At

1 U

4TU+4(AU + EU + 2TU)

5F2u

5 Tu

5TU + 5(AU +EU+ 2TU)

6 Gu

6 Au + 6 r„

6(AU + Tu) + 6 (Au +EU + 3 T„)

1HU

1EU + 1TU

1(EU + TU) + 1(AU+EU+ATU)

"The modes associated with the dopant atoms in doped solid Qo are not included in this listing and must be accounted for separately. The modes associated with translations and rotations (librations) of the center of mass of the C60 molecules also are not included in this table.

"The modes associated with the dopant atoms in doped solid Qo are not included in this listing and must be accounted for separately. The modes associated with translations and rotations (librations) of the center of mass of the C60 molecules also are not included in this table.

with translations and rotations of the center of mass is found by taking the direct product of the equivalence transformation (Ag + Tg) with (Tg + Tu) for the rotations and translations, to yield

(Ag + Tg) ® (Tg + Tu) Ag + Eg + 3Tg + AU+EU + 3Tu (7.3)

in which one of the Tu modes is the acoustic mode. These symmetries are used in §11.4 to guide the interpretation of the intermolecular vibrational mode spectra in crystalline C60.

7.1.3. Low-Temperature Phases

Below a characteristic temperature of T01 = 261 K, the Qq molecules in solid C60 lose two of their three degrees of rotational freedom. Thus the rotational motion below Tm occurs around the four (111) axes and is a hindered rotation whereby adjacent C60 molecules develop strongly correlated orientations [7.11,40,53,54], In the ordered phase below T01, the orientation of the C60 molecules relative to the cubic crystalline axes becomes important. In this section we consider the structural properties of crystalline C60 as a function of temperature with particular emphasis given to structural changes associated with phase transitions. The group theoretical issues associated with these phases are reviewed in §7.1.2.

Whereas the structure of solid C60 above Tm is the fee structure Fm3m discussed in §7.1.1, the structure below ~261 K is a simple cubic structure (space group or Pa3) with a lattice constant a0 = 14.17 A and four C60 molecules per unit cell, since the four molecules within the fee structure (see Fig. 7.2) become inequivalent below T01, as molecular orientation effects become important [7.11,40], From a physical standpoint, the lowering of the crystal symmetry as the temperature is reduced below T0l is caused by the assignment of a specific (111) direction (vector) to each of the four molecules within a unit cell as shown in Fig. 7.5. In this figure the four distinct molecules in the unit cell are shown, as well as the rotation axis for each of the molecules [7.55]. Thus the degrees of rotational freedom are greatly reduced from essentially free rotation above T01 to rotations about a specific threefold axis below T01. This reduction in the degrees of freedom is the cause of the phase transition at T01 and affects many of the physical properties of C60 in the crystalline state.

Supporting evidence for a phase transition at Tm from the fee structure (above T01) to the simple cubic structure below T01 is provided by anomalies in many property measurements, such as x-ray diffraction [7.56], differential scanning calorimetry [7.57], specific heat [7.15,58,59], electrical resistivity [7.60], NMR [7.32,33,35,57,61-64], inelastic neutron scattering

Fig. 7.5. The four molecules in the unit cell of C60 showing the same standard orientation, with twofold axes aligned parallel to the cube edges. Starting from this orientation, molecules at (0,0,0), (1,1,0), (5,0,5), (0,|,±) are rotated by the same angle about local axes (111), (111), (111), and (111), respectively. The sense of rotation about each {111} axis is indicated in the figure [7.55],

[7.65-68], electron diffraction [7.69], sound velocity and ultrasonic attenuation [7.7,70], elastic properties [7.46], Raman spectroscopy [7.71-74], thermal conductivity [7.17], and the thermal coefficient of lattice expansion [7.11,75],

As T is lowered below Tm, orientational alignment begins to take place, although the structural alignment occurs over a relatively large temperature range. The crystal structure of the ordered phase below temperature Tm can be understood by referring to Fig. 7.6(a), where one standard orientation of a C60 molecule is shown with respect to a cubic coordinate system [7.48]. In this standard orientation of the C60 molecule, the maximum point group symmetry (Th) common to the icosahedral point group Ih and the fee space group Fm3m is maintained. If all the C60 molecules in the lattice were to maintain the same standard orientation shown in Fig. 7.6(a), then the space group would be Fm3, as discussed in §7.1.2. Here all the {100} axes pass through three mutually orthogonal twofold molecular axes (the centers of the electron-rich double-bond hexagon-hexagon edges), and four (111) axes pass through diagonally opposite corners of the cube shown in Fig. 7.6. A second alternate standard orientation is shown in Fig. 7.6(b), which is obtained from Fig. 7.6(a) through rotation by 7r/2 about the [001] axis. In the case of M6C60 (M = K, Rb, Cs), all the C60 molecules are similarly oriented with respect to a single set of x,y, z axes. However, this is not the case in crystalline C60, where the molecules order with nearly equal probability in the two standard orientations. Merohedral disorder refers to

Fig. 7.6. (a) One "standard" orientation for the Cartesian axes in a cubic crystal so that these axes pass through three orthogonal twofold axes, (b) The other "standard" orientation, related to that in (a) by a 90° rotation about any of the Cartesian axes, (c) A view of the (b) "standard" orientation, showing twofold and threefold molecular axes aligned with the cubic axes of the C60 crystal. Placing the inscribed icosahedron (c) into (a) or (b) has mirror planes perpendicular to the three (100) directions. In this orientation, six of the icosahedral edges lie in planes parallel to the mirror planes, so that when the icosahedron is circumscribed by a minimal cube, these edges lie in the cube faces, as shown. Each (111) direction is a threefold axis [7.21,48],

Fig. 7.6. (a) One "standard" orientation for the Cartesian axes in a cubic crystal so that these axes pass through three orthogonal twofold axes, (b) The other "standard" orientation, related to that in (a) by a 90° rotation about any of the Cartesian axes, (c) A view of the (b) "standard" orientation, showing twofold and threefold molecular axes aligned with the cubic axes of the C60 crystal. Placing the inscribed icosahedron (c) into (a) or (b) has mirror planes perpendicular to the three (100) directions. In this orientation, six of the icosahedral edges lie in planes parallel to the mirror planes, so that when the icosahedron is circumscribed by a minimal cube, these edges lie in the cube faces, as shown. Each (111) direction is a threefold axis [7.21,48], the random choice between one of the two possible standard orientations shown in Fig. 7.6(a) and (b) for each C60 molecule in the solid (see §7.1.4).

In the low-temperature ordered phase, the centers of four C60 molecules have coordinate locations (0,0,0), (|, |,0), (|,0,|), and (0,|,|), and each molecule, respectively, is rotated by 22°-26° from the standard orientation shown in Fig. 7.6(a) about the threefold (1,1,1), (1,1,1), (1,1,1), and (1,1,1) axes [7.11]. These rotations of the {111} axes are shown in Fig. 7.5 [7.55]. This rotation angle of 22°-26° is determined experimentally and is governed by the intermolecular interactions between C50 molecules, as described below. Since the rotation angle is not fixed by symmetry [7.8], it must be measured experimentally.

In the idealized ordered structure, the relative orientation of adjacent molecules is stabilized by aligning an electron-rich double bond on one molecule opposite the electron-poor pentagonal face of an adjacent C60 molecule [see Fig. 7.7(a) and (c)]. This orientation corresponds to the 22°-26° rotation angle mentioned above. The preferred orientation of the C60 molecules is measured through an angular dependence of the Bragg peaks, using x-ray and neutron diffraction techniques [7.76]. As the temperature is lowered, the molecules align preferentially along these favored directions. A second "phase" transition at a lower temperature T02 is associated with this alignment in the sense that many physical properties also show an anomaly in their temperature dependence at this lower temperature T02. However, the value of T02 depends on the property being measured, and especially the time required for the excitation necessary to make the measurement, as discussed below.

Fig. 7.7. Electron-rich double bond on one C60 molecule opposite an electron-poor (a) pentagonal face, (b) hexagonal face on the adjacent molecule [7.48]. (c) is a view along an axis joining centers of nearest-neighbor Cm molecules, (d) shows a rotation of (c) by 60° about a threefold axis to bring the hexagonal face of one molecule adjacent to the twofold axis of a second molecule [7.76]. In (c) and (d), the black circles refer to the molecule in back and the open circles refer to the molecule in front.

Fig. 7.7. Electron-rich double bond on one C60 molecule opposite an electron-poor (a) pentagonal face, (b) hexagonal face on the adjacent molecule [7.48]. (c) is a view along an axis joining centers of nearest-neighbor Cm molecules, (d) shows a rotation of (c) by 60° about a threefold axis to bring the hexagonal face of one molecule adjacent to the twofold axis of a second molecule [7.76]. In (c) and (d), the black circles refer to the molecule in back and the open circles refer to the molecule in front.

Another structure (called the "defect structure"), with only slightly higher energy, places the electron-rich double bond of one C60 molecule opposite an electron-poor hexagonal face [see Figs. 7.7(b) and (d)]. This orientation can be achieved from the lower-energy orientation described in Fig. 7.5 in one of two ways, as shown in Fig. 7.8: by rotation of the C60 molecule by 60° around a (111) threefold axis or the rotation of the molecule by ~ 42° about any one of the three twofold axes normal to the body diagonal [111] direction in the lattice [7.53,54]. The lower energy configuration is believed to lie only ~ 11.4 meV below the upper energy configuration, and the two are separated by a potential barrier of ~ 290 meV [7.46,53,54,78,79], In Fig. 7.8, we see the dark rods denoting the six {110} directions where the double bonds of the indicated atom are adjacent to pentagonal faces of the nearest-neighbor C60 molecules. The light rods denote three of the six {110} directions for which the pentagons of the indicated C60 molecule are adjacent to double bonds on the nearest-neighbor molecules. For the

Fig. 7.8. Three possible orientations of a C^ molecule with respect to a fixed set of axes. The plane of each drawing is normal to a [111] direction. Thin shaded (unshaded) rods represent [110] directions that are normal (inclined at 35.26°) to the (111) direction. The 12 nearest-neighbors of the molecule are in the twelve {110} directions. At (a), which represents the majority orientation, pentagons face neighbors in the "inclined" (110) directions, and double bonds face neighbors in the "normal" (110) directions. At (b) and (c), which represent the same minority orientation, hexagons have replaced pentagons in the "inclined" (110) directions. The transformation from (a) to (b) involves a 60° rotation about the [111] direction, and from (a) to (c) involves a ~ 42° rotation about the (110) direction [7.77].

Fig. 7.8. Three possible orientations of a C^ molecule with respect to a fixed set of axes. The plane of each drawing is normal to a [111] direction. Thin shaded (unshaded) rods represent [110] directions that are normal (inclined at 35.26°) to the (111) direction. The 12 nearest-neighbors of the molecule are in the twelve {110} directions. At (a), which represents the majority orientation, pentagons face neighbors in the "inclined" (110) directions, and double bonds face neighbors in the "normal" (110) directions. At (b) and (c), which represent the same minority orientation, hexagons have replaced pentagons in the "inclined" (110) directions. The transformation from (a) to (b) involves a 60° rotation about the [111] direction, and from (a) to (c) involves a ~ 42° rotation about the (110) direction [7.77].

defect orientation, the double bonds are opposite a hexagonal face as in Figs. 7.7(b) and (d) [7.77].

As the temperature T is lowered below Tm = 261 K, the probability of occupying the lower energy configuration increases [7.53,54]. This model for two orientational configurations (see Figs. 7.7 and 7.8) with nearly the same energy is called the David model [7.53,54]. On the basis of studies of the temperature dependence of the 13C-NMR spin correlation time r, which is obtained from measurement of the NMR spin-lattice relaxation time a ratchet-type motion was proposed between these correlated orientations below Tm. These ratchet-type rotations are described by an activation-type behavior in which the diffusion constant D is given by D = D0 exp(—TA/T) [7.36]. Here the activation temperature TA for ratchet motion below T01 is T™1 = 2100 ± 600 K. As discussed in §7.1.1, some ratchet-type motion is also observed above Tou and for this regime, the activation temperature is much lower, T™1 = 695 ± 45 K [7.36], indicative of a motion much closer to free rotation of the molecule. The corresponding physical picture is that of a molecule that jumps via thermal excitation from one local potential minimum to another; these jumps are made quicker at high temperature and slower at low temperature, as discussed in more detail in §7.1.5.

Evidence for residual structural disorder in the low-temperature molecular alignment is found from neutron scattering experiments [7.53,54], specific heat [7.15,58,59,80], NMR motional narrowing studies [7.32,33,35, 57,63,64], thermal conductivity measurements [7.17,81], and others. The mechanism by which the orientational alignment is enhanced is by the ratcheting motion of the C60 molecules around the (111) axes as the molecules execute a hindered rotational motion. Very rapid ratcheting motion has been observed from about 300 K (room temperature) down to the T0l = 261 K phase transition. Slower ratcheting motion is the dominant molecular motion below Tm and this motion continues down to low temperatures. Below Tm, the rotational reorientation time r is much slower than above T0l (e.g., r ~ 2 ns at 250 K and 5 ps at 340 K) [7.35,36,45,63]. In the solid phase, librational motion takes place below T01 and also plays an important role in the molecular alignment (see §11.4).

The reorientation of C60 molecules is governed by an orientational potential V(0)

where 0hop is the angle through which the molecule librates. The energy VA is related to the librational energy ¿?nb, the moment of inertia / of the C60 molecule, and 0hop by showing that librations enable the molecules to overcome the potential barrier and hop to an equivalent potential minimum. Using values of Zslib = 2.5 meV (see §11.4) and / = 1.0 x 10~43 kg m2 (see Table 3.1), Fig. 7.9 is obtained showing V(0) plotted for ratcheting motion by 0hop = 60° about the (111) axes (solid curves) and by 0hop = 42° about the (110) axes (dashed lines) [7.76]. The experimental measurements favor rotations of the molecules about (110) axes with regard to the barrier height (~250 meV) and the presence of a second local minimum at 6 = 42° corresponding to the orientation of the double bond opposite a hexagonal face (the defect orientation), with an energy ~11 meV higher than F(0) [7.82]. Model calculations of the single-molecule orientational potential have been carried

Fig. 7.9. Schematic diagram of the single-particle potential V(6) in the low-temperature Pa3 phase. Here 6 = 0 corresponds to the equilibrium position of the C60 molecule, the solid line represents the potential for rotations of the molecule about the threefold axis aligned along a (111) direction, and the dashed line is the potential for rotations about the (110) direction, which corresponds to a twofold axis of the molecule. The "defect" orientation can be reached by a 60° rotation about [111] or a ~42° rotation about [110] as shown in Fig. 7.8 [7.76],

(degrees)

out, giving good agreement with the general results of Fig. 7.9, but with some differences in the barrier heights [7.43,83-85].

Evidence for another "phase" transition at lower temperatures (90 K to 165 K) has been provided by a number of experimental techniques, including velocity of sound and ultrasonic attenuation studies [7.7,70], specific heat measurements [7.16,59], thermal conductivity [7.17,81], elasticity [7.7,46,70], high-resolution capacitance dilatometry [7.75], dielectric relaxation studies [7.78], electron microscopy observations [7.86], neutron scattering measurements [7.53,54] and Raman scattering studies [7.71,74], A wide range of transition temperatures T02 has been reported in the literature for this lower-temperature phase transition. A detailed study shows that the temperature T02 obtained for this transition depends on the time scale of the measurement [7.46], Thus the temperature dependence of the relaxation time is given by the relation r, = r,0 exp(EAt/kBT), where EA, is the activation energy for the transition and t,0 is a characteristic relaxation time with values of r,0 = 4 ± 2 x 10-14 s and EAt = 300 ± 10 meV [7.46]. The dependence of the transition temperature on the time scale of the perturbation is shown in Fig. 7.10, where the temperature dependence of the real part of the elastic constant C'ai is plotted for various ac frequencies (o [7.46]. A comprehensive plot covering 11 orders of magnitude in w is shown in Fig. 7.11, where the frequencies of many experimental probes are related to their respective transition temperatures T02 shown in Fig. 7.10 [7.46]. The plots in Figs. 7.10 and 7.11 show that the microscopic origins of the phase transition over the temperature range 80 < T <182 K have the same physical origin. The very low frequency points in this plot

Fig. 7.10. Measured (o) and calculated (solid and dashed lines) real (C;f() and imaginary (Qff) parts of the complex elastic constant Ccff(7') vs. temperature at five frequencies between 0.6 and 50 Hz. The corresponding plot for the loss component Qtf(T) is shown at 0.6 Hz. The peak in Q, determines the transition temperature Tal. Different scales for Cci[ (arbitrary units) are used at different frequencies for clarity. The solid lines represent a least squares fit of a Debye relaxational response function with a single thermally-activated relaxation time, yielding good agreement with the experimental data. The dashed lines at 0.6 Hz were calculated with a Gaussian distribution of activation energies and a width of 40 meV; this calculation does not fit the data points. Inset: AQ( = Q„(T « T02) - C't„(T » T02) as a function of the transition temperature T02, which follows a straight line [7.46].

Fig. 7.10. Measured (o) and calculated (solid and dashed lines) real (C;f() and imaginary (Qff) parts of the complex elastic constant Ccff(7') vs. temperature at five frequencies between 0.6 and 50 Hz. The corresponding plot for the loss component Qtf(T) is shown at 0.6 Hz. The peak in Q, determines the transition temperature Tal. Different scales for Cci[ (arbitrary units) are used at different frequencies for clarity. The solid lines represent a least squares fit of a Debye relaxational response function with a single thermally-activated relaxation time, yielding good agreement with the experimental data. The dashed lines at 0.6 Hz were calculated with a Gaussian distribution of activation energies and a width of 40 meV; this calculation does not fit the data points. Inset: AQ( = Q„(T « T02) - C't„(T » T02) as a function of the transition temperature T02, which follows a straight line [7.46].

are determined from slow cooling rate experiments, where the sample is in equilibrium throughout the measurements [7.46]. For high-frequency measurement probes, the molecules cannot reorient quickly enough to follow the probe frequency, so that the molecules become frozen into positions corresponding to the two energy minima separated by 11.4 meV and de-

7. Crystalline Structure of Fullerene Solids 180 80 r02 (k)

Fig. 7.11. Plot of ln(or') vs. Tm[ for different measurement frequencies o). The points, triangles, full squares, and open circles are taken from elastic constant [7.46], thermal expansion [7.75], dielectric [7.78], and sound velocity [7.7] measurements, respectively. The two columns on the right relate representative probe frequencies to their T02 values [7.46],

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