Electron-electron correlations in fullerene ${\mathrm{C}}_{\mathbf{60}}$ probed by incoherent scattering of x rays

The static structure factor of electrons in fullerene has been obtained by a properly designed x-ray diffraction experiment. Due to the intense diffuse scattering caused by the orientational disorder of the ${\mathrm{C}}_{60}$ molecules in fullerene, the experiment presented several difficulties. Nonetheless, the data were accurate enough to provide an estimate of the exchange-correlation energy and to make a valuable comparison of the static structure factors of fullerene and diamond. The observed value of the exchange-correlation energy, which is larger in fullerene than in diamond, suggests a clear link between carbon polymorphism and dynamic electron correlations. The experimental data of static structure factor of fullerene, together with the derived exchange-correlation contribution to the cohesion and the pair-correlation function, offer an interesting test on the validity of the local-density treatment of the electron correlations in solids.

Figure 1

Rocking curve of the (222) reflection of ${\mathrm{C}}_{60}$ (lower panel): the raw intensity data are shown versus the rocking angle $\omega$. Coupled $\left(\omega \text{−}2\vartheta \right)$ scan along the [110] direction of the ${\mathrm{C}}_{60}$ crystal (upper panel): the raw intensity data are plotted vs the scattering angle $2\vartheta$.

Figure 2

Raw-intensity data measured along the reciprocal lattice line (shown in the inset) vs scattering angle 2$\vartheta$ in ${\mathrm{C}}_{60}$ (dots). The attenuation-corrected background to be subtracted from the intensity data is also shown (empty triangles). The dashed line is a guide to the eye. Inset: $\left(1\overline{1}0\right)$ reciprocal lattice section of ${\mathrm{C}}_{60}$ showing the path line followed by one of the $2\vartheta$ scans.

Figure 3

Electronic static-structure factor $S\left(Q\right)$ vs exchanged wave vector $Q$ in ${\mathrm{C}}_{60}$ and diamond. Circles: spherical average of the ${\mathrm{C}}_{60}$ experimental data (see text). Full line: best fit curve to the ${\mathrm{C}}_{60}$ experimental data, according to Eq. (3). Dot-dashed line: calculated core-electron contribution in ${\mathrm{C}}_{60}$. Short-dashed line: spherical average of the diamond experimental data (Ref. 7). Dashed line: calculated free-carbon atom (Ref. 15).

Figure 4

Static-structure factor $S^{dis}\left(Q\right)$ arising from the diffuse scattering associated to TDS and molecular orientational disorder in ${\mathrm{C}}_{60}$. Continuous line: data measured in the neutron-diffraction experiment of Ref. 13. Dots: data derived from the present x-ray experiment as an internal consistency check of the data-analysis procedure (see text).

Figure 5

Experimental static structure factor $S_{val}\left(Q\right)$ of the valence electrons in ${\mathrm{C}}_{60}$ (circles, present experiment) and diamond (empty triangles, experiment of Ref. 7). Calculated static structure factor of the valence electrons in ${\mathrm{C}}_{60}$: dotted line, homogeneous interacting electron gas model at the best fitting value $r_s$ = 0.99; dashed line, LDA for $w(Q;n(\mathbf{r}\left)\right)$; continuous line, beyond-LDA for $w(Q;n(\mathbf{r}\left)\right)$ (see Eq. (6)). Inset: model pair correlation function $g\left(r\right)$ in ${\mathrm{C}}_{60}$ (continuous line) and diamond[7] (dashed line).