Brillouin Flow Research Articles

Two-dimensional small signal analysis of backward-wave oscillation in a helix traveling-wave tube under Brillouin-flow, periodic permanent magnetic focusing

The phenomenon of electron beam rotation in the axial magnetic field of a magnetically focused helix TWT results in more complicated parasitic backward-wave oscillation conditions compared to the case when the beam rotation is not considered. Accounting for this phenomenon allows for further improvement of TWT stability to the parasitic backward-wave oscillation. In this paper, we examine the typical super-broad bandwidth TWT that has an anisotropically-loaded circuit-e.g., a helical structure supported by dielectric rods in a metal shell provided with metal longitudinal vanes projecting radial inward. The 2D small-signal model is developed that takes into account the effects of the beam's rotation and focusing magnetic field parameters under Brillouin flow on backward-wave oscillation conditions in TWT. The theory permits an investigation of the effects of amplitude and axial periodicity of permanent periodic magnetic (PPM) focusing magnetic flux density and of Pierce's relative electron velocity parameter on the start oscillation length. Physical and numerical analyses are presented for a TWT's stability to backward-wave oscillation under permanent magnetic focusing, which is longitudinally periodic and homogeneous across the beam, for the case when the backward-wave consists of two nonazimuthally symmetric harmonics. A particular numerical example reveals the dependence of start oscillation length of a TWT on the PPM focusing parameters as well, as Pierce's relative electron velocity parameter. The theory predicts that by tailoring of the magnetic field period it is possible to increase the start-oscillation length by a factor of 1.25 and the power output by the factor 2.6 compared to the case when the beam rotation is not taken into account. The 2D model developed in this paper provides sufficient data for more correct and precise choice between signal amplification and backward-wave oscillation suppression in the anisotropically-loaded helix TWT than the existing 1D theories.

Parametric interactions inside a magnetron

The purpose of this work is to understand theoretically what are the possible noise levels in a magnetron or a crossed-field amplifier (CFA), due to parametric three-wave interactions in the electron plasma, at various operating parameters. Our approach is to use the cold-fluid equations and their Fourier decomposition into a background (DC) mode, a pump (RF) mode, and two other noise (RF) modes. The two RF noise modes are assumed to interact parametrically with the large RF pump mode, and to satisfy the standard resonance conditions for the sum of the wave vectors and sum of the frequencies. We use our previous results to determine the background mode and the RF pump mode. Any strong RF electric field propagating in a crossed-field electron vacuum device can drive a Brillouin sheath unstable by means of a Rayleigh instability, whenever a wave–particle resonance can be found inside the sheath. What happens physically is that, at the resonance, the laminar flow of the electrons is strongly disturbed, and a diffusion process ensues, whereby the electrons diffuse away from the resonance region. This upsets the balance in the Brillouin flow, causing the electrons to redistribute into a new average DC density profile – which may be far from the original Brillouin profile, but is a stationary solution of a nonlinear diffusion equation. Using these stationary density profiles, we can then study the propagation of small RF signals on such a DC background, as well as their parametric interactions with the RF pump wave, at various DC voltages and magnetic fields. In addition to being able to predict the operating regime and the DC current flow, these studies demonstrate that parametric interactions probably limit the operating voltage range of a typical magnetron or crossed-field amplifier to about 20% above the Hartree voltage.

Existence and uniqueness of elliptic periodic solutions of the Brillouin electron beam focusing system

We find conditions for the existence of an elliptic periodic solution of a singular equation that governs the motion of a magnetically focused axially symmetric electron beam with Brillouin flow by using a monotone iterative scheme starting from a couple of upper and lower solutions on the reversed order. Also, a uniqueness result is proved by using Brouwer degree and index of solutions. Copyright © 2000 John Wiley & Sons, Ltd.

Collapse of cycloidal electron flows induced by misalignments in a magnetically insulated diode

The effect of a slight misalignment in the magnetic field on a magnetically insulated diode is investigated. It is found that a slight tilt in the magnetic field, with a minute component along the dc electric field, completely destabilizes the cycloidal electron flow in the crossed-field gap. The final state consists of the classical Brillouin flow superimposed by a turbulent background, together with a slow electron drift across the gap. This disruption of the cycloidal flow is quite insensitive to the emission current density, and is due to the accumulation of space charge in the gap caused by the magnetic misalignment. This result was obtained from a one-dimensional simulation code. It reinforces the notion that the turbulent, near Brillouin-like states are generic in ALL vacuum crossed-field devices.

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.

Periodic permanent magnet focusing of an annular electron beam and its application to a 250 MW ubitron free-electron maser

Stable electron beam flow is shown to exist for an annular electron beam within a coaxial periodic permanent magnet wiggler and transmission line with zero guide field. When the beam is properly centered near the magnetic minimum of the wiggler, a condition exists where the magnetic radial force balances the space-charge repulsion of the beam. The stability condition relating the electron beam current to the magnetic field strength in terms of the plasma frequency, wiggler cyclotron frequency and relativistic Lorentz factor is ω2p=(γ/2)Ω2w, which is the same stability condition found for Brillouin flow in a uniform magnetic field and for focused beam flow along periodic dipole magnetic fields. The radial scallop of an annular electron beam satisfying this condition can be extremely low. Although this propagation technique can be utilized for any high power coaxial linear-beam device, it is most appropriate for free electron masers. Using parameters satisfying the stability condition, a 500 kV, 1 kA, TE01 ubitron free-electron maser design is presented that is predicted to generate 250 MW at 11.4 GHz.

Nonneutral ion plasmas and crystals in Penning traps

Experimental work which uses laser-cooled ions in Penning traps is reviewed. With laser-cooling the ions are strongly coupled and exhibit spatial correlations characteristic of a liquid or crystal. In plasmas with dimensions less than 10–15 interparticle spacings, the observed correlations are strongly affected by the finite size and shape of the trapping potential. Plasmas with greater than 60 interparticle spacings should exhibit correlations characteristic of an infinite one-component plasma. Radiation pressure from a laser is also used to apply a torque to the plasma and change the plasma density. This permits access to all possible thermal equilibria, including the maximum density state where the plasma undergoes Brillouin flow. If the size of the plasma is small compared to the trap dimensions, Penning traps produce plasmas with simple shapes whose normal modes can be calculated exactly. The modes provide a nondestructive diagnostic technique for obtaining information on the plasma density and shape.

Numerical study of relativistic magnetrons

The relativistic multiresonator magnetron is analyzed in full cylindrical geometry, using the two-dimensional particle-in-cell simulation code magic. Detailed comparisons are made between the simulation results and the classic experiments by Palevsky and Bekefi for the relativistic A6 magnetron configuration [Phys. Fluids 22, 986 (1979)]. The computer simulations show the same dependence of microwave power on axial magnetic field as measured in the experiment. It is found that the electron flow in the preoscillation regime differs substantially from ideal Brillouin flow, and that the nonlinear regime is characterized by large-amplitude spoke formation. By analyzing a class of relativistic magnetrons for a wide range of operating voltage and axial magnetic field, it is found that the optimal microwave power exhibits an approximately cubic dependence on the applied diode voltage. Moreover, the simulations indicate that relativistic magnetrons with small aspect ratio oscillate predominantly in the π mode, whereas magnetrons with large aspect ratio can oscillate in various modes which depend upon the system parameters. Finally, it is shown that a finite power-supply impedance can strongly modify the resonance condition, thereby significantly degrading the magnetron performance.

Brillouin limit and beyond: A route to inertial-electrostatic confinement of a single-species plasma.

The physics limiting the density of a cold non-neutral plasma is presented. It is shown that when a flow has no strain, the instantaneous maximum number of charges that can be stored within a fixed boundary is equal to the total magnetic field energy within divided by the relativistic rest energy of a single charge. A higher limit can be supported by the presence of a deviatoric strain in the flow. Brillouin flow equilibria with arbitrarily high values of the Brillouin ratio, leading to the possibility of pure inertial-electrostatic confinement of non-neutral plasmas, are demonstrated.

The beam temperature and energy broadening of a charged-particle beam in an axially symmetric magnetic field

The beam temperature and energy broadening of a charged-particle beam in an axially symmetric magnetic field are investigated in consideration of the space charge effect. The result shows that this kind of energy broadening consists of three parts: the thermal motion broadening, the space charge broadening, and the magnetic-field broadening. For the nonscalloping beam (including the space charge equilibrium flow and Brillouin flow) held by the equilibrium between the magnetic-field convergence and the space charge divergence, the energy broadening ΔE is directly proportional to the square of the equilibrium magnetic induction intensity Be and the beam radius R, and inversely proportional to the mass of beam particles. For the immersed flow (magnetic shield coefficient M = 1), the magnetic field does not result in an energy broadening, and the energy broadening caused by space charge is dominant. The beam temperature is defined by the mean square deviation of the velocity of beam particles. The beam temperature in the θ direction is much higher, whereas the beam temperature in the meridian direction is much lower than the temperature of the thermal cathode.

Formation and stability of polarization sheaths of a cross-field beam

A layer of plasma (such as that comoving with the space shuttle) can travel across a magnetic field if it is flanked by ‘‘polarization sheaths’’ that create the E field necessary for E×B drift. Both the spontaneous evolution and the stability of these positive and negative sheaths can be studied by their analogy with the charge configuration in magnetron-type electron devices. That is, conservation of canonical momentum results in ‘‘Brillouin flow,’’ i.e., shear flows in a pure ion sheath on one side and a pure electron sheath on the other side of the neutral plasma beam. The equilibrium parameters of this flow were obtained and the stability of the equilibria was analyzed. Both the ion and electron sheaths were found to be one-dimensionally stable and two-dimensionally unstable in a mode variously known as ‘‘magnetron,’’ ‘‘slipping stream,’’ or ‘‘diocotron.’’ This mode connects up with the Kelvin–Helmholtz instability in neutral matter. The analytical results were confirmed both qualitatively and quantitatively by numerical simulations. The saturation of the unstable modes, observed in the simulation, allows one to estimate how the polarization sheaths eventually diffuse.

Rotational equilibria and low-order modes of a non-neutral ion plasma.

We study rotational equilibria and low-order electrostatic modes of a magnetically confined, non-neutral ion plasma. The plasma rotation rate is controlled with radiation pressure from a laser beam and is continuously varied over the entire allowed range, including Brillouin flow. Excitation of an asymmetric plasma mode by a static field asymmetry is observed. The symmetric quadrupole mode is also studied; its behavior is characteristic of a strongly magnetized plasma at low density, and of an unmagnetized plasma at Brillouin flow.

Design of a LaB6 gun using EGN2 and INTMAG

Abstract In order to launch a high-density electron beam to be focused in the 5 T superconducting solenoid of the Frankfurt EBIS [R. Becker et al., Nucl. Instr. and Meth. B24 (1987) 838], an electron gun has been designed, with a 0.5 mm diameter LaB 6 cathode (FEI Comp., Beaverton, USA) in a 70 mm diameter electrode geometry. The emitting surface is placed in the axial fringing field of the solenoid, modified by an axial shielding disk and a bucking coil, to provide either immersed flow or Brillouin flow conditions for the focused beam. Since the cathode diameter is small as compared to the electrodes, a new feature of EGN2 [W.B. Herrmannsfeldt, SLAC-331 (1988)] had to be used in order to have a sufficient number of meshes along the emitting surface. By starting a field line in the large geometry, a curved Neumann boundary is found for a subdivided part of the gun, which represents the influence of the larger part. EGN2 writes the coordinates of this field line on a file, which can be used by POLYGON [R. Becker, Nucl. Instr. and Meth. B42 (1989) 162] (a boundary setup program for EGN2) to define a curved Neumann boundary. By this procedure, it becomes possible to get a reliable simulation of the emission properties of a small cathode in large gun electrodes. The magnetostatic field calculations have been performed with INTMAG [R. Becker, Nucl. Instr. and Meth. B42 (1989) 303], which is a new program of the boundary element method type. Due to the integration calculus, the results do not need smoothing or “Maxwellisation” for the use in EGN2, where the off-axis fields are evaluated by radial expansion. INTMAG provides an output file, which is suitably formatted to be read in by EGN2. The gun design is based on space-charge-limited emission, but no Pierce-type electrode has been provided in the vicinity of the cathode; instead a Wehnelt electrode on negative bias with respect to the cathode is used to create the correct Pierce-type equipotential in free space, ending on the cathode edge with the correct angle. This gives an additional adjustment tool, if the axial position of the gun is not perfect and it relaxes the radial tolerance requirements considerably.

Computer simulation of relativistic multiresonator cylindrical magnetrons

The relativistic multiresonator magnetron is analyzed in cylindrical geometry, using the two-dimensional particle-in-cell simulation code magic. Detailed comparisons are made between the simulation results and the experiments by A. Palevsky and G. Bekefi [Phys. Fluids 22, 986 (1979)] using the A6 magnetron configuration. Within a constant scale factor, the computer simulations show a similar dependence of microwave power on magnetic field, with dominant excitations in the π and 2π modes. In the preoscillation regime, the electron flow in the simulations differs substantially from the ideal Brillouin flow model. In the nonlinear regime, the saturation is dominated by the formation of spokes.

An electrostatically and a magnetically confined electron gun lens system

Focal properties, electron trajectory calculations, and geometries are given for two electron ‘‘gun’’ lens systems that have a variety of applications in, for example, electron-neutral and electron-ion scattering experiments. One nine-lens system utilizes only electrostatic confinement and is capable of focusing electrons onto a fixed target with extremely small divergence angles, over a range of final energies 1–790 eV. The second gun lens system is a simpler three-lens system suitable for use in a uniform, solenoidal magnetic field. While the focusing properties of such a magnetically confined lens system are simpler to deal with, the system does illustrate features of electron extraction and Brillouin flow that have not been suitably emphasized in the literature.

Kinetic description of betatron oscillation instability for nonrelativistic nonneutral electron flow

The linearized Vlasov–Poisson equations are used to investigate the electrostatic stability properties of nonrelativistic nonneutral electron flow in a planar diode with cathode located at x=0 and anode at x=d. The electron layer is immersed in a uniform applied magnetic field B0êz, and the equilibrium flow velocity V0yb(x) is in the y direction. Stability properties are calculated for perturbations about the choice of self-consistent Vlasov equilibrium f0b (H,Py) =(n̂b/2πm) δ(H) δ(Py), which gives an equilibrium with uniform electron density (n̂b =const) extending from the cathode (x=0) to the outer edge of the electron layer (x=xb). Assuming flute perturbations, the eigenvalue equation is simplified and solved analytically for long-wavelength, moderately high-frequency perturbations satisfying k2x2b ≪1 and (ω−kVd)2 ≊ω2v =ω2c −ω̂2pb. The present analysis is restricted to electron densities below the Brillouin flow (ω̂2pb <ω2c) and the nonzero electric field at the cathode. The eigenvalue equation leads to a fourth-order algebraic dispersion relation for the complex eigenfrequency. The dispersion relation is solved numerically, and detailed stability properties are investigated as a function of system parameters for both the upshifted branch (ω−kVd ≊+ωv) and the downshifted branch (ω−kVd≊−ωv). For a sufficiently thin electron layer, it is found that only the upshifted branch exhibits instability. Typically, instability exists for a range of ω̂2pb/ω2c.

Stochastic electron motion in magnetically insulated diodes

The transition to stochasticity is investigated for electron motion in the field configuration of a magnetically insulated diode. The model system is planar and periodic. The equilibria studied are nonrelativistic with constant electron density across the sheath. This class of equilibria includes the Brillouin flow equilibrium as a limit. Action-angle variables are introduced that facilitate the nonlinear analysis. It is found that small periodic perturbations of the equilibrium potential may lead to stochastic motion of the sheath electrons. It is shown that the critical perturbation amplitude for global stochasticity goes to zero as the Brillouin limit is approached for a variety of perturbations. A global stochasticity diagram is presented that gives a simple characterization of the electron motion in the allowed phase space of a magnetically insulated diode. A piecewise linear map is introduced that allows an efficient study of the electron dynamics in the stochastic regions.

Time evolution for different geometrical configurations of charged particles in a time-varying magnetic field

Through rescaling and obtaining invariant equations, we study the expansion of charged particles immersed in a time-varying magnetic field B(t) = B0(1+Ωt)–β. We develop the theory for a configuration of point particles confined in a plane with an initial cylindrical symmetric configuration. It is found that the system expands according to (1+Ωt)2β/3 for β ≤ 1 and it is conjectured that in a properly rescaled space and time, it reaches asymptotically an inhomogeneous (in space) Brillouin flow with rigid rotation at half the cyclotron frequency (ω = ½ωc). Particular attention is given to the ‘pivot’ case β = 1. It is shown then that the radius of the rescaled configuration is now dependent on the ratio Ω/ωc, and grows indefinitely with time for Ω/> 3.2–3/2;. Numerical experiments are presented which confirm these theoretical predictions and conjectures. Finally, the properties of this system are compared with those of single-particle and ‘rod’ electrostatic models.

Cross-field free electron laser instability for a tenuous electron beam

The free electron laser instability is investigated for a tenuous circulating electron beam propagating perpendicular to a uniform magnetic field B0êz and transverse wiggler field modeled by Bw sin k0yêx in planar geometry. Unlike the rippled-field magnetron which operates at Brillouin flow, the present analysis assumes a low-density electron beam with ω2p ≪Ω2c. Making use of a macroscopic cold-fluid model for the electrons coupled with Maxwell’s equations for the fields, it is found that wave perturbations with ordinary-mode polarization (δE∥B0 and δB⊥B0) amplify with characteristic maximum growth rate Im(δω)=ωp (Ωw/2ck0) and emission frequency ωr =(1+βE)γ2Ek0VE. Here, Ωw =eBw/γEmc, βE =VE/C, γE =(1−β2E)−1/2, and VE =−cE0/B0, where E0 is the applied electric field across the anode–cathode gap. Depending on the size of Ωw/ck0, the characteristic exponentiation time ω−1p(Ωw/2ck0)−1 for the cross-field free electron laser instability can be relatively short in units of ω−1p.

Two-dimensional beam in a time-varying magnetic field. the brillouin flow as an asymptotic attractor

Through rescaling methods and invariance laws, we study the expansion of a one-species, two-dimensional beam immersed into a time-varying magnetic field B( t)∝ t - α . It is conjectured that the only possible steady state (the Brillouin flow with 2 ω p 2/ ω p 2 = 1) will be reached. Computer simulations support the hypothesis for 0 < β ⪕ 1.