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]
The Goldschmidt tolerance factor (
t
t={rA+rO\over\sqrt{2}(rB+rO)} | |||
| 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:
a=\sqrt{2}(rA+rO)=2(rB+rO) | |||
| 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). | |
The perovskite structure has the following tolerance factors (t):
| Goldschmidt tolerance factor (t) | Structure | Explanation | Example | Example lattice | |
|---|---|---|---|---|---|
| >1 | A ion too big or B ion too small. |
| - | ||
| 0.9-1 | A and B ions have ideal size. | ||||
| 0.71 - 0.9 | A ions too small to fit into B ion interstices. |
| |||
| <0.71 | Different structures | A ions and B have similar ionic radii. | - |