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Icosahedral symmetry breaking: \(\mathrm C_{60}\) to \(\mathrm C_{78}\), \(\mathrm C_{96}\) and to related nanotubes. (English) Zbl 1341.82120

Summary: Exact icosahedral symmetry of \(\mathrm C_{60}\) is viewed as the union of 12 orbits of the symmetric subgroup of order 6 of the icosahedral group of order 120. Here, this subgroup is denoted by \(A_2\) because it is isomorphic to the Weyl group of the simple Lie algebra \(A_2\). Eight of the \(A_2\) orbits are hexagons and four are triangles. Only two of the hexagons appear as part of the \(\mathrm C_{60}\) surface shell. The orbits form a stack of parallel layers centered on the axis of \(\mathrm C_{60}\) passing through the centers of two opposite hexagons on the surface of \(\mathrm C_{60}\). By inserting into the middle of the stack two \(A_2\) orbits of six points each and two \(A_2\) orbits of three points each, one can match the structure of \(\mathrm C_{78}\). Repeating the insertion, one gets \(\mathrm C_{96}\); multiple such insertions generate nanotubes of any desired length. Five different polytopes with 78 carbon-like vertices are described; only two of them can be augmented to nanotubes.

MSC:

82D80 Statistical mechanics of nanostructures and nanoparticles
81R40 Symmetry breaking in quantum theory
20F55 Reflection and Coxeter groups (group-theoretic aspects)
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