Structure of composites A1+x(A′xB1–x)O3 related to the 2H hexagonal perovskite: relation between composition and modulation

Abstract

An idealized structural model is proposed for materials closely related to the 2H hexagonal perovskite and with general formula A_{3n + 3m}A′_nB_{3m + n}O_{9m + 6n} {or A_{1 + x}(A′_xB_{1 – x})O₃, x = [0, 1/2]}. The structures of all these compounds, considered as commensurate or incommensurate modulated composites, can be described in a first approximation by a unique structure in the superspace formalism with occupational Crenel functions and sawtooth displacive modulations along the trigonal axis. The structural modulation has only two variable parameters: the modulation wave vector or misfit parameter, which is fixed by the compound composition, and the height difference between the octahedra and triangular prisms of O₆ present in the trigonal [A′,B]O₃ columns of the structure. Just by varying these parameters, the basic structural features of any compound with arbitrary composition parameter x are reproduced, including the two limiting cases ABO₃ (2H perovskite) and A₃A′BO₆. For instance, the sequence of octahedra and prisms along the [A′,B]O₃ columns can be considered a generalized Fibonacci chain and its dependence on composition follows a Farey tree rule which comes out directly from the model. A comparison with available experimental data, and in particular with some fully determined structures, demonstrates the soundness of the proposed scheme as a reference for such structures and the best starting point for their refinement. The model, although developed within the superspace formalism, is closely related to the polytype layer picture in direct space. The present structural analysis constitutes a new example of the efficiency of the 4-dimensional superspace formalism for describing in a unified form the structures of so-called ‘composition flexible’ systems.