Goldschmidt tolerance factor: Difference between revisions

Factor used to determine the compatibility of an ion with a crystal structure

Goldschmidt’s tolerance factor (from the German word Toleranzfaktor) is an indicator for the stability and distortion of crystal structures.^[1] It was originally only used to describe the perovskite ABO₃ structure, but now tolerance factors are also used for ilmenite.^[2]

Alternatively the tolerance factor can be used to calculate the compatibility of an ion with a crystal structure.^[3]

The first description of the tolerance factor for perovskite was made by Victor Moritz Goldschmidt in 1926.^[4]

The Goldschmidt tolerance factor (${\displaystyle t}$) is a dimensionless number that is calculated from the ratio of the ionic radii:^[1]

${\displaystyle t={r_{A}+r_{O} \over {\sqrt {2}}(r_{B}+r_{O})}}$
rA is the radius of the A cation.	rB is the radius of the B cation.	rO is the radius of the anion (usually oxygen).

In an ideal cubic perovskite structure, the lattice parameter (i.e., length) of the unit cell (a) can be calculated using the following equation:^[1]

${\displaystyle a={\sqrt {2}}(r_{A}+r_{O})=2(r_{B}+r_{O})}$
rA is the radius of the A cation.	rB is the radius of the B cation.	rO is the radius of the anion (usually oxygen).

The perovskite structure has the following tolerance factors (t):