Equivalence of superspace groups

• Published in: Acta crystallographica. Section A, Foundations of crystallography Foundations of crystallography, 2013-01, Vol.69 (1), p.75-90
• Main Authors: van Smaalen, Sander, Campbell, Branton J., Stokes, Harold T.
• Format: Article
• Language: English
• Subjects: Algorithms; Crystal lattices; Crystallography; Lattices; Mathematical analysis; Modulation; Research Papers; superspace groups; symmetry; Three dimensional; three-dimensionally modulated crystals; Transformations; two-dimensionally modulated crystals; Vectors (mathematics)
• ISSN: 0108-7673; 1600-5724; 2053-2733
• DOI: 10.1107/S0108767312041657

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 on , providing the transformation of any user‐given superspace group to the standard setting of this superspace group in . 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.