Space Groups for Solid State Scientists

Chapter 1. Point Symmetry Operations

What is Symmetry?

1.1 Symmetry Operations

1.2 Point Symmetry Operations

1.3 Hexagonal Coordinates

Chapter 2. Crystal Systems

Haüy’s Legacy

2.1 Lattice

2.2 Unit Cell

2.3 Crystal Structure

2.4 Crystal Systems

2.5 Summary

Chapter 3. Bravais Lattices

Symmetry and Lattices

3.1 Centering of Lattices

3.2 The 14 Bravais Lattices

3.3 Primitive Cells of the 14 Bravais Lattices

3.4 The Wigner–Seitz Unit Cell

3.5 Two-Dimensional Lattices

Chapter 4. Crystallographic Point Groups

Introduction to Groups

4.1 Development of Crystallographic Point Groups

4.2 The Point Groups for Each Crystal System

4.3 The 32 Point Groups from Holohedries

4.4 Laue Classes and Groups

4.5 Point Group Notation

Chapter 5. Development of Space Groups

Space Group Operators

5.1 The Symmorphic Space Groups

5.2 Non-Symmorphic Operations

5.3 Point Group of a Space Group

5.4 Space Groups

5.5 Derivation of Space Groups

5.6 Space Group Classifications

5.7 Two-Dimensional Space Groups

5.8 Subperiodic Groups

Problems

Chapter 6. Reading the Tables

What Does the ITA Tell Us?

6.1 Crystal Structure and Space Groups

6.2 ‘Typical’ Pages of the ITA

6.3 Example Pages from the ITA

6.4 Subgroups and Supergroups

6.5 Space Group Symmetry Operations

6.6 Hall Space Group Symbols

Chapter 7. Space Group Applications

And Now Atoms

7.1 Face-Centered Cubic Structures

7.2 Primitive Cubic Structures

7.3 Body-Centered Cubic Structures

7.4 Diamond Structure

7.5 Spinel Structure

7.6 Zinc Sulphide Structure

7.7 Chalcopyrite

7.8 Semiconductor Superlattices

7.9 Structural Phase Transitions in Crystals

7.10 Displacive SPTs

7.11 Proteins

7.12 Crystallographic Information File

7.13 Ferroic Phase Transitions

7.14 Surface Structure Plane and Layer Groups

7.15 Diffusion, Disordered Structures and Point Defects

7.16 Euclidean normalizers

7.17 Non-Crystallographic Symmetry

7.18 Structures with Z′ > 1

7.19 Icosahedral Symmetry

7.20 Incommensurate Modulations

7.21 Charge Density Wave

7.22 Quasicrystals

Chapter 8. Antisymmetry

Bicolor Symmetry

8.1 Black and White Antisymmetry Groups

8.2 Effect on Vectors

8.3 Magnetic Point Groups

8.4 Translational Subgroups of Magnetic Groups

8.5 Black and White Space Groups

8.6 Magnetic Space Groups

8.7 Examples of Magnetic Structures

8.8 Representation Method

8.9 OG/BNS Magnetic Group Symbols

Appendix 1. Matrices Representing the Symmetry Operations

Jones’ Faithful Representation Symbols

Appendix 2. Crystal Families, Systems, and Bravais Lattices

Appendix 3. The 14 Bravais Lattices

24 Wigner–Seitz Cells

Appendix 4. The 32 Crystallographic Point Groups

Appendix 5. Diagrams for the 32 Point Groups

Stereograms

Some Shapes Illustrating the 32 Point Groups

Appendix 6. Symbols

Symbols of Symmetry Planes

Symbols of Symmetry Axes

Order of Symbols

Three-dimensional lattices

Two-dimensional lattices

Appendix 7. The Space Groups

11 Enantiomorphic Space Group Pairs

The 230 Space Groups

Triclinic System

Monoclinic System

Orthorhombic System

Orthorhombic System

Tetragonal System

Tetragonal System

Trigonal System Hexagonal System

Cubic System

Appendix 8. The Reciprocal Lattice and Diffraction

Scattering from Disordered Structures

Appendix 9. Some Interesting Structures

A9-1 CeM2Si2

A9-2 Rutile

A9-3 Nickel Arsenide

A9-4 Cuprite

A9-5 Nb3Sn

A9-6 Perovskites and Their Superstructures

A9-7 Perovskite-like Phases

A9-8 Strukturbericht Notation

Appendix 10. Translational Subgroups of Magnetic Space Groups

Appendix 11. Cubic Space Group Diagrams

Appendix 12. Pitfalls

Bibliography

Solutions

Chapter 1

Chapter 2

Chapter 3

Chapter 4

Chapter 5

Chapter 6

Chapter 7

Chapter 8

Formula Index