Formation of radiation spectrum at the planar channeling of a positron bunch with allowance for the medium inhomogeneity

The radiation of a positron in a planar channel of a crystal is studied, considering the dependence of the medium polarization on the transverse coordinate. The expression for the spectral distribution of the number of radiated photons is obtained for a single positron. It is shown that because of the medium polarization in the soft and hard regions not all positrons make a contribution to the radiation formation. An analytical formula for the total spectral distribution of the number of photons is obtained.