Heating mechanism in one-dimensional bounded magnetized inductively coupled plasma

A self-consistent nonlocal kinetic model has been established to investigate the electron heating mechanism in an one-dimensional bounded magnetized inductively coupled plasma (MICP) under low pressures. The interaction function of electrons with rf electric field and electron energy distribution function (EEDF) are determined by solving the linearized Boltzmann equation incorporating with the Maxwell equations. The numerical results show that the presence of the direct-current (dc) magnetic field plays an important role in the EEDF and high-energy electrons are efficiently heated by the Azbel–Kaner resonances under the anomalous skin-effect conditions.     

Low-pressure RF discharge in the free-flight regime

The self-consistent equations system for low-pressure RF discharge in the free-flight regime is formulated. The expressions for the electron energy diffusion coefficient due to electron-neutral collisions and to the electron collisions with the plasma-space charge moving boundary (stochastic heating) are derived. If the electron-neutral elastic collisions frequency exceeds the inelastic one, the conventional two-term approximation for the electron distribution function (EDF) can be generalized, and the space-time-averaged electron kinetic equation can be reduced to the one-dimensional energy diffusion one. The fast electrons attached to the electrode surface can also be accounted for in this equation. It is shown that in the cases of (a) spatially uniform ion profile, (b) for frequencies that are small compared with the electron bounce frequency, and (c) for frequencies exceeding the electron plasma one in the sheath, the stochastic heating vanishes     

Instability-triggered phase transition to a dusty-plasma condensate

Highly charged dust grains in plasma discharges reside at the sheath edge, where the ions stream toward the electrode at speed approximately c(s). Above a critical pressure P(crit), the grains lose their kinetic energy and reach a strongly coupled crystalline state, but for P P(crit) by the combined effect of ion-molecule and grain-molecule collisions.     

Heat transfer in a two-dimensional crystalline complex (dusty) plasma

Heating and heat transfer were studied in a two-dimensional crystalline complex plasma at the kinetic level. The lattice was formed of microspheres levitated in a plasma sheath. One half of the crystal was heated anisotropically to obtain higher kinetic temperatures in one direction and heat conduction was observed in real time. It was found that the longitudinal phonons conduct heat better than the transverse. The thermometric conductivity coefficient was measured to be 53 mm2/s for longitudinal heating and 30 mm2/s for transverse heating. Heat decay lengths and energy exchange times between the temperature components were determined.     

Viscous effects on motion and heating of electrons in inductivelycoupled plasma reactors

A transport model is developed for nonlocal effects on motion and heating of electrons in inductively coupled plasma reactors. The model is based on the electron momentum equation derived from the Boltzmann equation, retaining anisotropic stress components which in fact are viscous stresses. The resulting model consists of transport equations for the magnitude of electron velocity oscillation and terms representing energy dissipation due to viscous stresses in the electron energy equation. In this model, electrical current is obtained in a nonlocal manner due to viscous effects, instead of Ohm's law or the electron momentum equation without viscous effects, while nonlocal heating of electrons is represented by the viscous dissipation. Computational results obtained by two-dimensional numerical simulations show that nonlocal determination of electrical current indeed is important, and viscous dissipation becomes an important electron heating mechanism at low pressures. It is suspected that viscous dissipation in inductively coupled plasma reactors in fact represents stochastic heating of electrons, and this possibility is exploited by discussing physical similarities between stochastic heating and energy dissipation due to the stress tensor     

Comment on Schöner-Haken systematic adiabatic approximation for stochastic differential equations

The Antoniewicz model is used to study the stimulated desorption of argon and krypton neutrals from Ru(001). We solve the Schrödinger equation semiclassically and find good agreement with the experiment for both the desorption yield and the kinetic energy distribution of the argon and krypton atoms. We use a reneutralization rate which consists of a single exponential inside a certain distance and is equal to zero outside. In contrast to earlier theories, we obtain in agreement with the experiment a relatively high yield, a low most probable kinetic energy and slowly decaying high energy tails of the kinetic energy distribution of the desorbing rare gas neutrals. We show explicitly that the single exponential form previously used for the reneutralization rate cannot account for the three relevant experimental findings simultaneously.Dedicated to Professor W. Brenig on the occasion of his 60th birthday     

A hybrid fluid–kinetic neutral model based on a micro–macro decomposition in the SOLPS-ITER plasma edge code suite

We present a hybrid fluid–kinetic model for the hydrogenic atoms in the plasma edge that is implemented in SOLPS-ITER. A micro–macro decomposition of the kinetic equation leads to a fluid model with a continuity and parallel momentum equation (implemented in B2.5) coupled to a kinetic correction equation (implemented in EIRENE). We assess the hybrid model for a high recycling fixed background plasma. The hybrid approach leads to a reduction of the Central Processing Unit(CPU) time required to obtain the same statistical error as the full kinetic Monte Carlo (MC) simulation with approximate factors of 1.7, 4.9, and 1.9 for the particle, parallel momentum, and electron energy source, respectively. However, there is an increase in CPU time for the ion energy source. By comparing the results with our in-house plasma edge code, we conclude that the hybrid performance can be improved by adapting some default MC features in EIRENE.     

Stochastic electron heating in a capacitive RF discharge withnon-Maxwellian and time-varying distributions

In capacitively coupled radio frequency discharges, the electrons gain and lose energy by reflection from oscillating, high voltage sheaths. When time-averaged, this results in stochastic heating, which at low pressure is responsible for most of the electron heating in these discharges. Previous derivations of stochastic heating rates have generally assumed that the electron distribution is a time-invariant, single-temperature Maxwellian, and that the sheath motion is slow compared to the average electron velocity, so that electrons gain or lose a small amount of energy in each sheath reflection. Here we solve for the stochastic heating rates in the opposite limit of fast sheath motion and consider the applicability of the slow and fast sheath equations in the intermediate region. We also consider the effect of a two-temperature Maxwellian distribution on particle balance and the effect of a time-varying temperature on the heating rates and densities     

Collisionless bounce resonance heating in dual-frequency capacitively coupled plasmas

We present the experimental evidence of the collisionless electron bounce resonance heating (BRH) in low-pressure dual-frequency capacitively coupled plasmas. In capacitively coupled plasmas at low pressures when the discharge frequency and gap satisfy a certain resonant condition, the high energy beamlike electrons can be generated by fast sheath expansion, and heated by the two sheaths coherently, thus the BRH occurs. By using a combined measurement of a floating double probe and optical emission spectroscopy, we demonstrate the effect of BRH on plasma properties, such as plasma density and light emission, especially in dual-frequency discharges.     

Effect of a high-frequency field on the reflection of ion acousticwaves from the sheath at a grid

Experiments are described which show that the reflection coefficient for ion acoustic waves (IAW) from the sheath at a grid is affected by an HF electric field with a frequency f_HF≲5f_pi(f_pi =ion plasma frequency). For peak-to-peak amplitudes of the HF voltage drop across the sheath Φ₀≲k_B T_e/e and f_HF>f _pi, the energy distribution of the ions passing through the grid develops a hot tail and the reflected wave suffers enhanced Landau damping. If Φ₀≳k_BT_e/e and f_HF<f_pi, a large-amplitude IAW is excited at the grid; a well-defined ion beam is formed; and local growth of the reflected wave is observed. Test waves launched from the grid show the same propagation characteristics as the reflected waves     

Brownian system in energy space

The main goal of this article is to present a simple way to describe non-equilibrium systems in energy space and to obtain new spacial solution that complements recent results of B.I. Lev and A.D. Kiselev, Phys. Rev. E 82 , (2010) 031101. The novelty of this presentation is based on the kinetic equation which may be further used to describe the non-equilibrium systems, as Brownian system in the energy space. Starting with the basic kinetic equation and the Fokker-Plank equation for the distribution function of the macroscopic system in the energy space, we obtain steady states and fluctuation relations for the non-equilibrium systems. We further analyze properties of the stationary steady states and describe several nonlinear models of such systems.     

The space-time-averaging procedure and modeling of the RF dischargeII. Model of collisional low-pressure RF discharge

For pt.I see ibid., vol.19, p.130-40 (1991). A self-consistent equations system for the low-pressure RF discharge is formulated and qualitatively analyzed. If the plasma and sheath dimensions exceed the electron-energy relaxation length, a simple spatially averaged kinetic equation can be derived that resembles the conventional one for the local case. Since the energy-diffusion coefficient for the slow electrons that are trapped by the average electric field in the discharge center is small, the distribution function slope decreases significantly with the energy growth. Analytic estimates are derived and reasonable agreement with the experiments of Godyak (1976, 1979, 1986, 1990) is obtained     

Particle density and non-local kinetic energy density functional for two-dimensional harmonically confined Fermi vapors

We evaluate analytically some ground state properties of two-dimensional harmonically confined Fermi vapors with isotropy and for an arbitrary number of closed shells. We first derive a differential form of the virial theorem and an expression for the kinetic energy density in terms of the fermion particle density and its low-order derivatives. These results allow an explicit differential equation to be obtained for the particle density. The equation is third-order, linear and homogeneous. We also obtain a relation between the turning points of kinetic energy and particle densities, and an expression of the non-local kinetic energy density functional. Received 27 March 2001 and Received in final form 12 June 2001     

Tonks-Langmuir problem for a bi-Maxwellian plasma

An analytical solution of the Tonks-Langmuir (TL) problem with a bi-Maxwellian electron energy distribution function (EEDF) is obtained for a plasma slab. The solution shows that the ambipolar potential, the plasma density distribution, and the ion flux to the wall are mainly governed by the cold electrons, while the ionization rate and voltage drop across the wall sheath are governed by the hot electrons. The ionization rate by direct electron impact is found to be spatially rather uniform, contrary to the T-L solution where it is proportional to the plasma density distribution. The temperature of hot electrons defined by the ionization balance is found to be close to that of the T-L solution for a mono-Maxwellian EEDF, and is in reasonable agreement with experiments carried out in a low pressure capacitance RF discharge. The energy balance for cold electrons in this discharge shows that their heating by hot electrons via Coulomb interaction is equalized by the cold electrons' escape to the RF electrodes during collapse of the RF sheath     

Effects of gas pressure on plasma characteristics in dual frequency argon capacitive glow discharges at low pressure by a self-consistent fluid model

A self-consistent fluid model for dual radio frequency argon capacitive glow discharges at low pressure is established.Numerical results are obtained by using a finite difference method to solve the model numerically, and the results are analyzed to study the effect of gas pressure on the plasma characteristics. It shows that when the gas pressure increases from 0.3 Torr(1 Torr = 1.33322×10~2 Pa) to 1.5 Torr, the cycle-averaged plasma density and the ionization rate increase;the cycle-averaged ion current densities and ion energy densities on the electrodes electrode increase; the cycle-averaged electron temperature decreases. Also, the instantaneous electron density in the powered sheath region is presented and discussed. The cycle-averaged electric field has a complex behavior with the increasing of gas pressure, and its changes take place mainly in the two sheath regions. The cycle-averaged electron pressure heating, electron ohmic heating, electron heating, and electron energy loss are all influenced by the gas pressure. Two peaks of the electron heating appear in the sheath regions and the two peaks become larger and move to electrodes as the gas pressure increases.     

Hot Ion Production in a Modified Penning Discharge

Ions with Maxwellian energy distributions and kinetic temperatures up to seven keV have been observed in a modified Penning discharge. Investigation of the plasma revealed two distinct spoke-like concentrations of charge, consisting respectively of ions and electrons, rotating with different velocities in the sheath between the plasma and the anode ring. Theoretical expressions are derived for the frequency of the ion and electron spoke rotation, for the ion kinetic temperature resulting from the ion spoke velocity, and for the ion heating efficiency. An extensive series of experimental measurements were made to check these theoretical expressions, and approximate agreement was obtained. It is shown that the ion kinetic temperature in the modified Penning discharge scales according to the relation Vi ~ Vani1/4/B1/2 where Va is the applied anode voltage, ni is the ion density in the sheath, and B is the magnetic field strength. The observed data demonstrate that the ion heating efficiency can be as high as several tens of percent.     

On energy transfer and generation of an electric current in the vicinity of an electrode dipped in an electrolyte and strongly heated by a current passing through it

Processes occurring when a metal electrode dipped in an electrolyte is heated by intense evaporation of the electrolyte are considered in terms of a physically rigorous model. Based on the Onsager principle of least energy dissipation rate in nonequilibrium processes, the fractions of thermal energy that are spent on heating and evaporating the electrolyte and on heating the vapor are found. The energy is released within the vapor-gas sheath when an electric current flows between the electrode and electrolyte surface. It is found that the electrolyte vapor temperature exceeds 1300 K. Analytical expressions are derived for the vapor-gas sheath thickness, the electrolyte vapor pressure, and the velocity of the vapor escaping the discharge zone. It is shown that field evaporation of thermally activated negative ions from the electrolyte surface cannot provide an electric current with densities found in experiments but is responsible for the generation of free electrons near the electrolyte surface. These electrons arise when the ions decay via collisions with excited molecules.     

Heating mechanism affects equipartition in a binary granular system

Two species of particles in a binary granular system typically do not have the same mean kinetic energy, in contrast to the equipartition of energy required in equilibrium. We investigate the role of the heating mechanism in determining the extent of nonequipartition of kinetic energy. In most experiments, different species are unequally heated at the boundaries. We show by event-driven simulations that differential boundary heating affects nonequipartition even in the bulk of the system. This conclusion is fortified by studying numerical and solvable stochastic models without spatial degrees of freedom. In both cases, even in the limit where heating events are rare compared to collisions, the effect of the heating mechanism persists.     

Scaling laws for the spatial distributions of the plasma parameters in the positive column of a dc oxygen discharge

Comprehensive self-consistent simulations of the positive column plasma of a dc oxygen discharge are performed with the help of commercial CFDRC software (), which enables one to carry out computations in an arbitrary 3D geometry using fluid equations for heavy components and a kinetic equation for electrons. The main scaling laws for the spatial distributions of charged particles are determined. These scaling laws are found to be quite different in the parameter ranges that are dominated by different physical processes. At low pressures, both the electrons and negative ions in the inner discharge region obey a Boltzmann distribution; as a result, a flat profile of the electron density and a parabolic profile of the ion density are established there. In the ion balance, transport processes prevail, so that ion heating in an electric field dramatically affects the spatial distribution of the charged particles. At elevated pressures, the volume processes prevail in the balance of negative ions and the profiles of the charged particle densities in the inner region turn out to be similar to each other.     

Thermodynamic equilibrium for nitrogen species discharge: Comparison with global model

The equilibrium process of plasma nitrogen species by chemical kinetic reactions along various pressures is successfully investigated. The equilibrium process is required in industrial application to obtain the stable condition when heating up the material for having homogenous reaction. Nitrogen species densities is modeled by a continuity equation and extended Arrhenius form. These equations are used to integrate the change of density over the time. The integration is to acquire density and the reaction rate of each reaction where temperature and time dependence are imposed. A comparison is made with global model within pressure range of 1-100 mTorr and the temperature of electron is set to be higher than other nitrogen species. The results show that the chemical kinetic model only agrees for high pressure because of no power imposed; while the global model considers the external power along the pressure range then the electron and nitrogen species give highly quantity densities by factor of 3-5.