By H. Wondratschek, U. Müller
This quantity provides a scientific remedy of the maximal subgroups and minimum supergroups of the crystallographic airplane teams and area teams. it's an extension of and a complement to quantity A, Space-group symmetry, during which purely easy facts for sub- and supergroups are supplied.
Group-subgroup kin, except their theoretical curiosity, are the foundation of a couple of vital purposes in crystallographic research:
(1) In solid-state section transitions there frequently exists a group-subgroup relation among the symmetry teams of the 2 levels. in keeping with Landau concept, this can be actually essential for displacive (continuous, second-order) section transitions. Group-subgroup kinfolk also are imperative in situations the place the symmetry teams of the 2 stages aren't without delay comparable yet percentage a standard subgroup or supergroup.
(2) Group-subgroup kin offer a concise and robust device for revealing and elucidating relatives among crystal buildings. they could hence aid to maintain with the ever-increasing volume of crystal-structure info. Their software calls for wisdom of the kinfolk of the Wyckoff positions of group-subgroup similar buildings.
(3) Group-subgroup kin are of significant significance within the learn of twinned crystals, area buildings and area boundaries.
(4) those kinfolk can also aid to spot error in space-group project and crystal-structure choice.
(5) Subgroups of area teams offer a invaluable method of educating crystallographic symmetry.
Volume A1 involves 3 parts:
Part 1 provides an advent to the idea of area teams at quite a few degrees and with many examples. It contains a bankruptcy at the mathematical conception of subgroups.
Part 2 supplies for every aircraft workforce and house crew a whole directory of all maximal subgroups and minimum supergroups. The remedy contains the turbines of every subgroup in addition to any beneficial adjustments of the coordinate process. Maximal isomorphic subgroups are given in parameterized shape as limitless sequence a result of countless quantity for every crew. a different characteristic of the presentation is graphs that illustrate the group-subgroup family members.
Part three lists the kinfolk among the Wyckoff positions of each area team and its subgroups. back, the limitless variety of maximal isomorphic subgroups of every house team are coated via parameterized sequence. those facts for Wyckoff positions are provided the following for the 1st time.
Audience: the quantity is a important addition to the library of scientists engaged in crystal-structure decision, crystal physics or crystal chemistry. it truly is crucial for these attracted to part transitions, the systematic compilation of crystal constructions, twinning phenomena and similar fields of crystallographic research.
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Additional info for International Tables for Crystallography, Vol.A1: Symmetry relations between space groups
Domain structures of a general phase transition Up to now, the examples have been concerned either with translationengleiche or with klassengleiche transitions only. e. a transition where H is a general subgroup of G. General subgroups are not listed in this volume but have to be derived from the maximal subgroups of each single step of the group–subgroup chain between G and H. 2, the chain has two steps. The results obtained under the PCA and without it are different and, therefore, will be discussed in some detail.
In the continuum description, we consider the point groups and |42m : mm2| = 2. There is only one subgroup of 42m of the type mm2. 1 of AuCu3 , G = M1 because H1 is a klassengleiche subgroup of G. Therefore, |G : M1 | = 1 and all four domain states belong to the same orientation state. This is obvious visually, because, as stated above, all four domain states are parallel and only shifted against each other. 4. Because of the isomorphism (G : M) ∼ = (PG : PH ) between the factor groups (G : M) and (PG : PH ), the 21 1.
143; . . ; N = 4 : 24 − 1 = 15 subgroups of index 2 for p2mm, No. 6, and P1, No. 2; N = 6 : 26 − 1 = 63 subgroups of index 2 for Pmmm, No. 47. An index of p2 , p > 2, occurs only for isomorphic subgroups of tetragonal, trigonal and hexagonal space groups when the basis vectors are enlarged to pa, pb. An index of p3 occurs for and only for isomorphic subgroups of cubic space groups with cell enlargements of pa, pb, pc (p > 2). This lemma means that a subgroup of, say, index 6 cannot be maximal.