Acta Crystallographica Section A

Foundations of Crystallography

Volume 69, Part 1 (January 2013)

research papers

Acta Cryst. (2013). A69, 75-90    [ doi:10.1107/S0108767312041657 ]

Equivalence of superspace groups

S. van Smaalen, B. J. Campbell and H. T. Stokes

Abstract: An algorithm is presented which determines the equivalence of two settings of a (3 + d)-dimensional superspace group (d = 1, 2, 3). The algorithm has been implemented as a web tool {\tt findssg} on {\tt SSG(3+d)D}, providing the transformation of any user-given superspace group to the standard setting of this superspace group in {\tt SSG(3+d)D}. It is shown how the standard setting of a superspace group can be directly obtained by an appropriate transformation of the external-space lattice vectors (the basic structure unit cell) and a transformation of the internal-space lattice vectors (new modulation wavevectors are linear combinations of old modulation wavevectors plus a three-dimensional reciprocal-lattice vector). The need for non-standard settings in some cases and the desirability of employing standard settings of superspace groups in other cases are illustrated by an analysis of the symmetries of a series of compounds, comparing published and standard settings and the transformations between them. A compilation is provided of standard settings of compounds with two- and three-dimensional modulations. The problem of settings of superspace groups is discussed for incommensurate composite crystals and for chiral superspace groups.

Keywords: symmetry; superspace groups; two-dimensionally modulated crystals; three-dimensionally modulated crystals.

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[ doi:10.1107/S0108767312041657/ ]
Output of SSG(3+d)D for different settings of superspace groups

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[ doi:10.1107/S0108767312041657/pc5018sup2.pdf ]
Algorithm for determining the transformation S between two settings of (3+d)-dimensional superspace groups (d=1, 2, 3)


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