An Introduction to Environmental Chemistry

The Chemistry of Continental Solids 71

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.:

eqn. 4.1

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 0.414r. 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

Radius ratio=rrcation anion

(a) (b)

(c) Lower layer

Upper layer

Octahedron with

apexes lying at

centres of

surrounding balls

showing largest

sphere accommodated

Tetrahedron

showing largest

sphere accommodated

in tetrahedral site

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.