Alternative approaches to onion-like icosahedral fullerenes

The fullerenes of the C60series (C60, C240, C540, C960, C1500, C2160etc.) form onion-like shells with icosahedralIhsymmetry. Up to C2160, their geometry has been optimized by Dunlap & Zope from computations according to the analytic density-functional theory and shown by Wardman to obey structural constraints derived from an affine-extendedIhgroup. In this paper, these approaches are compared with models based on crystallographic scaling transformations. To start with, it is shown that the 56 symmetry-inequivalent computed carbon positions, approximated by the corresponding ones in the models, are mutually related by crystallographic scalings. This result is consistent with Wardman's remark that the affine-extension approach simultaneously models different shells of a carbon onion. From the regularities observed in the fullerene models derived from scaling, an icosahedral infinite C60onion molecule is defined, with shells consisting of all successive fullerenes of the C60series. The structural relations between the C60onion and graphite lead to a one-parameter model with the same Euclidean symmetryP63mcas graphite and having ac/a= τ2ratio, where τ = 1.618… is the golden number. This ratio approximates (up to a 4% discrepancy) the value observed in graphite. A number of tables and figures illustrate successive steps of the present investigation.

AFFINE EXTENSION OF SUPERGRAVITY

The affine extension of supergravity is investigated. We discuss only the loop algebra since the affine extension of the super-Poincaré algebra appears inconsistent. The construction of the affine supergravity theory has been carried out by the group manifold method and leads to an action describing infinite towers of gravitons and gravitinos that interact subject to the symmetries of the loop algebra. The equations of motion satisfy the usual consistency check.

AFFINE EXTENSION OF SUPERSYMMETRIC FIELD THEORY

It is shown that there is no obstacle to the affine extension of supersymmetric Yang-Mills theories. This result holds equally for loop and Kac-Moody algebras. It also holds for 4, 6, and 10 dimensions. By dimensional reduction of the 10-dimensional theory one obtains a supersymmetric action describing interacting Kac-Moody fields in four dimensions.