  The evolution of low-energy spin dynamics in the iron-based superconductor
LaFeAsO$_{1-x}$F$_x$ was studied over a broad doping, temperature, and magnetic
field range (x = 0 - 0.15, T up to 480K, H up to 30T) by means of As nuclear
magnetic resonance (NMR). An enhanced spin-lattice relaxation rate divided by
temperature, 1/T1T, in underdoped superconducting samples (x = 0.045, 0.05 and
0.075) suggests the presence of antiferromagnetic spin fluctuations, which are
strongly reduced in optimally-doped (x = 0.10) and completely absent in
overdoped (x = 0.15) samples. In contrast to previous analysis, Curie-Weiss
fits are shown to be insufficient to describe the data over the whole
temperature range. Instead, a BPP-type model is used to describe the occurrence
of a peak in 1/T1T clearly above the superconducting transition, reflecting a
progressive slowing down of the spin fluctuations down to the superconducting
phase transition.
