The closest packing of equal spheres

Three spheres are obviously most closely packed in a triangular arrangement, and a fourth sphere will make the maximum number of contacts (3) if it completes a 122 

Since the most symmetrical arrangement of 12 neighbours (the icosahedral coordination group) does not lead to the densest possible 3D packing of spheres we have to enquire which of the infinite number of arrangements of twelve neighbours lead to more dense packings and what is the maximum density that can be attained 

(a) Relation between cuboctahedron and icosahedron, (b) Icosahedral packing of equal spheres, showing the third layer (n = 3). (AC 1962 15 916). 

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Figure 3.34. Holes in the closest packing of equal spheres. Two superimposed layers of spheres are shown (continuous and dotted lines). Tetrahedral (T) and octahedral (O) holes are indicated.

Three structures of this type were described in our discussion of the closest packing of equal spheres, namely, those in which certain proportions of the tetrahedral and/or octahedral holes are occupied ... 

These structures provide an elegant example of the interrelations of nets, open packings of polyhedra, space-filling arrangements of polyhedra, and the closest packing of equal spheres. 

It is well known in crystallography that, when spheres of equal radius are packed together, the closest type of packing is one in which each sphere has 12 other spheres in contact with it. In Sec. 24 it was mentioned that in water at room temperature each molecule has, on the average, only 4.4 other molecules in contact with it. If we wanted to place one or two additional H20 molecules in contact with any H20 molecule, there would be plenty of room to do this without seriously disturbing the neighbors that are already in contact with this molecule. Similarly, if this molecule is replaced by a solute particle of the same size, the same remark could be made about placing molecules in contact with the solute particle. 

Both the hexagonal closest packing of spheres and the cubic closest packing of spheres result in the (equally) densest packing of the spheres. The fraction of the total volume occupied by the spheres, when they touch, is 0.7405. 

Two metals that are chemically related and that have atoms of nearly the same size form disordered alloys with each other. Silver and gold, both crystallizing with cubic closest-packing, have atoms of nearly equal size (radii 144.4 and 144.2 pm). They form solid solutions (mixed crystals) of arbitrary composition in which the silver and the gold atoms randomly occupy the positions of the sphere packing. Related metals, especially from the same group of the periodic table, generally form solid solutions which have any composition if their atomic radii do not differ by more than approximately 15% for example Mo +W, K + Rb, K + Cs, but not Na + Cs. If the elements are less similar, there may be a limited miscibility as in the case of, for example, Zn in Cu (amount-of-substance fraction of Zn maximally 38.4%) and Cu in Zn (maximally 2.3% Cu) copper and zinc additionally form intermetallic compounds (cf. Section 15.4). 

In three-dimensional closest packing, the spherical atoms are located in position 4(a). There are two types of interstices octahedral and tetrahedral holes which occupy positions 4(b) and 8(c), respectively. The number of tetrahedral holes is twice that of the spheres, while the number of octahedral holes is equal to that of the spheres. The positions of the holes are shown in Fig. 10.1.1. 

A metallic crystal can be pictured as containing spherical atoms packed together and bonded to each other equally in all directions. We can model such a structure by packing uniform, hard spheres in a manner that most efficiently uses the available space. Such an arrangement is called closest packing (see Fig. 16.13). The spheres are packed in layers in which each sphere is surrounded by six others. In the second layer the spheres do not lie direotlv over those in the first layer. Instead, each one occupies an indentation (or dimple) formed by three spheres in the first layer. In the third layer the spheres can occupy the dimples of the second layer in two possible ways. They can occupy positions so that each sphere in the third layer lies directly over a sphere in the first layer (the aba arrangement), or they can occupy positions... 

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