Crystalline solids exhibit a regular and repeating pattern of constituent particles. The diagrammatic representation of the three dimensional arrangement of constituent particles in a crystal, in which each particle is depicted as a point in space is known as crystal lattice. The crystal lattice of a solid can be described in terms of its unit cell. A crystal lattice is made up of a very large number of unit cells where every lattice point is occupied by one constituent particle.The unit cell can be seen as a three dimension structure containing one or more atoms. We can determine the volume of this unit cell with the knowledge of the dimensions of the unit cell. For example: if we have a unit cell of edge “a”, the volume of the unit cell can be given as “\(a^3\)

Mass of unit cell = number of atoms in unit cell × mass of each atom = z × m

Where, z = number of atoms in unit cell,

m = Mass of ach atom

Mass of an atom can be given with the help of Avogadro number and molar mass as:

m = \( \frac { M}{N_A} \)

Where, M = molar mass

\(N_A\)

Volume of unit cell, V = \(a^3\)

=> Density of unit cell = \( \frac {mass~ of~ unit~ cell}{volume~ of~ unit ~cell}\)

=> Density of unit cell = \( \frac {m}{V} \)

Thus, with the knowledge of number of atoms in a unit cell, edge length and molar mass we can determine the density of a unit cell.

## A general expression for density of unit cell for various cases has been derived below:

**Primitive unit cell**: In a primitive unit cell, number of atoms in a unit cell, z is equal to one. Hence density is given as:

Density of unit cell = \( \frac {1~×~M }{a^3~×~N_A} \)

**Body-centered cubic unit cell**: In body-centered cubic unit cell, number of atoms in a unit cell, z is equal to two. Hence density is given as:

Density of unit cell = \( \frac {2~×~M }{a^3~×~N_A} \)

**Face-centered cubic unit cell**: In face-centered cubic unit cell, number of atoms in a unit cell, z is equal to four. Hence density of unit cell is given as:

Density of unit cell = \( \frac {4~×~M }{a^3~×~N_A} \)

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