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International
Tables for Crystallography Volume A Space-group symmetry Edited by Th. Hahn © International Union of Crystallography 2006 |
International Tables for Crystallography (2006). Vol. A. ch. 8.2, pp. 726-731
https://doi.org/10.1107/97809553602060000515 Chapter 8.2. Classifications of space groups, point groups and lattices
a
Institut für Kristallographie, Universität, D-76128 Karlsruhe, Germany |
Footnotes
1
These space-group types are often denoted by the word `space group' when speaking of the 17 `plane groups' or of the 219 or 230 `space groups'. In a number of cases, the use of the same word `space group' with two different meanings (`space group' and `space-group type' which is an infinite set of space groups) is of no further consequence. In some cases, however, it obscures important relations. For example, it is impossible to appreciate the concept of isomorphic subgroups of a space group if one does not strictly distinguish between space groups and space-group types: cf. Section 8.3.3![[link]](/graphics/greenarr.gif)
and Part 13
.
2 According to the `Theorem of Bieberbach', in all dimensions the classification into affine space-group types results in the same types as the classification into isomorphism types of space groups. Thus, the affine equivalence of different space groups can also be recognized by purely group-theoretical means: cf. Ascher & Janner (1965, 1968/69).
3 The classes defined here have been called `crystal families' by Neubüser et al. (1971). For the same concept the term `crystal system' has been used, particularly in American and Russian textbooks. In these Tables, however, `crystal system' designates a different classification, described in Section 8.2.8. To avoid confusion, the term `crystal family' is used here.
4 `Lattice systems' were called `Bravais systems' in editions 1 to 4 of this volume. The name has been changed because in practice `Bravais systems' may be confused with `Bravais types' or `Bravais lattices'.
and Part 13
.2 According to the `Theorem of Bieberbach', in all dimensions the classification into affine space-group types results in the same types as the classification into isomorphism types of space groups. Thus, the affine equivalence of different space groups can also be recognized by purely group-theoretical means: cf. Ascher & Janner (1965
, 1968/69).3 The classes defined here have been called `crystal families' by Neubüser et al. (1971)
. For the same concept the term `crystal system' has been used, particularly in American and Russian textbooks. In these Tables, however, `crystal system' designates a different classification, described in Section 8.2.8. To avoid confusion, the term `crystal family' is used here.4 `Lattice systems' were called `Bravais systems' in editions 1 to 4 of this volume. The name has been changed because in practice `Bravais systems' may be confused with `Bravais types' or `Bravais lattices'.