EFFECTIVE BOUNDARY CONDITIONS FOR A PLASMA IN A MAGNETIC FIELD, ADJACENT WITH AN ELECTRODE

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Submitted on 1 Jan 1979

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EFFECTIVE BOUNDARY CONDITIONS FOR A PLASMA IN A MAGNETIC FIELD, ADJACENT

WITH AN ELECTRODE

V. Rozhansky

To cite this version:

V. Rozhansky. EFFECTIVE BOUNDARY CONDITIONS FOR A PLASMA IN A MAGNETIC

FIELD, ADJACENT WITH AN ELECTRODE. Journal de Physique Colloques, 1979, 40 (C7), pp.C7-

515-C7-516. �10.1051/jphyscol:19797249�. �jpa-00219233�

JOURNAL DE PHYSIQUE CoZloque C7, suppZ4ment au n07, Tome 40, JuiZZet 1979, page C7- 515

EFFECTIVE BOUNDARY €ONDITIONS FOR A PLASMA IN A MAGNETIC FIELD, ADJACENT WITH AN ELECTRODE

V.A. Rozhansky.

KaZinin PoZytechnicaZ I n s t i t u t e , RadiophysicaZ Departmenl, Leningrad, U.S. S.R.

The boundary c o n d i t i o n s f o r a p l a s - m a , a d j a c e n t w i t h an e l e c t r o d e u s u a l l y

i n c l u d e t h e s u r f a c e p o t e n t i a l d i s t r i b u - t i o n . S i n c e t h e p o t e n t i a l d i f f e r e n c e i n t h e s h e a t h must be i m p o r t a n t , i t ' s impos- s i b l e t o a p p l y t h e s e c o n d i t i o n s f o r t h e q u a s i - n e u t r a l plasma e q u a t i o n s d i r e c t l y . In o r d e r t o r e c e i v e t h e e f f e c t i v e bounda- r y c o n d i t i o n s which a r e v a l i d f o r t h e plasma e q u a t i o n s i t ' s necessary t o e x p r e s t h e p o t e n t i a l d i f f e r e n c e i n t h e s h e a t h a s a f u n c t i o n of t h e p a r t i c l e f l u x e s t o a n e l e c t r o d e .

t h e abscence of magnetic i i e l d t h i s r e s u l t vras o b t a i n e d i n / I / f o r c o l l i s i o n l e s s s h e a t h and i n / 2 , 3 / ' numeri- c a l l y f o r c o l l i s i o n dominated one.

1. C o l l i s i o n l e s s s h e a t h i n a f u l l y i o n i z - ed magnetoplasma,

\Ye s h a l l c o n s i d e r t h e c a s e of a n e l e c t r o d e n e g a t i v e l y b i a s e d w i t h r e s p e c t t o plasma, t h e s h e a t h s c a l e small compa- r e d w i t h e l e c t r o r l c y c l o t r o n radius &Ie and p o t e n t i a l d i f f e r e n c e i n t h e s h e a t h

(Qc 7 T e / e . E l e c t r o n s t h a t r e a c h an e l e c t r o d e must have normal energy a t t h e

..

s h e a t h edge ( x = o , HIIZ-is p a r a l l e l t o a n e l e c t r o d e ) exeeding eQc, s o t h a t t h e i n - f l y e n c e of t h e plasma e l e c t r i c f i e l d on such e l e c t r o n s can be n e g l e c t e d , The cor- responding o r b i t s a r e almast empty, be-

cause t h e p a r t i c l e l i f e time on t h e s e o r b i t s Q,-'= rn,c/eIi i s m a l l compared w i t h t h e time of Coulomb c o l l i s i o n s , So

t h e f l u x t o an e l e c t r o d e i s equal t o t h e f a s t e l e c t r o n ' s f l u x t o t h e empty r e g i o n i n t h e gamma space. The k i n e t i c e q u a t i o n f o r t h e f a s t e l e c t r o n s /4/ w i t h t h e z e r o boundary c o n d i t i o n was s o l v e d i n / 5 / . The corresponding f l u x i s g i v e n by:

w i t h t h e energy eye.

2, C o l l i s i o n l e s s s h e a t h i n a p a r t i a l l y i o n i z e d magnetoplasma.

E l e c t r o n - n e u t r a l c o l l i s i o l l s t r a n s f e r t h e e l e c t r o n s t o t h e empty o r b i t s and consequently l e a d t o a n a d d i t i o n a l f l u x t o an e l e c t r o d e , Vie s h a l l c o n s i d e r t h e c a s e when t h e t o t a l c r o s s - s e c t i o n i s fi- n i t e , s o t h e l a r g e - a n g l e s c a t t e r i n g g i v e s t h e main c o n t r i b u t i o n t o - t h e f l u x . The d i s t r i b u t i o n f u n c t i o n f a r from t h e empty o r b i t s i s h k x ~ l e l l i a n one, s o vJe o b t a i n / 5 /

d 8 r d q l d ~ \aE/mh E x

-00

dE

( 7 1

Ita-neutral d e n s i t y , 6 - c r 0 s s - s e c t i ~ ~ , Q =

-mR ^x. A f t e r i n t e r g r a t i n g we have Py e e

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19797249

The e x p r e s s i o n bf t h e smie o r d e r was ob- t a i n e d i n / 6 / , but t h e d i s t r i b u t i o n fun- c t i o n was c o n s i d e r e d t o be $laxwellian on a l l o r b i t s . R e s u l t v ~ h i c h coirlcides v ~ h i t h ( 3 ) vJas obtaioed r e s e n t l y

/7/ by ano- t h e r method. The e l e c t r o n f l u x caused by e l e c t r o n - n e u t r a l c o l . l i s i o n s exeeds t h e f l u x connected w i t h Coulolnb c o l l i s i o n s ( 4 1

Expressions ( 2 , 3 ) t a k e p l a c e i f t h e Max- w e l l i a n d i s t r i b u t i o n f u n c t i o n f a r from

t h e empty o r b i t s i s supported by t h e e l e - c t r o n - e l e c t r o n c o l l i s i o n s . T h i s l e a d s t o t h e c o n d i t i o n s :

Jeo(Jam,i)Z&.. ^{L% ~ ( m ( 5 ) c}

If t h e c o l l i s i o n frequency.)) i s l e s s t h a n g i v e n by ( 5 ) t h e d i s t r i b u t i o n f u n c t i o n i s d e p l e t e d on a l l o r b i t s a t E , e9, and t h e f l u x c a n be c a l c u l a t e d a s a d i f f u s i o n f l u x of f a s t e l e c t r o n s from l e n g t h of energy r e l a x a t i o n / I / .

3 . C o l l i s i o n dominated s h e a t h i n a s l i g h t l y i o n i z e d plasma.

Near t h e e l e c t r o d e t h e Boltzmann d i s t r i b u t i o n f o r e l e c t r o n s i s v a l i d and t h e i o n i c f l u x i s determined by l i n e a r d e n s i t y p r o f i l e , A f t e r s u b s t i t u t i n g t h e q u a s i - n e u t r a l s o l u t i o n s i n t o P o i s s o n ' s e q u a t i o n we o b t a i n t h e d i s t a n c e where t h e space charge becomes s i g n i f i c a n t

n

= The

T e f';n 2

I" ⁽⁷

a:, ^{( I +} T = / T ~ )24qe2

p o t e n t i a l d i s t r i b u t i o n i n t h e s h e a t h can be o b t a i n e d s e g l e c t i n g e l e c t r o n den- s i t y :

( 8 ) The s i z e of t h e s h e a t h i s determined by c o n d i t i o l i t h a t t h e e l e c t r o n d i f f u s i o n term becomes dominant. The p o t e n t i a l d i f - f e r e n c e i n t h e s h e a t h i s g i v e n by:

-. -

+ 9 ^{& &[2 r ~ e a e m q} /( ^pen a i n ~ e )]

The p o t e n t i a l . d i f f e r e n c e between t h e p o i n t x i n p l a m a and s h e a t h edge is:

& p r e s s i o n s (9,10) c o i n c i d e wit'n t h e nwnerical r e s u l t /3/ w i t h ali accurancy

5 10% even when e y C o ~ e . References:

1. Therinoemission energy t r a n s f e r and low temperature plasma, ed. by B.Ya.l,loy-

zhes and G.E.Pikus, &I. ,IVauka, 1973, 2. C,H,Su, S.H.Lam, Phys,Fluids 5,1479,

1963.

3. 1 .&~.Cohen,Phys,Fluids,6,1492,1963 ; 8,2097,1965.

-

4. D.V.Sivukhin,Voprosy t e o r i i plasmy,

Din-normal d i f f u s i o n c o e f f i c i e n t , r i n -

ilo~rnal io n i c f l u x , The d e n s i t y i n t h i s

p o i n t i s g i v e n by: