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.

Fig. 4.2 Structure of SiO4 tetrahedron. (a) Silicon and oxygen packing. The shaded silicon atom lies below the central oxygen atom, but above the three oxygens that lie in a single plane. (b) SiO4 tetrahedron with bond length exaggerated.

Lower layer Upper layer.

Lower layer Upper layer.

Tetrahedron showing largest sphere accommodated in tetrahedral site

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

Fig. 4.3 (a) Spheres in planar layers showing hexagonal symmetry. (b) An upper layer of spheres (shaded) is stacked on the layer in (a), such that each upper sphere fits into the depression between three spheres in the lower layer. (c) Enlargement of (b), where heavy lines show coordination polyhedra, joining the centres of adjacent spheres, delineating two geometries, tetrahedra and octahedra. After McKie and McKie (1974) and Gill (1996), with kind permission of Kluwer Academic Publishers.

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.:

where r = ionic radius.

To fit exactly into an octahedral site delineated by six spheres of radius r, a cation must have a radius of 0.414 r. 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 length). In real crystals, radius ratios are usually smaller or larger than this critical value of 0.414. If smaller, the optimum bond length is exceeded, and the structure collapses into a new stable configuration where the cation maintains optimum bond length with fewer, more closely packed anions. If the radius ratio is larger than 0.414, octahedral coordination is maintained, but the larger cation prevents the anions from achieving their closest possible packing. The upper limit for octahedral coordination is the next critical radius ratio of 0.732, at which point the cation is large enough to simultaneously touch eight equidistant anion neighbours, reachieving optimum bond length.

In silicate minerals the layered stack of spheres is formed by oxygen anions (O2-) and the radius ratio rule can be defined as:

Radius ratio = rcation/rO2- eqn. 4.2

Radius ratio values relative to O2- are given in Table 4.3. The table shows that silicon (Si) exists in four-fold (tetrahedral) coordination with oxygen (O), i.e. it will fit into a tetrahedral site. This explains the existence of the SiO4 tetrahedron. Octahedral sites, being larger than tetrahedral sites, accommodate cations of larger radius. However, some cations, for example strontium (Sr2+) and caesium (Cs+) (radius ratio >0.732), are too big to fit into octahedral sites. They exist in eight-fold or 12-fold coordination and usually require minerals to have an open, often cubic, structure.

Table 4.3 Radius ratio values for cations relative to O2 . From Raiswell et al. (1980).

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