Packing Efficiency Calculator

Packing efficiency, or atomic packing factor, is the fraction of a unit cell filled by atoms modelled as touching hard spheres. It equals Z times four thirds pi r cubed divided by the cell volume, where r is the radius set by whichever direction the spheres actually touch. In simple cubic they touch along the edge so a = 2r, in body-centred cubic along the body diagonal so a = 4r divided by root 3, and in face-centred cubic along the face diagonal so a = 2 root 2 times r. The resulting values are exact constants independent of element: 52.36 percent for simple cubic, which is pi over 6, 68.02 percent for body-centred cubic, 74.05 percent for both face-centred cubic and hexagonal close packed, which is pi over 3 root 2, and only 34.01 percent for the diamond cubic structure of silicon and carbon. Higher packing goes with higher coordination number, 6 for simple cubic, 8 for body-centred, and 12 for the two close-packed arrangements. Ideal hexagonal close packing requires a c to a ratio of root 8 over 3, or 1.633, but real metals deviate: magnesium sits at 1.624 while zinc reaches 1.856 and cadmium 1.886, so their true packing falls below 74 percent. The hard-sphere picture also breaks down wherever bonding is strongly directional, which is exactly why diamond cubic is so open, and it does not apply directly to ionic solids where cation and anion radii differ. For comparison, random close packing of equal spheres, as in a poured powder, saturates near 64 percent.

The question of how densely equal spheres can be packed was raised by Johannes Kepler in his 1611 booklet on the six-cornered snowflake, where he asserted that the face-centred cubic stacking used by greengrocers for cannonballs is the densest possible arrangement. The claim became known as the Kepler conjecture and resisted proof for centuries. Carl Friedrich Gauss narrowed it in 1831 by proving that no lattice packing beats the face-centred cubic density of pi over 3 root 2, but the possibility of an irregular arrangement doing better remained open, and David Hilbert included the general sphere-packing problem in the eighteenth of his famous 1900 list of unsolved problems. Crystallography arrived at the same structures from the physical side. William Barlow enumerated close-packed sphere arrangements in the 1880s and correctly predicted several crystal structures years before diffraction could confirm them, and the Bragg structure determinations after 1913 showed that most metallic elements do adopt face-centred cubic, hexagonal close-packed or body-centred cubic arrangements. The mathematics was finally settled by Thomas Hales, who announced a computer-assisted proof of the Kepler conjecture in 1998 and published it in the Annals of Mathematics in 2005. Because referees could not fully verify the computations by hand, Hales led the Flyspeck project to produce a formal machine-checked proof, which was completed in 2014.