  We have probed the superconducting proximity effect through long high-quality
monocrystalline Ag nanowires, by realizing Josephson junctions of different
lengths, with different superconducting materials. Thanks to the high number of
junctions probed, both the contact resistance and electron diffusion constant
could be determined, enabling a comparison of the measured critical current to
theoretical expectation, over the entire regime from short to long diffusive
junction. Although the length dependence of the critical current is as
expected, the amplitude of the $R_{N}I_c$ product is smaller than predicted by
theory. We also address the magnetic field dependence of the critical current.
The quasi-gaussian decay of the critical current with field expected of a long
narrow junction is observed for all superconducting contacts we used except for
aluminum. We present the striking non-monotonous effect of field on the
critical current of junctions with aluminum contacts, and analyze it in terms
of improved quasiparticle thermalization by a magnetic field.
