Goldschmidt tolerance factor

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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[edit]

The Goldschmidt tolerance factor (t) is a dimensionless number that is calculated from the ratio of the ionic radii:[1]

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]

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[edit]

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. -
0.9-1[3] Cubic A and B ions have ideal size.
Cubic perovskite structure.png
0.71 - 0.9[3] Orthorhombic/Rhombohedral A ions too small to fit into B ion interstices.
CaTiO3 perovskite structure.png
<0.71[3] Different structures A ions and B have similar ionic radii. -

See also[edit]


  1. ^ a b c d e f g 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. ^ a b c d e f 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. 

External links[edit]