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-182 21 Prague 8
We present a dimension-independent classification of domain pairs arising from distinct
structural phase transitions. The classification is based on an identification of a domain
pair by the symmetry of one state within the pair, the twinning group [1] and the
stabilizing groups [3] of the pair. Invariants and relative invariants of the twinning
group and stabilizing groups
yield such variables (e.g. definite projections of tensors) that coincide, or differ only
in sign in both states of the pair [3,4].
The twinning group, in particular, determines those order parameters which can distinguish
between the two states, and which coincide in the pair. Using the point groups of the
groups characterizing a domain pair we have introduced permutational, rotational and
crystallographic types of pairs for equitranslational phase transitions [2]. We provide a
generalization of these types to include pairs resulting from non-equitranslational
transitions as well.
Domain pairs can be divided into four classes according to the complexity of the mutual
relationship of the two states [2], i.e. the respective twinning law. To visualize this
law we represent it by a lattice formed by relevant transitivity sets of stabilizing
groups of the pair; the inclusions between such sets show the relationships between
invariants. While the first two classes contain pairs that have at most one non-trivial
stabilizing group, pairs in two other classes can have two, or more stabilizing groups. To
identify a pair by means of its stabilizing groups one will need a unique symbol for each
subgroup of a space group. We give a proposal of modified (`orientational')
Hermann-Mauguin symbols that will solve
the problem of notation for individual subgroups of space groups.
The determination of stabilizing groups of domain pairs is facilitated by a C-program that
produces the lattice of subgroups of a given space group and displays it on screen. The
classification of pairs and the use of the program
is demonstrated on an illustrative example of a non-equitranslational phase a sample
non-equitranslational phase transition.
This work was supported by grant no. A1010611 of the Grant Agency of the Academy of Sciences of the Czech Republic.