Cubic Close Packing

There are three types of cubic lattices corresponding to three types of cubic close packing.

lattice type	basis vectors	packing density
simple cubic (SC)	, ,
face-centered cubic (FCC)	, ,
body-centered cubic (BCC)	, ,

Simple cubic close packing consists of placing spheres centered on integer coordinates in Cartesian space.

Arranging layers of close-packed spheres such that the spheres of every third layer overlying one another gives cubic close packing. To see where the name comes from, consider packing six spheres together in the shape of an equilateral triangle and place another sphere on top to create a triangular pyramid. Now create another such grouping of seven spheres and place the two pyramids together facing in opposite directions. A cube emerges (Steinhaus 1999, pp. 203-204). Connecting the centers of these 14 spheres gives a stella octangula.

Consider the cube defined by 14 spheres in cubic close packing, as illustrated above. This "unit cell" contains eight -spheres (one at each polygon vertex) and six hemispheres. The total volume of spheres in the unit cell is therefore

			(1)
			(2)

The diagonal of the face is , so each side is . The volume of the unit cell is therefore

(3)

and the packing density is

(4)

(Conway and Sloane 1993, p. 2). Now that the Kepler conjecture has been established, hexagonal close packing and cubic close packing, both of which have the same packing density of , are known to be the densest possible packings of equal spheres.

In cubic close packing, each sphere is surrounded by 12 other spheres. Taking a collection of 13 such spheres gives the cluster illustrated above. Connecting the centers of the external 12 spheres gives a cuboctahedron (Steinhaus 1999, pp. 203-205; Wells 1986, p. 237).

If spheres packed in a cubic lattice, face-centered cubic lattice, and hexagonal lattice are allowed to expand uniformly until running into each other, they form cubes, hexagonal prisms, and rhombic dodecahedra, respectively. In particular, if the spheres of cubic close packing are expanded until they fill up the gaps, they form a solid rhombic dodecahedron (left figure above), and if the spheres of hexagonal close packing are expanded, they form a second irregular dodecahedron consisting of six rhombi and six trapezoids (right figure above; Steinhaus 1999, p. 206). The latter can be obtained from the former by slicing in half and rotating the two halves with respect to each other. The lengths of the short and long edges of the rotated dodecahedron have lengths 2/3 and 4/3 times the length of the rhombic faces. Both the rhombic dodecahedron and squashed dodecahedron are space-filling polyhedra.

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