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* [[Ilmenite]], FeTiO<sub>3</sub> ([[Hexagonal crystal family|Trigonal]]) <ref name=”Schinzer 1998″ /> |
* [[Ilmenite]], FeTiO<sub>3</sub> ([[Hexagonal crystal family|Trigonal]]) <ref name=”Schinzer 1998″ /> |
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Latest revision as of 02:59, 9 November 2025
Factor used to determine the compatibility of an ion with a crystal structure
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]
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
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The Goldschmidt tolerance factor ( t {\displaystyle t} ) is a dimensionless number that is calculated from the ratio of the ionic radii:[1]
| t = r A + r O 2 ( r B + r O ) {\displaystyle t={r_{A}+r_{O} \over {\sqrt {2}}(r_{B}+r_{O})}} | ||
| 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]
| a = 2 ( r A + r O ) = 2 ( r B + r O ) {\displaystyle a={\sqrt {2}}(r_{A}+r_{O})=2(r_{B}+r_{O})} | ||
| 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
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The perovskite structure has the following tolerance factors (t):
