Fig. 11.12. Raman spectra for solid Cm films (~7000 A thick) taken at temperatures T = 523 and 20 K, showing overtones, combination modes, and modes arising from iso-topic symmetry-breaking effects. The data were taken using 488.0 nm Ar laser radiation. First-order and second-order mode frequencies are marked. Note that the intense structure near 1500 cm-1 has been reseated [11.20], integral multiples of the modes in Table 11.1) and combination modes (i.e., sums and differences of the modes in this table).

If we assume that the strongest second-order Raman features involve only modes which are also observed in first order, then the most intense second-order lines with Ag symmetry total 39, and the corresponding second-order lines with Hg symmetry total 88. To find this subset of overtones and combination modes, we calculate the number of modes with Ag and Hg symmetry that are contained in the direct product (2Ag + 8Hg) <g> (2Ag + 8Hg) and taking care not to double count. These special second-order modes have been marked on Fig. 11.12 and account for most of the highest-intensity features in the second-order spectrum. Somewhat weaker features in the second-order Raman spectrum are expected to arise from combination modes where only one of the components relates to the first-order Raman-active lines. In this case, modes with Ag or Hg symmetry arise from the direct product (2Ag + 8Hg) <g> nfTf where the nf vibrational modes with Tf symmetry are silent in the first-order Raman spectrum. In this category, there are no modes with Ag symmetry in the second-order Raman spectrum, whereas 152 modes with Hg symmetry are expected.

Thus the study of the second-order Raman spectra of Qq provides a powerful technique for the determination of the frequencies of the 32 silent modes. Even weaker lines in the second-order Raman spectra could arise from either overtones or combination modes associated with terms in the direct product /j,Ti ® nfYf with Ag and Hg symmetry, where r, and are both silent modes in the first-order Raman spectrum with the same parity.

Experimental studies of higher-order Raman lines show a large number of sharp features (see Fig. 11.12). Taking into account the strong, well-established features in the first-order Raman and infrared spectra, the higher-order features of the Raman spectra [11.20] are identified, as shown in Table 11.6 [11.21,42]. In addition, some of the features in Fig. 11.12 are identified with symmetry-lowering effects associated with the random presence of the 13 C isotope (natural abundance 1.1%) in approximately 50% of the C60 molecules (see §4.5). From an analysis of the higher-order Raman and infrared spectra for C60, values for the 32 silent mode frequencies and their symmetries have been obtained (see Table 11.1). Further refinements of these mode frequencies and assignments are needed as more experimental data become available. One check on the consistency of these mode frequencies is provided by a comparison of the spectroscopic results used to obtain the entries in Table 11.1 with the experimental determination of mode frequencies obtained by other techniques, such as photoluminescence [11.14,89,90], high-resolution electron energy loss spectroscopy [11.91], and inelastic neutron scattering [11.52] (see Table 11.6). The consistency of the mode assignments in Table 11.1 needs to be checked further by theoretical models, such as force constant models or first principles pseudopotential models, and by experimental studies of the temperature and pressure dependence of the spectral features in the higher-order Raman and infrared spectra.

11.5.4. Higher-Order Infrared Modes in C60

In addition to the first-order infrared spectra discussed in §11.5.2, well-resolved higher-order infrared spectral features extending to ~3200 cm"1 can be observed in thick films (see Fig. 11.13). Whereas isolated molecules generally exhibit narrow higher-order infrared features, most crystalline solids show broad first-order vibrational infrared features and only continuum features in the second-order spectra. Thus the observation of many well-resolved higher-order infrared lines in the high-resolution infrared spectrum of C60 films in Fig. 11.13 is unusual for studies of crystalline solids [11.21],

From group theoretical arguments, the expected number of second-order lines in the infrared molecular spectrum should be very large (a total of 380 combination modes). Since infrared-active modes have odd parity, no infrared overtone modes are expected in the second-order infrared spec-

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