Chapters / Chapter 1 · Solid state / Topic 13 of 14
Goldschmidt tolerance factor
Chapter 1 · Solid state and crystal structures
The Goldschmidt tolerance factor (t) is a dimensionless number used to predict the stability of a crystal and its symmetry in ABO3 perovskites. It relates the ionic radii of the A-site cation (rA), the B-site cation (rB) and the anion (rO).
t = (rA + rO) / [ √2 (rB + rO) ]
When t deviates from the ideal value 1, the crystal lattice experiences a steric mismatch, driving structural distortion to lower-symmetry phases.
| t | Structure | Why | Example |
|---|---|---|---|
| > 1 | Hexagonal or tetragonal | A-site cation too large; A site overbonded | BaTiO3 (t ≈ 1.06), BaNiO3 |
| 0.9 to 1 | Cubic | A, B and O packed ideally; BO6 octahedra untilted | SrTiO3 (t ≈ 1.00) |
| 0.71 to 0.9 | Orthorhombic / rhombohedral | A-site cation too small; BO6 octahedra tilt and rotate | CaTiO3, GdFeO3 |
| < 0.71 | Different structures | Perovskite framework cannot be sustained | Ilmenite, FeTiO3 |