Goldschmidt's tolerance factor (from the German word ''Toleranzfaktor'') is an indicator for the stability and distortion of crystal structures.
It was originally only used to describe the
perovskite ABO3 structure, but now tolerance factors are also used for
ilmenite
Ilmenite is a titanium-iron oxide mineral with the idealized formula . It is a weakly magnetic black or steel-gray solid. Ilmenite is the most important ore of titanium and the main source of titanium dioxide, which is used in paints, printing ...
.
Alternatively the tolerance factor can be used to calculate the compatibility of an ion with a crystal structure.
The first description of the tolerance factor for perovskite was made by
Victor Moritz Goldschmidt
Victor Moritz Goldschmidt (27 January 1888 in Zürich – 20 March 1947 in Oslo) was a Norwegian mineralogist considered (together with Vladimir Vernadsky) to be the founder of modern geochemistry and crystal chemistry, developer of the Goldsch ...
in 1926.
Mathematical expression
The Goldschmidt tolerance factor (
) is a dimensionless number that is calculated from the ratio of the
ionic radii
Ionic radius, ''r''ion, is the radius of a monatomic ion in an ionic crystal structure. Although neither atoms nor ions have sharp boundaries, they are treated as if they were hard spheres with radii such that the sum of ionic radii of the cation ...
:
In an ideal cubic perovskite structure, the lattice parameter (i.e., length) of the
unit cell
In geometry, biology, mineralogy and solid state physics, a unit cell is a repeating unit formed by the vectors spanning the points of a lattice. Despite its suggestive name, the unit cell (unlike a unit vector, for example) does not necessaril ...
(a) can be calculated using the following equation:
Perovskite structure
The perovskite structure has the following tolerance factors (t):
See also
*
Goldschmidt classification
The Goldschmidt classification,
developed by Victor Goldschmidt (1888–1947), is a geochemical classification which groups the chemical elements within the Earth according to their preferred host phases into lithophile ( rock-loving), siderophi ...
*
Victor Goldschmidt
Victor Moritz Goldschmidt (27 January 1888 in Zürich – 20 March 1947 in Oslo) was a Norwegian mineralogist considered (together with Vladimir Vernadsky) to be the founder of modern geochemistry and crystal chemistry, developer of the Golds ...
References
{{Reflist
Crystallography
Mineralogy