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Fig. 19.54. Dependence of the Raman-active (7,4) chiral nanotube mode frequencies on tubule diameter d, [19.83,87], which is found from the (n,m) index pairs by Eq. (19.2). The value of d, for the (n,m) = (7,4) nanotube is 7.56 A, and for (n,m) = (28,16) is 4(7.56) = 30.24 A.

Finally, we show in Figs. 19.54 and 19.55 the dependence on tubule diameter d, of the Raman-active and infrared-active modes for a (7,4) chiral nanotube. This nanotube belongs to the category n — m = 3, which is a conducting nanotube, with d = 1, dR = 3, d, = 7.56 A, and 6 = tan-1 [2-s/3/9] (see Table 19.2). Of interest is the difference of the spectral frequencies for nanotubes of different («, m) values and particularly the absence of modes in the 1350 cm-1 region in Fig. 19.54.

Although the number of Raman-active and infrared-active modes is basically the same for the symmorphic and nonsymmorphic carbon nanotubes, Figs. 19.54 and 19.55 show more bunching of the optically active mode frequencies in comparison to the behavior for the symmorphic nanotubes shown in Figs. 19.52 and 19.53. This bunching effect arises from the very large real space unit cell of the ID lattice for the chiral nanotubes, which results in a very small unit cell in reciprocal space. For example, the (7,4) nanotube has N — 62, with 372 phonon branches. The zone folding of the graphene reciprocal space unit cell into the very small reciprocal space unit cell of the nanotube gives rise to many zone foldings so that the frequen-

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