the atomic packing factor
Fill a box with marbles: no matter how neatly you arrange them, gaps always remain between the round shapes. The atomic packing factor is the fraction of the box actually filled by atoms — treating each atom as a hard sphere — with the rest being empty space.
The formula is APF = (atoms per cell times the volume of one atom) divided by the volume of the unit cell. For FCC: 4 atoms, each of volume (4/3) pi R^3, over a cell of volume (2R sqrt(2))^3 = 16 sqrt(2) R^3, works out to 0.74. Repeat the arithmetic and BCC gives 0.68, simple cubic 0.52, and the open diamond structure only 0.34. The value 0.74 is the densest possible packing of identical spheres.
Why it matters: combined with atomic mass, the packing factor lets you compute a material's theoretical density. It also hints at how much room exists for small foreign atoms to squeeze into the gaps (interstitial sites) — and interestingly, looser BCC iron has fewer but oddly shaped gaps than denser FCC iron, a subtlety at the very heart of how steel holds carbon.
APF for FCC: 4 atoms times (4/3) pi R^3, divided by cell volume (2R sqrt(2))^3 = 16 sqrt(2) R^3, gives 0.74. Repeat for BCC and you get 0.68; for a simple cubic, only 0.52. Nature favours the tightest packings for most metals.
Packing factor = atom volume in the cell divided by cell volume.
The packing factor treats atoms as hard, touching spheres — a useful idealization, not literal truth; real electron clouds are fuzzy and bonding shifts the spacing slightly.