Atoms within solids are arranged in one of several periodic structures known as a lattice. There are seven lattice systems in total. Examples of these include the facecentered cubic, body centered cubic and simple cubic arrangements. The proportion of volume that the atoms take up with in a given latttice is known as the packing fraction. You can calculate the packing fraction of a material such as diamond with some material parameters and simple mathematics.
Things You'll Need
 paper

Write down the equation for packing fraction. The equation is:
Packing fraction = Natoms x Vatom / Vunitcell
Where Natoms is the number of atoms in a unit cell, Vatom is the volume of the atom, and Vunitcell is the volume of a unit cell.

Substitute the number of atoms per unit cell into the equation. Diamond has eight atoms per unit cell so the formula now becomes:
Packing fraction = 8 x Vatom / Vunitcell

Substitute the volume of the atom into the equation. Assuming atoms are spherical, the volume is:
V = 4/3 x pi x r^3
The equation for packing fraction now becomes:
Packing fraction = 8 x 4/3 x pi x r^3 / Vunitcell

Substitute the value for the unit cell volume. Since the unit cell is cubic, the volume is
Vunitcell = a^3
The formula for packing fraction then becomes:
Packing fraction = 8 x 4/3 x pi x r^3 / a^3
The radius of an atom r is equal to sqrt(3) x a / 8
The equation is then simplified to : sqrt(3) x pi / 16 = 0.3401
References
 Photo Credit Zedcor Wholly Owned/PhotoObjects.net/Getty Images