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77-30569/330069 Constructions of algebraic geometry and simple polyhedra which are not stereohedra for clathrates and have 24 vertices and 14 faces
Engineering Education # 03, March 2012
The authors show the existence of special polyhedra (clusters) and combinations of polyhedral, which are the structural implementation of algebraic geometry constructions. An algorithm of graph derivation corresponding to a special class of simple polyhedra with 24 vertices and 14 faces was considered for the case when these polyhedra have 4-, 5- and 6-gonal faces. The graph was derived for the first time for the polyhedron with one tetragonal face and which is not a parallelohedron. Polyhedra of this class allowed to determine the possibility for 14-verticed canonic and non-canonic Frank-Kasper polyhedra to be presented in the structure of intermetallics. This approach allowed to determine the symmetry of wide spectrum of ordered (not necessary crystalline ones) tetra-coordinated (diamond-like, clathrates) and tetrahedral (metallic) structures larger, including also nanostructured states.
77-30569/327112 Ordered structures of clathrates as combinations of special helicoids
Engineering Education # 02, February 2012
The symmetry of a clathrate crystal can be adequately displayed within the apparatus based on constructions. The apparatus of algebraic geometry and topology was proposed for fundamental description of tetrahedral and tetra-coordinated structures. Such a description allows to represent any structure as a combination of helices (rods) consisting of polyhedrons i.e. cavities of clathrates or coordination polyhedrons of intermetallic semiconductors. The crystal lattice can arise only as a variant of such a combination. Crystal structures of clathrates (gas-hydrates and intermetallic semiconductors) are described as a combination of helicoids with nonintegral swivelling axes 10/3, 12/5 и 15/4. The structure of clathrate IX (Ba24Ge100) for the first time was determined as dual to the β-Mn structure.
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