Method for calculating the density of relativistic charged particles moving in a time-dependent electric field

We propose a method for calculating the density of relativistic charged particles that come from a planar boundary with a known velocity distribution function and move collisionless in the half-space under a time-dependent electric field of arbitrary form. Unlike the case when particles move in a stationary field, the density expression contains two additional functions: G and Q, which are associated with the field changeover time. Explicit expressions as integrals of field derivatives with respect to time are obtained for them. For the special case when particles move in a field that varies weakly in time, we get explicit expressions for G and Q functions as linear operators acting on small field perturbation amplitude. The results obtained can be used to study steady-states stability of diodes with relativistic charged particle flows. Stability features of a relativistic Bursian diode steady states for a regime without electron reflection from a potential barrier are studied as an example. The dispersion equation is obtained and solved. The eigen mode growth rates and frequencies as functions of the electron current density are built, and the relativistic factor effect on these dependences is studied.