  The Large Hadron electron Collider (LHeC) is a proposed future
particle-physics project colliding 60 GeV electrons from a six-pass
recirculating energy-recovery Linac (ERL) with 7 TeV protons stored in the LHC.
The ERL technology allows for much higher beam current and, therefore, higher
luminosity than a traditional Linac. The high-current, high-energy electron
beam can also be used to drive a free electron laser (FEL). In this study, we
investigate the performance of an LHeC-based FEL, operated in the
self-amplified spontaneous emission mode using electron beams after one or two
turns, with beam energies of, e.g., 10, 20, 30 and 40 GeV, and aim at producing
X-ray pulses at wavelengths ranging from 8~\AA\ to 0.5~\AA . In addition, we
explore a possible path to use the 40 GeV electron beam for generating photon
pulses at much lower wavelengths, down to a few picometre. We demonstrate that
such ERL-based high-energy FEL would have the potential to provide orders of
magnitude higher average brilliance at \AA\ wavelengths than any other FEL
either existing or proposed. It might also allow a pioneering step into the
picometre wavelength regime.
