On the polyhedral volume ratios V_A/V_B in perovskites ABX₃

This paper presents analytical expressions for the calculation of ratios of cation coordination polyhedra volumes (V_A/V_B) for perovskites ABX₃ of the Stokes–Howard diagram directly from atomic coordinates. We show the advantages of quantifying perovskite structure distortion with polyhedral volume ratios rather than with tilting angles, and discuss why space groups with multiple crystallographically inequivalent A or B sites (I4/mmm, Immm, P4₂/nmc etc.) are much less common than those with a single A and B site (I4/mcm, Rc, Pnma etc.). Analysis of crystallographic data for approximately 1300 perovskite structures of oxides, halides and chalcogenides from the Inorganic Crystal Structure Database revealed that the most highly distorted perovskites belong to the space group Pnma and formally lower-symmetry perovskites (I2/m, I2/a) are less distorted geometrically. Critical values of the V_A/V_B ratios for the most common phase transitions Pnma ↔ I4/mcm and Pnma↔ Rc are estimated to be ∼ 4.85 with the possible intermediate space group Imma stable in the very narrow range of V_A/V_B ≃ 4.8–4.9. Transitions to post-perovskite CaIrO₃-type structures may be expected for V_A/V_B < 3.8.