Shubnikov 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, pp. 821-822

Section 12.2.4. Shubnikov 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

12.2.4. Shubnikov symbols

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For the description of the mutual orientation of symmetry elements, the same symbols as for point groups are applied. In space groups, however, the symmetry elements need not intersect. In this case, the orientational symbols · (dot), : (colon), / (slash) are modified to $[\bigodot, \def\circcol{\mathop{\bigcirc\hskip -6pt{\raise.05pt\hbox{:}} \ }} \circcol, // ]$ . The space-group symbol starts with a description of the lattice defined by the basis a, b, c. For centred cells, the vectors to the centring points are given first. The same letters are used for basis vectors related by symmetry. The relative orientations of the vectors are denoted by the orientational symbols introduced above. The description of the lattice given in parentheses is followed by symbols of the generating elements of the related point group. If necessary, the symbols of the symmetry operations are modified to indicate their glide/screw parts . The first generator is separated from the lattice description by an orientation symbol. If this generator represents a mirror or glide plane, the dot connects the plane with the last two vectors whereas the colon refers only to the last vector. If the generator represents a rotation or a rotoreflection, the colon orients the related axis perpendicular to the plane given by the last two vectors whereas the dot refers only to the last vector. Two generators are separated by the symbols mentioned above to denote their relative orientations and sites. To make this description unique for space groups related to point group $[O_{h} \equiv \tilde{6} / 4]$ with Bravais lattices cP and cF, it is necessary to use three generators instead of two: $[4/\tilde{6} \cdot m]$ . For the sake of unification, this kind of description is extended to the remaining two space groups having Bravais lattice cI.

Example: Shubnikov symbol for the space group with Schoenflies symbol $[D_{2h}^{26}\it{\ (72)}]$

The Bravais lattice is oI (orthorhombic, body-centred). Therefore, the symbol for the lattice basis is $[\left({a+b+c \over 2} \bigg/ c : (a : b) \right),]$ indicating that there is a centring vector $[1/2({\bf a + b + c})]$ relative to the conventional orthorhombic cell. This vector is oblique with respect to the basis vector c, which is orthogonal to the perpendicular pair a and b. The basis vectors have independent lengths and are thus indicated by different letters a, b and c in arbitrary sequence.

To complete the symbol of the space group, we consider the point group $[D_{2h}]$ . Its Shubnikov symbol is $[m:2\cdot m]$ . Parallel to the (a, b) plane, there is a glide plane $[\widetilde{ab}]$ and a mirror plane m. The latter is chosen as generator. From the screw axis $[2_{1}]$ and the rotation axis 2, both parallel to c, the latter is chosen as generator. The third generator can be a glide plane c perpendicular to b. Thus the Shubnikov symbol of $[D_{2h}^{26}]$ is $[\left({a+b+c \over 2} \bigg / c : (a:b) \right) \cdot m : 2\cdot \tilde{c}.]$

The list of all Shubnikov symbols is given in column 3 of Table 12.3.4.1 .

References

International Tables for Crystallography (2006). Vol. A. ch. 12.2, pp. 821-822