An Introduction to Environmental Chemistry

72 Chapter Four

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 neigh-

bours, reachieving optimum bond length.

In silicate minerals the layered stack of spheres is formed by oxygen anions

(O^2 - ) and the radius ratio rule can be defined as:

eqn. 4.2

Radius ratio values relative to O^2 - 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 SiO 4 tetrahedron.

Octahedral sites, being larger than tetrahedral sites, accommodate cations of

larger radius. However, some cations, for example strontium (Sr^2 +) 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.

Radius ratio=rrcation O^2 -

Table 4.3Radius ratio values for cations relative to O^2 -. From Raiswell et al. (1980).

Commonly

observed

Critical radius Predicted Ion Radius ratio coordination

ratio coordination rk/rO^2 - numbers

3C^4 + 0.16 3

0.225

3B^3 + 0.16 3, 4

4Be^2 + 0.25 4

4Si^4 + 0.30 4

0.414

4Al^3 + 0.36 4, 6

6Fe^3 + 0.46 6

6Mg^2 + 0.47 6

6Li+ 0.49 6

6Fe^2 + 0.53 6

6Na+ 0.69 6, 8

0.732

6Ca^2 + 0.71 6, 8

8Sr^2 + 0.80 8

8K+ 0.95 8–12

1.000

8Ba^2 + 0.96 8–12

12 Cs+ 1.19 12