International Tables for Crystallography

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Relating crystal structures by group–subgroup relations
Ulrich Müller. International Tables for Crystallography (2011). Vol. A1, ch. 1.6, pp. 44-56  [ doi:10.1107/97809553602060000795 ]

Abstract

The relations between crystal structures that are related by symmetry can be set forth in a concise manner with a tree of group–subgroup relations of their space groups (called a Bär­nig­hausen tree). At its top, the tree starts from the space-group symbol of an aristotype, i.e. a simple, highly symmetrical crystal structure. Arrows pointing downwards depict symmetry reductions that result from structural distortions or partial substitutions of atoms; each arrow represents the relation from a space group to a maximal subgroup. In the middle of each arrow the kind of the subgroup is marked by a t (translationengleiche), k (klassengleiche) or i (isomorphic) followed by the index of the symmetry reduction. In addition, changes of the basis vectors and origin shifts are marked. Each step of the symmetry reduction may involve moderate changes of the atomic coordinates that have to be monitored carefully. An aristotype can be at the head of a large family of structures. From the kinds of subgroups it can be deduced what and how many kinds of domains can result at a phase transition or topotactic reaction involving a symmetry reduction.


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About International Tables for Crystallography

International Tables for Crystallography is the definitive resource and reference work for crystallography. The multi-volume series comprises articles and tables of data relevant to crystallographic research and to applications of crystallographic methods in all sciences concerned with the structure and properties of materials.