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. 4.3, p. 74

Section 4.3.5.5. Additional symmetry elements

E. F. Bertauta‡

a Laboratoire de Cristallographie, CNRS, Grenoble, France

4.3.5.5. Additional symmetry elements

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Parallel axes 2 and [2_{1}] occur perpendicular to the principal symmetry axis. Examples are space groups R32 (155), P321 (150) and P312 (149), where the screw components are [{1 \over 2},{1 \over 2},0] (rhombohedral axes) or [{1 \over 2},0,0] (hexagonal axes) for R32; [{1 \over 2},0,0] for P321; and [{1 \over 2},1,0] for P312. Hexagonal examples are P622 (177) and [P\bar{6}2c] (190).

Likewise, mirror planes m parallel to the main symmetry axis alternate with glide planes, the glide components being perpendicular to the principal axis. Examples are P3m1 (156), P31m (157), R3m (160) and P6mm (183).

Glide planes c parallel to the main axis are interleaved by glide planes n. Examples are P3c1 (158), P31c (159), R3c (161, hexagonal axes), [P\bar{6}c2] (188). In R3c and [R\bar{3}c], the glide component [0,0,{1 \over 2}] for hexagonal axes becomes [{1 \over 2},{1 \over 2},{1 \over 2}] for rhombohedral axes, i.e. the c glide changes to an n glide. Thus, if the space group is referred to rhombohedral axes, diagonal n planes alternate with diagonal a, b or c planes (cf. Section 1.4.4[link] ).

In R space groups, all additional symmetry elements with glide and screw components have their origin in the action of an integral lattice translation. This is also true for the axes [3_{1}] and [3_{2}] which appear in all R space groups (cf. Table 4.1.2.2[link] ). For this reason, the `rhombohedral centring' R is not included in Table 4.1.2.3[link] , which contains only the centrings A, B, C, I, F.








































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