Fig. 11.7. Dispersion of the intermolecular translational modes of single-crystal Cm above (a) and below (b) the first-order phase transition temperature at 261 K. The solid curves are the result of a fit to the experimental points with a two-parameter force constant model. The results for 200 K are shown in an extended zone scheme corresponding to an fee lattice [11.12].

Born-von Karman model used to fit the data via the adjustment of first and second nearest-neighbor force constants. Rapid convergence for the force constant model was obtained, with the second neighbor force constant being only 10% of that of the nearest neighbor. As can be seen, this simple "rigid molecule" model describes very successfully the measured phonon dispersion in the orientationally disordered phase, with ~50 cm"1 (or 6.2 meV) being the highest frequency observed. Referring to Fig. 11.7(a), we see that two Debye frequencies are needed to describe the acoustic modes in the high-T, orientationally disordered phase for C60. Shown in Fig. 11.7(b) are the translational modes observed at 200 K by inelastic neutron scattering on the same sample as in Fig. 11.7(a) [11.13]. These data show clear evidence for additional phonon branches upon cooling below T0l, as explained below in more detail.

At temperatures well below the orientational ordering temperature T01 ~ 261 K, where there are four inequivalent molecules in the simple cubic cell of space group Pa3 (see §7.1.3), both orientational and translational restoring forces are effective. In this regime, the three rotational and three translational degrees of freedom per molecule lead to 6 x 4 = 24 degrees of freedom per primitive unit cell. The symmetries of these 24 modes at the zone center can be obtained by taking the direct product of the molecular site symmetry in the T* space group (see Table 7.8), that is, (Ag + Tu), with those of the rotations (Tg) and translations (Tu). This direct product yields (Ag + Tu) ® (Tu + Tg) = (Ag + Eg + 3Tg) + (Au + Eu + 3Tu) [11.45]. Thus 10 distinct zone center frequencies (including the modes at to = 0 which are the Tu-symmetry acoustic phonons) should be observable in this low-temperature structure. The gerade modes and ungerade modes are identified, respectively, with zone center (long-wavelength) librations and translational phonon modes. One of the Tu modes at the zone center corresponds to a pure translation, whereby w = 0 at q — 0, while the other two Tu optic phonon modes are IR active; all the gerade librational modes are Raman active. However, the observation of these particular modes by optical spectroscopy should be quite difficult, because of their low frequency (see Table 11.3).

We show in Fig. 11.8(a), the experimental phonon/libron dispersion relations for C60 in the low-temperature Pa3 phase obtained from inelastic neutron scattering studies of a small single crystal (6 mm3) along high-symmetry directions following the notation shown in Fig. 11.8(b) [11.13], The data in Fig. 11.8(a) were taken along the T - A - R direction in reciprocal space, where 16 branches are allowed, and along the zone edge (R - T - M), where, because of mode degeneracies, only 6 branches are allowed [11.11] (see Table 11.4). Seven of the ten frequencies expected at

Fig. 11.8. (a) Experimentally determined low-frequency phonon (closed circles) and libron (open circles) dispersion relations, obtained from inelastic neutron scattering measurements in the simple cubic (sc) low-temperature phase [11.13]. The solid and dashed lines connecting the experimental points are guides to the eye. (b) Symmetry labels for high symmetry points and axes of the Brillouin zone for the low-temperature Pa3 structure, consistent with labels in (a). Also shown are several half reciprocal lattice vectors, namely g,/2, g2/2, and g,/2 [11.51], the T-point on the basis of group theoretical considerations have thus far been reported. The modes in Fig. 11.8 that are assigned primarily to libra-tional or translational phonons are indicated by the open and closed circles, respectively. As can be seen, the acoustic mode dispersion is broad enough to overlap all the optic branches and exhibits a maximum frequency of 6.7 meV (55 cm-1). Furthermore, the librational modes (open circles) are the four lowest optic branches observed, with frequencies lower than would be expected from an analysis of specific heat measurements [11.64,65], These librational modes are seen to exhibit almost no dispersion. The upper two optic branches, which exhibit zone center frequencies of 5 and 6.7 meV, are in very good agreement with mode frequencies observed at 41 cm-1 (5.1 meV) and 55 cm-1 (6.8 meV), respectively, using IR spectroscopy on powder samples at T = 1.5 K [11.56]. Therefore, these upper two branches should be assigned to Tu symmetry. These features in the far-IR spectra [11.56] were first found at about 261 K, as degassed powder

Table 11.4

Point group symmetries and branches of the various dispersion curves along high-symmetry axes [see Fig. 11.8(b)] [11.51]

Table 11.4

Point group symmetries and branches of the various dispersion curves along high-symmetry axes [see Fig. 11.8(b)] [11.51]

Points |

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