Technical Aspects of Supervision

ContentsPreface How to Use This Book Cross-References and Formulae Heading Markings Bullets Exercises and Problems Proofs Notation Warning. Asterisks 1. Revision of Quantum Mechanics 1. Basic Experimental Facts 2. Heisenberg^ Uncertainty Principle 3. The State Function 4. Stationary States 5. Normalization 6. Operators and Eigenvalue Equations 7. The Operator p 8. The Operator x 9. The Hamiltonian and the Schrödinger Equation 10. Boundary Conditions: Quantization 11. Degeneracy 12. Commuting Operators 13. Physical Meaning of Degeneracy 14. Exercise: Electron in A Box 75. Time Dependence 16. Wave and Group Velocities 17. Mean Values 18. The Variational Method 19. Exercises: The Hermitian Property. Orthogonality 2. Free-Electron Theory of Metals 1. The One-Particle Approximation 2. Core and Metal Electrons. Pauli Principle 3. The Free-Electron Model 4. Periodic (Born-Von Karman) Boundary Conditions 5. The Wave Function: Normalization 6. The Eigenfunctions in Three Dimensions 7. Degeneracy of The Levels 8. The Fermi Energy 9. The Density of States 10. Soft X-Rays 11. Heat Capacities 3. The Effect of the Crystal Field in One Dimension: Bloch Functions 1. The Use of Translational Symmetry 1. Symmetry Operations 2. Symmetry Operators 3. The Eigenfunctions of The Symmetry Operators Are Eigenfunctions of the Hamiltonian 4. S-Degeneracy 5. The Importance of Continuity 2. S-Degeneracy of The Eigenfunctions of The Translations 3. Symmetry Operations of The Linear Chain: Translations and Inversion 4. The Eigenfunctions of The Translations 1. Bloch Functions 2. Continuity: Bands 5. Quantization of k 6. Physical Considerations 1. The Weak-Field Limit 2. The Free-Atoms Limit 3. Energy Gaps 7. The Number of Eigenfunctions 1. The Number of Different Eigenvalues is N 2. Periodicity in k 3. The Number of Bloch Functions is N 8. The Effect of the Inversion 1. The Inversion Changes k Into —k: Proof 2. The Role of the Bloch Functions 9. Degeneracy of the Bloch Functions 10. Periodicity of the Energy 11. Change of Interval 12. Further Properties of the E(k) Curve 13. Extended and Reduced Band Schemes: Brillouin Zones 14. Band Crossings 15. Filling in of The Energy States 16. Propagation of An Electron in The Lattice: The Periodically Repeated Scheme 17. Bragg Reflections 18. Conductors and Insulators 19. Effective Mass. Holes 20. The Wave Functions 21. Lattice with Basis 22. Exercise: The Energy Gap 1. Method 2. Remarks 23. The Nearly Free-Electron Approximation 24. Fourier Coefficients of the Potential 4. Bloch Functions and Brillouin Zones in Three Dimensions 1. Crystal Periodicity 7. Primitive and Unit Cells 2. Bloch Functions in Three Dimensions 3. Scalar Product and Reciprocal Vectors 4. Reciprocal Lattice 5. The Bloch Functions in Vector Notation 6. The Wave Vector 7. Symmetry in The Reciprocal Space 8. E(k) = E(—k): Complex Conjugation 1. Complex Conjugation 9. The First Brillouin Zone 10. Brillouin Zones of Higher Order 11. Number of States in The Brillouin Zone 12. Conditions For The Faces of The Brillouin Zones 13. Symmetry Elements and The Faces of The Brillouin Zone 14. Energy Contours, Filling-In of Zones, and Fermi Surfaces 15. Bands, Overlaps, Conductors, and Insulators 16. Velocity 17. The Brillouin Zone For Cubic Lattices 1. Properties of The Faces of The F.C.C. Brillouin Zone 18. Hexagonal Close-Packed Lattice. Jones Zones 19. Sticking Together of Bands in The H.C.P. Lattice 20. Fourier Series in Three Dimensions 1. The Orthonormal Functions 2. Expansion of The Periodic Functions 21. Lattice with Basis. Structure Factor 22. The Nearly Free-Electron Approximation 1. The Basic Expansion 2. Condition For The Matrix Elements 3. Form of The Expansion 4. The Energy Gaps 5. The Structure Factor and Energy Gaps 5. Some Applications of Brillouin Zone Theory 1. The Jones Theory of The Hume-Rothery Rules 2. Further Theories of Phase Stability 3. Peaks in The Density of States Curves 4. Effect of The Fermi Energy On Lattice Parameters 6. The Calculation of Band Structures and Fermi Surfaces 1. The Problem and The Basic Equations 1. The Bloch Sums 2. The Energy Eigenvalues 1. The secular determinant 3. Exercise: Tight-Binding Treatment of A Linear Chain 1.Formulae 2. The Bloch Sums 3. Computation of the Energy 4. S and D bands 5. Computation of the Energy 3. The Nearly Free-Electron and Orthogonalized Plane Waves Method 4. The Cellular Method 1. The Potential Muffin Tins 5. The Augmented Plane Wave Method (APW) 6. Density of States and Fermi Surfaces 7. Comparison with Experiment: Heat Capacities and De Haas-Van Alphen Effect 1. Density of States and Heat Capacity 2. The De Haas-Van Alphen Effect 8. The Band Structure and Fermi Surface of Copper 1. The Band Structure 2. The Density of States 3. The Fermi Surface and Comparison with De Haas-Van Alphen Results General ReferencesIndex