Derivation of the full symbol from the short symbol

Burzlaff, H.; Zimmermann, H.

doi:10.1107/97809553602060000526

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. 12.3, pp. 823-825

Section 12.3.2. Derivation of the full symbol from the short symbol

H. Burzlaff^a and H. Zimmermann^b ^*

^a Universität Erlangen–Nürnberg, Robert-Koch-Strasse 4a, D-91080 Uttenreuth, Germany, and ^bInstitut für Angewandte Physik, Lehrstuhl für Kristallographie und Strukturphysik, Universität Erlangen–Nürnberg, Bismarckstrasse 10, D-91054 Erlangen, Germany
Correspondence e-mail: helmuth.zimmermann@krist.uni-erlangen.de

12.3.2. Derivation of the full symbol from the short symbol

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If the geometrical point of view is again considered, it is possible to derive the full international symbol for a space group. This full symbol can be interpreted as consisting of symmetry elements. It can be generated from the short symbol with the aid of products between symmetry operations. It is possible, however, to derive the glide/screw parts of the elements in the full symbol directly from the glide/screw parts of the short symbol.

The product of operations corresponding to non-parallel glide or mirror planes generates a rotation or screw axis parallel to the intersection line. The screw part of the rotation is equal to the sum of the projections of the glide components of the planes on the axis. The angle between the planes determines the rotation part of the axis. For 90°, we obtain a twofold, for 60° a threefold, for 45° a fourfold and for 30° a sixfold axis.

Example: $[Pbcn = D_{2h}^{14}\ (60)]$

The product of b and c generates a screw axis $[2_{1}]$ in the z direction because the sum of the glide components in the z direction is $[\textstyle{1\over 2}]$ . The product of c and n generates a screw axis $[2_{1}]$ in the x direction and the product between b and n produces a rotation axis 2 in the y direction because the y components for b and n add up to $[1 \equiv 0]$ modulo integers.

Thus, the full symbol is $[P {2_{1} \over b} {2 \over c} {2_{1} \over n}.]$

In most cases, the full symbol is identical with the short symbol; differences between full and short symbols can only occur for space groups corresponding to lattice point groups (holohedries) and to the point group $[m\overline{3}]$ . In all these cases, the short symbol is extended to the full symbol by adding the symbol for the maximal purely rotational subgroup. A special procedure is in use for monoclinic space groups. To indicate the choice of coordinate axes, the full symbol is treated like an orthorhombic symbol, in which the directions without symmetry are indicated by `1', even though they do not correspond to lattice symmetry directions in the monoclinic case.

References

International Tables for Crystallography (2006). Vol. A. ch. 12.3, pp. 823-825