Superradiant parametric X-ray emission

We compute a spectrum of parametric X-ray radiation (PXR) inside a crystal from a bunch of electrons, which is periodically modulated in density. We consider that the bunch of electrons is exiting from a XFEL channel. We demonstrate that in the case of a resonance between the frequency of parametric X-ray radiation and a frequency of modulation of an electron bunch the sequence of strong quasi-monochromatic X-ray pulses is formed -- superradiant parametric X-ray emission (SPXE) with frequencies multiples of the modulation frequency. The number of photons in the impulse of SPXE in the case of an extremely asymmetric diffraction is comparable with the photon number in the impulse of a XFEL. Moreover the SPXE is directed under the large angle to the electron velocity and every harmonic in the spectrum is emitted under its own angle.