Plasma-Wall Interaction: Sheath and Pre-sheath

Transcription

1 Plasa-Wall Interaction: Sheath and Pre-sheath Under ost conditions, a very thin negative sheath appears in the vicinity of walls, due to accuulation of electrons on the wall. This is in turn necessitated by the need to reduce the electron flux well below the un-inhibited theral flux n e c e/, which would lead to very high negative current densities toward the wall. In the liit of an insulating wall, we ust have Γ e =Γ i n e c e/, and the flux equality iposes a specific sheath potential drop (see later. But even where there is net negative current (electron capture its agnitude is still low enough copared to the theral value that a negative sheath fors (soewhat against intuition. Any ion arriving at the sheath s edge will be accelerated by the sheath and captured by the wall. This non-zero ion flux requires, by continuity, a eans of accelerating ions fro conditions far fro the wall, unless these conditions already iply a strong wall-directed ion flux. Thus, there is norally a Pre-sheath where the plasa is still neutral, but the potential is already falling toward the wall. The specifics of the pre-sheath vary depending on the situation, and in ost cases conditions are D or D. In contrast, the sheath is alost always so thin as to be 1D, and its physics is sipler and universal, largely independent of outside conditions. For this reason we analyze first the sheath, and it will turn out that its entry condition provides the required wall boundary condition to use for the uch larger scale of the pre-sheath. The Sheath Because the sheath thickness (a few Debye lengths is usually uch less than any collisional ean free path, we ignore collisions in it. With no agnetic field (or with B not nearly parallel to the wall, the electron force balance is between pressure and potential gradients, leading to Boltzann equilibriu: e(φ φs n e = n es e e (1 Where s is the sheath s edge. Actually, this balance is also valid in the pre-sheath, although less rigorously. For ions with no scattering or ionizing effects: n i u i = n is u is ( u i i + eφ = u i is + eφ s ( and eliinating u i between ( and (: n i n is = ( 1+ e(φs φ i u is The net charge density ρ ch = e(n i n e can then be used in Poisson s equation d φ = dx ρ ch : ɛ 0 1

2 [ ] d φ = e n dx es e e(φ φs n is e (5 ɛ 0 1+ e(φs φ is and, actually, n es = n is since this ust connect with outside, neutral zone. We can siplify notation by defining non-diensional variables, i u ϕ = e(φ s φ 0 ; ξ = x ; λ Ds = e λ Ds ɛ0 e e n es ; M is = u is e (6 i This yields, d ϕ = e ϕ 1 + (7 dξ 1+ ϕ A first integral of (7 is obtained by ultiplication ties dϕ and integration of the RHS with dξ respect to ϕ: ( 1 dϕ = +e ϕ M + is dξ 1+ ϕ C (8 Mis The constant of integration C is obtained by iposing zero slope at s (but only on the agnified sheath scale!. At s we also have ϕ =0,so M is C = 1+M is (9 When taking the square root in (8, the ( sign is appropriate since dφ < 0 in the sheath: dξ ( dϕ dξ = M ϕ is 1+ 1 (1 e φ (10 Unfortunately, a second integration is not analytically feasible, but iportant inforation can be obtained by exaining the radicand of (10 near the sheath s edge, i.e. when ϕ 1: ( [ Mis 1+ ϕ 1 (1 e φ = M ϕ Mis is 1 ( ϕ + 1 ( ϕ ] (11 M is 8 Mis 16 Mis ( ϕ ϕ + ϕ 6 = 1 ( 1 1 ( 1 1 ϕ + ϕ + Mis Mis 6 The leading ter (in ϕ ust be positive or zero for real solutions near the sheath edge: is M is 1 (1 and this is called the Boh sheath criterion. Supersonic ion entry is uncoon, so we will accept the weaker criterion. M is = 1 (1 When this is satisfied, the next ter in (11 is 1 ϕ > 0, and the sheath potential profile is real. Thus, the ions enter the sheath at their (isotheral speed of sound, u is = e i (1

3 and this is the boundary condition needed for the outer (pre-sheath solution. exceeds unity (φ s φ e, the expression (10 siplifies (roughly to which can be integrated: dϕ dξ = M is M is Once ϕ ϕ = / ϕ 1/ (15 M is = 1 ϕ 1 dϕ = / dξ ϕ / = / ξ + D where, using ϕ =0ats, D = / ξ s 5/ ξ s ξ = ϕ / (16 In particular, at the wall ξ = ξ 5/ w, ϕ = ϕ w, so the sheath thickness is ξ s ξ w = ϕ / w. In physical variables, ( x s x w = 5/ / eδφsh (17 λ Ds e so the sheath increases in thickness with the / power of the sheath potential drop. Putting also in (17 the definition of λ Ds, we can reorganize it in the for e Δφ / sh j i = en es u is = ɛ 0 (18 9 i (x s x w which is the Child-Languir equation for space-charge liited current. The Collisional Pre-sheath An alternative derivation of the Boh criterion could be arrived at in the following way. Start with steady 1D flow and ipose continuity of ion flow and quasineutrality, d (ne u i = 0 and n e = n i dx And include oentu conservation for both ions and electrons, du i n e i u i dx + dp i dx = n eee x i n e ν in u i du e dp e n e i u e + = n e ee x e n e ν en u e dx dx Adding these equations and cobining with continuity, we can write, d dx ( in e u i + p i + p e n ek(t e+ti = 0 i n e ν in u i en e ν en u e

4 or, with T e T i, (n d e u i ( i u i + e i ν in (n e u i dx u i So, we get the sae sheath entry velocity u is = e i = u Boh Particle and energy flux to a wall (A Rando particle flux across a plane Assue a Maxwellian distribution: {}}{ ( (wx + wy + w / z f = n e (19 π The forward directed flux across a plane (say, the yz plane is, Γ x = dw y dw z w y= w z= 0 w w x f(w dw x dw y dw z (0 It is actually better to use spherical velocity conditions, where w x = wcosθ and dw x dw y dw z = d w =πw sinθdwdθ: Γ x = π f(wπw sinθcosθdθdw (1 w=0 θ=0 π/ π/ sin θ Integrate first on θ : sinθcosθdθ = d( = 1 ( 0 0 Γ x = π w=o f(ww dw = πn( π / 0 e w w dw Change variables: w = ζ, wdw = dζ Γ x = πn( π / 0 } ζe ζ dζ {{ 1 } = 1 n π

5 Γ x = n ( π This is usually written in ters of the ean value of the velocity c 1 π wf(wπw sinθdθdw, which, with a siilar calculation gives, n 0 0 c = 8 π ( By division of (5 and (6, we obtain Γ x = n c (note x is an arbitrary direction in this case (5 (B Rando flux of kinetic energy across a plane We now want to calculate: Γ E = π/ 1 w=0 θ=0 ( w f(w(wcosθπw sinθdθdw (6 π = w=0 w 5 f(wdw = π n( Using the nae change of the variable of before, π / w=0 w 5 e w dw (7 Γ E ( π = n π / 1 ( ζ e ζ dζ 0 ζ e ζ + ζe ζ dζ= 0 0 Γ E = (8 π and coparing to (6, Γ E = Γ x (9 So the average particle that crosses the plane carries an energy. This is ore than the average energy per particle ( because faster particles cross ore often. (C Particle flux to a repelling wall 5

6 This could be electrons approaching a negatively charged wall, with a sheath in front. We do the calculation outside the sheath, and, if there are no collisions, the sae flux will cross any plane, including the wall plane. The calculation is like in part (A, but now we notice that not all particles reach the wall: they ust have enough x-directed energy to overcoe the repulsion. If qφ w > 0 (for exaple, electrons, with q = e to a negative wall, we need, (wcosθ qφ w (0 and for a fixed w, and so the inner integration (on θ now gives, θ MAX = cos 1 qφw (1 w θmax sinθcosθdθ = 0 0 θ MAX ( cos θ d = 1 (1 cos θ MAX qφ and for the integration (on w, w MIN = w, fro (1 Using again ζ = w : Γ x Γ = qφw ( / = πn π qφw ( / ( 1 = πn π n c f(ww (1 qφw dw w ( e w w 1 qφw dw w qφw ( e ζ ζ qφ 1 w = 1 ( 1 qφ w w 1 dζ ζ ( ( For the last integral, integrate by parts (e ζ dζ = d(e ζ, [ ( nc Γ = ζ qφ w e ζ qφw 0 + qφw ] e ζ dζ e qφw Γ = n c qφ e w n c ( 6

7 Clearly, the wall repulsion reduces the flux (strongly if qφ w >. (D Energy flux to a repelling wall The sae considerations apply now. We start with (8, but with the liits odified as in part C. The algebra is straightforward, although tedious, and the result is: nc Γ E = e qφw ( + qφ w } {{} Γ (5 So now the average crossing particle carries an energy +qφ w. The extra qφ w per particle is what is required for it to actually reach the wall, while the part is what is would carry in theral for with no repulsion. 7

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