|j Locii Symmetry Clusicrc C5 (12-fold)

C3 120-fold 1

|j Locii Symmetry Clusicrc C5 (12-fold)

C3 120-fold 1

Possible Hyperfine splittings. (May exceed superfine splittings)

Possible Hyperfine splittings. (May exceed superfine splittings)

Fig. 11.6. Possible rotational-vibrational structure of an infrared-active Flu fundamental mode of Cm. IR fundamental modes exist at 527, 576, 1183, and 1429 cm"1 (see Table 11.1), and for each Flu mode, additional structure is expected, associated with icosahedral symmetry-lowering effects (fine structure) and the nuclear hyperfine interaction (superfine structure). Highlighted in this figure is the difference between the ^C«, and ^C^ spectra [11.47].

Fig. 11.6. Possible rotational-vibrational structure of an infrared-active Flu fundamental mode of Cm. IR fundamental modes exist at 527, 576, 1183, and 1429 cm"1 (see Table 11.1), and for each Flu mode, additional structure is expected, associated with icosahedral symmetry-lowering effects (fine structure) and the nuclear hyperfine interaction (superfine structure). Highlighted in this figure is the difference between the ^C«, and ^C^ spectra [11.47].

selection rule on allowed J values. These differences in the 12C60 and nC60 rotational-vibrational spectra should be important in observations of these spectra at low temperatures.

To show the effect of the icosahedral symmetry on the rotational energy levels, the decomposition of a high angular momentum level (e.g., J ~ 50)

in full rotational symmetry into many states in the lower-symmetry icosa-hedral group is shown in Fig. 11.6 in the spectra labeled "fine structure." A summary of the decomposition of the irreducible representations of the full rotation group into the irreducible representations of the Ih group is contained in Table 4.3 up to / = 34. All of the icosahedral symmetries are permitted for rotational states in 13C60, but only modes with Ag symmetry are allowed for 12C60. If the icosahedral symmetry is explicitly considered in writing the wave functions for the angular momentum states J, then fine structure splittings result as shown in Fig. 11.6 [11.47]. These fine structure splittings vary as the sixth or higher power of the angular momentum J (where J = L 4- S) and could range in frequency from a few kHz to several GHz (1 THz = 33.356 cm-1). The accidental degeneracies that remain in the fine structure for "Qq are associated with the 12 C5 local clusters and the 20 C3 local clusters of the regular truncated icosahe-dron.

Since the high-angular-momentum lines for 12C60 contain only states with Ag symmetry (e.g., the state / = 50 contains only two Ag states when considered as a reducible representation of the Ih group), it is the splitting of only these two Ag states that is seen in the fine structure of 12CM.

In addition to the effect of the icosahedral symmetry on the rotational levels of the molecule which results in the fine structure, there is an interaction due to the nuclear spin which gives rise to superfine splittings, which are distinguished from the hyperfine interaction because the J values for the angular momentum in the superfine interaction are considered in terms of the irreducible representations of the icosahedral group Ih. Since the energy associated with the icosahedral distortion is much larger than that for the hyperfine interaction, arising from the magnetic field at the position of the nuclei associated with the rotational angular momentum J, the icosahedral distortion is considered first in perturbation theory. The superfine splittings vary over several orders of magnitude, from a few Hz to several MHz, and are particularly important for 13C60.

In 13C60, each 13C atom has a nuclear spin J = 1/2 and each molecule thus has 260 nuclear spin states, with Jlol values ranging from Jtot = 0 to J^tot = 30. The decomposition of 260 nuclear spin states into irreducible representations of all possible Jtol states has been examined by Harter and Reimer [11.47], noting that the totally antisymmetric states of 13C60 belong to the Ag irreducible representation of Ih, because all operations of Ih are even permutations, and the totally antisymmetric wave function does not change sign under even permutations. Thus the statistical weight for each irreducible representation of Ih for all 260 states of "Qq is well approximated by the dimension of the irreducible representation [11.50].

In the superfine structure for 13C60 in Fig. 11.6 we see two effects. The first is the splitting of the 12-fold and 20-fold accidentally degenerate levels of the fine structure when the nuclear spin is included, since the nuclear spin can be up or down at each 13C atomic site. For the example shown in Fig. 11.6, we see a 12-fold level split into the four irreducible representations of group Ih that are contained. For each of these four spectral lines, a second effect appears, explicitly associated with the hyperfine interaction between the orbital angular momentum and the total spin Jlot of the molecule. The large number of very closely spaced lines appearing with each fine structure line in the superfine spectrum is due to the large number (0 < Jtot < 30) of values that Jtol may have for "Cm, each Jxox value having its own weight, as discussed above.

In contrast to the complex superfine spectrum for 13C60 in Fig. 11.6, there are no superfine splittings for ^C^ because it has no nuclear spin [11.47,50],

For ordinary C60 molecules, the distribution of the 12C and 13C isotopes is in proportion to their natural abundance (i.e., 1.108% of 13C), so that approximately half the Qq molecules have one or more 13C isotopes and approximately half have only 12C isotopes (see §4.5). High isotopic purity is needed to study the vibrational spectrum of 12C60, since a purity of 99.9%, 99.99%, or 99.999% in the 12C atom content yields a purity of only 94.1%, 99.4%, or 99.9%, respectively, for the 12C60 molecule (see §4.5). For the molecules containing both 12C and 13C isotopes, there are no symmetry restrictions for the allowed J values. Since approximately half of the molecules in ordinary (i.e., without isotopic enrichment) CM samples are 12C60 molecules, the isotope effects discussed above are expected to affect the line intensities of the rotational and rotational-vibrational modes significantly in ordinary C60 samples at very low temperatures.

Isotope effects also apply to other situations [11.50], For both the 12C60 and the 13C60 molecules, the intensity ratios of both infrared and Raman transitions between the restricted excited rotational states are expected to be strongly affected by the isotope effect at very low temperatures. In addition to the rotational levels for the free molecules, the librational levels of 12C 60 and 1 C60 crystalline solids should experience symmetry selection rules, with restrictions also placed on the intensities for the excitation of specific librational states. Specifically, the rotation-vibration spectra in the solid state are expected to differ from the spectra observed for the free molecules, because of differences between the rotational and librational levels (see §11.4) [11.50].

Whereas the 12C60 molecule exhibits the highest degree of symmetry and the most stringent selection rules, and the 13C60 molecule shows the next highest stringency regarding selection rules, lower-symmetry molecules, such as 13Cj12C59 and 13C212C58, are still expected to show some isotope-dependent behavior [11.50]. If the vibrational and rotational levels and their occupations are modified by the isotope effect, it is expected that the corresponding infrared spectra, Raman spectra, and specific heat [11.50] would also be modified at very low temperature (below 5 K), where the required boson character of the wave function constrains the allowed states to an even number of phonons in 12C60, since the vibrational wave function must have even parity.

Also of interest in relation to vibrational spectra is the predicted 8% increase in the transition temperature Tm for pure 12C60 relative to that for the "On molecule, because of the higher-mass and therefore larger moment of inertia of 13C60 relative to 12C60. However, the difference between T01 values for 12C5913C! and 12C60 is probably too small to be observed experimentally. The isotope effect might, however, be a source of line broadening for this phase transition. Closely related to this isotope effect is the softening of a given intramolecular vibrational mode by virtue of the effect of the isotopic mass on the (k/M)1/2 factor, which determines force constants in the dynamical matrix. Such an effect has been reported for the Ag(2) Raman-active mode for "C^Qq.j for various values of x [11.25], as discussed in §11.5.5.

11.4. Intermolecular Modes

Because of the molecular nature of solid C60 and the large difference in frequency between the inter- and intramolecular modes, one expects a decoupling of these two types of vibrational modes. Consistent with this view is the experimental observation of a large frequency gap between the lowest frequency intramolecular [//^(l)] mode at 273 cm-1 and the highest frequency intermolecular or "phonon" mode at ~ 60 cm-1. Thus, lattice dynamics models in which molecules are allowed to translate and rotate as rigid units about their equilibrium lattice positions would be expected to be reasonably successful for describing the intermolecular vibrations, as shown below.

Inelastic neutron scattering provides a powerful and almost unique experimental technique for probing the wave vector dependence (i.e., dispersion) of the vibrational modes in solids [11.12,51-55]. These scattering experiments require large single crystals, and in the case of C60, suitable samples have only become available recently (see §9.1.1). Information on the intermolecular mode vibrations has also been provided by infrared [11.56] and Raman scattering [11.57] experiments. In principle, Brillouin scattering experiments [11.58] should be possible and would provide information on the acoustic branch for the intermolecular modes. However, the laser power needed to do these experiments may be so large that it causes phototransformation of the C60.

Most of the inelastic neutron scattering studies on librational excitations in C60 have been on polycrystalline samples. From this work a number of important findings have resulted. The major peak in the neutron scattering occurs at ~2.7 meV (21.8 cm-1) and a smaller peak is observed at ~4 meV (32 cm-1) [11.12,13]: the more intense peak is identified with 9 of the intermolecular modes in Table 11.3 and the smaller peak with the remaining 3 intermolecular modes. The temperature dependence shows that the librational energy softens by ~35% in going from 20 K to 250 K and the peak broadens by a factor of 6 over this temperature range. Above Tou no scattering intensity associated with librations is found, as expected. The large broadening effect of the librational line with temperature is attributed to a

Table 11.3

Zone center (r point) translational and librational mode frequencies for CM below 261 K." In most cases, symmetry assignments were not provided by the original references.



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