SPHERE PACKINGS

These layers of spheres can be stacked in two principal ways to generate the structures. In the first of these, a second layer fits into the dimples in the first layer, and the third layer is stacked in dimples on top of the second layer to lie over the 

The structural repeat normal to the stacking is two layers of spheres. The spacing of the close packed layers, d, is  

The ratio of the hexagonal lattice parameters, c/a, in this ideal sphere packing is thus 1.633. This model forms an idealised representation of the A3 structure of magnesium, (Chapter 1). 

Although this structure can be described in terms of a hexagonal unit cell, the structure turns out to be cubic, and this description is always chosen. In terms of the cubic unit cell, 

The relationship between the spacing of the close packed planes of spheres, d, the cubic unit cell 

The structure is identical to the A3 structure, described in Section 5.3.5. In most real structures the Co/flo ratio departs from this ideal value of 

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

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 

Stable arrangements with smaller numbers of neighbors, meaning lower coordination numbers, when directed bonds are present. In our discussion, however, the existence of chemical bonds is not a prerequisite at all. 

It may often be convenient to describe the crystal structure in terms of the domains of the atoms [9-26]. The domain is the polyhedron enclosed by planes drawn midway between the atom and each neighbor, these planes being perpendicular to the lines connecting the atoms. The number of faces of the polyhedral domain is the coordination number of the atom, and the whole structure is a space-filling arrangement of such polyhedra. 

The packing based on the sequence ABAB. .. is called hexagonal clo.sest 

Alternatively, continue placing spheres around a central one, all spheres having the same radius. The maximum number that can be placed in contact with the first sphere is 12. However, there is a little more room around the central sphere than just for 12, but not enough for a 13th sphere. Because of the extra room, there are an infinite number of ways of arranging the 12 spheres [9-26]. 

Alternatively, continue placing spheres around a central one, all spheres having the same radius. The maximum number that can be 

The question of densest packing of spheres has been an intriguing problem in mathematics for centuries it has been labeled one of the oldest math problems in the world [45], 

Buckminster Fuller recognized early the importance of icosahedral construction and its great stability in geodesic shapes as well as in viruses. He may have not had the rigorous scientific bases about nucleic acids and about the viruses, but had a fertile imagination and connected seemingly distant pieces of information about structures. This is what he wrote [48]  

This simple formula governing the rate at which balls are agglomerated around other balls or shells 

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Buckminsterfullerene is an allotrope of carbon in which the carbon atoms form spheres of 60 atoms each (see Section 14.16). In the pure compound the spheres pack in a cubic close-packed array, (a) The length of a side of the face-centered cubic cell formed by buckminsterfullerene is 142 pm. Use this information to calculate the radius of the buckminsterfullerene molecule treated as a hard sphere, (b) The compound K3C60 is a superconductor at low temperatures. In this compound the K+ ions lie in holes in the C60 face-centered cubic lattice. Considering the radius of the K+ ion and assuming that the radius of Q,0 is the same as for the Cft0 molecule, predict in what type of holes the K ions lie (tetrahedral, octahedral, or both) and indicate what percentage of those holes are filled. 

Fig. 5.2.4 Plot of the liquid holdup as a function of increasing liquid velocity. An increase in the liquid holdup is observed with increasing liquid velocity. The gas superficial velocity is constant at 66 mm s-1. Data are shown for a bed of 5-mm diameter glass spheres packed within a column of inner diameter 40 mm [40]. Reproduced from Ref. [40], with kind permission from Elsevier, Copyright Liquid velocity (mm s-1) (2001).

Other stacking sequences than these are also possible, for example AaBpAaCy... or statistical sequences without periodic order. More than 70 stacking varieties are known for silicon carbide, and together they are called a-SiC. Structures that can be considered as stacking variants are called polytypes. We deal with them further in the context of closest-sphere packings (Chapter 14). 

The geometric principles for the packing of spheres do not only apply to pure elements. As might be expected, the sphere packings discussed in the preceding chapter are also frequently encountered when similar atoms are combined, especially among the numerous alloys and intermetallic compounds. Furthermore, the same principles also apply to many compounds consisting of elements which differ widely. 

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). 

Although the space filling of the body-centered cubic sphere packing is somewhat inferior to that of a closest-packing, the CsCl type thus turns out to be excellently suited for compounds with a 1 1 composition. Due to the occupation of the positions 0,0,0 and with different kinds of atoms, the structure is not... 

The required local charge balance between cations and anions which is expressed in Pauling s rule causes the distribution of cations and anions among the octahedral and tetrahedral interstices of the sphere packing. Other distributions of the cations are not compatible with Pauling s rule. 

Body-centered cubic sphere packing => CsCl type => superstructures of the CsCl type... 

F. C. Frank, J. S. Kasper, Complex alloy structures regarded as sphere packings. I Definitions and basic principles, Acta Crystallogr. 11 (1958) 184. II Analysis and classification of representative structures, Acta Crystallogr. 12 (1959) 483. 

G. O. Brunner, An unconventional view of the closest sphere packings. Acta Crystallogr. A 27 (1971) 388. 

F. C. Frank and J. S. Kasper, Complex Alloy Structures Regarded as Sphere Packings. II. Analysis and Classification of Representative Structures, Acta Cryst., 12, 483 (1959). 

C. G. Wilson and F. J. Spooner, A Sphere-Packing Model for the Prediction of Lattice Parameters and Order in o Phases, Acta Cryst., 39A, 342 (1973). 

In this section a short description of a comparison between experimental and simulation results for heat transfer is illustrated (Nijemeisland and Dixon, 2001). The experimental set-up used was a single packed tube with a heated wall as shown in Fig. 8. The packed bed consisted of 44 one-inch diameter spheres. The column (single tube) in which they were packed had an inner diameter of two inches. The column consisted of two main parts. The bottom part was an unheated 6-inch packed nylon tube as a calming section, and the top part of the column was an 18-inch steam-heated section maintained at a constant wall temperature. The 44-sphere packed bed fills the entire calming section and part of the heated section leaving room above the packing for the thermocouple cross (Fig. 8) for measuring gas temperatures above the bed. 

J.H. Conway and N.J.A. Sloane, Sphere Packing. Lattice and Groups, Springer-Verlag, New York, 1988. 

Quantification of Disorder in Materials Distinguishing Equilibrium and Glassy Sphere Packings. 

Statistical Geometry of Particle Packings. I. Algorithm for Exact Determination of Connectivity, Volume, and Surface Areas of Void Space in Monodisperse and Polydisperse Sphere Packings. 

Figure 3.46. A non-crystallographic sphere packing with five-fold symmetry.

For a discussion on the symmetry principle , its alternative formulations and the history of its development, papers by Brunner (1977) and by Baminghausen (1980) may be consulted. In these papers a number of statements are reported which may be considered equivalent. When considering close sphere packings, the following statements are especially worthy of mention. 

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