THE USE OF LATTICES OF SUBGROUPS OF SPACE GROUPS IN DOMAIN DISTINCTION AND CLASSIFICATION OF DOMAIN PAIRS. THE DEMONSTRATION OF A C-PROGRAM TO VISUALIZE SUCH LATTICES
P. Engles1 and J. Fuksa2
1Lab. f. chem. u. mineral.
Kristallographie, Univ. Bern, Freiestrasse 3, CH-3012 Bern
2Institute of Physics AS CR, Na
Slovance 2, CZ-180 40 Prague 8
One of the objectives of domain structure analysis is to solve the domain distinction task: Given a phase transition G -->F, for any two single domain states to find which properties are the same, and which are different in both the states. The task can be solved with the use of permutational equivalence [1,3] of domain state pairs; by definition, any two equivalent pairs exhibit similar distinction.
A classification of domain pairs that will apply to pairs arising from distinct transitions must be based on a unique identification of a pair. From the analysis of permutational properties of domain states follows that for that purpose we can use the symmetry of one state within the pair, the twinning group [1] and the stabilizing groups [3] of this pair. For equitranslational phase transitions a classification of domain pairs into rotational and crystallographic types [2] was introduced as a generalization of the permutational equivalence. This classification is modified to include pairs resulting from non-equitranslational phase transitions.
Domain pairs can be divided into four classes according to their complexity expressed via their stabilizing groups by the respective twinning law [2]. While the first two classes contain pairs that have at most one non-trivial stabilizing group, pairs in the other classes can have a number of stabilizing groups. The twinning group of a domain pair determines its `ferroic' type, i.e. those order parameters which can distinguish between the two states, and which coincide in the pair [3,4]. Both the twinning group and stabilizing groups yield invariants and relative invariants of the pair, i.e. those variables (definite projections of tensors in particular) that coincide, or differ only in sign in both states of the pair.
The determination of stabilizing groups is facilitated by a C-program that produces the lattice of subgroups of a given space group and displays it on screen. To demonstrate the classification of pairs and the use of the program we give illustrative examples of non-equitranslational phase transitions.
This work was supported by grant no. A1010611 of the Grant Agency of the Academy of Sciences of the Czech Republic.