- Title Pages
- Preface
- Part I Introductory Topics
- 1 Elastic Behavior of Solids
- 2 Electric Behavior of Insulators
- 3 Metals and the Drude–Lorentz Model
- 4 Elementary Theories of the Thermal Properties of Solids
- 5 Elementary Theories of Magnetism
- 6 The Non-interacting Fermi Gas
- 7 Elementary Theories of Crystal Bonding
- Part II Crystal Structure and its Determination
- 8 Lattices and Crystal Structures
- 9 X-ray Diffraction
- Part III Electronic Structure of Periodic Solids
- 10 Electrons in a Periodic Solid
- 11 The Nearly Free Electron, OPW, Pseudopotential, and Tight Binding Methods
- 12 The Parameterization of Band Structures: Applications to Semiconductors
- 13 Augmented Plane Wave and Green’s Function Methods*
- Part IV Electron–Electron Interaction
- 14 The Self-consistent Dielectric Function
- 15 Hartree–Fock and Density Functional Theory
- Part V Lattice Dynamics
- 16 Harmonic Lattice Dynamics: Classical and Quantum
- 17 Thermal Expansion, Phonon–Phonon Interactions, and Heat Transport
- Part VI Electron Transport and Conduction Electron Dynamics
- 18 Motion of Electrons and Holes in External Electric and Magnetic Fields
- 19 Electronic Transport Properties Governed by Static Scattering Centers
- 20 Measuring the Electronic Energy Spectrum On and Off the Fermi Surface
- 21 The Interacting System of Metallic Electrons and Phonons
- Part VII Semiconductors
- 22 Homogeneous Semiconductors
- 23 Inhomogeneous Semiconductors
- Part VIII Electric and Magnetic Properties of Insulators
- 24 Electric and Magnetic Susceptibilities
- 25 Piezoelectricity, Pyroelectricity, and Ferroelectricity
- Part IX Magnetism
- 26 Ferromagnetism and Antiferromagnetism
- 27 Dynamic Properties of Magnetic Materials
- 28 Magnetic Resonance28
- Part X Optical Properties
- 29 Optical Responses
- 30 Polaritons, Excitons, and Plasmons
- 31 Behavior Under Intense Illumination: NLO, the e–h Liquid and Excitonic BEC
- Part XI Superconductivity and Superfluidity
- 32 A Phenomenological Theory of Superconductivity: The London Equations
- 33 A Phenomenological Theory of Superconductivity: The Ginzburg–Landau Theory and the Josephson Effects
- 34 The Microscopic Theory of Superconductivity: Cooper Pairing and the Bardeen–Cooper–Schrieffer Theory
- 35 Elementary Excitations and the Thermodynamic Properties of Superconductors
- 36 Superfluid<sup>4</sup>He
- 37 Landau’s Theory of a Fermi Liquid
- 38 Superfluid<sup>3</sup>He
- Part XII Disordered Materials
- 39 Alloys
- 40 Defects and Diffusion in Crystalline Solids
- 41 Dislocations and Grain Boundaries
- 42 Quantum Theory of Electrical Transport in Dilute Alloys
- 43 Electrical Transport in Highly Disordered Media: Localization/Interaction Effects
- 44 Magnetic Impurities and their Interactions: The Anderson Model, the Kondo Effect, and the RKKY Interaction
- Part XIII Special Topics
- 45 Strongly Correlated Systems
- 46 High Temperature Superconductors
- 47 Artificially Structured and Patterned Materials; Surfaces and Interfaces
- 48 The Quantum Hall Effects
- 49 Graphene, Carbon Nanotubes, and Fullerenes
- Appendix A The Calculus of Variations
- Appendix B The Symmetry of Many–Particle Wave Functions; The Occupation Number RepresentationB
- Index

# Lattices and Crystal Structures

# Lattices and Crystal Structures

- Chapter:
- (p.107) 8 Lattices and Crystal Structures
- Source:
- The Physics of Solids
- Author(s):
### J. B. Ketterson

- Publisher:
- Oxford University Press

This chapter begins with a definition of space lattice. For most pure materials, the lowest energy state at low temperatures is a crystalline solid, where the atoms or molecules making up the substance bind together in a manner which periodically repeats itself in space. The atoms are not points but have a finite spatial extent, and they execute thermal and quantum mechanical zero-point motions. If the crystal structure involves only one kind of atom and if there is only one atom per unit cell, then we may locate each atom at the origin of a unit cell. However, if there are multiple atoms per unit cell we must specify their location within the unit cell. The collection of atom coordinates is referred to as a *basis* and together with the lattice defines a crystal structure: lattice + basis = crystal structure. The remainder of the chapter covers point groups, Bravais lattices, and space groups in two dimensions; point groups, Bravais lattices, and space groups in three dimensions; common crystal structures; Miller indices; and the Wigner–Seitz polyhedra and coordination polyhedra.

*Keywords:*
space lattice, crystalline solid, atoms, crystal structure, point group, Bravais lattice, space group, Miller indices, Wigner–Seitz polyhedra, coordination polyhedra

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- Title Pages
- Preface
- Part I Introductory Topics
- 1 Elastic Behavior of Solids
- 2 Electric Behavior of Insulators
- 3 Metals and the Drude–Lorentz Model
- 4 Elementary Theories of the Thermal Properties of Solids
- 5 Elementary Theories of Magnetism
- 6 The Non-interacting Fermi Gas
- 7 Elementary Theories of Crystal Bonding
- Part II Crystal Structure and its Determination
- 8 Lattices and Crystal Structures
- 9 X-ray Diffraction
- Part III Electronic Structure of Periodic Solids
- 10 Electrons in a Periodic Solid
- 11 The Nearly Free Electron, OPW, Pseudopotential, and Tight Binding Methods
- 12 The Parameterization of Band Structures: Applications to Semiconductors
- 13 Augmented Plane Wave and Green’s Function Methods*
- Part IV Electron–Electron Interaction
- 14 The Self-consistent Dielectric Function
- 15 Hartree–Fock and Density Functional Theory
- Part V Lattice Dynamics
- 16 Harmonic Lattice Dynamics: Classical and Quantum
- 17 Thermal Expansion, Phonon–Phonon Interactions, and Heat Transport
- Part VI Electron Transport and Conduction Electron Dynamics
- 18 Motion of Electrons and Holes in External Electric and Magnetic Fields
- 19 Electronic Transport Properties Governed by Static Scattering Centers
- 20 Measuring the Electronic Energy Spectrum On and Off the Fermi Surface
- 21 The Interacting System of Metallic Electrons and Phonons
- Part VII Semiconductors
- 22 Homogeneous Semiconductors
- 23 Inhomogeneous Semiconductors
- Part VIII Electric and Magnetic Properties of Insulators
- 24 Electric and Magnetic Susceptibilities
- 25 Piezoelectricity, Pyroelectricity, and Ferroelectricity
- Part IX Magnetism
- 26 Ferromagnetism and Antiferromagnetism
- 27 Dynamic Properties of Magnetic Materials
- 28 Magnetic Resonance28
- Part X Optical Properties
- 29 Optical Responses
- 30 Polaritons, Excitons, and Plasmons
- 31 Behavior Under Intense Illumination: NLO, the e–h Liquid and Excitonic BEC
- Part XI Superconductivity and Superfluidity
- 32 A Phenomenological Theory of Superconductivity: The London Equations
- 33 A Phenomenological Theory of Superconductivity: The Ginzburg–Landau Theory and the Josephson Effects
- 34 The Microscopic Theory of Superconductivity: Cooper Pairing and the Bardeen–Cooper–Schrieffer Theory
- 35 Elementary Excitations and the Thermodynamic Properties of Superconductors
- 36 Superfluid<sup>4</sup>He
- 37 Landau’s Theory of a Fermi Liquid
- 38 Superfluid<sup>3</sup>He
- Part XII Disordered Materials
- 39 Alloys
- 40 Defects and Diffusion in Crystalline Solids
- 41 Dislocations and Grain Boundaries
- 42 Quantum Theory of Electrical Transport in Dilute Alloys
- 43 Electrical Transport in Highly Disordered Media: Localization/Interaction Effects
- 44 Magnetic Impurities and their Interactions: The Anderson Model, the Kondo Effect, and the RKKY Interaction
- Part XIII Special Topics
- 45 Strongly Correlated Systems
- 46 High Temperature Superconductors
- 47 Artificially Structured and Patterned Materials; Surfaces and Interfaces
- 48 The Quantum Hall Effects
- 49 Graphene, Carbon Nanotubes, and Fullerenes
- Appendix A The Calculus of Variations
- Appendix B The Symmetry of Many–Particle Wave Functions; The Occupation Number RepresentationB
- Index