  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.
