By generalizing the concepts of the wavevector star channel in the Brillouin zone and its symmetry group [Kim et al. (2017). Acta Cryst. A73, 403-413; Kim et al. (2021). Acta Cryst. A77, 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 Pm3m[k11(1)k13] and Pm3m[k11(1)k12k13] has been carried out to show that the complex ordered phases observed experimentally in many perovskite crystals, including CaTiO3, SrZrO3, KMnF3, CsPbCl3 and Pb1-xCaxTiO3, 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 Pm3m.
Nanocrystals exhibit size-dependent structural behaviour because surface and interface effects become increasingly important as the characteristic domain size decreases. In this work, we investigate the size dependence of three diffraction-derived structural parameters - lattice parameter, isotropic Debye-Waller coefficient and microstrain - by combining molecular dynamics simulations of spherical Pd, Fe and Ti nanocrystals with powder diffraction analysis and comparison with representative literature data. The atomistic results show that small nanocrystals are characterized by a mean compressive state together with pronounced surface-stress inhomogeneity, while enhanced atomic displacements are concentrated in the outer coordination shells. On this basis, three simple nanoscale hypotheses are proposed. The lattice parameter is described as an ideal capillarity-driven reference trend with leading 1/D (where D is the particle diameter) behaviour, although comparison with experimental data confirms that this quantity is not universal and may be strongly modified by surface chemistry, defects, non-stoichiometry and morphology. By contrast, the Debye-Waller coefficient follows a more general 1/D decrease, consistent with a surface-shell picture of enhanced vibrational and static disorder. Microstrain arises from surface-stress gradients and, in the small-particle limit, exhibits a natural 1/D2 dependence associated with surface-stress heterogeneity; over broader size ranges, a mixed 1/D + 1/D2 form provides a more effective description. These results support a heuristic surface-driven interpretation of nanoscale structural disorder and clarify the different degrees of generality of the three scaling laws.
