# Calculation of density of unit cell

Definition of Unit Cell

Atom’s smallest group in which there’s complete crystal symmetry, and from which whole of the lattice can be formed in 3 dimensions by the repetition process is defined as the Unit Cell. A repeating and regular pattern is displayed by Crystalline Solids of constituent particles.

What is a Lattice?

A lattice is defined as the framework, which resembles a 3-D, a periodic array of points, on which the crystal is formed.

M. A. Bravais in 1850 demonstrated that relative to space, similar points can be arranged to create 14 regular patterns types. And these fourteen (14) space lattices are called, Bravais lattices.

Solid’s crystal lattice can be illustrated in the context of its unit cell. A crystal lattice is composed of unit cells, present in huge number, where each of the lattice point is in use by a single constituent particle. Having one or more than one atoms in a unit cell can be witnessed as a 3-D structure.

We can find out this unit cell’s volume by having the knowledge of the unit cell dimensions. For instance: if there’s a unit cell having edge “a”, then unit cell’s volume can be represented by “a3”. The calculation of density of the unit cell is defined as the ratio of mass “M” and volume “V” of unit cell. Unit cell’s mass is equal to multiplication of number of atoms present in a unit cell, given by “z” and the mass “m” of each atom present in the unit cell.

Unit cell mass = number of atoms × mass of each atom in unit cell = z × m….eq 1

Where,

z = number of atoms in unit cell,

m = Mass of each atom, as mentioned above also

A mass of the atom can be given using molas mass and Avogadro’s number as: M/Na

Where,

M = molar mass

Unit cell’s volume, ‘V’ = a3 ……..eq 2

=> Density of the unit cell = mass of unit cell / volume of unit cell

=> Density of the unit cell = m/V = z × m/a3 ….. From eq 1, m = z × m and from eq 2 V = a3

therefore, m/V = z × M / a3 × Na

Hence, by having the knowledge of unit cell’s number of atoms, molar mass, and the edge length, we can easily figure out the “density of a unit cell”.

A common expression used for unit cell’s density in different cases are derived below:

1. Primitive unit cell: in this unit cell, the number of atoms in a given unit cell, z is equal to 1. Therefore, density of the unit cell is given as:

Density of unit cell = 1 × Ma3 × NA

2. Body-centered cubic unit cell: In the number of atoms present in a unit cell, z is equal to 2. Therefore, density is specified as:

Density of unit cell = 2 × Ma3 × NA

3. Face-centered cubic unit cell: In this unit cell, the number of atoms, z is equal to 4 in the unit cell. Therefore, the unit cell density is as follows:

Density of unit cell = 4 × Ma3 × NA