Geometrical requirements in the close-packed structures

Clearly, the balls will be packed in layers, and each layer must be closely packed. We begin by arranging the layer as shown in Fig. 27.2. In the layer, each sphere has six nearest neighbors. To build the crystal in three dimensions, we stack the layers one on top of the other in a regular way. Two possibilities exist after the second layer is put on. Consider sphere A in the figure and suppose that the spheres in the next layer nestle in the notches at the positions marked with dots. Three of these will make contact with A. In this second layer there is a notch over A, but there are also notches over the positions marked with crosses. The third layer may then repeat the arrangement in the first layer or not. If the third layer repeats the first, then there are only two kinds of layers, denoted x and y, and the 

The arrangement of close-packed layers in the pattern xyxy... is the hexagonal close-packed structure (hep) the pattern xyzxyz... is the cubic close-packed (cep) or face-centered cubic (fee) structure. In each of these structures, every sphere is in contact with twelve others six in its own layer, three in the layer above, and three in the layer below. The twelve coordination in these structures is shown by an exploded view in Fig. 27.3(a) and (b), and in a different view in Fig. 27.3(c) and (d). The hep and fee structures are the typical structures encountered in metals. The high coordination number (twelve) in these structures results in a crystal of comparatively high density. 

In some structures the repetition pattern of the close-packed layers is more complicated, and we find the CH notation is useful to describe the structure. If we choose any close-packed layer, and if the two layers on each side of it are identical, the layer is designated by H, since repetition of these layers will build the hexagonal close-packed structure. Thus the alternation xyxyxy... is denoted hy HHHH. If the two layers on each side of the chosen layer are not identical, the layer is designated by C, since repetition of layers of this kind will build the cubic close-packed (the fee) structure. Thus the repetition pattern, 

Another common arrangement of spheres that occurs in a few metals is the body-centered cubic (bcc), which is built up of layers having the arrangement shown in Fig. 27.4. The second layer fits in the notches of the first and the pattern of layers repeats, xyxy. In these layers the number of nearest neighbors around any sphere is four, as compared with six in the close-packed layers. In the body-centered structure the overall coordination number is eight there are four nearest neighbors within the most closely packed layer, two 

Twenty or thirty marbles or coins can be helpful in working out these arrangements. 

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