A general expression is derived for the dipolar NMR spin-lattice relaxation rate $1/T_1$ of a system exhibiting Brownian dynamics in a discrete and finite configuration space. It is shown that this approach can be particularly useful to model the proton relaxation rate in molecular rotors.
Former extensive studies of superconductivity in the textit{A}$_{3}$C$_{60}$ compounds, where textit{A} is an alkali, have led to consider that Bardeen Cooper Schrieffer (BCS) electron-phonon pairing prevails in those compounds, though the incidence
of electronic Coulomb repulsion has been highly debated. The discovery of two isomeric fulleride compounds Cs$_{3}$C$_{60}$ which exhibit a transition with pressure from a Mott insulator (MI) to a superconducting (SC) state clearly re-opens that question. Using pressure ($p$) as a single control parameter of the C$_{60}$ balls lattice spacing, one can now study the progressive evolution of the SC properties when the electronic correlations are increased towards the critical pressure $p_{c}$ of the Mott transition. We have used $^{13}$C and $^{133}$Cs NMR measurements on the cubic phase A15-Cs$_{3}$C$_{60}$ just above $p_{c}=5.0(3)$ kbar, where the SC transition temperature $T_{c}$ displays a dome shape with decreasing cell volume. From the $T$ dependence below $T_{c}$ of the nuclear spin lattice relaxation rate $(T_{1})^{-1}$ we determine the electronic excitations in the SC state, that is $2Delta$, the SC gap value. We find that $2Delta $ increases with decreasing $p$ towards $p_{c}$, where $T_{c}$ decreases on the SC dome, so that $2Delta /k_{B}T_{c}$ increases regularly upon approaching the Mott transition. These results bring clear evidence that the increasing correlations near the Mott transition are not significantly detrimental to SC. They rather suggest that repulsive electron interactions might even reinforce elecron-phonon SC, being then partly responsible for the large $T_{c}$ values, as proposed by theoretical models taking the electronic correlations as a key ingredient.
We have performed sound velocity measurements in (V$_{1-x}$Cr$_x$)$_2$O$_3$ in the vicinity of the critical point of the first order Mott transition line. The pressure sweeps at constant temperature reveal a large dip in the $c_{33}$ compression modu
lus, this dip sharpens as the critical point is approached. We do not observe signs of criticality on the shear modulus $c_{44}$ which is consistent with a transition governed by a scalar order parameter, in accordance with the DMFT description of the transition. However, the amplitude of the effect is an order of magnitude smaller than the one obtained from DMFT calculations for a single band Hubbard model. We analyze our results using a simple model with the electronic response function obtained from the scaling relations for the conductivity.
