Sphere , efficient packing

Some metals do not adopt a close-packed structure but have a slightly less efficient packing method this is the body-centred cubic structure (bcc), shown in Figure 1.8. (Unlike the previous diagrams, the positions of the atoms are now represented here—and in subsequent diagrams—by small spheres which do not touch this is merely a device to open up the structure and allow it to be seen more clearly—the whole question of atom and ion size is discussed in Section 1.6.4.) In this structure an atom in the middle of a cube is surrounded by eight identical and equidistant atoms at the corners of the cube—... 

As was described in Chapter 3, the structures of metals are determined by the various ways to efficiently pack spheres. With regard to structures, efficiency refers to minimizing the amount of free (empty) space in the structure and maximizing the number of atoms that are in simultaneous contact (the extent of metallic bonding). Three of the most common ways to arrange spherical atoms are shown in Figure 18.1. 

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... 

It seems reasonable to suppose that the more symmetrical structures (a) are consistent with the efficient packing of the ions, regarded as incompressible spheres having spherical charge distributions. If the ions are to pack as closely as possible the determining factor will be the number of the larger ions that can pack around one of the smaller ions (usually the cation A). The c.n. of the larger ion follows from the fact that in A X the c.n. s of A and X must be in the ratio n m. 

If identical hard spheres such as ball bearings are placed in a flat box to form a single layer, each sphere is soon surrounded by six others. This arrangement provides the most efficient packing model. In general, in structural chemistry the number of nearest neighbours around an atom is... 

A potential advantage in toughness results from the more efficient packing of spheres in a bimodal system (5). However, even when the volume of the... 

In summary, a method has been developed for the placement of bimodal sphere distributions within three-dimensional boundaries. The bimodal distribution is created from the combination of two sphere populations, where each population represents a distinct distribution. The efficient packing of a bimodal distribution of spheres can produce a high volume of the discrete phase in a toughened plastic and a corresponding small interparticle distance. However, combining two materials containing equal discrete-phase volumes of monosized spheres to make a bimodal system does not decrease interparticle... 

Clearly there is more empty space in the simple cubic and body-centered cubic cells than in the face-centered cubic cell. Closest packing, the most efficient arrangement of spheres, starts with the structure shown in Figure 11.20(a), which we call layer A. Focusing on the only enclosed sphere we see that it has six immediate neighbors in that layer. In the second layer (which we call layer B), spheres are packed into the depressions between the spheres in the first layer so that all the spheres are as close together as possible [Figure 11.20(b)]. 

The most efficient packing results in the greatest possible density. The density is the fraction of the total space occupied by the packing units. Only those packings are considered in which each sphere is in contact with at least six neighbors. The densities of some packings are given in Table 9-5. There are... 

The structures adopted by crystalline solids are those that bring particles in closest contact to maximize the attractive forces between them. In many cases the particles that make up the solids are spherical or approximately so. Such is the case for atoms in metallic solids. It is therefore instructive to consider how equal-sized spheres can pack most efficiently (that is, with the minimum amount of empty space). 

The hexagonal and face-centered cubic unit cells. Spheres are packed most efficiently in these cells. First, in the bottom layer (labeled a, orange), we shift every other row laterally so that the large diamond-shaped spaces become smaller triangular spaces. Then we place the second layer (labeled b, green) over these spaces (Figure 12.24D). 

Treating the individual atoms as spheres of the same radius, we would find that the most efficient packing, filling 74% of the available space, occurs for two patterns, shown in Fig. 13.8 the face-centered cubic (fee) or cubic close-packed (cep) lattice, and the hexagonal close-packed (hep) lattice. The hep lattice can be visualized by arranging one set of atoms in staggered rows on a plane, then adding an identical plane of atoms on top of the first one... 

More space-efficient packing can be achieved by aligning neighboring rows of atoms in a pattern with one row offset from the next by one-half a sphere, as shown here ... 

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