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

International Tables for Crystallography (2016). Vol. A, ch. 3.3, p. 779

Section 3.3.2.3. The role of translation parts in the Shubnikov and Hermann–Mauguin symbols

H. Burzlaffa and H. Zimmermannb*

aUniversität Erlangen–Nürnberg, Robert-Koch-Strasse 4a, D-91080 Uttenreuth, Germany, and bInstitut für Angewandte Physik, Lehrstuhl für Kristallographie und Strukturphysik, Universität Erlangen–Nürnberg, Bismarckstrasse 10, D-91054 Erlangen, Germany
Correspondence e-mail:  helmuth.zimmermann@krist.uni-erlangen.de

3.3.2.3. The role of translation parts in the Shubnikov and Hermann–Mauguin symbols

| top | pdf |

A crystallographic symmetry operation [\ispecialfonts\sfi W] (cf. Chapter 1.2[link] ) is described by a pair of matrices [({\bi W}, {\bi w}) = ({\bi I}, {\bi w})({\bi W}, {\bi o}).]W is called the rotation part, w describes the translation part and determines the translation vector w of the operation. The translation part w can be decomposed into a glide/screw part [{\bi w}_{g}] and a location part [{\bi w}_{l}{:}\ {\bi w} = {\bi w}_{g} + {\bi w}_{l}]; here, [{\bi w}_{l}] determines the location of the corresponding symmetry element with respect to the origin. The glide/screw part [{\bi w}_{g}] may be derived by projecting w on the space invariant under W, i.e. for rotations and reflections w is projected on the corresponding rotation axis or mirror plane. With matrix notation, [{\bi w}_{g}] is determined by [({\bi W}, {\bi w})^{k} = ({\bi I}, {\bi t})] and [{\bi w}_{g} = (m/k){\bi t}_1], where k is the order of W, the integers m are restricted by [0 \le m\,\lt\, k] and t1 is the shortest lattice vector in the direction of t (for details, cf. Sections 1.2.2.4[link] and 1.5.4.1[link] ). Space groups contain sets of screw and rotation axes or glide and mirror planes. A screw rotation is symbolized by km. The Shubnikov notation and the international notation use the same symbols for screw rotations. The symbols for glide reflections in both notations are listed in Table 3.3.2.1[link].

Table 3.3.2.1| top | pdf |
Symbols of glide planes in the Shubnikov and Hermann–Mauguin space-group symbols

Glide plane perpendicular toGlide vectorShubnikov symbolHermann–Mauguin symbol
b or c [{1 \over 2}{\bf a}] [\tilde{a}] a
a or c [{1 \over 2}{\bf b}] [\tilde{b}] b
a or b or ab [{1 \over 2}{\bf c}] [\tilde{c}] c
c [{1 \over 2}({\bf a} + {\bf b})] [\widetilde{\!ab}] n
a [{1 \over 2}({\bf b} + {\bf c})] [\widetilde{bc}] n
b [{1 \over 2}({\bf c} + {\bf a})] [\widetilde{ac}] n
ab [{1 \over 2}({\bf a} + {\bf b} + {\bf c})] [\widetilde{abc}] n
c [{1 \over 4}({\bf a} + {\bf b})] [{1 \over 2}\>\widetilde{\!ab}] d
a [{1 \over 4}({\bf b} + {\bf c})] [{1 \over 2}\widetilde{bc}] d
b [{1 \over 4}({\bf c} + {\bf a})] [{1 \over 2}\widetilde{ac}] d
ab [{1 \over 4}({\bf a} + {\bf b} + {\bf c})] [{1 \over 2}\widetilde{abc}] d
a + b [{1 \over 4}(- {\bf a} + {\bf b} + {\bf c})]   d

If the point-group symbol contains only one generator, the related space group is described completely by the Bravais lattice and a symbol corresponding to that of the point group in which rotations and reflections are replaced by screw rotations or glide reflections, if necessary. If, however, two or more operations generate the point group, it is necessary to have information on the mutual orientations and locations of the corresponding space-group symmetry elements, i.e. information on the location components [{\bi w}_l]. This is described in the following sections.








































to end of page
to top of page