Self amplification of channeling radiation at Bragg diffraction conditions

X-ray radiation of relativistic electrons passing through the crystal lattice (in case of channeling radiation) requires electrons with energies just from tens up to hundreds MeV, but has a drawback: low number of emitted X-ray quanta. However, the brightness of such a source could be potentially increased if the current density of the electron bunch is high enough to initiate the self amplified spontaneous emission (SASE) process. The conditions at which this phenomenon could take place for the case of axial channeling are analyzed in the present paper within the first order perturbation theory. The transition from spontaneous to SASE regime is described, the requirements for bunch current parameters initiating SASE process are determined taking into account the periodic structure of the crystal medium. It is also shown that satisfying Bragg diffraction conditions could enhance self amplification. Numerical results for the case of axial channeling in Si, Ge and C crystals are presented as an example.