Efficiency of packing in body centered cubic structures is found to be:

68%

Body centered cubic unit cell (BCC)

Atom at the centre will be in touch with the other two atoms diagonally arranged.

In ∆ EFD,

b² = a² + a² = 2a²

b = \(\sqrt{2}\)a

Now in ∆ AFD

c² = a² + b² = a² + 2a² = 3a²

c = \(\sqrt{3}\)a

The length of the body diagonal c is equal to 4r, where r is the radius of the sphere (atom), as all the three spheres along the diagonal touch each other.

Therefore, \(\sqrt{3}\)a = 4r

a = \(\frac{4r}{\sqrt{3}}\)

Also we can write, r = \(\frac{\sqrt{3}}{4}\)a

In this type of structure, total number of atoms is 2 and their volume is 2 x \(\frac{4}{3}\)πr³

Volume of the cube, a³ will be equal to (\(\frac{4}{\sqrt{3}}\)r)³ or a³ = (\(\frac{4}{\sqrt{3}}\)r)³

Packing efficiency = \(\frac{\text{Volume occupied by four spheres in the unit cell}}{\text{Total volume of the unit cell}}\) x 100%

P.E. = \(\frac{2 × \frac{4}{3}πr^3}{(\frac{4}{\sqrt{3}})^3}\) x 100 = 68%

% of free space in BCC unit cell i.e., Void efficiency = 100 - 68 = 32%