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 face-centered 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.
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