Three component non-symmetric counter-streaming instabilities: longitudinal electron plasma modes

The counter-streaming instabilities arising in three-component electron plasmas are investigated analytically and numerically in the general non-symmetric case, i.e. when Here n₁ n₂ and n_a represent the electron particle density in the first and second beams and in the background (ambient) stationary plasma, respectively; U₁ and U₂ represent the streaming velocities of the two counter-streaming electron beams. No magnetic or temperature effects are considered; consequently the three components interact only through the electric collective fields and only longitudinal modes are present. The positive ions here represent a stationary neutralizing background. Combined analytical and numerical solutions of the dispersion equation indicate that the basic properties of the unstable plasma modes may change significantly, depending on the values of the dimensionless parameters e, a and g. Thus, the standing wave spectrum (Re ω = 0) which occurs in the symmetric case (ɛ = a = 1) without background (gr = 0) may be replaced by a mixed travellingstanding wave spectrum having a rather complex structure; the maximum growth rate could be also strongly affected. The transformation of the instability from ‘absolute’ into mixed ‘convective and absolute’ may have significant physical implications, especially for finite size plasma systems or finite length unstable interaction regions. The results are relevant for laboratory and (especially) astrophysical situations in which counter-streaming electron beams having unequal streaming velocities (and particle densities) penetrate plasma regions with significant relative particle concentration.