A rational view of the beam instabilities

Constructing the dielectric tensor of a beam-component plasma necessarily introduces various kinds of singularities. Here, we present a rational singularity-removal method for constructing the dielectric tensor and solving for the dispersion relation in a cold plasma with beam components. We retain the finite electron mass. The method makes extensive use of the polynomial expression of the dielectric tensor. The wave equation is arranged into a rational expression by removing the singularities and the unphysical modes prior to finding the roots of the equation. The singularity removal method has the advantages that the seed frequency is no longer needed and that all the physically relevant modes are obtained. The method is successfully tested against the right-hand beam instability, in which both single and triple solutions may appear, depending on the beam velocity. The method is also applied to the oblique beam instability, which incorporates the full 3-by-3 dispersion matrix. Our algorithm is computationally inexpensive and can be used as a dispersion solver as well as an instability analysis tool for various setups of beams and instability scenarios.