Il A

where A = 0,1,..., d - 1 and a = [p(m + 2 n) + q(n + 2 m)]{d/dR) (19.13)

in which the integers (p, q) determine the vector R (see Fig. 19.24). Thus (p, q) denotes the coordinates reached when the symmetry operation (i//|r) acts on an atom at (0,0), i.e., (i/f|r)(0,0) = (p, q).

If (i/Hr) is a symmetry operation for the tubule, then (i/*|r)2, («/>|r)3,... (i/*|r)JV/d are all distinct symmetry operations where {<p\t)n'd = E is the identity operation, bringing the lattice point O to an equivalent lattice point B", where

Referring to Fig. 19.24, we see that after N/d symmetry operations (ip\r), the vector (N/d)R along the zigzag direction reaches a lattice point, which we denote by B" (not shown in figure). Correspondingly, after N/d symmetry operations (ifi\r)Nld, the translation (N/d)r yields one translation of the lattice vector T of the tubule and (il/d2) revolutions of 2tt around the tubule axis. Although il/d is an integer, il/d2 need not be. In Table 19.2 we list the characteristic parameters of carbon tubules specified by (n, m), including d, the highest common divisor of n and m, and the related quantity dR given by Eq. (19.8). Also listed in Table 19.2 are the tubule diameter d, in units of A, the translation repeat distance T of the ID lattice in units of the lattice constant a0 — V3ac_c for a 2D graphene sheet, the number N

Table 19.2

Values for characterization parameters" for selected carbon nanotubes labeled by

Table 19.2

Values for characterization parameters" for selected carbon nanotubes labeled by

(n,m)

d

d*

d,( A)

L/a0

T/a0

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