A SIMPLIFIED SCHEME FOR DEVELOPING THE TOLERANCE FACTORS FOR SOME CRYSTAL STRUCTURES

Abstract

The tolerance factor, developed and proposed by Goldschmidt to estimate the stability limits of the cubic perovskite ABX3 structural type for a given chemical composition, has been widely used in solid state chemistry and physics as well as in the related fields of science. Based on the idealized cubic perovskite structure and the ionic radii of the constituents r(A), r(B), and r(X), Goldschmidt derived the formula for the tolerance factor as the ratio between the sums of the ionic radii of cations and anions, {[r(A) + r(X)] / (√2 × [r(B) + r(X)])}, which is equal to 1 in the ideal perovskite structures. From the obtained value of the tolerance factor, it is possible to make reliable predictions about the possibility of the existence of a cubic perovskite phase of a given chemical composition. The convenience of using the Goldschmidt tolerance factor in the studies of perovskite structures prompted some researchers to begin searching for similar indicators for representatives of other scientifically and technologically important structural families. However, despite the mathematical correctness of the final formulas of the alternative tolerance factors, the derivation of these formulas in a number of works seems to be unnecessarily complicated. In the present work, a simplified approach has been proposed for developing the alternative tolerance factors from the geometric characteristics of regular coordination polyhedra – namely, from the relationships between the edges and the circumscribed sphere radii of such polyhedra. The proposed approach can be used to derive the tolerance factors formulas for any crystal structures containing regular or nearly regular coordination polyhedra of different types with the same edge lengths.