Calculate the packing efficiency in a body-centred cubic $(BCC)$ structure.

(N/A) In a body-centred unit cell,the spheres located at the corners do not touch each other but they are in contact with the body-centred atom.
In $\Delta EFD,$
$b^2 = a^2 + a^2 = 2 a^2$
$b = \sqrt{2} a$
In $\Delta AFD,$
$c^2 = a^2 + b^2 = a^2 + 2 a^2 = 3 a^2$
$c = \sqrt{3} a$
The length of the body diagonal $c$ is equal to $4 r$,where $r$ is the radius of the sphere,as all three spheres along the diagonal touch each other.
Therefore,$\sqrt{3} a = 4 r$
$a = \frac{4 r}{\sqrt{3}}$
Number of atoms in a $BCC$ unit cell = $2$.
Volume of two spheres = $2 \times \frac{4}{3} \pi r^3 = \frac{8}{3} \pi r^3$
Volume of the unit cell = $a^3 = (\frac{4 r}{\sqrt{3}})^3 = \frac{64 r^3}{3 \sqrt{3}}$
Packing efficiency = $\frac{\text{Volume of two spheres}}{\text{Volume of the unit cell}} \times 100$
Packing efficiency = $\frac{\frac{8}{3} \pi r^3}{\frac{64 r^3}{3 \sqrt{3}}} \times 100 = \frac{\sqrt{3} \pi}{8} \times 100 \approx 68\%$