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[en] The results of the quantum theory of radiation of the 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. The conditions of suppression of spectral density of Cherenkov radiation are found too. The gotten results can be used for creation intensive, compact, short-wave (up to x-ray) FEL.