The packing fraction in a face-centered cubic cell of crystals is

The edge length of a fcc crystal, $\text{a = 2}\sqrt{2}\mathrm{r}$

Volume of unit atom= $\frac{4}{3}{\mathrm{\pi r}}^3$

We are aware that each unit cell in the fcc structure consists of four spheres. Total volume of four spheres is equal to $4\times \left(\frac{4}{3}\right){\mathrm{\pi r}}^3$ and volume of the cube is ${\mathrm{a}}^3$ or $(2\sqrt{2}\mathrm{r}{)}^3$.

$\text{ Packing efficiency }=\frac{\text{ Volume occupied by four spheres in the unit cell }}{\text{ Total volume of the unit cell }}$$=\frac{4\times (4/3){\mathrm{\pi r}}^3}{(2\sqrt{2}\mathrm{r}{)}^3}=\frac{\sqrt{2}}{6}\mathrm{\pi }$