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-\r\ntype magnetic trap with a high-frequency field in a cylindrical resonator is studied. The averaged equations\r\nof motion of electrons in such a trap are obtained taking into account the terms of the first order of\r\nsmallness in the high-frequency field amplitude. An equation for the resonant phase, which has the form of\r\nan equation for a nonlinear oscillator with a constant force, is derived to change the magnetic field with time\r\naccording to a linear law in a weakly relativistic approximation. Based on the analysis of its solutions, a general\r\ncriterion for the electron capture into the gyromagnetic autoresonance mode is obtained. Using the Bogolyubov\r\nmethod, the change in the energy of particles is studied, taking into account the time dependence of the\r\nparameters of the combined trap. It is shown that during autoresonance, the change in the electron energy\r\nwith time occurs synchronously with the change in the magnetic field.