Electron Plasma Period Research Articles

Direct implicit and explicit energy-conserving particle-in-cell methods for modeling of capacitively coupled plasma devices

Achieving large-scale kinetic modeling is a crucial task for the development and optimization of modern plasma devices. With the trend of decreasing pressure in applications, such as plasma etching, kinetic simulations are necessary to self-consistently capture the particle dynamics. The standard, explicit, electrostatic, momentum-conserving particle-in-cell method suffers from restrictive stability constraints on spatial cell size and temporal time step, requiring resolution of the electron Debye length and electron plasma period, respectively. This results in a very high computational cost, making the technique prohibitive for large volume device modeling. We investigate the direct implicit algorithm and the explicit energy conserving algorithm as alternatives to the standard approach, both of which can reduce computational cost with a minimal (or controllable) impact on results. These algorithms are implemented into the well-tested EDIPIC-2D and LTP-PIC codes, and their performance is evaluated via 2D capacitively coupled plasma discharge simulations. The investigation reveals that both approaches enable the utilization of cell sizes larger than the Debye length, resulting in a reduced runtime, while incurring only minor inaccuracies in plasma parameters. The direct implicit method also allows for time steps larger than the electron plasma period; however, care must be taken to avoid numerical heating or cooling. It is demonstrated that by appropriately adjusting the ratio of cell size to time step, it is possible to mitigate this effect to an acceptable level.

Axion excitation by intense laser fields in a plasma

We study the excitation of axions by intense laser pulses propagating in a plasma. We assume that the pulses propagate along the direction of a static magnetic field. We examine two different configurations. One is that of a long pulse, with a duration Δt much longer than the electron plasma period, Δt ≫ 1/ωp. In this case, the axion field couples with the electron plasma waves (or plasmons) which are excited by the laser pulse, due to a modulational instability. The dispersion relation of the coupled axion-plasmon modes, the unstable regimes and the instability growth rates are established. The other configuration is that of a very short laser pulse, with a duration of the order of (or shorter than) the electron plasma period. In this case, the modulational instability is absent, but a laser wakefield can be excited. The latter consists of both electron density and axion field perturbations. The amplitude of this wakefield is described with a simple one-dimensional model. The two configurations (long and short pulse) can eventually be used to create axions in the laboratory and we suggest that laser plasma experiments could shed some light on the existence of these hypothetical dark matter particles.

Electron acceleration by Bessel–Gaussian laser pulse in a plasma in the presence of an external magnetic field

The wakefield acceleration of electron by Bessel–Gaussian laser pulse in a homogeneous plasma in the presence of an external magnetic field is investigated and compared with results of electron acceleration by Gaussian pulse. For this purpose, the analytical expressions for the longitudinal wakefield behind the pulse and electron energy-gain are obtained and discussed for these types of pulses by making the pulse duration the same as the electron plasma period. The results show that the wakefields excited by laser pulses are strongly dependent on the plasma electron density, pulse intensity, pulse frequency and pulse length. Moreover, the results of the present analysis indicate that the wakefield and the electron energy-gain get enhanced for the larger external magnetic field, but with increasing laser intensity, the external magnetic field is unaffected on the electron acceleration. Using typical parameters and comparing the results obtained for two different laser pulses, the Bessel–Gaussian pulse is found to be significant as in this case the wakefield amplitude and electron energy-gain can be received up to 250 GV/m and 150 MeV, respectively when I=3×1022Wm2, Bo=40T and no=7.5×1024m−3.

Effects of electron temperature on the ion extraction characteristics in a decaying plasma confined between two parallel plates

In this paper, a one-dimension particle-in-cell (PIC) code (EDIPIC) is employed to simulate the parallel-plate ion extraction process under an externally applied electrostatic field, focusing on the analysis of the influence of the initial electron temperature on the extracted ion fluxes to the metal plates during the ion extraction process. Compared with previously published results, the plasma oscillations on a timescale of the electron plasma period, and the excitation of the ion acoustic rarefaction waves resulting from the plasma oscillations originating from both the negative and positive electrodes, are studied for the first time. The modeling results show that both the negative and positive extractors can collect ions due to the plasma oscillations and the propagation of the ion acoustic rarefaction waves. With the increase of the initial electron temperature achieved by keeping other parameters unchanged, on the one hand, both the ion speed and flux to the negative and positive plates increase, which leads to a significant decrease of the ion extraction time, while on the other hand, the ion flux to the positive plate after the formation of a Child–Langmuir sheath is much more sensitive to an increase of the initial electron temperature than that to the negative plate. The PIC simulation results provide a deeper physical understanding of the influence of the initial electron temperature on the characteristics of the entire ion extraction process in a decaying plasma.

Influence of Gaussian, Super-Gaussian, and Cosine-Gaussian Pulse Properties on the Electron Acceleration in a Homogeneous Plasma

In this paper, the process of electron acceleration in a homogeneous plasma is determined for the Gaussian pulse (GP), super-GP, and cosine-GP (CGP). In addition to the wakefield ( $E_{w}$ ) behind the laser pulses, the expressions for the electron energy gain and density perturbation are obtained and compared with each other for all these types of laser pulses by making the pulse duration the same as the electron plasma period. The analytical results show that the amplitude of wakefield depends on the laser intensity, pulse wavelength, pulselength, and plasma electron density. By comparing the results, the CGP is found to be most significant as in this case the electron can be accelerated up to 800 MeV when $I=3\times 10^{22}$ (W/m2), $\lambda =820$ nm, and $n=3.4\times 10^{24}\text{m}^{-3}$ .

Electron acceleration in a homogeneous plasma by Bessel‐Gaussian and Gaussian pulses

The process of electron acceleration by both Bessel‐Gaussian (BG) and Gaussian (G) laser pulses has been investigated comparatively in a homogeneous plasma. Starting with the hydrodynamics fluid and the Maxwell's equations, the three corresponding equations could be acquired which allow us to evaluate electron density perturbations (), wakefield (Ew) and electron energy‐gain (ΔW) respectively for both pulses. Here, the pulse duration and the electron plasma period are taken the same, and by utilizing the perturbation theory the proposed equations have been solved. The analytical results show that the wakefield which leads to electron acceleration, depends on pulse intensity (I), pulse length (L), pulse wavelength (λ) and plasma density (n). Furthermore, by comparing the outcomes, the BG pulse is found to be most suitable for wakefield excitation. The pertinent electron could be accelerated up to 90 MeV if I = 4 × 1021W/m2, L = 20 µm and n = 5.318 × 1024 m−3. Our results are in a good agreement with experimental data reported in literature.

Self-similar space-time evolution of an initial density discontinuity

The space-time evolution of an initial step-like plasma density variation is studied. We give particular attention to formulate the problem in a way that opens for the possibility of realizing the conditions experimentally. After a short transient time interval of the order of the electron plasma period, the solution is self-similar as illustrated by a video where the space-time evolution is reduced to be a function of the ratio x/t. Solutions of this form are usually found for problems without characteristic length and time scales, in our case the quasi-neutral limit. By introducing ion collisions with neutrals into the numerical analysis, we introduce a length scale, the collisional mean free path. We study the breakdown of the self-similarity of the solution as the mean free path is made shorter than the system length. Analytical results are presented for charge exchange collisions, demonstrating a short time collisionless evolution with an ensuing long time diffusive relaxation of the initial perturbation. For large times, we find a diffusion equation as the limiting analytical form for a charge-exchange collisional plasma, with a diffusion coefficient defined as the square of the ion sound speed divided by the (constant) ion collision frequency. The ion-neutral collision frequency acts as a parameter that allows a collisionless result to be obtained in one limit, while the solution of a diffusion equation is recovered in the opposite limit of large collision frequencies.

A comparison of fluid and kinetic models of steady magnetic reconnection

For steady magnetic reconnection in a current sheet, a two-fluid model that includes tensor pressure is compared with kinetic simulations. The kinetic simulations model a Harris sheet in two dimensions with prescribed inflow and free outflow. The two-fluid model provides solutions along the symmetry line from the inflow boundary to the flow stagnation point. The model assumes incompressibility and neglects the Hall current. Profiles from the inflow boundary to the X-point of the out-of-plane electric field and the terms in Ohm’s law that contribute to it from the simulation and from the fluid model agree in almost every detail. The best agreement is obtained when relaxation of the tensor pressure to isotropy occurs on time scales comparable to the electron plasma period. On ion scales, the two-fluid and kinetic simulations give similar results when finite Larmor radius physics is included. The ion pressure relaxation scales linearly with the ion/electron mass ratio.

Vlasov modelling of parallel transport in a tokamak scrape-off layer

A one-dimensional Vlasov–Poisson model is used to describe the parallel transport in a tokamak scrape-off layer. Thanks to a recently developed ‘asymptotic-preserving’ numerical scheme, it is possible to lift numerical constraints on the time step and grid spacing, which are no longer limited by, respectively, the electron plasma period and Debye length. The Vlasov approach provides a good velocity-space resolution even in regions of low density. The model is applied to the study of parallel transport during edge-localized modes, with particular emphasis on the particles and energy fluxes on the divertor plates. The numerical results are compared with analytical estimates based on a free-streaming model, with good general agreement. An interesting feature is the observation of an early electron energy flux, due to suprathermal electrons escaping the ions' attraction. In contrast, the long-time evolution is essentially quasi-neutral and dominated by the ion dynamics.

Transverse instability and perpendicular electric field in two‐dimensional electron phase‐space holes

A multidimensional electron phase‐space hole (electron hole) is considered to be unstable to the transverse instability. In this paper, we perform two‐dimensional (2D) particle‐in‐cell (PIC) simulations to study the evolution of electron holes at different plasma conditions; we find that the evolution is determined by combined actions between the transverse instability and the stabilization by the background magnetic field. In very weakly magnetized plasma (Ωe ≪ ωpe, where Ωe and ωpe are the electron gyrofrequency and plasma frequency, respectively), the transverse instability dominates the evolution of the electron holes. The parallel cut of the perpendicular electric field (E⊥) has bipolar structures, accompanied by the kinking of the electron holes. Such structures last for only tens of electron plasma periods. With the increase of the background magnetic field, the evolution of the electron holes becomes slower. The bipolar structures of the parallel cut of E⊥ in the electron holes can evolve into unipolar structures. In very strongly magnetized plasma (Ωe ≫ ωpe), the unipolar structures of the parallel cut of E⊥ can last for thousands of electron plasma periods. At the same time, the perpendicular electric field (E⊥) in the electron holes can also influence electron trajectories passing through the electron holes, which results in variations of charge density along the direction perpendicular to the background magnetic field outside of the electron holes. When the amplitude of the electron hole is sufficiently strong, streaked structures of E⊥ can be formed outside of the electron holes, which then emit electrostatic whistler waves because of the interactions between the streaked structures of E⊥ and vibrations of the kinked electron holes.

Electrostatic solitary waves in current layers: from Cluster observations during a super-substorm to beam experiments at the LAPD

Abstract. Electrostatic Solitary Waves (ESWs) have been observed by several spacecraft in the current layers of Earth's magnetosphere since 1982. ESWs are manifested as isolated pulses (one wave period) in the high time resolution waveform data obtained on these spacecraft. They are thus nonlinear structures generated out of nonlinear instabilities and processes. We report the first observations of ESWs associated with the onset of a super-substorm that occurred on 24 August 2005 while the Cluster spacecraft were located in the magnetotail at around 18–19 RE and moving northward from the plasma sheet to the lobes. These ESWs were detected in the waveform data of the WBD plasma wave receiver on three of the Cluster spacecraft. The majority of the ESWs were detected about 5 min after the super-substorm onset during which time 1) the PEACE electron instrument detected significant field-aligned electron fluxes from a few 100 eV to 3.5 keV, 2) the EDI instrument detected bursts of field-aligned electron currents, 3) the FGM instrument detected substantial magnetic fluctuations and the presence of Alfvén waves, 4) the STAFF experiment detected broadband electric and magnetic waves, ion cyclotron waves and whistler mode waves, and 5) CIS detected nearly comparable densities of H+ and O+ ions and a large tailward H+ velocity. We compare the characteristics of the ESWs observed during this event to those created in the laboratory at the University of California-Los Angeles Plasma Device (LAPD) with an electron beam. We find that the time durations of both space and LAPD ESWs are only slightly larger than the respective local electron plasma periods, indicating that electron, and not ion, dynamics are responsible for generation of the ESWs. We have discussed possible mechanisms for generating the ESWs in space, including the beam and kinetic Buneman type instabilities and the acoustic instabilities. Future studies will examine these mechanisms in more detail using the space measurements as inputs to models, and better relate the ESW space measurements to the laboratory through PIC code models.

Implicitly charge-conserving solver for Boltzmann electrons

An implicitly charge-conserving algorithm has been developed for solving the nonlinear Poisson equation that results from the use of Boltzmann electrons. The new algorithm solves for the Boltzmann density parameter and, in the case of a Neumann boundary condition, the surface-charge density, simultaneously as it solves for the discretized electrostatic potential. Numerical stability is demonstrated for time steps exceeding the electron plasma period and spatial resolutions much coarser than the Debye length.

Perpendicular electric field in two‐dimensional electron phase‐holes: A parameter study

Previous multi‐dimensional particle simulations have shown that electron phase‐holes can be formed during the nonlinear evolution of bi‐stream instability. In these holes, the parallel cut of the parallel electric field (E∥) has bipolar structures while the parallel cut of the perpendicular electric field (E⊥) has unipolar structures. In this paper, two‐dimensional (2D) electrostatic particle‐in‐cell simulations are performed to investigate the evolution of E⊥ in such electron holes for different plasma conditions, and the generation mechanism of the unipolar structures of E⊥ is also discussed. The electrons trapped in electron holes bounce in the parallel direction, which leads to transverse instability (Muschietti et al., 2000). At the same time, they gyrate in the background magnetic field, which tends to stabilize electron holes. In this way, the trapped electrons are forced to accumulate locally, and the charge density has variations along the perpendicular direction inside the electron holes. The balance between these two effects leads to the following results: in weakly magnetized plasma (Ωe < ωpe, but Ωe is comparable to ωpe. Where Ωe and ωpe are the electron cyclotron frequency and electron plasma frequency, respectively), electron holes have two‐dimensional structures (isolated along both the parallel and perpendicular directions). Within such holes the parallel cut of E⊥ has unipolar structures. In strongly magnetized plasma (Ωe > ωpe), electron holes have one‐dimensional structures along the direction perpendicular to the background magnetic field within which a series of islands (with alternate positive and negative E⊥) develop because of the variations of the charge density along the perpendicular direction. Therefore one recovers that a parallel cut of E⊥ has unipolar structures at the location of the holes. Present results show that the unipolar structure of E⊥ in electron holes is attributed to the balance between the electron transverse instability and the stabilization of the background magnetic field. The unipolar structures of E⊥ in electron holes last for hundreds to thousands of electron plasma periods. They are destroyed and the streaked structures are formed in the whole simulation domain after the electrostatic whistler waves are excited and have sufficiently large amplitude. The influences of the initial perpendicular thermal velocity of electrons (via temperature anisotropy) and the drift speed of electron beam on the structures of E⊥ are also analyzed in details. At last, the relevance between our simulation results and the unipolar structures of the parallel cut of E⊥ observed in the auroral region is discussed.

Analytical calculations of wake field generated by microwave pulses in a plasma filled waveguide for electron acceleration

Analytical expressions are obtained for the longitudinal field (wake field), density perturbation, and the potential behind microwave pulse propagating in a plasma filled rectangular waveguide with the pulse duration half of the electron plasma period. A feasibility study on wake field is carried out with rectangular pulse and its combination with Gaussian and triangular pulses under the effects of microwave pulse parameters and waveguide dimensions. It is inferred that the wake field in the waveguide cannot be attained when the length of rectangular microwave pulse is exactly equal to the plasma wavelength. A 1 ns short rectangular pulse with intensity of 250 kW/cm2 at the frequency of 5.03 GHz can excite the wake field of 1.0 MV/m in a waveguide with width of 6 cm and height of 4 cm. However, enhanced field is obtained when rectangular-triangular pulse (combination of rectangular and triangular pulses) is used. The field of wake gets weakened at higher microwave frequency and larger dimensions of the waveguide for other fixed parameters. However, a larger field is achieved when the pulse length of the microwave pulses is made shorter and/or intensity of the pulses is increased. A comparative study of the pulses shows that better results can be obtained with rectangular pulse (rectangular-Gaussian pulse: combination of rectangular and Gaussian pulses) if the microwave of shorter pulse duration (higher intensity) is available.

Electron acceleration by laser produced wake field: Pulse shape effect

Analytical expressions are obtained for the longitudinal field (wake field: E x ), density perturbations ( n e ′ ) and the potential ( ϕ) behind a laser pulse propagating in a plasma with the pulse duration of the electron plasma period. A feasibility study on the wake field is carried out with Gaussian-like (GL) pulse, rectangular–triangular (RT) pulse and rectangular–Gaussian (RG) pulse considering one-dimensional weakly nonlinear theory ( n e ′ / n 0 ≪ 1 ), and the maximum energy gain acquired by an electron is calculated for all these three types of the laser pulse shapes. A comparative study infers that the RT pulse yields the best results: In its case maximum electron energy gain is 33.5 MeV for a 30 fs pulse duration whereas in case of GL (RG) pulse of the same duration the gain is 28.6 (28.8)MeV at the laser frequency of 1.6 PHz and the intensity of 3.0 × 10 18 W/m 2. The field of the wake and hence the energy gain get enhanced for the higher laser frequency, larger pulse duration and higher laser intensity for all types of the pulses.

Effects of finite pulse length, magnetic field, and gas ionization on ion beam pulse neutralization by background plasma

This paper presents a survey of the present theoretical understanding of plasma neutralization of intense heavy ion beams. Particular emphasis is placed on determining the degree of charge and current neutralization. We previously developed a reduced analytical model of beam charge and current neutralization for an ion beam pulse propagating in a cold background plasma. The model made use of the conservation of generalized fluid vorticity. The predictions of the analytical model agree very well with numerical simulation results. The model predicts very good charge neutralization during quasi-steady-state propagation, provided the beam pulse duration is much longer than the electron plasma period. In the opposite limit, the beam pulse excites large-amplitude plasma waves. If the beam density is larger than the background plasma density, the plasma waves break, which leads to electron heating. The reduced-fluid description provides an important benchmark for numerical codes and yields useful scaling relations for different beam and plasma parameters. This model has been extended to include the additional effects of a solenoidal magnetic field, gas ionization and the transition regions during beam pulse entry and exit from the plasma. Analytical studies show that a sufficiently large solenoidal magnetic field can increase the degree of current neutralization of the ion beam pulse. However, simulations also show that the self-magnetic field structure of the ion beam pulse propagating through background plasma can be complex and non-stationary. Plasma waves generated by the beam head are greatly modified, and whistler waves propagating ahead of the beam pulse are excited during beam entry into the plasma. Accounting for plasma production by gas ionization yields a larger self-magnetic field of the ion beam compared to the case without ionization, and a wake of the current density and self-magnetic field are generated behind the beam pulse. Beam propagation in a dipole magnetic field configuration and background plasma has also been studied.

Wake field excitation and electron acceleration by triangular and sawtooth laser pulses in a plasma: an analytical approach

The amplitude of wake fields generated by the finite-width laser pulse propagating in an underdense plasma are determined using weakly nonlinear equations (n′e≪n0) for triangular (TR) and two types of sawtooth (SD and SI, defined in section 2) pulses by making the pulse duration the same as the electron plasma period. In addition to density perturbations ne′, the potential ϕ and the wake field Ez behind the laser pulses, the expression for the change in relativistic factor γ or energy gain for an electron is obtained and discussed for all these types of laser pulses under the effect of laser frequency f, pulse duration τ (corresponding to plasma density n0) and laser intensity I0. The TR pulse is found to be most significant as in this case the electrons can be accelerated up to hundreds of MeV (471.6 MeV) when τ = 6 ps, I0 = 1.0×1012 W m−2 and f = 282 THz (corresponding to Nd:YAG laser of wavelength 1.064 μm); even more acceleration is possible with prudently selected parameters.

Relation between magnetic fields and electric currents in plasmas

Abstract. Maxwell's equations allow the magnetic field B to be calculated if the electric current density J is assumed to be completely known as a function of space and time. The charged particles that constitute the current, however, are subject to Newton's laws as well, and J can be changed by forces acting on charged particles. Particularly in plasmas, where the concentration of charged particles is high, the effect of the electromagnetic field calculated from a given J on J itself cannot be ignored. Whereas in ordinary laboratory physics one is accustomed to take J as primary and B as derived from J, it is often asserted that in plasmas B should be viewed as primary and J as derived from B simply as (c/4π)∇×B. Here I investigate the relation between ∇×B and J in the same terms and by the same method as previously applied to the MHD relation between the electric field and the plasma bulk flow vmv2001: assume that one but not the other is present initially, and calculate what happens. The result is that, for configurations with spatial scales much larger than the electron inertial length λe, a given ∇×B produces the corresponding J, while a given J does not produce any ∇×B but disappears instead. The reason for this can be understood by noting that ∇×B≠4π/c)J implies a time-varying electric field (displacement current) which acts to change both terms (in order to bring them toward equality); the changes in the two terms, however, proceed on different time scales, light travel time for B and electron plasma period for J, and clearly the term changing much more slowly is the one that survives. (By definition, the two time scales are equal at λe.) On larger scales, the evolution of B (and hence also of ∇×B) is governed by ∇×E, with E determined by plasma dynamics via the generalized Ohm's law; as illustrative simple examples, I discuss the formation of magnetic drift currents in the magnetosphere and of Pedersen and Hall currents in the ionosphere. Keywords. Ionosphere (Electric fields and currents) – Magnetospheric physics (Magnetosphere-ionosphere interactions) – Space plasma physics (Kinetic and MHD theory)

Particle-in-cell simulation of an electron shock wave in a rapid rise time plasma immersion ion implantation process

A one-dimensional Monte Carlo collision‐particle-in-cell plasma computer code was used to simulate plasma immersion ion implantation by applying a negative voltage pulse to the substrate while the reactor wall is grounded. The results presented here show the effect of short rise time pulses: for rise times shorter than the electron plasma period stypically 5 ns/ kVd, an electron shock wave is observed where a rapidly expanding sheath heats the electrons up to high energies. Many of these fast electrons are expelled from the plasma leading to a high plasma potential and thus to a high surface electric field on the earthed electrode which could give rise to non-negligible electron field emission. © 2005 American Institute of Physics . fDOI: 10.1063/1.1872894g

Ion beam pulse neutralization by a background plasma in a solenoidal magnetic field

Ion beam pulse propagation through a background plasma in a solenoidal magnetic field has been studied analytically. The neutralization of the ion beam current by the plasma has been calculated using a fluid description for the electrons. This study is an extension of our previous studies of beam neutralization without an applied magnetic field. The high solenoidal magnetic field inhibits radial electron transport, and the electrons move primarily along the magnetic field lines. For high-intensity ion beam pulses propagating through a background plasma with pulse duration much longer than the electron plasma period, the quasi-neutrality condition holds, n e = n b + n p , where n e is the electron density, n b is the density of the ion beam pulse, and n p is the density of the background plasma ions (assumed unperturbed by the beam). For one-dimensional electron motion, the charge density continuity equation ∂ ρ / ∂ t + ∇ · j = 0 combined with the quasi-neutrality condition [ ρ = e ( n b + n p - n e ) = 0 ] yields j = 0 . Therefore, in the limit of a strong solenoidal magnetic field, the beam current is completely neutralized. Analytical studies show that the solenoidal magnetic field starts to influence the radial electron motion if ω ce ⩾ ω pe β (where ω ce = eB / mc is the electron gyrofrequency, ω pe is the electron plasma frequency, and β = V b / c is the ion beam velocity relative to the speed of light). This condition holds for relatively small magnetic fields. For example, for a 100 MeV, 1 kA Ne + ion beam ( β = 0.1 ) and a plasma density of 10 11 cm - 3 , B corresponds to a magnetic field of 100 G.