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

References

1. Parkin, editors-in-chief, Helmut Kronmller, Stuart; Mats Johnsson; Peter Lemmens (2007). Handbook of magnetism and advanced magnetic materials ([Online-Ausg.] ed.). Hoboken, NJ: John Wiley & Sons. ISBN 978-0-470-02217-7. Retrieved 17 May 2012.
2. Liu, XiangChun; Hong, Rongzi; Tian, Changsheng (24 April 2008). "Tolerance factor and the stability discussion of ABO3-type ilmenite". Journal of Materials Science: Materials in Electronics. 20 (4): 323–327. doi:10.1007/s10854-008-9728-8. S2CID 96085518.
3. Schinzer, Carsten. "Distortion of Perovskites". Retrieved 17 May 2012.
4. Goldschmidt, Victor M. (1926). "Die Gesetze der Krystallochemie". Die Naturwissenschaften. 14 (21): 477–485. Bibcode:1926NW.....14..477G. doi:10.1007/bf01507527. S2CID 33792511.