RESUMEN
Phase transitions that result in incommensurate structural modulations are widely observed in crystalline solids and are relevant to a broad range of physical phenomena in magnetic, electronic, optical and structural materials. While the (3+1)-dimensional superspace-group symmetries associated with one-dimensional modulations have been tabulated, the order parameters that produce these modulations have not been explored in detail. Here, using group-theoretical methods, we present a unique and exhaustive enumeration of the isotropy subgroups (and their corresponding order-parameter directions) belonging to irreducible representations of the (3+1)-dimensional superspace extensions of the 230 crystallographic space groups at all incommensurate k points. The vast majority of experimentally observed incommensurately modulated structures have order parameters belonging to one of these subgroups.
RESUMEN
Group-theoretical methods are used to analyze perovskite structures where both ferroelectric cation displacements and simple tilting of octahedral units are present. This results in a list of 40 different structures, each with a unique space-group symmetry. The list is compared with that of Aleksandrov & Bartolomé [Phase Transit. (2001), 74, 255-335] and a number of differences are found. The group-subgroup relationships between the structures are also determined, along with an indication of those phase transitions that must be first order by Landau theory.