  We report on a future multi-purpose fixed-target experiment with the proton
or lead ion LHC beams extracted by a bent crystal. The multi-TeV LHC beams
allow for the most energetic fixed-target experiments ever performed. Such an
experiment, tentatively named AFTER for "A Fixed-Target ExperRiment", gives
access to new domains of particle and nuclear physics complementing that of
collider experiments, in particular at RHIC and at the EIC projects. The
instantaneous luminosity at AFTER using typical targets surpasses that of RHIC
by more than 3 orders of magnitude. Beam extraction by a bent crystal offers an
ideal way to obtain a clean and very collimated high-energy beam, without
decreasing the performance of the LHC. The fixed-target mode also has the
advantage of allowing for spin measurements with a polarised target and for an
access over the full backward rapidity domain up to xF ~ - 1. Here, we
elaborate on the reachable luminosities, the target polarisation and a
selection of measurements with hydrogen and deuterium targets.
