Discussion
Consider a BCC lattice with one atom per lattice point.
The maximum packing fraction is close to:
Answer: B
The packing fraction of BCC is close to 68 %.
A BCC cell has 2 atoms (8 corners shared eight ways, plus 1 at the centre). The atoms touch along the body diagonal, so \(4r = \sqrt{3}\,a\):
\[ \begin{aligned} \text{PF} &= \frac{2 \times \frac{4}{3}\pi r^{3}}{a^{3}}, \quad r = \frac{\sqrt{3}}{4}a \\ &= \frac{\sqrt{3}\,\pi}{8} \approx 0.680 \end{aligned} \]
74 % is the close-packed value for FCC and HCP, and 53 % is near the simple cubic value of \(\pi/6 \approx 52\) %. 81 % exceeds the close-packing limit for equal spheres, so no lattice of identical atoms can reach it.
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