species (see §7.1.5). The experimental results are shown in Fig. 11.29 and are related to structural considerations which are determined by the orien-tational potential. When the size of the alkali metal ion in a tetrahedral site is equal to or less than the size of the site (1.12 A), the behavior of the li-brations is similar to that of C60, consistent with observations on Na2Rb1C60 [11.142] and their Pa3 crystal structure.
However, for K3C60 and Rb2 6K0 4C60, where the ionic size exceeds 1.12 A, the crystal structure remains fee. For these compounds the inelastic neutron scattering studies show larger librational energies which increase with the size of the tetrahedral ion, demonstrating that the repulsive part of the alkali metal-carbon (A-C) interaction is crucial for determining the orien-tational potential for K3C60 and Rb3C60. In contrast, this repulsive term is not so important for the Na^Q,, compounds, and the orientational potential is most sensitive to the interaction between the CWI molecular anions as in the case of C60 itself, favoring a nesting of the double bonds between two hexagons on one C60 molecule opposite pentagons and hexagons on adjacent C60 molecules. When the alkali-carbon (A-C) repulsive interaction (due to the large size of the alkali metal ion) becomes important, the minimum in the orientational potential is achieved when the hexagonal faces of the C60 molecules face the tetrahedral sites, thereby maximizing the distance between the alkali metal and carbon sites for a given lattice constant, in agreement with molecular dynamics calculations [11.71].
11.7. Vibrational Spectra for C70 and Higher Fullerenes
Except for C70, relatively little is known about the detailed vibrational spectra of the higher fullerenes. Because of their generally lower symmetry, larger number of degrees of freedom, and the many possible isomers, little systematic experimental or theoretical study has been undertaken on high-mass fullerene molecules or crystalline solids. The lack of adequate quantities of well-separated and characterized samples of higher fullerenes has curtailed experimental studies. For these reasons, we focus our attention on vibrations in C70 which have now been studied both theoretically and experimentally by Raman and infrared spectroscopy and by inelastic neutron scattering studies. Most of the studies on C70 thus far have been on the intramolecular vibrations, although a few studies on the intermolecular vibrations of C70 have more recently been carried out. The symmetry of the C70 molecule and of its molecular vibrations is discussed in §4.4.1 and symmetry tables useful for the discussion of molecular vibrations are Tables 4.12—4.24. Section §4.4.1 also provides symmetry information useful for describing the vibrations in higher-mass fullerenes.
The Raman and infrared spectra for C70 are much more complicated than for C60 because of the lower symmetry and the large number of Raman-active modes (53) and infrared-active modes (31), out of a total of 122 possible vibrational mode frequencies for C70.
A number of ab initio calculations have been carried out using several different approaches [11.23,24,37,143-146], reporting results for the 122 eigenfrequencies and normal modes. A phenomenological force constant model [11.88,147] has also been published and the results for the predicted eigenfrequencies have been used in the interpretation of the experimental Raman and infrared spectra. In all models, the D5h symmetry of C70 is explicitly treated.
Using the simplest possible approximation, the phenomenological treatment [11.147] assumes the same set of force constants for C70 as was used in the case of C60 [11.22], except that a perturbation is introduced to reduce the shear forces between the 10 additional belt atoms and the atoms on the adjacent rings. This perturbation is shown to lead to an improved set of calculated eigenfrequencies for C70, while introducing only two additional parameters. This simple model can account for many of the observed Raman and infrared spectral features [11.147].
The Dsh symmetry for the C70 molecule implies that there are five in-equivalent atomic sites on the C7o molecule (see § 3.2), which, in turn, allows for the existence of eight different bond lengths as well as 12 different angles between the bonds connecting nearest-neighbor atoms [11.143]. Thus, if only nearest-neighbor interactions are considered in constructing the dynamical matrix, the lower symmetry of the C70 molecule requires consideration of eight different bond-stretching and 12 different angle-bending force constants. In the simplified phenomenological treatment described above, only two distinct bond-stretching and two distinct angle-bending force constants are considered for bonds connecting nearest-neighbor atoms in C70 [11.147]. In this approximation, each bond is classified as either long or short, depending on whether its length is longer or shorter than the average bond length. Similarly, an angle is classified as belonging to either a pentagon or a hexagon. These simple approximations are sufficient to provide a qualitative description of the vibrational frequencies in C70.
Well-resolved infrared spectra [11.8,88] and Raman spectra [11.44,81,88, 148-150] have been observed experimentally for C70, as shown in Fig. 11.30. For both the infrared and Raman spectra, many more lines are observed for C70 relative to C60 (see Fig. 11.11), consistent with the lower symmetry and larger number of degrees of freedom for C70. Using polarization studies and a force constant model calculation [11.88,147], a preliminary assignment of the mode symmetries has been made. A listing of the calculated C70 mode frequencies is given in Table 11.8 [11.147].
In this table, the mode frequencies are labeled by their irreducible representation of group D5h, and modes associated predominantly with belt atom displacements are also labeled. It is found that mode frequencies below ~ 900 cm-1 tend to have predominantly radial displacements, while the higher-frequency modes have predominantly tangential displacements.
Inelastic neutron scattering measurements similar to those discussed in §11.3 for Cm have been carried out on C70 to study the librational spectra for solid C70 [11.72,152] below the orientational phase transition at r01. The study in C70 is more difficult to carry out because of the anisotropy of the C70 molecule, the lower crystal symmetry of the low-temperature phase, and the difficulty in obtaining single-crystal C70 samples of sufficient purity and size for detailed studies. The measurements thus far have been done on polycrystalline samples of reasonably good quality.
The inelastic neutron scattering measurements at 10 K show a broad peak centered at 1.82 meV (14.7 cm-1) for the lowest librational energy with a full width at half-maximum (FWHM) linewidth of 1.8 meV [11.152]. This librational frequency is significantly lower than for the corresponding mode at 2.77 meV (22.4 cm-1) for C60, which has a much smaller FWHM linewidth of 0.38 meV (3.1 cm-1) at 20 K. The softening of the librational mode between the lowest temperature (10-20 K) and 250 K (near the orientational phase transition) is much smaller in C7q (~17%) than in (~35%). Since the authors could not explain the observed linewidth for
11.7. Vibrational Spectra for C,0 and Higher Fullerenes 1400
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