the atomic packing factor
The atomic packing factor answers a very physical question: if atoms are hard spheres, what fraction of the crystal's volume is actually solid ball, and what fraction is empty gap? A packing factor of 0.74 means 74 percent of the space is filled and 26 percent is void. It is a single number that tells you how efficiently a structure uses space.
The recipe is: APF = (number of atoms in the cell times the volume of one atom) divided by the volume of the cell. Take FCC: there are 4 atoms per cell, each of volume (4/3) pi r cubed, and the cell edge is a = 2 sqrt(2) r so the cell volume is (2 sqrt(2) r) cubed. Dividing gives APF = 4 times (4/3) pi r^3 divided by (2 sqrt(2) r)^3 = 0.74. Running the same arithmetic gives 0.68 for BCC, 0.52 for simple cubic, and only 0.34 for the open diamond structure.
The packing factor matters because the empty 26 percent is exactly where the action happens. Those voids are the interstitial holes that host carbon in steel or cations in oxides; a low packing factor like diamond's 0.34 signals a structure held open by strong directional bonds rather than by dense sphere stacking. And 0.74 is not merely typical, it is the proven ceiling for identical spheres, so no arrangement of equal atoms can ever exceed it.
FCC: APF = 4 x (4/3) pi r^3 / (2 sqrt(2) r)^3 = 0.74. Diamond, held open by tetrahedral bonds, manages only 0.34.
The number is dimensionless and radius-independent: r cancels top and bottom.
APF assumes atoms are rigid touching spheres of one size. Real atoms are squishy and bonds are directional, so the hard-sphere APF is a useful idealisation, not a measured density.