Theory of the resistive hose instability in relativistic electron beams

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Abstract
A theory of the resistive hose instability is developed for an infinitely long relativistic electron beam propagating parallel to an applied axial magnetic field. Complete space‐charge neutralization by the ambient plasma and paraxial flow (p2z≳≳p2r+p2ϑ) are assumed. An integro‐differential eigenvalue equation is derived from Vlasov–Maxwell theory for the general case, and this equation is reduced to an ordinary differential equation by using a model in which the class of beam electrons with a given transverse energy displaces as a rigid component. The model introduced spread betatron frequency in a natural way that includes correlations to radial position. Using a variational technique, an approximate dispersion relation is found for arbitrary density profile and is evaluated in closed form for either the Bennett or square profile. Stability properties are illustrated and discussed in detail for a square profile, including the influence of the applied magnetic field (stabilizing), proximity to a conducting guide (stabilizing), and partial current neutralization (destabilizing).