Electromagnetic Particle-in-cell Research Articles

Particle-in-cell and Weak Turbulence Simulations of Plasma Emission

Abstract The plasma emission process, which is the mechanism for solar type II and type III radio burst phenomena, is studied by means of particle-in-cell (PIC) and weak turbulence (WT) simulation methods. “Plasma emission” is meant as a loose description of a series of processes, starting from the solar flare-associated electron beam exciting Langmuir and ion-acoustic turbulence, and subsequent partial conversion of beam energy into radiation energy by nonlinear processes. PIC simulation is rigorous but the method is computationally intense, and it is difficult to diagnose the results. The numerical solution of equations of WT theory, known as WT simulation, on the other hand, is efficient and naturally lends itself to diagnostics because various terms in the equation can be turned on or off. Nevertheless, WT theory is based upon a number of assumptions. It is therefore desirable to compare the two methods, which we do for the first time with numerical solutions of the complete set of equations of WT theory and a two-dimensional electromagnetic PIC simulation. Upon making quantitative comparisons, it is found that WT theory is largely valid, although some discrepancies are also found. The present study also indicates that large computational resources are required in order to accurately simulate the radiation emission processes, especially for low electron beam speeds, such that it may be more advantageous to employ the WT method in order to describe the radiation emission itself. Findings from the present paper thus imply that both methods may be useful for the study of solar radio emissions, as they are complementary.

Polynomial Finite-Size Shape Functions for Electromagnetic Particle-in-Cell Algorithms Based on Unstructured Meshes

In this article, we describe a novel implementation of finite-size (super) particles described by polynomial-based spatial shape factors in electromagnetic particle-in-cell (EM-PIC) algorithms for kinetic plasma simulations based on unstructured meshes. The proposed implementation is aimed at mitigating (spurious) numerical Cherenkov radiation effects while preserving exact charge conservation. This is achieved by employing a representation of Maxwell's equations based on the exterior calculus of differential forms and a discretization of twisted differential forms associated with the primal mesh by Whitney forms, while ordinary differential forms are associated with the dual mesh. The set of full-discrete Maxwell's equations is then obtained by using a finite-element Galerkin method and a leap-frog time integration, leading to a mixed D-H finite-element time-domain Maxwell field solver. In the proposed EM-PIC implementation, the gather and scatter steps of the algorithm yield integrals of polynomials over polyhedral domains in 3-D space (or polygonal domains in 2-D space) that can be evaluated analytically. We provide numerical examples confirming charge conservation down to machine precision on an unstructured mesh.

Design and Simulation of a T-Vane Relativistic Inverted Magnetron

A design for a high-power relativistic inverted magnetron oscillator (IMO) is presented and modeled numerically using the massively parallel electromagnetic particle-in-cell code, improved concurrent electromagnetic particle in cell (ICEPIC). The inverted magnetron presented here is designed to operate in the $L$ -band for magnetic fields ranging from 0.07 to 0.1 T with voltages ranging from ~250 to 480 kV. Simulations in that voltage/magnetic field range demonstrate that the inverted magnetron is capable of fast startup, $\pi $ -mode dominance with little-to-no mode competition, as well as high output power, exceeding 2.5 GW at the high end of voltages examined. End-loss current, a common source of energy loss in relativistic magnetrons, is not present for this IMO design. The IMO radiates RF energy axially through the use of a novel dual-ring radiator that excites the TM01 mode in the downstream waveguide. This approach yields nominal RF output power efficiencies near 20%. ICEPIC simulations predict that the above features combined with the IMO’s stable, robust, and reliable performance in the desired mode over a large voltage range yield a class of high-power microwave source notable for its inverted geometry and T-vane slow-wave structure.

High-Power Recirculating Planar Crossed-Field Amplifier Design and Development

The recirculating planar crossed-field amplifier (RPCFA) was designed and simulated using the finite-element frequency-domain code ANSYS HFSS and the particle-in-cell (PIC) code MAGIC. The RPCFA is a high-power microwave device adapted from the recirculating planar magnetron, developed at the University of Michigan, Ann Arbor, MI, USA. Electromagnetic (EM) PIC simulations of a planar, meander line, and slow wave structure demonstrated 13.5-dB amplification of a 1.3-MW, 3-GHz signal to approximately 29 MW. The RPCFA is designed to be driven by pulsed power from the Michigan electron long beam accelerator-ceramic insulator, which is currently configured to deliver pulses at −300 kV, 1–10 kA, with 0.3– $1~\mu \text{s}$ pulse lengths. The RF input-drive signal will be provided by an MG5193 magnetron which delivers 5- $\mu \text{s}$ pulses up to 2.6 MW at 3 GHz. EM PIC simulations also demonstrated zero-drive stability of the design and were used to evaluate changes in performance resulting from variations of several experimental parameters. Variation of the drive frequency suggested that the RPCFA is expected to have a 3-dB amplification bandwidth of 300 MHz or 10%.

Electrical Oscillations of the UNU-ICTP Plasma Focus Device in the Early Breakdown Phase

Recent circuit driven electromagnetic particle in cell simulation of the United Nations University-International Centre for Theoretical Physics plasma focus device revealed regular oscillations in both the voltage and current profiles. The simulated voltage waveform agreed well with the experimental profile, where similar unaccounted oscillations were also observed. In this paper, the oscillations are attributed to the plasma inductance, whose value was calculated by circuit analysis and agreed reasonably well with the computed experimental value. A circuit simulation of the prebreakdown and postbreakdown phases, with the plasma inductance incorporated, reproduced with sufficiency accuracy the electromagnetic particle in cell generated waveforms and consequently confirmed our finding.

Performance analysis and implementation details of the Energy Conserving Semi-Implicit Method code (ECsim)

We present in this work the implementation of the Energy Conserving Semi-Implicit Method in a parallel code called ECsim. This new code is a three-dimensional, fully electromagnetic particle in cell (PIC) code. It is written in C/C++ and uses MPI to allow massive parallelization. ECsim is unconditionally stable in time, eliminates the finite grid instability, and has the same cycle scheme as the explicit code with a computational cost comparable to other semi-implicit PIC codes. All these features make it a very valuable tool to address situations which have not been possible to analyze until now with other PIC codes. In this work, we show the details of the algorithm implementation and we study its performance in different systems. ECsim is compared with another semi-implicit PIC code with different time and spectral resolution, showing its ability to address situations where other codes fail.

Surface effect investigation on multipactor in microwave components using the EM-PIC method

Multipactor poses a great risk to microwave components in space and its accurate controllable suppression is still lacking. To evaluate the secondary electron emission (SEE) of arbitrary surface states on multipactor, metal samples fabricated with ideal smoothness, random roughness, and micro-structures on the surface are investigated through SEE experiments and multipactor simulations. An accurate quantitative relationship between the SEE parameters and the multipactor discharge threshold in practical components has been established through Electromagnetic Particle-In-Cell (EM-PIC) simulation. Simulation results of microwave components, including the impedance transformer and the coaxial filter, exhibit an intuitive correlation between the critical SEE parameters, varied due to different surface states, and multipactor thresholds. It is demonstrated that it is the surface micro-structures with certain depth and morphology that determine the average yield of secondaries, other than the random surface relieves. Both the random surface relieves and micro-structures have a scattering effect on SEE, and the yield is prone to be identical upon different elevation angles of incident electrons. It possesses a great potential in the optimization and improvement of suppression technology without the exhaustion of the technological parameter.

Axisymmetric charge-conservative electromagnetic particle simulation algorithm on unstructured grids: Application to microwave vacuum electronic devices

We present a charge-conservative electromagnetic particle-in-cell (EM-PIC) algorithm optimized for the analysis of vacuum electronic devices (VEDs) with cylindrical symmetry (axisymmetry). We exploit the axisymmetry present in the device geometry, fields, and sources to reduce the dimensionality of the problem from 3D to 2D. Further, we employ ‘transformation optics’ principles to map the original problem in polar coordinates with metric tensor diag(1,ρ2,1) to an equivalent problem on a Cartesian metric tensor diag(1,1,1) with an effective (artificial) inhomogeneous medium introduced. The resulting problem in the meridian (ρz) plane is discretized using an unstructured 2D mesh considering TEϕ-polarized fields. Electromagnetic field and source (node-based charges and edge-based currents) variables are expressed as differential forms of various degrees, and discretized using Whitney forms. Using leapfrog time integration, we obtain a mixed E−B finite-element time-domain scheme for the full-discrete Maxwell's equations. We achieve a local and explicit time update for the field equations by employing the sparse approximate inverse (SPAI) algorithm. Interpolating field values to particles' positions for solving Newton–Lorentz equations of motion is also done via Whitney forms. Particles are advanced using the Boris algorithm with relativistic correction. A recently introduced charge-conserving scatter scheme tailored for 2D unstructured grids is used in the scatter step. The algorithm is validated considering cylindrical cavity and space-charge-limited cylindrical diode problems. We use the algorithm to investigate the physical performance of VEDs designed to harness particle bunching effects arising from the coherent (resonance) Cerenkov electron beam interactions within micro-machined slow wave structures.

Simulation of beam-induced plasma in gas-filled rf cavities

Processes occurring in a radio-frequency (rf) cavity, filled with high pressure gas and interacting with proton beams, have been studied via advanced numerical simulations. Simulations support the experimental program on the hydrogen gas-filled rf cavity in the Mucool Test Area (MTA) at Fermilab, and broader research on the design of muon cooling devices. SPACE, a 3D electromagnetic particle-in-cell (EM-PIC) code with atomic physics support, was used in simulation studies. Plasma dynamics in the rf cavity, including the process of neutral gas ionization by proton beams, plasma loading of the rf cavity, and atomic processes in plasma such as electron-ion and ion-ion recombination and electron attachment to dopant molecules, have been studied. Through comparison with experiments in the MTA, simulations quantified several uncertain values of plasma properties such as effective recombination rates and the attachment time of electrons to dopant molecules. Simulations have achieved very good agreement with experiments on plasma loading and related processes. The experimentally validated code SPACE is capable of predictive simulations of muon cooling devices.

Local, Explicit, and Charge-Conserving Electromagnetic Particle-In-Cell Algorithm on Unstructured Grids

We present a charge-conserving electromagnetic particle-in-cell (EM-PIC) algorithm on unstructured grids based on a finite element (FE) time-domain methodology with explicit field update, i.e., requiring no linear solver. The proposed explicit EM-PIC algorithm attains charge conservation from first principles by representing fields, currents, and charges by differential forms of various degrees, following the methodology put forth in reference [25]. The need for a linear solver is obviated by constructing a sparse approximate inverse (SPAI) for the FE system matrix, which also preserves the locality (sparsity) of the algorithm. We analyze in detail the residual error caused by SPAI on the motions of charged particles and beam trajectories and show that this error is several orders of magnitude smaller than the inherent error caused by the spatial and temporal discretizations.

Conformal Electromagnetic Particle in Cell: A Review

Conformal (or body-fitted) electromagnetic particle-in-cell (EM-PIC) numerical solution schemes are reviewed. Included is a chronological history of relevant particle physics algorithms often employed in these conformal simulations. Brief mathematical descriptions of particle-tracking algorithms and current weighting schemes are provided, along with a brief summary of major time-dependent electromagnetic solution methods. Several research areas are also highlighted for recommended future development of new conformal EM-PIC methods.

Simulation of multipactor on the rectangular grooved dielectric surface

Multipactor discharge on the rectangular grooved dielectric surface is simulated self-consistently by using a two-and-a-half dimensional (2.5 D) electrostatic particle-in-cell (PIC) code. Compared with the electromagnetic PIC code, the former can give much more accurate solution for the space charge field caused by the multipactor electrons and the deposited surface charge. According to the rectangular groove width and height, the multipactor can be divided into four models, the spatial distributions of the multipactor electrons and the space charge fields are presented for these models. It shows that the rectangular groove in different models gives very different suppression effect on the multipactor, effective and efficient suppression on the multipactor can only be reached with a proper groove size.

A multi-dimensional, energy- and charge-conserving, nonlinearly implicit, electromagnetic Vlasov–Darwin particle-in-cell algorithm

For decades, the Vlasov–Darwin model has been recognized to be attractive for particle-in-cell (PIC) kinetic plasma simulations in non-radiative electromagnetic regimes, to avoid radiative noise issues and gain computational efficiency. However, the Darwin model results in an elliptic set of field equations that renders conventional explicit time integration unconditionally unstable. Here, we explore a fully implicit PIC algorithm for the Vlasov–Darwin model in multiple dimensions, which overcomes many difficulties of traditional semi-implicit Darwin PIC algorithms. The finite-difference scheme for Darwin field equations and particle equations of motion is space–time-centered, employing particle sub-cycling and orbit-averaging. The algorithm conserves total energy, local charge, canonical-momentum in the ignorable direction, and preserves the Coulomb gauge exactly. An asymptotically well-posed fluid preconditioner allows efficient use of large cell sizes, which are determined by accuracy considerations, not stability, and can be orders of magnitude larger than required in a standard explicit electromagnetic PIC simulation. We demonstrate the accuracy and efficiency properties of the algorithm with various numerical experiments in 2D–3V.

On the numerical dispersion of electromagnetic particle-in-cell code: Finite grid instability

The Particle-In-Cell (PIC) method is widely used in relativistic particle beam and laser plasma modeling. However, the PIC method exhibits numerical instabilities that can render unphysical simulation results or even destroy the simulation. For electromagnetic relativistic beam and plasma modeling, the most relevant numerical instabilities are the finite grid instability and the numerical Cherenkov instability. We review the numerical dispersion relation of the Electromagnetic PIC model. We rigorously derive the faithful 3-D numerical dispersion relation of the PIC model, for a simple, direct current deposition scheme, which does not conserve electric charge exactly. We then specialize to the Yee FDTD scheme. In particular, we clarify the presence of alias modes in an eigenmode analysis of the PIC model, which combines both discrete and continuous variables. The manner in which the PIC model updates and samples the fields and distribution function, together with the temporal and spatial phase factors from solving Maxwell's equations on the Yee grid with the leapfrog scheme, is explicitly accounted for. Numerical solutions to the electrostatic-like modes in the 1-D dispersion relation for a cold drifting plasma are obtained for parameters of interest. In the succeeding analysis, we investigate how the finite grid instability arises from the interaction of the numerical modes admitted in the system and their aliases. The most significant interaction is due critically to the correct representation of the operators in the dispersion relation. We obtain a simple analytic expression for the peak growth rate due to this interaction, which is then verified by simulation. We demonstrate that our analysis is readily extendable to charge conserving models.

Simulation investigation of multipactor in metal components for space application with an improved secondary emission model

Effects of the secondary electron emission (SEE) phenomenon of metal surface on the multipactor analysis of microwave components are investigated numerically and experimentally in this paper. Both the secondary electron yield (SEY) and the emitted energy spectrum measurements are performed on silver plated samples for accurate description of the SEE phenomenon. A phenomenological probabilistic model based on SEE physics is utilized and fitted accurately to the measured SEY and emitted energy spectrum of the conditioned surface material of microwave components. Specially, the phenomenological probabilistic model is extended to the low primary energy end lower than 20 eV mathematically, since no accurate measurement data can be obtained. Embedding the phenomenological probabilistic model into the Electromagnetic Particle-In-Cell (EM-PIC) method, the electronic resonant multipacting in microwave components can be tracked and hence the multipactor threshold can be predicted. The threshold prediction error of the transformer and the coaxial filter is 0.12 dB and 1.5 dB, respectively. Simulation results demonstrate that the discharge threshold is strongly dependent on the SEYs and its energy spectrum in the low energy end (lower than 50 eV). Multipacting simulation results agree quite well with experiments in practical components, while the phenomenological probabilistic model fit both the SEY and the emission energy spectrum better than the traditionally used model and distribution. The EM-PIC simulation method with the phenomenological probabilistic model for the surface collision simulation has been demonstrated for predicting the multipactor threshold in metal components for space application.

Elimination of the numerical Cerenkov instability for spectral EM-PIC codes

When using an electromagnetic particle-in-cell (EM-PIC) code to simulate a relativistically drifting plasma, a violent numerical instability known as the numerical Cerenkov instability (NCI) occurs. The NCI is due to the unphysical coupling of electromagnetic waves on a grid to wave–particle resonances, including aliased resonances, i.e., ω+2πμ/Δt=(k1+2πν1/Δx1)v0, where μ and ν1 refer to the time and space aliases and the plasma is drifting relativistically at velocity v0 in the 1ˆ-direction. We extend our previous work Xu et al. (2013) by recasting the numerical dispersion relation of a relativistically drifting plasma into a form which shows explicitly how the instability results from the coupling modes which are purely transverse electromagnetic (EM) modes and purely longitudinal modes in the rest frame of the plasma for each time and space aliasing. The dispersion relation for each μ and ν1 is the product of the dispersion relation of these two modes set equal to a coupling term that vanishes in the continuous limit. The new form of the numerical dispersion relation provides an accurate method of systematically calculating the growth rate and location of the mode in the fundamental Brillouin zone for any Maxwell solver for each μ and ν1. We then focus on the spectral Maxwell solver and systematically discuss its NCI modes. We show that the second fastest growing NCI mode for the spectral solver corresponds to μ=ν1=0, that it has a growth rate approximately one order of magnitude smaller than the fastest growing μ=0 and ν1=1 mode, and that its location in the k space fundamental Brillouin zone is sensitive to the grid size and time step. Based on these studies, strategies to systematically eliminate the NCI modes for a spectral solver are developed. We apply these strategies to both relativistic collisionless shock and LWFA simulations, and demonstrate that high-fidelity multi-dimensional simulations of drifting plasmas can be carried out with a spectral Maxwell solver with no evidence of numerical Cerenkov instability.

An External Circuit Model for 3-D Electromagnetic Particle-In-Cell Simulations

In this paper, for the first time, second-order accurate algorithms for coupling external circuit elements to 3-D electromagnetic (EM) particle-in-cell (PIC) simulations are developed and introduced. The circuit equation including an external voltage or current source, resistance, inductance, capacitance, and a dynamic EM and/or plasma load is solved simultaneously and self-consistently with the EM PIC updaters through an instant measured voltage $V$ across the system to obtain the supplied current $I$ for feeding into the system. This external circuit model has been demonstrated and implemented in a 3-D conformal finite-difference time-domain PIC code, Vorpal. The integrated simulation model can be widely used in the vacuum electronics community as well as in the plasma processing community if one wants to consider the effects of a power supply on operation of microwave tubes or plasma reaction chambers.

Wave-particle-interaction in kinetic plasmas

Resonant scattering of energetic protons off magnetic irregularities is the main process in cosmic ray diffusion. The typical theoretical description uses Alfvén waves in the low frequency limit. We demonstrate that the usage of Particle-in-Cell (PiC) simulations for particle scattering is feasible. The simulation of plasma waves is performed with the relativistic electromagnetic PiC code ACRONYM and the tracks of test particles are evaluated in order to study particle diffusion. Results for the low frequency limit are equivalent to those obtained with an MHD description, but only for high frequencies results can be obtained with reasonable effort. PiC codes have the potential to be a useful tool to study particle diffusion in kinetic turbulence.

Electromagnetic particle in cell modeling of the plasma focus: Current sheath formation and lift off

The shaping and formation of the current sheath takes place in the breakdown phase of a plasma focus device and critically controls the device performance. Electrostatic particle in cell codes, with magnetic effects ignored, have been used to model the breakdown phase. This Letter reports the successful development and implementation of an electromagnetic particle in cell (EMPIC) code, including magnetic effects self-consistently, to simulate the breakdown phase; from the ionization, localization and gliding discharge along the insulator to the time instant of current sheath lift off. The magnetic field was found to be appreciable from the time the current sheath came into contact with the anode with increased local current, initiating the voltage breakdown of the device as a result.

Analysis and PIC simulation of a Gyrotron travelling wave tube amplifier

The analysis of a Ka-band gyrotron Travelling Wave Tube (gyro-TWT) amplifier using a uniform cylindrical waveguide as its interaction circuit has been presented for the TE01 mode of operation a self-consistent nonlinear analysis in the large signal regime. The analysis predicts that the saturated peak output power of ~ 134 kW with a power conversion efficiency of ~ 22.7 %. The saturated gain has been calculated as ~ 41.3 dB for the amplifier driven by 72 kV, 8.2 A electron beam of a pitch factor 1.05. The critical interaction length of smooth wall uniform metal guide is found 10.7 cm for the stable amplifier operation. The form factor and the norm factor have also been estimated with the large signal non-linear code. Further, for 5% spread the amplifier develops the peak power of ~ 128 kW with an electronic efficiency of ~ 21.7 % and the gain of ~ 41.07 dB. These results are found to be good harmony when the beam wave interaction is studied with a commercial 3-D electromagnetic particle in cell (PIC) code. The behaviour of the beam all over the length of the interaction circuit has been monitored by calculating its energy, momentum, phase, etc. with the help of a commercial PIC code and which also in good agreement with analytical results with 1% deviation.