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Linear theory of the parametric beam instability or the self-amplification ofparametric x-ray radiation (PXR) from relativistic electrons in a crystal isconsidered taking into account finite emittance of the electron beam andabsorption of the radiation. It is shown that these factors change essentiallythe estimation of threshold parameters of the electron bunches for the coherentX-ray generation. The boundary conditions for the linear theory of the effectis analyzed in details and it is shown that the grazing incidence diffractiongeometry is optimal for the growth of instability. Numerical estimations ofamplification and coherent photon yield in dependence on the electron currentdensity are presented for the case of mm-thickness Si crystal and 100 MeVelectrons. Possible improvements of the experimental scheme for optimization ofthe coherent radiation intensity are discussed.

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