# 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]

 ${\displaystyle t={r_{A}+r_{0} \over {\sqrt {2}}(r_{B}+r_{0})}}$ 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]

 ${\displaystyle a={\sqrt {2}}(r_{A}+r_{0})=2(r_{B}+r_{0})}$ 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 or Tetragonal 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.
-