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

International Tables for Crystallography (2016). Vol. A, ch. 3.5, pp. 826-827

Section 3.5.1. Introduction and definitions

E. Koch,a W. Fischera and U. Müllerb

3.5.1. Introduction and definitions

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3.5.1.1. Introduction

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The mathematical concept of normalizers forms the common basis for the solution of several crystallographic problems:

It is generally known, for instance, that the coordinate description of a crystal structure trivially depends on the coordinate system used for the description, i.e. on the setting of the space group and the site symmetry of the origin. It is less well known, however, that for most crystal structures there exist several different but equivalent coordinate descriptions, even if the space-group setting and the site symmetry of the origin are unchanged. The number of such descriptions varies between 1 and 24 and depends only on the type of the Euclidean normalizer of the corresponding space group. In principle, none of these descriptions stands out against the others.

In crystal-structure determination with direct methods, the phases of some suitably chosen structure factors have to be restricted to certain values or to certain ranges in order to specify the origin and the enantiomorph. The information necessary for a correct selection of such phases and for their appropriate restrictions follows directly from the Euclidean normalizer of the space group. Similar examples are the positioning of the first atom(s) within an asymmetric unit when using trial-and-error or Patterson methods, the choice of a basis system for indexing the reflections of a diffraction pattern or the indexing of the first morphological face(s) of a crystal.

For the following problems, normalizers also play an important role: They supply information on the interchangeability of Wyckoff positions and their assignment to Wyckoff sets (cf. Section 1.4.4[link] and Chapter 3.4[link] ), needed e.g. for the definition of lattice complexes. They are important for the comparison of crystal structures, for their assignment to structure types and for the choice of a standard description for each crystal structure (Parthé & Gelato, 1984[link], 1985[link]). They allow the derivation of `privileged origins' for each space group (Burzlaff & Zimmermann, 1980[link]) and facilitate the complete deduction of subgroups and supergroups of a crystallographic group. They enable an easy classification of magnetic (black–white or Shubnikov) space groups and of colour space groups. They may also be used to reduce the parameter range in the study of geometrical properties of point configurations, e.g. their eigensymmetry or their sphere packings and Dirichlet partitions (cf. e.g. Koch, 1984a[link]).

In the past, most of these problems have been treated by crystallographers without the aid of normalizers, but the use of normalizers simplifies the solution of all these problems and clarifies the common background (for references, see Fischer & Koch, 1983[link]).

3.5.1.2. Definitions

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Any pair, consisting of a group [\cal G] and one of its supergroups [\cal S], is uniquely related to a third intermediate group [\cal N_S(G)], called the normalizer of [\cal G] with respect to [\cal{S}]. [\cal N_S(G)] is defined as the set of all elements [\ispecialfonts{\sfi s} \in \cal S] that map [\cal G] onto itself by conjugation (cf. Section 1.1.8[link] ):[\ispecialfonts{\cal N_S(G)} := \{{\sfi s} \in {\cal S}\;|\;{\sfi s}^{\rm -1}{\cal G}{\sfi s} = {\cal G}\}.]The normalizer [\cal N_S(G)] may coincide either with [\cal G] or with [\cal S] or it may be a proper intermediate group. In any case, [\cal G] is a normal subgroup of its normalizer.

For most crystallographic problems, three kinds of normalizers are of special interest:

  • (i) The normalizer of a space group (plane group) [\cal G] with respect to the group [\cal E] of all Euclidean mappings (motions, isometries) in [{\bb E}^3] [({\bb E}^2)], called the Euclidean normalizer of [\cal G]:[\ispecialfonts{\cal N_E(G)} := \{{\sfi s} \in {\cal E}\;|\;{\sfi s}^{\rm -1}{\cal G}{\sfi s} = {\cal G}\} .]

  • (ii) The normalizer of a space group (plane group) [\cal G] with respect to the group [\cal A] of all affine mappings in [{\bb E}^3] [({\bb E}^2)], called the affine normalizer of [\cal G]:[\ispecialfonts{\cal N_A(G)} := \{{\sfi s} \in {\cal A}\;|\;{\sfi s}^{\rm -1}{\cal G}{\sfi s} = {\cal G}\}. ]

  • (iii) The normalizer of a space group [\cal G] with respect to the group [{\cal E}^+] of all chirality-preserving Euclidean mappings in [{\bb E}^3], i.e. of all translations and proper rotations (including screw rotations), but excluding symmetry operations of the second kind (viz. inversions, reflections, glide reflections and rotoinversions). We call it the chirality-preserving Euclidean normalizer of [\cal G]:[\ispecialfonts {\cal N_{E^+}(G)} := \{{\sfi s} \in {{\cal E^+}}\;|\;{\sfi s}^{\rm -1}{\cal G}{\sfi s} = {\cal G}\} .]

[\cal N_{E^+}(G)] exists only if [\cal G] is a Sohncke space group. The 65 Sohncke space-group types are those space-group types that have no symmetry operations of the second kind (Flack, 2003[link]).1 They include the eleven pairs of types of enantiomorphic space groups; these eleven pairs are the only ones where the space groups themselves are chiral, i.e. which have an Euclidean normalizer containing only isometries of the first kind. The space groups of the remaining 43 Sohncke types are not chiral but do allow chiral crystal structures. A rigid object (or spatial arrangement of points or atoms) is chiral if it is nonsuperposable by pure rotation or translation on its image formed by inversion through a point. A chiral crystal structure is compatible only with a Sohncke space group.

The Euclidean normalizers of the space groups were first derived by Hirshfeld (1968[link]) under the name Cheshire groups. They have been tabulated in more detail by Gubler (1982a[link],b[link]) and Fischer & Koch (1983[link]). The Euclidean normalizers of triclinic and monoclinic space groups with specialized metric of the lattice were determined by Koch & Müller (1990[link]). The affine normalizers of the space groups have been listed by Burzlaff & Zimmermann (1980[link]), Billiet et al. (1982[link]) and Gubler (1982a[link],b[link]). They were also used for the derivation of Wyckoff sets and the definition of lattice complexes by Koch & Fischer (1975[link]), even though there the automorphism groups of the space groups were tabulated instead of their affine normalizers. The chirality-preserving Euclidean normalizers are tabulated in this volume for the first time.

References

Billiet, Y., Burzlaff, H. & Zimmermann, H. (1982). Comment on the paper of H. Burzlaff and H. Zimmermann. `On the choice of origin in the description of space groups'. Z. Kristallogr. 160, 155–157.
Burzlaff, H. & Zimmermann, H. (1980). On the choice of origin in the description of space groups. Z. Kristallogr. 153, 151–179.
Fischer, W. & Koch, E. (1983). On the equivalence of point configurations due to Euclidean normalizers (Cheshire groups) of space groups. Acta Cryst. A39, 907–915.
Flack, H. D. (2003). Chiral and achiral crystal structures. Helv. Chim. Acta, 86, 905–921.
Gubler, M. (1982a). Über die Symmetrien der Symmetriegruppen: Automorphismengruppen, Normalisatorgruppen und charakteristische Untergruppen von Symmetriegruppen, insbesondere der kristallographischen Punkt- und Raumgruppen. Dissertation, University of Zürich, Switzerland.
Gubler, M. (1982b). Normalizer groups and automorphism groups of symmetry groups. Z. Kristallogr. 158, 1–26.
Hirshfeld, F. L. (1968). Symmetry in the generation of trial structures. Acta Cryst. A24, 301–311.
Koch, E. (1984a). A geometrical classification of cubic point configurations. Z. Kristallogr. 166, 23–52.
Koch, E. & Fischer, W. (1975). Automorphismengruppen von Raumgruppen und die Zuordnung von Punktlagen zu Konfigurationslagen. Acta Cryst. A31, 88–95.
Koch, E. & Müller, U. (1990). Euklidische Normalisatoren für trikline und monokline Raumgruppen bei spezieller Metrik des Translationengitters. Acta Cryst. A46, 826–831.
Parthé, E. & Gelato, L. M. (1984). The standardization of inorganic crystal-structure data. Acta Cryst. A40, 169–183.
Parthé, E. & Gelato, L. M. (1985). The `best' unit cell for monoclinic structures consistent with b axis unique and cell cell choice 1 of International Tables for Crystallography (1983). Acta Cryst. A41, 142–151.








































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