## 1 Introduction  
Synchrotron radiation (SR) sources based on low-emittance electron storage rings, as well as Free Electron Lasers (FELs) driven by linear electron accelerators (Linacs), allow the fine analysis of matter, with applications extending from life sciences to material physics. Most FELs worldwide operate in the Self-Amplified Spontaneous Emission (SAE) mode [1, 2, 3, 4, 5, 6, 7, 8], providing extremely brilliant and short pulses with more than 10\({}^{12}\) photons per pulse, and shot-to-shot time and intensity titers determined by the intrinsic fluctuations of the emission process. These Ultra-Violet (UV) or X-ray flashes are used to probe matter in highly excited states, to study nonlinear processes or to test before destroying individual objects such as macromolecules constituting proteins, thus replacing crystallography with single object imaging. In order to remain below the linear response threshold and to collect adequate statistics in a short time, spectroscopic probes, as well as pump-probe photoemission experiments in advanced atomic, molecular, nano- and solid- state physics, instead require ultra-short photon pulses (order 10 fs), MHz-class repetition rates and moderate fluxes of photons/pulse. Currently available FEL sources produce a number of photons per pulse exceeding by 2-4 orders of magnitude the linear response level: severe attenuation of the pulses, with a huge waste of energy, is therefore required in photoemission or X-ray absorption spectroscopy. The pulses' repetition rate is determined by the acceleration technology, and is much higher in Super-Conducting facilities, exceeding the 10-100 Hz range of Warm Linacs up to MHz-class [4, 5]. Moreover, spectroscopic applications with X-ray FELs are severely limited by SASE fluctuations and full seeding, successfully conducted at FERMI (Free-Electron laser for Multidisciplinary Investigations) in the XUV-soft X-ray range [9], should ideally be extended to X-ray energies. There is therefore a scientific need, and ample room, for a novel type of source that is able to provide 10 fs coherent pulses of 10\({}^{7}\)-10\({}^{8}\) photons at 0.5-2 MHz in the tender X ray range, bridging the gap between the most advanced SR and the current FEL sources. These demanding requests about structure, intensity, repetition rate, reduced jitters and true coherence of the pulses are addressed by conceiving a tailored seeded or self-seeded FEL driven by a linac based on Super Conducting cavities, providing \(10^{7}\)-\(10^{10}\) coherent photons at 2-5 keV, at 1 MHz of repetition rate. In the seeded amplifier FEL configuration, the pulses' stability and coherence is strongly increased by an external coherent source imprinting its temporal phase on the electron beam at the undulator entrance, thus enabling us to reach a high degree of temporal coherence within a short distance. The direct seeding [10] implementation is limited by the lack of high-power coherent seeds in the Vacuum-UV regime and below. High Gain Harmonic Generation (HGHG) multistage cascades [11], seeded with harmonics of an IR laser generated in crystals [11; 12; 13], were studied in the optical-UV range [14; 15; 16] and were demonstrated at FERMI up to a few nm wavelengths [9]. The implementation and reliability of this scheme in the tender/hard X-ray spectral range is highly demanding and has ye to be proven. In perspective, the X-ray range could be reached with the technique of the Echo Enabled Harmonic Generation (EEHG) [17; 18; 19; 20], in which two coherent radiation pulses seed the electron beam in two sequential modulators interspersed by a strong dispersive section. Both cascaded schemes allow us to reach the boundary between Extreme Ultra-Violet and soft X-ray frequencies, and may be seeded by laser harmonics generated in gases or UV oscillators [21; 22; 23; 24].