In very simple terms, the random close packing – jammed configuration where each sphere is confined by its neighbours – is invariably at .64 of volume filling. The ordered hexagonally close packed can achieve higher ratios, of about 0.74, and one would think that the special number of 0.64 can be derived and/or proven mathematically – not so.
Kamien and Liu show that 0.64 is the number for which the number of available configuration states disappears (goes to zero).
Here they used a very clever “wedge” design by making a wedge of CrV material, where the thickness varies along one direction and composition (V doping) varies along the perpendicular direction. This allows mapping out the magnetic phase diagram (C-SDW, I-SDW AFM phase vs. Paramagnetic phase) as a function of temperature, thickness of the film and V composition without having to change the samples.