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We report on a future multi-purpose fixed-target experiment with the protonor lead ion LHC beams extracted by a bent crystal. The multi-TeV LHC beamsallow for the most energetic fixed-target experiments ever performed. Such anexperiment, tentatively named AFTER for "A Fixed-Target ExperRiment", givesaccess to new domains of particle and nuclear physics complementing that ofcollider experiments, in particular at RHIC and at the EIC projects. Theinstantaneous luminosity at AFTER using typical targets surpasses that of RHICby more than 3 orders of magnitude. Beam extraction by a bent crystal offers anideal way to obtain a clean and very collimated high-energy beam, withoutdecreasing the performance of the LHC. The fixed-target mode also has theadvantage of allowing for spin measurements with a polarised target and for anaccess over the full backward rapidity domain up to xF ~ - 1. Here, weelaborate on the reachable luminosities, the target polarisation and aselection of measurements with hydrogen and deuterium targets.

A Fixed-Target ExpeRiment at the LHC (AFTER@LHC) : luminosities, target polarisation and a selection of physics studies