Coordination of ions and the radius ratio rule

In crystals where bonding is largely ionic (see Section 2.3.2), the densest possible packing of equal-sized anions (represented by spheres) is achieved by stacks of regular planar layers, as shown in Fig. 4.3. Spheres in a single layer have hexagonal symmetry, i.e. they are in symmetrical contact with six spheres. The layers are stacked such that each sphere fits into the depression between three other spheres in the layer below. 

Octahedron with apexes lying at centres of surrounding balls showing largest sphere accommodated 

The gaps between neighbouring spheres have one of two possible three-dimensional geometries. The first geometry is delineated by the surfaces of four adjacent spheres. A three-dimensional shape constructed from the centre of each adjacent sphere (Fig. 4.3) has the form of a tetrahedron consequently these gaps are called tetrahedral sites. The second type of gap is bounded by six adjacent spheres and a three-dimensional shape constructed from the centre of these spheres has the form of a regular octahedron. These are called octahedral sites. In ionic crystals, cations occupy some of these tetrahedral and octahedral sites. The type of site a cation occupies is determined by the radius ratio of the cation and anion, i.e.  

To fit exactly into an octahedral site delineated by six spheres of radius r, a cation must have a radius of 0.414r. With this radius ratio the cation touches all six of the surrounding anions in octahedral coordination. The short distance between ions means that the bond length is short and strong (optimum bond 

In silicate minerals the layered stack of spheres is formed by oxygen anions (02 ) and the radius ratio rule can be defined as  

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