Answer: 34%
The atoms in a simple cubic crystal are located at the corners of the units cell, a cube with side $a$.
For a diamond shaped crystal, the radius = $\large\frac{\sqrt 3 a }{8}$
Note: In this type of structure, total number of atoms is 8 per unit cell.
Packing Density $ = \large\frac{\text{Volume of atoms}}{\text{Volume of unit cell}}$
Volume of atoms $ =8 \large\frac{4}{3}$$\pi r^3$ and Volume of unit cell $= a^3$
Substituting $r = \large\frac{\sqrt 3 a }{8}$, we get:
Packing Density $ = \Large \frac{ 8 \frac{4\pi}{3} (\frac{\sqrt 3 a }{8})^3}{a^3}$$ = \large\frac{\pi \sqrt 3}{16}$$ = 0.34$