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. 2.2, p. 40
Section 2.2.17. Crystallographic groups in one dimension^{a}Institut für Kristallographie, Rheinisch-Westfälische Technische Hochschule, Aachen, Germany, and ^{b}Laboratorium voor Chemische Fysica, Rijksuniversiteit Groningen, The Netherlands |
In one dimension, only one crystal family, one crystal system and one Bravais lattice exist. No name or common symbol is required for any of them. All one-dimensional lattices are primitive, which is symbolized by the script letter ; cf. Chapter 1.2 .
There occur two types of one-dimensional point groups, 1 and . The latter contains reflections through a point (reflection point or mirror point). This operation can also be described as inversion through a point, thus for one dimension; cf. Chapters 1.3 and 1.4 .
Two types of line groups (one-dimensional space groups) exist, with Hermann–Mauguin symbols and , which are illustrated in Fig. 2.2.17.1. Line group , which consists of one-dimensional translations only, has merely one (general) position with coordinate x. Line group consists of one-dimensional translations and reflections through points. It has one general and two special positions. The coordinates of the general position are x and ; the coordinate of one special position is 0, that of the other . The site symmetries of both special positions are . For , the origin is arbitrary, for it is at a reflection point.
The one-dimensional point groups are of interest as `edge symmetries' of two-dimensional `edge forms'; they are listed in Table 10.1.2.1 . The one-dimensional space groups occur as projection and section symmetries of crystal structures.