The compositional dependence of octahedral tilting in orthorhombic and tetragonal perovskites

A new parameterization is defined for the quantitative description of octahedral tilting in orthorhombic and tetragonal perovskites. It contains six parameters, s₁, s₂, s₃, θ_x, θ_y and θ_z. s₁, s₂ and s₃ refer to the lengths of the lines or `stalks' joining pairs of opposite octahedral vertices, and θ_x, θ_y and θ_z to the angles subtended by these stalks with pseudo-cubic axes x, y and z. An equation is derived for the dependence of polyhedral volume ratio, V_a / V_B, on these parameters: V_a / V_B = 6cos²θ_m cos θ_z − 1, where θ_m = (θ_x + θ_y)/2. To a good approximation, lengths s₁, s₂ and s₃ do not affect V_a / V_B. The validity of this equation is tested by reference to the known crystal structures of 48 ternary oxide and fluoride perovskites, and its versatility demonstrated by application to the structures of some ternary palladium and platinum hydrides. The relationship of the approach to Glazer's system of nomenclature for octahedral tilting in perovskites is considered, in particular concerning the numbers of tilts operative in a given structure. A comparison is also made with the parameterization proposed for octahedral tilting and distortion in rhombohedral perovskites [Thomas & Beitollahi (1994). Acta Cryst. B50, 549–560]. Factors governing the choice of rhombohedrai or orthorhombic symmetry are discussed, with the significance of rhombohedral symmetry in obtaining ferroelectric properties brought out. Through the compilation of a table of AO₁₂ and BO₆ polyhedral volumes, the prospect is identified of predicting both the degree of octahedral tilting and the likelihood of ferroelectric behaviour for novel, hypothetical oxide compositions.