The crystallographic chameleon: when space groups change skin

Abstract

Volume A of International Tables for Crystallography is the reference for space-group information. However, the content is not exhaustive because for many space groups a variety of settings may be chosen but not all of them are described in detail or even fully listed. The use of alternative settings may seem an unnecessary complication when the purpose is just to describe a crystal structure; however, these are of the utmost importance for a number of tasks, such as the investigation of structure relations between polymorphs or derivative structures, the study of pseudo-symmetry and its potential consequences, and the analysis of the common substructure of twins. The aim of the article is twofold: (i) to present a guide to expressing the symmetry operations, the Hermann-Mauguin symbols and the Wyckoff positions of a space group in an alternative setting, and (ii) to point to alternative settings of space groups of possible practical applications and not listed in Volume A of International Tables for Crystallography.

The chromatic symmetry of twins and allotwins

“…2. The first polytype crystallizes in a space group of type P12 1 /a1, which is however better described in the unconventional setting B12 1 /e1 (Nespolo & Aroyo, 2016b), where the presence of a metric orthorhombic lattice becomes evident (a = À2c cos in the primitive cell, corresponding to = 90 in the B-centred cell); the second polytype crystallizes in a space group of type Pca2 1 . The first polytype crystallizes in a space group of type P12 1 /a1, which is however better described in the unconventional setting B12 1 /e1 (Nespolo & Aroyo, 2016b), where the presence of a metric orthorhombic lattice becomes evident (a = À2c cos in the primitive cell, corresponding to = 90 in the B-centred cell); the second polytype crystallizes in a space group of type Pca2 1 .…”

“…The conventional basis is hexagonal (Nespolo & Kogure, 1998). One can, however, always use a orthohexagonal basis in which the a and b vectors are the same in length as those of the single layer, although this would lead to a non-standard setting of the space group, with symbol C123 1 or C123 2 (Nespolo & Aroyo, 2016b).…”

“…An allotwin of malonamide, C 3 H 6 N 2 O 2 , has been reported by Strand et al (2016). The first polytype crystallizes in a space group of type P12 1 /a1, which is however better described in the unconventional setting B12 1 /e1 (Nespolo & Aroyo, 2016b), where the presence of a metric orthorhombic lattice becomes evident (a = À2c cos in the primitive cell, corresponding to = 90 in the B-centred cell); the second polytype crystallizes in a space group of type Pca2 1 . 8 The a and b cell parameters of the two polytypes are identical; the c parameter of the oB cell of the monoclinic polytype (transformation a, b, a + 2c from the mP cell) is twice the c parameter of the oP cell of the orthorhombic polytype within one standard uncertainty, so that the unit cell of the allotwin lattice coincides with the oB unit cell of the monoclinic polytype and is twice the unit cell of the orthorhombic polytype.…”

The Tricks of the Chameleon. Unexpected Symmetry of the Diffraction Pattern

“…The choice of a non‐conventional basis is always possible and can be preferable in some cases, like in the study of phase transitions between polymorphs showing a group–subgroup relation, in the analysis of the relations between structure types, in the presence of a high degree of structural pseudo‐symmetry, or in the case of twins, when the lattice common to the twinned individuals (domains) requires a space‐group description corresponding to an alternative setting (a series of examples can be found in ref. []). The Hermann–Mauguin symbol depends on the choice of the basis vectors and reflects the setting of the space group; as a result, the number of possible settings, and therefore of the possible Hermann–Mauguin symbols, even when restricting ourselves to simple and straightforward changes of basis, is much larger than the number of space‐group types.…”

International Tables for Crystallography, Volume A, Space-group symmetry. 6th edition. Edited by Mois I. Aroyo. Wiley, 2016. Pp. xxi + 873. Price GBP 295.00, EUR 354.00 (hardcover). ISBN 978-0-470-97423-0.

“…The tables are still partly incomplete, because for monoclinic groups the settings corresponding to a non-conventional unit cell with double volume (S instead of P; F instead of C) are absent. These are, however, present in the literature (Nespolo & Aroyo, 2016) and may perhaps be included in a future revision.…”