Fig. 11.9. Theoretical calculations of the low-frequency librations (a) and translational branches (b) of the intermolecular phonon dispersion relations [11.11], See Fig. 11.8(b) for labels of symmetry points in the Brillouin zone.

Tu Eu

Fig. 11.9. Theoretical calculations of the low-frequency librations (a) and translational branches (b) of the intermolecular phonon dispersion relations [11.11], See Fig. 11.8(b) for labels of symmetry points in the Brillouin zone.

the pentagonal face, centered on the short C-C bonds around the pentagonal faces, centered on the C atoms, or even positioned at the center of the C60 molecule. These point charges were introduced phenomeno-logically to harden the orientational potential, so that the resulting dispersion curves would reproduce qualitatively the features of the neutron data. However, the calculated librational frequencies are noticeably low, indicating that the potentials are still too soft. Models based on Lennard-Jones 6-12 potentials [11.71], similar to the point charge models, also underestimate the librational frequencies by about a factor of 2. Finally, it should be noted that the model of Yildirim and Harris [11.11] results in a small frequency difference between the Au, Eu, and lowest Tu translational phonon branches [Fig. 11.9(b)], as do also other models. This suggests that the separate contributions from these branches may be difficult to obtain quantitatively by inelastic neutron studies until larger single crystals are available.

By comparing experimental data with theoretical calculations for the intermolecular modes (see Table 11.3), we can make several observations. Using level ordering arguments and Fig. 11.8, we tentatively assign the Eu mode to 38 cm-1. The assignment of the symmetries for the librational modes is more difficult than for the odd-parity vibrational modes, because so many of the librational modes lie close to one another. Using the calculated intermolecular phonon model, some assignments of the experimental data to specific intermolecular modes can be made with some confidence (see Table 11.3). It is also of interest to note that fairly good agreement is obtained between the experimental dispersion relations (Fig. 11.8) and the corresponding calculations (Fig. 11.9) away from the zone center, as for example near the M and R points in the Brillouin zone, regarding the number of modes, their parity, and mode frequencies (after multiplying the calculated librational modes by a factor of 2, as mentioned above). It should also be mentioned that inelastic neutron scattering measurements on polycrystalline samples [11.51,52,55,72] give a broad peak at ~2.3 meV (19 cm-1), consistent with observations on single crystals (see Fig. 11.8).

Experimental specific heat studies suggest phonon bands at 21, 26, and 40 cnr1 [11.64], where the 21 cm-1 phonon band is identified with Ag + Eg + 2Tg librations, the 26 cm-1 band with Tu phonons, and the 40 cm-1 band with Au + Eu + 2TU + Tg modes [11.51]. These assignments appear to be consistent with other experiments (see Table 11.3).

A theoretical calculation of the specific heat [11.73], using the density of states of the 24 librational and intermolecular vibrational modes, reproduces the experimental specific heat very well, except for the excess heat which appears below Tm (see §7.1.1 and §14.8.1). An excess entropy is associated with the orientational disorder of the 30 double bonds (C=C) for a C60 molecule. The entropy for N molecules with random orientations is

which is very close to the experimental value of 30 JKr'moH [11.74], The difference between these two values may be due to the contribution from the additional ratcheting states that are coupled to the nuclear spin states, because of the presence of the 13 C isotope in its natural abundance.

In addition to the detailed studies available for the intermolecular vibrations in C60, a limited amount of information is also becoming available for C70 (see §11.7.2) and for doped C60 (see §11.6.4).

11.5. Experimental Results on C60 Solids and Films

In this section we gather experimental results for the intermolecular and intramolecular modes for solid C60 from Raman scattering, infrared spectroscopy, photoluminescence, and inelastic neutron scattering. The results are compared with theory. Raman and infrared spectroscopies provide the most quantitative methods for determining the vibrational mode frequencies and symmetries. In addition, these methods are sensitive for distinguishing C60 from higher-molecular-weight fullerenes with lower symmetry (e.g., C70 has D5h symmetry as discussed in §4.4). Since most of the higher-molecular-weight fullerenes have lower symmetry as well as more degrees of freedom, they have many more infrared- and Raman-active modes.

11.5.1. Raman-Active Modes

In Fig. 11.10 we display the polarized Raman spectra of a vacuum-deposited C60 film (on glass) [11.19], and similar Raman spectra for C60 have been reported by many groups [11.3,4,8,15,75,76], The spectra in Fig. 11.10 were taken with 514 nm laser excitation at room temperature, and the spectra show both the (||,||) and (||, _L) scattering geometries, where the symbols indicate the optical E field direction for the incident and scattered light, respectively. Ten strong lines identified with first-order scattering were observed and identified with the 10 Raman-active modes (2Ag + 8Hg) expected for the isolated molecule. The frequencies of these lines are in good agreement with those listed in Table 11.1. Two of the lines in the spectra are identified with polarized (Ag) modes, and therefore they should disappear from the spectrum collected with crossed polarizers (||,-L). As can be seen in the (||,-L) spectrum, the purely radial Ag mode (496 cm-1) appears

Fig. 11.10. Polarized Raman spectra for C^. The upper trace is for the (||, ||) polarization and shows both Ag and Hg modes. The lower trace is for the (||,-L) polarization and shows primarily Hg modes, the weak intensities for the Ag modes being attributed to small polarization leakage [11.19].

to become extinguished, yet the high-frequency, tangential Ag mode ("pentagonal pinch mode") at 1469 cm-1 exhibits considerable intensity. Complete extinction of the pentagonal pinch mode has since been reported [11.19]. Although the 1469 cm-1 line disappears in cross-polarizers, the nearby shoulder at 1458 cm-1 does not, confirming that 1469 cm-1 is the intrinsic Ag mode frequency. Early Raman studies by several groups [11.76— 79] incorrectly identified the unpolarized 1458 cm-1 mode with the intrinsic pentagonal pinch mode, and later studies have shown that this mode should be identified with the photochemically induced polymeric state of Qq (see §11.8) [11.80].

11.5.2. Infrared-Active Modes

The first-order infrared spectrum for C60 contains only four strong lines, each identified with an intramolecular Flu mode (see Table 11.5 and Fig. 11.11). The four strong Flu lines at 526, 576, 1182, 1428 cm-1 in Fig. 11.11 [11.82] provide a convenient spectral identification for the molecule as C60. This infrared spectrum, in fact, provided a historical identification of C60 in the first publication on solid CM [11.82]. The infrared spectra also provide a sensitive characterization tool for measuring the compositional purity of C60 samples, especially with regard to their contamination with C70. Because of its lower symmetry and larger number of carbon atoms, C70 shows a much more complex infrared spectrum [11.88] than C60 (see §11.7.1), with 31 distinct infrared-active mode frequencies

Table 11.5

Experimental normal mode vibrational frequencies of the Cm molecule from Raman, IR, photoluminescence (PL), neutron inelastic scattering (NIS), and high-resolution electron energy loss spectroscopy (HREELS) data."

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