Diocotron Instability Research Articles

Resistive destabilization of cycloidal electron flow and universality of (near-) Brillouin flow in a crossed-field gap

It is shown that a small amount of dissipation, caused by current flow in a lossy external circuit, can produce a disruption of steady-state cycloidal electron flow in a crossed-field gap, leading to the establishment of a turbulent steady state that is close to, but not exactly, Brillouin flow. This disruption, which has nothing to do with a diocotron or cyclotron instability, is fundamentally caused by the failure of a subset of the emitted electrons to return to the cathode surface as a result of resistive dissipation. This mechanism was revealed in particle simulations, and was confirmed by an analytic theory. These near-Brillouin states differ in several interesting respects from classic Brillouin flow, the most important of which is the presence of a microsheath and a time-varying potential minimum very close to the cathode surface. They are essentially identical to that produced when (i) injected current exceeds a certain critical value [P. J. Christenson and Y. Y. Lau, Phys. Plasmas 1, 3725 (1994)] or (ii) a small rf electric field is applied to the gap [P. J. Christenson and Y. Y. Lau, Phys. Rev. Lett. 76, 3324 (1996)]. It is speculated that such near-Brillouin states are generic in vacuum crossed-field devices, due to the ease with which the cycloidal equilibrium can be disrupted. Another novel aspect of this paper is the introduction of transformations by which the nonlinear, coupled partial differential equations in the Eulerian description (equation of motion, continuity equation, Poisson equation, and the circuit equation) are reduced to an equivalent system of very simple linear ordinary differential equations.

Three-dimensional, particle-in-cell simulations of applied-B ion diodes on the particle beam fusion accelerator II

We have used the three-dimensional, particle-in-cell code QUICKSILVER [J. P. Quintenz, et al., Lasers and Particle Beams 12, 283 (1994)] to simulate radial applied-B ion diodes on the particle beam fusion accelerator II at Sandia National Laboratories. The simulations agree well with experiments early in the beam pulse, but differ substantially as the ion-beam current increases. This is attributed to the oversimplified ion emission model. We see the same instabilities seen in earlier simulations with idealized diode geometries; Early in time there is a diocotron instability, followed by a transition to an ‘‘ion mode’’ instability at much lower frequency. The instability-induced beam divergence for the ∼10 MeV beam during the diocotron phase is <10 mrad, significantly less than the total beam divergence in experiments early in the pulse, but increases to ≳25 mrad after the transition. The ion mode has a distinct harmonic structure along the applied field lines, making the instability transition sensitive to the diode geometry. The ion mode instability in our latest simulations is consistent with evidence of instabilities from recent experiments.

Ion divergence in magnetically insulated diodes

Magnetically insulated ion diodes are being developed to drive inertial confinement fusion. Ion beam microdivergence must be reduced to achieve the very high beam intensities required to achieve this goal. Three-dimensional particle-in-cell simulations [Phys. Rev. Lett. 67, 3094 (1991)] indicate that instability-induced fluctuations can produce significant ion divergence during acceleration. These simulations exhibit a fast growing mode early in time, which has been identified as the diocotron instability. The divergence generated by this mode is modest, due to the relatively high-frequency (≳1 GHz). Later, a low-frequency low-phase-velocity instability develops with a frequency that is approximately the reciprocal of the ion transit time. This instability couples effectively to the ions, and can generate unacceptably large ion divergences (≳30 mrad). Linear stability theory reveals that this mode has structure parallel to the applied magnetic field and is related to the modified two-stream instability. Measurements of ion density fluctuations and energy-momentum correlations have confirmed that instabilities develop in ion diodes and contribute to the ion divergence. In addition, spectroscopic measurements indicate that lithium ions have a significant transverse temperature very close to the emission surface. Passive thin-film lithium fluoride (LiF) anodes have larger transverse beam temperatures than laser-irradiated active sources. Calculations of the ion beam source divergence for the LiF film due to surface roughness and the possible loss of adhesion and fragmentation of this film are presented.

Diocotron instability of a warm electron beam in crossed fields

The warm-fluid equation derived from the drift kinetic equation is solved numerically to investigate the electrostatic low-frequency stability of an electron ribbon beam drifting in the crossed-fields of a planar magnetron. The temperature effect is manifested only for oblique propagation with respect to the drifting beam direction. The dispersion relation takes the form ω = ω(k⊥/ks∥) = ω(ks∥/k⊥), where k⊥ and k⊥ are respectively the components of the surface wave vector k parallel and perpendicular to the magnetic field. The obliqueness of the propagation direction and the non-zero temperature give rise to a resonant instability, in addition to the diocotron instability, and the wavenumber corresponding to the maximum diocotron growth rate shifts as the temperature changes.

Diocotron instability of the electron beam in the drift tube of a gyrotron

A cold fluid model is used to investigate instabilities associated with a velocity shear due to the self fields of a relativistic electron beam inside the beam tunnel of a gyrotron. General statements concerning the stability of an electron beam like the frequency and the growth rate are possible. The growth is expressed as a length that can be compared with the geometry in the gun-tube-resonator system.

Dynamics of non-neutral plasmas

In this paper the focus is on the dynamics of two-dimensional cylindrical non-neutral plasmas. After reviewing some highlights of the non-neutral plasma dynamics, some recent two-dimensional results are described: vortex dynamics, diocotron instabilities of hollow profiles, collisionless damping of modes and fluid trapping by modes, fluid echoes, the cyclotron center of mass modes and warm plasma Bernstein modes, and temperature determination from fluctuation measurements. Attention is called to some unsolved problems.

Resonant diocotron instability in a non-neutral plasma with finite mass

The condition for a resonant diocotron instability in a column of non-neutral plasma is generalized to include the effect of plasma inertia. Near the Brillouin limit, for modes with large azimuthal mode number, instability is shown to occur for a nonzero plasma density at the resonant surface, regardless of the density gradient. In the presence of inertia the existence of another singularity is pointed out. Its effect on stability is shown to be small near the Brillouin limit for modes with large azimuthal mode number. For certain values of the number density below the Brillouin limit this singularity can exist in the region of large plasma density, for a density profile close to a step function profile, and is shown to resonantly drive an instability if the average plasma density decreases radially outward at the resonant surface.

Diocotron instability of a relativistic sheet electron beam propagating through a rectangular conducting wall

Properties of the diocotron instability in a relativistic sheet electron beam propagating through a rectangular conducting wall are investigated within the framework of a macroscopic cold fluid model. The electron beam is assumed to be partially neutralized by the positive immobile ions with the fractional charge neutralization f. The eigenvalue equation is obtained for low-frequency perturbations in standing waves. The dispersion relation of the diocotron instability is derived and used to investigate stability properties for a broad range of system parameters including the ratio a/d of the beam thickness (2a) to the conductor gap (2d) and the charge neutralization f. The dispersion relation indicates that the system is stabilized by increasing the neutralization f to 1/γ2b, where γb is the characteristic value of the beam relativistic factor. It is also shown that the diocotron perturbations are completely stabilized by increasing the beam thickness to more than one-half the conductor gap (i.e., a/d≳0.5) for f=0. The growth rate of instability is a substantial fraction of the diocotron frequency if the system is unstable.

Periodic focusing and ponderomotive stabilization of sheet electron beams.

Particle simulations compare the behavior of nonrelativistic sheet electron beams in uniform static and nonuniform time-harmonic magnetic fields. The time-harmonic fields are equivalent to periodically cusped magnetic (PCM) focusing. While the sheet beam in a uniform field exhibits diocotron instability, the PCM-focused beam is stabilized by ponderomotive forces, in agreement with recent analytic predictions [J. Appl. Phys. 73, 4140 (1993)]. Mismatched Pcm-focused beams exhibit envelope oscillation and initially rapid emittance growth followed by a region of slower increase, in agreement with a recent semianalytic Fokker-Planck model.

Diocotron instability in curved magnetic field

The diocotron instability is studied in a plasma confined in between two concentric cylinders by an azimuthal magnetic field. In this geometry the perturbations are compressible while equilibrium is incompressible. The Rayleigh criterion for compressible perturbations is shown to be modified. Marginal stability curves (Im ω=0) for different mode numbers are plotted. The quasilinear evolution of this instability is associated with a mean inward motion of the plasma column (inward pinch) apart from the usual diffusion to the marginal stability.

Localized density clumps generated in a magnetized nonneutral plasma

The final, nonlinear, saturated state of the diocotron instability is investigated experimentally in a magnetized, annular, nonneutral plasma. As the wave grows to large amplitude, the plasma is organized into vortices. In the limit of high magnetic fields, and moderate central conductor biases, the diocotron modes compress the plasma into dense ( n ∼ 3.5 n 0) localized clumps. Two-dimensional particle-in-cell computer simulations confirm the physical picture of vortex formation by resonant particle trapping.

Diocotron instability for relativistic non-neutral electron flow in planar magnetron geometry

Diocotron stability properties of relativistic non-neutral electron flow in a planar magnetron are investigated within the framework of the cold-fluid-Maxwell equations. The eigenvalue equation for the extraordinary-mode waves in a relativistic velocity-sheared electron layer is obtained, and is solved in the massless, guiding-center approximation. Approximating the electromagnetic field in the anode resonator by the lowest-order mode, the dispersion relation for the diocotron instability is obtained. Although the tenuous beam approximation is assumed, the eigenvalue equation and corresponding dispersion relation are both fully electromagnetic, and valid for relativistic electron flow. The dispersion relation is numerically investigated for a broad range of system parameters. From numerical calculations of the dispersion relation, it is shown that the typical growth rate of the diocotron instability indicates a strong instability. The early evolution of the diocotron instability as an important precursor to the evolution of the full magnetron oscillation is discussed.

The diocotron instability of a general relativistic electron beam

The diocotron instability of a general relativistic electron beam is studied using a macroscopic, cold fluid model. In contrast with the previous treatments where the theoretical analyses are carried out for an annular electron beam in a guided magnetic field, such that plasma frequency ≪ cyclotron frequency, the restriction on the magnitude of the beam density and guiding magnetic field is removed in deriving and solving the general eigenvalue equation. In the limit of long axial wavelengths, a dispersion relation is extracted for the special case of a sharp boundary density profile. The stability properties are investigated for a broad range of beam parameters. The results show that the kink mode can be unstable as the plasma frequency approaches the cyclotron frequency.

The effect of a virtual cathode on the electromagnetic stability of high-power ion diodes

Previous stability analyses have shown magnetically insulated ion diode equilibria to be unstable. However, important equilibrium features such as a virtual-cathode and a charge–neutral region were not included. A stability analysis including these features is presented, which indicates that the stability behavior is strongly affected by the equilibrium model. In particular, the diocotron instability displaces the ion resonance instability found in the previous analyses. The calculated growth rates for the diocotron mode are consistent with a fast growing mode that has been observed in recently reported three-dimensional numerical simulations. These simulations exhibit a transition from the diocotron to an ion resonance mode and a subsequent increase in ion divergence due to the much lower frequency of the ion mode. The stability analysis presented in this paper demonstrates how the evolution of the electron sheath reduces the diocotron growth rate resulting in unstable ion modes, thus explaining the transition to ion modes observed in the simulations.

Effects of electrostatic confinement fields and finite gyroradius on an instability of hollow electron columns

Exponentially growing diocotron instabilities with azimuthal mode number l=1 arise in hollow, cylindrical electron columns when the continuity equation includes small terms corresponding to finite gyroradius or axial variation of the radial electric field. The modes are similar in spatial form to the algebraically growing perturbations of the theory without these effects.

Physics issues associated with low- beta plasma generators

Kinetic aspects of MHD generators are explored by examining the propagation of dense, low- beta streams of plasma. Three situations are considered: the basic principles of plasma-stream propagation, the propagation of plasma streams into vacuum, and the propagation of plasma streams into ambient plasmas. These three situations are analogous to plasma generators, plasma generators with vacuum loads, and plasma generators with plasma loads. Kinetic (microphysics) aspects include oscillations of the generator plasma, the effects of diocotron instabilities, the acceleration of particles, the starvation of current systems, and plasma-wave production. >

Equilibrium and stability properties of intense non-neutral electron flow

Non-neutral plasmas, like electrically neutral plasmas, exhibit a broad range of collective properties, such as plasma waves and instabilities, and the ability to support long-lived, large-amplitude coherent structures. This paper reviews the equilibrium and stability properties of intense non-neutral electron flow in crossed electric and magnetic fields. Following a description of equilibrium properties for magnetically insulated electron flow in planar geometry, extraordinary-mode stability properties are investigated for relativistic non-neutral electron flow between planar conductors. Particular emphasis is placed on the magnetron and diocotron instabilities, and detailed stability behavior is shown to exhibit a sensitive dependence on the self field intensity (as measured by the dimensionless parameter ${s}_{e}=\frac{{\ensuremath{\gamma}}_{e}^{0}{\ensuremath{\omega}}_{\mathrm{pe}}^{2}}{{\ensuremath{\omega}}_{\mathrm{ce}}^{2}}$) as well as on the shape of the equilibrium profiles. The influence of cylindrical effects (such as the centrifugal and Coriolis accelerations of an electron fluid element) on stability behavior is then investigated for rotating electron flow in cylindrical geometry. Finally, the properties of large-amplitude coherent structures in non-neutral plasmas with circulating electron flow are investigated. Topics covered in this area include particle-in-cell computer simulations of dense (${s}_{e}\ensuremath{\sim}1$) electron flow in relativistic magnetrons which show large-amplitude spoke formation in the circulating electron density, and application of a cold-fluid guiding-center model to investigate large-amplitude vortex structures in low-density (${s}_{e}\ensuremath{\ll}1$) non-neutral plasma. The accessibility and stability of such stationary structures (in the rotating frame) remain important topics for future investigation.

Experiments on vortex dynamics in pure electron plasmas

Magnetically confined columns of electrons are excellent experimental manifestations of two-dimensional (2-D) vortices in an inviscid fluid. Surface charge perturbations on the electron column (diocotron modes) are equivalent to surface ripples on extended vortices; and unstable diocotron modes on hollow electron columns are examples of the Kelvin–Helmholtz instability. Experiments demonstrate that the stable and unstable modes are distinct and may coexist, having different frequencies and radial eigenfunctions. For azimuthal mode number l=1, an exponentially unstable mode is observed on hollow columns, in apparent contradiction to 2-D fluid theory. For l=2, a similar unstable mode is observed, consistent with fluid theory. These diocotron instabilities on hollow columns saturate with the formation of smaller vortex structures, and radial transport is determined by the nonlinear interaction of these secondary vortices. The vortex pairing instability has been observed for isolated, well-controlled vortices, and the instability is found to depend critically on the vortex separation distance.

Computer simulations of finite plasma streams convected across a magnetized vacuum

A two-dimensional electrostatic particle-in-cell code is used to simulate the convection of a finite stream of initially neutral plasma across a uniform magnetic field. The simulations show that the stream loses momentum with distance as a result of two erosion mechanisms that have greater effects for denser plasmas: (1) erosion of the charge layers at the sides of the stream as a result of velocity shear and (2) erosion of the head of the stream as a result of charge separation where ions travel ahead of the electrons. The electron charge layer exhibits a velocity shear that excites the diocotron instability. This instability occurs earlier for denser plasmas but it does not appear when the length of the stream is shorter than four wavelengths. The charge separation at the head of the stream causes the eroded plasma to drift with a sheared velocity. A flutelike instability develops at the head of the plasma stream for sufficiently dense plasmas. The simulations show that the plasma is eroded faster by the head erosion mechanism. Electric field fringe effects cause the plasma head to broaden and the tail of the plasma to narrow. The simulations show that although the plasma configuration is changed a great deal by erosion and fringe effects, the stream is convected across the magnetic field with a constant velocity for sufficiently dense plasmas and with a velocity that approaches the injection for denser plasmas. The simulations also show that the convection velocity for a partially and for a completely eroded plasma is the same.

Influence of the return current effects on the diocotron instability of a relativistic hollow electron beam

Influence of the return current effects on the diocotron instability of a relativistic hollow electron beam propagating through a background plasma is investigated within the framework of a cold fluid model. The return current density induced in the background plasma is taken to be steadily proportional to the axial electron beam current density. By making use of the linearized fluid-Maxwell equations, a dispersion relation for the eigenfrequencies of the system is derived and used to examine instabilities. It is found that as the fraction of current neutralization increases, lower mode instabilities become dominant while higher mode perturbations are stabilized except in the case of very thin beams. It is also observed that for all values of the fraction of current neutralization, increasing the plasma density gradient or decreasing the beam thickness causes a destabilizing influence.