Sphere cubic-closest

Figure 14 ABC stacking of spheres (cubic closest packing). The bottom layer is shown as solid grey spheres, the middle layer as empty circles, and the top layer as dotted circles with a grey fill.

The cubic symmetry of the ABC arrangement of spheres (cubic closest packing) is more difficult to visualize. A portion of four layers of the ABC arrangement is shown in Figure 33. 

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... 

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. 

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

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). 

Table 17.1 Crystallographic data of the hexagonal and cubic closest-packings of spheres. +F means +(j,0), +(j,0, j), +(0, j, j) (face centering). Values given as 0 or fractional numbers are fixed by space-group symmetry (special positions)...

Unit cells for hexagonal (left) and cubic closest-packing of spheres. Top row projections in the stacking direction. Spheres are dr awn smaller than then actual size. Spheres with the same coloring form hexagonal layers as in Fig. 14.1... 

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