Canonical polyhedra

Our polygon description of the structure of 2D dense random packings of hard disks parallels Bernal s description of three-dimensional (3D) dense random packings of hard spheres as space-filling arrays of elementary polyhedral units ( Bernal holes, or canonical polyhedra ) [2-5]. Bernal s approach to 3D liquid structure is discussed in more detail in Section IV.A. 

The atomic packing in disordered solids was investigated first by Bernal (1964), who considered the problem in the context of a model of a simple liquid that consisted of randomly close-packed hard spheres of uniform size and described the structure as a distribution of five different canonical polyhedra with well-defined volume fractions. 

For example, the bond valence (b) of Ge94 is b = 5(8n — g) = [9 x 8 -(9x4+4)] = 16, and the monocapped square-antiprismatic anion is stabilized by six MMM 3c-2e bonds and four M-M 2c-2e bonds. Bonding within the polyhedron involves a resonance hybrid of several canonical forms of the type shown in Fig. 14.6.2. The structures of the polyatomic anions are summarized in Table 14.6.1. 

The COR is constructed to calculate those extra structure constants which may be used to correct the ASA for approximate treatment of the region between the sphere and the atomic polyhedron, and for the neglect of higher components as described in Sect.6.9. The programme produces and stores on disk or tape a set of correction-term matrices distributed on a suitable grid in the irreducible wedge of the Brillouin zone. Whenever requested the correction matrices may be retrieved by LMTO and used together with the canonical structure constants to set up the corrected LMTO matrices. 

When individual bond networks do not match the symmetry of the polyhedron in question, resonance between plausible canonical forms needs to be invoked. 

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