— 01

(Ih symmetry) would split under Th point group symmetry according to Eq. (7.1), \G]ih -> [A + T]Th. Now, taking into account the four molecules per unit cell for the low-temperature cubic phase (^as = Ag + Tg), we obtain seven distinct mode frequencies for the splitting of each Gg mode, in accordance with

using Tables 4.15 and 4.16. Thus the 6 fourfold degenerate Gg modes for the C60 molecule in icosahedral Ih symmetry (see §11.3.1) are transformed into 24 x 4 Raman-active modes, associated with 42 distinct frequencies (lines) which would be symmetry allowed in the low-temperature simple cubic symmetry phase. Since their Raman activity would arise only through the crystal field interaction, the Gg-derived Raman lines would be expected to give rise to weakly Raman-active spectral features. _ Another way to obtain the splitting of a given mode V is to consider the 3 site symmetry for each of the molecules, as given in Table 7.6. Using the 3 site symmetry, the mode splitting for each of the intramolecular modes T can be found in analogy with the symmetry lowering discussed above for the case of the m3 site symmetry involving the Th group. The mode splittings for all possible mode symmetries I in icosahedral symmetry are given in the last column of Table 7.6 for the case of Th point group symmetry. If each of the four molecules is properly aligned in Pa3 symmetry, the direct product for the transformation of the four molecules *as in the unit cell (Ag + Tg) with the mode symmetries in the column labeled m3 (Th) in Table 7.6 gives the splittings listed in the right-hand column (labeled T point in T^). A comparison of the entry for the Gg mode in Table 7.6 yields the same result as is given in Eq. (7.2). If merohedral disorder is present in the lattice (see §7.1.4), and the four molecules per unit cell become uncorrelated, but the 3 site symmetry is preserved, then the splitting of the Gg line would yield 12 lines with symmetries 4(2Ag + Eg) in accordance with Table 7.6.

Using crystal field and zone-folding arguments such as illustrated in Table 7.6, we see that the total number of Raman-allowed modes for C60 in the low-temperature structure becomes very large. Table 7.10 lists the jymmetries^ of the normal modes for C60 in the crystal field for the Fm3m and Pa3 structures above and below Tm, the latter including many Raman-active modes: 29 one-dimensional Ag modes, 29 two-dimensional Eg modes, and 87 three-dimensional Tg modes. Table 7.10 assumes that every molecule is properly oriented in the crystal lattice and that no disorder exists. The one-electron band calculations typically make this assumption, although the molecules in actual crystals are not ordered in this way (see §7.1.4).

Group theoretical considerations must also be taken into account in treating the intermolecular vibrational modes (see §11.4). For the case of one molecule per unit cell, the intermolecular rotational and translational motions, respectively, give rise to modes with Tg + Tu symmetries, one of which (Tu) is the acoustic mode. For the T^ space group (Pa3) with four C60 molecules per unit cell, the number of intermolecular modes associated

Table 7.10

Symmetries of Brillouin zone-center intramolecular vibrational modes for carbon atoms" in C60, first with icosahedral symmetry (as in solution), then in Th point group symmetry (the fee structure above Tm), and finally in space group symmetry (for T<Tm).



n (r point)



2Ag + 2Tg


3 T,


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