NMR studies of slow motion in the organic superconductors k-(BEDT-TTF)2Cu[N(CN)2]X, X= Br, Cl

The $^{13}$C NMR absorption lines in the organic superconductors $\kappa$-(BEDT-TTF)$_2$Cu[N(CN)$_2$]X, X~=~Br ($T_C=11.6$~K), Cl ($T_C=12.8$~K at 0.3~kbar) show an abrupt increase in the linewidth at 170~K at 9.4~T. This line broadening transition is believed to be caused by the abrupt slowing down of some ``motion'' as suggested by a sharp peak in the exponential component of the transverse relaxation rate ($1/T_{2E}$) at a slightly lower temperature at 9.4~T. To investigate the characteristics of this motion, we applied the multi-pulse ``S-shape'' sequence. Two preparation pulses label the spins according to their precession frequencies and modulate the NMR spectrum. We then allow the spins to evolve for a time $T_{ev}$, after which inspection pulses are applied to observe changes in the spectrum. The modulation in the NMR spectrum shows a decay as $T_{ev}$ increases. The decay is described by 2 time constants, $R_\mathrm{fast}(T)$ and $R_\mathrm{slow}(T)$. Each rate shows a sharp peak at some temperature near to, but not at, the linewidth transition. Outside the transition region, $R_\mathrm{fast}$ is independent of $T$ and is close to the Gaussian component of the transverse relaxation rate $1/T_{2G}$. $R_\mathrm{fast}$ represents the relaxation of the spectrum modulations by dipolar coupling. $R_\mathrm{slow}$ is also largely $T$ independent outside the transition region. Unlike $R_\mathrm{fast}$, $R_\mathrm{slow}$ shows a strong $T_p$ dependence, namely, $R_\mathrm{slow}$ varies as $T_p^2$. $T_p$ is the delay between the 2 preparation pulses, and $\pi/T_p$ is equal to the separation between the adjacent peaks and troughs in the spectrum modulations. The $T_p^2$ dependence of $R_\mathrm{slow}$ reflects the diffusive nature of the motion relaxing the modulations. It has been proposed that the motion causing the linewidth transition is the rotation of the ethylene end groups. But a second moment calculation reveals that the ethylene end groups contribute only a small fraction of the total measured $^{13}$C linewidth below the linewidth transition. An anisotropic Knight shift calculation suggests that the (ET)$_2$ dimers only need to deviate from its equilibrium position by $2.5^\circ$ to produce a large enough spread in Knight shifts to account for the total measured linewidth below the linewidth transition. We believe a diffusive rotational motion of the (ET)$_2$ dimers together with a BPP style relaxation model is the cause of the peak of $R_\mathrm{slow}$, the $T_p^2$ dependence of $R_\mathrm{slow}$, the linewidth transition, as well as the peak in $1/T_{2E}$