The new approach to creation of sources of intensive coherent ultra-violet and x-ray radiation is offered. This approach is based on that physical fact, that at presence even weak periodic heterogeneous of dielectric or potential, in which nonrelativistic charged particles are moving, the spectrum of their radiation can, practically, coincide with a spectrum of relativistic particles. In our previous works this physical mechanism was studied within the framework of classical electrodynamics.
In the present work the results of the quantum theory of radiation of nonrelativistic charged particles, which are moving in spatial – periodic potential and in an external electromagnetic field are stated. The conditions of excitation of radiation, which frequency considerably surpasses frequency of an external electromagnetic wave are found. It is shown, that the efficiency of such radiation is close to efficiency of Cherenkov radiation. The maximum in a spectrum of this radiation is near to high numbers of harmonics of an external wave. This maximum coincides with a maximum of radiation of oscillators, which are moving in periodic – nonuniform dielectric (at equality of the periods of heterogeneity of potential and dielectric). The conditions of suppression of spectral density of Cherenkov radiation are found too. The gotten results are interesting for understanding of mechanisms of radiation of the charged particles which are oscillating in crystals. The crystal grating has as periodic heterogeneity of dielectric permeability, so periodic heterogeneity of potential. The periods of these heterogeneities are coincide. The crystals can be a basic element of sources for coherent radiation. At this the grating plays a role of wiggler. Free electrons at acting on them of an external laser field turn in radiating oscillators. The radiating system similar to FEL is formed. And the period such FEL appears minimal of all real periods. Density of emitters also appears maximal – density of electrons of solid.
The gotten results can be used for creation intensive, compact, short-wave (up to x-ray) FEL.