Close Packing and the PTOT System

Laves’ principles, (a) the space principle, (b) symmetry principle, and connection principle, are very important in packing spheres: the spheres are expected to pack together in a crystal structure of highest symmetry, highest coordination, and with densest packing. If we let hard spheres of uniform diameter, such as marbles or ball bearings, arrange themselves in one layer on the bottom of a box, they form a two-dimensional close-packed layer. Each sphere is surrounded by a hexagon of six spheres at equal distance, and each of these spheres is at the center of such a hexagon (except at edges, of course). Figure 3.1a shows this close-packed layer expanded so we can see the positions for added layers. The atoms touch in a close-packed layer. We can begin a second close-packed layer by dropping an identical sphere on the first layer. This sphere will roll into an indentation—any one, they are all equivalent. To keep track of positions in subsequent layers, let us label the positions of the spheres in the first layer A positions. Following the alphabetical order, we call the position of the sphere beginning the second layer position B. These letters label the positions as viewed from above the layers as in Figure 3.1. Adding more spheres to complete the second layer, without moving the first one at position A, gives the B layer. It is identical to the first layer except shifted. The choice of the position of the first sphere of the second layer is arbitrary.