This gives the critical radius ratio representing the triangular void.
Example 3 Show that the critical radius ratio for a tetrahedral coordination is
0.225.
Solution: Let us consider a tetrahedral void surrounded by four spheres of radius R
(Fig. 1.32a). The centers of these spheres lie at corners of a regular tetrahedron of
side a = 2R and height h (Fig. 1.32b). For a regular tetrahedron of side “a,” asin60°
is the median of any of the four bounding faces. Let BP is the median of a side face,
drawn from the apex B of the tetrahedron and BQ be the perpendicular from B to
the base of the tetrahedron. The triangle BPQ is shown in Fig. 1.32c. As the point P
is the midpoint of one side of the base and Q is its centroid, then
BP ¼ a sin 60
; PQ ¼
1
3 a sin 60
and (BP)
2
¼ (BQ)
2
þ (QP)
2
or a
2 sin
2 60
¼ h
2
þ a
2
9 a sin
2 60
or h
a ¼
ffiffi
2
p ffiffi
3
p ¼ 0.8165
or c
a ¼ nh
a ¼ 0.8165 Â n
Let r be the radius of the sphere that jut fits into this void whose center is at P and
is equidistant from all corners of the tetrahedron (Fig. 1.33a). Let the bond length
p = r + R. As shown above, the height BQ = h is related to the side of the tetrahedron “a” as
h =
ffiffi ffi
2
p
ffiffi ffi
3
p a
Hence, from the right angled triangle PQR shown in Fig. 1.33b , we have
Fig. 1.32 a Tetrahedral voids in a close packing b Tetrahedron formed by the centers of the
spheres c Section BPQ of (b)
30
1 Unit Cell Composition