Capabilities and limitations of a (3 + d)-dimensional incommensurately modulated structure as a model for the derivation of an extended family of compounds: example of the scheelite-like structures

The previously reported incommensurately modulated scheelite-like structure KNd(MoO₄)₂ has been exploited as a natural (3 + 1)-dimensional superspace model to generate the scheelite-like three-dimensional structure family. Although each member differs in its space-group symmetry, unit-cell parameters and compositions, in (3 + 1)-dimensional space, they share a common superspace group, a common number of building units in the basic unit cell occupying Wyckoff sites with specific coordinates (x, y, z) and specific basic unit-cell axial ratios (c/a, a/b, b/c) and angles. Variations of the modulation vector q, occupation functions and t₀ are exploited for the derivation. Eight topologically and compositionally different known structures are compared with their models derived from the KNd(MoO₄)₂ structure in order to evaluate the capabilities and limitations of the incommensurately modulated structure to act as a superspace generating model. Applications of the KNd(MoO₄)₂ structure as a starting model for the refinement and prediction of some other modulated members of the family is also illustrated. The (3 + 1)-dimensional presentation of the scheelite-like structures reveals new structural relations, which remain hidden if only conventional three-dimensional structure descriptions are applied.