Arrangement of spheres

Arrangement of spheres in a hexagonal layer and the relative position of the layer positions A, B and C... 

In each layer of a closest packing arrangement of spheres, there are six spheres... 

Let us call the arrangement of spheres shown in Figure 9.33 as A. The next layer may be arranged as B or C. Let us consider the case of B. The third layer will be A or C, etc. In the former case, alternation of the layers gives a scheme ABABAB... (hcp-packing) in the latter it gives AB-CABC... (fcc-packing). More complicated alternations are possible, for example, ABCBABCB..., ABCABABCAB..., etc. 

Figure 6.1 Sectional view of a close-packed hexagonal arrangement of spheres.

Equation (48) has been derived under the assumption that the volume fraction can reach unity as more and more particles are added to the dispersion. This is clearly physically impossible, and in practice one has an upper limit for , which we denote by max. This limit is approximately 0.64 for random close packing and roughly 0.71 for the closest possible arrangement of spheres (face-centered cubic packing or hexagonal close packing). In this case, d in Equation (46) is replaced by d(j>/[ 1 — (/m[Pg.169]

Figure 1.3. Arrangements of spheres for (a) ccp, (b) hep, (c) NaCl, and (d) CsCl as described by W. Barlow.

Figure 5.3. Top view of volume averaging over a hexagonal arrangement of spheres (a) Cocentrated with a sphere (b) Centered in a pyramid of spheres.

Corundum is one of the many polimorphs of alumina (A1203). The corundum-type structure is the structural shape of hematite (aFe203). In Figure 2.12, layers A and B of a close-packed arrangement of spheres, and the formation of the corresponding octahedral and tetrahedral sites are shown [51,52]. 

Next in importance for a proper understanding of size-distribution and particle-measurement is the manner in which fine material packs. It is well known that a column filled with sand may be shaken so that it will occupy less space. Similarly, certain soils may be pressed or tamped to become denser. Compaction is achieved at the expense of the void space, which may vary from a theoretical minimum of 26 percent to as high as 48 percent for spheres. Unfortunately there is no adequate method of describing a packing in terms of partide-orientation. We can only deal with it in terms of the free space present or the ease with which a liquid flows through it. Except in a statistical sense it is never possible to make two packings precisely identical unless we make a systematic arrangement of spheres. 

Computation of Voids—Except for ideal arrangements of spheres, voids can be determined only experimentally. Calculations of voids are in terms of the true and apparent specific densities of the material used in the packing, in accordance with the formula... 

Consider the section x-y in the rhombohedral arrangement of spheres having a diameter d as shown in Figure 34, the dotted lines indicating... 

Figure 76. Geometric Arrangement of Spheres in Most-open and Closest Packings.

The body-centered arrangement of spheres is not a closest packed structure, as we can show by calculating the fraction of space occupied by the spheres. To express the volume of the cube in terms of the radius of the packed spheres, we must use the Pythagorean theorem twice. First, we express the face diagonal f in terms of the edge e (see Fig. 16.20) ... 

The HCP structure is so called because it is one of the two ways in which spheres can be packed together in space with the greatest possible density and still have a periodic arrangement. Such an arrangement of spheres in contact is shown in Fig. 2-15(d). If these spheres are regarded as atoms, then the resulting picture of an HCP metal is much closer to physical reality than is the relatively open... 

Although Bernal found no evidence of crystalline regions in DRPs, he did find evidence of so-called pseudonuclei, which are exceptionally dense local arrangements of spheres consisting of helical chains of... 

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)]. 

A fourth layer added so that it, similarly, is different from the second and third must fall vertically above the first, and if further layers are added they will form the sequence ABC ABC... indefinitely continued (fig. 5.01 d). It is not by any means immediately apparent that this arrangement has cubic symmetry and is, in fact, an arrangement of spheres at the corners and face centres of a cubic unit cell (fig. 5.03a), so that it constitutes cubic close packing. This point, however, is made clear by fig. 5.036, which represents 27 unit cells of the structure. Here the face-centred unit cells are apparent by a comparison with fig. 5.03 a, but the close-packed nature of the layers normal to one of the cube... 

Outline the similarities and differences between cubic and hexagonal close-packed arrangements of spheres, paying particular attention to (a) coordination numbers, (b) interstitial holes and (c) unit cells. 

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