Goldschmidt tolerance factor

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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]

Contents

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 () 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.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]

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)StructureExplanationExampleExample 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 structuresA ions and B have similar ionic radii.-

See also

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References

  1. 1 2 3 4 5 6 7 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.{{cite book}}: |first1= has generic name (help)
  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.{{cite journal}}: CS1 maint: multiple names: authors list (link)
  3. 1 2 3 4 5 6 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.