# Band Theory of Metals

## 1st Edition

### The Elements

**Authors:**Simon L. Altmann

**eBook ISBN:**9781483158990

**Imprint:**Pergamon

**Published Date:**1st January 1970

**Page Count:**264

## Description

Band Theory of Metals: The Elements focuses on the band theory of solids.
The book first discusses revision of quantum mechanics. Topics include Heisenberg’s uncertainty principle, normalization, stationary states, wave and group velocities, mean values, and variational method. The text takes a look at the free-electron theory of metals, including heat capacities, density of states, Fermi energy, core and metal electrons, and eigenfunctions in three dimensions. The book also reviews the effects of crystal fields in one dimension. The eigenfunctions of the translations; symmetry operations of the linear chain; use of translational symmetry; degeneracy of the Bloch functions; and effects of inversion are described.
The text also focuses on Bloch functions and Brillouin zones in three dimensions. Concerns include symmetry in the reciprocal space; scalar product and reciprocal vectors; Brillouin zones of higher order; and conditions for the faces of the Brillouin zones.

The book is a good source of data for readers interested in the band theory of solids.

## Table of Contents

Preface

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's 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

15. 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 Kármán) 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

1. 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

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

18. Hexagonal Close-Packed Lattice. Jones Zones

19. Sticking Together of Bands in the H.C.P. Lattice

20. Fourier Series in Three Dimensions

21. Lattice With Basis. Structure Factor

22. The Nearly Free-Electron Approximation

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 t Fermi Energy On Lattice Parameters

6. The Calculation of Band Structures and Fermi Surfaces

1. The Problem and the Basic Equations

2. The Tight-Binding Method

3. The Nearly Free-Electron and Orthogonalized Plane Waves Method

4. The Cellular Method

5. The Augmented Plane Wave Method (APW)

6. Density of States and Fermi Surfaces

7. Comparison With Experiment: Heat Capacities and de Haas-Van

8. The Band Structure and Fermi Surface of Copper

General References

Index

## Details

- No. of pages:
- 264

- Language:
- English

- Copyright:
- © Pergamon 1970

- Published:
- 1st January 1970

- Imprint:
- Pergamon

- eBook ISBN:
- 9781483158990