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.     

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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。     

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.     

The effects of ion-neutral collision frequency on the plasma sheath dynamics for oblique entrance of ions into the sheath

Recently it has been shown that the magnetic field has significant effects on the characteristics of ions which enter the collisionless plasma sheath with an oblique incidence angle. Here, we have investigated these effects in collisional plasma sheath. Considering the ion-neutral collision, the basic equations of fluid model in a plasma sheath have been numerically solved in the presence of an external magnetic field where the ion collision frequency has a power law dependency to the ion flow velocity. The results show that the effects of magnetic field on the plasma sheath dynamics strongly depend on the ion-neutral collision frequency and its dependency to ion flow velocity.     

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。     

Application of the Child-Langmuir law to magnetron dischargeplasmas

In order to investigate the influence of magnetic field on magnetron plasmas, the dependence of discharge-current density (J_d ) on magnetic flux densities (B) perpendicular to the electric field over the whole discharge space in a magnetron apparatus with flat multi-targets has been studied. The dependencies on working gas pressures (P) and voltages (V) have been also studied in the configuration. From the experimental results, it was found that J_d was approximately proportional to B², P² and V^3/2. The sheath structure in the apparatus has been investigated as a function of B by both a probe method and a spectroscopic one. The cathode-fall thickness was roughly proportional to B^-1. From a numerical analysis on kinetics of γ electrons emitted from the cathode surface, it was found that the dependence of J_d on B could be explained by the Child-Langmuir law, in which the sheath thickness was replaced by the equivalent electron Larmor radius     

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.     

Comparison of the directional characteristics of swift ion excitation for two small biomolecules: glycine and alanine

The characteristics of ions that enter the plasma sheath with an oblique incident angle have been investigated in the presence of an external magnetic field. The ion dynamics in a collisional and collisionless magnetized plasma sheath have been numerically calculated by using a fluid model. Several values for the ion velocity at the sheath edge, orientation and strength of the magnetic field and the ion-neutral collision frequency have been considered. The results show that in a collisionless magnetized plasma sheath, the behaviour of ions that obliquely enter the sheath with some specific velocities at the sheath edge and at some specific orientations and strengths of magnetic field, is more complicated than that of ions with normal entrance angles. For the oblique entrance of ions, the weak magnetic fields cause some fluctuations in ion velocity around its boundary value, i.e. the ion velocity does not accelerate. However, the numerical calculations show that the ion dynamics in a collisional magnetized plasma sheath are the same for both normal and inclined entrance of ions into the sheath.     

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).     

基于库仑定律的二极管空间电荷限制效应研究

真空二极管是电子束源装置中的一个关键部件, 其阴极发射的电子束进入二极管阴阳极间隙区会产生很强的空间电荷限制效应. 本文针对平板真空二极管中的空间电荷限制效应物理模型, 从库仑定律的积分形式出发, 选取了不同参数的阴极半径与阴阳极间隙距离之比, 计算得到了真空二极管阴阳极间隙区的电场分布情况, 从而避免了采用Poisson方程求解时需要处理复杂的非线性偏微分方程这一问题. 结果表明, 基于库仑定律的计算结果与从Child-Langmuir定律分析得到的结果仅相差一个与二 极管极板边界条件有关的修正量. 当二极管阴极半径与阴阳极间隙距离之比 R/D>10时,一维空间电荷限制定律将是一个较好的近似结果,当 R/D较小时, 用该定律计算误差较大,在实际物理问题中需要谨慎使用.     

脉冲束流引出时的等离子体鞘层变化

采用一维无碰撞的动力学鞘层模型计算了脉冲等离子体在恒压引出时的等离子体鞘层厚度变化,分别对短脉冲和长脉冲放电时的离子源发射面演变进行了分析。结果表明:对于短脉冲放电,发射面位置的变化相对等离子体密度的变化存在一定时间的延迟;对于长脉冲的上升沿和直流放电的开启阶段,鞘层厚度变化的速度与离子初始速度相关,稳定后发射面的位置与离子初始速度和等离子体密度的乘积相关。     

Experimental studies of plasma sheath near meshes of different transmissivity

<正>The measurements of the potential distributions in the boundary layer near meshes with different mesh spacing were conducted in weakly collisional plasmas using a fine-structured emissive probe and the results of the sheath thickness and electric field at the sheath-presheath edge were compared with theoretical models of collisional presheath and collisionless sheath.It was shown that,because the meshes are partially transparent to ions,the sheath is thinner and the electric field is stronger for the mesh of higher transmissivity,owing to the increased ion density in the sheath contributed from the ions transmitted from the other side of the mesh.However,the potential profiles in the presheath remain almost the same for different meshes except for the shift of the sheath-presheath edge.The thickness of the sheath decreases while the electric field at the edge increases with the increase of the neutral gas pressure.Furthermore, depending on the pressure,the measured electric fields at the edge are close to that from the models of a transition region.     

Analytic expression for the electric potential in the plasma sheath

An expression for the spatial dependence of the electric potential in a collisionless and source-free planar plasma sheath is presented. This expression is derived by analogy with Child's law and approaches Child's law asymptotically as the potential drop φ_W across the sheath becomes large, |eφ_W/kT_e|>10⁴. Here k is Boltzmann's constant, T_e is the electron temperature, and e is the electronic charge. Comparison with numerical solutions of the model equations indicates that the sheath thickness and potential variation predicted by this improved Child's law are accurate for |eφ_W/kT_e|>10. In contrast, the authors find that Child's law is accurate only when |e φ_W/kT_e|>10⁴     

射频等离子体鞘层动力学模型

在流体力学方程的基础上建立了一种自洽的无碰撞射频等离子体鞘层动力学模型.这种自洽性包含两个方面:一方面,由于考虑了瞬时鞘层电场对离子运动的影响,因此该模型适用于描述任意频率段的射频鞘层演化过程;另一方面,在模型中采用等效电路方法来自洽地确定极板上的瞬时电位与瞬时鞘层厚度之间的关系.采用数值方法模拟出鞘层的瞬时厚度及极板的瞬时电位变化、鞘层内离子密度和电场强度等物理量的时空变化.结果表明,当射频场的频率小于或等于离子等离子体频率时,离子流密度明显地随时间变化 关键词： 射频 离子 鞘层 流体力学     

Anomalous capacitive sheath with deep radio-frequency electric-field penetration

A novel nonlinear effect of anomalously deep penetration of an external radio-frequency electric field into a plasma is described. A self-consistent kinetic treatment reveals a transition region between the sheath and the plasma. Because of the electron velocity modulation in the sheath, bunches in the energetic electron density are formed in the transition region adjacent to the sheath. The width of the region is of order V(T)/omega, where V(T) is the electron thermal velocity, and omega is the frequency of the electric field. The presence of the electric field in the transition region results in a collisionless cooling of the energetic electrons and an additional heating of the cold electrons.     

Transition region between a nonequilibrium plasma and a negative electrode

A model is developed that describes the transition region between a quasineutral plasma and a planar negative electrode and in which the electron velocity distribution is represented as the sum of two Maxwellian distributions with different temperatures or as the sum of a Maxwellian distribution and distribution corresponding to an electron beam directed toward the electrode. Criteria for the formation of a sheath of positive space charge and a secondary plasma in the transition region are derived. An analysis is made of the dependence of the structure of the transition region on the parameters of the electron distribution, the space charge density distribution in the sheath, and the density of the ion current to the electrode. The criteria obtained are compared with the Bohm criterion.     

Presheath and Space Charge Sheath Characteristics in Front of a Floating Wall

The sheath thicknesses and numbers of collisions from an infinite fluid model of the space between an atomic cold gas plasma and a planar wall are reported in dependence on the normalized ionisation, ion elastic and charge exchange collision frequences. The Bohm criterion validity, ion heating in double layer electric field, volume radiative recombination and related questions are discussed.     

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)     

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.