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For the case of a (6, 6) armchair nanotube 72 phonon branches are expected and Eq. (2) gives a total of 48 distinct mode frequencies. Appropriate character tables can be used to determine which modes are Raman active. Thus, it can be easily demonstrated that there are 16 Raman active frequencies (4A1g + 4E1g and 8E2g). Notice that the number of active modes depends just on the SWNT chirality but is independent of nanotube diameter.

Detailed characterization of SWNTs by Raman spec-troscopy started with the initial joint efforts of Dresselhaus et al. and Eklund et al. [86], who obtained the Raman spectra of SWNT bundles using several laser excitation energies. They reported a strong enhancement of the Raman intensity due to the diameter-selective resonance Raman effect. This work confirmed the theoretical predictions of the diameter-selective Raman scattering that is a particularly important for the Raman bands associated with the A1g radial breathing mode (RBM) [75, 87-89]. In fact, the presence of the RBM feature in the Raman spectrum is used today as a signature of the presence of SWNTs in the sample. Variations in RBM band shape and position provide information on the nanotube diameters present in a given sample.

Following the seminal work of Dresselhaus et al., the unusual response of this line to changes in excitation energy has been discussed in many articles. Ab initio calculations confirmed the inverse proportionality of the A1g mode frequency with the diameter and yielded an appropriate proportionality factor [90, 91]. The RBM band appears in the Raman spectrum below 300 cm-1 and has been proposed to be independent of chirality [92]. Bandow et al. calculated the RBM frequencies of all types of SWNTs and found that all frequencies fall on a common line according to the expression [93]

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