Photon emission by electrons on four-wave diffraction in a monocrystal

Some properties of the electromagnetic waves emitted by electrons on Laue four-wave scattering by two sets of mutually perpendicular reflecting crystal planes are considered. It is shown that in the case under consideration, along with electric dipole transitions there also appear mixed electric quadrupole and magnetic dipole transitions which result in the emission of waves with frequencies ωsub(+-)=sub(1)+-ωsub(2)ω (ωsub(1, 2) are the electric dipole radiation frequencies). The corresponding multipole moments exceed the atomic values by several orders of magnitude. The multipole structure of the matrix elements arises as a result of expansion of the Bloch electron functions in the parameter of deviation from the Bragg condition in the final state and reflects the nature of the electron wave beat in the crystal. The usual approach, which is based on the expansion in terms of the radiation field multipoles, is not valid in the given case since it implies a restricted (e. g. by the channel boundaries or on channeling) region of motion of the partcle. As a result the radiation from a diffracted electron is found to be similar to the radiation from macroscopic multipoles moving in a refracting medium and oscillating eith different frequencies. This seems to be a unique situation from the viewpoint of experimental investigation of multipole radiation of multipoles moving with ''near-light'' or ''super-light'' velocities. Formulas for the radiation intensities are obtained