Electron distribution function relaxation in monatomic gases

The authors discuss an analytic solution of the Boltzmann equation which describes the relaxation in time of the electron distribution function for electrons in a plasma derived from the monatomic gases He, Ne, Ar, and Xe. It is assumed that there are no perturbing forces on the electrons and that at t=0 they have a Maxwellian distribution function corresponding to an average energy of 2 eV. The electrons then lose energy through elastic collisions with neutrals and eventually energy-equilibrate with the neutrals, which are assumed to be cold. The evolution of the electron distribution function in time and velocity space is calculated for each gas. This model is approximately correct for the afterglow period of an electrical discharge in a monatomic gas. It is possible to calculate a time which is a measure of the decay time of the electron energy in an afterglow plasma     

Harmonics Analysis of the Electron Component of rf Bulk Plasmas. I. Theoretical Base

rf discharges are increasingly used in low pressure plasma processing, i.e. for etching, film deposition and sputtering. The modelling of such discharges is a very complex task, especially dependent on discharge conditions, however of large importance for the insight into the main physical processes and thus for their control to improve the final results. One main important aspect is the determination of the electron velocity distribution function and of relevant rate and transport coefficients. The paper contributes to the treatment of this problem. In the first part a systematic Fourier expansion of the kinetic equation and of the consistent particle, energy and momentum balance equation is described. Then, a mathematical analysis of the resulting ordinary differential equation system for the coefficients of the Fourier expansion is performed. Based upon this we succeeded to develop a numerical approach to calculate the physical relevant solution of this system. By this approach in addition to the harmonics of the distribution function that of relevant macroscopic quantities, as transport coefficients and collision frequencies, can be determined. In the second part of this paper this method will be applied to investigate the bulk plasma of a rf discharge in molecular hydrogen.     

Electron clouds around charged particulates

Poisson’s equation is used to derive an expression for the characteristics of the Debye electron cloud around a multiply charged particle. It is shown that the limiting dimension of the Debye cloud (for an infinitely large charge of the particles) varies from 0.7 to 2.2 Debye radii, depending on the geometry of the problem. A first-principles modeling of the dynamics of many electrons moving around an immobile charged center is carried out. It is shown that a metastable state which relaxes at least one thousand times more slowly than would follow from the kinetic theory is established. Calculations show that in this metastable state (which is far from thermodynamic equilibrium) there is a detailed balance of electron transitions from a state with one energy to another. The distribution of electrons over kinetic energy has a Maxwellian form, while the distribution over total energy is radically different from the Boltzmann distribution and is close to that which was established previously by the authors for a plasma of singly charged ions. The potential distribution around the immobile charge in the metastable plasma is obtained. Zh. Tekh. Fiz. 69, 53–57 (January 1999)     

The effect of sawtooth oscillations on the alpha particle distribution and energy balance in the ITER plasma

The mixing of toroidal plasma under the conditions of sawtooth oscillations is considered using the Kadomtsev model. A new mixing formula for the averaged distribution function of fast transit and trapped particles is proposed in the methodology of a kinetic equation averaged over drift trajectories. The proposed formula generalizes the known results for the case of non-circular magnetic surfaces, an arbitrary aspect ratio, and charged particle drift trajectories significantly deviating from the magnetic surfaces. The formula is applicable for a sufficiently wide class of instabilities. The 3D kinetic equation is numerically solved using the FPP- 3D computation code for parameters close to the ITER inductive scenario. The alpha particle distribution function and the power introduced by alpha particles in plasma when sawtooth oscillations occur are calculated. It is shown that such oscillations may change the energy input of a thermonuclear reaction in certain areas by several times.     

Kinetic theory of stochastically heated RF capacitive discharges

Stochastic sheath heating is the dominant heating mechanism at low pressures for radio frequency (RF) capacitive discharges. It produces an electron energy probability distribution function (EEPF) that approximates a two-temperature Maxwellian, as seen in both experiments and numerical simulations. We have used the fundamental kinetic equation to obtain a space- and time-averaged kinetic equation. We assume that electrons with the x component kinetic energy lower than a certain threshold Φ are prevented from interacting with the sheath heating fields. With these approximations and either a knowledge of the central density or an ansatz on Φ, we obtain a self-consistent solution for the quasiequilibrium discharge parameters valid for low pressures in argon. The results are compared to those found in experiments, yielding reasonable agreement     

Modelling of Secondary Electron Impact on the Self-Consistent RF Bulk Plasma Description

The impact of a secondary electron beam, generated at the electrodes and accelerated in the sheaths, on the self-consistent treatment of the electron behaviour in an rf bulk plasma has been investigated by a parametric study. Source of electrons in the plasma are collisional ionization and secondary electron injection. Electrons are lost by ambipolar diffusion to the electrodes of a parallel plate rf discharge configuration. The non-stationary Boltzmann equation is used to determine self-consistently the rf field amplitude necessary for maintaining the steady-state rf bulk plasma as well as the time resolved behaviour of the electron energy distribution function and of all contributions to the electron particle and power balance, at given source rate and energy distribution of secondary electron injection.     

射频辉光放电CH4等离子体中电子的能量分布

通过求解Lorentz简化的玻尔兹曼方程,得到射频放电CH₄等离子体中电子的能量分布函数.求解过程中使用一个简化的射频电场模型代替泊松方程求解放电电场.共计包含6类环境气体及27种电子碰撞反应.通过EEDF对等离子体中的电子反应率系数、电子平均能量、电子的传输率系数等进行求解分析.结果表明,在等离子体鞘层区域电子能量具有Maxwell分布形式,在正柱区域具有Druyvesteyn分布形式.最高电子能量和最大反应率系数出现在鞘层区域.电子的迁移率系数和扩散率系数随射频周期的演化时空分布不均匀.     

Electron boltzmann kinetic equation averaged over fast electron bouncing and pitch-angle scattering for fast modeling of electron cyclotron resonance discharge

The electron distribution function (EDF) in an electron cyclotron resonance (ECR) discharge is far from Maxwellian. The self-consistent simulation of ECR discharges requires a calculation of the EDF on every magnetic line for various ion density profiles. The straightforward self-consistent simulation of ECR discharges using the Monte Carlo technique for the EDF calculation is very computer time expensive, since the electron and ion time scales are very different. An electron Boltzmann kinetic equation averaged over the fast electron bouncing and pitch-angle scattering was derived in order to develop an effective and operative tool for the fast modeling (FM) of low-pressure ECR discharges. An analytical solution for the EDF in a loss cone was derived. To check the validity of the FM, one-dimensional (in coordinate) and two-dimensional (in velocity) Monte Carlo simulation codes were developed. The validity of the fast modeling method is proved by comparison with the Monte Carlo simulations. The complete system of equations for FM is presented and ready for use in a comprehensive study of ECR discharges. The variations of plasma density and of wall and sheath potentials are analyzed by solving a self-consistent set of equations for the EDF.     

Electron Energy Distribution and Optical Characteristics of a Hollow Cathode Discharge

A model is proposed for the formation of the electron energy distribution in a hollow cathode discharge. On the basis of this model, an integral equation has been derived to calculate the electron distribution function over the entire cathode cavity volume. The equation holds true for both the isotropic distribution, and the case when the local distribution function is anisotropic. Solutions of the kinetic equation obtained are presented, for electron energies over 2–3 eV and up to the cathode fall potential. It is shown that the electron energy distribution function in this interval determines the optical characteristics of the hollow cathode discharge. A comparison is given of the calculated and measured radiation powers for the cases of the hollow cathode discharge in xenon and carbon dioxide. The discrepancy between the theoretical and experimental data does not exceed 20%.     

Electron temperature measurement in a surface-wave-produced argon plasma at intermediate pressures

The main objective of this work is to obtain the electron temperature in an argon surface-wave-produced plasma column at intermediate gas pressures. After proving that argon upper excited states remain in Excitation Saturation Balance, the value of electron temperature along the plasma column has been obtained using a modified Saha equation and a corrected Boltzmann-plot. Moreover, the electron energy distribution function has been verified to be nearly Maxwellian in a 0.8-2.8 torr intermediate pressure range. Received 24 July 2000 and Received in final form 19 January 2001     

Mikrowellendiagnostik der Rauschtemperatur im Stickstoffplasma der Schwachstromsäule

This paper covers the calculation and measurement of the noise temperature for a stationary and axial homogeneous column plasma of a nitrogen discharge. The calculations are based on an electron energy distribution function which was obtained by the solution of the Boltzmann equation for a nitrogen plasma. In the low current column of this plasma the noise temperature was measured, using a Dicke-radiometer. The comparison between the calculated and measured values of the noise temperature shows a good agreement in the considered parameter region. Using a Maxwellian distribution, which was fitted to the real nitrogen plasma by the aid of the formulation of an energy balance for the electrons, the noise temperature was additionally determined in a approximate way and compared to the first calculations.     

Velocity distributions in magnetron sputter

Results of the particle simulation of magnetron sputter are presented. Using a kinetic code, we obtain the spatial profiles of plasma density, potential, and velocity distribution function, along with the electron temperature, the ion density, the current density, and the deposition profiles at the anode surface. The result of simulation is compared with the Child-Langmuir law applied to the magnetron discharge and the global model. The velocity distribution function of electrons is Maxwellian, but that of ions is non-Maxwellian near the cathode with the majority in the energy range below 50 eV     

Temperature anisotropy relaxation of the one‐component plasma

The relaxation rate of a Maxwellian velocity distribution function that has an initially anisotropic temperature (T _‖≠T _⊥) is an important physical process in space and laboratory plasmas. It is also a canonical example of an energy transport process that can be used to test theory. Here, this rate is evaluated using molecular dynamics simulations of the one‐component plasma. Results are compared with the predictions of four kinetic theories; two treating the weakly coupled regime, namely (a) the Landau equation, and (b) the Lenard–Balescu equation, and two that attempt to extend the theory into the strongly coupled regime, namely (c) the effective potential theory and (d) the generalized Lenard–Balescu theory. The role of dynamic screening is studied, and is found to have a negligible influence on this transport rate. Oscillations and a delayed relaxation onset in the temperature profiles are observed at strong coupling, which are not described by the kinetic theories.     

DC column plasma kinetics in a longitudinal magnetic field

The cylindrical column plasma of a neon dc glow discharge under the influence of a weak longitudinal magnetic field is studied. An extended, fully self-consistent model of the column plasma has been used to determine the kinetic quantities of electrons, ions and excited atoms, the radial space charge field, and the axial electric field for given discharge conditions. The model includes a nonlocal kinetic treatment of the electrons by solving their spatially inhomogeneous kinetic equation, taking into account the radial space charge field and the axial magnetic field. The treatment is based on the two-term expansion of the velocity distribution and comprises the determination of its isotropic and anisotropic components in the axial, radial, and azimuthal direction. A transition from a distinctly nonlocal kinetic behavior of the electrons in the magnetic-field-free case to an almost local kinetic behavior has been found by increasing the magnetic field. The establishment of the electron cyclotron motion around the column axis increasingly restricts the radial electron energy transport and reduces the radial ambipolar current. The complex interaction of these transport phenomena with the alterations in the charge carrier production leads finally to a specific variation of the electric field components. The axial field increases by applying weak magnetic fields, however, decreases with increasingly higher magnetic fields. At higher magnetic fields, the radial space-charge field is considerably reduced     

Substantiation of the two-temperature kinetic model by comparing calculations within the kinetic and fluid models of the positive column plasma of a dc oxygen discharge

Results from kinetic and fluid simulations of the positive column plasma of a dc oxygen discharge are compared using commercial CFDRC software (), which enables one to perform numerical simulations in an arbitrary 3D geometry with the use of both the fluid equations for all the components (fluid model) and the kinetic equation for the electron energy distribution function (kinetic model). It is shown that, for both the local and nonlocal regimes of the formation of the electron energy distribution function (EEDF), the non-Maxwellian EEDF can satisfactorily be approximated by two groups of electrons. This allows one to take into account kinetic effects within the conventional fluid model in the simplest way by using the proposed two-temperature approximation of the nonequilibrium and nonlocal EEDF (2T fluid model).     

The Impact of Direct Ionization by Beam Electrons on the Electron Kinetics of the Beam Discharge Plasma in Molecular Hydrogen

In a recent paper the stationary beam plasma discharge in partially dissociated hydrogen was investigated where the electron component was described by the Boltzmann equation for a mixture of atomic and molecular hydrogen and the main heavy charged and neutral particles by balance equations. It was assumed that, via the quasilinear beam plasma interaction, the electron beam produces only the turbulent electric field whilst an additional production of plasma electrons due to direct ionization by the beam and thus a direct influence on the balances of charge carriers were neglected. Now the additional production of plasma electrons due to direct ionization by the beam is studied on the basis of a generalized Boltzmann equation but for the simpler model of a purely molecular hydrogen plasma. For experimentally obtainable values of the turbulence energy density, beam energy, beam ionization degree and electron life time the calculation of the electron energy distribution function and of the direct beam contribution to the electron particle balance shows a marked influence of the direct beam ionization with increasing degree of beam ionization.     

The Nonstationary Behaviour of the Electron Current Density in the Weakly Inonized Plasma at High Field Frequencies

Based upon former studies concerning the nonstationary electron kinetics in collision dominated, weakly ionized plasmas the phase delay between the ac electric field component and the resultant ac electron current density has been analysed, theoretically and experimentally, under the specific conditions of a microwave field superimposed to a dc discharge plasma column. The complex plasma conductivity and thus the phase delay has been calculated by solving an appropriate electron kinetic equation. The same quantities have been experimentally determined by using the microwave cavity which operates with different resonator modes. A comprehensive comparison between calculated and measured quantities for different neon discharge plasmas leads to a complete confirmation of the theoretical expectations.     

The influence of metastable atoms and the effect of the nonlocal character of the electron distribution on the characteristics of the positive column in an argon discharge

Commercial CFDRC software () is used to self-consistently simulate the plasma of the positive column (PC) of a dc discharge in argon. The software allows simulations in an arbitrary 3D geometry by using Poisson’s equation for the electric potential and fluid equations for the heavy components and by solving a nonlocal kinetic equation for electrons. It is shown that, in calculating the electron distribution function, the local approximation is almost always inapplicable under real conditions of a diffuse PC usually met in practice (pR<(5–10) cm Torr). The influence of metastable atoms, which can substantially affect the parameters of the PC plasma, is considered. It is shown that superelasic collisions play an important role in enriching the fast component of the electron distribution function and that the Penning ionization can result in an ascending volt-ampere characteristic of the positive column.