Calculate the packing efficiency in a metal crystal for a face-centred cubic $(FCC)$ unit cell (assuming that atoms are touching each other).

(N/A) For a face-centred cubic $(FCC)$ unit cell:
Let the edge length of the unit cell be $a$ and the length of the face diagonal $AC$ be $b$.
From $\triangle ABC$,we have:
$AC^{2} = BC^{2} + AB^{2}$
$\Rightarrow b^{2} = a^{2} + a^{2}$
$\Rightarrow b^{2} = 2a^{2}$
$\Rightarrow b = \sqrt{2}a$
Let $r$ be the radius of the atom.
From the figure,it can be observed that the face diagonal $b = 4r$.
$\Rightarrow \sqrt{2}a = 4r$
$\Rightarrow a = \frac{4r}{\sqrt{2}} = 2\sqrt{2}r$
Volume of the unit cell = $a^{3} = (2\sqrt{2}r)^{3} = 16\sqrt{2}r^{3}$.
An $FCC$ unit cell contains $4$ atoms.
Volume of $4$ atoms = $4 \times \frac{4}{3}\pi r^{3} = \frac{16}{3}\pi r^{3}$.
Packing efficiency = $\frac{\text{Volume of 4 atoms}}{\text{Total volume of unit cell}} \times 100\%$
$= \frac{\frac{16}{3}\pi r^{3}}{16\sqrt{2}r^{3}} \times 100\%$
$= \frac{\pi}{3\sqrt{2}} \times 100\% \approx 74\%$