Kinetic approach to deriving the envelope equation for a relativistic electron beam propagating in the ion focusing regime in the presence of an ion channel with an arbitrary radial profile

Kinetic methods are applied to derive the transport equations, the virial equations, the dynamic-equilibrium equation, and the envelope equation for an axially symmetric paraxial relativistic electron beam propagating in the ion focusing regime in the presence of an ion plasma channel with a radial density profile differing significantly from the corresponding profile of the beam current density. The resultant equations contain the terms that take into account this difference. In addition, the equations obtained account for the case when the electron component of the background plasma is only partially removed from the beam region.