Past exam of the physics A course of the University of Cambridge / 2010 / Paper 1 / Section A / 4 / Solution

For hard spheres of radius r in a face-centered cubic lattice, contact occurs along a face diagonal: $4r=\sqrt{2}a$. Four spheres belong to a conventional cell, so the hard-sphere cubic packing fractions giveFor a body-centered cubic lattice, contact occurs along a body diagonal, $4r=\sqrt{3}a$, and there are two spheres per conventional cell:Their ratio $8/(3\sqrt{6})=1.0887>1$ proves that FCC packs more densely. Coordination alone would not establish the occupied volume fraction.