Self-consistent interaction between the plasma wake field and the driving relativistic electron beam

It is shown that the self-consistent interaction between wake fields and the driving electron bunch in a collisionless, unmagnetized, overdense (${\mathit{n}}_{\mathit{p}}$≫${\mathit{n}}_{\mathit{b}}$) plasma is governed by three coupled equations. In the long-beam limit, they reduce to a pair consisting of an appropriate nonlinear Schrödinger equation for the a/Ibeam wave function Ψ, and an equation for the wake-field potential that is driven by the transverse profile of the beam density, which is proportional to ‖Ψ${\mathrm{\Vert }}^2$. The pair of equations are suitable for studying the beam self-focusing (self-pinching equilibrium) for the case in which the beam-spot size is larger (smaller) than the wavelength of the wake fields. It is demonstrated that our self-consistent theory, which is based on the recently proposed thermal wave model for relativistic charged-particle beam propagation, is capable of reproducing the main results for the beam-filamentation threshold and the self-pinching equilibrium condition that are already known in the conventional theory of the beam self-interaction in collisionless plasmas.