(v) Each octahedral void is surrounded by six spheres while each tetrahedral
void is surrounded by four spheres.
Therefore, the number of octahedral voids (n o ) and the number of tetrahedral
voids (n t ) belonging to a sphere, respectively, are:
n o ¼
Number of octahedral voids around a sphere
Number of spheres around an octhedral void
¼
6
6
¼ 1
n t ¼
Number of tetrahedral voids around a sphere
Number of spheres around a tetrahedral void
¼
8
4
¼ 2
This follows that:
(i) There are as many octahedral voids as there are spheres, and
(ii) There are twice as many tetrahedral voids as there are spheres.
Example 2 Show that the critical radius ratio for a triangular coordination is 0.155.
Solution: Let us consider a triangular void surrounded by three spheres of radius R
touching each other. Let a small sphere of radius r is placed within the triangular
void such that the central sphere just touches the three coordinating spheres as
shown in Fig. 1.31. From this simple geometrical construction, we have
LM
LO ¼ R
R þ r ¼ cos 30
or r ¼ R
cos 30
À R ¼
2 ffiffi
3
p R À R ¼ 1.155R À R
or r ¼ 0.155R
r
R ¼ 0.155
Fig. 1.31 A planar void
1.5 Close Packing of Identical Atoms (Spheres)
29