  The magnetic interactions between the spin of an unpaired electron and the
surrounding nuclear spins can be exploited to gain structural information, to
reduce nuclear relaxation times as well as to create nuclear hyperpolarization
via dynamic nuclear polarization (DNP). A central aspect that determines how
these interactions manifest from the point of view of NMR is the timescale of
the fluctuations of the magnetic moment of the electron spins. These
fluctuations, however, are elusive, particularly when electron relaxation times
are short or interactions among electronic spins are strong. Here we map the
fluctuations by analyzing the ratio between longitudinal and transverse nuclear
relaxation times T1 and T2, a quantity which depends uniquely on the rate of
the electron fluctuations and the Larmor frequency of the involved nuclei. This
analysis enables rationalizing the evolution of NMR lineshapes, signal
quenching as well as DNP enhancements as a function of the concentration of the
paramagnetic species and the temperature, demonstrated here for LiMgMnPO4 and
Fe(3+) doped Li4Ti5O12, respectively. For the latter, we observe a linear
dependence of the DNP enhancement and the electron relaxation time within a
temperature range between 100 and 300K.
