Upper bound for the space-charge limited current of relativistic electron beams in finite-length coaxial drift tubes

Abstract

A two-dimensional theory for calculating an upper bound for the space-charge limited current of relativistic electron beams is extended to coaxial drift tubes. The theory applies to annular beams of arbitrary radius, thickness and length travelling in a grounded coaxial drift tube of circular cross section. Poisson’s equation for space-charge limited flow is replaced with a Sturm–Liouville-type eigenvalue problem. The problem reduces to numerically solving a determinantal equation rather than a partial differential equation. Under limiting cases, closed-form analytical expressions are derived.

Relativistic Charged-Particle Beam Space-Charge Limited Current in Finite Length Coaxial Drift Tube

“…Then, for another series of (0th-order) annular (difference) Bessel functions Series of the type (20) appear in solutions to the Poisson equation by the method of Green's function in coaxial cylindrical and finite thickness spherical geometries (see, for example, [29]). However, there series of annular (difference) Bessel functions of order ν ≥ 1 are involved.…”

“…We proposed simple but reliable analytical and numerical estimates of SCL current of annular relativistic finite thickness charged-particle beams propagating in long (unbounded) coaxial drift tubes, which may also have dielectric inserts lining the tube coaxial conductors and a bias voltage applied between the inner and outer conductors of the drift tube, in the strong magnetic field approximation [8], [11]. Some researchers have found analytical and developed numerical estimates of the charged-particle beam SCL current in finite in the (longitudinal) direction of the beam propagation (i.e., bounded) cylindrical drift tubes [12], [16]- [20]. In this paper, we aim to establish an analytical estimate of the finite thickness relativistic charged-particle beam SCL current in a coaxial drift tube bounded by annular conducting end plates (grids) transparent for the propagating in the strong guide magnetic field finite thickness relativistic chargedparticle beam.…”