RELATING CARBON TUBULES

Two methods for relating different carbon tubules are described. A sin gle graphene ribbon of infinite length but finite width can be coiled to form an infinite set of tubules, each of which has a unique pitch o r helicity. This maps in a one-to-one fashion the translation/rotation operations of the group of each tubule in the set. Within the set all irreducible representations are collected into the same number of ban ds. Alternatively, tubules can be imagined to be pressed flat so that centers of six-member rings lie along the crease. The direction of the ir creases on a graphene sheet relate tubules having the same helicity but different numbers of identical rotationally symmetric subunits ar ound their circumference. These sets help to reconcile the different e xpressions for band structure of tubules. These sets also help sort th e various ways to join semi-infinite tubules. A perfect infinite tubul e is composed entirely of hexagons. Adding one heptagon and one pentag on transforms half of the tubule into a different tubule. The two ways to group tubules suggest that the least distortion of neighboring hex agons occurs if the heptagon and pentagon are joined together or are s eparated to opposite sides of the tubule. In the latter case, the tubu le could be imagined to be flattened so that the heptagon and pentagon are folded in half, one along each crease. This heptagon-pentagon def ect best connects sets of tubules in a pairwise fashion. The paired se ts of tubules have axis vectors that meet at a 30-degrees angle on a g raphene sheet. This analysis and experimental considerations suggest t hat the ideal bend in a tubule caused by a heptagon-pentagon pair is l ikely to be 30-degrees. Because entire sets of tubules are joined in s imilar fashion, tailoring of tubule electronic properties can be imagi ned.