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. 60-61

Section 1.4.3.1. Selected order for non-translational generators

H. Wondratscheke

1.4.3.1. Selected order for non-translational generators

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For the non-translational generators, the following sequence has been adopted:

  • (a) In all centrosymmetric space groups, an inversion (if possible at the origin O) has been selected as the last generator.

  • (b) Rotations precede symmetry operations of the second kind. In crystal classes [\bar{4}2m] and [\bar{4}m2] and [\bar{6}2m] and [\bar{6}m2], as an exception, [\bar{4}] and [\bar{6}] are generated first in order to take into account the conventional choice of origin in the fixed points of [\bar{4}] and [\bar{6}].

  • (c) The non-translational generators of space groups with C, A, B, F, I or R symbols are those of the corresponding space group with a P symbol, if possible. For instance, the generators of [I2_12_12_1] (24) are those of [P2_12_12_1] (19) and the generators of Ibca (73) are those of Pbca (61), apart from the centring translations.

    Exceptions: I4cm (108) and I4/mcm (140) are generated via P4cc (103) and P4/mcc (124), because P4cm and P4/mcm do not exist. In space groups with d glides (except [I\bar{4}2d], No. 122) and also in [I4_1/a] (88), the corresponding rotation subgroup has been generated first. The generators of this subgroup are the same as those of the corresponding space group with a lattice symbol P.

    Example

    [F4_1/d\bar{3}2/m] (227):

    [P4_132\ (213)\ \longrightarrow\ F4_132\ (210)\ \longrightarrow\ F4_1/d\bar{3}2/m].

  • (d) In some cases, rule (c) could not be followed without breaking rule (a), e.g. in Cmme (67). In such cases, the generators are chosen to correspond to the Hermann–Mauguin symbol as far as possible. For instance, the generators (apart from centring) of Cmme and Imma (74) are those of Pmmb, which is a non-standard setting of Pmma (51). (A combination of the generators of Pmma with the C- or I-centring translation results in non-standard settings of Cmme and Imma.)

For the space groups with lattice symbol P, the generation procedure has given the same triplets (except for their sequence) as in IT (1952[link]). In non-P space groups, the triplets listed sometimes differ from those of IT (1952[link]) by a centring translation.

References

International Tables for X-ray Crystallography (1952). Vol. I, Symmetry Groups, edited by N. F. M. Henry & K. Lonsdale. Birmingham: Kynoch Press. [Abbreviated as IT (1952).]








































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