## Unit – IV B |

## 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 = a
^{3}= (2r)^{3}= 8r^{3} - 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

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