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\section{Analysis} |
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Frank first proposed icosahedral arrangement of atoms as a model for structure supercooled atomic liquids.\cite{19521106} The ability to cool simple liquid metals well below their equilibrium melting tempatures was attributed to this icosahedral local ordering. Frank further showed that a 13-atom icosahedral cluster has a 8.4\% higher binding energy the either a face center cubic or hexagonal close packed crystal structure. Icosahedra also have six fivefold symmetry axes that cannot be extended indefinitely in three dimensions making them incommensurate with long-range positional crystallographic order. This does not preclude icosahedral clusters from posessing long-range orientational order. The "frustrated" packing of these icosahedral structures into dense clusters has been proposed as a model for glass formation.\cite{19871127} The size of the icosahedral clusters increase until frustration prevents any further growth near the glass .\cite{HOARE:1976fk} Molecular Dynamics calculations of a Lennard-Jones binary glass shows that a two component glass has clusters of face-sharing icosahedra that are distributed throughout the material.\cite{PhysRevLett.60.2295} |
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Frank first proposed icosahedral arrangement of atoms as a model for structure supercooled atomic liquids.\cite{19521106} The ability to cool simple liquid metals well below their equilibrium melting temperatures was attributed to this icosahedral local ordering. Frank further showed that a 13-atom icosahedral cluster has a 8.4\% higher binding energy the either a face center cubic or hexagonal close packed crystal structure. Icosahedra also have six fivefold symmetry axes that cannot be extended indefinitely in three dimensions making them incommensurate with long-range positional crystallographic order. This does not preclude icosahedral clusters from possessing long-range orientational order. The "frustrated" packing of these icosahedral structures into dense clusters has been proposed as a model for glass formation.\cite{19871127} The size of the icosahedral clusters increase until frustration prevents any further growth near the glass .\cite{HOARE:1976fk} Molecular Dynamics calculations of a Lennard-Jones binary glass shows that a two component glass has clusters of face-sharing icosahedra that are distributed throughout the material.\cite{PhysRevLett.60.2295} Molecular Dynamics simulations of freezing of single component metalic nanoclusters have shown a tendency for icosohedral structure formation particularly at the surface.\cite{Gafner:2004bg,PhysRevLett.89.275502,Ascencio:2000qy,Chen:2004ec} |
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Various structural probes have been used to characterize structural order in systems including common neighbor analysis, voronoi-analysis and orientational bond-order parameters.\cite{HoneycuttJ.Dana_j100303a014,Iwamatsu:2007lr,hsu:4974,nose:1803} Experimentally, the splitting (or shoulder) on the second peak of the X-ray structure factor in binary metal glasses has been attributed to the formation of face-sharing tetrahedra.\cite{Waal:1995lr} These tetraherda form structural units that are linked by sharing of an icosohedron creating face sharing icosohedron linked by tetrahedral structures. |
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One method of analysis that has been used extensively for determining local and extended orientational symmetry of a central atom with its surrounding neighbors is that of bond-orientational analysis as formulated by Steinhart et.al.\cite{Steinhardt:1983mo} |
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In this model, a set of spherical harmonics is associated with its near neighbors as defined by the first minimum in the radial distribution function forming an association that Steinhart et.al termed a "bond". More formally, this set of numbers is defined as |
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\begin{equation} |
