Flute-Like Instabilities in Plasma with Cylindrical Relativistic Electron Beam

The linearized, relativistic Vlasov equations are analyzed for the stability of flute-like modes in an infinite, collisionless plasma with a cold background and a relativistic annular electron beam situated in a uniform external magnetic field. Neglecting self-fields, a dispersion equation is obtained for small thickness beams. It is found that oscillations with frequency near harmonics of the gyrofrequency of the relativistic electrons are unstable. The most unstable oscillations are shown to be those with long wavelengths relative to the thickness of the beam. Growth rates and conditions for instability are given for systems where the beam particles are charge neutralized by cold background ions, and when the beam particles are dilute compared with the background species. For rarefied beams, an instability occurs at the hybrid frequency of the background species where the growth rate depends on the beam thickness. As the background density increases, a critical value can be reached where the long-wavelength oscillations are stabilized; and short-wavelength oscillations become most unstable. For these modes growth rates are maximized with respect to the harmonic number ℓ, and beam velocity β = υ/c.