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, s(1), s(2), s(3), theta(x), theta(y) and theta(z). s(1), s(2) and s(3) refer to the lengths of the lines or 'stalks' joining pairs of opposite octahedral vertices, and theta(x), theta(y) and theta(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 = 6 cos(2) theta(m) cos theta(z) - 1, where theta(m) = (theta(x) + theta(y))/2. To a good approximation, lengths s(1), s(2) and s(3) 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 rhombohedral or orthorhombic symmetry are discussed, with the significance of rhombohedral symmetry in obtaining ferroelectric properties brought out. Through the compilation of a table of AO(12) 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.