Kinetic Treatment of the Stability of a Relativistic Particle Beam Passing through a Plasma

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Abstract
The equilibrium configuration consists of a uniform particle beam of circular cross section and infinite extent streaming at highly relativistic velocity through a uniform, dense background plasma. The plasma is characterized by a scalar conductivity, and the beam is described by a collisionless Boltzmann equation in which a two mass approximation to the relativistic dynamics has been made. The stability problem for this configuration is formulated as a set of three linear, coupled integral equations for three field variables (certain Hankel transforms of the perturbed electric field), and a formal solution of the equations is obtained by iteration. The dispersion relation appears as a solvability condition. The treatment gives a full account of the betatron orbits of the beam particles. Asymptotic results are obtained for low-frequency, long-wavelength disturbances and for high-frequency, highly localized disturbances.