Octahedron with apices
lying at the centres of six
surrounding balls, showing
the largest sphere that can
Tetrahedron showing the
largest sphere that can be
accommodated in a tetra-
Figure 7.2 (a) Two layers in a close-packed array of spheres. The heavy lines show the co-ordination polyhedra of the
tetrahedral and octahedral interstitial voids between the spheres. (b) 3D model showing tetrahedral co-ordination of anions
around a small cation; the cation (ball bearing) can be seen nestling between the four anions. (c) Octahedral co-ordination
around a larger cation. (d) Octahedron with top anion layer removed to show the octahedrally co-ordinated cation within.
(Sources: Adapted from McKie & McKie 1974; K. d'Souza.)
The second type of hole in a close-packed array of
identical spheres is bounded by six neighbouring spheres,
whose centres lie at the six apexes of a regular octahedron 1
(Figure 7.2a,c,d). Such holes are called octahedral sites . In
terms of the size of the largest interstitial sphere that each
hole can accommodate (analogous to the glass marble in
Figure 7.2d), octahedral sites are 'bigger' than tetrahedral
sites. Both are substantially larger than the cavity between
three adjacent spheres in the same layer.
In many ionic crystals the anions are assembled in a
more or less close-packed array, the cations occupying
some of the tetrahedral and/or octahedral sites
between them. The kind of site a given cation occupies
is determined by the value of the ratio r cation : r anion , known
as the radius ratio . Using three-dimensional trigonom-
etry, it is not difficult to show that, to fit exactly into an
octahedral site between six identical 'anion' spheres of
radius R, a 'cation' sphere must have a radius of 0.414 R .
A cation in this position in a crystal is said to have oct-
ahedral co-ordination . However, the likelihood of a real ion
pair having a radius ratio of exactly 0.414 is negligible,
so we must consider the effect on the co-ordination
number if the radius ratio deviates from this value.
A radius ratio of exactly 0.414 allows the 'cation'
sphere to touch all of the surrounding spheres at once,
An octahedron (meaning 'eight sides') has six apexes (points)
but eight faces.