Hexagonal and cubic closest packing of equal spheres

Hexagonal and cubic closest packing of equal spheres 

We now consider the sphere packings in which close-packed plane layers are stacked in the closest possible way. If we label the positions of the spheres in one layer as (Fig. 4.12) then an exactly similar layer can be placed above the first so that the centres of the spheres in the upper layer are vertically above the positions B. It is obviously immaterial whether we choose the positions B or the similar positions C, as may be seen by inverting Fig. 4.12. When the third layer is placed above the second (B) layer there are alternatives the centres of the spheres may lie above either the C ot A positions. The two simplest sequences of layers are evidently 

Three successive layers of cubic closest packing. 

. and ABCABC. but there is also an indefinite number of more complex sequences. In all such sphere packings the volume occupied per sphere is 5-66 for spheres of radius a. Alternatively, the mean density is 0-7405 if that of the spheres is unity and that of the space between the spheres is zero. 

The sequence AB AB. .. is referable to a hexagonal unit cell (Fig. 4.13) and is called hexagonal closest packing (h.c.p.). In this arrangement, also illustrated in Fig. 4.14(d), the twelve neighbours of an atom are situated at the vertices of the coordination polyhedron of Fig. 4.5(a) ( twinned cuboctahedron ). 

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