where the subscripts ID and 2D refer, respectively, to the one-dimensional tubule and the two-dimensional graphene sheet, k is a wave vector in the direction K2 along the reciprocal space ID periodic lattice of the tubule, (i is a nonnegative integer used to label the wave vectors or the states along the K} reciprocal space direction normal to the tubule axis, and N, given by Eq. (19.9), denotes the number of hexagons in the ID unit cell of the tubule. The result given by Eq. (19.39) includes both symmorphic and nonsymmorphic space groups.

In this section, we first present explicit dispersion relations for armchair and zigzag carbon nanotubes corresponding to the symmorphic space groups. We then extend the discussion to chiral nanotubes corresponding to nonsymmorphic space groups. In this discussion, we assume the length of the tubules is so much larger than their diameters that the tubules can be described in the ID limit where the nanotubes have infinite length and the contributions from the carbon atoms in the caps can be neglected. In this discussion, we first enumerate the symmetry types of the various vibrational modes of the tubules and in the next subsection (§19.7.2) we comment on their optical activity.

The appropriate symmetry groups for the symmorphic armchair tubules (n — m) are Dnh or Dnd, respectively, depending on whether n is even or odd [19.4]. For armchair tubules with Dnh symmetry (n is even), the vibrational modes are decomposed according to the following irreducible representations (assuming that n/2 is even):

+ 4 E2u + --. + 4 EW2_i)g + 8£(„/2_I)„. (19.40)

If n/2 is odd [such as for (n, m) = (6,6)], the 4 and 8 are interchanged in the last two terms in Eq. (19.40). The vibrational mode which describes rotation about the cylindrical axis has A2g symmetry; the translation mode along the cylinder axis has A2u symmetry, while the corresponding translations along the directions perpendicular to this axis have Elu symmetry. For example, the (6,6) armchair tubule for which N = 12 has 12 hexagons in the ID unit cell and is described by 72 phonon branches. From Eq. (19.40) we see that the (6,6) tubule has 48 distinct mode frequencies at the Bril-louin zone center k = 0, while the (8,8) armchair tubule has ¿V = 16 and 96 phonon branches with 60 distinct mode frequencies at k = 0. The modes that transform according to the Alg, Elg, or E2g irreducible representations are Raman active, while those that transform as Alu or Elu are infrared active. Hence there are 16 Raman-active mode frequencies (4Ais + 4Elg + 8£2g) and 8 distinct infrared-active nonzero frequencies (Alu + 7Elu) for tubules with Dnh symmetry (« even). As an example, of the 48 distinct mode frequencies for the (6,6) armchair tubule, 8 are IR active, 16 are Raman active, 2 have zero frequency, and 22 are silent.

Although the number of vibrational modes increases as the diameter of the carbon nanotube increases, the number of Raman-active and infrared-active modes remains constant. This is illustrated by comparing the normal modes of the (8,8) and the (6,6) armchair tubules. For the 60 distinct frequencies of the (8,8) armchair tubule, only 8 are IR-active modes, 16 are Raman-active modes, 34 are silent modes, and 2 have zero frequency, yielding the same number of Raman-active and infrared-active modes as for the (6,6) tubule. The 4EJg + 8£3h modes, present in the (8,8) tubule but absent in the (6,6) tubule, are all silent modes. The constant number of Raman- and infrared-active phonon tubule modes for a given symmetry group, or for a given chirality, is common to each category of carbon nanotube discussed below.

For armchair tubules with Dnd symmetry (n = odd integer), the vibrational modes of the carbon nanotube contain the following symmetries:

As mentioned above, the zone center k = 0 vibrational frequencies are zero for one A2u mode and for one Elu mode, corresponding to the acoustic branches at wave vector k = 0. We therefore have 15 Raman-active modes and 7 infrared-active modes with distinct nonzero frequencies for armchair tubules with Dnd symmetry. As an example of a tubule with Dnd symmetry, the (5,5) armchair tubule has N — 10, and 60 degrees of freedom per ID unit cell, 12 nondegenerate phonon branches, 24 doubly degenerate phonon branches, with 7 nonvanishing IR-active mode frequencies, 15 that are Raman active, 2 that are of zero frequency and 12 silent mode frequencies, thereby accounting for the 36 distinct phonon branches.

To illustrate the phonon dispersion relations for carbon nanotubes, we show in Fig. 19.51 explicit results for the 36 phonon dispersion relations for

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