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

International Tables for Crystallography (2015). Vol. A, ch. 1.4, pp. 42-73
https://doi.org/10.1107/97809553602060000922

Chapter 1.4. Space groups and their descriptions

B. Souvignier,d H. Wondratschek,e M. I. Aroyo,a* G. Chapuisb and A. M. Glazerc

aDepartamento de Física de la Materia Condensada, Universidad del País Vasco (UPV/EHU), Bilbao, Spain,bÉcole Polytechnique Fédérale de Lausanne, BSP/Cubotron, CH-1015 Lausanne, Switzerland,cDepartment of Physics, University of Oxford, Parks Road, Oxford, United Kingdom,dRadboud University Nijmegen, Faculty of Science, Mathematics and Computing Science, Institute for Mathematics, Astrophysics and Particle Physics, Postbus 9010, 6500 GL Nijmegen, The Netherlands, and eLaboratorium für Applikationen der Synchrotronstrahlung (LAS), Universität Karlsruhe, Germany
Correspondence e-mail:  mois.aroyo@ehu.es

Footnotes

Deceased.

1 According to the recommendations of the International Union of Crystallography Ad Hoc Committee on the Nomenclature of Symmetry (de Wolff et al., 1992[link]), the characters appearing after the lattice letter in the HM symbol of a space group should represent symmetry elements, which is reflected, for example, in the introduction of the `e-glide' notation in the HM space-group symbols. To avoid misunderstandings, it is worth noting that in the following discussion of the HM symbolism, the author preferred to keep strictly to the original idea according to which the characters of the HM symbols were meant to represent (generating) symmetry operations of the space group, and not symmetry elements.
2 The numbers listed are those for bipolar directions, for which direction and opposite direction are equivalent. For the corresponding polar directions in cubic space groups only the four equivalent polar directions [\langle111\rangle] or [\langle\overline{1} \overline{1}\overline{1}\rangle] of the tetrahedron occur.
3 The `symmetry-element' interpretation of the constituents of the HM symbols (cf. footnote 11) results in the following modification of the `simplest-operation' rule [known as the `priority rule', cf. Section 4.1.2.3[link] of International Tables for Crystallography, Volume A (2002[link]) (referred to as IT A5)]: When more than one kind of symmetry element exists in a given direction, the choice of the corresponding symbols in the space-group symbol is made in order of descending priority [m\,\gt\,e\,\gt\,a,b, c\,\gt\,n], and rotation axes before screw axes.
4 In the original HM symbols the constituent `1' was avoided by the use of different centred cells.
5 One usually chooses [[1\overline{1}0]] as the representative direction and not the equivalent direction [110], in analogy to the cases of trigonal and hexagonal space groups where [[1\overline{1}0]] is the representative of the set of tertiary symmetry directions, while [[\overline{1}\overline{1}0]] (or [110]) belongs to the set of secondary symmetry directions, cf. Table 2.1.3.1[link] .
6 Note: `3' or `[\overline{3}]' directly after the lattice symbol denotes a trigonal or rhombohedral space group; `3' or `[\overline{3}]' in the third position (second position after the lattice symbol) is characteristic for cubic space groups.