The radius ratio for an ionic solid is defined as the ratio of the radius of the smaller ion, r_{s}, to the radius of the larger ion r_{l}. An ionic solid with a 8:8 coordination structure (e.g. CsCl) has the following radius ratio:
where a is the length of a side of the cube. Using Pythagoras’ theorem, the body diagonal of the cube, d, is:
Substituting eq1 in eq2
The body diagonal is also given by:
Substituting eq3 in eq4 and rearranging,
The radius ratio of 0.732 is the minimum ratio required for a 8:8 coordination compound. If the radius of the cation is smaller, it will not be in contact with the anions, resulting in a compound that has a coordination of less than 8:8, i.e. a 6:6 coordination compound.
The diagram above shows one of the six surfaces of an ionic solid with a 6:6 coordination structure (e.g. NaCl). Its radius ratio is calculated as follows:
where f is the face diagonal. Substituting eq5 in eq6,
The face diagonal is also given by:
Combining eq7 and eq8 and simplifying,
Again, the radius ratio of 0.414 is the minimum ratio required for a 6:6 coordination compound. If the radius of the cation is smaller, it will not be in contact with the anions, resulting in a compound that has a coordination of less than 6:6, i.e. a 4:4 coordination compound.
Finally, for a 4:4 coordination structure like zinc blende (see diagram above), its radius ratio is calculated by selecting the appropriate ions (diagram IIf):
Since the tetrahedral angle is 109.5°,
Substituting eq10 in eq11 and simplifying,
Once again, the radius ratio of 0.225 is the minimum ratio required for a 4:4 coordination compound.
Therefore, assuming that ions in crystal are rigid spheres and that cations are in contact with anions, we can use the three calculated radius ratios to predict structures of ionic compounds as follows:-
Radius Ratio |
|||
Coordination |
4:4 | 6:6 |
8:8 |
Structure |
Zinc blende type | NaCl type |
CsCl type |