In a BCC unit cell: - Number of atoms per unit cell = 2 - Atoms touch along the body diagonal Body diagonal = $\sqrt{3}a$ = $4r$ (where $a$ is edge length, $r$ is atomic radius) So $a = \frac{4r}{\sqrt{3}}$ Volume of unit cell = $a^3 = \left(\frac{4r}{\sqrt{3}}\right)^3 = \frac{64r^3}{3\sqrt{3}}$ Volume of atoms = $2 \times \frac{4}{3}\pi r^3 = \frac{8\pi r^3}{3}$ Packing efficiency = $\frac{\text{Volume of atoms}}{\text{Volume of unit cell}} \times 100$ $$= \frac{\frac{8\pi r^3}{3}}{\frac{64r^3}{3\sqrt{3}}} \times 100 = \frac{8\pi \sqrt{3}}{64} \times 100$$ $$= \frac{\pi\sqrt{3}}{8} \times 100 = \frac{3.14 \times 1.732}{8} \times 100 = 68\%$$