Radius Ratio Rules

In the ionic model, the bonding is described as the electrostatic interaction between charged spheres, whose sizes are given by the ionic radius.

In determining an ionic radius, it is necessary to split up the internuclear separation into a contribution from the anion and a contribution from the cation. This is most often done by assuming the value of the radius of one ion, and then calculating the radii of other ions from this basis. This standard ion is generally the oxide ion, as it occurs in combination with many other elements.

Also, it is a relatively unpolarizable ion, and so its size changes little with changing counterion.

The use of ionic radii to predict aspects of crystal structure like lattice parameters, the lengths of the axes of the unit cells, is often useful, but only when the values of the ionic radii are taken form the same source, i.e. they use the same reference ion and so have the correct relative sizes.

It should also be noted that the ionic radius of a given ion changes with coordination number: As the coordination number increases, the ions must get further away from the central ion in order to accommodate more of them, and hence the interionic separation increases, and the short ranged repulsion decreases, and the electron cloud on the central ion can expand, and hence the central ion increases in size.

Hence, ionic radius increases with coordination number.

The sizes of the ions can be used to predict the structure that will be adopted when they are combined. In a cubic close packed array of anions, for example, the octahedral and tetrahedral holes have different sizes, and so cation might be expected to occupy the holes which are just big enough to hold them. This is examined in terms of the radius ratio.

The Radius Ratio

The radius ratio of a given pair of ions is defined at the ionic radius of the smaller ion divided by the ionic radius of the larger ion, ie. ρ = rs/rl.

Often the smaller ion is the cation (as the reduced repulsion brought about by the missing electron tends to contract the electron cloud), and the larger ion is the anion (as the extra repulsion from the negative charge tends to make the ion expand).

Coodination Number Radius ratio Ionic sizes
8 > 0.7
6 0.4-0.7
4 0.2-0.4
3 0.1-0.2

Consider a simple cubic arrangement of anions, with a cation in the center of the cubic cell (as in CsCl, which is (8,8)-coordinate). As the cation decreases in size, it will reach a point when the anions begin to touch, which unfavourable electrostatically, due to the repulsion between like-charged species. At this point, the structure changes so the anions are again separated by oppositely charged cations, an arrangement which is electrostatically favourable, and the (6,6)-coordinate NaCl structure is adopted. as this trend is continued, there will be a switch to the (4,4)-coordinate ZnS structure. Similar arguments hold for structures of stoichiometry AB2, and others.

Therefore, as the radius ratio decreases, there is a trend towards structures of lower coordination numbers. The Radius ratio rules are the prediction of structure adopted by a given set of ions based on the radius ration of those ions.

The radius ratio rules are not universally successful. As the degree of covalency in the bonding increases, the deviation from the ionic model increases and the less reliable the choice of structure based on the radius ratio becomes. The rules are least reliable for simple compounds like alkali metal halides and alkaline earth metal oxides, and are most reliable for complex fluorides and the salt of oxoanions: in general, as the degree of ionicity increases, so does the accuracy of the rules.