Carbon nanotori

Fig. 34.10. Optimized stmctures of various types of carbon nanotori and their stability compared with graphene, fullerenes, and nanotubes. (Reproduced with permission from American Physical Society [147].)...

Carbon nanotori spontaneously self assemble, in the same type of experiments, which lead to carbon fUllerenes and nanotubes. We have shown a generalization of the Descartes-Euler formula, called the Euler-Poincare... 

It should be mentioned here that the very large size of these carbon-based materials precludes the use of high-level quantum methods. One has to therefore take recourse to the use of semi-empirical or tight-binding methods. As can be seen from Fig. 34.10, the stmctures and electronic properties of smallest nanotori exhibit interesting metal, semiconductor, and insulator characteristics depending on nanotube building blocks. 

Shape Analysis of Carbon Nanotubes, Nanotori and Nanotube Junctions... 

The Euler-Poincare formula invokes the use of Betti numbers [10] which may be calculated as the count of the number of critical points, of various types, associated with the geometrical structure of nanotori. The theory of Morse flmctions [11] relates critical points to topological structure. We shall show, an alternating sum of Betti numbers defines the Euler characteristic of a torus to be zero. This connects the topology of a nanotorus, nanotube, and plan sheet, which have the same Euler characteristic. We show that for every possible carbon nanotorus there is a geometrical dual boron nanotorus. 

The Euler-Poincare formula implies a duality which we have explored in the context of molecular nanotori. We show that for every possible carbon nanotorus there is a topological dual boron nanotorus. Moreover the alternating sum of Betti numbers is also zero for an open nanotube, and a planar network, which is consistent with the possibility of converting a nanotorus into an open nanotube, and thence into a planar network. 

Table VI. Carbon and Boron Dual Nanotori. Listed are the factors of the Euler-Poincare formula for a torus, viz., P - C + F = 0.

D. J. Klein and A. T. Balaban, Clarology for conjugated carbon nanostructures Molecules, polymers, graphene, defected graphene, fractal henzenoids, fullerenes, nanotuhes, nanocones, nanotori, etc.. Open Org. Chem. J. (Suppl. 1-M3) (2011) 27-61. 

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