Acceleration of Electrons in a Combined Magnetic Trap in the Synchronous Mode

Electron capture into the mode of synchronous gyromagnetic autoresonance in a combined mirror-type magnetic trap with a high-frequency field in a cylindrical resonator is studied. The averaged equations of motion of electrons in such a trap are obtained taking into account the terms of the first order of smallness in the high-frequency field amplitude. An equation for the resonant phase, which has the form of an equation for a nonlinear oscillator with a constant force, is derived to change the magnetic field with time according to a linear law in a weakly relativistic approximation. Based on the analysis of its solutions, a general criterion for the electron capture into the gyromagnetic autoresonance mode is obtained. Using the Bogolyubov method, the change in the energy of particles is studied, taking into account the time dependence of the parameters of the combined trap. It is shown that during autoresonance, the change in the electron energy with time occurs synchronously with the change in the magnetic field.