Sheath thickness evaluation for collisionless or weakly collisionalbounded plasmas

The widely used Child-Langmuir law for sheath thickness evaluation in semi-infinite collisionless plasmas makes the assumptions of quasi-neutrality (n_e=n_i) and zero electric field intensity E=0 at the sheath edge, as well as applying the Bohm criterion for ions entering the sheath. However, through a whole region fluid model, Poisson's equation has been solved numerically for the steady-state solution through the sheath and presheath without these assumptions. With the sheath edge defined, as in the Child-Langmuir law, at the place where the ion velocity is equal to the Bohm velocity, the sheath thickness of a bounded collisionless or weakly collisional plasma has been found with this model in some cases to be much larger than that obtained with the Child-Langmuir Law. The sheath thickness discrepancy is significant under conditions found in low pressure high density plasma (HDP) tools for plasma processing. Results presented indicate that the sheath thickness is very sensitive to the electric field and space charge density at the sheath edge. The electric field and space charge density can be successfully estimated by an intermediate scale matching method, and are used to derive a modified expression for the potential in the sheath that can be solved numerically for sheath thickness. With these results, the matching problem, arising when sheath and plasma are modeled separately, can be overcome     

Plasma-sheath transition in the magnetized plasma-wall problem for collisionless ions

A model for the collisionless plasma-wall problem under the action of an applied magnetic field is developed. The behavior of its solution is examined and found to be qualitatively consistent with experiment. The plasma and the sheath are then modeled separately to obtain the position of the quasi-neutral plasma boundary and the position of the edge of the electron-free sheath. It is shown that the plasma boundary can be specified as the point where the component of the ion velocity normal to the wall reaches the ion sound speed (Bohm criterion), and the sheath edge is specified as the point corresponding to Godyak's condition for the electric field. Studying the behavior near the plasma boundary and the sheath edge, the plasma solution and the solution of the space charge region are patched together to approximate the solution of the plasma-wall problem.     

Comments on "On asymptotic matching and the sheath Edge"

In Sternberg and Godyak (2003), the authors claim that the sheath edge obtained through asymptotic matching is the edge of the electron free ion sheath characterized by Godyak's "strong" electric field |E|=kT/sub e//(e/spl lambda//sub D/). I present a careful re-analysis of the same problem and show that the paper is incorrect. The "intermediate region" of asymptotic analysis has an extremely narrow validity range in potential space and does not contain the ion-electron sheath. Consequently, in asymptotic theory, the sheath edge is uniquely defined by the transition from the quasi-neutral plasma to the ion-electron sheath. It may equivalently be characterized by the Bohm criterion or by a "medium" electric field |E|/spl sim/kT/sub e//(eL/sup 3/5//spl lambda//sub D//sup 2/5/) mediating between strong sheath fields |E|/spl sim/kT/sub e//(e/spl lambda//sub D/) and weak plasma fields |E|/spl sim/kT/sub e//(eL).     

Theory of self-organized dust sheaths in edge plasmas

A concept of self-organisation is used to derive the structure of dust sheath close to the wall taking into account self-consistently dust charges and dust distribution together with electron, ion distributions and ion drift. It is shown that instead of Bohm criterion the dust sheath is determined by other criteria allowing Mach number to be in restricted value ranges. The total structure of the dust sheath in dimensionless variables is determined by the Mach number only. Dust sheaths can contain definite number of dust particles per unit area which are determined by the value of the Mach number. The dust sheaths change drastically the electrostatic potential distributions and are usually much thicker than the usual plasma sheaths. A theoretical model together with numerical computations gives all dust sheath parameter distributions and the global characteristics such as the total drop of the potential, the thickness of the sheath, the number of dust particles in the sheath and the criterion of dust levitation. The recent observations on “Textor 94” indicating the long time dust survival in edge region of the present tokamaks, the dust levitation and agglomeration in the volume of edge tokamak plasma are discussed.     

电子和负离子的反射运动对碰撞电负性磁鞘的影响

研究了鞘层中电子和负离子的反射运动对碰撞电负性磁鞘玻姆判据和鞘层结构的影响.通过理论推导得到了考虑鞘层中电子和负离子的反射运动时鞘层玻姆判据表达式,并通过数值模拟得到了电子和负离子采用玻尔兹曼模型和反射运动模型时离子马赫数的下限随参数的变化曲线以及鞘层中带电粒子密度的分布曲线.结果表明,电子和负离子的反射运动模型和玻尔兹曼模型离子马赫数的上限完全相同,下限表达式不同,反射运动模型中下限还与基板电势有关,且随着基板电势值的增加而增大,达到与玻尔兹曼分布中相同值后保持不变,随着鞘边负离子浓度和温度的不同达到最大值的速度不同;离子马赫数的下限在玻尔兹曼和反射运动模型中都随鞘边负离子浓度的增加和温度的降低而减小,只是在反射运动模型中的最大值要小;两种模型中离子马赫数的下限都随鞘边电场的增加而增加,但在玻尔兹曼模型中增加得更快最终值更大;两种模型离子马赫数的下限都随碰撞参数或磁场角度的增加而降低,但在玻尔兹曼模型中降低更快,随着碰撞参数或者磁场角度的增加两种模型中离子马赫数的下限趋于一致;当基板电势值较小时,电子和负离子的反射运动对鞘层结构影响较大,当基板电势值较大时电子和负离子反射运动对鞘层中带电粒子密度分布的影响很小.     

����������ɢ���н���ƽ�������ε��ʲ�ṹ

利用发射探针诊断了处于扩散等离子体流场中平板两侧的鞘层电势分布并给出鞘层厚度及边界电场。结合Child-Langmuir(CL)定律并利用一维流体模型,分析对比了鞘厚及边界电场。结果显示,上下游的鞘层呈现不对称结构,下游比上游厚,边界电场下游比上游弱,并且下游的预鞘还呈现类似虚阳极的结构。上下游CL鞘的厚度满足线性比例关系,比例系数约为1.2,且实验值与理论值非常接近。上下游CL鞘的边界电场与模型Te/(eλD)比较接近,上游Bohm鞘边界电场的实验结果与过渡区模型 较符合,但下游Bohm鞘的边界电场与现有模型有较大差异。     

等离子体中热阴极鞘层的结构

结合热阴极电子发射理论和等离子体鞘层流体方程研究了热阴极附近存在虚阴极结构时的等离子体鞘层问题,采用Sagdeev势的方法讨论了鞘层解和广义Bohm判据.结果表明,不同于普通的Bohm鞘,由于热阴极附近存在大量发射电子,影响整个等离子体鞘层结构,使得进入鞘层的离子临界Mach数不是独立的常数,而是与鞘层电位降等参数有关的物理量.临界Mach数随着鞘层电位降（从鞘边缘到虚阴极）先增大后降低,并且随着热阴极温度的升高单调增大.此外,在平板模型下有相当可观的残余热发射电子越过虚阴极鞘层进入等离子体. 关键词： 鞘层 热阴极发射 Bohm判据     

两种带电尘埃颗粒的等离子体鞘层玻姆判据

采用流体方程和尘埃充电自洽模型研究了鞘边含有两种尘埃颗粒的等离子体玻姆判据. 通过拟牛顿法数值模拟了鞘边两种尘埃颗粒的存在对尘埃自身充电以及离子马赫数的影响. 两种尘埃颗粒中含量较少的尘埃颗粒数密度的增加, 导致两种尘埃颗粒表面悬浮势一个降低, 一个升高. 含量较少的尘埃颗粒的数密度越多和半径越小, 都会导致离子马赫数增大. 另外鞘边无论何种尘埃颗粒的速度增加, 鞘边离子马赫数都将减小. 关键词： 等离子体鞘层 尘埃颗粒 玻姆判据     

Analytical solution for capacitive RF sheath

A self-consistent solution for the dynamics of a high voltage, capacitive radio frequency (RF) sheath driven by a sinusoidal current source is obtained under the assumptions of time-independent, collisionless ion motion and inertialess electrons. Expressions are obtained for the time-average ion and electron densities, electric field and potential within the sheath. The nonlinear oscillation motion of the electron sheath boundary and the nonlinear oscillating sheath voltage are also obtained. The effective sheath capacitance and conductance are also determined. It was found that: (1) the ion-sheath thickness S _m is √50/27 larger than a Child's law sheath for the DC voltage and ion current density; (2) the sheath capacitance per unit area for the fundamental voltage harmonic is 2.452 ϵ₀ /S_m, where ϵ₀ is the free space permittivity; (3) the ratio of the DC to peak value of the oscillating voltage is 54/125; (4) the second and third voltage harmonics are, respectively, 12.3 and 4.2% of the fundamental; and (5) the conductance per unit area for stochastic heating by the oscillating sheath is 2.98 (λ_D/S_m)^2/3 (e ²n₀/mu_e), where n ₀ is the ion density, λ_D is the Debye length at the plasma-sheath edge, and u_e is the mean electron speed     

具有非广延分布电子的碰撞等离子体磁鞘的结构

很多关于等离子体鞘层的研究工作都是基于电子满足经典的麦克斯韦速度分布函数,而等离子体中的粒子具有长程电磁相互作用,使用Tsallis提出的非广延分布来描述电子更为恰当.本文建立一个具有非广延分布电子的碰撞等离子体磁鞘模型,理论推导出受非广延参数q影响的玻姆判据,离子马赫数的下限数值会随着参数q的增大而减小.经过数值模拟,发现与具有麦克斯韦分布(q=1)电子的碰撞等离子体磁鞘对比,具有超广延分布(q<1)和亚广延分布(q>1)电子的碰撞等离子体磁鞘的结构各有不同,包括空间电势分布、离子电子密度分布、空间电荷密度分布.模拟结果显示非广延分布的参数q对碰撞等离子体磁鞘的结构具有不可忽略的影响.希望这些结论对相关的天体物理、等离子体边界问题的研究有参考价值.     

Ar+ and Xe+ velocities near the presheath-sheath boundary in an Ar/Xe discharge

The velocities of Ar+ and Xe+ ions near the presheath-sheath boundary in an Ar/Xe discharge are studied by particle-in-cell Monte Carlo simulation. For a pure argon discharge the argon ion has almost the same velocity profile as it does in the mixture of argon and xenon. Similarly, for a xenon discharge the xenon ion has almost the same velocity profile as it does in the mixture of argon and xenon. The ion speed at the sheath-presheath boundary is the same for an ion in a pure argon or xenon discharge and for the same ion in a mixture of argon and xenon. We conclude that, in our simulation, each ion reaches its own Bohm speed at the presheath-sheath interface.     

Two Electron Transfer and Stabilization in Slow O^6＋ and Rare-Gas Collisions

The stabilization ratios R for double-electron transfer, i.e., the cross section ratios of true double capture to total double-electron transfer, are measured in O^6＋ ＋He, Ne and Ar collisions at 6 keV/u. A high R value about 68% is obtained for the He target, while for the Ar target, the R value is only 8%. The high R value for the He target is due to the significant direct population of the （2l, nl′ ） configurations with high n. For the Ar target, the （quasi）symmetric configurations （3l, nl′） lead to the much lower R value. Neglecting the core effects, the O^6＋ ion can be taken as a bare ion C^6＋ except the occupied ls shell, and then the measured R values are compared with previous experimental results of C^6＋ projectile ions at Ne and Ar target, while the occupied ls shell for the C^6＋ ＋He collisions. similar impact velocity. It yields good agreement with the O^6＋ ＋He system results in a higher R value than that in     

The Double Sheath at the Plasma‐Wall Boundary

The relation between plasma parameters determining a steady‐state charged sheath at a plasma‐wall boundary was derived and analyzed using a fluid model, which accounted for inertia and partial pressure of both plasma components. The relation generalizes the well‐known Bohm criterion and, in particular, shows that a steadystate sheath may be formed in some cases where the Bohm criterion is not satisfied. Conditions allowing formation of a double sheath structure were formulated and analyzed. It was shown that the structure may be formed only if a) the generalized Bohm criterion is satisfied, and b) the Sagdeev potential has a minimum. The double sheath structure consists of two sub‐sheath of different polarity, which depends on the wall potential polarity and the relation between the plasma component temperatures and inertia. For positive wall potential, the sub‐sheath adjacent to the plasma (the sheath edge) is enriched by negative particles, and the sub‐sheath adjacent to the wall is enriched by positive particles, while the sub‐sheath polarities are reversed for negative wall potential. An analytical theory was formulated and illustrated by numerical solutions of the Poisson equation for special cases. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

尘埃等离子体鞘层的玻姆判据

采用流体方程和自洽电荷变化模型研究了尘埃等离子体鞘层的玻姆判据. 讨论了离子临界马赫数和尘埃粒子临界马赫数随尘埃密度变化的关系, 以及尘埃表面势随尘埃密度变化的趋势. 由于鞘边尘埃粒子的存在, 离子必需以大于声速的速度进入鞘层; 随尘埃密度的持续增加, 离子的临界马赫数增加到一个最大值后开始逐渐减小. 数值计算得到的结果满足Sagdeev势的定性分析. 关键词： 等离子体 鞘层 玻姆判据     

Dust particles in collisionless plasma sheath with arbitrary electron energy distribution function

Dust particles often appear in industrial plasmas as undesirable product of the plasma-wall interactions. Large particles of several micrometers in diameter are concentrated in a thin layer (the sheath) above the lower electrode of the rf driven parallel plate device, where the electric force is strong enough to compensate particle’s gravity. Experimental and theoretical uncertainties are significantly increased in the plasma sheath. Common models of dust charging in the plasma sheath suppose the Maxwellian electron energy distribution function (EEDF) in conjunction with a flux of cold ions satisfying classical Bohm criterion at the sheath edge. In this paper we generalize this model to arbitrary EEDF with adapted Bohm criterion. We limit our considerations to collisionless or slightly collisional plasma, where the EEDF inside the sheath is expressed through the EEDF in the plasma bulk. Derived theoretical formulas are incorporated into numerical model, describing collisionless radio frequency (rf) plasma sheath together with the electrical charge, various kinds of forces, balancing radius and oscillation frequency of particles.     

Comments on the Bohm Criterion and on the Determination of the Ion Velocity at the Sheath Edge in Non‐Maxwellian Plasma

This work is devoted to the study of the Bohm criterion in the general case of the electron energy distribution function (EEDF). Investigations are performed by means of a Monte Carlo integration method. We resolve the cold fluid equation system describing the ion motion within the sheath, assuming collisionless conditions, singly charged and mono kinetic incoming ions (BOHM model). Results confirm that the limit ion velocity at the sheath edge to assure a monotone electric field with a positive charge over the entire sheath is v_i ≥ (kT_e/M_i) or ε_i ≥ 1/3 <ε_e> in the case of Maxwellian electrons. We show that in the case of a Druyvesteyn electron energy distribution, this limit is larger, it is ε_i ≥ 0.6 <ε_e>. The study is also extended to other distributions functions. Because of the large controversy in recent publications, concerning the boundary conditions at the sheath entrance, we discuss the collisionless conditions at the sheath edge according to the plasma parameters. It is shown that in a collisionless sheath, the condition n_i(χ) ≥ n_e(χ) can be used to determine the limit ion velocity at the sheath edge of the negatively biased collector (Langmuir probe for instance) (© 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

New Applications of Langmuir Probes

In this work, two new possibilities for standard probe diagnostics are described. The first one can be used to study isotropic, collisionless low-pressure plasma in which the electron energy distribution function is close to a Maxwellian one. In such plasmas, the Boltzmann law, Bohm effect, and 3/2 power law are valid. Use of corresponding system of equations for cylindrical Langmuir probes allowed for measurements of probe sheath thicknesses and the mean ion mass. The solution of this task was provided by accurate probe diagnostics of inductive xenon plasma at pressure p = 2 mTorr that resulted in the determination of the Bohm coefficient C_BCyl = 1.22. The second possibility of probe diagnostics includes a method and device for evaluation of ion current density to a wall under a floating potential using a radially movable plane wall Langmuir probe simulator. This measurement in the same xenon plasma served as the basis for development of an ion source in which the given wall was represented by an ion extracting electrode of the ion extraction grid system.     

Numerical solution of momentum balance equations for plasmas with two ion species

In plasmas bounded by material surfaces the Bohm criterion has to be satisfied at the entrance of the Debye sheath near the surface. With a single ion species this constraint prescribes a boundary condition for the momentum balance equation governing the ion mass velocity. If, however, several ion species are present a generalization of the Bohm criterion does not provide enough number of boundary conditions. Additional “intermediate” conditions follow from the requirement that spatial derivatives of the ion velocities are finite everywhere within the plasma volume. The amount of such independent conditions is sufficient to determine, in an iterative way, also the position in the plasma where they have to be imposed. A numerical approach to find unique regular solutions of fluid motion equations, satisfying the generalized Bohm criterion at the plasma boundary, is elaborated and realized for the case of two ion species.     

Effect of Cathode and Anode Voltage on an Ion Sheath Thickness in a Magnetically Confined Diffusion Plasma

This article reports about the ion sheath thickness variation occurring in front of a negatively biased plate immersed in the target plasma region of a double plasma device. The target plasma is produced due to the local ionization of neutral gas by the high energetic electrons coming from the source region (main discharge region). It is observed that for an increase in cathode voltage (filament bias voltage) in the source region, the ion flux into the plate increases. As a result, the sheath at the plate contracts. Again, for an increase in source anode voltage (magnetic cage bias), the ion flux to the plate decreases. As a result, the sheath expands at the plate. The ion sheath formed at the separation grid of the device is found to expand for an increase in cathode voltage and it contracts for an increase in the anode voltage of the main discharge region. One important observation is that the applied anode bias can control the Bohm speed of the ions towards the separation grid. Furthermore, it is observed that the ion current collected by the separation grid is independent of changes in plasma density in the diffusion region but is highly dependent on the source plasma parameters. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

尘埃颗粒与等离子体鞘层相互作用的数值模拟

通过求解一维稳态的尘埃等离子体鞘层模型,得到了等离子体鞘层势、正离子密度、电子密度和尘埃颗粒密度随一维横向的分布,Bohm判据及鞘层边界无量纲Bohm速度随尘埃密度的变化曲线,尘埃颗粒的带电量和尘埃密度的关系,尘埃颗粒的温度对尘埃颗粒自身在鞘层中分布的影响。结果表明,随着尘埃密度的增加,鞘层的厚度在减小,鞘层内的电子密度在下降,而且尘埃颗粒的带电量也在逐渐减少;随着尘埃温度的增加,鞘层的厚度减小,电子密度下降,而且鞘层附近的尘埃颗粒在逐渐增多。