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INVARIANTS : program for obtaining a list of invariant polynomials of the order‐parameter components associated with irreducible representations of a space group
Author(s) -
Stokes Harold T.,
Hatch Dorian M.
Publication year - 2003
Publication title -
journal of applied crystallography
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.429
H-Index - 162
ISSN - 1600-5767
DOI - 10.1107/s0021889803005946
Subject(s) - invariant (physics) , mathematics , group (periodic table) , irreducible representation , order (exchange) , algebra over a field , space (punctuation) , pure mathematics , combinatorics , computer science , physics , mathematical physics , finance , quantum mechanics , economics , operating system
In a crystalline phase transition, the space-group symmetry of the highand low-symmetry phases often exhibits a group±subgroup relationship: L H. In the Landau theory of phase transitions, this transition is described by a primary order parameter (OP). Such an OP is an n-dimensional vector g in the space de®ned by an irreducible representation (IR) of H. The action of the symmetry transformations induces corresponding transformations on the basis of the IR. The most general invariant free energy (`Landau potential'), which describes the energy change at the transition, is expressed as homogeneous sets of polynomials of the OP components of degree p, each polynomial being invariant under the symmetry group H. The high-symmetry group H corresponds to g = 0 and the low-symmetry group L to the minimum of for non-zero g. In addition to the primary OP there are also secondary OPs which correspond to other IRs of H and which couple to the primary OP. These secondary OPs also play a role in the transition as additional distortions and thus invariants corresponding to these IRs, along with their coupling with the primary OP, are of interest. The aim of the program is to generate symmetry-allowed invariants associated with a given high-symmetry group H and the primary OP, along with invariants and coupling polynomials of the secondary OPs.