Fig. 11.13. Experimental trace of second-order infrared spectrum for a Qo film in the frequency range (a) of the first-order vibrations and (b) above the highest first-order vibration. The frequencies predicted by the second-order model described in the text are marked on the figure [11.21], tram, in contrast to the situation for the higher-order Raman spectra, where overtones (harmonics) are commonly observed. The total number of combination modes in the infrared spectra can be found by counting the number of times the symmetry Flu is contained in the direct product n,T, ig) nfTf, where T, denotes the symmetry of one of the irreducible representations of the Ih icosahedral group, and nt is the number of vibrational modes with symmetry T,, and likewise for the second irreducible representation Yf. The values for n, and nf are contained in Table 11.1 for the vibrational modes for C60 and the direct products T, <g> containing Flu can be found from Table 4.7. For example, since the direct product Flg <g> Flu — Au + Flu + Hu contains Flu once, one infrared-active combination mode arising from the Fu and Flg vibrations is predicted for the second-order infrared spectrum. Not all mode combinations are possible; for example, the direct product of Hg and Flu does not result in infrared-active combination modes in the second-order spectrum, since Hg (g> Flu = F2u + Gu + Hu does not contain Flu. If we assume that the strongest second-order vibrations must involve one of the four Flu modes, then the most intense second-order infrared lines would be expected to arise by taking the direct product 4F,„ ® /i;Ty. Table 4.7 shows that strong infrared combination modes occur only for equal to either Ag, Flg, or Hg, thereby giving rise to a total of 52 combination modes. These modes, which are expected to be more intense, account well for ~75% of the observed higher-order IR modes. Since the Ag and Hg mode frequencies are well known from Raman spectroscopy, it is only the silent modes with Flg symmetry that are not as well established. The detailed analysis of the higher-order infrared spectra for thick C60 films (see Fig. 11.13) has, together with the corresponding higher-order Raman spectra (see Fig. 11.12), provided a complete determination of all the silent modes for C60. A summary of the mode frequencies and symmetries thus obtained for C60 is given in Table 11.1. As further experiments and calculations are carried out, some refinements to the mode frequencies and their symmetry identifications are expected.

More recent work on single crystals [11.92-96] has yielded over 200 well-resolved features in the higher-order infrared spectrum for C60. Use of synchrotron radiation sources provides a higher photon flux and therefore enhances the weak features of the higher-order infrared spectra [11.93,95]. Temperature-dependent studies have been used to indicate that difference combination modes are not important for explaining the low-frequency features in the higher-order spectrum [11.93,95]. Temperature-dependent infrared studies may also be useful for identifying specific features in the infrared spectra with (1) an isotope-induced symmetry-lowering effect, (2) the onset of crystal field effects below T0l, and (3) combination modes that give rise to infrared activity [11.92] (see §11.5.10).

In addition to the symmetry-lowering phenomena associated with the presence of the 13C isotope discussed in §4.5, §11.3.3, and §11.5.3, there is a direct isotopic effect on the mode frequencies arising from their dependence on (k/M)1/2, where k and M are the force constant and atomic mass, respectively. Thus if the fullerene mass is increased by having one or two 13C atoms on a fullerene molecule nCtnCm_x, the vibrational frequency is expected to be lowered by an amount proportional to the mass difference between l2Cm and 13Q12C 60-*, which can be simply written as

where the atomic mass of 12C60 is 720 a.u. and x is the number of 13C atoms on a C60 molecule [11.25].

By preparing high-purity Cm in a CS2 solution, the natural Raman linewidth could be reduced, so that the Raman lines associated with 12C60, 13CV2C59, and 13C212C58 could be resolved with a separation of 1.00 ± 0.02 cm-1 between these Raman peaks, as shown in Fig. 11.14 for the spectrum taken at 30 K. Data taken at higher temperatures are shown in the inset [11.25], By taking both the mass spectra and Raman spectra for a sample that was isotopically enriched with 13C, but still containing many 12C60 molecules, quantitative confirmation of this perturbative interpretation of the isotopic composition of the Ag(2) Raman line could be verified [11.25]. A more detailed analysis of the mode shift in terms of first-order nondegenerate perturbation theory provides a sensitive probe of the mode mixing induced by the isotope effect. To fit the experimental data, it is predicted that the Gg(6) mode should lie ~20 cm-1 above the Ag(2) mode at 1470 cm-1 [11.25]. This may be the best determination of the mode frequency at 1490.0 cm-1 for Gg(6) presently available (see Table 11.1).

In addition to Raman and infrared spectroscopy, several other experimental techniques such as inelastic neutron scattering (see §11.5.8), electron energy loss spectroscopy (see §11.5.9), and photoluminescence (see §11.5.7) provide information on the lattice modes of Qq (see Tables 11.5 and 11.6). These techniques traditionally provide information about the silent modes, not observed in the first-order Raman or infrared spectra, and also provide complementary information (although usually at lower resolution than Raman and infrared spectroscopy) about the optically active modes.

As discussed in §11.5.3 and §11.5.4, the higher-order Raman and infrared spectra for C60 provide an even more powerful means for determining the

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