# Physics:Goldschmidt tolerance factor

Factor used to determine the compatibility of an ion with a crystal structure

Goldschmidt's tolerance factor 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

The Goldschmidt tolerance factor ($\displaystyle{ t }$) is a dimensionless number that is calculated from the ratio of the ionic radii:[1]

 $\displaystyle{ t={r_A+r_O \over \sqrt{2}(r_B+r_O)} }$ 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]

 $\displaystyle{ a=\sqrt{2}(r_A+r_O)=2(r_B+r_O) }$ 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

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.
• BaNiO3[1]
• BaTiO3 (t=1.0617）
-
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.
• GdFeO3 (Orthorhombic)[1]
• CaTiO3 (Orthorhombic)[1]
<0.71[3] Different structures A ions and B have similar ionic radii.
-