Electron Boltzmann Equation Research Articles

Kinetic study of the linear stage of thermocurrent instability in the framework of the Boltzmann equation for a model gas

The linear stage of thermocurrent instability is investigated for a model gas in which the integral of inelastic collisions of electrons with gas particles has a divergent form and the frequencies of elastic and inelastic collisions are independent of the electron velocity. The proposed approach consists in the reduction of the Boltzmann equation for electrons in an inhomogeneous plasma to a set of equations for the moments of the electron velocity distribution function. The instability growth rate and the wave phase velocity as functions of the perturbation wavenumber are calculated, the maximum growth rate and the corresponding wavenumber are determined, and the dependence of these quantities on the degree of plasma quasineutrality is examined. It is demonstrated that the model satisfactorily (both qualitatively and quantitatively) describes the linear stage of thermocurrent instability in helium.

Microwave N2–Ar plasma torch. I. Modeling

The spatial structure of a microwave plasma torch driven by an azimuthally symmetric surface wave operating in a N2–Ar mixture at atmospheric pressure is investigated. A two-dimensional (2D) self-consistent theoretical model is developed to investigate the entire spatial structure of the source, including the discharge zone, sustained by the field of the surface TM00 mode, and the postdischarge plasma. Maxwell’s equations, the rate balance equations for the most important excited species—vibrationally and electronically excited states, ions and nitrogen atoms N(S4)—and the Boltzmann equation for electrons are consistently solved. Model calculations of the 2D spatial distributions of species of interest such as charged particles (electrons and positive ions), N2(Χ Σ1g+,v) vibrationally excited molecules, N2(A Σ3u+) metastable molecules, and N(S4) ground state atoms are presented and discussed.

Barium transport in the hot spot region of fluorescent lamps

The transport of barium atoms and ions in the vicinity of the hot spot in fluorescent lamps operating at 25 kHz is investigated by a combined experimental and theoretical approach. By laser-induced fluorescence, the particle densities of barium atoms and ions were measured time-resolved at different distances from the spot centre. In addition, the time-dependent cathode fall voltage was measured using an improved band method. The model combines a kinetic part for the electrons with a fluid part for the barium atoms and ions. Both parts are spatially resolved in spherically symmetric geometry. The space-dependent electron Boltzmann equation yields the electron density and the ionization rate coefficient of barium as functions of the cathode fall voltage. These results are used to solve the time-dependent particle balance equations of barium atoms and ions which include the ionization of barium as gain and loss terms, respectively. Good agreement between the measured and calculated particle densities of barium atoms is obtained. A sensitive dependence of the ionization frequency and of the barium particle densities on the cathode fall voltage was found.

Electron-phonon hydrodynamical model for semiconductors

A hydrodynamical model for the electron-phonon system in semiconductor is developed by closing the moment system arising from the coupled Boltzmann equations for electrons and phonons with the maximum entropy principle. Limiting models are obtained under appropriate scaling, and comparisons with the existing models of charge transport in semiconductors including the thermal effects of the crystal lattice are presented.

Effect of high Xe-concentration in a plasma display panel with a SrCaO cold cathode

We present here measurements of high Xe-contents plasma display panel (PDP) with SrCaO cold cathode. Luminous efficacy (η) shows a two-step increase with Xe-concentration in Ne/Xe gas mixture: η drastically increases up to Xe-concentration of 30% (Xe: 30%), and then attains 5 lm/W at the highest Xe-concentration of Xe: 100%. The high performance PDP with Xe: 100% can be operated at low applied voltage between 230 and 377 V due to the high secondary electron emission from the SrCaO cathode. Emission measurements clearly show the change in discharge characteristics at Xe: 30%, where the discharge changes from a Ne/Xe mixture discharge to an almost pure Xe discharge, and the vacuum ultraviolet (VUV) radiation from the combination of resonance and excimer radiations to only excimer radiation. Theoretical analysis solving Boltzmann equation for electron demonstrates that increasing Xe-concentration enhances the collision frequency for electron impact excitation directly from ground state to lower levels concerned with the VUV radiation, resulting in a drastic increase in luminous efficacy up to Xe: 30%. Also, one-dimensional fluid simulation of a Ne/Xe dielectric barrier discharge clearly shows that a combination of high secondary electron emission cathode and high Xe-concentration is quite effective for high VUV radiation efficiency because it induces a drastic increase in electron-heating efficiency.

Vibrational kinetics of air plasmas induced by sprites

The vibrational kinetics of air plasmas produced by the presence of sprites in the mesosphere of the Earth is studied in detail. The present model solves the coupled electron Boltzmann equation and rate equations for electrons, neutrals (ground and excited), and ions. Special attention is paid to the vibrational kinetics of N2 (X1Σg+) and to that of the N2 triplet states (A3Σu+, B3Πg, C3Πu, W3Δu, and B′3Σu+). The results presented are for an altitude of 78 km over sea level where the model‐predicted vibrational distribution function (VDF) of N2 (B3Πg) is in agreement with available VDF derived from sprite spectra (640–820 nm) obtained using spectrographs coupled to high‐speed video cameras (300 frames per second (fps)). In addition, the model predictions indicate that the sprite plasma VDFs of N2 (B3Πg) and N2 (C3Πu) are almost in thermal equilibrium and exhibit a maximum at v = 0 followed by lower values for v = 1 and v = 2. Finally, it is also predicted that the N2 (B3Πg) and N2 (C3Πu) VDFs recorded with very high speed video cameras (up to 10000 fps) should not differ much from the N2 (B3Πg) and N2 (C3Πu) VDFs obtained with cameras working at lower speeds (30 and 300 fps).

Kinetic processes for laser induced plasma diagnostic: A collisional-radiative model approach

A zero-dimensional collisional–radiative (CR) model, coupled self-consistently with the electron Boltzmann equation, has been applied to the description of a metallic-laser induced plasma at experimental conditions typical of LIBS experiment. To take in account expansion effects, the experimental temperature and total number density as function of time have been used as input data. Plasma composition and the simultaneous time evolution of both heavy particle level distributions and the electron energy distribution function have been calculated by taking into account the most relevant collisional and radiative processes. This approach estimates the hierarchy of the elementary processes during the expansion and possible deviations from LTE conditions. The comparison of the experimental and theoretical results shows a good agreement, but at the same time new questions arise on the analysis of spectroscopic results and on the assumption generally made in LIBS.

An efficient energy-conserving numerical model for the electron energy distribution function in the presence of electron–electron collisions

An efficient algorithm to calculate the contribution of electron–electron collisions in the Boltzmann equation for free electrons, in the two-term approximation is presented. The electron–electron collision term must be energy-conserving, while, due to non-linearity, commonly used algorithms do not satisfy this requirement. The efficiency of the algorithm make feasible the use of a non-linear iterative solver to conserve electron energy in electron–electron collisions. The performance of the proposed algorithm has been compared with standard numerical schemes obtaining: 1) considerable gain in computational time; 2) the conservation of the total electron energy density in e–e collisions under the required tolerance.

Kinetic models of nonequilibrium nitrogen and hydrogen plasma for diagnostics of gas discharges

Computation-experimental methods for plasma diagnostics of nonequilibrium nitrogen and hydrogen gas discharges are presented. These methods are based on a combined use of collisional-radiative models and of experimental results on optical plasma diagnostics. Nonstationary self-consistent level-to-level collisional-radiative models include computer codes: for the processing and the simulation of the emission and the coherent anti-Stokes Raman scattering spectra; for the solution of the kinetic Boltzmann equation for plasma electrons, the thermal conductivity equation, and the balance equations for most of plasma neutral, excited, and charged particles. Banks of data were created for the processing and the simulation of the spectra, and the solution of the kinetic equations. The models were validated by the comparison of obtained results with numerous known results of experiments and with calculations for different discharges.

Fluid Modeling and Analysis of the Constriction of the DC Positive Column in Argon

The glow-to-arc transition of the positive column of a dc discharge in argon in the course of constriction has been investigated on the basis of a self-consistent 1-D axisymmetric fluid model. The model adopts the nonlocal moment method, i.e., the system of balance equations resulting from the moments of the radially dependent Boltzmann equation is solved. The electron transport and rate coefficients are applied in dependence on the mean energy of the electrons, the gas temperature, and the ionization degree. Investigations have been performed for currents from 0.6 to 70 mA and pressures from 100 to 500 torr. The model predictions are compared with experimental and other available modeling results, and they show good agreement with these data in general. The pronounced nonlocal features of the mean electron energy balance are found, and their influence on the constricted argon positive column is analyzed. Different assumptions concerning the electron velocity distribution function have been considered in the present model. In particular, the impact of using a Maxwellian distribution instead of solutions of the steady-state spatially homogeneous electron Boltzmann equation is discussed where the assumption of a Maxwellian distribution for the electrons was found to be inappropriate for describing the constriction effect.

Spectral dependence of the ultrafast optical response of nonspherical gold nanoparticles

The extension of the discrete dipole approximation (DDA) model to the calculation of the ultrafast transient optical response of metal nanoparticles, whatever their shape, is presented. For this, the DDA is combined with the resolution of the Boltzmann equation for the conduction electrons within the low-perturbation limit and the Rosei model for the metal band structure. This approach thus takes into account the athermal regime for the electron gas. It is then illustrated in the cases of gold nanospheres and nanopeanuts. The spectral dependence of the ultrafast response is shown to be very sensitive to the particle morphology, the surface plasmon resonance enhancing the transient modification of the interband transitions. These results are used to interpret spectrally resolved pump-probe measurements performed on a dense nanoparticle assembly, demonstrating selective-excitation and selective-detection processes.

Mechanism of pulse discharge production of iodine atoms from CF3I molecules for a chemical oxygen–iodine laser

The pulsed chemical oxygen–iodine laser (COIL) development is aimed at many new applications. Pulsed electric discharge is most effective in turning COIL operation into the pulse mode by instant production of iodine atoms. A numerical model is developed for simulations of the pulsed COIL initiated by an electric discharge. The model comprises a system of kinetic equations for neutral and charged species, electric circuit equation, gas thermal balance equation and the photon balance equation. Reaction rate coefficients for processes involving electrons are found by solving the electron Boltzmann equation, which is re-calculated in a course of computations when plasma parameters changed. The processes accounted for in the Boltzmann equation include excitation and ionization of atoms and molecules, dissociation of molecules, electron attachment processes, electron–ion recombination, electron–electron collisions, second-kind collisions and stepwise excitation of molecules. The last processes are particularly important because of a high singlet oxygen concentration in gas flow from the singlet oxygen chemical generator. Results of numerical simulations are compared with experimental laser pulse waveforms. It is concluded that there is satisfactory agreement between theory and the experiment. The prevailing mechanism of iodine atom formation from the CF3I donor in a very complex kinetic system of the COIL medium under pulse discharge conditions, based on their detailed numerical modelling and by comparing these results both with experimental results of other authors and their own experiments, is established. The dominant iodine atom production mechanism for conditions under study is the electron-impact dissociation of CF3I molecules. It was proved that in the conditions of the experiment the secondary chemical reactions with O atoms play an insignificant role.

Inhomogeneous microwave discharge in oxygen

Results are given of one-dimensional quasi-static simulation of steady microwave discharge in a system of electrodes with spherical symmetry in oxygen at pressures of 1 to 8 torr. The model includes the equation for electric field intensity in quasi-static approximation, Poisson equation, balance equations defining the kinetics of charged and neutral particles of plasma, and steady-state homogeneous Boltzmann equation for electrons. The well-known analytical expression for vibrational distribution of molecules, derived in diffusion approximation, is used for including the processes which involve vibrationally excited particles. It is demonstrated that the presence of negative ions does not result in differences of the electrodynamics of discharge from those of discharge in nitrogen. The composition of heavy component of plasma is investigated and, in particular, it is demonstrated that the concentration of molecules in the O2(a1Δ) state may be as high as 20%.

Backscattering of secondary electrons to the cathode in the oblique electric field in dielectric barrier discharge systems

In dielectric barrier discharge systems, where the cathode is covered with the dielectric layer, electric field lines may cross the cathode surface at an oblique angle θ. The secondary electrons emitted from this surface either return to the cathode due to collisions with atoms of the background gas or escape from the cathode vicinity. Using the two-term approximation to the Boltzmann equation for electrons, we have found that, for Ramsauer gases, the electron escape factor fes as a function of the angle θ unexpectedly exhibits non-monotonic behaviour. Monte Carlo simulations of backscattering of electrons performed for xenon and argon have confirmed the theoretical results and shown similar trends for non-Ramsauer gases.

Nonequilibrium dynamics of laser-generated plasma channels

A time-dependent nonequilibrium kinetics model based upon the time-dependent electron Boltzmann equation coupled with an extensive air chemistry model accounting for gas heating and vibrational kinetics is developed. The model is applied to the temporal evolution of femtosecond laser-generated air plasma channels at atmospheric pressure in an external electric field. The plasma channel dynamics depend upon the initial free electron density, the initial electron energy of the plasma, and upon the externally applied electric field strength. The model predicts an electric breakdown field strength of 5–10kV∕cm with a delay time of hundreds of nanoseconds when the electron density drops to the optimum value of ∼1012–1013cm−3. The experimentally observed breakdown field is ∼5.7kV∕cm with a statistical breakdown delay time of ∼200ns. The reduction in the breakdown field strength in natural air from ∼30to5kV∕cm is attributed to a combination of processes such as enhanced ionization due to relaxation of the initial electron energy distribution function toward a Maxwellian distribution, strong electron detachment, and gas heating. The calculated electron density decay of the laser-generated plasma channel in both pure nitrogen and dry air is in good agreement with the NRL experiments. The derived rate constant for recombination in dry air is bBair=3.9×10−8cm3s−1 and in pure nitrogen it is bBN2=4.4×10−8cm3s−1. The attachment rate coefficient in dry air is ηBair=7.5×106s−1.

Electron-molecule collision cross sections needed for breakdown electric field calculations of hot dissociated SF6

The critical electric fields of hot SF6 are calculated for large temperature and pressure ranges (300 K to 3000 K from 1 bar to several bars). Calculations are based on a multi-term electron Boltzmann equation solution which needs the knowledge of electron-gas collision cross sections for ten SF6 dissociation products. The collision cross sections are fitted using an electron-swarm unfolding technique. These critical fields are then used to predict the circuit breaker behaviours during the SF6 recovery phase.

New Breakdown Electric Field Calculation for SF6 High Voltage Circuit Breaker Applications

The critical electric fields of hot SF6 are calculated including both electron and ion kinetics in wide ranges of temperature and pressure, namely from 300 K up to 4000 K and 2 atmospheres up to 32 atmospheres respectively. Based on solving a multi-term electron Boltzmann equation the calculations use improved electron-gas collision cross sections for twelve SF6 dissociation products with a particular emphasis on the electron-vibrating molecule interactions. The ion kinetics is also considered and its role on the critical field becomes non negligible as the temperature is above 2000 K. These critical fields are then used in hydrodynamics simulations which correctly predict the circuit breaker behaviours observed in the case of breaking tests.

Modeling of the electrode microwave discharge in nitrogen

Self-consistent two-dimensional modeling of a steady microwave discharge initiated at the end of the central electrode in nitrogen is presented. The discharge parameters are calculated at a gas pressure of 1 Torr and incident power of 30 W. The computational model includes the time-dependent Maxwell's equations, the balance equations describing the kinetics of charged and neutral plasma particles and the time-independent homogeneous Boltzmann equation for electrons. The processes involving vibrationally excited ground state molecular nitrogen are taken into account by the well-known analytic expression for the vibrational distribution of molecules in the diffusion approximation. It is shown that the spatially non-uniform microwave field causes the difference in plasma particles kinetics in different parts of the discharge. Results of numerical simulations for space distribution of electronically excited molecules have been found in good qualitative agreement with those taken from spectral measurements of first and second positive systems of nitrogen. Results confirm the concept according to which such a discharge comprises a self-sustained and a non-self-sustained discharge.

Monte Carlo evaluation of an analytical model for nonequilibrium-phonon-induced electron velocity saturation in GaN

The ensemble Monte Carlo method is used to solve the Boltzmann equation for electrons together with the equation for phonons treated in the relaxation-time approximation. The effect of nonequilibrium longitudinal optical phonons on hot-electron energy distribution, mean energy, drift velocity and power dissipation is considered for bulk GaN subjected to electric fields in a range up to 100 kV cm−1. In particular, the suppression of the high-energy tail of the hot-electron distribution due to the phonon accumulation is resolved. The results of the simulation are compared with those obtained within the recently proposed analytical model for the saturation of electron drift velocity at high electric fields. The saturated velocity decreases as the electron gas density increases. Values around 107 cm s−1 are reached at an electric field of 100 kV cm−1 for an electron gas density of 1018 cm−3. The analytically obtained results are in a better agreement with those of Monte Carlo simulation if the electron gas density is higher and the electric field is stronger.

Influence of hot Phonons on the Transport Properties of Single-Wall Carbon Nanotubes

We investigate the effects of hot phonons on the transport properties of metallic single-wall carbon nanotubes lying on a substrate. A deterministic numerical scheme is applied to solve the system of Boltzmann equations for electrons and phonons. The simulations demonstrate that the optical phonons are strongly driven out of thermal equilibrium and considerably influence the electron transport at high bias. Comparisons of the theoretical results with experimental current-voltage characteristics exhibit very good agreement.