Parametric instabilities in density modulated relativistic electron beam plasma interaction

Parametric coupling between negative energy beam synchronous modes and positive energy plasma oscillation modes of a uniform beam-plasma system, due to the presence of sinusoidal density modulation in the monoenergetic, relativistic electron beam, have been theoretically investigated on the basis of linearized Vlasov--Poisson equations using a multiple-time-scale perturbation theoretical approach. It has been found that, for a given beam-to-plasma density ratio, there exists an upper limit to the initial beam energy for which the instabilities can be excited. Within the allowed range, maximum linear growth rate exhibits a minimum as the beam energy increases from its nonrelativistic value. To determine the effectiveness of the parametric interaction process, feedback effects of the instabilities on the beam-plasma distribution functions have also been considered and estimates of energy transfers have been made. For typical parameters, the beam is found to lose about 15% of its initial kinetic energy during the ''hydrodynamic'' stage and the time required is about 4.72 nsec