Particle-in-cell Research Articles

Two-plasmon decay of microwaves in low temperature magnetized laboratory plasmas

Observations from magnetically confined fusion experiments have in the past decades revealed instances of two-plasmon decay instabilities (TPDIs) between injected 2nd harmonic X-mode waves and trapped upper hybrid (UH) waves near half frequency of the injected wave. We present and compare growth rates of TPDIs computed from reduced analytical models and particle-in-cell (PIC) simulations in low-temperature plasmas, emulating conditions found in more accessible controllable laboratory devices. The growth rates obtained from simulations agree with our modeling, confirming that we can model TPDIs using both analytical models and PIC simulations in low-temperature plasmas. Furthermore, the growth rates are on the order of 0.01 ns−1 to 0.1 ns−1 for injected waves with intensity of 103 W/cm2, which indicates that we can expect to see signals of TPDIs in controllable laboratory plasma devices.

Over-30-GeV intense laser phase-locked direct electron acceleration

When an intense laser obliquely irradiates a solid, a pre-pulse will first ionize the solid surface, followed by the main pulse interacting with the plasma and ultimately being reflected by the plasma. Simultaneously, certain electrons within the plasma will be trapped in the accelerating phase of the laser field, subsequently gaining effective acceleration within the field, this phenomenon is known as phase-locked electron acceleration. Given the current intense lasers' electric field intensity nearing the TV/m range, electrons could potentially acquire energy levels on the order of hundreds of GeV or even TeV, provided they stay in the accelerating phase of the laser field long enough. Here, we initially use PIC (Particle-in-Cell) simulations to simulate the interaction process between laser pulses and plasma, thereby obtaining the properties of phase-locked electrons. In order to reduce computational demands, we turn to use a three-dimensional (3D) test particle model to calculate the subsequent interactions of these electrons with the reflected laser field. By this model, we obtain the data of the locked-phase electrons after having interacted with the reflected laser (Fig. (a)). Furthermore, we use this model to calculate the dynamical behavior of electrons under different initial conditions (Fig. (b)). Under the laser intensity of <inline-formula><tex-math id="M1">\begin{document}$ {a}_{0}=350 $\end{document}</tex-math></inline-formula>(<inline-formula><tex-math id="M2">\begin{document}$ {a}_{0} $\end{document}</tex-math></inline-formula> is the normalized laser vector potential), the energy of the electrons directly accelerated by the laser is enhanced to 32 GeV. In contrast, under the same laser intensity, the energy of the electrons accelerated by ponderomotive force is only 0.35 GeV. The research findings indicate that the strong laser with peak power around 10 PW can directly accelerate electrons to approximately 30 GeV. Additionally, this study outlines the optimal initial conditions for injecting electrons into the laser field and the final electron energy within the phase-locked acceleration mechanism, thereby establishing a calibration relationship with the laser field intensity. Given the continual enhancement of laser intensity and the potential application of the laser phase-locked electron acceleration mechanism to positron acceleration, this research holds promise for its implementation in fields such as miniaturized positron-electron colliders and high-energy gamma-ray sources.

Particle acceleration by sub-proton cyclotron frequency spectrum of dispersive Alfven waves in inhomogeneous solar coronal plasmas

ABSTRACT The problem of explaining observed soft X-ray fluxes during solar flares, which invokes acceleration of large fraction of electrons, if the acceleration takes places at the solar coronal loop-top, can potentially be solved by postulating that flare at loop-top creates dispersive Alfven waves (DAWs) which propagate towards the foot-points. As DAWs move in progressively denser parts of the loop (due to gravitational stratification) the large fraction of electrons is no longer needed. Here, we extend our previous results by considering f−1 frequency spectrum of DAWs and add He++ ions using fully kinetic particle-in-cell (PIC) simulations. We consider cases when transverse density gradient is in the range 4–40c/ωpe and DAW driving frequency is 0.3–0.6ωcp. We find that (i) The frequency spectrum case does not affect electron acceleration fraction in the like-to-like cases, but few times larger percentage of He++ heating is seen due to ion cyclotron resonance; (ii) In cases when counter propagating DAWs collide multiple-times, much larger electron and ion acceleration fractions are found, but the process is intermittent in time. This is because intensive heating (temperature increase) makes the-above-thermal-fraction smaller; Also more isotropic velocity distributions are seen; (iii) Development of kink oscillations occurs when DAWs collide; (iv) Scaling of the magnetic fluctuations power spectrum steepening in the higher-density regions is seen, due to wave refraction. Our PIC runs produce much steeper slopes than the orginal spectrum, indicating that the electron-scale physics has a notable effect on DAW spectrum evolution.

Enhancement of electron energies in the laser wakefield acceleration using a density up-ramp capillary gas-cell

In the laser wakefield acceleration (LWFA), the dephasing problem is a serious energy-limiting factor, which is caused by the velocity difference between the accelerated electrons and the laser wake wave. To overcome the dephasing problem, we developed a special capillary gas-cell with a density up-ramp along the laser propagation direction and used it for electron acceleration experiments. Our experiments, which were performed with a peak laser power of 15 TW at GIST, show that the electron beam energy was enhanced to 260 MeV compared with 98 MeV from a flat density profile, which is more than two fold by the density up-ramp due to the increase of the dephasing length in the density up-ramp. This is the first experimental demonstration for electron energy enhancement using the density up-ramp in a capillary gas-cell. Two-dimensional particle-in-cell (PIC) simulations were also performed to confirm the effect of the density up-ramp, which shows a good agreement with the experimental results. Compared with a gas jet, the capillary gas-cell can provide a more stable and longer plasma so that the density up-ramp method in the capillary gas-cell may produce high quality electron beams of energy up to GeV range.

Time-resolved optical shadowgraphy of solid hydrogen jets as a testbed to benchmark particle-in-cell simulations

Particle-in-cell (PIC) simulations are a widely-used tool to model kinetics-dominated plasmas in ultrarelativistic laser-solid interactions (dimensionless vectorpotential a0 > 1). However, interactions approaching subrelativistic laser intensities (a0 ≲ 1) are governed by correlated and collisional plasma physics, calling for benchmarks of available modeling capabilities and the establishment of standardized testbeds. Here, we propose such a testbed to experimentally benchmark PIC simulations of laser-solid interactions using a laser-irradiated micron-sized cryogenic hydrogen-jet target. Time-resolved optical shadowgraphy of the expanding plasma density, complemented by hydrodynamics and ray-tracing simulations, is used to determine the bulk-electron-temperature evolution after laser irradiation. We showcase our testbed by studying isochoric heating of solid hydrogen induced by laser pulses with a dimensionless vectorpotential of a0 ≈ 1. Our testbed reveals that the initial surface-density gradient of the target is decisive to reach quantitative agreement at 1 ps after the interaction, demonstrating its suitability to benchmark controlled parameter scans at subrelativistic laser intensities.

Influence of the magnetic field curvature on the radial–azimuthal dynamics of a Hall thruster plasma discharge with different propellants

The topology of the applied magnetic field is an important design aspect of Hall thrusters. For modern Hall thrusters, the magnetic field topology most often features curved lines with a concave (negative) curvature upstream of the field's peak and a convex (positive) curvature downstream. Additionally, the advent of the magnetic shielding technique has resulted in Hall thruster designs with non-conventional field topologies that exhibit high degrees of concavity upstream of the field's peak. In this article, a detailed study is carried out on the effects that the magnetic field curvature has on the plasma discharge in a 2D configuration representative of a Hall thruster's radial–azimuthal cross section. The analyses are performed for discharges of three propellants of high applied interest: xenon, krypton, and argon. For each propellant, high-fidelity electrostatic reduced-order particle-in-cell (PIC) simulations are performed with various degrees of positive and negative curvatures of the magnetic field. Corresponding 1D radial PIC simulations are also performed for xenon to compare the observations against the 2D results. Most notably, it is observed that the instability spectra in the positive-curvature cases are mostly dominated by electron cyclotron drift instability, whereas the modified two stream instability is dominant in the negative-curvature cases. The distributions of electron and ion temperatures, in particular, as well as the contribution of various mechanisms to electrons’ cross-field transport show notable variations between the positive and negative curvature values. Finally, the field curvature is observed to majorly influence the ion beam divergence along the radial and azimuthal coordinates.

Moment tracking and their coordinate transformations for macroparticles with an application to plasmas around black holes

Particle-in-cell (PIC) codes usually represent large groups of particles as a single macroparticle. These codes are computationally efficient but lose information about the internal structure of the macroparticle. To improve the accuracy of these codes, this work presents a method in which, as well as tracking the macroparticle, the moments of the macroparticle are also tracked. Although the equations needed to track these moments are known, the coordinate transformations for moments where the space and time coordinates are mixed cannot be calculated using the standard method for representing moments. These coordinate transformations are important in astrophysical plasma, where there is no preferred coordinate system. This work uses the language of Schwartz distributions to calculate the coordinate transformations of moments. Both the moment tracking and coordinate transformation equations are tested by modelling the motion of uncharged particles in a circular orbit around a black hole in both Schwarzschild and Kruskal–Szekeres coordinates. Numerical testing shows that the error in tracking moments is small, and scales quadratically. This error can be improved by including higher order moments. By choosing an appropriate method for using these moments to deposit the charge back onto the grid, a full PIC code can be developed.

Fully kinetic study of facility pressure effects on RF-source magnetic nozzles

A fully kinetic 2D axisymmetric Particle-in-Cell (PIC) model is used to examine the effects of background facility pressure on the plasma transport and propulsive efficiency of magnetic nozzles. Simulations are performed for a low-power (150 W class) cathode-less radio-frequency (RF) plasma thruster, operating with xenon, between background pressures up to 10−2 Pa and average electron discharge temperatures of 4–16 eV. When the electron temperature within the near-plume region reaches 8 eV, a decisive reduction in performance occurs: at 10−2 Pa, in-plume power losses surpass 25% of the discharge energy flux. Given that the ionisation energy for Xe is 12 eV, the 8 eV threshold indicates that a consistent percentage of electrons has energy enough to trigger ionisation. On the other hand, when the temperature is below such threshold, the primary collisions are charge-exchange and inelastic ion scattering, and the power loss remains less than 10%. It is established that losses in the considered thruster are significant if the facility pressure is greater than 10−3 Pa, at absorbed powers larger than 130 W. At the nominal 150 W, this results in a 15% thrust reduction. When facility pressure is taken into consideration over ideal vacuum simulations, numerical error is reduced to <30% when compared to experimental thrust measurements at 10−3 Pa.

First Simulations for the EuAPS Betatron Radiation Source: A Dedicated Radiation Calculation Code

X-ray production through betatron radiation emission from electron bunches is a valuable resource for several research fields. The EuAPS (EuPRAXIA Advanced Photon Sources) project, within the framework of EuPRAXIA, aims to provide 1–10 keV photons (X-rays), developing a compact plasma-based system designed to exploit self-injection processes that occur in the highly nonlinear laser-plasma interaction (LWFA) to drive electron betatron oscillations. Since the emitted radiation spectrum, intensity, angular divergence, and possible coherence strongly depend on the properties of the self-injected beam, accurate preliminary simulations of the process are necessary to evaluate the optimal diagnostic device specifications and to provide an initial estimate of the source’s performance. A dedicated tool for these tasks has been developed; electron trajectories from particle-in-cell (PIC) simulations are currently undergoing numerical analysis through the calculation of retarded fields and spectra for various plasma and laser parameter combinations. The implemented forward approach evaluation of the fields could allow for the integration of the presented scheme into already existing PIC codes. The spectrum calculation is thus performed in detector time, giving a linear complex exponential phase; this feature allows for a semi-analitical Fourier transform evaluation. The code structure and some trajectories analysis results are presented.

Plasma plume simulation of an atomic oxygen-fed ion thruster in very-low-earth-orbit

The plasma plume flow of an atomic oxygen-fed (AO-fed) ion thruster is numerically investigated as a simplification of the atmosphere-breathing electric propulsion (ABEP). A predictive analysis is conducted focusing on the ion backflow phenomenon and plume-background interaction in very-low-earth-orbit (VLEO). The computational framework employs two sequentially integrated numerical methods: a zero-dimensional (0-D) analytical model for the radio-frequency ion thruster and a hybrid method of the particle-in-cell (PIC) and direct simulation Monte Carlo (DSMC) techniques. The 0-D analytic model is employed for the prediction of exhaust conditions, while the hybrid PIC-DSMC method adopts these predictions to conduct the plasma plume simulations. A generalized collision cross-section model is introduced to enable consistent kinetic simulations for both AO and xenon propellants in VLEO atmosphere. The plasma plume simulations are conducted in an axisymmetric domain, including a cylindrical satellite body to consider wake flow. The exhaust ions exhibit diffusive transport transverse to the ion beam direction, implying the ion backflow. The backflowing ion current density can be increased in AO-fed thrusters, which require a high propellant flow rate to achieve a practical thrust. The AO-fed ion thruster shows a more active interaction between its plasma plume and the VLEO atmosphere compared to conventional xenon-based thrusters. The intensified plume-background interaction modifies the backflowing ion current density and the kinetic energy of individual ions, factors related to the spacecraft’s surface contamination.

Particle-in-cell simulations of high frequency capacitively coupled plasmas including spatially localised inductive-like heating

High frequency (HF) capacitively coupled plasmas (CCPs) are ubiquitous, having several industrial applications, especially in the semiconductor industry. Inductive heating effects within these plasmas play an important role and therefore understanding them is key to improve industrial applications. For this purpose kinetic research, using particle-in-cell (PIC) codes, offers significant opportunity to study, and improve, industrial plasma processes that operate at the atomic level. However, PIC codes commonly used for CCPs are electrostatic and thus cannot be used to simulate electromagnetically induced currents. Therefore we have developed EPOCH-LTP, a 1D PIC code with a current heating model, that enables the simulation of inductive heating effects in HF CCPs. First simulation results, from an HF CCP (60 MHz) operated at 1 mTorr of argon, show that inductive currents couple most of their power to the electrons at the interface between the bulk plasma and the sheath. Furthermore, the simulation of a dual-frequency CCP, where a HF inductive current and a low-frequency (LF) voltage waveform at 400 kHz are applied, have shown a synergy between the HF and LF waveforms that increase the inductive heating rate.

Geometric Particle-In-Cell discretizations of a plasma hybrid model with kinetic ions and mass-less fluid electrons

We explore the possibilities of applying structure-preserving numerical methods to a plasma hybrid model with kinetic ions and mass-less fluid electrons satisfying the quasi-neutrality relation. The numerical schemes are derived by finite element methods in the framework of finite element exterior calculus (FEEC) for field variables, Particle-In-Cell (PIC) methods for the Vlasov equation, and splitting methods in time based on an anti-symmetric bracket proposed. Conservation properties of energy, quasi-neutrality relation, positivity of density, and divergence-free property of the magnetic field are given irrespective of the used resolution and metric. Local quasi-interpolation is used for dealing with the current terms in order to make the proposed methods more efficient. The implementation has been done in the framework of the Python package STRUPHY [1], and has been verified by extensive numerical experiments.

Effective thermo-electric-mechanical modeling of capacitively coupled plasma in low-pressure conditions: Modeling and application in dry etching

In semiconductor dry etching process, capacitively coupled plasma (CCP) is widely used in low-pressure environments. However, the kinetics of species particles in low-pressure conditions are different from the general environments. Traditional continuum models are popular in industry due to their simplicity, but they struggle to account for the kinetics of particles in low-pressure conditions. Although particle-in-cell (PIC) modeling is suitable for simulating the behavior of a single particle, it is difficult to perform such industrial simulations due to the high computational cost. To account for both kinetics and engineering applicability, this study presents an effective thermo-electric-mechanical modeling structure of CCP at low-pressure conditions. This modeling structure includes balance laws of continua derived from the global energy equilibrium containing thermal, electric and mechanical energies. The ion momentum, which is ignored in traditional plasma continuum models, is considered in the balance laws, and electron energy distributions are obtained by solving the Boltzmann equation to obtain the properties of electron transport. In addition, the model variables are calculated based on mixed kinetic theory accounting for the fraction of each species. The results of the proposed modeling are compared to those of a PIC model to validate its ability to describe the distribution of electrons with respect to pressure conditions. Based on the proposed numerical formulations, this study built a MATLAB-based program to guide the design of a screen baffle in dry etching equipment. This program captures the critical aspects of the practical application of screen baffle design in the context of the dry etching process. The results of this paper showed that the proposed model can be applied to engineering applications under low-pressure conditions.

An integrative mathematical model of microwave ion thruster for mHz-frequency thrust noise analysis

Many high-precision space missions need thrusters to produce thrust with low noise to compensate for disturbances and ensure satellite platform stability. Microwave ion thruster is characterized with a wide thrust range and potential for these missions. A cost-effective and accurate mathematical model is crucial for mHz-frequency thrust noise analysis and feedback controller design. The Particle-In-Cell (PIC) and global models are two common simulation tools. The PIC model is characterized with high accuracy but huge computation cost, which is difficult to analyze long-time performance characteristics. Now, the global model is only used for the discharge chamber with low accuracy and cannot reflect ion extraction properties. In this paper, an integrative mathematical model is built for a 1-cm microwave ion thruster and can reflect ion beamlet divergence and impingement on the Accelerator Grid (AG). Simulation results show good agreement with experiments at 0.06 sccm. However, the model demonstrates worse consistency with experiments when the flux increases to 0.1 sccm, which may be because the influence of neutral gas on the Electron Cyclotron Resonance (ECR) is not considered in the model. A long-time (1000 s) simulation is conducted with this model under 35 μN. It takes 3 hrs, and the thrust noise reaches 1 μN/Hz0.5 at 1 mHz.

Dynamic mode decomposition for data-driven analysis and reduced-order modeling of E × B plasmas: I. Extraction of spatiotemporally coherent patterns

The advent of data-driven/machine-learning based methods and the increase in data available from high-fidelity simulations and experiments has opened new pathways toward realizing reduced-order models for plasma systems that can aid in explaining the complex, multi-dimensional phenomena and enable forecasting and prediction of the systems’ behavior. In this two-part article, we evaluate the utility and the generalizability of the dynamic mode decomposition (DMD) algorithm for data-driven analysis and reduced-order modeling of plasma dynamics in cross-field E × B configurations. The DMD algorithm is an interpretable data-driven method that finds a best-fit linear model describing the time evolution of spatiotemporally coherent structures (patterns) in data. We have applied the DMD to extensive high-fidelity datasets generated using a particle-in-cell (PIC) code based on the cost-efficient reduced-order PIC scheme. In this part, we first provide an overview of the concept of DMD and its underpinning proper orthogonal and singular value decomposition methods. Two of the main DMD variants are next introduced. We then present and discuss the results of the DMD application in terms of the identification and extraction of the dominant spatiotemporal modes from high-fidelity data over a range of simulation conditions. We demonstrate that the DMD variant based on variable projection optimization (OPT-DMD) outperforms the basic DMD method in identification of the modes underlying the data, leading to notably more reliable reconstruction of the ground-truth. Furthermore, we show in multiple test cases that the discrete frequency spectrum of OPT-DMD-extracted modes is consistent with the temporal spectrum from the fast Fourier transform of the data. This observation implies that the OPT-DMD augments the conventional spectral analyses by being able to uniquely reveal the spatial structure of the dominant modes in the frequency spectra, thus, yielding more accessible, comprehensive information on the spatiotemporal characteristics of the plasma phenomena.

GPIC: A set of high-efficiency CUDA Fortran code using gpu for particle-in-cell simulation in space physics

Previous implementation of the full kinetic particle-in-cell (PIC) simulation on the Graphical Processing Unit (GPU) device has shown advantages compared to traditional Central Processing Unit (CPU) computing. However, conventional PIC simulations using GPU computing have faced limitations in terms of low performance when the simulation box size or the number of particles per cell increases. In this study, we boost computing efficiency by designing novel schemes of kernels with the combination of numerical and technical aspects. The time expenses of the global data transferring and duplicate data fetching processes have been massively reduced by the utilization of the on-chip memory. The reduction treatment on the particles, the graded current computing scheme, and the 2.5D thread launch strategy are designed to accelerate the simulation iterations. The new scheme can reach up to about 5.5 times the acceleration ratio than the old one and attain the highest to 734 times faster than the program using CPUs. Our new scheme can realize large-scale PIC simulations with both decent performances on the legacy GPUs and the increasing trend of the acceleration ratio on the lasted GPUs.

An alternative two-way coupled Euler-Lagrange scheme to model the performance of finite-size particle in supersonic flow

In Euler-Lagrange (EL) framework, the Gauss kernel (GK) is usually introduced to avoid Dirac source disturbance of traditional particle-in-cell (PIC) method in subsonic regime. This work investigates the applicability and limitations of PIC-two-way coupling and GK for finite-size particles subjected to supersonic and dilute regime. Results indicate that the source disturbance of GK always peaks at the symmetric center of GK causing a suboptimal shock and ignoring wake in supersonic regime. Meanwhile, classical PIC-interpolation underestimates the drag force on particles due to shock and finite-size effects. To circumvent those issues, a modified coupling method utilizing upstream weighting averaged method and mix-gaussian kernel is proposed to optimize Euler-to-Lagrange coupling and Lagrange-to-Euler coupling, respectively. The modified method not only allows for accurate tracking of arbitrary finite-size particles in supersonic flows but also restores the undisturbed laden flow quantities exerting on particles. Shock and wake zone are reproduced before and after the particle based on recommended parameters. Furthermore, the modified method is validated using a particle-pair exposed to supersonic flow referred from Utkin's experiment(Utkin et al., 2021), and shows excellent agreement.

PHARE: Parallel hybrid particle-in-cell code with patch-based adaptive mesh refinement

Modeling multi-scale collisionless magnetized processes constitutes an important numerical challenge. By treating electrons as a fluid and ions kinetically, the so-called hybrid Particle-In-Cell (PIC) codes represent a promising intermediary between fully kinetic codes, limited to model small scales and short durations, and magnetohydrodynamic codes used large scale. However, simulating processes at scales significantly larger than typical ion particle dynamics while resolving sub-ion dissipative current sheets remain extremely difficult. This paper presents a new hybrid PIC code with patch-based adaptive mesh refinement. Here, hybrid PIC equations are solved on a hierarchy of an arbitrary number of Cartesian meshes of incrementally finer resolution dynamically mapping regions of interest, and with a refined time stepping. This paper presents how the hybrid PIC algorithm is adapted to evolve such mesh hierarchy and the validation of the code on a uniform mesh, fixed refined mesh and dynamically refined mesh.

Quantitative proton radiography and shadowgraphy for arbitrary intensities

Charged-particle radiography and shadowgraphy data can be directly inverted to obtain a line-integrated transverse Lorentz force or a line-integrated transverse refractive index gradient if intensity modulations due to scattering and absorption are negligible, and angular deflections are small. We develop a new direct-inversion algorithm based on plasma physics and compare it to a new Monge–Ampère code and an existing power diagram code (Kasim et al., 2017). The measured or source intensity is represented by electrons subject to drag, and the other intensity by fixed ions. The decrease in kinetic plus electrostatic energy determines convergence. The displacement of the electrons from their initial to their equilibrium positions determines the line-integrated force or refractive index gradient. We have implemented two approaches: PIC (particle in cell) and Lagrangian fluid, in 1-D and 2-D. The PIC code works for arbitrary intensities, can work efficiently in parallel, and can make use of existing codes. The Lagrangian code requires less memory and is faster than the PIC code without massively parallel processing, but fails in 2-D for large intensity modulations. The Monge–Ampère code is by far the fastest in 2-D, without massively parallel processing, but fails for intensities with large voids, high contrast ratios and large deflections across the boundaries, and could not obtain the degree of convergence possible with the PIC code. The power diagram code was by far the slowest and most memory intensive, and failed for large peaks in the measured intensity.

An energy-conserving Fourier particle-in-cell method with asymptotic-preserving preconditioner for Vlasov-Ampère system with exact curl-free constraint

We present an efficient and accurate energy-conserving implicit particle-in-cell (PIC) algorithm for the electrostatic Vlasov system, with particular emphasis on its high robustness for simulating complex plasma systems with multiple physical scales. This method consists of several indispensable elements: (i) the reformulation of the original Vlasov-Poisson system into an equivalent Vlasov-Ampère system with divergence-/curl-free constraints; (ii) a novel structure-preserving Fourier spatial discretization, which exactly preserves these constraints at the discrete level; (iii) a preconditioned Anderson-acceleration algorithm for the solution of the highly nonlinear system; and (iv) a linearized and uniform approximation of the implicit Crank-Nicolson scheme for various Debye lengths, based on the generalized Ohm's law, which serves as an asymptotic-preserving preconditioner for the proposed method. Numerical experiments are conducted, and comparisons are made among the proposed energy-conserving scheme, the classical leapfrog scheme, and a Strang operator-splitting scheme to demonstrate the superiority of the proposed method, especially for plasma systems crossing physical scales.