A

Fig. 11.3. Schematic diagram for the displacements corresponding to (a) the radial Ag(l) breathing mode where all displacements are in the radial direction, and (b) the tangential Ag(2) "pentagonal pinch" mode where all displacements are tangential. For both Ag modes all displacements are of equal magnitude and phase.

metry type, and only modes of the same symmetry are coupled. Thus the only eigenvector that can be uniquely specified is the one associated with the single Au mode at the Brillouin zone center.

Just as it is instructive to consider the electronic states of a C60 molecule in a simple approximation where the 360 electrons (i.e., six electrons per carbon atom) are confined to a sphere of radius 7 A (see §12.1.2), it is similarly instructive to consider the vibrational modes of a C60 molecule as elastic deformations of a thin spherical shell. This calculation has been carried out for the axisymmetric modes of C60 [11.28] and the calculation applies the classical results of Lamb for an elastic spherical shell published in 1883 [11.29]. For this shell model, there are two distinct families of axi-ally symmetric mode frequencies displayed vs. mode number n in Fig. 11.4, an upper branch denoted by a„ and a lower branch denoted by b„. The frequencies of these modes can be computed analytically in terms of a single parameter A which can be shown to depend on Young's modulus, the Poisson ratio y (taken as 1/3), the shell radius, and the mass density [11.28]. To fit the continuum shell model to experiment, a value for A is obtained by setting the theoretical expression for the radial breathing mode frequency <i>i(Ag) equal to the experimental value of 497 cm-1. In Fig. 11.4, the agreement between experimental Raman and IR frequencies (filled circles) [11.6,30] and the model (open circles) is reasonably good, considering the simplicity of the model. In Fig. 11.5, the classical normal modes of the de-

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