The wavevector star channel and symmetry group

Abstract

The concepts of `wavevector star channel' and `wavevector star channel group' are newly defined, which allow the effective study of phase transitions considering directly the translational symmetry breaking in crystals. A method is suggested by which the wavevector star channels can be found using the image of the representation of the translational group. According to this method, the wavevector star channels are found for the 80 Lifschitz stars in the reciprocal lattice. The wavevector star channel group is defined as the set of symmetry elements of the parent phase which leave the star channel invariant, and the wavevector star channel groups with one, two, three and four arms are calculated. It is shown that the complicated symmetry changes in the pyroelectric crystal Pb1−x Ca x TiO3 (PCT) can be described using the new five-component reducible order parameter transformed according to the representation of the wavevector star channel group, rather than the nine-component one based on the theory of the full irreducible representation of the space group.