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The tolerance factor inventend for the perovskite structure by Goldschmidt takes simply into account, that three types of spheres in a perovskitestructure, 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 anioncation contacts in the ideal cubic pervoskite and their relation to the lattice parameter are shown here:
The anions separate both, the Acations and the Bcations. The lattice constant is determined by the sum of the ionic diameters of the anion and the Bcation, as is shown in the left figure. At the same time, the sum of the diameters of anion and Acation determines the plane diagonal (which equals —2) times the lattice constant; cf. right figure).
Thus, a condition to yield the ideal perovskite is
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  = 


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
tvalue  Effect  Possible Structures 

> 1  Acations are too large to fit into their interstices  Hexagonal perovskite polytypes 
~0.9 ... 1.0  Ideal conditions  Cubic perovskites 
0.71 ... 0.9  Acations are too small to fit into their interstices  Several possible structures. Among them:

< 0.71  Acations are of same size as Bcations ( 1/—2 = 0.71 )  Possible closepacked structures

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