Particle-in-cell Research Articles

Self Consistent Modeling of Relativistic Runaway Electron Beams Giving Rise to Terrestrial Gamma‐Rays Flashes

AbstractTerrestrial Gamma Ray Flashes (TGFs) are short bursts of gamma rays occurring during thunderstorms. They are believed to be produced by Relativistic Runaway Electron Avalanches (RREAs). In this paper, we present a new numerical model based on the Particle‐In‐Cell (PIC) method to simulate the interactions between the electromagnetic fields and the electron avalanche self‐consistently. The code uses a cylindrical Yee lattice to numerically solve the electromagnetic fields, a Monte Carlo approach to simulate collisions with air molecules, and a plasma fluid model to calculate the effects of low‐energy electrons and ions. The model is first tested through the reproduction of dispersion relations in a hot plasma. RREAs propagating under a homogeneous background electric field are then simulated. Owing to the self‐consistent nature of this description, we report here new physical properties such as saturation processes in the electron density and in the number of high‐energy electrons, detailed dynamical screening of the electric field in the ion trail of the avalanche, and the associated electric currents. We find that the saturation of RREAs is obtained when the numbers of high‐energy electrons and photons is consistent with those believed to be representative of TGF sources.

A numerical approach for nonlinear transmission line analysis with bidirectional coupling to lumped-element and particle-in-cell models

In this article, an iterative method for nonlinear transmission lines (TLs) based on the Lax-Wendroff method has been established to describe a bidirectional coupling model of nonlinear lumped-element models, nonlinear transmission line models, and particle-in-cell (PIC) models. The proposed self-consistent global circuit model is achieved through the introduction of the charge conservation equation at the upper plate as the Robin boundary condition and its coupling with the PIC method used for plasma simulation. The effectiveness of the proposed model is validated through five numerical experiments, including nonlinear transmission line coupling to lumped circuits, and the most general case of plasma-TLs-lumped elements model (LEM) coupling systems. This non-linear global circuit coupling model provides a powerful tool for simulating the complex behavior of various physical systems, including but not limited to capacitively coupled plasmas, vacuum electronic devices, and Z-pinches.

Proton energy enhancement by optimizing a laser pulse profile

Based on current laboratory laser parameters and the low density target that is induced by the inevitable prepulse, we propose what we believe to be a new scheme to enhance the proton energy by employing a laser pulse with two different peak intensities. Initially, the lower-intensity peak of the laser pulse P1, irradiates the low-density plasma target induced by the prepulse to form a significantly denser plasma target. Such a compressed high-density target is critical for supporting the subsequent main pulse P2 with higher peak intensity to drive proton acceleration. As an example, particle-in-cell (PIC) simulations reveal that when using a circularly polarized (CP) flat-top P1 with a peak intensity of approximately 1.71 × 10 19 W/cm2, full-width at half-maximum(FWHM) duration of 325 fs and a CP P2 with a peak intensity of 1.54 × 10 22 W/cm2, FWHM duration of 26.5 fs, and focal spot radius of 4 µm successively acting on a target with an initial density of 8nc, protons with cut-off energy of 940 MeV can be obtained from the cascaded acceleration scheme. Compared with the case without P1, the cutoff energy increased by 340 MeV. Owing to the intervention of P1, this scheme overcomes the limitation of laser contrast and is more feasible to be implemented experimentally.

Polarized QED cascades over pulsar polar caps

ABSTRACT The formation of e± plasmas within pulsar magnetospheres through quantum electrodynamics (QED) cascades in vacuum gaps is widely acknowledged. This paper aims to investigate the effect of photon polarization during the QED cascade occurring over the polar cap of a pulsar. We employ a Monte Carlo-based QED algorithm that accurately accounts for both spin and polarization effects during photon emission and pair production in both single-particle and particle-in-cell (PIC) simulations. Our findings reveal distinctive properties in the photon polarization of curvature radiation (CR) and synchrotron radiation (SR). CR photons exhibit high linear polarization parallel to the plane of the curved magnetic field lines, whereas SR photons, on average, demonstrate weak polarization. As the QED cascade progresses, SR photons gradually dominate over CR photons, thus reducing the average degree of photon polarization. Additionally, our study highlights an intriguing observation: the polarization of CR photons enhances e± pair production by approximately 5 per cent, in contrast to the inhibition observed in laser–plasma interactions. Our self-consistent QED-PIC simulations in the corotating frame reproduce the essential results obtained from single-particle simulations.

Particle-in-cell Simulations of the Magnetorotational Instability in Stratified Shearing Boxes

Abstract The magnetorotational instability (MRI) plays a crucial role in regulating the accretion efficiency in astrophysical accretion disks. In low-luminosity disks around black holes, such as Sgr A* and M87, Coulomb collisions are infrequent, making the MRI physics effectively collisionless. The collisionless MRI gives rise to kinetic plasma effects that can potentially affect its dynamic and thermodynamic properties. We present 2D and 3D particle-in-cell (PIC) plasma simulations of the collisionless MRI in stratified disks using shearing boxes with net vertical field. We use pair plasmas, with initial β = 100 and concentrate on sub-relativistic plasma temperatures (kBT ≲ mc2). Our 2D and 3D runs show disk expansion, particle and magnetic field outflows, and a dynamo-like process. They also produce magnetic pressure dominated disks with (Maxwell stress dominated) viscosity parameter α ∼ 0.5 − 1. By the end of the simulations, the dynamo-like magnetic field tends to dominate the magnetic energy and the viscosity in the disks. Our 2D and 3D runs produce fairly similar results, and are also consistent with previous 3D MHD simulations. Our simulations also show nonthermal particle acceleration, approximately characterized by power-law tails with temperature dependent spectral indices −p. For temperatures kBT ∼ 0.05 − 0.3 mc2, we find p ≈ 2.2 − 1.9. The maximum accelerated particle energy depends on the scale separation between MHD and Larmor-scale plasma phenomena in a way consistent with previous PIC results of magnetic reconnection-driven acceleration. Our study constitutes a first step towards modeling from first principles potentially observable stratified MRI effects in low-luminosity accretion disks around black holes.

Diffusive shock acceleration in relativistic, oblique shocks

Cosmic rays are charged particles that are accelerated to relativistic speeds by astrophysical shocks. Numerical models have been successful in confirming the acceleration process for (quasi-)parallel shocks, which have the magnetic field aligned with the direction of the shock motion. However, the process is less clear when it comes to (quasi-)perpendicular shocks, where the field makes a large angle with the shock-normal. For such shocks, the angle between the magnetic field and flow ensures that only highly energetic particles can travel upstream at all, reducing the upstream current. This process is further inhibited for relativistic shocks, since the shock can become superluminal when the required particle velocity exceeds the speed of light, effectively inhibiting any upstream particle flow. In order to determine whether such shocks can accelerate particles, we use the particle-in-cell (PIC) method to determine what fraction of particles gets reflected initially at the shock. We then use this as input for a new simulation that combines the PIC method with grid-based magnetohydrodynamics to follow the acceleration (if any) of the particles over a larger time-period in a two-dimensional grid. We find that quasi-perpendicular, relativistic shocks are capable of accelerating particles through the DSA process, provided that the shock has a sufficiently high Alfvénic Mach number.

High-resolution particle-in-cell simulations of two-dimensional Bernstein–Greene–Kruskal modes

We present two-dimensional (2D) particle-in-cell (PIC) simulations of 2D Bernstein–Greene–Kruskal modes, which are exact nonlinear steady-state solutions of the Vlasov–Poisson equations, on a 2D plane perpendicular to a background magnetic field, with a cylindrically symmetric electric potential localized on the plane. PIC simulations are initialized using analytic electron distributions and electric potentials from the theory. We confirm the validity of such solutions using high-resolutions up to a 20482 grid. We show that the solutions are dynamically stable for a stronger background magnetic field, while keeping other parameters of the model fixed, but become unstable when the field strength is weaker than a certain value. When a mode becomes unstable, we observe that the instability begins with the excitation of azimuthal electrostatic waves that ends with a spiral pattern.

Quasi-perpendicular shocks of galaxy clusters in hybrid kinetic simulations

Context. Cosmic ray acceleration in galaxy clusters is still an ongoing puzzle, with relativistic electrons forming radio relics at merger shocks and emitting synchrotron radiation. These shocks are also potential sources of ultra-high-energy cosmic rays, gamma rays, and neutrinos. Our recent work focuses on electron acceleration at low Mach number merger shocks in the hot intracluster medium which is characterized by high plasma beta. Using particle-in-cell (PIC) simulations, we previously showed that electrons are energized through the stochastic shock-drift acceleration process, which is facilitated by multi-scale turbulence, including ion-scale shock surface rippling. For the present work, we performed hybrid-kinetic simulations in a range of various quasi-perpendicular foreshock conditions, including plasma beta, magnetic obliquity, and the shock Mach number. Aims. We study the ion kinetic physics, which is responsible for the shock structure and wave turbulence, that in turn affects the particle acceleration processes. We cover the spatial and temporal scales, which allow the development of large-scale ion turbulence modes in the system. Methods. We applied a recently developed generalized fluid-particle hybrid numerical code that can combine fluid modeling for both electrons and ions with an arbitrary number of kinetic species. We limited this model to a standard hybrid simulation configuration with kinetic ions and fluid electrons. The model utilizes the exact form of the generalized Ohm’s law, allowing for an arbitrary choice of mass and energy densities, as well as the charge-to-mass ratio of the kinetic species. Results. We show that the properties of ion-driven multi-scale magnetic turbulence in merger shocks are in agreement with the ion structures observed in PIC simulations. In typical shocks with the sonic Mach number Ms = 3, the magnetic structures and shock front density ripples grow and saturate at wavelengths reaching approximately four ion Larmor radii. Only shocks with Ms ≳ 2.3 develop ripples. At very weak shocks with Ms ≲ 2.3, weak turbulence is formed downstream of the shock. We observed a moderate dependence of the strength of magnetic field fluctuations on the quasi-perpendicular magnetic field obliquity. However, as the field obliquity decreases, the shock front ripples exhibit longer wavelengths. Finally, we note that the steady-state structure of Ms = 3 shocks in high-beta plasmas shows evidence that there is little difference between 2D and 3D simulations. The turbulence near the shock front seems to be a 2D-like structure in 3D simulations.

Analysis of single particle and collective beam effects in high intensity beams in a periodic quadrupole channel

In high intensity proton accelerators, there are two main mechanisms that can cause beam degradation: incoherent and coherent effects due to nonlinear space charge forces. The incoherent effects represent the single particle dynamics while the coherent effects represent the collective response of the beam. Of particular interest is the region above a zero current phase advance (σ0) of 90° where the (coherent) second-order envelope instability and the (incoherent) fourth-order particle resonance are seen to lead to emittance growth. Large emittance growth is also seen below the envelope instability region as the full current beam phase advance (σ) decreases. In the present study, we have studied the nonlinear effects in a high intensity beam propagating through a focusing-defocusing (FD) quadrupole channel for σ0 greater than 90°, both analytically, by studying the solutions of Kapchinsky-Vladimirsky (KV) equation and the particle core model, and through detailed particle-in-cell (PIC) simulations using the code. The KV envelope equation gives the collective response of the beam while the particle core model gives the contribution of the single particle effects. With pic simulations, which resemble the behavior of the real beams more closely as compared to envelope calculations or the particle core model, it is possible to study the evolution of the beam in a self-consistent manner. Our studies show, that it is possible to identify the specific process responsible for beam degradation in high intensity beams. It is also possible to identify which process dominates under different conditions. We further show that the width of the emittance increase stop band calculated from PIC simulations is wider than that calculated by the envelope equations and that the width depends on the length of the channel being studied. Published by the American Physical Society 2024

3-D numerical simulation of the heat transfer of a fluidized bed with a horizontal tube bundle and Geldart D particles

In the present study, a numerical simulation is performed to analyse hydrodynamic and heat transfer of a fluidized bed in a vertical channel. As a novelty, MP−PIC method is applied to 3-D tube bundles immersed in a fluidized bed with Geldart D particles. The main results are compared and validated using the experimental results published in the literature.In both tube bundle distributions, the lowest heat transfer coefficients were obtained in the top region of the tube, which represents approximately 58% of the time average heat transfer coefficient of the surface tube. This behaviour differs at the bottom and sides, where particles are continuously being replaced by other particles, resulting in low particle volume fractions and higher heat transfer coefficients.Analysing each tube arrangement for the cases with the same gas velocity and particle diameter, the results numerically obtained were consistent with the experimental findings. The present study found that the time average heat transfer coefficient for the staggered arrangement was approximately 6%, 5% and 1.6% higher than that in-line, for the particle diameter of 1400, 1600 and 1850 μm, respectively. Some discrepancies can be highlighted when comparing the experimentally and numerically obtained time average heat transfer coefficients, with maximum differences close to 18%, which correspond to the cases with the highest gas velocity and particle diameter (1.71 m/s, 1850 μm).

From linear to nonlinear Breit-Wheeler pair production in laser-solid interactions.

During the ultraintense laser interaction with solids (overdense plasmas), the competition between two possible quantum electrodynamics (QED) mechanisms responsible for e^{±} pair production, i.e., linear and nonlinear Breit-Wheeler (BW) processes, remains to be studied. Here, we have implemented the linear BW process via a Monte Carlo algorithm into the QED particle-in-cell (PIC) code yunic, enabling us to self-consistently investigate both pair production mechanisms in the plasma environment. By a series of two-dimensional QED-PIC simulations, the transition from the linear to the nonlinear BW process is observed with the increase of laser intensities in the typical configuration of a linearly polarized laser interaction with solid targets. A critical normalized laser amplitude about a_{0}∼400-500 is found under a large range of preplasma scale lengths, below which the linear BW process dominates over the nonlinear BW process. This work provides a practicable technique to model linear QED processes via integrated QED-PIC simulations. Moreover, it calls for more attention to be paid to linear BW pair production in near future 10-PW-class laser-solid interactions.

When should PIC simulations be applied to atmospheric pressure plasmas? Impact of correlation heating

Molecular dynamics simulations are used to test when the particle-in-cell (PIC) method applies to atmospheric pressure plasmas. It is found that PIC applies only when the plasma density and macroparticle weight are sufficiently small because of two effects associated with correlation heating. The first is the physical effect of disorder-induced heating (DIH). This occurs if the plasma density is large enough that a species (typically ions) is strongly correlated in the sense that the Coulomb coupling parameter exceeds one. In this situation, DIH causes ions to rapidly heat following ionization. PIC is not well suited to capture DIH because doing so requires using a macroparticle weight of one and a grid that well resolves the physical interparticle spacing. These criteria render PIC intractable for macroscale domains. The second effect is a numerical error due to artificial correlation heating (ACH). ACH is like DIH in that it is caused by the Coulomb repulsion between particles, but differs in that it is a numerical effect caused by a macroparticle weight larger than one. Like DIH, it is associated with strong correlations. However, here the macroparticle coupling strength is found to scale as Γw2/3 , where Γ is the physical coupling strength and w is the macroparticle weight. So even if the physical coupling strength of a species is small, as is expected for electrons in atmospheric pressure plasmas, a sufficiently large macroparticle weight can cause the macroparticles to be strongly coupled and therefore heat due to ACH. Furthermore, it is shown that simulations in reduced dimensions exacerbate these issues.

Suppression of multipacting growth in a superconducting spoke resonator by the use of higher order modes

Abstract Multipacting is a vacuum discharge phenomenon that constrains the performance of accelerator structures and various microwave devices significantly. The different ways to mitigate multipacting discharge have been a central research goal in Accelerator Physics & Technologies for a prolonged period. In this work, both Monte-Carlo and particle-in-cell (PIC) simulations show that the multipactor avalanche can be mitigated, by the use of Higher Order Modes (HOMs). The intuition behind the approach of using HOMs to suppress multipacting has been explained first. Then, the multipacting suppression has been demonstrated in a simple 1-D metallic gap, when it is exposed to a second carrier frequency in addition to fundamental rf mode, with the help of a Monte-Carlo simulation code. Finally, a detailed PIC simulation has been carried out using CST Microwave studio, to investigate the effect of various HOMs on the multipactor growth in a superconducting spoke cavity. It is observed that a slight excitation of some particular modes can drastically diminish the multipactor growth. Simulation has been done first without any HOMs, to calculate the growth rate and also to spot the discharge location. Next, among various HOMs (more specifically, quadrupole modes), only those have been selected, whose magnetic field locations coincide with the multipacting prone zone of the cavity. In a simulation study, the presence of these preselected quadrupole modes has been found to exhibit a substantial suppression effect on the multipacting growth of the cavity.

One-dimensional General Relativistic Particle-in-cell Simulations of Stellar-mass Black Hole Magnetospheres: A Semianalytic Model of Gamma-Rays from Gaps

In the absence of a sufficient amount of plasma injection into the black hole (BH) magnetosphere, the force-free state of the magnetosphere cannot be maintained, leading to the emergence of strong, time-dependent, longitudinal electric fields (i.e., spark gaps). Recent studies of supermassive BH magnetospheres using analytical methods and particle-in-cell (PIC) simulations propose the possibility of efficient particle acceleration and consequent gamma-ray emission in the spark gap. In this work, we perform 1D general relativistic PIC simulations to examine the gamma-ray emission from stellar-mass BH magnetospheres. We find that intermittent spark gaps emerge and particles are efficiently accelerated in a similar manner to the supermassive BH case. We build a semianalytic model of the plasma dynamics and radiative processes, which reproduces the maximum electron energies and peak gamma-ray luminosities of the simulation results. Based on this model, we show that the gamma-ray signals from stellar-mass BHs wandering through the interstellar medium could be detected by gamma-ray telescopes such as the Fermi Large Area Telescope or the Cherenkov Telescope Array.

Impedance of electric field sensors in magnetized plasmas: particle-in-cell simulations

To obtain both the intensities and phases of electric fields, it is necessary to calibrate the data observed by the sensor by consulting the sensor characteristics. The frequency profile of the sensor impedance obtained from the simulation is roughly consistent with that obtained from the theory. However, some differences are identified in the frequency profile of the sensor impedance obtained from the simulation compared to the theory. We conducted simulations using the particle-in-cell (PIC) simulation tool to analyze the impedance of electric field sensors immersed in magnetized space plasmas. The simulation model comprised a dipole sensor placed in a three-dimensional simulation box filled with electrons and ions. The dipole sensor was represented by perfect conductor rods. We carefully selected simulation parameters and employed an appropriate feeding technique to accurately evaluate the sensor impedance. We observed significant changes in the frequency dependence of the sensor impedance as cyclotron frequency varied. To understand the simulation results, we introduced the linear dispersion relation. By comparing the real part of the sensor impedance with the ω-k\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\omega-k$$\\end{document} diagram, we found that the impedance peaks corresponded to frequencies at which branches of plasma waves were present, with k=khalf\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k=k_{\ ext{half}}$$\\end{document}, representing half the wavelength equal to the tip-to-tip sensor length, i.e., the sensor can be assumed as the half-wavelength dipole antenna.Graphical

A binary filter inspired from the PIC sparse grid technique – Illustration on the XTOR-K code

It is known [1,2] that the sparse grid method for Particle-In-Cell (PIC) solvers acts as a filter to reduce the PIC noise. In this paper, a simple rule to discard or keep modes in Fourier space (a binary filter with values either 0 or 1) is derived using the sparse grid combination formula. Its relation to the standard sparse grid filter, which is characterized quantitatively, is explained. The relations between the sparse grid filters on grids of arbitrary levels are also outlined. Namely, in two (resp. three) dimensions and for bi-linear (resp. tri-linear) moment deposition, it is proven rigorously that the sparse grid filter, for a grid of size equal to an arbitrary power of two, can be expressed in terms of two (resp. three) unique real valued functions. The advantage of the binary filter over the standard sparse grid filter is the reduction of signal deformation introduced by the latter, for the same noise reduction capability. By applying the filter to moments of a marker distribution coming from the XTOR-K code, it appears the noise could be significantly reduced, with a moderate overhead in the moment deposition part of the algorithm.

Study of the breathing mode development in Hall thrusters using hybrid simulations

We use a 2.5D hybrid simulation to study the breathing mode (BM) dynamics in Hall thrusters (HTs). This involves a 1D Euler fluid simulation for neutral dynamics in the axial direction, coupled with a 2D axial–azimuthal Particle-in-Cell (PIC) simulation for charged species. The simulation also includes an out-of-plane virtual dimension for wall losses. This setup allows us to replicate the BM’s macroscopic features observed in experiments. A comprehensive analysis of plasma parameters in BM’s phases divides it into two growth and two decay sub-phases. Examining 1D axial profiles of electron temperature, gas and plasma densities, and particle creation rate shows that an increase in electron temperature alone cannot sustain ionization. Ionization seems to be influenced by the spatial correlation between electron and gas densities and the ionization rate coefficient. Investigating ion back-flow reveals its impact on modulating neutral flux entering the ionization region. The hybrid simulation’s outcomes let us assess the usual 0D predator–prey model’s validity and identify its limitations. The ionization and ion convection term approximations hold, but the gas convective term approximation does not. Introducing an alternative gas convective term approximation involving constant density ejection from the ionization region constructs an unstable BM model consistent with simulation results. In addition, this paper explores how varying the imposed voltage and mass flow rate impacts the BM. The BM frequency increases with imposed voltage, aligning with theoretical predictions. The mass flow rate variation has a limited effect on BM frequency, following the theoretical model’s trend.

PICturing coronary pathology postmortem: forensic cardiac imaging with Postmortem Infrared Coronary angiography of human heart ex situ

The diagnostic assessment of sudden and unexpected cardiovascular deaths remains intricate in forensic medicine. Building upon the foundational technique introduced by P. Fais et al.(2018), we present modifications to the post-mortem infrared coronary angiography (PIC) tailored specifically for the assessment of human hearts. Refinements to PIC encompass the integration of a 3D-printed clamp for catheter stabilization and the procedural alteration of utilizing warm water injections, negating the need for additional cardiac cooling. An enhanced imaging modality is achieved using the FLIR Thermal Lepton 3.5 camera, embedded within the robust Cat S62 Pro mobile device, ensuring optimal resolution and suitability for an autopsy environment. The advanced PIC technique provides superior visualization of the coronary arteries, frequently correlating with subsequent autopsy and histological assessments. Notably, the method allows immediate continuation to the autopsy without compromising the cardiac structure. Nevertheless, certain anatomical variances, such as muscular bridging or pronounced pericardial fat, might reduce locally what appears to be an otherwise excellent specificity. The refined PIC method emerges as a pivotal diagnostic adjunct in forensic evaluations of sudden cardiovascular fatalities. Its ability to preserve cardiac integrity and facilitate uninterrupted autopsy progression underscores its potential utility. However, rigorous validation is imperative to ascertain its comprehensive applicability and inherent limitations.

Acceleration characteristics of hot electrons resulted from magnetic reconnection process driven by double-beam intense laser pulses in near critical density plasmas

In this work, we investigated the magnetic annihilation and reconnection and the resulted hot electron acceleration driven by double-beam intense laser pulses in two-layer near critical density (NCD) plasma target. The results are obtained by performing two-dimensional (2D) particle-in-cell (PIC) simulations. It is found that a quasi-mono-energetic peak can be formed in the energy spectrum of electrons accelerated by the process of magnetic field annihilation (MA) at cutoff energy. Electron spectra feature depends on the length of the second low-density layer. This suggests that the process of relativistic magnetic annihilation may be controlled in experiments by target design.

Anomalous hot electron generation via stimulated Raman scattering in plasma with up-ramp density profiles

We investigate the evolution and propagation of the electron plasma waves (EPWs) excited by stimulated Raman scattering (SRS) in the inhomogeneous plasma theoretically and numerically with particle-in-cell (PIC) simulations. A theoretical model of EPWs in inhomogeneous plasmas is presented, which shows that the evolution of the EPW wavenumber is mainly related to the plasma density profile rather than the plasma electron temperature, in agreement with PIC simulations. When the density gradient is positive along the propagation direction of an EPW, its wavenumber decreases with time and consequently its phase velocity increases continuously, causing the trapped electrons to be accelerated to anomalous high energy. Furthermore, it is found that the Langmuir decay instability tends to reduce the levels of SRS saturation and electron acceleration and produce hot electrons in the opposite direction. This work provides a new understanding of electron heating due to SRS excitation.