Calculation of the electric field in space plasma near a conducting sphere injecting an electron beam

The problem is considered of determining the electric field induced in the vicinity of a conducting spherical body that is at rest in a collisionless plasma and at the surface of which there is a prescribed sink of negative charge. The problem is solved for the general relativistic case under the assumption that the electron velocity in the neutralizing current is comparable with the speed of light. An integrodifferential equation is derived that describes the radial behavior of the electric field potential in the vicinity of the injector. A simplified method for determining the potential in the perturbed region is developed. The method implies that the problems of the potentials in a space charge region of radius R* (with a prescribed boundary potential ?*) adjacent the body and in the outer region r>R* are solved separately and then the solutions obtained are matched at boundary between these regions.     

Nonlinear dynamics and chaotization of oscillations of a virtual cathode in an annular electron beam in a uniform external magnetic field

Results are presented from a numerical study of the effect of an external magnetic field on the conditions and mechanisms for the formation of a virtual cathode in a relativistic electron beam. Characteristic features of the nonlinear dynamics of an electron beam with a virtual cathode are considered when the external magnetic field is varied. Various mechanisms are investigated by which the virtual cathode oscillations become chaotic and their spectrum becomes a multifrequency spectrum, thereby complicating the dynamics of the vircator system. A general mechanism for chaotization of the oscillations of a virtual cathode in a vircator system is revealed: the electron structures that form in an electron beam interact by means of a common space charge field to give rise to additional internal feedback. That the oscillations of a virtual cathode change from the chaotic to the periodic regime is due to the suppression of the mechanism for forming secondary electron structures.     

Low-frequency instability of an ion beam in the field of a high-current REB

A study is made of the linear and nonlinear stages of the low-frequency instability of an ensemble of ions that execute radial oscillations in the electric field of the space charge of an unneutralized high-current relativistic electron beam. Nonlinear mechanisms for stabilizing the low-frequency ion instability are considered. It is shown, in particular, that, under certain conditions, the development of the low-frequency instability can lead to the ejection of ions onto the walls of the drift chamber.     

Unsteady processes during stimulated emission from a relativistic electron beam in a quasi-longitudinal electrostatic pump field

The problem of stimulated emission from a relativistic electron beam in an external electrostatic pump field is studied. A set of nonlinear time-dependent equations for the spatiotemporal dynamics of the undulator radiation amplitude and the amplitude of the beam space charge field is derived. The beam electrons are described by a modified version of the macroparticle method. The regimes of the single-particle and collective Cherenkov effects during convective and absolute instabilities are considered. The nonlinear dynamics of radiation pulses emitted during the instabilities of the beam in its interaction with the forward and backward electromagnetic waves is investigated.     

Plasma dynamics near an earth satellite and neutralization of its electric charge during electron beam injection into the ionosphere

A study is made of the dynamics of the ionospheric plasma in the vicinity of an earth satellite injecting an electron beam. The time evolution of the electric charge of the satellite is determined. The electric potential of the satellite is found to be well below the beam-cutoff potential. It is shown that, under conditions typical of active experiments in space, the plasma electrons are capable of neutralizing the satellite’s charge.     

Dynamics of beam instability in a finite plasma volume: Numerical experiment

Results are presented from numerical simulations of the dynamics of beam instability in a finite plasma volume (plasma-filled cavity) in a weak magnetic field. It is shown that, in such a system, the low group velocity of the plasma waves excited by an electron beam can result in the generation and amplification of an electric field; strong electron heating in the axial region; and, as a consequence, the generation of a high potential at the axis. The quasistatic radial electric field so produced accelerates ions toward the periphery of the plasma column, forming a directed ion beam with an energy much higher than the thermal energy of the bulk plasma electrons.     

Nonlinear relativistic plasma resonance: Renormalization group approach

An analytical solution to the nonlinear set of equations describing the electron dynamics and electric field structure in the vicinity of the critical density in a nonuniform plasma is constructed using the renormalization group approach with allowance for relativistic effects of electron motion. It is demonstrated that the obtained solution describes two regimes of plasma oscillations in the vicinity of the plasma resonance— stationary and nonstationary. For the stationary regime, the spatiotemporal and spectral characteristics of the resonantly enhanced electric field are investigated in detail and the effect of the relativistic nonlinearity on the spatial localization of the energy of the plasma relativistic field is considered. The applicability limits of the obtained solution, which are determined by the conditions of plasma wave breaking in the vicinity of the resonance, are established and analyzed in detail for typical laser and plasma parameters. The applicability limits of the earlier developed nonrelativistic theories are refined.     

Simulation of oscillatory processes in a beam-plasma system with a virtual cathode in gas-filled interaction space

Physical processes occurring in an intense electron beam with a virtual cathode in an interaction space filled with neutral gas are studied in a two-dimensional model. A mathematical model is proposed for investigating complicated self-consistent processes of neutral gas ionization by the beam electrons and the dynamics of an electron beam and heavy positive ions in the common space charge field with allowance for the two-dimensional motion of charged particles. Three characteristic dynamic regimes of the system are revealed: complete suppression of oscillations of the virtual cathode as a result of neutralizing its space charge by positive ions; the pulsed generation regime, in which the ions dynamics repeatedly suppresses and restores the virtual cathode oscillations; and the continuous generation regime with an anomalously high level of noisy oscillations.     

Formation and nonlinear dynamics of the squeezed state of a helical electron beam with additional deceleration

Results of numerical simulations and analysis of the formation and nonlinear dynamics of the squeezed state of a helical electron beam in a vircator with a magnetron injection gun as an electron source and with additional electron deceleration are presented. The ranges of control parameters where the squeezed state can form in such a system are revealed, and specific features of the system dynamics are analyzed. It is shown that the formation of a squeezed state of a nonrelativistic helical electron beam in a system with electron deceleration is accompanied by low-frequency longitudinal dynamics of the space charge.     

Effect of the external magnetic field on the critical current for the onset of a virtual cathode in an electron beam

The critical current at which an unsteady oscillating virtual cathode forms in an electron beam is studied as a function of the external magnetic field guiding the beam electrons. It is shown that the critical beam current decreases with external magnetic field and that there is an optimum magnetic induction at which the critical current for the onset of an oscillating virtual cathode in the beam is minimum. For a strong guiding magnetic field, the critical beam current is described by relationships derived under the assumption that the motion of the beam electrons is one-dimensional. Such behavior is explained by the characteristic features of the dynamics of the beam electrons in longitudinal and radial directions in the interaction space at different inductions of the external magnetic field.     

On the nonlinear theory of collective Cherenkov interaction of a high-density relativistic beam with a plasma

The nonlinear dynamics of the instability of a straight high-density relativistic electron beam under the conditions of the stimulated Cherenkov effect in a plasma waveguide is studied both analytically and numerically. It is shown that, for a beam of sufficiently high density such that the stabilizing factors are nonlinear frequency shifts and for a plasma described in a linear approximation, the basic equations have soliton-like solutions and the electron beam after saturation of the instability relaxes to its initial, weakly perturbed state, provided that only one harmonic of the plasma and the beam density is taken into account. The analytical solutions obtained here for this case correlate well with the numerical ones. A more general model that accounts for the generation of higher harmonics of the plasma and the beam density does not yield soliton-like solutions for the time evolution of the amplitudes of the plasma and beam waves. In such a model, the instability will be collective again: it can be described analytically (at least, up to the time at which it saturates) by using equations with cubic nonlinearities and the method of expansion of the electron trajectories and momenta.     

Nonlinear theory of the collective Cherenkov interaction of a dense relativistic electron beam with a dense plasma in a waveguide

A nonlinear theory of the instability of a straight relativistic dense electron beam in a plasma waveguide is derived for conditions of the stimulated collective Cherenkov effect. A study is made of a waveguide with a dense plasma such that the plasma wave excited by the beam during the instability can be escribed, with a good degree of accuracy, as a potential wave. General relativistic nonlinear equations are btained that describe the temporal dynamics of beam-plasma instabilities with allowance for plasma nonlinearity and the generation of harmonics of the initial perturbation. Under the assumption that the resonant interaction between the beam waves and the plasma waves is weak, the general equations are reduced to relativistic equations with cubic nonlinearities by using the method of expansion in small perturbations of the trajectories and momenta of the beam and plasma electrons. The reduced equations are solved analytically, the time scales on which the instability saturates are determined, and the nonlinear saturation amplitudes are obtained. A comparison between analytical solutions to the reduced equations and numerical solutions to the general nonlinear equations shows them to be in good agreement. Nonlinear processes caused by the relativistic nature of the beam are found to prevent stochastization of the system in the nonlinear stage of the well-developed instability. In contrast, a nonrelativistic electron beam is found to be subject to significant anomalous nonlinear stochastization.     

Process of compensation of the space charge of a negative ion beam in a gas

The process of compensation of the space charge of a negative ion beam propagating through a neutral gas is investigated numerically. A comparison of the results obtained with experimental data unambiguously proves that, at high gas pressures, when the beam space charge is overcompensated, the electric field within the beam is determined by Coulomb collisions of the beam ions with plasma electrons. At low pressures, when the space charge is undercompensated, the field within the beam is determined by the dynamic processes related to oscillations of the beam current.     

Possible mechanism for radial ion acceleration in a beam-plasma discharge

A mechanism is proposed that can lead to radial ion acceleration in a plasma discharge excited by an electron beam in a relatively weak longitudinal magnetic field. The mechanism operates as follows. The beam generates an azimuthally asymmetric slow potential wave, which traps electrons. Trapped magnetized electrons drift radially with a fairly high velocity under the combined action of the azimuthal wave field (which is constant for them) and a relatively weak external longitudinal magnetic field. The radial electron flux generates a radial charge-separation electric field, which accelerates unmagnetized plasma ions in the radial direction. The ion flux densities and energies achievable in experiments with kiloelectronvolt electron beams in magnetic fields of up to 100 G are estimated.     

Influence of the electrons reflected from the collector on the parameters of a high-current relativistic electron beam

In plasma microwave oscillators, electrons fall onto the surface of a graphite collector, which leads to the generation of secondary electrons. The influence of the electrons reflected from the collector on the parameters of a high-current relativistic electron beam propagating in a strong longitudinal magnetic field was studied experimentally and by numerical simulations. It is shown that the penetration of the reflected electrons into the drift space can lead to a substantial increase in the depth of the potential well in the drift space, a decrease in the velocity of the beam electrons, and a broadening of the electron energy distribution function.     

Solution of a set of Maxwell-Lorentz equations for a ring relativistic electron beam

Nonlinear solutions to a set of Maxwell’s equations and the relativistic equations of electron motion are obtained that describe the equilibrium of a high-power ring relativistic electron beam against the background of immobile ions. By transforming the basic equations, a set of equations for a three-component vortex vector field is derived that describes ring beam configurations for plasma confinement. An example of a numerical calculation of the steady state of a compact beam torus of immobile ions and relativistic electrons is presented.     

Surface waves on the boundary of a conducting medium and their excitation by a relativistic electron beam

Excitation of surface waves by a relativistic electron beam propagating over a conducting cylindrical medium (metal or highly ionized plasma) is investigated theoretically. Dispersion relations describing the linear interaction of surface electromagnetic waves with a monoenergetic electron beam are derived, and the growth rates and spatial amplification factors of excited waves are determined. Condition for the nonlinear trapping of the beam electrons by a surface wave is used to determine the maximum amplitude of the excited wave and the optimal radiator length. The electric field of a surface wave excited by an electron beam is estimated for a particular case.     

Theoretical and experimental investigations of the excitation of high-frequency oscillations in an ion collective accelerator model based on the Doppler effect

Results are presented from studies of a two-beam scheme of ion acceleration by a high-frequency field excited by an electron beam due to the instabilities associated with anomalous and normal Doppler effects. The dynamics of the excitation of eigenmodes in a periodic slow wave structure (SWS) by a relativistic electron beam via the anomalous Doppler effect is investigated theoretically. Mechanisms for the saturation of the instability are considered, analytical expressions for the maximum field amplitude and the efficiency with which the energy of beam electrons is converted into the energy of the excited wave are derived, and the results of numerical simulations of such excitation are presented. An experimental stand designed to test the principles and possibility of proton acceleration up to an energy of 8 MeV at a current up to 3 A is described. A double resonance (associated with anomalous and normal Doppler effects) occurring in the interaction of an electron beam with a helical SWS is studied experimentally. In this case, an increase in the efficiency with which the accelerating high-frequency field is excited is observed.     

Azimuthally asymmetric eigenmodes of a magnetized plasma cylinder around a dielectric rod in a circular waveguide

The electrodynamics of a circular waveguide with a dielectric rod surrounded by a magnetized plasma layer is considered. A general dispersion relation for azimuthally asymmetric perturbations is derived, and its solutions describing slow waves—specifically, electromagnetic and plasma modes, as well as (and primarily) hybrid waves that combine the properties of both mode types—are investigated numerically. For the fundamental waveguide mode of the system—the HE₁₁ mode—the parameters of the plasma layer are determined at which the mode cannot be subject to Cherenkov interaction with a relativistic electron beam at a given frequency. For both waveguide and plasma modes, the radial profiles of the longitudinal components of the electric field and Poynting vector, the fractions of RF power carried within the dielectric and plasma regions and vacuum gap, and the coupling impedance are calculated as functions of the parameters of the plasma layer. The evolution of the field structure during the formation of asymmetric hybrid waves is traced. The results of calculating the dispersion and coupling impedance are analyzed as applied to an antenna-amplifier—a relativistic traveling-wave tube operating on the HE₁₁ mode of the dielectric rod: specifically, the implementability of the concept in the presence of a plasma at the rod surface is estimated, and the possible role of azimuthally asymmetric and symmetric plasma modes is examined.     

Specific features of virtual cathode formation and dynamics with allowance for the magnetic self-field of a relativistic electron beam

The conditions and mechanisms of virtual cathode formation in relativistic and ultrarelativistic electron beams are analyzed with allowance for the magnetic self-field for different magnitudes of the external magnetic field. The typical behavior of the critical current at which an oscillating virtual cathode forms in a relativistic electron beam is investigated as a function of the electron energy and the magnitude of the uniform external magnetic field. It is shown that the conditions for virtual cathode formation in a low external magnetic field are determined by the influence of the magnetic self-field of the relativistic electron beam. In particular, azimuthal instability of the electron beam caused by the action of the beam magnetic self-field, which leads to a reduction in the critical current of the relativistic electron beam, is revealed.