Frank-Kasper crystal structures

Then, we can derive a very simple rule on selecting crystal structures of the clathrate hydrates by employing Yarmolyuk-Kripyakevich s empirical rule on the coordination numbers of a Frank-Kasper alloy structure [79]. The rule denotes the relation between the number of 12-, 14-, 15-, and 16-coordinated atoms in an alloy structure. That is, a set of such numbers in any FK crystal structure is given by a linear combination of the three representative Frank-Kasper crystal structures, A15, C15, and Z. For example, the number of 12-, 14-, 15-, and 16-coordinated atoms in a structure, say, 4, , and Ajg, is 20,32,8, and 0,... 

Crystal approximants. Several crystalline phases contain more or less closely packed atomic assemblies (polyhedra, clusters) which have been considered fundamental constituents of several quasicrystals, metal glasses and liquids. Such crystalline phases (crystal approximants), as reported in the previous paragraph, are often observed in the same (or similar) systems, as those corresponding to the formation of quasicrystals and under similar preparation conditions. Crystalline phases closely related to the quasicrystals (containing similar building blocks) have generally complex structures as approximants to the ico-quasicrystals we may, for instance, mention the Frank-Kasper phases (previously described in 3.9.3.1). 

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). 

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