Electron Energy Equation Research Articles

Modeling of ultrafast phase changes in metal films induced by an ultrashort laser pulse using a semi-classical two-temperature model

A semi-classical two-temperature (2T) model, together with the extended Drude model for temperature-dependent optical properties, is proposed to describe ultrafast thermal transport and phase changes in gold films irradiated by an ultrashort laser pulse. This model reduces to a classical 2T model when the electron momentum equation and the two terms related to electron drifting velocity in the electron energy equation are dropped off. It is found that the significant changes in reflectivity and absorption coefficient due to rapid temperature rise could drastically alter laser energy deposition and in turn, the thermal response in metal films. It is also found that the semi-classical 2T model could predict lower electron and lattice temperature and less severe solid–liquid and liquid–vapor phase change than the classical 2T model. The difference could be more distinct with increase of laser fluence and/or increase of laser pulse duration.

Electron energetics in the Enceladus torus

[1] The inner magnetosphere of Saturn contains both neutral gas and plasma whose composition is dominated by water group ion species. This gas is distributed in a torus-shaped region located near the orbit of Enceladus. The source of this gas is the predominately water vapor plume located in the south polar region of Saturn's icy satellite Enceladus. The electron distribution in the torus comprises two populations: a relatively cold ( 10 eV) suprathermal tail. This paper describes model calculations of the electron energy distribution and energetics in the torus in order to explore how the observed electron distributions can be explained. A thermal electron energy equation is numerically solved, and electron temperatures are calculated. A separate electron energy deposition model determines suprathermal electron fluxes as functions of energy. The main suprathermal electron population is due to photoelectron production from the absorption of solar radiation by water and other neutral species, and this is demonstrated by a comparison of the model results with recently published Cassini data. The model includes heating of thermalized electrons by fresh photoelectrons and by hot pickup ions and also includes cooling by collisions with water and other neutral species. Calculated electron temperatures for the thermal population are about 104 − 2 × 104 K (i.e., 1–2 eV), values which are somewhat lower than recently published Cassini Langmuir probe electron temperature values of 2–3 eV.

Rise and fall of electron temperatures: Ohmic heating of ionospheric electrons from underdense HF radio wave pumping

Here we present electron temperature variations observed with incoherent scatter radar during a European Incoherent Scatter Scientific Association Heating experiment with high‐frequency (HF) radio wave transmission at frequencies above the peak ionospheric critical frequency. The electron temperature increased from 2000 K up to 2800 K during the HF transmission periods. During the experiment both pump frequency and polarization were altered between pump pulses. The observed temperature variation is compared with numerical solutions to the electron energy equation with ohmic heating modeling the effect of the radio wave heating of the plasma. Agreement between observations and model is found to be good.

Fluid model of inductively coupled plasma etcher based on COMSOL

Fluid dynamic models are generally appropriate for the investigation of inductively coupled plasmas. A commercial ICP etcher filled with argon plasma is simulated in this study. The simulation is based on a multiphysical software, COMSOL™, which is a partial differential equation solver. Just as with other plasma fluid models, there are drift–diffusion approximations for ions, the quasi-neutrality assumption for electrons movements, reduced Maxwell equations for electromagnetic fields, electron energy equations for electron temperatures and the Navier–Stokes equation for neutral background gas. The two-dimensional distribution of plasma parameters are shown at 200 W of power and 1.33 Pa (10 mTorr) of pressure. Then the profile comparison of the electron number density and temperature with respect to power is illustrated. Finally we believe that there might be some disagreement between the predicted values and the real ones, and the reasons for this difference would be the Maxwellian eedf assumption and the lack of the cross sections of collisions and the reaction rates.

2D DC Subnormal Glow Discharge in Argon

A two-dimensional time-dependent fluid model was developed and used to describe a DC subnormal glow discharge in argon with Cartesian geometry. This configuration allows us to take into account the transverse expansion of the discharge. A hydrodynamic fluid model used in this paper is based on the moments of the Boltzmann transport equation. The resultant set of governing equations consists of continuity equations (fluxes and densities) for electrons and ions, an energy equation for electrons, and Poisson's equation. Simulation results are presented for the densities of charged particles, the electric voltage, the electric field, and the electron temperature of the discharge. The results were compared with those obtained in the literature.

Research of Ionospheric Instability Excited in Mid and Low Latitudes HF Heating

AbstractThe results of 2D simulation of ionospheric HF pump heating indicate that although the simulation results are affected by the background ionospheric conditions, the electron density and temperature tend towards stationary after heating simulation begins under most undisturbed circumstance. This result is proved by many experiments. Nevertheless on occasion the temporal and spatial distribution of the electron density and temperature near the reflection region of HF pump waves appear to be instable after turning on/off the HF heating equipment. This phenomenon is simulated and interpreted in this paper. The excitation of this instability is explained by a positive feedback mechanism as follows, when the electron temperature is heating up by the pump waves, the reduction of electron density in this region is generated by the plasma expansion. The electron energy decreases in the electron energy equation due to the collision between electron and ion which is influenced by the electron temperature and density. In consequence the positive feedback mechanism is set up as a result of electron temperature increase. It can be concluded that a more obvious change of electron density and temperature in HF pump heating is generated by an instability mechanism if three conditions are satisfied, (1) an evident two‐peak structure of ionospheric electron temperature; (2) a high electron temperature in the heated region; (3) electron and ion collision is the main proportion of the energy loss factor. These conditions are more easily to be satisfied in the afternoon. Key words Hight Frequency (HF) heating, Mid and low latitude ionosphere, Instability, Simulation

On the accuracy and limitations of fluid models of the cathode region of dc glow discharges

This paper compares the performance and limitations of different models of the cathode region of cold-cathode low-pressure dc glow discharges: (i) we review known modelling approaches, (ii) develop our own simulation codes based on these approaches, (iii) perform calculations using these codes for reference sets of discharge conditions, which allows a critical comparison of the models and (iv) for a further check of the simulation results we carry out Langmuir probe measurements of electron densities in abnormal Ar glow discharges. The theoretical approaches include fluid models both neglecting and including the electron energy balance equation, as well as hybrid models, which combine the fluid treatment of slow plasma species with the kinetic simulation of fast electrons. We also test the effect of the choice of the ionization source term in fluid models. We find that the electron densities calculated from the fluid models are far (several orders of magnitude) below the experimental values even if the electron energy equation is considered in the calculations. This weakness of fluid models clearly points out the importance of an accurate calculation of the ionization source term, which can only be accomplished by a kinetic approach under the conditions of highly nonlocal electron transport in the cathode region of glow discharges. In hybrid models Monte Carlo simulation is used for this purpose, and indeed, this approach gives electron densities comparable to our experimental data.

Electron and proton heating by solar wind turbulence

Previous formulations of heating and transport associated with strong magnetohydrodynamic (MHD) turbulence are generalized to incorporate separate internal energy equations for electrons and protons. Electron heat conduction is included. Energy is supplied by turbulent heating that affects both electrons and protons and is exchanged between them via collisions. Comparison to available Ulysses data shows that a reasonable accounting for the data is provided when (1) the energy exchange timescale is very long and (2) the deposition of heat due to turbulence is divided, with 60% going to proton heating and 40% into electron heating. Heat conduction, determined here by an empirical fit, plays a major role in describing the electron data.

Study on Similarity Criteria of Hall Effect Thrusters

AbstractThe scaling design of Hall effect thrusters is based on similarity criteria. Up to now, few of the similarity criteria proposed concern about the inside physical processes of the thruster except Melikov‐Morozov similarity criterion which embodies the ionization of propellant. As many other significant processes, such as electron conduction, ion acceleration and energy exchange, are out of consideration, it is far from enough to direct the thruster design appropriately at present. Therefore, in this paper, we have deduced out many new similarity criteria by analyzing the neutral continuity equation, the ion/electron momentum equations and the electron energy equation with the equation analysis method. By further comparing the magnitudes of source terms of those equations, we obtain certain similarity criteria which are primary and should be guaranteed for modeling design of thrusters. These dominant similarity criteria are finally verified in experiments (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Discretization of the Joule heating term for plasma discharge fluid models in unstructured meshes

The fluid (continuum) approach is commonly used for simulation of plasma phenomena in electrical discharges at moderate to high pressures (>10’s mTorr). The description comprises governing equations for charged and neutral species transport and energy equations for electrons and the heavy species, coupled to equations for the electromagnetic fields. The coupling of energy from the electrostatic field to the plasma species is modeled by the Joule heating term which appears in the electron and heavy species (ion) energy equations. Proper numerical discretization of this term is necessary for accurate description of discharge energetics; however, discretization of this term poses a special problem in the case of unstructured meshes owing to the arbitrary orientation of the faces enclosing each cell. We propose a method for the numerical discretization of the Joule heating term using a cell-centered finite volume approach on unstructured meshes with closed convex cells. The Joule heating term is computed by evaluating both the electric field and the species flux at the cell center. The dot product of these two vector quantities is computed to obtain the Joule heating source term. We compare two methods to evaluate the species flux at the cell center. One is based on reconstructing the fluxes at the cell centers from the fluxes at the face centers. The other recomputes the flux at the cell center using the common drift-diffusion approximation. The reconstructed flux scheme is the most stable method and yields reasonably accurate results on coarse meshes.

Modeling of a dielectric barrier discharge used as a flowing chemical reactor

Our aim is to develop and optimize a model for a dielectric barrier discharge used as a chemical reactor for gas treatment. In order to determine the optimum operating conditions, we have studied the influence of the gas flow rate, reactor geometry and applied voltage parameters on the discharge characteristics.For this purpose, a two-dimensional time-dependent fluid model has been applied to an atmospheric pressure DBD in helium with nitrogen impurities, in a cylindrical geometry. The numerical model is based on the continuity and flux equations for each type of particles treated, the electron energy equation and the Poisson equation. The gas flow is incorporated in the flux equations as a source term. The set of coupled partial differential equations is solved by the so-called modified strongly implicit method. The background gas flow is numerically treated separately, assuming in the model that there is no influence of the plasma on the flow. Indeed, the gas convection velocity is calculated using the commercial code Fluent and it is used as input into the 2D fluid model. The plasma characteristics have been studied in terms of gas flow rate, applied voltage amplitude and frequency, and geometrical effects. The electric currents as a function of time for a given applied potential have been obtained, as well as the number densities and fluxes of plasma species.

Modeling characteristics of nonequilibrium processes during breakdown of capacitive rf argon glow discharge

A self-consistent, diffusion-drift approximated fluid model including ion and electron continuity equations and an electron energy equation is presented for simulating the nonequilibrium characteristics during breakdown of argon rf glow discharge. The nonlinear partial differential equations of the model are solved numerically by using a so-called finite volume method. The numerical results indicate that there exist two different phases on the breakdown curves, i.e., the low p⋅d phase and high p⋅d phase. The breakdown voltage, analyzed for three different gaps, varies little in low p⋅d phase while in the high p⋅d phase, it increases linearly with pressure increase. Time evolution of Ohmic heating and energy loss during breakdown are presented in detail. Analysis yields that the heating mechanisms are the dominant factors in breakdown. Moreover, the steady discharge characteristics are also studied specifically under breakdown voltage of 10Torr. The charged particle densities are on the order of 1015∕m3 and the electron energy has two characteristic values at different rf phases.

Interaction of intense ultra-short laser pulses with dielectrics

The interaction of intense ultra-short laser pulses with dielectrics is studied theoretically using a one-dimensional simulation model. The curl Maxwell's equations are solved coupled to the electron continuity, momentum and energy equations. What is new and innovative in our approach is the application of the ‘forest fire’ method to describe the optical field ionization. The cumulative action of the applied laser field and holes in the dielectric, created during ionization events, enhances the optical field ionization rate by orders of magnitude speeding up the electron avalanche development in the early times of the ionization dynamics. For peak laser intensity of about 1017 W m−2 (1013 W cm−2) the ‘forest fire’ model plays a major role in the ionization dynamics and increases the electron density by two orders of magnitude. The underlying mechanisms leading to electron avalanche in dielectrics are investigated for peak laser intensities between 1016 and 3 × 1018 W m−2 and laser pulse durations between 10 fs and 1 ps. It was established that the dominant process for electron multiplication depends only on the laser fluence: for F < 4 kJ m−2 (F < 0.4 J cm−2) the cumulative optical field ionization rate exceeds the collisional ionization rate, while for F > 4 kJ m−2 (F > 0.4 J cm−2) the collisional ionization is the dominant ionization mechanism. Since for laser pulse duration exceeding 20 fs the threshold for dielectric damage of SiO2 is at least Fthr > 15 kJ m−2 (Fthr > 1.5 J cm−2) (Lenzner et al 1998 Phys. Rev. Lett. 80 4076), the primary cause for dielectric damage is an electron avalanche due to collisional ionization.

Nonthermal Atmospheric RF Plasma in 1-D Spherical Coordinates: A Parametric Study

We present 1-D simulations of atmospheric-pressure RF-excited plasma with two concentric spherical electrodes and the inner electrode powered. The gap distance between the inner and outer electrodes is 1 mm. The gas used is helium with a small amount of nitrogen as an impurity. The coupled continuity equations and electron energy equation are solved with Poisson's equation using the finite element method. In this paper, we particularly focus on the influence of a blocking capacitor, nitrogen concentration, and secondary electrons. In the presence of a blocking capacitor, the self-bias voltage alters sign from negative to positive as the power increases. Reduction of relative nitrogen concentration from 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-3</sup> to 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-6</sup> leads to significant changes in the gas-phase chemistry as well as species flux impacting the outer electrode. Secondary electrons turned out to play a minor role in the transition from the corona to glow modes. We also show that the directed ion kinetic energy distribution to the outer electrode largely depends on the discharge mode.

Nonthermal atmospheric rf plasma in one-dimensional spherical coordinates: Asymmetric sheath structure and the discharge mechanism

We present one-dimensional simulations of atmospheric pressure rf-excited plasma with two concentric spherical electrodes and the inner electrode powered. The gas used is helium with 0.1% nitrogen addition. The gap distance between the inner and outer electrodes is 1mm. The coupled continuity equations and electron energy equation are solved with Poisson’s equation using the finite element method. A mode transition is observed in the discharge power-voltage curve between 1 and 1000mW. In the low power mode, ionization rate peaks only near the inner electrode. The electron-impact excitation and ionization rates peak in the local cathodic phase. In the high power mode, the rate of ionization peaks near the outer electrode as well as the inner electrode. The inner sheath significantly shrinks and the direct electron-impact ionization is the primary ionization reaction near the inner electrode. The ionization rate near the outer electrode is due to Ohmic sheath oscillation heating of electrons, resulting in a peak in metastable helium creation. Penning ionization is the major ionization reaction near the outer electrode. Thus, two different ionization mechanisms coexist near the inner and outer electrodes. Electron heating near the outer electrode may have implications for surface processing in atmospheric pressure microdischarges. The local field approximation (LFA) in high power mode fails to predict the ionization rate peak near the outer electrode due to its inability to properly account for electron diffusion in the presence of both a strong electric field and electron density gradient. However, use of the LFA is adequate to model the low power mode and it correctly predicts the existence of the mode transition.

Estimation of Width of Electron Energy Loss Region in Cusp Magnetic Field in Negative Ion Sources

The loss width of electron energy in the cusp magnetic field in negative ion sources is investigated analytically. Energy equation and continuity equation for electrons are solved with heat flux and particle flux parallel and perpendicular to the magnetic field. Electron energy and density perpendicular to the magnetic field are assumed to decrease exponentially with each decay length and the cusp loss width is defined by the doubling of the decay length. An expression of the ratio of the cusp loss width of electron energy to that of electron density is obtained. The ratio of the cusp loss width depends on electron collision frequencies across the magnetic field and a heat transmission coefficient defined by the ratio of the heat flux to the particle flux multiplied by the electron temperature along the magnetic field. The heat transmission coefficient is related to the sheath potential on the wall surface. Both parameter dependence and the calculated value of the cusp loss width of electron energy derived from our analysis are in good agreement with Horiike et al.'s experimental result for the magnetic multipole ion source.

A new paradigm for solving plasma fluid modeling equations

A new paradigm for solving plasma fluid modeling equations is proposed and verified in this paper. Model equations include continuity equations for charged species with drift-diffusion approximation, electron energy equation, and Poisson’s equation. Resulting discretized equations are solved jointly by the Newton–Krylov–Schwarz (NKS) [1] scheme by means of a parallelized toolkit called PETSc. All model equations are nondimensionalized and are discretized using fully implicit finite-difference method with the Scharfetter–Gummel scheme for the fluxes. At electrodes, thermal flux is considered for electrons, while both thermal and drift fluxes are considered for ions. A quasi-1D argon gas discharge with a radio frequency power source (13.56 MHz, Vp−p = 200 Volts), gap distance = 20 mm and 20 mm × 20 mm (100 × 100 mesh points) in size is used as the test case. Results of evolution of potential and plasma number density are shown Fig. 1, which are comparable to previous studies. Table 1 lists all the resulting timings of the present parallelized code using different combination of preconditioners (Additive

ホールスラスタにおける放電振動の一次元数値解析

A one-dimensional numerical simulation was conducted inside an anode layer type Hall thruster in order to clarify the nature of low frequency discharge oscillation in Hall thrusters. The model assumed quasi-neutrality, Maxwellian electron distribution, and classical and Bohm diffusions across magnetic field lines. Heavy species were simulated using Particle-in-Cell (PIC) method, and electrons were dealt as fluid continuum. Generalized Ohm’s law was used to determine the electric potential, and an electron energy equation was solved to determine the electron temperature. As a result, we obtained discharge oscillations which were caused by ionization instability. The calculated current, its oscillations, and the stable operation range agreed with measured ones. We also succeeded to simulate the transition from the classical diffusion mode to Bohm diffusion mode by the calculation. In the discussion, it is shown that the electron energy given by the electric field has the large effect on the current oscillation.

Electron temperature of Titan's sunlit ionosphere

Titan's upper atmosphere is ionized by solar radiation and particle bombardment from Saturn's magnetosphere. The induced ionosphere plays a key role in the coupling of Titan's atmosphere with the Kronian environment. It also provides unique signatures for identifying energy sources upon Titan's upper atmosphere. Here we focus on observations from the first, close flyby by the Cassini spacecraft and assess the ionization and electron heating sources in Titan's sunlit ionosphere. We compare CAPS electron spectra with spectra produced by an electron transport model based on the INMS neutral densities and a MHD interaction model. In addition, we compare RPWS electron temperature against the models. The important terms in the electron energy equation include loss through excitation of vibrational states of N2 and CH4, Coulomb collisions with suprathermal electrons, and thermal conduction. Our analysis highlights the important role of the magnetic field line configuration for aeronomic studies at Titan.

A numerical study of the response of ionospheric electron temperature to geomagnetic activity

The response of ionospheric electron temperatures to geomagnetic activity has been simulated using the Thermosphere‐Ionosphere Nested Grid (TING) model. The cause of this response has been analyzed using a new postprocessor for the TING model that looks at the individual physical terms that drive the electron energy equation. It is found that (1) electron temperatures are significantly enhanced in regions of depleted electron densities, especially inside the middle‐latitude electron density trough. The most pronounced electron temperature enhancement occurs at the equatorward edge of the trough; (2) this enhancement is produced by heat flux from the plasmasphere, coupled with the effect of the relatively low thermal electron gas heat capacity there and the inefficient heat conduction in the bottomside of the F region, where electron densities have almost vanished; (3) in regions of enhanced electron densities, electron temperatures are significantly decreased as a result of enhanced energy loss to the ions; this prevents the electron temperature “morning overshoot” from occurring in the middle and low latitudes; (4) upwelling of molecular‐species‐rich air from the lower thermosphere to higher altitudes during the storm increases mixing ratios of the neutral molecular species in the F region, thus enhances the relative contribution of the neutral molecular species to the overall electron cooling there; (5) ion frictional heating increases high‐latitude F region ion temperatures during geomagnetic storms, causing the ions to transfer energy to the electrons and thus enhancing electron temperatures; and (6) there are no significant electron temperature increases in the E region during the storm because of the rapid energy loss from the electrons to the neutrals.