EP1_2010 Sec3b

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INTRODUCTION TO PLASMA PHYSICS 3. Plasma in an Electric Field

Prof. Mariano Andrenucci - AY 2009-2010 3.20

The Sheath

Let us now study what happens in the vicinity of a wall. Consider a

simple 1D model with no magnetic field. Quasi-neutrality ensuresthat the electric potential is zero in the plasma bulk. However, as

vthe

>> vthi , elettrons are lost more quickly to the wall than ions, andthe wall acquires a negative potential (or if you prefer, the plasma is

left with a net positive charge).

Because of Debye shielding, the potential variation takes place over a

layer a few Debye-lengths thick, called the sheath. The sheath forms

a potential barrier wich tends to confine the escaping electronselectrostatically. The height of this barrier self-adjustes so as tosatisfy ambipolarity (

i =

e).

The situation is schematically depicted in Fig. 3.9. The plasma

potential distribution is shown in (a); the corresponding densitydistribution of ions and electrons is shown in (b). The density of ions

is higher than electrons near the wall due to the negative electric field

established there by the escaping electron flux.

Fig. 3.9 Electrostatically confinedplasma

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The Sheath (contd)We shall now use the fluid equations to analyse the electric potential variation in the sheath region. Let us

adopt a few simplifying assumptions:

1-D, no magnetic field no collisions (ei~ 1 m >>D at n~ 1018 m3,Te~ 5eV ) cold drifting ions u

i >> v

thi(a beam like distribution with ion temperature close to that of the neutrals)

Boltzmann distributed electronsAt equilibrium, and neglecting ionization and recombination in the plasma bulk (so that t= 0) the ion

particle flux must be constant from the centre to the wall. Even if we have ignored ionization, the ion flow u0

away from the sheath, though small, must be finite in order to satisfy particle balance between source

(ionization) and sink (walls). The ions gain kinetic energy as they are accelerated through the sheath to the

negatively charged wall1

2m

iui

2+ Z e =

1

2m

iu0

2 (3.46)

where we have ignored the potential 0 in the body of the plasma. This can be solved for the ion drift

velocity in the region of non-zero potential

ui= u

0

2 2Ze

mi

1 2

(3.47)

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The Sheath (contd)Using continuity we can obtain the plasma density n

i x( ) with respect

to the density in the plasma body n0

ni x( ) = n

0 1

2Ze

miu

0

2

1 2

(3.48)

The electrons are Boltzmann distributed

ne x( ) = n

0exp

e

KTe

(3.49)

so that we can solve for the potential using Poissons equation

2 =

0

d2

dx2 =

e

0

(ni n

e)

=

e n0

0

expe

KTe

1 2

Ze

miu0

2

1 2

(3.50)

This is a nonlinear differential equation for x( ) . To solve it approximately, we look at two limits: near theplasma edge of the sheath and near the wall (see Fig. 3.10).

Fig. 3.10 The sheath region

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Plasma Edge of the SheathAs we shall see later, the potential at the wall

w is a few times the thermal energy, so that e

w~ KT

e.

Within the plasma edge of the sheath, we shall take the potential variation e > v

thi, we have e KT

em

i( )

1 2

(3.53)

then X2> 0 and the decay length is comparable with the Debye length. If the condition (3.53) were violated,

Xwould be imaginary, and the electric potential would become an oscillating function of potential near the

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Plasma Edge of the Sheath (contd)wall. This would trap particles in the steady state potential well, which cannot happen as dissipative processes

would destroy such an ordered state. Therefore, the condition (3.53), known asBohm sheath criterion, must

be satisfied, stating that ions must enter the sheath with a velocity greater than the acoustic velocity (thermal

speed). To obtain a directed velocity u0, there must be a small accelerating field in the body of the plasma.

The assumption that = 0 at x = 0 in obtaining (3.52) is only approximate and is made possible by

d>>D

. Ultimately, u0is fixed by the ion production rate (ionization).

The choice of the boundary at which ui = u

0 is somewhat arbitrary. In reaching this

position in the plasma the ions have fallen throughsome overall potential drop 0(earlier

assumed small and ignored). We hereafter take our starting position to the plasma edge ofthe sheath, which we define to commence when the electrostatic potential energy is equal

to the electron thermal energy, e0 = m

iu

0

22= K T

e 2 . In this case the initial ion drift

velocity is

u0

= uBohm

=

K Te

mi

1/2

(3.54)

and where uBohm

is known as Bohm speed. It is often the case that Te> T

i, so that

ui > u

0> v

thias was assumed earlier in the analysis. The region in the plasma over which the potential drops

slowly from the centre to the edge of the sheath is known aspre-sheath.

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The Wall Edge of the Sheath

Near the wall, the electric potential is very negative and the electron density is very low so that for the charge

density we can write

= e ne+ e Z n

i e Z n

i= n

0u0

Z e ui (3.55)

where we have used continuity. The Poisson equation gives

d2

dx2 =

n0u

0Z e

0 2Ze

0

( ) m

i[ ]

1 2 (3.56)

where we have substituted e0 = m

iu0

22 in Eq. (3.47). Introducing the potential drop V =

0allows Eq.

(3.56) to be written as

d2V

dx 2 =

g

V1 2 con (3.57)

with g =n

0u

0Z e

0

2Z e mi

( )1 2

J

0

2Z e mi

( )1 2

(3.58)

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Wall Edge of the Sheath (contd)To obtain the solution for Vwe now proceed as follows

V =gV1 2

2 V 2 V V =g V

2V1 2 4

integr.w.r.t.x V( )2

= 4gV1 2

V = 2 g V1 4

V V1 4

= 2 g

integr.w.r.t.x 4

3V

3 4= 2 g x

g =4V

3 2

9x2

(3.59)

whereJis the ion current density in the sheath and Eq. (3.59) describes the variation of the plasma potential

in the region close to the wall. This variation is expressed explicitly in the Child-Langmuir Law

V J2 3x

4 3 (3.60)

Fig. 3.11 compares the variation expressed by Eq. (3.60) with the variation that would be expected for

uniform (V x2 ) and point source charge distribution (linear).

Fig. 3.11 Potential variation at thewall edge of the sheath

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Plasma Potential and Wall Potential

The thermal flux of particles across an interface for a given species is (see Appendix 3.1)

s

= n

KT

2m

1 2

(3.61)

For ne = ni , se >> siand therefore w is negative. The sheath potential is thus established by the mobilityof the electrons due to their thermal motion. As the electrons are Maxwellian (non drifting) we have

se

= nev 4 = n

0v 4( ) exp e KTe( ) (3.62)

The ions on the contrary are drifting

i n

0vBohm (3.63)

To determine the electric potential at the wall, we impose the equilibrium condition se

= i, to find

n0

KTe

mi

1 2

n0

KTe

2mi

1 2

expe

w

KTe

ew

KTe

1

2ln

2me

mi

w =

0ln

mi

2me

(3.64)

The right hand side factor takes values of 3 to 6 (typically 4.7) for Argon plasma, and 2 to 4 (typically 2.8) forHydrogen plasma (depending on choice of constants). The plasma wall potential is thus several times the

electron temperature. This potential is necessary to ensure ambipolarity. Note that both 0(the presheath

drop) and w

are negative. The total plasma potential at centre is therefore P =

0 +

w

w. Thus, for

example, for a 1 eV Argon plasma, the ions with an initial energy KTe /2 at the sheath boundary reach the

wall with an energy of0.5 + 4.7 = 5.2

times the electron temperature.

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Langmuir Probes

Let us now study the behaviour of a conducting probe inserted into a plasma. Initially, we assume the probe

to be biased negative such that all ions striking the probe are collected and the electrons repelled. Thecollected ion current density is j = q , and the ion current is (assumingZ=1)

I = n0u0eA (3.65)

where A is the collection area and u0 = u

Bohm. Now, there is a potential drop

00

0 in the presheath to

accelerate the ions to the Bohm speed

es =

e 0

00( )

=

1

2miu0

2=

1

2KTe where

sis the sheath edge potential. The electron density at the sheath edge is

n0 = n

00 exp e

s K T

e( ) = n

00exp 0.5( ) = 0.61n

00 0.5n

00

and the ion saturation currentis

Isi

= n0

u0

e A =1

2n

00e A

K Te

mi

1 2

(3.66)

Thus, given the electron temperature, we can use probes to measure the particle density in a plasma.

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(Sheath-wall boundary)More generally, consider some probe potential VVp = V with respect to the plasma potential V

p =

w. The

electron current arriving at the probe is

I = Aeseexp eV K T

e( ) =Ise exp eV K Te( ) (3.67)

where Ise

= Aen0

v 4 is the electron saturation current.

For a positive probe bias, essentially all the electronsarriving at the probe are collected and I = I

se. Note that

Ise

Isi

~ mi

me

( )1 2

>>1 . As shown in Fig. 3.12, the

location of the knee of the V - I characteristic gives the

plasma potential V = Vp( ) .

Fig. 3.12 The Langmuir probe I-V characteristic

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EP1_2010 Sec3b