Comment on 'On the consistency of the collisionless

Response to “Comment on ‘On the consistency of the collisionless sheath model’” [Phys. Plasmas 10, 3437 (2003)] Valery Godyak and Natalia Sternberg Citation: Phys. Plasmas 10, 3439 (2003); doi: 10.1063/1.1589487 View online: http://dx.doi.org/10.1063/1.1589487 View Table of Contents: http://pop.aip.org/resource/1/PHPAEN/v10/i8 Published by the AIP Publishing LLC.

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PHYSICS OF PLASMAS

VOLUME 10, NUMBER 8

AUGUST 2003

Response to ‘‘Comment on ‘On the consistency of the collisionless sheath model’ ’’ †Phys. Plasmas 10, 3437 „2003…‡ Valery Godyak Osram Sylvania, Beverly, Massachusetts 01915

Natalia Sternberg Clark University, Worcester, Massachusetts 01610

共Received 12 May 2003; accepted 13 May 2003兲 关DOI: 10.1063/1.1589487兴

Our Letter1 is concerned with the internal inconsistency of the classical collisionless sheath model given by d 2 ␩ /d ␰ 2 ⫽ 共 1⫹2 ␩ 兲 ⫺1/2⫺exp共 ⫺ ␩ 兲

共A兲

with the zero-field boundary condition for ␭ D1 ⫽0. In Ref. 1, we have shown that Eq. 共A兲 yields an infinitely large sheath consisting mainly of quasi-neutral plasma. Furthermore, ionization is infinitely larger in that sheath than in the adjacent plasma. Thus, the sheath described by Eq. 共A兲 contradicts the common perception of a sheath and the main assumption the sheath model is based upon, namely, the absence of ionization in the sheath. The comment by Tskhakaya and Shukla, does not address any of those critical issues. Instead, they use blurry rhetoric and unsubstantiated claims which misrepresent our Letter:1

M v d v /dx⫹edV/dx⫹M zn e /n p ⫽0,

共D兲

d 2 V/dx 2 ⫽4 ␲ e 共 n e ⫺n p 兲 .

共E兲

Ld/ 共 ˜n p u 兲 /dx⫽0.57 ˜n e ,

共3⬘兲

˜ e u/n ˜ p ⫽0. Ludu/dx⫺Ld ␩ /dx⫹0.57n

共4⬘兲

There is no more ‘‘small parameter ␭ D1 /L,’’ and the only way to neglect the ionization terms in (3 ⬘ ) and (4 ⬘ ) is neglecting the electron density (n ˜ e ⫽0). The authors of the comment also fail to see that their Eqs. 共2兲–共4兲 together with the Poisson equation describe the full plasma-wall problem from the center ␰⫽0 up to the wall ␰ ⫽ ␰ w ⫽L/␭ D1 . For ␭ D1 /L⫽0 these equations are completely equivalent to our Eqs. 共B兲–共E兲. Therefore, neglecting

The authors of the comment ignore our analysis1 that shows inconsistency in the derivation of the sheath model 共A兲 from the plasma-wall equations:

1070-664X/2003/10(8)/3439/2/$20.00

共C兲

According to Eq. 共C兲, the assumption of a constant ion flux, d(n p v )/dx⫽0 for the sheath region requires the neglect of the electron density in the sheath, since the ionization frequency z⬇0.57v s /L is the eigenvalue of the plasma-wall problem and cannot be neglected. In our plasma-wall model, which assumes dT e /dx⫽0, the ionization frequency z is the same for both the plasma and the sheath, since z is defined solely by the electron temperature T e , gas pressure and the ionization cross section. The sheath model 共A兲 is obtained from Eqs. 共B兲–共E兲 by neglecting electrons in the continuity Eq. 共C兲 and in the momentum Eq. 共D兲, but retaining them in the Poisson Eq. 共E兲. This results in an inconsistent sheath Eq. 共A兲. A consistent derivation of the sheath equations from Eqs. 共B兲–共E兲 requires the neglect of the electron density not only in the continuity and the momentum equations, but also in the Poisson equation. Such approach is fully consistent with the generally accepted sheath definition given by Bohm: ‘‘the sheath region, characterized by negligible electron density.’’ 2 The authors of the comment argue that in the continuity and momentum equations written in the sheath coordinate ( ␰ ⫽x/␭ D1 ) one can neglect the terms that contain ␭ D1 /L ⬇0.01, in order to obtain the sheath model 共A兲. Concentrating on the ‘‘small parameter ␭ D1 /LⰆ1’’ in the ionization terms in their Eqs. 共3兲 and 共4兲, they do not notice this ‘‘small parameter’’ in all other terms of these equations. Remind, ␰ ⫽x/␭ D1 . Indeed, dividing their Eqs. 共3兲 and 共4兲 by ␭ D1 ⫽0, one obtains

共1兲 The authors of the comment claim that in our Letter,1 we are ‘‘contradicting our own results,’’ but nowhere in the comment is this claim substantiated. 共2兲 Our results are based on rigorous mathematical proofs, which the authors of the comment do not attempt to disprove. 共3兲 The authors of the comment support the claim that their Eq. 共5兲 ‘‘is perfectly valid for the investigation of the collisionless sheath’’ for ␭ D1 /LⰆ1 by quoting Allen’s comment 共their Ref. 2兲, yet in his comment, Allen insists that this equation ‘‘does not apply to the case of nonzero ␭ D .’’ Thus, Allen’s comment is not relevant to our letter where ␭ D1 ⫽0, nor is it relevant to the comment where ␭ D1 /L⫽0.01. 共4兲 Equation 共6兲 in the comment shows that, in sheath coordinates, the sheath edge is at ⫺⬁, but the authors of the comment fail to see that for ␭ D1 ⫽0 共the case they are discussing兲, the sheath coordinates and the real coordinates are equivalent. This leads precisely to our result, which states that any point within the sheath given by model 共A兲 is infinitely remote from the sheath edge. One can obtain our result by integrating their Eq. 共5兲.

n e ⫽n 0 exp共 eV/kT e 兲 ,

d 共 n p v 兲 /dx⫽zn e ,

共B兲 3439

© 2003 American Institute of Physics

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3440

Phys. Plasmas, Vol. 10, No. 8, August 2003

V. Godyak and N. Sternberg

␭ D1 /L in their continuity equation 共3兲 would mean that the ion flux is constant everywhere, from the center up to the wall, including the bulk plasma. This certainly does not hold for the plasma-wall problem considered here, as can be seen from Eq. 共C兲. Thus, for ␭ D1 /L⫽0, the term ␭ D1 /L can never be neglected, no matter how small it is. The only region where the ion flux could become constant is the region where the electron density is negligible. This region is precisely the ion sheath where n e Ⰶn p . Beyond the ion sheath, the ion and electron densities are comparable (n e ⬇n p ), and the sheath model 共A兲 becomes inconsistent, as it was shown in Ref. 1. The inconsistency of the sheath model 共A兲 is not a matter of judgment, interpretation or taste, but a matter of mathematics. Rigorous mathematical analysis1 of the sheath model 共A兲 with the zero-field boundary condition reveals the striking result that has been overlooked for a long time: the total ionization in the sheath Q is infinitely larger than the total ionization P in the adjacent plasma: Q/ P⫽

␭ D1 n 1 L具n典

冕

␩w

0

exp共 ⫺ ␩ 兲

冉 冊 d␩ d␰

⫺1

d ␩ →⬁,

small but nonzero, and any such ␭ D1 /L yields Q/ P→⬁ if Eq. 共A兲 is used. On the other hand, the sheath model 共A兲 is based on the assumption Q/ P⫽0. This contradiction was pointed out in our Letter, but not mentioned in the comment at all. Instead, the authors of the comment divert to a number of disjoint statements about asymptotic matching, which show that they are not quite familiar with the subject. In our Letter,1 we clearly stated that ␭ D1 /L⫽0, while asymptotic matching deals with the case ␭ D1 ⫽0. The two cases are fundamentally different, since for ␭ D1 ⫽0, Eqs. 共2兲–共4兲 in sheath coordinates are not equivalent to the corresponding equations 共B兲–共D兲 in real coordinates, while for ␭ D1 ⫽0 the corresponding equations are equivalent. The equivalency in the sheath coordinates ( ␰ ⫽x/␭ D1 ) is lost in the limiting process ␭ D1 →0. We see no point in discussing the asymptotic approach here and instead refer to our latest paper3 where we extensively studied the subject and its application to the plasma-sheath problem. V. Godyak and N. Sternberg, Phys. Plasmas 9, 4427 共2002兲. D. Bohm, The Characteristics of Electrical Discharges in Magnetic Fields, edited by A. Guthrie and R. K. Wakerlink 共McGraw–Hill, New York, 1949兲, Chap. 3. 3 N. Sternberg and V. Godyak, IEEE Trans. Plasma Sci., Special Issue on Modeling of Collisional and Near-Collisional Low Temperature Plasmas 共to be published兲. 1

共F兲

where 具 n 典 is the averaged plasma density, n 1 is the ion density at the plasma-sheath interface, (n 1 / 具 n 典 ⬇0.8), and ␩ w is the normalized wall potential. In the real world, ␭ D1 /L is

2

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