Lester E. Thode

Linear theory of a cold relativistic beam propagating along an external magnetic field

The linear theory of a cold relativistic electron beam propagating parallel to an external magnetic field and through a cold, homogeneous plasma is investigated. The electromagnetic dispersion relation is solved numerically and compared with analytical predictions based on the electrostatic approximation. It is found that electromagnetic effects are important for determining the entire unstable spectrum. However, except for the strong magnetic field regime, the maximum growth rates and corresponding frequencies are in agreement with those predicted by the electrostatic approximation. In the strong magnetic field regime the two−stream spectrum is found to be much narrower in angle than predicted by the electrostatic approximation. In the moderate and strong magnetic field regime the growth rate of waves propagating at large angles with respect to the beam are independent of beam energy.

Vacuum propagation of solid relativistic electron beams

An investigation of equilibrium, stability, and space‐charge‐ limiting current of a solid relativistic beam propagating along a finite external magnetic field has been carried out and compared with experiments. The concept of a rigid‐rotor equilibrium is only approximately valid when the beam current is much less than the Alfven critical current. Typically, low‐voltage beams rotate fastest at the beam edge whereas high‐voltage beams rotate fastest near the beam axis. A limited investigation of the rigid rotor stability condition indicates, at worst, a weak instability may be present if the rigid‐rotor equilibrium is artificially imposed, and no instability at all for a self‐consistent equilibrium. Numerical solutions for the space‐charge limiting current and relativistic factor on axis are presented and compared with two‐dimensional, cylindrical, space‐ and time‐dependent simulations. Analytical expressions for the limiting current are valid for ωc/ωb≳5, where ωc is the cyclotron frequency and ωb is the b...

Plasma heating by scattered relativistic electron beams: correlations among experiment, simulation, and theory

The streaming instability is the primary heating mechanism in most, if not all, experiments in which the beam is injected into partially or fully ionized gas. In plasma heating experiments the relativistic beam must traverse an anode foil before interacting with the plasma. The linear theory for such a scattered beam is discussed, including a criterion for the onset of the kinetic interaction. A nonlinear model of the two‐stream instability for a scattered beam is developed. Using this model, data from ten experiments are unfolded to obtain the following correlations: (i) for a fixed anode foil the dependence of the plasma heating on the beam‐to‐plasma density ratio is due to anode foil scattering, (ii) for a fixed beam‐to‐plasma density ratio the predicted change in the magnitude of plasma heating as a function of the anode foil is in agreement with experiment, and (iii) the plasma heating tentatively appears to be proportional to the beam kinetic energy density and beam pulse length.

Simple criteria for the absence of the beam‐Weibel instability

Rigorously sufficient and approximately necessary conditions for the absence of the beam‐Weibel instability are derived. These conditions include previously known stability criteria and resolve the seeming contradiction that these modes can be stabilized by beam temperature when the plasma is cold, but they cannot be stabilized by beam temperature when the plasma has infinitesimally small temperature.

Formation of virtual cathodes and microwave generation in relativistic electron beams

Simulation of the generation of a relativistic electron beam in a foil diode configuration and the subsequent intense microwave generation resulting from the formation of the virtual cathode is presented. The oscillating virtual cathode and the trapped beam electrons between the real and the virtual cathodes were found to generate microwaves at two distinct frequencies. Generation of high‐power microwaves with about 10% efficiency might reasonably be expected from such a virtual‐cathode configuration.

Energy lost by a relativistic electron beam due to two‐stream instability

The two‐stream instability between a monoenergetic relativistic electron beam of energy γ0 mc2 and density nb and an unmagnetized, homogeneous plasma of density ne is investigated. For weak interactions, S=β20γ0(nb/2ne)1/3<0.45, the two‐stream spectrum is predominantly one‐dimensional, and the fractional beam energy loss is ΔE=S (1+S)−5/2. However, for strong interactions, S≳0.45, efficient coupling occurs to waves with perpendicular wavenumbers 0≳k⊥< (nb/ne)1/3 ωe/c. Independent of S ΔE≃0.18 for strong interactions. The analysis is supported by one‐ and two‐dimensional computer simulations. In addition, it is observed that nonlinear wave particle interactions lead to rapid heating of the plasma in two‐dimensional space. Symmetric bulk heating and high energy tail formation of plasma electrons is observed perpendicular to the beam direction. Plasma heating parallel to the beam is asymmetric (v≳0), since the large amplitude waves are generated as a result of beam trapping in the forward direction.

Analytical and numerical studies of foilless diodes

The generation of intense annular beams by foilless diodes is studied through analytic equilibrium models and particle‐in‐cell simulation. In the high‐voltage regime, the foilless diode operates below the space‐charge limit and the impedance is nearly independent of the voltage. The current density is proportional to the current and to the square of the external magnetic field. From a kinetic theory equilibrium model, the beam scattering angle is found to be inversely proportional to the diode voltage, external magnetic field strength, and cathode radius. The predicted scaling is in good agreement with the simulation results. Criteria for adiabatic expansion and cooling of the beam, supported by simulation results, have also been considered.

Electromagnetic two-stream and filamentation instabilities for a relativistic beam--plasma system

Previous investigations on the two‐stream and filamentation instabilities are based on either the electrostatic or the ordinary‐mode approximation. A general relativistic dispersion formulation is presented to study these two instabilities for a scattered electron beam propagating a collisional, bi‐Maxwellian plasma. New analytical results that apply to a general beam distribution are obtained for the stability boundary of the filamentation instability. The general dispersion relation uncovers the inadequacy in applying the ordinary‐mode approximation in a frame other than the rest frame of the plasma. Analytical expressions for the the growth rates of the filamentation modes in various parameter regimes are obtained. Finally, numerical comparisons are made between the general dispersion results and the earlier results based on the electrostatic and ordinary‐mode approximations.

Intense annular relativistic electron beam generation in foilless diodes

The generation of intense annular electron beams in cylindrical foilless diodes is analyzed. A beam equilibrium model is used to determine the diode current. At low voltage the current is very nearly that of the space-charge limited beam. However, at higher voltages the current can be substantially less than the space-charge limit. The analysis is compared to detailed particle-in-cell simulations.

Effect of electron-ion collisions on the nonlinear state of the relativistic two-stream instability

The analysis outlines conditions for the hydrodynamic interaction, including foil scattering and beam self‐magnetic field effects. For the hydrodynamic interaction, the oblique coupling coefficient model is extended to include the effect of a finite electron‐ion collision rate on the nonlinear state. Partial numerical simulation results are found to be in agreement with the model. Limitations due to temperature inhomogeneity along the deposition length and to increased deposition length are also discussed. It is found that an intense relativistic electron beam can couple its energy via the two‐stream instability to a high density 1017–1020 cm−3 plasma. It is required that the plasma be fully ionized, with an initial electron temperature of ≳10 eV for 1017 cm−3, 10–20 eV for 1018 cm−3, 15–50 eV for 1019 cm−3, and ∼100 eV for 1020 cm−3.