Collisionless self-consistent trapping of electrons into a nonstationary potential well: Dynamics of trapped electrons

The subject of investigation is the early stage of self-consistent trapping of electrons into a potential well that forms during the development of aperiodic Pierce instability. An analytical estimate for threshold gap δ_th = d _th/λ_D (λ_D is the Debye beam length) above which the trapping begins is derived. The nonlinear dynamics and distribution function of trapped electrons are studied in detail using a numerical method ((E, K) code). It is found that the trapped particles produce a localized steep-edge bunch, which “dangles” around between the electrodes, causing potential oscillations. Trapped electrons render the well shallower. Some of the particles in the bunch are shown to periodically escape to the electrodes. As a result, the potential oscillation amplitude fades away and the mean depth of the well increases.