On nonlinear dynamics of a sheet electron beam

The present paper studies a relativistic electron beam with parabolic current density profile with the aim to predict a behaviour of the beam envelope with time, that would be useful for numerous projects of electron accelerator development. The nonlinear dynamics of the sheet electron beam is studied in collisionless approximation. The kinetic distribution function is dependent on the particle motion integral, so the distribution function automatically satisfies the Vlasov equation. Nonuniform transverse distribution of the beam current density is shown to lead to essentially nonlinear particle transverse oscillations. The oscillations can be described by Duffing equation with zero right part. Depending on nonlinearity power the growth of effective emittance can be observed at a time corresponding to about a quarter of the plasma wavelength