Constraints on Small Fullerene Helices
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be ignored and the tubule considered to be a straight cylinder made entirely of hexagons having helical, rotational, and translational symmetry [6,7]. A tubule can be bent. If the bend is sufficiently large, the strain energy can be relieved through rebonding individual carbon atoms. By Euler's theorem the number of bonds between three-fold coordinated carbon atom is preserved if the total number of extra sides in all the polygons larger than hexagons equals the number of deficient sides in all the polygons smaller than hexagons. The minimal such transformation of a tubule is caused by a single heptagon and a single pentagon [8,9]. The optimal position of the pentagon is to supply positive curvature at the outside of the bend. The optimal position of the heptagon is to supply negative curvature at the inside of the bend. In order to introduce the heptagon and pentagon, the helicity of an entire half of the tubule must be altered; the heptagon and pentagon can only connect tubules of compatible helicities [10]. Thus the helicities of every second tubule segment must be identical. Each tubule segment must have roughly the same diameter, because it costs energy to flatten the larger tubule to have an oval cross section. The heptagon and pentagon defects are located far apart on opposite sides of the tubule. The twist angle between successive tubule bends can only assume a finite number of values [11]. These constraints could drive the high regularity seen in nanoscale graphitic helices [121. In fact a continuum model of differential creation rates has been described that would drive helix formation [13].
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Mat. Res. Soc. Symp. Proc. Vol. 359 01995 Materials Research Society
RESULTS AND DISCUSSION The helicity of a tubule can be defined as the smallest angle that its circumference makes to any line of edge-sharing rows of hexagons on its surface. It is possible that tubules of different helicities could twist different amounts due to differences in electronic structure and differences in relative curvature of the three symmetry-inequivalent bonds. To be independent of twist angle, it is better to define the topological helicity as the minimal number of steps along one or two different rows of edge-sharing hexagons necessary to return to the same or another point on the circumference. This definition of helicity can be reduced to a set of two nonnegative numbers that have no common factors other than unity. Tubules can be indicated by the lattice vector [L, M] of points on a graphene sheet that when rolled up to overlap create the tubule [14]. The set of all tubules of a given helicity axe [ni, nj] where (i,j) is the helicity of the class and n is a positive integer. The axis of the nth member of each class is an n-fold rotational axis, thus the tight-binding electronic structures of the members of each tubule class are related, neglecting the effects of curvature [7]. The members of a tubule class have uniformly spaced circumferences and radii. Thus each tubule class has a density of radii, which is the ratio of
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