Linear Theory Analysis of Self-Amplified Parametric X-ray Radiation from High Current Density Electron Bunches

Linear theory of the parametric beam instability or the self-amplification of parametric x-ray radiation (PXR) from relativistic electrons in a crystal is considered taking into account finite emittance of the electron beam and absorption of the radiation. It is shown that these factors change essentially the estimation of threshold parameters of the electron bunches for the coherent X-ray generation. The boundary conditions for the linear theory of the effect is analyzed in details and it is shown that the grazing incidence diffraction geometry is optimal for the growth of instability. Numerical estimations of amplification and coherent photon yield in dependence on the electron current density are presented for the case of mm-thickness Si crystal and 100 MeV electrons. Possible improvements of the experimental scheme for optimization of the coherent radiation intensity are discussed.