closepacking

Close packing describes the arrangement of equal spheres in three dimensions that yields the highest possible density. In crystallography and materials science, close-packed structures are usually realized as hexagonal close packing (HCP) and cubic close packing, the latter commonly referred to as face-centered cubic (FCC). Both achieve the theoretical maximum packing density of π/(3√2) ≈ 0.74048, leaving about 26% of space as void.

In hexagonal close packing, layers are stacked ABAB… within a hexagonal lattice. Each sphere contacts 12 neighbors. The ideal HCP unit cell has a c/a ratio of about 1.633. In cubic close packing, layers are stacked ABCABC… in a face-centered cubic lattice. Each sphere also coordinates 12 neighbors, and the conventional cubic unit cell contains four atoms.

Close-packed lattices feature interstitial voids of tetrahedral and octahedral types. These voids can accommodate smaller atoms or ions, influencing diffusion, alloying behavior, and other material properties. The distribution and size of voids affect how atoms migrate through the crystal and how alloys form.

Close-packed metal structures occur widely. Aluminum, copper, and nickel adopt FCC arrangements, while magnesium and titanium commonly exhibit HCP structures under appropriate conditions. The arrangement impacts density, slip systems and ductility, and contributes to characteristic mechanical behavior at different temperatures. Close-packed models also serve as reference points in studies of colloidal suspensions and theoretical packing problems.

Historically, the concept relates to the pursuit of the densest possible sphere packing, a problem associated with Kepler’s conjecture, which identifies FCC and HCP as achieving the maximum theoretical density for equal spheres.