- 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

# Basics of crystallography, part 2

# Basics of crystallography, part 2

- Chapter:
- (p.41) 4 Basics of crystallography, part 2
- Source:
- Symmetry Relationships between Crystal Structures
- Author(s):
### Ulrich Müller

- Publisher:
- Oxford University Press

A space group consists of an infinity of crystallographic symmetry operations which are represented by matrix-column pairs *W*,*w*. However, the number of different matrices *W* always is finite. Rotations are restricted to rotation angles of 360°/*N* with the orders of *N* = 1, 2, 3, 4, and 6. Symmetry operations are designated by Hermann-Mauguin symbols. These include: 1 for the identity; the number *N* for rotations; *N _{p} * for screw rotations;

*N̄*for rotoinversions;

*m*for reflections;

*a*,

*b*,

*c*,

*d*,

*e*, and

*n*for glide reflections. A plane perpendicular to an axis is specified by a fraction sign like 4/

*m*. The geometric meaning of a matrix-column pair

*W*,

*w*can be inferred from the determinant and the trace of

*W*.

*Keywords:*
matrix-column pair, crystallographic symmetry operation, Hermann-Mauguin symbol

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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