When spheres are placed in any kind of arrangement, they can not occupy the whole space because some voids such as the one created between three spheres touching one another are unavoidable. The ratio of the volume occupied by the spheres to the total volume available is called the packing fraction. The highest packing fraction possible is for the ababab or the abcabc packing described earlier.
Let us consider the packing fraction of a simple cubic lattice first. In this structure, the packing is neither ababab nor abcabc but it is aaaaaa. Eight spheres at the corners of a cube constitute the unit cell whose edge length is equal to the diameter of the sphere.
Figure 17.5 Packing fraction in a) simple cubic lattice, b) BCC lattice and c) FCC lattice.
If r is the radius of each sphere, the volume of each sphere is 4 r 3 / 3. Since 1/ 8th volume of each of the eight spheres is inside the cube of edge length 2r, the packing fraction, n is
=
= 4 / 24 = / 6 = 0.5236
(17.1)
This means that in a simple cubic lattice, 52.36 % volume is occupied by the spheres. In the void space available, other spheres of smaller radii can easily be fitted, resulting in a much higher packing fraction.
r 3 / 3. Since 1/ 8th volume of each of the eight spheres is inside the cube of edge length 2r, the packing fraction, n is 
=
