closepacked

Close packing refers to the arrangement of equal spheres in three dimensions that minimizes void space and maximizes the volume occupied by the spheres. The densest regular packings have a packing fraction of pi/√18, approximately 0.740. The two most common close-packed structures are hexagonal close packing and cubic close packing, the latter also known as face-centered cubic.

In cubic close packing, layers stack in an ABCABC sequence, with each sphere resting in the hollows of the layer below. Hexagonal close packing uses an ABAB sequence, repeating every two layers. Both arrangements achieve the same coordination number of 12, meaning each sphere touches 12 neighbors, but they differ in symmetry and stacking order. The cubic close-packed form is cubic in symmetry, whereas hexagonal close packing is hexagonal.

Unit cells and composition: The conventional unit cell of cubic close packing contains four spheres, and the lattice constant a relates to the sphere diameter d by a ≈ d√2. Hexagonal close packing has a hexagonal unit cell, typically described with a primitive cell containing two atoms and a conventional cell containing six.

Occurrence and properties: Many metals crystallize in close-packed structures. Copper, aluminum, silver, gold, and nickel adopt the cubic close-packed (face-centered cubic) arrangement, while magnesium and zinc adopt hexagonal close packing. Close-packed metals generally have high density and multiple close-packed directions, contributing to specific mechanical properties, such as ductility in FCC metals and comparatively more limited ductility in HCP metals due to fewer available slip systems.

History and context: The concept of close packing is linked to Kepler’s conjecture that the densest sphere packing in three dimensions is a close-packed arrangement, a result proven only many centuries after Kepler’s proposal.