Oriented site-symmetry symbols

Hahn, Th.; Looijenga-Vos, A.

doi:10.1107/97809553602060000505

International
Tables for
Crystallography
Volume A
Space-group symmetry
Edited by Th. Hahn

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International Tables for Crystallography (2006). Vol. A. ch. 2.2, pp. 28-29

Section 2.2.12. Oriented site-symmetry symbols

Th. Hahn^a ^* and A. Looijenga-Vos^b ^‡

^a Institut für Kristallographie, Rheinisch-Westfälische Technische Hochschule, Aachen, Germany, and ^bLaboratorium voor Chemische Fysica, Rijksuniversiteit Groningen, The Netherlands
Correspondence e-mail: hahn@xtl.rwth-aachen.de

2.2.12. Oriented site-symmetry symbols

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The third column of each Wyckoff position gives the Site symmetry⁴ 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

(1) In the tetragonal space group $[P4_{2}2_{1}2\ (94)]$ , Wyckoff position 4f has site symmetry ..2 and position 2b has site symmetry 2.22. The easiest way to interpret the symbols is to look at the dots first. For position 4f, the 2 is preceded by two dots and thus must belong to a tertiary symmetry direction. Only one tertiary direction is used. Consequently, the site symmetry is the monoclinic point group 2 with one of the two tetragonal tertiary directions as twofold axis.

Position b has one dot, with one symmetry symbol before and two symmetry symbols after it. The dot corresponds, therefore, to the secondary symmetry directions. The first symbol 2 indicates a twofold axis along the primary symmetry direction (c axis). The final symbols 22 indicate two twofold axes along the two mutually perpendicular tertiary directions $[[1\bar{1}0]]$ and [110]. The site symmetry is thus orthorhombic, 222.
(2) In the cubic space group I23 (197), position 6b has 222.. as its oriented site-symmetry symbol. The orthorhombic group 222 is completely related to the primary set of cubic symmetry directions, with the three twofold axes parallel to the three equivalent primary directions [100], [010], [001].
(3) In the cubic space group $[Pn\bar{3}n\ (222)]$ , position 6b has 42.2 as its site-symmetry symbol. This `cubic' site-symmetry symbol displays a tetragonal site symmetry. The position of the dot indicates that there is no symmetry along the four secondary cubic directions. The fourfold axis is connected with one of the three primary cubic symmetry directions and two equivalent twofold axes occur along the remaining two primary directions. Moreover, the group contains two mutually perpendicular (equivalent) twofold axes along those two of the six tertiary cubic directions $[\langle 110 \rangle]$ that are normal to the fourfold axis. Each pair of equivalent twofold axes is given by just one symbol 2. (Note that at the six sites of position 6b the fourfold axes are twice oriented along a, twice along b and twice along c.)
(4) In the tetragonal space group $[P4_{2}/nnm\ (134)]$ , position 2a has site symmetry $[\bar{4}2m]$ . The site has symmetry for all symmetry directions. Because of the presence of the primary $[\bar{4}]$ axis, only one of the twofold axes along the two secondary directions need be given explicitly and similarly for the mirror planes m perpendicular to the two tertiary directions.

The above examples show:

(i) The oriented site-symmetry symbols become identical to Hermann–Mauguin point-group symbols if the dots are omitted.
(ii) Sets of symmetry directions having more than one equivalent direction may require more than one character if the site-symmetry group belongs to a lower crystal system than the space group under consideration.

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, $[\bar{4}2m]$ and [4/mmm] . 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 $[2/m\ 2/m\ 2/m]$ ), 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 $[\bar{4}2m]$ which can occur in two settings, $[\bar{4}2m]$ and $[\bar{4}m2]$ . Finally, for class [4/mmm] (full symbol $[4/m\ 2/m\ 2/m]$ ), 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.Google Scholar

International Tables for Crystallography (2006). Vol. A. ch. 2.2, pp. 28-29