At ISIS (Oxford neutron source)…

Spallation generation: High-energy protons (~800 MeV) hit a heavy target, releasing a wide spectrum of fast neutrons up to hundreds of MeV. These are then moderated down to useful energies for experiments.

It’s not the LHC, sure. But I don’t see any reason (apart from “why bother”) why they can’t do spallation in Geneva. OK maybe there’s a cooling problem…