The Hexagonal and Face Centered Cubic Arrangement

Forming the first and second layers. The first close-packed layer is built exactly as in the 2D ABAB... arrangement of Section 6.6.2 -- the spheres of the second row nestling into the depressions of the first -- and this whole layer is labelled 'a'. A second layer, labelled 'b', is then formed by placing its spheres into the depressions (voids) of layer a. Layer a actually has two distinct kinds of void, labelled x and y, and layer b can be built by using either set -- the choice is arbitrary and equivalent by symmetry, so the text follows the case where layer b occupies the x voids.

Tetrahedral and octahedral voids. Wherever a layer-b sphere sits directly above an x-void of layer a, a tetrahedral void is formed -- made up of 4 spheres (3 from the lower layer a, 1 from the upper layer b) whose centres, when joined, trace out a tetrahedron. At the same time, the y-voids of layer a are only partially covered by layer b's spheres; such a partially-covered void is called an octahedral void -- made up of 6 spheres (3 from layer a, 3 from layer b) whose centres, when joined, trace out an octahedron. New tetrahedral voids are simultaneously created the other way up too, from 3 layer-b spheres and 1 layer-a sphere. In general, if n is the number of close-packed spheres, the number of octahedral voids generated equals n, and the number of tetrahedral voids generated equals 2n.

Forming the third layer -- hcp vs ccp. The third layer of spheres can be added in two distinct ways to keep the packing closest:

1. aba stacking -- hexagonal close-packed (hcp). The third layer's spheres fit into the depressions so that the third layer sits directly over the first layer again -- i.e. the tetrahedral voids of the second layer are exactly covered by the spheres of the third layer. The stacking sequence is ababab...
2. abc stacking -- cubic close-packed (ccp). Alternatively, the third layer's spheres are placed so that they all fit into the octahedral voids of the layers below -- a position different from both layer a and layer b, so this third layer is labelled 'c'. Continuing the stacking as abcabcabc... gives the cubic close-packed (ccp) structure, which is built on the face-centred cubic (fcc) unit cell of Section 6.5.3. In BOTH hcp and ccp, every sphere has a coordination number of 12 -- 6 neighbours in its own layer, 3 in the layer above and 3 in the layer below -- making these the most efficient packing arrangements possible. Packing efficiency of ccp/fcc. From the fcc unit cell's geometry (Section 6.5.3), the face diagonal AC spans 4 sphere radii: AC = 4r. By Pythagoras on the face triangle, AC^2 = AB^2 + BC^2 = a^2 + a^2 = 2a^2, so AC = root-2 * a. Equating the two expressions, 4r = root-2 * a, so r = (root-2 / 4) a. The volume of one sphere is: …