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 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 () 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 TetragonalA ion too big or B ion too small. -
0.9-1[3]CubicA and B ions have ideal size.
0.71 - 0.9[3]Orthorhombic/RhombohedralA ions too small to fit into B ion interstices.
<0.71[3]Different structuresA ions and B have similar ionic radii. -
gollark: ```lualUtils.drawEditBox { parent = userbox, configThing = true}```
gollark: It sounds like you should have a table.
gollark: It would be really neat if I could get signing for that working because then (well, with more work) it would be possible to distribute potatOS update manifests (and the actual code with them) securely via *any* platform!
gollark: It says "EdDSA-like digital signatures", which implies that it may not actually be something available outside of CC.
gollark: It would be neat if they were cryptographically signed too, but it turns out I have no idea what actual algorithm the potatOS ECC library is implementing, oops.

See also

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.CS1 maint: extra text: authors list (link)
  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.CS1 maint: multiple names: authors list (link)
  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.
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