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International
Tables for Crystallography Volume E Subperiodic groups Edited by V. Kopský and D. B. Litvin © International Union of Crystallography 2006 |
International Tables for Crystallography (2006). Vol. E. ch. 1.2, pp. 20-22
Section 1.2.15.3. Minimal non-isotypic non-enantiomorphic supergroups
a
Department of Physics, University of the South Pacific, Suva, Fiji, and Institute of Physics, The Academy of Sciences of the Czech Republic, Na Slovance 2, PO Box 24, 180 40 Prague 8, Czech Republic, and bDepartment of Physics, Penn State Berks Campus, The Pennsylvania State University, PO Box 7009, Reading, PA 19610-6009, USA |
If G is a maximal subgroup of a group H, then H is called a minimal supergroup of G. Minimal supergroups are again subdivided into two types, the translationengleiche or t supergroups I and the klassengleiche or k supergroups II. For the t supergroups I of G, the listing contains the index [i] of G in H and the conventional Hermann–Mauguin symbol of H. For the k supergroups II, the subdivision between IIa and IIb is not made. The information given is similar to that for the subgroups IIb, i.e. the relations between the basis vectors of group and supergroup are given, in addition to the Hermann–Mauguin symbols of H. Note that either the conventional cell of the k supergroup H is smaller than that of the subperiodic group G, or H contains additional centring translations.
Example: G: Layer group ![[p2_1/m11]](/teximages/each1o2/each1o2fi90.svg)
(L15)
Minimal non-isotypic non-enantiomorphic supergroups:![[\eqalign{&{\bf I}\phantom{\bf I}\quad[2]\;pmam;\;[2]\;pmma;\;[2]\;pbma;\;[2]\;pmmn \cr &{\bf II}\quad[2]\;c2/m11;\;[2]\;p2/m11\;(2{\bf a}'={\bf a})}]](/teximages/each1o2/each1o2fd25.svg)
Block I lists [2] pmam, [2] pmma and [2] pmmn. Looking up the subgroup data of these three groups one finds [2] p21/m11. Block I also lists [2] pbma. Looking up the subgroup data of this group one finds [2] p121/m1 (p21/m11). This shows that the setting of pbma does not correspond to that of p21/m11 but rather to p121/m1. To obtain the supergroup H referred to the basis of p21/m11, the basis vectors a and b must be interchanged. This changes pbma to pmba, which is the correct symbol of the supergroup of p21/m11.
Block II contains two entries: the first where the conventional cells are the same with the supergroup having additional centring translations, and the second where the conventional cell of the supergroup is smaller than that of the original subperiodic group.
