The packing efficiency in a simple cubic cell system of crystals is

The edge length or side of the cube be 'a', and the radius of each particle be 'r'.
a=2r
Volume of cube =a³
(2r)³=8r³
In simple cube, each unit cell has only one sphere,
Volume of sphere =$\frac{4}{3}$​πr³
Packing efficiency = $\frac{\mathrm{volume} \mathrm{of} \mathrm{the} \mathrm{sphere}}{\mathrm{voume} \mathrm{of} \mathrm{the} \mathrm{cube}}$​×100%
=$\frac{{\displaystyle \frac{4}{3}}{\mathrm{\pi r}}^3}{8{\mathrm{r}}^3}$×100 = 52.4 %.