Calculate the packing efficiency in a simple cubic unit cell.

(N/A) In a simple cubic unit cell,two spheres at the corners of the cube touch each other along the edge.
Let the edge length of the cube be $a$.
Let the radius of the sphere be $r$.
The relationship between the radius of the sphere and the edge length of the cube is given by:
$a = 2r$
The volume of the cubic unit cell is $a^{3} = (2r)^{3} = 8r^{3}$.
Since a simple cubic unit cell contains only $1$ atom,the volume of the occupied space is $\frac{4}{3} \pi r^{3}$.
$\text{Packing efficiency} = \frac{\text{Volume of one atom}}{\text{Volume of cubic unit cell}} \times 100$
$\text{Packing efficiency} = \frac{\frac{4}{3} \pi r^{3}}{8r^{3}} \times 100 = \frac{\pi}{6} \times 100$
$\text{Packing efficiency} \approx 52.4 \%$