Frank-Kasper polyhedrons

Figure 3.87 The selenium substructure in 108. The central Se atom is surrounded by the Sens Frank-Kasper polyhedron, which is itself surrounded by the Se44 polyhedron in a second coordination sphere.

Fig. 1.6 Partial view along [111] of the structure of type I NagSi46, showing one of the two C.N.12 Frank-Kasper polyhedron of Na(2) atoms surrounding the central Si2o dodecahedron, itself centered on a Na(l) atom...

The molecular structure of 73, which contains the bulky PfBua ligand, is also shown in Figure 3-77. The fifteen peripheral tellurium atoms form a Frank-Kasper polyhedron with a central tellurium atom. This polyhedron is also very distorted due to Te-lb interactions. Copper atoms coordinated by PrBua ligands are situated above ten of the Te3 faces and sixteen Cu atoms are located below the remaining lb, faces. 

Frank-Kasper polyhedron supported by metal-oxygen bonds (not shown for clarity) in the crystal strnctnre of potassinm nraninm peroxide. The cylinders shown represent the polyhedral arrangements, not chemical bonds. 

An example of the relationship between concentric fullerene and Frank-Kasper polyhedra can be found in potassium uranium peroxide, in whose crystal structure a Ki6 Frank-Kasper polyhedron (h = 4) is supported by uranyl groups at the face centers, thus forming O28 and U28 fullerenes (Figure 14). 

Frank-Kasper polyhedron A polyhedron that has a duality relationship with a fullerene, whose faces are all triangles and whose vertices connect either five or six neighboring faces. Traditionally, this name was used only for those polyhedra with 14, 15, or 16 vertices. 

FRANK-KASPER polyhedra in MgCu2. The polyhedron around an Mg atom (c.n. 16) is composed of 12 Cu atoms and of four Mg atoms that form a tetrahedron by themselves the Cu atoms form four triangles that are opposed to the Mg atoms. The polyhedron around a Cu atom (c.n. 12) is an icosahedron in which two opposite faces are occupied by Cu atoms... 

In relation to the previously reported Frank-Kasper proposal, O Keeffe (1979) suggested that coordinating atoms contribute faces to the Voronoi polyhedron around the central atom, and their contributions are weighed in proportion to the solid angle subtended by that face at the centre. 

Several structures (.Frank-Kasper structures) can be considered in which all atoms have either 12 (icosahedral), 14, 15 or 16 coordinations. These can be described as resulting from the polyhedra included in Fig. 3.18. These polyhedra interpenetrate each other so that every vertex atom is again the centre of another polyhedron. 

The Frank Kasper method (Frank Kasper, 1958) defines as domain of an atom all points which are closer to the atom than to all other atoms. We draw lines joining the central atom to the surrounding atoms and then bisect these lines with perpendicular planes. The polyhedron limited by these planes is the domain of the atom (called also Voronoi polyhedron or Wigner-Seltz cell). All atoms emerging from the faces of the domain are coordinated with the central atom and define its coordination polyhedron. 

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