Bilbao Crystallographic Server SYMMODES Documentation

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The problem

The analysis of symmetry break in a structural phase transition.

The method

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:

1. 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 H_j 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 H_j. By specifying the relevant subgroup H_j of G the module returns the particular graph of maximal subgroups for G > H_j. 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).

2. For a given symmetry break G > H_j 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 > H_j graph.
   • The splitting schemes of the Wyckoff positions during the symmetry break G > H_j are calculated by the WYCKSPLIT module. The decomposition of an orbit O^G of G into O^H_i suborbits of H is achieved via the splitting of the general position of G.

The program

Input Data

1. The space groups ITA numbers.
2. The index of the transformation and the subgroup type or the matrix transformation that relates both space groups.
3. Wyckoff Positions of the atoms.

Output data

1. Group - Subgroup relations, matrix transformations and classification.
2. Order parameters, isotropy groups and k - vectors.
3. The primary and secondary modes according to the irreps of G for each of the Wyckoff positions.
4. Wyckoff positions splitting.

References

• 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.