## Stokes and Anti Stokes Spectra in Resonant Raman Scattering

So far, almost all of the resonance Raman scattering experiments have been carried out on the Stokes spectra. The metallic window is determined experimentally as the range of ^laser over which the characteristic Raman spectrum for metallic nanotubes is seen, for which the most intense Raman component is at 1540 cm-1 [28]. Since there is essentially no Raman scattering intensity for semiconducting nanotubes at this phonon frequency, the intensity I1540 provides a convenient measure for the metallic window. The normalized intensity of the dominant Lorentzian component for metallic nanotubes I1540 (normalized to a reference line) has a dependence on Elaser given by

M2/4

X [(EjM(dt) - Elaser ± Ephonon)2 + r2/^-1, where d0 and A dt are, respectively, the mean diameter and the width of the Gaussian distribution of nanotube diameters within the SWNT sample, Ephonon is the average energy (0.197eV) of the tangential phonons and the + ( —) sign in (19) refers to the Stokes (anti-Stokes) process, re is a damping factor that is introduced to avoid a divergence of the resonant denominator, and the sum in Eq. (19) is carried out over the nanotube diameter distribution. Equation (19) indicates that the normalized intensity for the Stokes process IH40 (d0) is large when either the incident laser energy is equal to E^^) or when the scattered laser energy is equal to E^f (dt) and likewise for the anti-Stokes process. Since the phonon energy is on the same order of magnitude as the width of the metallic window for nanotubes with diameters dt, the Stokes and the anti-Stokes processes can be observed at different resonant laser energies in the resonant Raman experiment. The dependence of the normalized intensity /1540(d0) for the actual SWNT sample on Elaser is primarily sensitive [27,28,29] to the energy difference E^^) for the various dt values in the sample, and the resulting normalized intensity I"1540(d0) is obtained by summing over dt.

In Fig. 16 we present a plot of the expected integrated intensities /1540(d0) for the resonant Raman process for metallic nanotubes for both the Stokes (solid curve) and anti-Stokes (square points) processes. This figure is used to distinguishes 4 regimes for observation of the Raman spectra for Stokes and anti-Stokes processes shown in Fig. 17: (1) the semiconducting regime (2.19eV), for which both the Stokes and anti-Stokes spectra receive contributions from semiconducting nanotubes, (2) the metallic regime (1.58eV), where metallic nanotubes contribute to both the Stokes and anti-Stokes spectra, (3) the regime (1.92eV), where metallic nanotubes contribute to the Stokes spectra and not to the anti-Stokes spectra, and (4) the regime (1.49eV), where the metallic nanotubes contribute only to the anti-Stokes spectra and not to the Stokes spectra. The plot in Fig. 16 is for a nanotube diameter distribution dt = 1.49 ± 0.20 nm assuming 7e = 0.04 eV. Equation (17) can be used to determine y0 from the intersection of the Stokes and anti-Stokes curves at 1.69 eV in Fig. 16, yielding a value of y0 = 2.94 ± 0.05 eV [55,60].

Fig. 16. Metallic window for carbon nanotubes with diameter of dt = 1.49±0.20 nm for the Stokes (solid line) and anti-Stokes (square points) processes plotted in terms of the normalized intensity of the phonon component at 1540 cm-1 for metallic nanotubes vs the laser excitation energy for the Stokes and the anti-Stokes scattering processes [60]. The crossing between the Stokes and anti-Stokes curves is denoted by the vertical arrow, and provides a sensitive determination of 70 [55,60]

Fig. 16. Metallic window for carbon nanotubes with diameter of dt = 1.49±0.20 nm for the Stokes (solid line) and anti-Stokes (square points) processes plotted in terms of the normalized intensity of the phonon component at 1540 cm-1 for metallic nanotubes vs the laser excitation energy for the Stokes and the anti-Stokes scattering processes [60]. The crossing between the Stokes and anti-Stokes curves is denoted by the vertical arrow, and provides a sensitive determination of 70 [55,60]

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