E.Kroumova, M.I. Aroyo, J. M. Pérez-Mato, H.T. Stokes and D. Hatch
A software package for group-theoretical analysis of structural phase transitions
Dept. Física de la Materia Condensada
, Universidad del País Vasco,
Apdo 644, 48080 Bilbao, Spain
Dept. of Physics and Astronomy, Brigham Young University, Provo, Utah 84602 USA.
Keywords: Phase transitions, symmetry break, primary and secondary modes.
The analysis of symmetry break in a structural phase transition.
Consider a structural phase transition with a symmetry change
G > H, where the low-symmetry space
group H is a subgroup of the high-symmetry group G.
The main steps of the symmetry analysis carried out by SYMMODES
can be summarized as follows:
- Given the space-group types of G and H, and their
index, the SUBGROUPGRAPH module constructs the lattice of maximal subgroup
s relating G and H. All possible
subgroups Hj of the type of H are listed, and their
distribution into classes of conjugate subgroups is indicated. In addition, the
program also supplies the corresponding transformation matrices relating
the (conventional) bases of G to each of the subgroups
Hj. By specifying the relevant subgroup Hj
of G the module returns the
particular graph of maximal subgroups for G > Hj.
The results on the group-subgroup relations for the chain G >
H obtained by the program SUBGROUPGRAPH are based on the data of maximal subgroups of space groups available in
International Tables for Crystallography, vol.A1 (2003).
- For a given symmetry break G > Hj
and a crystal structure specified by the Wyckoff positions of the occupied
atomic orbits, the program calculates:
- The number and the patterns of the primary and
secondary modes. The symmetry of
each mode is characterized by an irrep of the
high-symmetry group, the direction of the order parameter in the
representation space, and the corresponding isotropy subgroup from the
G > Hj graph.
- The splitting schemes of the Wyckoff positions during the symmetry break G > Hj are calculated by the WYCKSPLIT module.
The decomposition of an orbit OG of G into OHi
suborbits of H is achieved via the splitting of the general
position of G.
- The space groups ITA numbers.
- The index of the transformation and the subgroup type or the matrix transformation that relates both space groups.
- Wyckoff Positions of the atoms.
- Group - Subgroup relations, matrix transformations and classification.
- Order parameters, isotropy groups and k - vectors.
- The primary and secondary modes according to the irreps of G for each of the Wyckoff positions.
- Wyckoff positions splitting.
- International Tables of Crystallography. Space Group
Symmetry (2002), Vol.A. Ed. T. Hahn, 5th ed. Dordrecht:
Kluwer Academic Publishers.
- International Tables of Crystallography. Maximal
Subgroups of Plane and Space Groups (2003), Vol. A1. Eds.
H. Wondratschek, U. Müller, Dordrecht:
Kluwer Academic Publishers.(to appear).
- Ivantchev, S., Kroumova, E., Madariaga, G., Perez-Mato, J.M.,
Aroyo, M.I. (2000). J. Appl. Cryst. 33, 1190-1191.
- Kroumova, E., Aroyo, M.I., Perez-Mato, J.M. (1998). J. Appl. Cryst. 31, 646-646.
- Kroumova, E., Aroyo, M.I., Perez-Mato, J.M., Kirov, A., Capillas,
C., Ivantchev, S., Wondratschek, H. (2002).
Phase Transitions, (in print).
- Miller, S.C., Love, W.F. (1967). Tables of Irreducible
Representations of Space Groups and Co-representations of Magnetic Space
Groups. Boulder: Prett.
- Stokes, H. T. ,Hatch, D. M. (1998). ISOTROPY. Department of Physics
and Astronomy, Bringham Young University, Provo, USA.