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71 SHEATH ENTRY CRITERION FROM THE ABSORBING BOUNDARY

Prabhat Kumar Dubey Christian Eminent College, Indore

Abstract - In the present paper the structure of a electrostatic sheath forming plasmas flowing into an absorbing boundary is explained using dispersion relation. We discuss the Bohm criterion and dispersion relation.

1 INTRODUCTION

Plasma is an ionized gas that consists of ions, electrons, and neutral atoms or molecules, they maintain charge neutrality. Electrons and ions are so close enough that each of them can influence many nearby charged particles within a radius called Debye screening length.

Due to these charged particles in plasma responds collectively to the external fields. It have high density of free moving ions and electrons, hence plasma is highly electrically conductive. Except at boundary regions between plasma and wall, plasma contains the same amount of positive and negative charges. There is no space charge within the bulk of the plasma.

The function of plasma sheath in a multiple ion species plasma is to ensures the quasineutrality of the bulk plasma by localizing the maximum potential drop close to the absorbing boundary. In various magnetized and unmagnetized conditions the structure of electrostatic sheath is governed by the dispersion properties of the plasma which are collectively determined by the parameters of all the ion species constituting the plasma.

2 BOHM CRITERION

The standard Bohm criterion by its formulation as an inequality, duly accounts for all possible cases of sheath formation covering the complete range of sheath thickness. While in its widely applied marginal (or equality) form, the criterion describes the cases where the sheath thicknessis sufficiently large in comparison with the Debye length, its inequality form refers to a range of more realistic cases where a vanishing Debye length is only an approximation. The latter form becomes important and must be applied when the exact values of the entry flow velocities, corresponding to a particular sheath scaling. It is however not readily clear that what prescription must be applied in order estimate exact entry value of ion flow velocity that simultaneously satisfy the Bohm criterion for a prescribed sheath dimension, for example, in an application where only limited flexibility of dimensions is allowed. This, moreover, needs to be determined in case of a multiple-ion- specie plasma using the generalized Bohm criterion such that expected relative ion velocities between ion species and the relative density concentrations of them can suitably be accommodated.

In many experiments analyzing, for example, the entry velocity of the ion species into the electrostatic sheath, the plasma conditions providing larger sheath thickness is preferred so that an increased spatial resolution is achievable and better measurements can be made in the laboratory conditions. While such a selection facilitates desired convenience in the measurements, it can easily be concluded that the corresponding results may already

72 be limited in its capacity if the marginal form of the criterion is still considered for the interpretation. This might at the same time also exclude reasonable analysis of many general cases where the inequality form of the criterion assumes importance and must be considered by finding corresponding entry flow velocities in order to explore more diverse allowable parameter regimes.

For the general cases, where the sheath structure extends only a few Debye lengths andλD

l_sis nonzero in comparison with the sheath thickness unlike marginal limit, the criterionmust be satisfied in its inequality form where the entry velocities are indeed determined bythe exact form of the left and right hand sides of the inequality. However out of these two only one, i.e., the limiting value is well defined. The entry velocities are thus correlated, particularly, to the value of the quantities entering the other side of the expression of the criterion which varies from its value in the cases when the criterion is satisfied in its marginal form. In this paper we discuss a prescription for the sheath structure in a multipleionspecies plasma that accounts for the cases where particular sets of plasma parameters are necessary to satisfy the exact form of the criterion for in the non- marginal (inequality) range. Using the dispersion relations of the electrostatic modes which are often detectable invarious experiments and computer simulations it is shown that for smaller sheath thickness the entry velocities would require to be larger in magnitude while the plasma Debye length essentially remains the limiting value of the sheath thickness . For this limiting case the required entry velocities tend to approach infinity and therefore no sheath structure thin nerthan the limiting value of Debye length would be possible. The present analysis is however limited to the smallness of the scale lengths below which kinetic effects, excluded from the present treatment, become important. The present analysis also uses only unmagnetizedplasma as a model, for the simplicity of the formulation, however the boundary layers forming due to objects moving in a magnetized plasma may also be addressable by its more general magnetized form.

3 DISPERSION RELATION

Here the generalized criterion is discussed with reference to its validity in the plasmas with ion species streaming with supersonic velocities in an electrostatic sheath. In this paper the sheath structure in plasmas with distinct flow velocities of ion species is discussed showing the dependence of sheath scale length in various entry velocity combinations (including the supersonic velocities) of a two-ion-species plasma. More common cases, of sheath forming with nearly same entry velocity values of two ion species, are discussed with some quantitative analysis.

∈ ω

, k = 1 −k N Cie2

i=1 A − Ui ic2α_ǁ² D − Ui _ic²d ǁ ǁ = 0

The above relation describes the dispersion of magneto-acoustic modes in a fluid plasma where functions A_iandD_i have following structure,

A = di iǁ ǁα_⊥²− d + diǁ α_{iǁ ⊥ ǁ}α_⊥+ di⊥ _⊥α_ǁ² And

D = d_i d_{iǁ ǁ i⊥⊥}− d d_{i ǁǁ⊥ i⊥ǁ} The matrix elements d_i are,

diǁ ǁ= U_i²− 3 + fiq α_ǁ²+ U_i² U_i²− U_ic² α_⊥²C_i²

d_{i⊥ ⊥} = U_i²− U_i²+ U_ic²

U_i²− U_ic² α_ǁ²+ 2 + U_i²+ 2U_ic² U_i²− 4U_ic² + f_iq α_⊥² C_i² diǁ ⊥ = −1 + U_i²+ U_ic² U_i²− U_ic² +fiqC_i²αǁα⊥

d_{i⊥ ǁ} = −1 + U_i² U_i²− U_ic² +f_iqC_i²α_ǁα_⊥ Where we have defined

Ui2= Ω k ²− ω − kViα 2 k²

73 Ωi= ω − 𝐤 ∙ 𝐕𝐢= ω − kV_iα ,

U_ic² = Ωic k ², P_iq² = 3U_i²− 2C_iq²

C_iq² = −2C_iq² α_ǁ²+ U_i² U_i²− U_ic² α⊥2 .

The Chodura’s ion flow condition at an absorbing boundary, defined as condition for outgoing characteristics in is closely related to the onset condition of the electrostatic sheath in front of the target. This onset condition may be obtained in two different limits (namely, the quasineutral and nonquasineutral potential variation), from the dielectric functionk of a small amplitude electrostatic wave with frequency ω and wave vector k = k in a magnetized plasma. The undisturbed plasma is assumed to be homogeneous on the length scale 1/k of the disturbance. The disturbance exp i(k·x− ωt) is assumed to be collisionless on this scale and quasineutral. Disturbed electrons are assumed to be isothermal. Linearization of the ion Equation for a small disturbance gives the dispersion relation.

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