# How to calculate the packing fraction of diamond lattice

Zedcor Wholly Owned/PhotoObjects.net/Getty Images

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-centred cubic, body centred cubic and simple cubic arrangements.

The proportion of volume that the atoms take up with in a given lattice 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:

- Atoms within solids are arranged in one of several periodic structures known as a lattice.
- The proportion of volume that the atoms take up with in a given lattice is known as the packing fraction.

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:

- 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.
- 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.

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:

- 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.

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

Writer Bio

Samuel Markings has been writing for scientific publications for more than 10 years, and has published articles in journals such as "Nature." He is an expert in solid-state physics, and during the day is a researcher at a Russell Group U.K. university.