I Minimal translationengleiche supergroups

Wondratschek, H.; Aroyo, M. I.

doi:10.1107/97809553602060000542

International
Tables for
Crystallography
Volume A1
Symmetry relations between space groups
Edited by Hans Wondratschek and Ulrich Müller

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International Tables for Crystallography (2006). Vol. A1. ch. 2.1, p. 53 | 1 | 2 |

Section 2.1.6.2. I Minimal translationengleiche supergroups

Hans Wondratschek^a ^* and Mois I. Aroyo^b

^a Institut für Kristallographie, Universität, D-76128 Karlsruhe, Germany, and ^bDepartamento de Física de la Materia Condensada, Facultad de Ciencias, Universidad del País Vasco, Apartado 644, E-48080 Bilbao, Spain
Correspondence e-mail: wondra@physik.uni-karlsruhe.de

2.1.6.2. I Minimal translationengleiche supergroups

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For each space group $[{\cal H}]$ , under this heading are listed those space-group types $[{\cal G}]$ for which $[{\cal H}]$ appears as an entry under the heading I Maximal translationengleiche subgroups. The listing consists of the index in brackets […], the conventional HM symbol and (in parentheses) the space-group number (…). The space groups are ordered by ascending space-group number. If this line is empty, the heading is printed nevertheless and the content is announced by `none', as in [P6/mmm] , No. 191.

The supergroups listed on the line I Minimal translationengleiche supergroups are realized only if the lattice conditions of $[{\cal H}]$ fulfil the lattice conditions for $[{\cal G}]$ . For example, if $[{\cal G}=P422]$ , No. 89, is a supergroup of $[{\cal H}=P222]$ , No. 16, two of the three independent lattice parameters a, b, c of [P222] must be equal (or in crystallographic practice, approximately equal). These must be a and b if c is the tetragonal axis, b and c if a is the tetragonal axis or c and a if b is the tetragonal axis. In the latter two cases, the setting of [P222] has to be adapted to the conventional c-axis setting of [P422] . For the cubic supergroup [P23] , No. 195, all three lattice parameters of [P222] must be (approximately) equal. Such conditions are always to be taken into consideration if the t-supergroup belongs to a different crystal family⁴ to the original group. Therefore, for $[{\cal H}=P222]$ there is no lattice condition for the supergroup $[{\cal G}=Pmmm]$ because [P222] and [Pmmm] belong to the same crystal family.

References

International Tables for Crystallography (2006). Vol. A1. ch. 2.1, p. 53