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]

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]

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 A ion too big or B ion too small. -
0.9-1[3] Cubic A and B ions have ideal size.
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. -

See also

References

  1. ↑ 1.0 1.1 1.2 1.3 1.4 1.5 1.6 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. ↑ 3.0 3.1 3.2 3.3 3.4 3.5 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 Structural Science 48 (6): 764–769. doi:10.1107/S010876819200510X.

External links