$^{1}$H-NMR spin-echo measurements of the static and dynamic spin properties in $\lambda$-(BETS)$_{2}$FeCl$_{4}$

$^{1}$H-NMR spin-echo measurements of the spin-echo decay $M(2\tau)$ with a decay rate 1/$T_{2}$ and the frequency shift $\Delta\nu/\nu_{0}$ under applied magnetic field $\mathbf{B}$$_{0}$ = 9 T along the a-axis over a temperature range 2.0$-$180 K are reported for a single crystal of the organic conductor $\lambda$-(BETS)$_{2}$FeCl$_{4}$. It provides the spin dynamic and static properties in the paramagnetic metal (PM) and antiferromagnetic insulator (AFI) states as well as across the PM$-$AFI phase transition. A large slow beat structure in the spin-echo decay is observed with a typical beat frequency of $f$ $\sim$ 7 kHz and it varies across the spectrum. Its origin is attributed to the $^{1}$H$-$$^{1}$H dipole interactions rather than to the much larger dipolar field contribution from the Fe$^{3+}$ electrons (spin $S$ = 5/2). A simple phenomenological model provides an excellent fit to the data. The dominant $^{1}$H-NMR frequency shift comes from the dipolar field from the 3d Fe$^{3+}$ ions, and the Fe$^{3+}$ $-$ Fe$^{3+}$ exchange interactions ($J_{0}$) ($J_{0}$ includes the d$-$d exchange interactions through the $\pi-$electrons) have a substantial effect to the local field at the proton sites expecially at low temperatures. A good fit is obtained with $J_{0}$ = - 1.7 K. The data of the spin-echo decay rate 1/$T_{2}$ indicates that there is a significant change in the slow fluctuations of the local magnetic field at the $^{1}$H-sites on traversing the PM to AFI phase. This evidence supports earlier reports that the PM$-$AFI phase transition in $\lambda$-(BETS)$_{2}$FeCl$_{4}$ is driven magnetically and first order.