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, pp. 28-29
Section 2.2.12. Oriented site-symmetry symbols^{a}Institut für Kristallographie, Rheinisch-Westfälische Technische Hochschule, Aachen, Germany, and ^{b}Laboratorium voor Chemische Fysica, Rijksuniversiteit Groningen, The Netherlands |
The third column of each Wyckoff position gives the Site symmetry^{4} of that position. The site-symmetry group is isomorphic to a (proper or improper) subgroup of the point group to which the space group under consideration belongs. The site-symmetry groups of the different points of the same special position are conjugate (symmetrically equivalent) subgroups of the space group. For this reason, all points of one special position are described by the same site-symmetry symbol.
Oriented site-symmetry symbols (cf. Fischer et al., 1973) are employed to show how the symmetry elements at a site are related to the symmetry elements of the crystal lattice. The site-symmetry symbols display the same sequence of symmetry directions as the space-group symbol (cf. Table 2.2.4.1). Sets of equivalent symmetry directions that do not contribute any element to the site-symmetry group are represented by a dot. In this way, the orientation of the symmetry elements at the site is emphasized, as illustrated by the following examples.
Examples
The above examples show:
To show, for the same type of site symmetry, how the oriented site-symmetry symbol depends on the space group under discussion, the site-symmetry group mm2 will be considered in orthorhombic and tetragonal space groups. Relevant crystal classes are mm2, mmm, 4mm, and . The site symmetry mm2 contains two mutually perpendicular mirror planes intersecting in a twofold axis.
For space groups of crystal class mm2, the twofold axis at the site must be parallel to the one direction of the rotation axes of the space group. The site-symmetry group mm2, therefore, occurs only in the orientation mm2. For space groups of class mmm (full symbol ), the twofold axis at the site may be parallel to a, b or c and the possible orientations of the site symmetry are 2mm, m2m and mm2. For space groups of the tetragonal crystal class 4mm, the twofold axis of the site-symmetry group mm2 must be parallel to the fourfold axis of the crystal. The two mirror planes must belong either to the two secondary or to the two tertiary tetragonal directions so that 2mm. and 2.mm are possible site-symmetry symbols. Similar considerations apply to class which can occur in two settings, and . Finally, for class (full symbol ), the twofold axis of 2mm may belong to any of the three kinds of symmetry directions and possible oriented site symmetries are 2mm., 2.mm, m2m. and m.2m. In the first two symbols, the twofold axis extends along the single primary direction and the mirror planes occupy either both secondary or both tertiary directions; in the last two cases, one mirror plane belongs to the primary direction and the second to either one secondary or one tertiary direction (the other equivalent direction in each case being occupied by the twofold axis).
References
Fischer, W., Burzlaff, H., Hellner, E. & Donnay, J. D. H. (1973). Space groups and lattice complexes. NBS Monograph No. 134. Washington, DC: National Bureau of Standards.