Infinite lattice, boron

Fig. 12.5 Part of one layer of the infinite lattice of a-rhombohedral boron, showing the Bn-icosahedral building blocks which are covalently linked to give a rigid, infinite lattice.

Fig. 12.6 The construction of the Bg4-unit, the main building block of the infinite lattice of P-rhombohedral boron.

VM Fig. 13.6 The construction of the Bg -unit, the main building block of the infinite lattice of P-rhombohedral boron, (a) In the centre of the unit is a B] 2-icosahedron, and (b) to each of these twelve, another boron atom is covalently bonded, (e) A B o-cage is the outer skin of the Bg4-unit. (d) The final Bg -unit can be described in terms of covalently bonded sub-units (Bi2)(Bi2)(Bgo). 

Figure 6.1 The icosahedron and some of its symmetry elements, (a) An icosahedron has 12 vertices and 20 triangular faces defined by 30 edges, (b) The preferred pentagonal pyramidal coordination polyhedron for 6-coordinate boron in icosahedral structures as it is not possible to generate an infinite three-dimensional lattice on the basis of fivefold symmetry, various distortions, translations and voids occur in the actual crystal structures, (c) The distortion angle 0, which varies from 0° to 25°, for various boron atoms in crystalline boron and metal borides.

In graphite, which can be considered as a giant two-dimensional molecule from the series of condensed rings, the bonding between the separate layers is very weak, being due, as in molecular lattices, to Van der Waals-London interaction. The now infinite system of n electrons results in metallic conduction, only, however, in the plane of the rings Boron nitride has perhaps also a diamond-like form as well as the common graphite-like modification (p. 235). 

The continuous reduction in size of a solid finally leads to a situation where the original solid state properties can be only partially observed or may be even completely lost, as these properties are exclusively the result of the cooperation between an infinite number of building blocks. Further reduction of size finally leads to typical molecular behavior. On the other hand, even here are structural relations to the bulk occasionally detectable. For instance, the arrangements of the sp hybridized carbon atoms in cyclohexane or in adamantane can easily be traced back to the diamond lattice, whereas benzene or phenanthrene represent derivatives of the graphite lattice. However, neither cyclohexane, benzene, nor phenanthrene have chemical properties which are comparable with those of the carbon modifications they originate from. The existence of the above mentioned Q, C]o or Ci4 units is otUy made possible by the saturation of the free valencies by hydrogen atoms. Comparable well known examples for other elements are numerous, for instance the elements boron, silicon, and phosphorous. Figure 1-1 illustrates some of the relations between elementary and molecular structures. 

The structures of the two rhombohedral forms of elemental boron (Table 5) are of interest in illustrating what can happen when icosahedra are packed into an infinite three-dimensional lattice. In these rhombohedral structures the local symmetry of a Bj2 icosahedron is reduced from //, to Dsd because of the loss of the 5-fold rotation axis when packing icosahedra into a crystal lattice. The 12 vertices of an icosahedron, which are all equivalent under //, local symmetry, are split under iXd local symmetry into two nonequivalent sets of six vertices each (Figure 19a). The six rhombohedral vertices (labeled R in Figure 19a) define the directions of the rhombohedral axes. The six equatorial vertices (labeled E in Figure 19a) lie in a staggered belt around the equator of the... 

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