P.D. Kravchenya  Post-graduant Student, Department of Physics Volgograd State Technical University (VSTU)
A.G. Shein  D.Sc. (Phys.-Math.), Professor, Department of Physics Volgograd State Technical University (VSTU)

Because of relativistic electron beams are the oscillatory system which can provide the powerful electromagnetic waves generation and amplification under the certain conditions the investigation of the different wave phenomena in them is an important scientific and technical problem. The relativistic beams moving in the crossed fields are of particular interest. The theoretical analysis in this paper is performed for the slim tape relativistic electron beams moving in the crossed fields. The laminar beams with space charge and self-magnetic field which are speed-modulated by the external high frequency signal are considered. The joint solving of both the charged beam particle movement equation and Maxwell equations in the small signal and the little beam height approximations are in the basis of the theory. All the variables dependencies on the time and longitudinal coordinate are supposed to be harmonic to bring all of differential equations to algebraic. The result of this analysis is the dispersive equation relating charge waves propagation constants with the speed of a beam, the frequency of its speed modulation, the space charge and the external magnetic field. It is an algebraic equation of the sixth order having four real and two complex conjugate roots at low beam current, weak fields in the interaction space and modulation frequency more then plasmic one. The paper examines the only real solutions of the dispersion equation. They correspond to two longitudinal space charge waves and two excitation electromagnetic waves. One space charge wave is fast and one is slow. Spreading in the beam they lead to the appearance of speed spatial beats which are confirmed by the numerical beam simulation. The fact of all theoretically predicted waves excitation in the experimental beams is confirmed by the spectral analysis of the dependence of the average particles speed in the beam cross section on the longitudinal coordinate.