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25 the solid line through the data points [14.102],

The linear term in the specific heat determines N(EF) for a simple metal, but in the case of doped fullerenes there are also many-body mass enhancement effects associated with both the electron-phonon interaction and spin fluctuations, so that the specific heat measurements and analysis carried out thus far do not yield a definitive determination of N(EF) [14.102]. Also, merohedral disorder is expected to broaden the peak structure in the density of states, thereby lowering the value of N(EF) expected for a perfectly ordered crystal. After subtracting the Einstein and Debye phonon contributions to the heat capacity, the residual heat capacity 8C(T) divided by T was obtained and the results for K3C60 are plotted vs. T in Fig. 14.19 at zero magnetic field and at a field of 6 T. The results show a specific heat jump at Tc [14.102], of magnitude (AC/TJ = 68 ± 13 mJ/mol K2, which was interpreted in terms of the expression where g(\ep) is a strong coupling correction term, \ep is the electron-phonon coupling coefficient, and As is a spin fluctuation contribution to the quasiparticle mass enhancement [14.102]. If the terms g(\ep), kep and ks in Eq. (14.16) are neglected, then N(EF) would become 28 states/eV-Qq. Since the values of g(\ep), \ep, and As are not accurately known, it is difficult at present to obtain a quantitative value for N(EF) directly from measurements of AC/T for K3C60. A comparison between the H = 0 and H = 6 T data in Fig. 14.19 yields dHc2/dT = -3.5 ± 1.0 T/K, in agreement with magnetization measurements [14.158-160], as discussed in §15.3.

Fig. 14.19. The specific heat contribution, 8C(T)/T, divided by temperature after subtraction of the phonon contribution shown in Fig. 14.18. The solid lines are to guide the eye and show a jump AC(T)/Tc due to the superconducting transition. Also shown are data taken in an applied magnetic field of 6 T. The two data sets are offset for clarity [14.102],

Fig. 14.19. The specific heat contribution, 8C(T)/T, divided by temperature after subtraction of the phonon contribution shown in Fig. 14.18. The solid lines are to guide the eye and show a jump AC(T)/Tc due to the superconducting transition. Also shown are data taken in an applied magnetic field of 6 T. The two data sets are offset for clarity [14.102],

14.9. Scanning Calorimetry Studies

Differential scanning calorimetry (DSC) has been widely applied to study phase transitions in C60, the main focus of this work being directed to establishing the order of the transition, the transition temperature, and the enthalpy changes associated with the transition [14.145,161-163]. The DSC results of Fig. 14.20 show a latent heat, indicating that the phase transition for C60 near 261 K is first order. For the particular sample that was studied in Fig. 14.20, the measured transition temperature was r01 = -15.34°C (257.66 K) and an enthalpy change of 9.1 J/g was measured [14.163,164]. Also shown in Fig. 14.20 is the effect of residual solvent in the sample, giving rise to a small downshift in the transition temperature T01 and a sizable decrease in the enthalpy change at 701. These results show the importance of sample quality for thermodynamic measurements on fullerenes. Calori-metric studies have also been used to establish the pressure dependence of the transition temperature Tm, which is observed to increase at a rate of 10.4-11.7 K/kbar (see §7.3) [14.111,165]. With this value of dTm/dp, the Clausius-Clapeyron equation has been used [14.165] to obtain a value for the isobaric volume expansion of the C60 lattice at T{n (see §14.10). Scanning calorimetric spectra of samples are often taken to test whether undoped C60 material is present in a given sample, by looking for the characteristic DSC structure for C60 in the vicinity of T(n, since the alkali metal-doped M_tC60 does not show structure in this temperature region.

DSC studies of C70 have also been carried out showing two first-order phase transitions (see §7.2), a heat of transition of 3.5 J/g at the lower

Fig. 14.20. Differential scanning calorimetry of chromatographed C«, samples, heated at 5°C/min under N2 flow. Top: scan performed immediately after room temperature evaporation of solvent and showing a broad precursor 10-20°C below the main peak. Bottom: scan performed after 200°C annealing for 18-24 h, showing a decrease in the magnitude of the precursor, an increase in transition temperature 7"0i, and an increase in area under the peak (measured between the fiducial marks) [14.163], transition temperature T02 (276 K) and of 2.7 J/g at the higher temperature T0l (337 K) [14.166],

14.10. Temperature Coefficient of Thermal Expansion

The temperature dependence of the lattice constant a0 for C60 (see Fig. 14.21) shows a large discontinuity at the structural phase transition temperature T0i = 261 K [14.153,167,168] (see §7.1.3 and §7.4). The change in a0 at T01 is found to be 0.044±0.004 A on heating [14.153], indicative of a first-order phase transition and consistent with other experiments, especially the heat capacity (see §14.8). From the Clausius-Clapeyron equation, measurement of the latent heat at the T0l phase transition yields a value for the fractional change in the volume of the unit cell of AF/K = 7.5 x 10-3 [14.165], which is in reasonable agreement with the direct structural measurement of 9.3 ± 0.8 x 10~3 [14.153],

The average isobaric volume coefficient of thermal expansion a both below and above the 261 K transition is 6.2 ± 0.2 x 10"5/K [14.153,167,168], which is relatively large compared to values for typical ionic crystals. Surprisingly, measurements of a taken to high temperature (1180 K) show a somewhat lower value of 4.7 x 10"5/K than for the lower temperature range [14.169]. The thermal expansion of K3C60 has also been measured i o u.

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