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 ABX(3) 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, P4(2)/nmc etc.) are much less common than those with a single A and B site (I4/mcm, R\bar 3c, 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<--> R\bar 3c are estimated to be approximately 4.85 with the possible intermediate space group Imma stable in the very narrow range of V(A)/V(B) approximately 4.8-4.9. Transitions to post-perovskite CaIrO(3)-type structures may be expected for V(A)/V(B) < 3.8.