E, e2

inversion group containing E and i. In addition to the identity class E, the classes of Dn in Table 19.3 constitute the nth roots of unity where ±<f>j rotations belong to the same class, and in addition there is a class (2j + 1 )C2 of n twofold axes at right angles to the main symmetry axis ,. Thus, Table 19.3 yields the character tables for D„d, or D5d, Dld, ... lor sym-morphic tubules with odd numbers of unit cells around the circumference [(5,5), (7,7), ... armchair tubules, and (9,0), (11,0), ... zigzag tubules]. Likewise, the character table for Dnh, or D6h, DSh, ... for even n is found from Table 19.4 by taking the direct product Dn ® i = Dnh. Table 19.4 shows two additional classes for group D2j because rotation by tt about the main symmetry axis is in a class by itself. Also the 2j twofold axes nC'2 form a class and are in a plane normal to the main symmetry axis C^ , while the nC2 dihedral axes, which are bisectors of the nC2 axes, also form a class for group Dn when n is an even integer. Correspondingly, there are two additional one-dimensional representations (5, and B2) in D2j, since the number of irreducible representations equals the number of classes. Table 19.5 lists the irreducible representations and basis functions for the various ID tubule groups and is helpful for indicating the symmetries that are infrared active (transform as the vector x, y, z) and Raman active (transform as the symmetric quadratic forms).

The symmetry groups for carbon nanotubes can be either symmorphic (as for the special case of the armchair and zigzag tubules) or nonsymmorphic for the general case of chiral nanotubes. For chiral nanotubes the chiral angle in Eq. (19.3) is in the range 0 < 9 < 30° and the space group operations (i^lr) given by Eqs. (19.11) and (19.12) involve both rotations and translations, as discussed in §19.4.1. Figure 19.24 shows the symmetry vector R which determines the space group operation (<//|r) for any carbon nanotubule specified by (n,m). From the symmetry operation R = (t//|r) for tubule (n, m), the symmetry group of the chiral tubule can be determined. If the tubule is considered as an infinite molecule, then the set of all operations R', for any integer j, also constitutes symmetry operators of the group of the tubule. Thus from a symmetry standpoint, a carbon tubule is a one-dimensional crystal with a translation vector T along the cylinder axis and a small number of carbon hexagons associated with the circumferential direction [19.82,86,87].

The symmetry groups for the chiral tubules are Abelian groups. All Abelian groups have a phase factor e, such that all h symmetry elements of the group commute, and are obtained from any symmetry element by multiplication of e by itself an appropriate number of times, such that eh = E, the identity element. To specify the phase factors for the Abelian group, we introduce the quantity ft of Eq. (19.13), where il/d is interpreted as the number of 2ir rotations which occur after N/d rotations of ip. The phase factor e for the Abelian group for a carbon nanotube then becomes e = cxp(2iriil/N) for the case where (n, m) have no common divisors (i.e., d = 1). If ft = 1 and d = 1, then the symmetry vector R in Fig. 19.24 reaches a lattice point after a 2v rotation. As seen in Table 19.1, many of the actual tubules with d = 1 have large values for ft; for example, for the (6,5) tubule, ft = 149, while for the (7,4) tubule ft = 17, so that many 2tt rotations around the tubule axis are needed to reach a lattice point of the ID lattice.

The character table for the Abelian group of a carbon nanotube is given in Table 19.6 for the case where (n, m) have no common divisors and is labeled by CN/a. The number of group elements is N, all symmetry elements commute with each other, and each symmetry operation is in a class by itself. The irreducible representation A in Table 19.6 corresponds to a 27t rotation, while representation B corresponds to a rotation by tt. Except for the irreducible representations A and B, all other irreducible representations are doubly degenerate. The E, representations correspond to two levels which stick together by time reversal symmetry, for which the corresponding eigenvectors are related to one another by complex conjugation. Since N can be quite large, there can be a large number of symmetry operations in the group CN/n, all of which can be represented in terms of a phase factor e = exp(iij/) = exp(2mfl/N) and the irreducible represen-

The character table for the group Cv/a for chiral nanotubes, where N and Q have no common divisor, corresponding to (n, m) having no common divisor."

The character table for the group Cv/a for chiral nanotubes, where N and Q have no common divisor, corresponding to (n, m) having no common divisor."

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