Derivation of the Relationship Between Radius r of the Tetrahedral Void and the Radius R of the Atoms in the Close Packing.
A tetrahedral void may be represented in a cube as shown in the below figure.
fig 2.12 - Tetrahedral void
The three spheres form the triangular base, the fourth lies at the top and the sphere E occupies the tetrahedral void.
Let the length of the side of the cube = a
From right angled triangle ACB, face diagonal.
As spheres A and B are actually touching each other face diagonal AB = 2R.
Again from the right angled triangle ABD body diagonal
But as shaded sphere touches other spheres evidently body diagonal
AD = 2(R + r)
Dividing equation (ii) by equation (i) we get
r = 0.225 R
Radius ratio of the tetrahedral void = 0.225 R
