Twinned cuboctahedron

If we have N hard spheres (of radius rs) forming a close-packed polyhedron, another sphere (of smaller radius rc) can fit neatly into the central hole of the polyhedron if the radius ratio has a well-defined value (see also 3.8.1.1). The ideal radius ratio (rc/rs) for a perfect fit is 0.225.. (in a regular tetrahedron, CN 4), 0.414.. (regular octahedron CN 6), 0.528.. (Archimedean trigonal prism CN 6), 0.645... (Archimedean square antiprism CN 8), 0.732.. (cube CN 8), 0.902... (regular icosahedron CN 12), 1 (cuboctahedron and twinned cuboctahedron CN 12). 

The sequence AB AB. .. is referable to a hexagonal unit cell (Fig. 4.13) and is called hexagonal closest packing (h.c.p.). In this arrangement, also illustrated in Fig. 4.14(d), the twelve neighbours of an atom are situated at the vertices of the coordination polyhedron of Fig. 4.5(a) ( twinned cuboctahedron ). 

When Li metal is cold-worked it transforms from body-centred cubic to cubic close-packed in which each atom is surrounded by 12 others in twinned cuboctahedral coordination below 78 K the stable crystalline modification is hexagonal dose-packed in which each lithium atom has 12 nearest neighbours in the form of a cuboctahedron. This very high coordination... 

Figure 21. Radial distribution functions calculated using a Fourier transform of scattering patterns produced with a Debye equation. Top Cuboctahedron (cluster with both octahedral 111 and cnbe 100 faces) and icosahedron (multiply twinned hep structure) clusters of the same size. Center Cuboctahedra of different sizes. Bottom Experimental and simulated cluster RDF of a Pt colloid. The fit is a 90 10 mixture of 55 and 147 cuboctahedral clusters, respectively. After Casanove et al. (1997).

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