Physics:Goldschmidt tolerance factor

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Factor used to determine the compatibility of an ion with a crystal structure

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 ([math]\displaystyle{ t }[/math]) is a dimensionless number that is calculated from the ratio of the ionic radii:[1]

[math]\displaystyle{ t={r_A+r_O \over \sqrt{2}(r_B+r_O)} }[/math]
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]

[math]\displaystyle{ a=\sqrt{2}(r_A+r_O)=2(r_B+r_O) }[/math]
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) Structure Explanation Example Example lattice
>1[3] Hexagonal or Tetragonal A ion too big or B ion too small.
  • BaNiO3[1]
  • BaTiO3 (t=1.0617)
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.
  • GdFeO3 (Orthorhombic)[1]
  • CaTiO3 (Orthorhombic)[1]
CaTiO3 perovskite structure.png
<0.71[3] Different structures A ions and B have similar ionic radii. -

See also


  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. 
  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. 

External links