Frank-Kasper

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... 

According to the Frank-Kasper definition, the coordination number is unambiguously 12 in the hexagonal close-packed metals and assumes the value 14 in a body-centred cubic metal. Generally in several complex metallic structures this definition yields reasonable values such as 14, even when the nearest-neighbour definition would give 1 or 2. 

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. 

Figure 3.18a. AET (According to Daams and Villars 1992,1993,1994,1997). The polyhedra corresponding to frequently observed AET are shown together with their codes. The Frank-Kasper (FK) polyhedra are indicated (see 3.9.3.1). Notice the same code 122 2 of the two polyhedra describing the cubic (c) as well as the hexagonal (h) atomic environments of the two ideal close-packed structures.

WoWj/2 the body-centred cubic structure of W (1 atom in 0, 0, 0 and 1 atom in A, A, /) corresponds to a sequence of type 1 and type 4 square nets at the heights 0 and A, respectively. Note, however, that for a fall description of the structure, either in the hexagonal or the tetragonal case, the inter-layer distance must be taken into account not only in terms of the fractional coordinates (that is, the c/a axial ratio must be considered). For more complex polygonal nets, their symbolic representation and use in the description, for instance, of the Frank-Kasper phases, see Frank and Kasper (1958) and Pearson (1972). 

An important contribution to the structure analysis of intermetallic phases in terms of the coordination polyhedra has been carried out by Frank and Kasper (1958). They described several structure types (Frank-Kasper structures) as the result of the interpenetration of a group of polyhedra, which give rise to a distorted tetrahedral close-packing of the atoms. Samson (1967, 1969) developed the analysis of the structural principles of intermetallic phases having giant unit cells (Samson phases). These structures have been described as arrangements of fused polyhedra rather than the full interpenetrating polyhedra. 

Tetrahedrally close-packed structures. Frank-Kasper structures. A... 

The four possible Frank-Kasper coordination polyhedra are included in Fig. 3.18 (and coded 12a, 14a, 15a, 16a) and correspond to the following properties ... 

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. 

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). 

Tetrahedrally close-packed phases, Frank-Kasper phases (cross-reference note)... 

In addition to the Frank-Kasper phases, other structures may be considered in which the same four types of coordination polyhedra prevail although some regularity is lost. A few notes about these phases are reported in 3.9.3.2. 

Figure 10.3 All ( 5,6, 3)-spheres that are 6Rq (all classical dual Frank - Kasper spheres), besides Dodecahedron...

We call the Frank-Kasper ( 5, >, 3)-map any ( 5, b, 3)-map that is bRo (in chemistry and crystallography Frank-Kasper polyhedra are just four polyhedra, dual to all four ( 5,6, 3)-polyhedra that are 6Rq). Note that the only oriented ( a, b], 3)-maps with a < 4 that is bRo are Prismb, Cube, Tetrahedron, and Bundle3. 

The classification of all Frank-Kasper ( 5, b, 3)-spheres can be done using the elementary polycycles decomposition exposed in Chapter 7. If G is a Frank-Kasper ( 5, b), 3)-sphere, then we remove all its Agonal feces and obtain a (5, 3)gen-polycycle. This (5, 3)ge -polycycle is decomposed into elementary (5,3)gen-polycycles along bridges (see Chapter 7 for definition of those notions). In this chapter a bridge is an edge, where the two vertices are contained in different Agonal feces. 

Theorem 13.23 (i) Forb < 11, the number of Frank-Kasper ( 5, b, 3)-polyhedra is finite they are (besides Dodecahedron) ... 

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