Intergrowth polytypoids as modulated structures: the example of the cation deficient oxides LaTi_1-xO₃

A new, essentially composition-independent structural model is proposed for the perovskite-related compound series La_nTi_n-δO_3n, with δ ≤ n/4, based on superspace formalism. In the standard 3-dimensional crystallographic approach, one must specify space group and cell parameters separately for each composition. In the superspace approach, the superspace group is invariant while the cell parameters and primary modulation wave-vector vary slowly and continuously with composition. To a first approximation, the structure at any particular composition can be described in terms of mixed cubic and hexagonal stacking of close-packed LaO₃ (and Ti) layers along the z-axis. The actual layer stacking sequence varies abruptly with composition, and the titanium vacancies occur as entire layers between neighbouring hexagonal close-packed LaO₃ layers. The continuously variable, composition-dependent character of reciprocal space indicates that the various layer stacking sequences must be intimately correlated and shows that all the compounds can be interpreted in terms of a structural modulation over a common average structure. It is shown that a very simple model in superspace is sufficient to describe the layer stacking sequences for any particular composition. The atomic surfaces representing the atoms in superspace are shown to be well described by means of crenel functions. The cation displacive modulations are approximately described by sawtooth functions. The only variable parameters are the composition-dependent (but smoothly varying) primary modulation wave-vector and associated Ti occupation domain width. Conventional space groups can be easily determined from the superspace group and the composition. From a practical point of view, this means that we have a very good single starting point to refine the actual atomic positions for any composition, including incommensurate or long period structures. Even more importantly, the superspace approach has the flexibility to represent in a simple manner the symmetry-allowed deviations from the ideal model. A comparison with the scarce number of refined structures in this system and in the homologous (Ba,La)_nTi_n-1O_3n system, demonstrates the suitability of the formalism for the analysis of these compound series and for the analysis of compositionally flexible systems in general.