Electron Boltzmann Equation Research Articles

Unraveling NO Production in N2-O2 Plasmas with 0D Kinetic Modeling and Experimental Validation.

This work presents a detailed investigation aimed at understanding the key mechanisms governing nitric oxide (NO) production in N2-O2 discharges by systematically comparing experimental results to modeling data. The experimental phase capitalizes on radiofrequency (13.56 MHz) discharges, sustained at 5 mbar pressure conditions, featuring varying concentrations of oxygen, ranging from pure N2 plasma to air-like mixtures. On the modeling front, we adopt an integrated approach that combines the solution of the Boltzmann equation for electrons with a system of rate balance equations for heavy species. To account for ground state NO(X) generation at the reactor wall, we combine the volume chemistry with a mesoscopic description of the surface, taking into account adsorption sites and various elementary surface phenomena. Comparisons between experiments and modeling demonstrate very good agreement, extending beyond NO(X) formation to encompass other species in the plasma such as N2O(X) and atomic nitrogen N(4S). Noteworthy findings include (i) the pivotal role of surface mechanisms in NO(X) production, particularly at low oxygen content values; (ii) the significance of accurately describing the postdischarge phase, where depletion of plasma species occurs at different time scales (millisecond range); and (iii) the importance of vibrationally and electronically excited states (e.g., O2(b)) in elucidating NO(X) formation dynamics, crucial for unraveling reaction pathways and energy transfer processes. This work makes an important step toward formulating a comprehensive reaction mechanism for N2 and N2-O2 plasmas applied to nitrogen fixation, covering both volume and surface mechanisms, and lays a robust foundation for future research on plasma-based NO(X) production, particularly in the presence of catalysts.

ChemPlasKin: A general-purpose program for unified gas and plasma kinetics simulations

This work introduces ChemPlasKin, a freely accessible solver optimized for zero-dimensional (0D) simulations of chemical kinetics of neutral gas in non-equilibrium plasma environments. By integrating the electron Boltzmann equation solver, CppBOLOS, with the open-source combustion library, Cantera, at the source code level, ChemPlasKin computes time-resolved evolution of species concentration and gas temperature in a unified gas–plasma kinetics framework. The model allows high fidelity predictions of both chemical thermal effects and plasma-induced heating, including fast gas heating and slower vibrational–translational relaxation processes. Additionally, a new heat loss model is developed for nanosecond pulsed discharges, specifically within pin–pin electrode configurations. With its versatility, ChemPlasKin is well-suited for a wide range of applications, from plasma-assisted combustion (PAC) to fuel reforming. In this paper, the reliability, accuracy and efficiency of ChemPlasKin are validated through a number of test problems, demonstrating its utility in advancing gas–plasma kinetic studies.

On fundamental inconsistencies in a commonly used modification of a fluid model for glow discharge

This work considers the fundamental contradictions in the concept of one of the most well-known and widely used modifications of the fluid model for simulation of a glow discharge (GD), the ‘local mean energy approximation’ (LMEA). In this model, it is proposed to determine the kinetic coefficients in the electron particle and energy balance equations as functions of the electron mean energy (temperature) rather than local electric field, using a one-to-one correspondence between these parameters through the electron Boltzmann equation. It is shown that the scope of applicability of this model, like any other modification of the fluid model, is limited by the local mode of formation of the electron energy distribution function (EEDF). Therefore, as demonstrated by the examples of typical 1D and 2D problems for a GD in argon, its extension to the region of nonlocal EEDF is in no way justified and leads not only to serious errors in the results, but also to a logically intractable situation in attempts to apply the main postulate of the LMEA model to the region of a weak (or even reverse) electric field in a negative glow plasma. At the same time, the apparent reliability of calculations within the framework of the LMEA model for a number of parameters, in our opinion, only slows down progress in modeling of gas discharge plasma.

Time-dependent electron Boltzmann equation for hypersonic plasmas

The electron kinetics in hypersonic plasmas is modeled by solving the time-dependent electron Boltzmann equation for the electron energy distribution function (EEDF). This plasma is created by strong shock compression of the gas in front of a vehicle moving with hypersonic speed. The main source of energy for the electrons is gas heating due to elastic collisions and second-kind collisions (de-excitation) from vibrationally excited states of N2. We established that the electron energy distribution function is most sensitive to vibrational level populations. At mid-altitudes (tens of kilometers), the electron temperature equilibrates with the vibrational temperature on a microsecond timescale. The electron distribution function reaches steady state on a comparable timescale. Numerical simulations of air plasma showed that the electron energy distribution function is far from Maxwellian and the collision rates differ by orders of magnitude from those computed with a Maxwellian distribution. The two most important parameters for the electron kinetics and the electron energy distribution function are the vibrational temperature and ionization degree.

An efficient solution of the multi-term multi-harmonic electron Boltzmann equation for use in global models

Solving the electron Boltzmann equation is an essential but costly step in simulating low-temperature plasma kinetics. This work addresses the problem by introducing a solution of the general multi-term multi-harmonic Boltzmann equation (MTMH-BE) optimized for electrons in time-dependent non-equilibrium gases and electric fields. This is accomplished by configuring the numerical Jacobian as the product of a time-independent sparse matrix and an array of time-dependent coefficients. As this approach requires a fixed energy grid, the MTMH-BE is discretized using a finite volume scheme with support for non-uniform cell size, enabling robust solutions across a wide range of reduced field strengths. The new MTMH-BE solver is tested against a range of benchmarks, demonstrating excellent agreement with existing methods. Performance with time-dependent electron energy is evaluated using LoKI-B's pulsed electric field model, with the new solver demonstrating improved wall time scaling and an improvement in grid convergence of over three orders of magnitude. Alternatively, performance with quasi-stationary electrons in an evolving non-equilibrium gas is tested using a nitrogen cross-section set with up to 58 vibrational states and 3,570 processes. Here, the MTMH-BE solver achieves a wall time reduction of up to 99.6% over MultiBolt for error-controlled simulations in the two-term limit, reducing the average solution time from 1-2 seconds to less than 10 ms. These results indicate that the new MTMH-BE solver can improve accuracy and performance in state-to-state global models without the use of grid refinement, cross-section pruning, and other time-saving procedures.

Self-Consistent State-to-State Kinetic Modeling of CO2 Cold Plasmas: Insights on the Role of Electronically Excited States

This study focuses on the role of CO2 and CO electronically excited states in the kinetics of CO2 cold non-equilibrium plasma discharges by means of a state-to-state OD kinetic model based on the simultaneous and self-consistent solution of the electron Boltzmann equation and the master equations describing the vibrationally and electronically excited state kinetics and the plasma composition. A new CO2 dissociation model based on the use of the Biagi electron impact excitation cross sections, considered as fully dissociative, of several CO2 electronic excited states, in the energy range from 6.5 to 25 eV, is tested and compared with the results obtained by using the Phelps database in typical glow and microwave discharge conditions. Moreover, a refinement of the kinetics of the COa3Π\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathrm{CO}\\left({\\mathrm{a}}^{3}\\Pi \\right)$$\\end{document} excited state is proposed by including new production and loss terms and the effect of the change of its time evolution density on the eedf, the electron temperature, the CO2 and CO vibrational distribution functions, electron impact and vibrational induced dissociation rates is investigated. Finally, the contribution of the COa3Π\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathrm{CO}\\left({\\mathrm{a}}^{3}\\Pi \\right)$$\\end{document} state to CO2 dissociation is examined in terms of production and recombination (or back-reaction) processes both in microwave and glow discharge conditions.

(Invited) Mechanism of H2 Synthesis Due to Plasma-Induced Water Splitting Via Free-Electron Impact

The electrified plasma-liquid interface (EPLI) has two major local reactors, the plasma phase and the liquid phase. Both local reactors enable water splitting, but their synthesis mechanism is different. While the mechanism of H2 synthesis in a liquid phase (plasma electrochemistry) was reasonably deciphered, the respective mechanism in the plasma phase does not (plasma chemistry). In this present contribution, the Chemical Reaction Network (CRN) analysis serves as a tool to decipher the H2 synthesis mechanism in the plasma reactor. For that, the datum from the laboratory of Sankaran serves as grounds for the CNR analysis. Sankaran team identified that the equilibrium vapor pressure of water introduces water vapor in the plasma phase; and that, fundamentally, there were eight elementary reactions involved in the plasma-chemistry mechanism of H2 synthesis, i.e., those reactions containing the largest constant rates (or electron rate coefficients). [1] The authors obtained the reaction rate constants via a numerical solution of the Boltzmann equation (BE) for electrons in weakly ionized gases considering the uniform electrical field. [2] Together, those eight elementary reactions form the following global reaction for the water splitting via electron-impact8 H2O → ●H+ ●OH + ●O + O– + O+ + OH– + OH+ + H– + H+ that is also rationalized in terms of global CRN, shown in figure 1. The water molecule splits into two major classes of fragments, the radical fragment set {●H, ●OH and ●O} and the ionic fragment set {O–, O+, OH–, OH+, H–, H+}. These two classes of fragments receive different treatments in this work because their kinetic relaxation is supposed to follow different time scales. The radical fragment is supposed to relax toward final products much faster than the ionic fragments. For this reason, only the chemical dynamics of the radical fragment are considered in the relaxation step toward the final products. The chemical dynamics of relaxation toward final products, thus, take into account only the set of radical fragments. This talk will present and discuss the chemical reaction network (CRN) for the relaxation of the radical fragment toward final products like hydrogen (H2), hydrogen peroxide (H2O2), water (H2O), and eventually oxygen (O2). The radical-relaxation CNR is not shown in this abstract but will be present/discussed in the talk.Roughly speaking, the radical-relaxation dynamics must deliver hydrogen as the major product and hydrogen peroxide as the major side-product, besides a stochiometric ratio of 4 to 1, respectively. Specifics on whether the radicals •OH relaxes toward H2O2 or H2O depend on the local dynamics, meaning that it mostly relies on the parameter set values that drive the CRN. Unfortunately, there is no experimental evidence already reporting the stoichiometry of products, which makes it unfeasible to evaluate the adherence of our analysis to the experimental pieces of evidence.

The Transport and Optical Characteristics of a Metal Exposed to High-Density Energy Fluxes in Compressed and Expanded States of Matter

This article presents a theoretical study of the optical and transport properties of metals. Iron, as an example, was used to discuss, through a theoretical description, the peculiarities of these properties in the compressed and expanded states under the influence of high-density energy fluxes. By solving the semi-classical Boltzmann equation for conduction electrons for a broad range of densities and temperatures, the expressions of electrical conductivity, electronic thermal conductivity, and thermoelectric coefficient calculations were derived. The real and imaginary parts of the iron permittivity and the energy absorption coefficient for the first and second harmonics of Nd:YAG laser radiation were obtained. The calculation peculiarities of the metal’s optical characteristics of matter in an expanded state in a broad range of densities and temperatures were considered. The analysis of the obtained results shows their agreement with the theoretical description for cases of ideal non-degenerate and dense degenerate electron plasmas. It is shown that the behavior of the electrical conductivity and optical characteristics in the critical and supercritical regions of density and temperature are in agreement with the known experimental results.

Scanning drift tube measurements and kinetic studies of electron transport in CO

We present scanning drift tube measurements of electron swarm transport coefficients in CO as a function of the reduced electric field at room temperature under time-of-flight conditions. The measurements are compared to modeling results and other available experimental data on swarm transport over the broad range of from 2 Td to 1603 Td. The modeling results are obtained in Monte Carlo simulations and by solving the electron Boltzmann equation using a multi-term approach and the density gradient expansion procedure. We find generally good agreement between the measured and calculated transport coefficients. We propose a strategy to improve the cross-section set used to explain certain discrepancies at lower values. Measurements and calculations of electron transport coefficients under hydrodynamic conditions are complemented by Monte Carlo simulations of electron transport in an idealized steady-state Townsend (SST) setup. The ionization coefficient is calculated as a function of from the spatial density profiles of the electrons and compared to the corresponding values evaluated from the knowledge of the effective ionization frequency, drift velocity and longitudinal diffusion coefficient. Contrary to the traditional views, according to which the spatial relaxation of the mean energy and other transport properties for electrons in molecular gases is most commonly monotonic or quasi-monotonic, we find a ‘window’ of where the SST transport properties of the electrons exhibit oscillatory behavior as they relax towards the equilibrium state far downstream from the electron emitting boundary.

Terahertz emission from a bounded plasma

The dynamics of electrons submitted to voltage pulses in a thin semiconductor layer is investigated using a kinetic approach based on the solution of the electron Boltzmann equation using particle-in-cell/Monte Carlo collision simulations. The results showed that due to the fairly high plasma density, oscillations emerge from a highly nonlinear interaction between the space-charge field and the electrons. The voltage pulse excites electron waves with dynamics and phase-space trajectories that depend on the doping level. High-amplitude oscillations take place during the relaxation phase and are subsequently damped over time-scales in the range 100–400 fs and decrease with the doping level. The power spectra of these oscillations show a high-energy band and a low-energy peak that were attributed to bounded plasma resonances and to a sheath effect. The high-energy THz domain reduces to sharp and well-defined peaks for the high doping case. The radiative power that would be emitted by the thin semiconductor layer strongly depends on the competition between damping and radiative decay in the electron dynamics. Simulations showed that higher doping level favor enhanced magnitude and much slower damping for the high-frequency current, which would strongly enhance the emitted level of THz radiation.

Non-equilibrium plasma kinetics of CO2 in glow discharges: a comparison with existing modeling and experimental results

We report results obtained by our 0D, time-dependent self-consistent model for the description of the CO2 plasma kinetics in glow discharge conditions, comparing our results with the simulation and experimental results reported by Grofulovic et al (2018 Plasma Sources Sci. Technol. 27 115009; 2019 PhD Thesis) and Klarenaar et al (2017 Plasma Sources Sci. Technol. 26 115008). Our model is based on the simultaneous solution of the kinetic equations describing the vibrational, the electronic excited states and the plasma chemistry and of the electron Boltzmann equation for the calculation of the electron energy distribution function (eedf). The results for the vibrational level densities of CO2 show a satisfactory agreement with the Grofulovic’s model results, despite the differences in the vibrational energy level scheme and in the kinetic processes included with the correspondent rate coefficients, with a good match also with the corresponding experimental results. Moreover, conditions characterized by higher power density (5–50 W cm−3) have been investigated to understand the behavior of the CO2 plasma discharge when a higher vibrational excitation is present. Large deviations of the vibrational distributions of CO2 and CO from equilibrium ones are predicted both in discharge and post discharge conditions. In particular, the CO2 vibrational distribution presents a behavior similar to a Treanor distribution for v < 15 while a deactivation of the plateau in the vibrational distribution function after v > 15 appears as a consequence of the dissociation induced by vibrational excitation mechanism, i.e. pure vibrational mechanism, becoming important at higher power densities. Finally, the results dependence on the selection of the CO2 electron molecule dissociation cross section, i.e. Phelps (1973 J. Appl. Phys. 44 4464 or Cosby (1993 Report No. AD-A266 464 WL-TR-93-2004 (Dayton, OH: Wright-Patterson Airforce Base)), has been investigated, showing that its more opportune choice is still a problem to be discussed for the description of conditions in which the electron impact dissociation dominates the kinetics, while once vibrational excitation is activated, CO2 dissociation is essentially driven by vibrational-induced dissociation, depending to a minor extent from that choice.

Analysis of coherent Thomson scattering from a low temperature plasma

The spectrum of coherent Thomson scattering (CTS) induced by a periodic ponderomotive perturbation in a low-density low temperature plasma is considered. The analysis is performed for the case when the period of the resulting optical lattice is less than the Debye screening length in the plasma by solving an electron Boltzmann equation, where the total force is the sum of the periodic force due to the optical lattice and the electrostatic force due to self-consistent electric field in the plasma. An analogy between the CTS spectra calculated here and coherent Rayleigh scattering spectra in a neutral gas is established. For relatively low intensity for the optical lattice, the calculated CTS spectra are nearly Gaussian with widths slightly wider than the incoherent Thomson widths. We demonstrate that at higher intensities the line shape narrows and saturates to a width approximately half of that found at low lattice intensities. The proportionality of the spectral width to the square root of the electron temperature allows one to extract the electron temperature from the saturated spectra. Possible application of CTS for remote measuring the electron temperature in plasma is discussed.

On the population density of the argon excited levels in a high power impulse magnetron sputtering discharge

Population densities of excited states of argon atoms in a high power impulse magnetron sputtering (HiPIMS) discharge are examined using a global discharge model and a collisional-radiative model. Here, the ionization region model (IRM) and the Orsay Boltzmann equation for electrons coupled with ionization and excited states kinetics (OBELIX) model are combined to obtain the population densities of the excited levels of the argon atom in a HiPIMS discharge. The IRM is a global plasma chemistry model based on particle and energy conservation of HiPIMS discharges. OBELIX is a collisional-radiative model where the electron energy distribution is calculated self-consistently from an isotropic Boltzmann equation. The collisional model constitutes 65 individual and effective excited levels of the argon atom. We demonstrate that the reduced population density of high-lying excited argon states scales with (p*)−6, where p* is the effective quantum number, indicating the presence of a multistep ladder-like excitation scheme, also called an excitation saturation. The reason for this is the dominance of electron impact processes in the population and de-population of high-lying argon states in combination with a negligible electron–ion recombination.

Formation of energy spectra of electrons in a dense weakly ionized plasma generated by fission fragments

AbstractFor a weakly ionized dense plasma irradiated by fission fragments, the coupled self‐consistent Boltzmann equations for fission fragments and electrons are defined. The evolution of the energy spectra of fast particles in a plasma is investigated on the basis of these equations. Studies of the energy distribution function of fission fragments and primary electrons for a helium‐3 plasma irradiated by neutron flux, as well as their non‐stationary and stationary analytical solutions, are found and analysed.

Effects of the N2 to O2 ratio in air on the removal rate and the degree of decomposition of 2-chloroethyl ethyl sulphide by atmospheric plasma

It is difficult to simultaneously satisfy all the essential requirements for the removal of chemical warfare agents from air through atmospheric air plasma chemistry. Such requirements include a high removal rate (RR) indicating the completeness of the transition into other substances and a high degree of decomposition (DD) indicating the toxicity of these substances. In this study, the N2 to O2 ratio (NOR) in air was adjusted to scrutinise the role of electrons and active oxidative particles in 2-chloroethyl ethyl sulphide removal. It was found that a high NOR led to a high RR, but a low DD. Spectral diagnosis was used to compare the electron temperature (Te) of the plasma at different NORs. Plasma kinetics based on the calculation of the electron Boltzmann equation and product analysis were conducted. The results showed that the Te increased with increasing NORs, while the production rate of active oxidative particles tended to decrease at higher NORs. Thus, this study suggests that high-energy electrons may dominate the RR, while active oxidative particles may play a dominant role in enhancing the DD. The results may lead to a novel methodology that modifies the NORs to achieve both optimum RR and DD.

Plasma characteristics in Ne/Xe/HCl gas mixtures: A parametric study

An important class of exciplex molecules consists of rare gas halides (RgH) such as the monochloride of xenon (XeCl). This excimer emits photons with energies of about eV at wavelength of 308 nm. The present work examines the effects of external circuit parameters and total gas pressure on the characteristic properties of excimer discharge plasma in Ne/Xe/HCl gas mixture. The numerical investigation is carried using a zero-dimensional model, in which the electric circuit, the kinetic processes in the active medium, and the electron Boltzmann equation are considered. We find that increasing values of capacitance, inductance, charging voltage, and gas pressure affects significantly the discharge plasma behavior.

Features of processes in a microwave discharge in water vapor

A zero-dimensional steady-state simulation of microwave discharge in water vapor at atmos-pheric and reduced pressures and a constant gas temperature has been carried out. A model of a continuous stirring reactor is used. A joint solution of the balance equations for neutral and charged plasma components, the Boltzmann equation for plasma electrons, and the equation for the stationary distribution of the microwave field in a volume filled with plasma is carried out. The dependences of various parameters of thedischarge (the magnitude of the microwave field, the concentrations of all components) on the input specific power WVare obtained. It is shown that at reduced pressure the magnitude of the microwave field in the plasma is signifi-cantly lower, and the electron concentration is higher than at atmospheric pressure at the same applied specific power. At atmospheric pressure the water plasma is electronegative, and quasi-neutrality is maintained by the negative OH-ion in the range of the considered WV values. Transition from electronegative to electropositive plasma occurs at pressure of 30 Torr and ap-plied specific power of 60–70 kW/cm3

Extracting electron scattering cross sections from swarm data using deep neural networks

Electron-neutral scattering cross sections are fundamental quantities in simulations of low temperature plasmas used for many technological applications today. From these microscopic cross sections, several macro-scale quantities (called ‘swarm’ parameters) can be calculated. However, measurements as well as theoretical calculations of cross sections are challenging. Since the 1960s, researchers have attempted to solve the inverse swarm problem of obtaining cross sections from swarm data; but the solutions are not necessarily unique. To address these issues, we examine the use of deep learning models which are trained using the previous determinations of elastic momentum transfer, ionization and excitation cross sections for different gases available on the LXCat website and their corresponding swarm parameters calculated using the BOLSIG+ solver for the numerical solution of the Boltzmann equation for electrons in weakly ionized gases. We implement artificial neural network (ANN), convolutional neural network (CNN) and densely connected convolutional network (DenseNet) for this investigation. To the best of our knowledge, there is no study exploring the use of CNN and DenseNet for the inverse swarm problem. We test the validity of predictions by all these trained networks for a broad range of gas species and we deduce that DenseNet effectively extracts both long and short term features from the swarm data and hence, it predicts cross sections with significantly higher accuracy compared to ANN. Further, we apply Monte Carlo dropout as Bayesian approximation to estimate the probability distribution of the cross sections to determine all plausible solutions of this inverse problem.

On the quasi-stationary approach to solve the electron Boltzmann equation in pulsed plasmas

This work analyzes the temporal evolution of the electron kinetics in dry-air plasmas (80% N2: 20% O2), excited by electric-field pulses with typical rise-times of 10−9 and 10−6 s, applied to a stationary neutral gaseous background at pressures of 105, 133 Pa and temperature of 300 K. The study is based on the solution of the electron Boltzmann equation (EBE), adopting either (i) a time-dependent formulation that considers an intrinsic time evolution for the electron energy distribution function (EEDF), assuming the classical two-term expansion and a space-independent exponential temporal growth of the electron density; or (ii) a quasi-stationary approach, where the time-independent form of the EBE is solved for different values of the reduced electric-field over the duration of the pulse. The EBE was solved using the LisbOn KInetics Boltzmann solver (LoKI-B), whose original capabilities were extended to accept time-dependent non-oscillatory electric fields as input data. The role of electron–electron collisions, under specific conditions, is also reported and discussed. The simulations show that the quasi-stationary approach gives solutions similar to the time-dependent formulation for rise-times longer than the characteristic evolution time of the EEDF, i.e. 20 ns at 105 Pa and 20 μs at 133 Pa, meaning that a quasi-stationary description is possible in a high-collisionality situation and long rise-times (e.g. microsecond pulses at atmospheric pressure), failing for faster rise-times (e.g. nanosecond pulses for both pressures considered here).

Modeling the thermalization of electrons in conditions relevant to atmospheric pressure He-O2 nanosecond pulsed discharges

The electron thermalization process is significant in nanosecond pulsed discharges due to the applied voltage pulse's short duration and rapid rise and fall times. In this contribution, a comparison was made between two approaches to modeling the electron kinetics of electron thermalization in atmospheric pressure helium plasma with an oxygen admixture. Modeling based on the direct solution of the local time-dependent electron Boltzmann equation was compared with modeling based on the commonly used but less general local mean energy approximation. For modeling based on the local time-dependent electron Boltzmann equation, a temporary faster decay in the population of electrons in the high energy tail and a slower decay in the population of intermediate energy electrons were observed while the electron swarm cooled from an average energy of above 8 eV, without an electric field present. During that period, the electron impact reaction rate coefficients of helium direct ionization and electronic excitation decreased by more than three orders of magnitude as compared to the modeling based on the local mean energy approximation. Global modeling of the evolution of plasma species densities in response to an electric field typical of atmospheric pressure pulsed discharges was performed with the two approaches to electron kinetics. Differences in the species densities were observed between the two approaches, with a 100% increase in the maximum density of electrons found with the modeling based on the local mean energy approximation.