High repetition rate and coherent Free-Electron Laser in the tender X-rays based on the Echo-Enabled Harmonic Generation of an Ultra-Violet Oscillator pulse.

Fine time-resolved analysis of matter - i.e. spectroscopy and photon scattering - in the linear response regime requires a fs-scale pulsed, high repetition rate, fully coherent X-ray source. A seeded Free-Electron Laser (FEL) driven by a Super-Conducting Linac, generating $10^{8}$-$10^{10}$ coherent photons at 2-5 keV with abou 0.5 MHz of repetition rate, can address this need. The seeding scheme proposed is the Echo-Enabled Harmonic Generation, alimented by a FEL Oscillator working at 13.6 nm with a cavity based on Mo-Si mirrors. The whole chain of the X-ray generation is here described by means of start-to-end simulations. Comparisons with the Self Amplified Spontaneus Emission and a fresh-bunch harmonic cascade performed with similar electron beams show the validity of this scheme.