## Distortion of Perovskites

### Goldschmidt's Tolerance Factor t

The tolerance factor inventend for the perovskite structure by Goldschmidt takes simply into account, that three types of spheres in a perovskite-structure, will have contact each other. Since the ionic radii are generally known quite well, a tolerance factor can be calculated, that indicates the "compatibitly" of a given set of ions with the ideal, cubic perovskite structure.

The anion-cation contacts in the ideal cubic pervoskite and their relation to the lattice parameter are shown here:

The anions separate both, the A-cations and the B-cations. The lattice constant is determined by the sum of the ionic diameters of the anion and the B-cation, as is shown in the left figure. At the same time, the sum of the diameters of anion and A-cation determines the plane diagonal (which equals 2) times the lattice constant; cf. right figure).

Thus, a condition to yield the ideal perovskite is

rX + rA = 2 * (rX + rB)

The introduction of factor t, that is simply the ratio of the equation's left side and it#s right side, leads to the definition for Goldsschmidt's tolerance factor:

t =
rX + rA
2 * (rX + rB)

Once we have introduced t, we can try to find out what happens, when it differs from the ideal value:

Tolerance Factor t ‚ Validity Ranges and Corresponding Perovskite Variants

t-value Effect Possible Structures
> 1 A-cations are too large to fit into their interstices Hexagonal perovskite polytypes
~0.9 ... 1.0 Ideal conditions Cubic perovskites
0.71 ... 0.9 A-cations are too small to fit into their interstices Several possible structures.
Among them:
• orthorhombic perovskites
• rhombohedral variants
Here is, where Glazer's concept comes in !
< 0.71 A-cations are of same size as B-cations ( 1/2 = 0.71 ) Possible close-packed structures
• corundum structure (disordered arrangement of cations)
• ilmenite structure (ordered arrangement within sheets)
• K Nb O3 structure (sheetwise ordered arrangement)

last changed: Jul 19, 1998. © 1997‚1998 Carsten Schinzer.