Intense non-neutral beam propagation in a periodic solenoidal field using a macroscopic fluid model with zero thermal emittance

Abstract

A macroscopic fluid model is developed to describe the nonlinear dynamics and collective processes in an intense high-current beam propagating in the z-direction through a periodic focusing solenoidal field Bz(z+S)=Bz(z), where S is the axial periodicity length. The analysis assumes that space-charge effects dominate the effects of thermal beam emittance, Krb2≫εth2, and is based on the macroscopic moment-Maxwell equations, truncated by neglecting the pressure tensor and higher-order moments. Here, K=2NbZi2e2/γ̂b3mβb2c2 is the self-field perveance, Nb is the number of particles per unit axial length, and rb is the characteristic beam radius. Assuming a thin beam with rb≪S, azimuthally symmetric beam equilibria with ∂/∂t=0=∂/∂θ are investigated, allowing for an axial modulation of the beam density nb(r,z) and macroscopic flow velocity Vrb(r,z)êr+Vθb(r,z)êθ+Vzb(r,z)êz by the periodic focusing field. To illustrate the considerable flexibility of the macroscopic formalism, assuming (nearly) uniform axial flow velocity Vb over the beam cross section, beam equilibrium properties are calculated for two examples: (a) uniform radial density profile over the interval 0⩽r<rb(z), and (b) an infinitesimally thin annular beam centered at r=rb(z). The analysis generally allows for the azimuthal flow velocity Vθb(r,z) to differ from the Larmor frequency, and the model is used to calculate the (leading-order) correction δVzb(r,z) to the axial flow velocity for the step-function density profile in case (a) above.