Packing Efficiency in Simple Cubic Lattice

In a simple cubic lattice the atoms sit only at the corners of the cube and touch each other along the edge (Fig. 1.26). So the edge length $a$ and the radius $r$ are related by:

The volume of the cubic unit cell is:

A simple cubic unit cell contains only 1 atom, whose volume is $\tfrac{4}{3}\pi r^3$. Hence:

$\text{Packing efficiency} = \frac{\frac{4}{3}\pi r^3}{8r^3} \times 100 = \frac{\pi}{6} \times 100 = 52.36\% \approx 52.4\%$ …