Harmonic band

Figure 10.4—Vibrational energy levels of a bond, a) For isolated molecules b) For molecules in the condensed phase. The transition from V — 0 to V = 2 corresponds to a weak harmonic band. Because of the photon energy involved in the mid IR, it can be calculated that the first excited state (V = 1) is 106 times less populated than the ground state. Harmonic transitions are exploited in the near IR.

The case of three and four electrons is more complicated, but the two characteristic features of the energy spectra observed for small coz, i.e., the nearly-degenerate multiplet structure of the energy levels of different spin multiplicities and the harmonic band structure of these levels, can be rationalized in a similar way. In the case of three electrons, for example, the internal space can be defined by the two correlated coordinates Zb and zc defined by Equation (11). The potential function becomes a sum of two harmonic-oscillator Hamiltonians for the Zb and zc coordinates plus three Coulomb-type potentials originating from the three electron-electron... 

Since many atmospheric pollutants such as ozone have absorption bands in the ultraviolet, measurements have also been performed in this spectral region. Recently, it was shown that phase locking within the filaments results in enhanced third harmonic generation [46]. Then, the build-up of the ultraviolet supercontinuum was characterized over both the laboratory and the atmospheric scales. The UV-visible part of the continuum measured in the laboratory with a single filament is presented in Fig. 15.7. At the beginning of filamentation, a third harmonic band with 20 nm bandwidth is generated around 270 nm. Two meters further, the intensity of the third harmonic is reduced and a plateau appears in the UV-Visible region be-... 

In the region of the second-order bands, a weak sum harmonic (i.e., Vd+Vg band) is observed (Fig. 7.6d) in addition to 2vd and 2vq harmonics. As the frequency of the laser excitation radiation Vl increases, the intensity of the harmonic bands as well as the intensity of the broadband noise also increases but not the intensities of the fundamental bands, which apparently remain the same. In order to separate useful vibration bands, the broadband noise was approximated with a linear or a quadratic power function and was subtracted from the corresponding Raman spectra. As discussed above, the intensity of the allowed transition of 2vd harmonic was significantly higher than the intensity of the fundamental D band (vd). It was also higher than the intensity of the 2vq harmonic despite the fact that in the region of the main bands the intensity of the fundamental G-band was dominating. 

Fig. 7.8 Comparison of harmonic bands 2vd (a) and 2vg (b) highlighted in the Raman spectra for SWCNT (curves 1) and graphite single crystal (curves 2)...

Fig. 7.9 Comparison of the experimentally observed SWCNT Raman spectra (curves 1) in the frequency range of harmonic bands 2vd (a), 2vg (b), and of the sum tone Vd+Vg (c) (laser excitation with the wavelength > l=476.5 nm) with theoretically calculated bands based on the main bands and Vg spectra (curves 2). All bands are normalized to the corresponding maxima and the calculated bands are shifted to the lower frequencies on values Av of 14 cm (a) and 24 cm (c) to achieve coincidence of the theoretical bands maxima positions with the experimental ones...

Due to a pronounced anharmonic behavior, the maximum position of the experimentally observed harmonic band 2vd is shifted to lower frequencies in comparison to the theoretically calculated value. Therefore, the calculated band of the harmonic vibration Id(2v) in Fig. 7.9a has been shifted 14 cm to lower frequencies to achieve a coincidence with the experimentally observed band. On the other hand, in the case of 2vq harmonic, theoretical band position matches well the experimentally observed maximum due much smaller anharmonicity of the G mode for the nanotubes. Therefore, G- and D-bands show very different anharmonic behavior as will be considered in more detail below. 

First of aU, the observed 2vd harmonic band is anomalously sharp in comparison to the one theoretically calculated from the main D-band and does not have such a pronounced internal structure as the latter (Fig. 7.9a). Due to a presence of two maxima (around 1570 cm and around 1592 cm ) in the main G-band, one should expect three peaks in the 2vq harmonic band (two peaks corresponding to a doubled frequency of these two peaks and one peak corresponding to their sum) with a spectral interval of 22 cm. However, this is not the case even with consideration of a possible fine structure for the constituting bands. Relatively... 

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 have compared the intensity of harmonic bands with one of G-bands, which has weak dependence on the wavelength of excitation. As can be seen from Fig. 7.11a, the intensity of 2vd harmonic band increases linearly under the excitation frequency Vl increase, and significantly exceeds the increase of the 2vq harmonic vibration. When A,l changes from 514.5 to 476.5 nm the intensity of the 2vd tone increases 1.69 times. This is significantly higher than the theoretical value for the Raman spectrum enhancement (1.36 in accordance to (6 law). These results cannot be explained only by the resonance Raman spectrum enhancement mechanism due to the fact that the multiplication factor is different for G- and D-band harmonics. We explain the anomalous enhancement of the 2vd and 2vq tones by an opposite frequency displacement of the main bands v, Vg and by a different variation of the anharmonicity of D and G modes under excitation frequency Vl increase. 

Fig. 7.11 Dependence of the ratios of the second-order to the first-order Raman band intensities (a) and of the experimentally observed harmonic bands (b) on the excitation frequency Vl (b). (c) Temperature dependencies of the G and D Raman bands of SWCNTs (1) and graphite crystals (2)...

Contrary to the above considered harmonic bands, the band of the sum tone Vq+Vrbm is much broader than the low frequency band. We explain this fact by a well-pronounced doublet structure of G-band, which has appearance in the considered sum tone. This is also confirmed by the same spectral range between the observed maxima and by similarity in the intensities of the corresponding constituting components of the doublet bands. More detailed analysis of the structure of all the observed bands is required in the future. 

We report on anomalous behavior of the observed anharmonicity for different vibrational modes of SWCNT. This peculiarity appears in a strong dependence of the anharmonicity on the wavelength of excitation as well as in the opposite trends observed for different vibration bands. While the anharmonicity of the D mode increases with the frequency of the excitation radiation (Vl), the same of the G mode decreases. The sum harmonic band VdH-Vg is characterized with the highest anharmonicity while the same for a composed tone Vq+Vrbm is negligible. The frequency dependence of the anharmonicity also supports the concept of variation of the electronic states under the light excitation. 

The structure of vibration bands of the first and the second order in SWCNT Raman spectra has also been studied for ordered and disordered forms of graphite. This was accomplished by decomposition of the complex spectral bands into constituting components. We found proximity of spectral positions in most of spectral components of the nanotubes and graphite and considerable variation of their intensities. This also demonstrates variation of the electronic polarizabilities and can explain anomalous shifts of the harmonic bands 2vq and 2vd for nanotubes in comparison to corresponding bands of a single crystalline graphite. Narrow width of the low frequency mode Vrbm 160 cm leads to reproduction of the G-band structure in the sum harmonic band Vg+Vrbm" 1750 cm while the complex stmcture of the broad Vp band is remarkably reproduced in the Vq+Vg sum tone. The narrow width of SWCNT s 2vd and 2vg harmonics in the Raman spectra may be related to group synchronism effects [72]. 

When the amplitude of modulation is small, i.e., A(B/B/ < /s, see Fig. 7 (i)(b), the time dependent change in the resistance SR under photoexcitation at frequency / shown in Fig. 7 (i)(a), reflects mostly the time variation of the magnetic field within a phase factor. This situation changes dramatically, however, when the modulation amplitude matches the period of the radiation induced resistance oscillations, see Fig. 6(c), and Fig. 7(ii)(a) and (b). Here, in Fig. 7(ii)(a), the time response of the specimen, i.e., Sl (t), exhibits a strong harmonic component, which is evident both in the Fourier transform (inset. Fig. 7(ii)) and the harmonic band-pass filtered portion of Si (t) (see Fig. 7(ii)(a)). A further increase in the modulation amplitude such that it corresponds to two periods of the radiation induced resistance oscillations (Figs. 6(d) and 7(iii)), leads to the disappearance of the 3 harmonic component, as a 5 harmonic component takes its place, see inset Fig. 7(iii). 

Vibration-rotation bands have been obtained for the halogen halides, for CO and NO. All the work has been done in absorption and hence refers to molecules originally in the vibrationless state " = 0. The spectra he in the near infra-red. For each value of v we get one band, for v ==l the fundamental band, for t = 2, 3, 4,. .., the first, second, third,. .., harmonic bands, which in accordance with the expectation lie at about twice, three times, four times,. .., the frequency of the fundamental... 

Higher-order Raman scattering (e.g., two-phonon or three-phonon, such as harmonic bands and combination bands) can also exist but will not be discussed here. 

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