, 4Ag + 4Eg + l2Tg
i
I 4Au + 4Eu + 12Tu
twofold axes of the crystalline solid. It is of significance that the group Th has no fourfold symmetry axes, necessitated by the absence of fourfold axes in group lh. In the Th point group, the vector transforms as the Tu irreducible representation, which is used to establish the lattice modes of solid C60, once the equivalence transformation for the atom sites within the unit cell is specified. Crystal field splitting effects, together with the zone folding for the lattice modes of crystalline C60, are further discussed in §11.3.2.
When the C60 molecules are in m3 crystal field sites, there are three inequivalent carbon atoms on each molecule. These are shown in Fig. 7.3 from the perspective of a threefold axis and a twofold axis. In Fig. 7.3(b) the 4 threefold axes and one of the twofold axes of the Th point group are shown [7.52].
According to Table 4.15, only the fourfold and fivefold vibrational modes in the molecular Ih symmetry will be split in an fee crystal field, and the parity of the modes in the Ih Th symmetry-lowering transformation is preserved. Thus suppressing the g and u parity subscripts, we can express the mode splitting in the Ih Th symmetry-lowering transformation by
where the subscripts are used to denote the pertinent point symmetry groups. A summary of the relations between the molecular modes in //, symmetry and the modes in various lower symmetries is given in Table 7.6. As is discussed in §11.3.2, the magnitudes of the crystal field splittings in crystalline C60 are small and are not easily resolved over the observed linewidths of the Raman and infrared spectral features.
The basis functions for the irreducible representations of the point group Th (see Table 7.4) imply that the Ag,Eg, and Tg modes are Raman active, with the Ag and Eg modes appearing only for the ||, || geometry and the
Table 7.6
Compatibility relations of the high symmetry group Ih with lower symmetry groups 3, 23, m3 and for the T point of the zone-folded T¡f space group.
Table 7.6
Compatibility relations of the high symmetry group Ih with lower symmetry groups 3, 23, m3 and for the T point of the zone-folded T¡f space group.
h |
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