Toroidal forms of graphitic carbon. II. Elongated tori

An elongated toroidal cage form of graphitic carbon ${\mathrm{C}}_{240}$ that has an elliptical cross section is proposed. The elongated toroidal form consists of pairs of pentagons and heptagons among the hexagons of carbon atoms. The rotational symmetry, the cohesive energies, and thermal stabilities of the elongated toroidal forms derived from ${\mathrm{C}}_{240}$, as well as other various toroidal forms are studied by molecular dynamics. It is found that the sevenfold rotational symmetry is the most stable for the elongated toroidal forms derived from ${\mathrm{C}}_{240}$, although both fivefold and sixfold rotational symmetries are the most stable for the toroidal forms derived from the tori ${\mathrm{C}}_{360}$ previously proposed by us and from ${\mathrm{C}}_{540}$ proposed by Dunlap. This indicates that five and seven pentagon (on the outer rim) -heptagon (on the inner rim) pairs might also appear in the turnover edge of carbon nanometer-sized tubes, even though only hexagonal pairs are experimentally observed by transmission electron microscopy. The distortion energies of the elongated tori are estimated from the cohesive energies of the series of the tori ${\mathrm{C}}_{\mathit{n}}$, where n ranges from 80 to 1680. The proposed structures are found to be thermodynamically stable by the finite-temperature simulations.