Deriving the two-component description of incommensurate structures from the superspace group

Abstract

McConnell & Heine [Acta Cryst. (1984), A40, 473-482] have shown that an incommensurate (IC) structure may be fully described as an average structure plus two pure component difference structures C1 and C2 modulated by cos (Q . r) and sin (Q . r) respectively, where the symmetries of C1 and C2 are related in a precise way. This result was derived from the conventional Landau theory where the symmetry is specified by an irreducible representation of the space group of the average or disordered structure. It has also been shown by de Wolff, Janssen & Janner [Acta Cryst. (1981), A37, 625-636] that an IC crystal has the symmetry of a four-dimensional space group; the papers discussing these superspace groups describe the modulation in terms of only a single component. It is proved here that the two descriptions are identical in content, showing that the structure of a superspace group implicitly requires the existence of both C1 and C2, and that their symmetries are uniquely related in this formulation as in the McConnell-Heine theory. Two one-dimensional examples are discussed and NaNO2 is considered in detail. Although the McConnell-Heine theory was formulated in terms of the sinusoidal modulation which occurs just below the transition temperature, it is shown that the symmetry properties derived in that theory continue to be valid as the modulation 'squares up' at lower temperatures.