Packing of spheres

Packing of spheres Polymorphism Alloys and intermetallic compounds Band theory Semiconductors Sizes of ions Prototype structures Lattice energy Born-Haber cycle Defects in solid state structures 

An anion is a negatively charged ion and a cation is a positively charged ion. 

Although metallic and ionic solids have 3-dimensional stmctures, it does not follow that 3-dimensional stmctures are necessarily metallic or ionic. Diamond, for example, is a non-metal (see Sections 6.11 and 6.12). In Sections 2.2 and 2.5, we considered the inclusion of ionic contributions to covalent bonding pictures. Later in this chapter we discuss how including some covalent character in a predominantly ionic model comes closer to reality for some so-called ionic compounds. 

Many readers will be familiar with descriptions of metal lattices based upon the packing of spherical atoms, and in this section we provide a resume of commOTi types of packing. 

Now consider the hollows that are visible in layer B in Fig. 6.2b. There are two distinct types of hollows. Of the four hollows between the grey spheres in layer B, one lies over a red sphere in layer A, and three lie over hollows in layer A. The consequence of this is that when a third layer of spheres is constmcted, two different close-packed arrangements are possible as shown in Figs. 6.2c and 6.2d. The arrangements shown can, of course, be extended sideways, and the sequences of layers can be repeated such 

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The general geometrical problem of the packing of spheres has not been solved. An example of closest packing of atoms with some variation in effective radius is the icosahedral packing found (13) in the intermetallic compound Mg3B(Al,Zn) (Fig. 1). The successive layers in this structure contain 1, 12, 32, and 117 spheres. These numbers are reproduced (to within 1) by the empirical equation (12)... 

Tc or c cubic closest-packing of spheres Th or h hexagonal closest-packing of spheres Ts stacking sequence AA... of hexagonal layers Qs stacking sequence AA... of square layers... 

The structure of iodine at four different pressures. The outlined face-centered unit cell in the 30-Gpa figure corresponds to that of a (distorted) cubic closest-packing of spheres. At 24.6 GPa four unit cells of the face-centered approximant structure are shown the structure is incommensurately modulated, the atomic positions follow a sine wave with a wave length of 3.89 x c. The amplitude of the wave is exaggerated by a factor of two. Lower left Dependence of the twelve interatomic contact distances on pressure... 

This bismuth-III structure is also observed for antimony from 10 to 28 GPa and for bismuth from 2.8 to 8 GPa. At even higher pressures antimony and bismuth adopt the body-centered cubic packing of spheres which is typical for metals. Bi-III has a peculiar incommensurate composite crystal structure. It can be described by two intergrown partial structures that are not compatible metrically with one another (Fig. 11.11). The partial structure 1 consists of square antiprisms which share faces along c and which are connected by tetrahedral building blocks. The partial structure 2 forms linear chains of atoms that run along c in the midst of the square antiprisms. In addition, to compensate for the... 

In crystalline C60 the molecules have a face-centered cubic arrangement, i.e. they are packed as in a cubic closest-packing of spheres as they are nearly spherical, the molecules spin in the crystal. The crystals are as soft as graphite. Similar to the intercalation com-... 

In a-B12 the icosahedra are arranged as in a cubic closest-packing of spheres (Fig. 11.16). In one layer of icosahedra every icosahedron is surrounded by six other icosahedra that are linked by three-center two-electron bonds. Every boron atom involved contributes an average of electrons to these bonds, which amounts to -6 = 4 electrons per icosahedron. Every icosahedron is surrounded additionally by six icosahedra of the two adjacent layers, to which it is bonded by normal B-B bonds this requires 6 electrons per icosahedron. In total, this adds up exactly to the above-mentioned 10 electrons for the inter-icosahedron bonds. 

Germanium forms the same kinds of modifications as silicon at similar conditions (Fig. 12.4). Tin, however, does not exhibit this diversity )3-tin transforms to a body-centered cubic packing of spheres at 45 GPa. Lead already adopts a cubic closest-packing of spheres at ambient pressure. 

The cluster condensation can be carried on the chains of octahedra sharing edges can be joined to double-strands and finally to layers of octahedra (Fig. 13.18). Every layer consists of metal atoms in two planes arranged in the same way as two adjacent layers of atoms in a closest-packing of spheres. This is simply a section from a metal structure. The X atoms occupy positions between the metal layers and act as insulating layers. Substances like ZrCl that have this structure have metallic properties in two dimensions. 

Unit cells for hexagonal (left) and cubic closest-packing of spheres. Top row projections in the stacking direction. 

The space filling in the body-centered cubic packing of spheres is less than in the closest packings, but the difference is moderate. The fraction of space filled amounts to ns/3 = 0.6802 or 68.02 %. The reduction of the coordination number from 12 to 8 seems to be more serious however, the difference is actually not so serious because in addition to the 8 directly adjacent spheres every sphere has 6 further neighbors that are only 15.5 % more distant (Fig. 14.3). The coordination number can be designated by 8 + 6. 

Corresponding to its inferior space filling, the body-centered cubic packing of spheres is less frequent among the element structures. None the less, 15 elements crystallize with this structure. As tungsten is one of them, the term tungsten type is sometimes used for this kind of packing. 

Unit cell of the body-centered cubic packing of spheres and the coordination around one sphere... 

Fig. 11.11, p. 112). Incommensurate structures related to bismuth-III are also observed for strontium and barium. Magnesium, calcium and strontium are remarkable in that they transform from the normal closest-packing of spheres to a body-centered packing upon exertion of pressure. Even more remarkable is the following decrease of the coordination number to 6 for calcium and strontium (Ca-III, a-Po type Sr-III, /3-tin type). 

The solid noble gases also adopt closest-packings of spheres at low temperatures Ne... Xe c helium becomes solid only under pressure (depending on pressure, c, h or i)... 

State the Jagondzinski and the Zhdanov symbols for the closest-packings of spheres with the following stacking sequences ... 

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. 

The CsCl type offers the simplest way to combine atoms of two different elements in the same arrangement as in body-centered cubic packing the atom in the center of the unit cell is surrounded by eight atoms of the other element in the vertices of the unit cell. In this way each atom only has adjacent atoms of the other element. This is a condition that cannot be fulfilled in a closest-packing of spheres (cf. preceding section). 

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