (2 pi/a)(1/2, 1/2, 1/2)
12.6 Elastic Properties of the Diamond Structure
12.7 Comparison with Experiment
12.8 Summary
Chapter 13. Symmetry of Molecular Wave Functions
13.1 Molecular Orbital Theory
13.2 Valence Bond Orbitals
13.3 Many-Body Wave functions and Chemical Structures
13.4 Hartree-Fock Wave Functions and Broken Symmetry
13.5 The Jahn-Teller Effect
13.6 Summary
Appendix A. Character Tables and Basis Functions for the Single and Double Point Groups
Appendix B. Schoenflies, International, and Herring Notations
Appendix C. Decomposition of D subscript J superscript plus/minus of Full Rotation Group into Point Group Representations
Appendix D. Orthogonality Properties of Eigenvectors of the Equation alpha PSI = lambda B PSI; Reciprocals of Singular Matrices
Appendix E. The Brillouin Zones
Appendix F. Multiplier Representations for the Point Groups
Appendix G. Wigner Mappings and the Fundamental Theorem of Projective Geometry
Appendix H. Generalized Mobility Theory
Author Index and Bibliography; Subject Index; Symbol Index