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关注:
1) 结构优化过程中子群到高对称性超群的转变
2) crystal structure determination
If G is a group and H is a subgroup of G, then G is a supergroup of H.
If H is a maximal subgroup of G, then G is a minimal supergroup of H.
题记:
Since occupation of some special positions in one space group makes it equivalent to one of its supergroups,
a full relaxation of ions in alower symmetry phase would degenerate itself into (or equivalently relax to) a higher symmetry phase (Figs. 1).
参考网址:
程序
http://www.cryst.ehu.es/cryst/supergroups.html
程序说明
http://www.cryst.ehu.es/cryst/super_theory.html
The crystallographic problem
The problem of the determination of the supergroups of a given space group is of rather general interest. It is useful for search of overlooked symmetries in crystal structure determination, or in the analysis of successive phase transitions, where the introduction of a hypothetical (supergroup) parent phase can be important.
Another important application is related with the detection of pseudosymmetries in crystal structures as a method of predicting structural phase transitions at higher temperatures (Igartua, Aroyo, Perez-Mato, 1996).
There are few papers treating the supergroups of space groups in some detail (Koch, 1984). The existing listings of minimal supergroups of space groups are not complete as they provide only a list of those space groups G which contain a space group H as a maximal non-isomorphic subgroup
(International Tables for Crystallography, vol. A, 1992 (ITA)).
This information is in general not sufficient because it does not include all the possible supergroups of H isomorphic to G. The presented program determines all possible supergroups Gi of H which belong to the space group type of G.
Method of solutionSUPERGROUPS determines all possible supergroups Gi systematically inverting the data on maximal subgroups of space groups (International Tables for Crystallography, vol. A1 (ITA1)). This data has been prepared in CIF-like format in computer-readable form.
Following the procedure based in normalizers of space groups (Koch 1984, Wondratschek 1996) the program calculates all supergroups Gi > H, Gi ~ G.
Derivation of supergroupsSupergroup: S; Subgroup: G: G <S
The decomposition of S with respect to G:
S = G + q2G + q3G + ... + qnG
Supergroups S(i) of G, isomorphic to S:
S(i) = ai-1 S ai, S(i) ~ S
Decomposition of S(i) with respect to G:
S(i) = ai-1 G ai + ai-1 q2 G ai + ai-1 q3 G ai + ... + ai-1 qn G ai =
= Gi + q2i Gi + q3i Gi + ... + qni Gi, qkj = aj-1 qk aj
S(i) decomposition with respect to G:
S(i) = G + qi,2 G + qi,3 G + ... + qi,n G
==> TWO sets of supergroups S(i) of G:
Supergroups conjugated under N(G) (the normalizer of G)
S(i) = ai-1 S ai, ai in N(G)
Supergroups not conjugated under N(G)
S(i) = ai-1 S ai, ai not in N(G)
http://www.ncbi.nlm.nih.gov/pubmed/11326116
The application of Hermann's group M in group-subgroup relations between space groups.This paper is devoted to the study of the group-subgroup relations U < G between space groups.
A procedure has been developed for the derivation of all subgroups U(j) < G which are
obtained from U by a transformation with a translation (T-equivalent subgroups).
All T-equivalent supergroups G(k) > U can be determined in the same way
from one supergroup G > U.
The decisive group in this procedure is the translation part of the (Euclidean) normalizer of Hermann's group M. The group M is the uniquely determined group U < or = M < or = G with the translations of G and the point group of U. The method is particularly useful in the search for supergroups of space groups and is based on several lemmata which are formulated and proven in this paper. The results suggest under special conditions the possibility of a transition with 'region' formation in some analogy to the well known domain formation. This transition could occur from high symmetry to low symmetry or from low symmetry to high symmetry or even both ways.
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