Diocotron Instability Research Articles

Generating spokes in direct current magnetron sputtering discharges by an azimuthal strong-to-weak magnetic field strength transition

Abstract Spokes are regions of enhanced ionization in magnetron sputtering discharges that are interesting because of their role for magnetron operation and their potential effect on deposition processes. Here, we show that spokes can intentionally be generated by introducing a strong-to-weak magnetic field strength transition along the racetrack. Spokes are triggered at the transition point from an accelerating electron drift when weakening the magnetic field strength. The spokes are then propagating against the electron drift into the strong magnetic field strength section of the racetrack. At the weak-to-strong magnetic field transition, we observe the inverse effect. The electron drift is decelerated at this point, creating a region of enhanced optical emission. From rectangular racetracks this is known as the cross-corner effect. Here, we show that a corner is not necessary for observing that effect. Pronounced spokes at low working gas pressure of 0.2 Pa exhibit a substructure that could be caused by the diocotron instability previously predicted by computer simulations.

An implicit, conservative and asymptotic-preserving electrostatic particle-in-cell algorithm for arbitrarily magnetized plasmas in uniform magnetic fields

We introduce a new electrostatic particle-in-cell algorithm capable of using large timesteps compared to particle gyro-period under a uniform external magnetic field. The algorithm extends earlier electrostatic fully implicit PIC implementations with a new asymptotic-preserving particle-push scheme that allows timesteps much larger than particle gyroperiods. In the large-timestep limit, the integrator preserves all particle drifts, while recovering the full orbit for small timesteps. The scheme allows for a seamless, efficient treatment of particles with coexisting magnetized and unmagnetized species, and conserves energy and charge exactly without spoiling implicit solver performance. We demonstrate by numerical experiment with several problems of variable species magnetization (diocotron instability, modified two-stream instability, and drift instability) that orders of magnitude wall-clock-time speedups vs. the standard fully implicit electrostatic PIC algorithm are possible without sacrificing solution accuracy.

MODULATION OF NEGATIVELY CHARGED PARTICLE FLUX FROM PENNING DISCHARGE WITH METAL HYDRIDE CATHODE

The features of modulation of negatively charged particle flux extracted along the magnetic field from Penning discharge with metal hydride cathode have been investigated. The separation of negative ions from coextracted electrons was performed by an electromagnetic filter with efficiency not worse than 90 %. It is shown that electrons in the extracted flow are deeply modulated with the frequency of the diocotron instability of the anode layer, while negative ions are not. An increase in the interelectrode distance and anode length resulted in a more pronounced expression of this effect due to the minimization of the influence of the magnetic field of the filter on the discharge.

Observation of the diocotron instability in a diode with split cathode

Diocotron instability has been observed in the pure electron plasma formed in a split cathode coaxial diode. This plasma consists of electrons, trapped in the longitudinal potential well between the two parts of the cathode. The mathematical model of the electron squeezed state, which allows the calculation of the equilibrium plasma density, is presented. The model is applied in a comprehensive analysis of experimental data, and the presence of the diocotron instability is unambiguously confirmed.

An Implicit, Conservative and Asymptotic-Preserving Electrostatic Particle-in-Cell Algorithm for Arbitrarily Magnetized Plasmas in Uniform Magnetic Fields

We introduce a new electrostatic particle-in-cell algorithm capable of using large timesteps compared to particle gyro-period under a uniform external magnetic field. The algorithm extends earlier electrostatic fully implicit PIC implementations with a new asymptotic-preserving particle-push scheme that allows timesteps much larger than particle gyroperiods. In the large-timestep limit, the integrator preserves all the averaged particle drifts, while recovering the full orbit for small timesteps. The scheme allows for a seamless, efficient treatment of particles with coexisting magnetized and unmagnetized species, and conserves energy and charge exactly without spoiling implicit solver performance. We demonstrate by numerical experiment with several problems of variable species magnetization (diocotron instability, modified two-stream instability, and drift instability) that orders of magnitude wall-clock-time speedups vs. the standard fully implicit electrostatic PIC algorithm are possible without sacrificing solution accuracy.

Sparse grid-based adaptive noise reduction strategy for particle-in-cell schemes

We propose a sparse grid-based adaptive noise reduction strategy for electrostatic particle-in-cell (PIC) simulations. By projecting the charge density onto sparse grids we reduce the high-frequency particle noise. Thus, we exploit the ability of sparse grids to act as a multidimensional low-pass filter in our approach. Thanks to the truncated combination technique [1–3], we can reduce the larger grid-based error of the standard sparse grid approach for non-aligned and non-smooth functions. The truncated approach also provides a natural framework for minimizing the sum of grid-based and particle-based errors in the charge density. We show that our approach is, in fact, a filtering perspective for the noise reduction obtained with the sparse PIC schemes first introduced in [4]. This enables us to propose a heuristic based on the formal error analysis in [4] for selecting the optimal truncation parameter that minimizes the total error in charge density at each time step. Hence, unlike the physical and Fourier domain filters typically used in PIC codes for noise reduction, our approach automatically adapts to the mesh size, number of particles per cell, smoothness of the density profile and the initial sampling technique. It can also be easily integrated into high performance large-scale PIC code bases, because we only use sparse grids for filtering the charge density. All other operations remain on the regular grid, as in typical PIC codes. We demonstrate the efficiency and performance of our approach with two test cases: the diocotron instability in two dimensions and the three-dimensional electron dynamics in a Penning trap. Our run-time performance studies indicate that our approach can provide significant speedup and memory reduction to PIC simulations for achieving comparable accuracy in the charge density.

Anomalous Beam Transport through Gabor (Plasma) Lens Prototype

An electron plasma lens is a cost-effective, compact, strong-focusing element that can ensure efficient capture of low-energy proton and ion beams from laser-driven sources. A Gabor lens prototype was built for high electron density operation at Imperial College London. The parameters of the stable operation regime of the lens and its performance during a beam test with 1.4 MeV protons are reported here. Narrow pencil beams were imaged on a scintillator screen 67 cm downstream of the lens. The lens converted the pencil beams into rings that show position-dependent shape and intensity modulation that are dependent on the settings of the lens. Characterisation of the focusing effect suggests that the plasma column exhibited an off-axis rotation similar to the m=1 diocotron instability. The association of the instability with the cause of the rings was investigated using particle tracking simulations.

An efficient trajectory tracking algorithm for the backward semi-Lagrangian method of solving the guiding center problems

In this paper, we develop an effective numerical algorithm that tracks the trajectory needed to solve guiding center models using the backward semi-Lagrangian method. In terms of numerical calculations, two appreciably fast algorithms for the departure points are designed. One is a completely explicit formula for numerical solutions of the discrete system for each Cauchy problem. This formula is characterized by numerical factors that are less than half the multiplication number used in the usual Gaussian elimination. The other finds the required departure points with an interpolation method that is at least 30% less expensive for heavy Cauchy problems. Last, we propose a method to modify the solution to improve estimation of physical quantities, such as conservation of mass, which can be lost in interpolation solutions calculated at the departure points. It turns out that the proposed method not only saves a great deal of computation time, but also preserves physical quantities such as mass and total kinetic energy much better than conventional methods. To demonstrate the numerical evidence, we use the proposed method to simulate several problems such as the incompressible Euler equation, Kelvin-Helmholtz instability, Diocotron instability and a three-dimensional guiding center model.

Current Limitations for the Axially Symmetric Hollow Electron Beam in the Drift Tubes of Millimeter-Wave Vacuum Electronic Devices

The conditions for ensuring the equilibrium steady-state of a high-density hollow electron beam transported in cylindrical drift tubes and focused by a uniform magnetic field are investigated. The results of numerical calculations of the currents limited due to the longitudinal deceleration of electrons by the space charge and the violation of equilibrium transverse magnetic focusing conditions when transporting beams of different configurations in klystron-type devices are presented. The limitations of the working length of the device associated with the excitation of diocotron instability are estimated. The calculation results were compared with the data obtained by the quasi-3D Arsenal—MSU computer code and by the approximate analytical formulas of other authors.

Limitations on Currents in Cylindrical Drift Tubes of Millimeter-Wave Vacuum Electronic Devices

The conditions providing the equilibrium stationary state of high-density solid and annular electron beams, transported in cylindrical drift tubes and focused by a homogeneous magnetic field, have been investigated. An efficient numerical model is proposed to determine the distributions of the electrostatic potential, charge density, and electron velocities in the stationary state of these beams, with allowance for the space-charge forces. The results of the numerical calculations for the limiting currents, determined by the longitudinal deceleration (caused by space-charge forces) and conditions for equilibrium transverse magnetic focusing for transported beams of different configurations on the length of a klystrontype vacuum electronic device, are presented. The limitations on the device working length, related to the excitation of diocotron instability, are estimated. The calculation results are compared with the data obtained using the quasi-three-dimensional program “Arsenal-MSU” and analytical formulas suggested by other researchers.

Modeling Instabilities in Relativistic Electronic Beams in the CST Particle Studio Environment

The paper is devoted to the development of the classical vircator model in the CST Particle Studio for the investigation of the physical mechanisms of the development, coexistence, and interaction of Pierce/Bursian and diocotron instabilities in relativistic electronic streams. Such a model is numerically analyzed by the CST Particle Studio. We find that a numerical instability develops, which leads to an additional small-scale space density modulation in the electron beam during its motion in the system. We find the character time scale of the instability and determine its relation with a spatial mesh step. We study the instability in detail and propose a method to suppress it by the CST Particle Studio. Also, important notes concerning the development of models of microwave devices in the CST Particle Studio are presented.

Control of the diocotron instability of a hollow electron beam with periodic dipole magnets

A method to control the diocotron instability of a hollow electron beam with periodic dipole magnetic fields has been investigated by a two-dimensional particle-in-cell simulation. At first, relations between the diocotron instability and several physical parameters such as the electron number density, the current and shape of the electron beam, and the solenoidal field strength are theoretically analyzed without periodic dipole magnetic fields. Then, we study the effects of the periodic dipole magnetic fields on the diocotron instability using the two-dimensional particle-in-cell simulation. In the simulation, we considered the periodic dipole magnetic field applied along the propagation direction of the beam, as a temporally varying magnetic field in the beam frame. A stabilizing effect is observed when the oscillating frequency of the dipole magnetic field is optimally chosen, which increases with the increasing amplitude of the dipole magnetic field.

Simulation of the development and interaction of instabilities in a relativistic electron beam under variation of the beam wall thickness

The development and interaction of Bursian and diocotron instabilities in an annular relativistic electron beam propagating in a cylindrical drift chamber are investigated analytically and numerically as functions of the beam wall thickness and the magnitude of the external uniform magnetic field. It is found that the interaction of instabilities results in the formation of a virtual cathode with a complicated rotating helical structure and several reflection regions (electron bunches) in the azimuthal direction. It is shown that the number of electron bunches in the azimuthal direction increases with decreasing beam wall thickness and depends in a complicated manner on the magnitude of the external magnetic field.

Development of diocotron instability in the squeezed state of a relativistic electron beam

Nonlinear nonstationary processes in a beam-plasma system containing a hollow intense relativistic electron beam with an overcritical current in the squeezed state are studied numerically. The emergence and development of diocotron instability in the squeezed state, which leads to formation of vortex structures, is analyzed in detail, and the effect the magnitude of the external magnetic field has on the processes of structure formation in the beam is investigated.

The development and interaction of instabilities in intense relativistic electron beams

We report on the physical mechanisms of development, coexistence and interaction of Pierce-Bursian and diocotron instabilities in the non-neutral relativistic electron beam (REB) in the classic vircator. The analytical and numerical analysis is provided by means of 3D electromagnetic simulation. We conducted an extensive study of characteristic regimes of REB dynamics determined by the instabilities development. As a result, a regime map has been obtained. It demonstrates sequential switching of the REB dynamics from the regime with N=1 to the regime with N = 7 electron bunches in the azimuth direction with the beam current growth for the different external magnetic fields. The numerical analysis of bunch equilibrium states has identified the physical causes responsible for the REB regime switchings.

Nonlinear shearing modes approach to the diocotron instability of a planar electron strip

The nonlinear evolution of the diocotron instability of a planar electron strip is investigated analytically by means of the nonlinear shearing mode for the solution of the initial and boundary value problems. The method is based on the sheared spatial coordinates which account for the motion of electron flow in the electrostatic field of the unstable diocotron modes in addition to the unperturbed sheared motion of the electron flow on the transformed shear coordinates. The time evolutions are studied by the solution of the initial and boundary value problems. The obtained solutions for the perturbed electrostatic potential include two nonlinear effects—the effect of the distortion of the boundaries of the planar electron strip and the effect of the coupling of the sheared nonmodal diocotron modes. It was proved by a two-dimensional particle-in-cell simulation that the developed theory is valid as long as the distortion of the boundaries of the basic shear flow does not change the frequency and growth rate of the linear diocotron instability in the transformed coordinates.

The stabilizing effect of a conducting boundary on the diocotron instability of nonneutral electron flow

The effect of the conducting boundary on the temporal evolution of the diocotron instability of a planar and a cylindrical electron flow in a uniform magnetic field has been investigated by using two-dimensional particle-in-cell (PIC) simulations for various boundary-wall locations. The growth rates measured in the PIC simulations match well those calculated by using analytic linear theories. The approach of conducting boundaries to the electron flow was observed not only to reduce the growth rate of the diocotron instability but also to change the mode structure. The nonlinear behaviors of the instability lead to a phase-locking configuration to generate vortices.

Numerical study of a dust-contaminated electron plasma

The collective behaviour of dusty plasmas is heavily affected by the presence of a small fraction of micrometric or sub-micrometric dust particles which collect a large surface charge. While dusty plasmas under study are usually quasi-neutral, we propose here an investigation on a magnetized nonneutral plasma (a situation found for example in Penning traps) where a conventional plasma with a single sign of charge (e.g. electrons) is contaminated by a dust population. We simulate the two-dimensional dynamics of such a plasma in the plane orthogonal to a homogeneous magnetic field with a tailored Particle-In-Cell code implementing a mass-less fluid (drift-Poisson) approximation for electrons and a kinetic description for the dust component, including gravity effects. Simulations with a range of initial conditions are performed to observe the influence of dust on the diocotron instability developing in the electron plasma. The early stage of the growth of diocotron modes is analyzed by Fourier decomposition. The fully non-linear evolution is studied by means of a statistical analysis of probability density functions and flatness of spatial vorticity increments in order to characterize the intermittency properties of the turbulence.

A new fully two-dimensional conservative semi-Lagrangian method: applications on polar grids, from diocotron instability to ITG turbulence

While developing a new semi-Lagrangian solver, the gap between a linear Landau run in 1Dx1Dand a 5D gyrokinetic simulation in toroidal geometry is quite huge. Intermediate test cases are welcomefor testing the code. A new fully two-dimensional conservative semi-Lagrangian (CSL) method is presentedhere and is validated on 2D polar geometries. We consider here as building block, a 2D guiding-centertype equation on an annulus and apply it on two test cases. First, we revisit a 2D test case previouslydone with a PIC approach [18] and detail the boundary conditions. Second, we consider a 4D drift-kineticslab simulation (see [10]). In both cases, the new method appears to be a good alternative to deal withthis type of models since it improves the lack of mass conservation of the standard semi-Lagrangian (BSL)method.

Theoretical and Experimental Investigation on Intense Sheet Electron Beam Transport With Its Diocotron Instability in a Uniform Magnetic Field

In this paper, the cold-fluid model theory of an intense sheet electron beam is developed to investigate the diocotron instability during its transport in a uniform magnetic field. The model shows that if the magnetic field strength and the beam filling factor are increased separately, the diocotron instability will be suppressed, which extends the sheet-beam transport distance effectively. To verify the above conclusion, a set of complicated instruments, the electron beam measuring and analyzing system (EBMAS), was developed to measure the beam cross section, beam current density, and the 3-D trajectory. The sheet electron beam transport process in a uniform magnetic field with its diocotron instability is investigated on the basis of EBMAS measurements. The measured results agree very well with theoretical calculation and numerical simulation. A sheet-electron-beam tube based on a uniform magnetic field was then manufactured to drive a future W-band sheet-beam klystron (WSBK). Tuning the sheet-beam parameters over a broad range, including a cathode voltage range of 20-82 kV, current range of 0.5-4.27 A, and with a beam cross section of about 10 mm $\times\,$ 0.5 mm, the experimental beam transmission rate is above 98% through a beam tunnel of 100 mm in length.