Reducible wavevector star channel groups and symmetry analysis of ordered phases in perovskite crystals

By generalizing the concepts of the wavevector star channel in the Brillouin zone and its symmetry group [Kim et al. (2017). Acta Cryst. A 73 , 403–413; Kim et al. (2021). Acta Cryst. A 77 , 572–592] to a reducible wavevector star containing several Lifshitz stars, new concepts of the reducible wavevector star channel and its symmetry group are defined and methods of finding them are proposed. It is shown that structural distortions, such as octahedral tilting, atomic displacement and Jahn–Teller ordering, characterized by ten reducible stars related to the Lifshitz ones Γ, M , R and X in ferroics and multiferroics belonging to 15 parent space groups of the simple cubic lattice, are described by 105 reducible star channels, allowed by symmetry, and reducible star channel groups are found. The symmetry analysis of ordered phases induced by the reducible representations of the reducible wavevector star channel groups Pm 3 m [ k 11 (1) k 13 ] and Pm 3 m [ k 11 (1) k 12 k 13 ] has been carried out to show that the complex ordered phases observed experimentally in many perovskite crystals, including CaTiO 3 , SrZrO 3 , KMnF 3 , CsPbCl 3 and Pb 1− x Ca x TiO 3 , can be described by much simpler order parameter models, rather than the ones associated with reducible representations corresponding to the reducible wavevector star of the space group Pm 3 m .