Superspace description of the crystal structures of Ca(n)(Nb,Ti)nO3n + 2 (n = 5 and 6)

The crystal structures of two members of the homologous series Ca(n)(Nb,Ti)(n)O(3n + 2), with n = 5 and 6, are presented within the superspace formalism. A common (3 + 1)-dimensional superspace model is used to describe the crystal structures of both compositions within a particular homologous series, where the primary modulation wavevector and the width of the atomic domains vary systematically with composition. The two crystal structures are characterized as commensurately modulated structures consisting of discontinuous atomic domains described by occupational crenel functions. The displacive modulation functions for the two compounds exhibit similarities, but they also show that the idea of a unified superspace model does not extend toward the precise atomic positions. For n = 6, the centrosymmetric (3 + 1)-dimensional superspace symmetry provides a natural explanation for the pseudo-symmetries that are present in the non-centrosymmetric (three-dimensional) superstructure of this compound. The efficiency of the superspace approach is demonstrated by structure refinements in (3 + 1)-dimensional superspace and by comparing these results with the refinements in their three-dimensional superstructures.