- Title Pages
- Title Pages
- Dedicated to <b>Hartmut Baärnighausen</b> and <b>Hans Wondratschek</b>
- Preface
- List of symbols
- 1 Introduction
- 2 Basics of crystallography, part 1
- 3 Mappings
- 4 Basics of crystallography, part 2
- 5 Group theory
- 6 Basics of crystallography, part 3
- 7 Subgroups and supergroups of point and space groups
- 8 Conjugate subgroups, normalizers and equivalent descriptions of crystal structures
- 9 How to handle space groups
- 10 The group-theoretical presentation of crystal-chemical relationships
- 11 Symmetry relations between related crystal structures
- 12 Pitfalls when setting up group-subgroup relations
- 13 Derivation of crystal structures from closest packings of spheres
- 14 Crystal structures of molecular compounds
- 15 Symmetry relations at phase transitions
- 16 Topotactic reactions
- 17 Group—subgroup relations as an aid for structure determination
- 18 Prediction of possible structure types
- 19 Historical remarks
- Appendices
- A Isomorphic subgroups
- B On the theory of phase transitions
- C Symmetry species
- D Solutions to the exercises
- References
- Glossary
- Index

# Introduction

# Introduction

- Chapter:
- (p.1) 1 Introduction
- Source:
- Symmetry Relationships between Crystal Structures
- Author(s):
### Ulrich Müller

- Publisher:
- Oxford University Press

This introductory chapter outlines the symmetry principle in crystal chemistry states: in crystal structures the arrangement of the atoms has a tendency towards the highest possible symmetry; counteracting factors may enforce a reduced symmetry; and during phase transitions and topotactic reactions which result in products of lower symmetry, the higher symmetry of the starting material is often indirectly preserved by the formation of oriented domains. Many crystal structures can be derived from a few simple, highly symmetrical crystal structures. A family of structures is headed by a most symmetrical structure — the aristotype — and the space groups of the derivative structures or hettotypes are subgroups of the space group of the aristotype. A Bärnighausen tree depicts the group-subgroup relations. An example is the relation diamond-zinc blende. Symmetry reduction to a subgroup can take place during a phase transition, and a twinned crystal can be the result.

*Keywords:*
symmetry principle, diamond-zinc blende, aristotype, hettotype, group-subgroup relations, Bärnighausen tree, phase transition, diamond, twinned crystal, calcium chloride

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- Title Pages
- Title Pages
- Dedicated to <b>Hartmut Baärnighausen</b> and <b>Hans Wondratschek</b>
- Preface
- List of symbols
- 1 Introduction
- 2 Basics of crystallography, part 1
- 3 Mappings
- 4 Basics of crystallography, part 2
- 5 Group theory
- 6 Basics of crystallography, part 3
- 7 Subgroups and supergroups of point and space groups
- 8 Conjugate subgroups, normalizers and equivalent descriptions of crystal structures
- 9 How to handle space groups
- 10 The group-theoretical presentation of crystal-chemical relationships
- 11 Symmetry relations between related crystal structures
- 12 Pitfalls when setting up group-subgroup relations
- 13 Derivation of crystal structures from closest packings of spheres
- 14 Crystal structures of molecular compounds
- 15 Symmetry relations at phase transitions
- 16 Topotactic reactions
- 17 Group—subgroup relations as an aid for structure determination
- 18 Prediction of possible structure types
- 19 Historical remarks
- Appendices
- A Isomorphic subgroups
- B On the theory of phase transitions
- C Symmetry species
- D Solutions to the exercises
- References
- Glossary
- Index