Fullerenes: Topology and Structure
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The discovery of fullerenes has disclosed a new wide area of research in fundamental condensed matter physics, as well as in chemistry and materials science, confirming carbon as the most versatile element of nature. Fullerenes constitute a family of cage-like carbon molecules where each carbon atom is threefold coordinated and forms covalent sp2 bonds with the three nearest neighbors like in a single graphite sheet. The atoms of fullerenes lie on a closed surface topologically equivalent to a sphere. The actual shapes of fullerene clusters are polyhedra whose structural features can be largely understood from pure topological arguments. The number N of atoms of a fullerene molecule vary from a minimum of 20 to indefinite large values, presumably in the range of several thousands, although the far most stable fullerene is made of 60 atoms (C60) and takes the shape of a soccer ball, otherwise known as a truncated icosahedron. The shape of this highly symmetric polyhedron, having 60 equivalent vertices that can be generated by the 60 operations of the icosahedral point group, was well known to ancient Greek mathematicians, notably to Archimedes who described the 13 possible semiregular polyhedra. Semiregular polyhedra, otherwise known as the Archimedean polyhedra, are made of two different kinds of regular polygons and are obtained by vertex truncation of the five regular (Platonic) polyhedra. The first known representations of the truncated icosahedron have been given by the celebrated Renaissance artists and scholars Piero della Francesca and Leonardo da Vinci.
In this article, the structural and topological features of graphite-like carbon are described, with special emphasis on the novel families of graphene materials that have been discovered and thoroughly investigated after the first identification of the fullerenes C60 and C70. In “Historical Background,” an overview is given on the historical development and the scientific and technical motivations of studies on the different forms of graphite-like carbon. The topology of graphene structures is presented in “Graphene Topology.” This section contains an elementary introduction to the general topological properties of graphenes, which allows for a classification of the graphene families in terms of topological connectivity. This is followed by four subsections describing the main graphene families: fullerenes, nanotubes, schwarzites, and amorphous sp2 carbon. “Topology vs. Total Energy and Growth Processes” shows that the energetics and the growth processes of graphene structures can also be related to and, to some extent, understood from general topological arguments.