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 perovskite 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 (t) 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. | r0 is the radius of the anion (usually oxygen). |
In an ideal cubic perovskite structure the axis of the unit cell (a) can be described with the following equation:[1]
 |
||
| rA is the radius of the A-cation. | rB is the radius of the B-cation. | r0 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 | A ion too big or B ion too small. |
|
- |
| 0.9-1[3] | Cubic | A and B ions have ideal size. | ||
| 0.71 - 0.9[3] | Orthorhombic/Rhombohedral | A ions too small to fit into B ion interstices. | ||
| <0.71[3] | Different structures | A ions and B have similar ionic radii. | - |
See also
References
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