"il is given by Eq. (19.13); T is given by Eq. (19.7) with a0 = V3ac_c\ L is the length of the chiral vector ird, given by Eq. (19.2); N is given by Eq. (19.9); ip is given by Eq. (19.12); r is given by Eq. (19.11); Cl/d is the number of complete 2ir revolutions about the tubule axis to reach a lattice point.

"il is given by Eq. (19.13); T is given by Eq. (19.7) with a0 = V3ac_c\ L is the length of the chiral vector ird, given by Eq. (19.2); N is given by Eq. (19.9); ip is given by Eq. (19.12); r is given by Eq. (19.11); Cl/d is the number of complete 2ir revolutions about the tubule axis to reach a lattice point.

of hexagons per unit cell of the ID tubule, the rotation angle ip expressed in units of Ch/2ir, and the translation r of the symmetry operation R = («//1 -r) expressed in units of a0. The length of the chiral vector L (L = Ch = ird,) is listed in Table 19.2 in units of a0. For cases where n and m have a common divisor d, the angle of rotation ip is defined modulo lir/d instead of 2tt, so that the group C'N/n has N/d elements. To illustrate use of Table 19.2 we refer to the (4,2) tubule illustrated in Fig. 19.2(a), which has N = 28, d = dR = 2, and T = 14r = V2la0 and also Ch = V28a0. Thus for the (4,2) tubule, {N/d)\p = (5/2)(27r) corresponds to 5tt rotations around the tubule axis, and (N/d) translations R reach a lattice point.

As a second example, we consider the case of tubule (7,4) for which the phonon spectrum has been calculated (see §19.7) [19.83], For this tubule, there are no common divisors, so d = 1, but since n — m = 3, we obtain dR = 3. Solution of Eq. (19.10) for (p,q) yields p = 2 and q = 1, which yields the rotation angle (in units of 27t) Aft/f = il/d = 17(27t), where N - 62. Thus, after 62 operations (*P\t)n, the origin O is transformed into a new lattice point a distance T from O along the T axis, after having completed 17 rotations of 2tt around the tubule axis. For the armchair tubule (5,5), the highest common divisor is 5, and since n — m = 0, we have dR = 3 x 5 = 15, yielding N — 10 and Nijj = 2v. Regarding translations, use of Eq. (19.11) yields r = a0/2 — T/2. Finally, we give the example of the smallest zigzag tubule (0,9), for which d — dR = 9 and N — 18. Using p = 1 and q = 0 and N = 18, we obtain Aty = 2v and t = v3a0/2 = T/2.

In this section we consider the symmetry properties of the highly symmetric armchair and zigzag carbon tubules, which can be described by symmorphic groups, and we then summarize the symmetry operations for the general chiral tubule.

For symmorphic groups, the translations and rotations are decoupled from each other, and we can treat the rotations simply by point group operations. Since symmorphic groups generally have higher symmetry than the nonsymmorphic groups, it might be thought that group theory plays a greater role in specifying the dispersion relations for electrons and phonons for symmorphic space groups and in discussing their selection rules. It is shown in §19.4.3 that group theoretical considerations are also very important for the nonsymmorphic groups, leading to important and simple classifications of their dispersion relations.

In discussing the symmetry of the carbon nanotubes, it is assumed that the tubule length is much larger than its diameter, so that the tubule caps, which were discussed in §19.2.3, can be neglected when discussing the electronic and lattice properties of the nanotubes. Hence, the structure of the infinitely long armchair tubule (n = m) or zigzag tubule (m = 0) is described by the symmetry groups Dnh or Dnd for even or odd n, respectively, since inversion is an element of Dnd only for odd n and is an element of D„h only for even n [19.84]. Character tables for groups Dsh and DSd are given in §4.4 under Tables 4.12 and 4.13. We note that group Dsd has inversion symmetry and is a subgroup of group Ih, so that compatibility relations can be specified between the lower symmetry group D5d and Ih, as given in Table 4.15. In contrast, D5h is not a subgroup of I h, although Ds is a subgroup of I. Character tables for D6h and D6d are readily available in standard group theory texts [19.85], For larger tubules, appropriate character tables can be constructed from the generalized character tables for the D„ group given in Table 19.3 (for odd n = 2j + 1) and in Table 19.4 (for even n = 2j), and the basis functions are listed in Table 19.5. These tables are adapted from the familiar character table for the semi-infinite group Dooh [19.85], and 91 denotes the irreducible representations.

The character table for group Dnd for odd integers n — 2j + 1 is constructed from Table 19.3 for group D„ [or group (n2) in the international notation] by taking the direct product Dn <8> i where i is the two element

Table 19.3

Character table for point group D(

Table 19.3

Character table for point group D(

m |
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