Goldschmidt tolerance factor

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]

Mathematical expression

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).

Perovskite structure

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

Goldschmidt tolerance factor (t)	Structure	Explanation	Example	Example lattice
>1[3]	Hexagonal or Tetragonal	A ion too big or B ion too small.	BaNiO3[1] BaTiO3 (t=1.0617）	-
0.9-1[3]	Cubic	A and B ions have ideal size.	SrTiO3[1]
0.71 - 0.9[3]	Orthorhombic/Rhombohedral	A ions too small to fit into B ion interstices.	GdFeO3 (Orthorhombic)[1] CaTiO3 (Orthorhombic)[1]
<0.71[3]	Different structures	A ions and B have similar ionic radii.	Ilmenite, FeTiO3 (Trigonal) [3]	-

See also

• Goldschmidt classification
• Victor Goldschmidt