International
Tables for
Crystallography
Volume A
Space-group symmetry
Edited by M. I. Aroyo

International Tables for Crystallography (2016). Vol. A, ch. 3.3, pp. 778-779

Section 3.3.1.4.3. Short symbols and generators

H. Burzlaffa and H. Zimmermannb*

aUniversität 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

3.3.1.4.3. Short symbols and generators

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If the symbols are not only used for the identification of a group but also for its construction, the symbol must contain a list of generating operations and additional relations, if necessary. Following this aspect, the Hermann–Mauguin symbols can be shortened. The choice of generators is not unique; two proposals were presented by Mauguin (1931[link]). In the first proposal, in almost all cases the generators are the same as those of the Shubnikov symbols. In the second proposal, which, apart from some exceptions (see Section 3.3.4[link]), is used for the international symbols, Mauguin selected a set of generators and thus a list of short symbols in which reflections have priority (Table 3.3.1.3[link], column 3). This selection makes the transition from the short point-group symbols to the space-group symbols fairly simple. These short symbols contain two kinds of notation components:

  • (i) components that represent the type of the generating operation, which are called generators;

  • (ii) components that are not used as generators but that serve to fix the directions of other symmetry elements (Hermann, 1931[link]), and which are called indicators.

The generating matrices are uniquely defined by (i)[link] and (ii)[link] if it is assumed that they describe motions with counterclockwise rotational sense about the representative direction looked at end on by the observer. The symbols 2, 4, [\bar{4}], 6 and [\bar{6}] referring to direction [001] are indicators when the point-group symbol uses three sets of lattice symmetry directions. For instance, in 4mm the indicator 4 fixes the directions of the mirrors normal to [100] and [\hbox{[}1\bar{1}0\hbox{]}].

Note: The generation of (a) point group 432 by a rotation 3 around [111] and a rotation 2 and (b) point group [\bar{4}3m] by 3 around [111] and a reflection m is only possible if the representative direction of the tertiary set is changed from [\hbox{[}1\bar{1}0\hbox{]}] to [110]; otherwise only the subgroup 32 or 3m of 432 or [\bar{4}3m] will be generated.

References

Hermann, C. (1931). Bemerkungen zu der vorstehenden Arbeit von Ch. Mauguin. Z. Kristallogr. 76, 559–561.
Mauguin, Ch. (1931). Sur le symbolisme des groupes de répetition ou de symétrie des assemblages cristallins. Z. Kristallogr. 76, 542–558.








































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