Definitions

Billiet, Y.; Bertaut, E. F.

doi:10.1107/97809553602060000528

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. 13.1, p. 836

Section 13.1.1. Definitions

Y. Billiet^a ^‡ and E. F. Bertaut^b ^§

^a Département de Chimie, Faculté des Sciences et Techniques, Université de Bretagne Occidentale, Brest, France, and ^bLaboratoire de Cristallographie, CNRS, Grenoble, France

13.1.1. Definitions

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A subgroup $[{\cal H}]$ of a space group $[{\cal G}]$ is an isomorphic subgroup if $[{\cal H}]$ is of the same or the enantiomorphic space-group type as $[{\cal G}]$ . Thus, isomorphic space groups are a special subset of klassengleiche subgroups. The maximal isomorphic subgroups of lowest index are listed under IIc in the space-group tables of this volume (Part 7 ) (cf. Section 2.2.15 ). Isomorphic subgroups can easily be recognized because the standard space-group symbols of $[{\cal G}]$ and $[{\cal H}]$ are the same [isosymbolic subgroups (Billiet, 1973)] or the symbol of $[{\cal H}]$ is enantiomorphic to that of $[{\cal G}]$ . Every space group has an infinite number of maximal isomorphic subgroups , whereas the number of maximal non-isomorphic subgroups is finite (cf. Section 8.3.3 ). For this reason, isomorphic subgroups are discussed in more detail in the present section.

If a, b, c are the basis vectors defining the conventional unit cell of $[{\cal G}]$ and $[{\bf a}',{\bf b}',{\bf c}']$ the basis vectors corresponding to $[{\cal H}]$ the relation $[({\bf a}',{\bf b}',{\bf c}') = ({\bf a},{\bf b},{\bf c}){\bi S} \eqno(13.1.1.1)]$ holds, where (a, b, c) and $[({\bf a}',{\bf b}',{\bf c}')]$ are row matrices and S is a $[(3 \times 3)]$ matrix. The coefficients $[S_{ij}]$ of S are integers.¹

The index of $[{\cal H}]$ in $[{\cal G}]$ is equal to $[|\det ({\bi S})|]$ ¹, which is the ratio of the volumes [ $[{\bf a}'{\bf b}'{\bf c}']$ ] and [abc] of the two cells. $[\det ({\bi S})]$ is positive if the bases of the two cells have the same handedness and negative if they have opposite handedness.

If O and O′ are the origins of the coordinate systems (O, a, b, c) and $[(O',{\bf a}',{\bf b}',{\bf c}')]$ , used for the description of $[{\cal G}]$ and $[{\cal H}]$ , the column matrix of the coordinates of O′ referred to the system (O, a, b, c) will be denoted by s. Thus, the coordinate system $[(O',{\bf a}',{\bf b}',{\bf c}')]$ will be specified completely by the square matrix S and the column matrix s, symbolized by $[\specialfonts{\bbsf S}: ({\bi S}, {\bi s})]$ .

An example of the application of equation (13.1.1.1) is given at the end of this chapter.

13.1.1.1. The mathematical expression of equivalence

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Let $[\specialfonts{\bbsf W} = ({\bi W}, {\bi w})]$ be the operator of a given symmetry operation of $[{\cal H}]$ referred to (O, a, b, c) and $[\specialfonts{\bbsf W}' = ({\bi W}', {\bi w}')]$ the operator of the same operation referred to $[(O',{\bf a}',{\bf b}',{\bf c})]$ . Then the following relation applies $[\specialfonts{\bbsf S}{\bbsf W}' = {\bbsf W}{\bbsf S}\quad\rm \hbox{or}\quad {\bbsf W}' = {\bbsf S} ^{\rm -1}{\bbsf W}{\bbsf S} \eqno(13.1.1.2)]$ (cf. Bertaut & Billiet, 1979). The latter expression is more conventional, the former is easier to manipulate. Identifying the rotational (matrix) and translational (column) parts of $[\specialfonts{\bbsf W}]$ , one obtains the following two conditions: $[\eqalign{{\bi S}{\bi W}' &= {\bi W}{\bi S},\cr {\bi s} + {\bi S}{\bi w}' = {\bi w} + {\bi W}{\bi s} &= \hat{{\bi w}} + {\bi t}_{\cal G} + {\bi W}{\bi s}} \eqno(13.1.1.2{a})]$ or $[{\bi S}{\bi w}' - \hat{{\bi w}} + ({\bi I} - {\bi W}){\bi s} = {\bi t}_{\cal G}. \eqno(13.1.1.2b)]$

Here we have split w into a fractional part $[\hat{{\bi w}}]$ (smaller than any lattice translation) and $[{\bi t}_{\cal G}]$ which describes a lattice translation in $[{\cal G}]$ .

The general expression of the matrix S is $[{\bi S} = \pmatrix{S_{11} &S_{12} &S_{13}\cr S_{21} &S_{22} &S_{23}\cr S_{31} &S_{32} &S_{33}\cr}. \eqno(13.1.1.3)]$ This general form, without any restrictions on the coefficients, applies only to the triclinic space groups P1 and $[P\bar{1}]$ ; P1 has only isomorphic subgroups (cf. Billiet, 1979; Billiet & Rolley Le Coz, 1980). For other space groups, restrictions have to be imposed on the coefficients $[S_{ij}]$ .

References

Bertaut, E. F. & Billiet, Y. (1979). On equivalent subgroups and supergroups of the space groups. Acta Cryst. A35, 733–745.Google Scholar

Billiet, Y. (1973). Les sous-groupes isosymboliques des groupes spatiaux. Bull. Soc. Fr. Minéral. Cristallogr. 96, 327–334.Google Scholar

Billiet, Y. (1979). Le groupe P1 et ses sous-groupes. I. Outillage mathématique: automorphisme et factorisation matricielle. Acta Cryst. A35, 485–496.Google Scholar

Billiet, Y. & Rolley Le Coz, M. (1980). Le groupe P1 et ses sous-groupes. II. Tables de sous-groupes. Acta Cryst. A36, 242–248.Google Scholar

International Tables for Crystallography (2006). Vol. A. ch. 13.1, p. 836