Half-band width definition

The half-widths of the second harmonic bands 2vd and 2vg are 56 and 64 cm, respectively (Fig. 7.9a, b). This is practically the same frequency shift as observed for the maxima of these bands relative to similar bands observed in the single crystalline graphite sample (Fig. 7.8a, b). The half-widths observed for the first-order vibration bands v and Vg are 56 and 21 cm correspondingly. The too narrow width of the observed tone bands does not allow for a more definite conclusion as concerning the appearance of cooperative effects in the vibration modes of the SWCNT. Nevertheless, a significant role of nonlinear interactions of their excitations shows the importance of their wave properties for the Raman spectra interpretation. 

We assume that (i) the 960 cm band does not saturate for the TS-1 sample with lowest Ti content and (ii) the true Full Width at Half Maximum (FWHM) of this band is constant over all set of TS-1 samples. The former assumption is supported by the fact that the intensity of the 960 cm band, for the x = 1.0 sample, is less than 1.5 in absorbance units, while the latter is exactly what expected for a band associated to different concentrations of a unique species. Assumption (i) allows us to measure the true FWHM of the 960 cm band in TS-1 samples (27 cm ). Now, following assumption (ii), we were able to estimate for all samples reported in Fig. 3, the absorbance (W) at the height where the width of the band corresponds to 27 cm. This method minimize the errors due to the instrument sensibility and allows a quantitative estimation of the band intensity. Such obtained values plotted against x in the Fig. 3d ( data), give a high linear correlation (r = 0.9998) and validate assumption (i) and (ii). A further, and definitive, validation comes from the similar linearity found for the intensities of the Raman band, which can not be affected by saturation problems, see Fig. 3d ( data). These results implies that the 960 cm band is well a fingerprint of the insertion of Ti in the zeolitic framework, even if not... 

It follows from the above discussions that the spectral width of the light emitted by the atoms should be infinitely small. Correspondingly, the atoms should absorb only light of a definite frequency. Actually, a band of intensity distribution close to the Gaussian (Figure 2) is emitted or absorbed, respectively. The profile of the line is characterized by the central frequency Vo, the peak intensity Ip, and the frequency distribution with a width of Aveff, generally quoted as the full-width at half-maximum Ip/2 (FWHM). 

For the constant C, the half width of the band is taken into consideration. The following two definitions are used for the half width (Fig. 2-5) ... 

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