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

Burzlaff, H.; Zimmermann, H.

doi:10.1107/97809553602060000525

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. 12.2, p. 821

Section 12.2.3. The role of translation parts in the Shubnikov and Hermann–Mauguin symbols

H. Burzlaff^a and H. Zimmermann^b ^*

^a Universität 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

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

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A crystallographic symmetry operation W (cf. Part 11 ) 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. 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 $[{\bi w}_{g} = (1/k){\bi t}]$ , where k is the order of the operation W. If $[{\bi w}_{g}]$ is not a symmetry translation, the space group contains sets of screw axes or glide planes instead of the rotation axis or the mirror plane of the related point group. A screw rotation is symbolized by $[{n}_{m}]$ , where $[{\bi w}_{g} = (m/n){\bi t}]$ , with t the shortest lattice vector in the direction of the rotation axis. 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 12.2.3.1.

Table 12.2.3.1 | top | pdf |
Symbols of glide planes in the Shubnikov and Hermann–Mauguin space-group symbols

Glide plane perpendicular to	Glide vector	Shubnikov symbol	Hermann–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 a − b	$[{1 \over 2}{\bf c}]$	$[\tilde{c}]$	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
a − b	$[{1 \over 2}({\bf a} + {\bf b} + {\bf c})]$	$[\widetilde{abc}]$	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
a − b	$[{1 \over 4}({\bf a} + {\bf b} + {\bf c})]$	$[{1 \over 2}\widetilde{abc}]$	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 $[{\bi w}]$ . This is described in the following sections.

References

International Tables for Crystallography (2006). Vol. A. ch. 12.2, p. 821