A re-examination of the relationship between lattice strain, octahedral tilt angle and octahedral strain in rhombohedral perovskites

Abstract

A new expression is proposed for the relationship between lattice strain 90°− αpc ( αpc: pseudo-cubic angle) and mean BO6 octahedral tilt angle <ω> in rhombohedral perovskites ABO3. It is derived from volumetric arguments, leading to a cubic equation which incorporates lattice strain 90° −αpc and octahedral elongation explicitly. Numerical solutions of this equation are derived for equally spaced values of octahedral strain, giving rise to a set of parametric curves which relate ω to 90° − αpc for different values of η. These curves can be represented as polynomials of the fourth degree, thereby enabling their routine use in the analysis of rhombohedral perovskite structures. It is anticipated that these parametric curves will supersede earlier work [Megaw & Darlington (1975). Acta Cryst. A31, 161–173], in which an analytical expression was derived linking 90° − αpc, ω and octahedral strain for positive lattice strains only. By comparison, the relationship proposed here accommodates both negative and positive lattice strains. Correlations between values of <ω>, η, 90° − αpc and space-group symmetry are found, with an analysis of known rhombohedral and orthorhombic Pnma structures revealing the importance of cationic charges in determining symmetry. Since the polyhedral volume ratio VA/VB may be quantitatively related to <ω>, allowed values of tilt angle, lattice strain and octahedral elongation may be inferred for a given composition, which has characteristic values of VA and VB .