Goldschmidt tolerance factor

Goldschmidt's tolerance factor is an indicator for the stability and distortion of crystal structures.^[1] It was originally only used to describe perovskite 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 (t) is a dimensionless number that is calculated from the ratio of the ionic radii:^[1]

${\displaystyle t={r_{A}+r_{0} \over {\sqrt {2}}(r_{B}+r_{0})}}$
rA is the radius of the A-cation.	rB is the radius of the B-cation.	r0 is the radius of the anion (usually oxygen).

In an ideal cubic perovskite structure the axis of the unit cell (a) can be described with the following equation:^[1]

${\displaystyle a={\sqrt {2}}(r_{A}+r_{0})=2(r_{B}+r_{0})}$
rA is the radius of the A-cation.	rB is the radius of the B-cation.	r0 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]	-
0.9-1[3]	Cubic	A and B ions have ideal size.	SrTiO3[1] BaTiO3[5]
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. Retrieved 17 May 2012.
3. Schinzer, Carsten. "Distortion of Perovskites". Retrieved 17 May 2012.
4. Goldschmidt, Victor M. (1926). "Die Gesetze der Krystallochemie". Die Naturwissenschaften. 21: 477–485. doi:10.1007/bf01507527.
5. Buttner, R. H.; Maslen, E. N. (30 November 1992). "Structural parameters and electron difference density in BaTiO3". Acta Crystallographica Section B. 48 (6): 764–769. doi:10.1107/S010876819200510X.