Packing efficiency of body centred cubic unit cell

Body centred cubic (BCC) unit cell is one of the three types of cubic unit cells. In this type of unit cell, the lattice points are located at the corners and in the centre of the cube. The coordination number of each lattice point is 8.

Packing efficiency is the ratio of the volume occupied by the atoms in a unit cell to the volume of the unit cell. It gives an idea of how closely the atoms are packed in the unit cell.

To calculate the packing efficiency of a BCC unit cell, we need to consider the following:

- Each corner atom contributes 1/8th of its volume to the unit cell.
- Each centre atom contributes its entire volume to the unit cell.

So, the total volume of atoms in a BCC unit cell can be calculated as:

Volume of atoms = (Number of corner atoms x Volume of each corner atom)/8 + (Number of centre atoms x Volume of each centre atom)

Volume of atoms = (8 x (4/3)πr³)/8 + (1 x (4/3)πr³)

Volume of atoms = (4/3)πr³ + (4/3)πr³

Volume of atoms = (8/3)πr³

The volume of the unit cell can be calculated as:

Volume of unit cell = a³

where a is the edge length of the cube.

Now, the packing efficiency can be calculated as:

Packing efficiency = (Volume of atoms/Volume of unit cell) x 100%

Packing efficiency = [(8/3)πr³/a³] x 100%

Packing efficiency = (8π/3√3)(r/a)³ x 100%

For a BCC unit cell, the ratio of r/a is √3/2. So, the packing efficiency can be calculated as:

Packing efficiency = (8π/3√3)(√3/2)³ x 100%

Packing efficiency = 68%

Therefore, the correct answer is option B, 68%.