## Density and Packing Efficiency of Unit Cell

• In whatever way the constituent particles atoms, molecules or ions are packed, there is always some free space in the form of voids.
• The fraction of the total space in unit cell occupied by the constituent particles is called packing fraction.
Mathematically
• Packing efficiency is the percentage of total space filled by the constituent particles in the unit cell.

Packing efficiency = Packing Factor x 100

• A vacant space not occupied by the constituent particles in the unit cell is called void space.

Fraction of void space = 1 – Packing Fraction
% Void space = 100 – Packing efficiency.

### Packing Efficiency of Simple Cubic Crystal Lattice (SCC):

• In a simple cubic lattice the atoms are located only on the corners of the cube.
• The particles touch each other along the edge as shown. Thus, the edge length or side of the cube ‘a’, and the radius of each particle, r are related as    a = 2r. Thus radius of an atom is half the side of the simple cubic unit cell.
• The volume of the cubic unit cell  = a3 = (2r)3 = 8r3
• Since a simple cubic unit cell contains only 1 atom The volume of the occupied space
• Now, packing efficiency
• The packing efficiency of simple cubic cell is 52.4 %. Thus 47.6 % volume is empty space(void space). i.e. almost half the space is empty. Hence the simple cubic crystalline solid is loosely bonded.

### Packing Efficiency of Body Centred Cubic Crystal Lattice (BCC):

• In a body centred cubic lattice the eight atoms are located on the eight corners of the cube and one at the centre of the cube.
• Let the edge length or side of the cube ‘a’, and the radius of each particle be r. The particles along body diagonal touch each other.

∴  AD = r  + 2r + r = 4r   ……. (1)

• Thus radius of an atom is  3/4 times the side of the body centred cubic unit cell.
• The volume of the cubic unit cell
• Since a body centred cubic unit cell contains 2 atoms The volume of the occupied space
• Now, packing efficiency
• The packing efficiency of body centred cubic cell is 68 %. Thus 32 % volume is empty space(void space).

Density and Packing Efficiency of Unit Cell

Back                              Next Topic FurtherDensity and Packing Efficiency of Unit Cell