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DIFFUSION THEORY OF THE POSITIVE COLUMN IN HYDROGEN GLOW DISCHARGES WITH
VARIABLE DEGREE OF DISSOCIATION
P. Michel, S. Pfau, A. Rutscher, R. Winkler
To cite this version:
P. Michel, S. Pfau, A. Rutscher, R. Winkler. DIFFUSION THEORY OF THE POSITIVE COLUMN
IN HYDROGEN GLOW DISCHARGES WITH VARIABLE DEGREE OF DISSOCIATION. Journal
de Physique Colloques, 1979, 40 (C7), pp.C7-133-C7-134. �10.1051/jphyscol:1979765�. �jpa-00219471�
JOURNAL DE PHYSIQUE Colloque C7, suppZ6ment au n07, Tome 40, J u i l Z e t 1979, page C7- 133
OFFUSION THEORY OF THE POSITIVE COLUMN IN HYDROGEN GLOW DIXHARGES WITH VARIABLE DEGREE OF DISSOCIATION
P. Michel, S. Pfau, A. Rutscher and R. winklerx.
E.M.A. U n i v e r s i t a t GreifswaZd, S e k t i o n Physik@Zektronik GreifswaZd, D.D.R.
'
Akademie der Wissenschaften der DDR, Z e n t r a Z i n s t i t u t fiir EZektronenphysik GreifswaZd, D.D.R.1IJTRODUCTION:In c o n n e c t i o n w i t h an i n c r e a s - i n g i n t e r e s t i n plasma chemical r e a c t i o n s under non-isothermal c o n d i t i o n s r e c e n t l y t h e c l a s s i c a l SCHOTTKY d i f f u s i o n t h e o r y of t h e p o s i t i v e column was extended t o g a s e s of v a r y i n g m a t e r i a l composition / I / . This e x t e n s i o n r e s u l t e d i n a new viewpoint f o r t h e u n d e r s t a n d i n g of one of t h e most im- p o r t a n t problems i n e v e r y column t h e o r y : t h e q u a n t i t a t i v e d e s c r i p t i o n of t h e e l e c - t r i c c h a r a c t e r i s t i c of t h e d i s c h a r g e p l a s - .-.Till now t h e f a l l i n g behaviour of t h e column c h a r a c t e r i s t i c VJaS a t t r i b u t e d main- l y t o such p r o c e s s e s a s d e v i a t i o n s from t h e q u a s i - n e u t r a l i t y and t h e occurence of t h e s t e p w i s e i o n i z a t i o n . ~ ~ o w i n t h e s p e c i a l c a s e of t h e hydrogen d i s c h a r g e i t was shown t h a t t h e n e g a t i v e d i f f e r e n t i a l re- s i s t a n c e of t h e column could be d e c l a r e d by t h e v a r y i n g degree of d i s s o c i a t i o n i n dependence on t h e d i s c h a r g e current.To cal- c u l a t e t h e e l e c t r i c c h a r a c t e r i s t i c on t h i s b a s i s a model of t h e d i s c h a r g e mechanism was used a s a f i r s t s t e p which n e g l e c t e d some a p p e a r i n g elementary processes.Eow we have improved t h e model and extended t h e c a l c u l a t i o n t o d i f f e r e n t g a s p r e s s u r e s and d i f f e r e n t recombination c o n d i t i o n s a t t h e tube w a l l .
THEORETICAL MODEL :The st a r t ing-point of our c a l c u l a t i o n s was t h e c o r r e c t energy d i s t r i b u t i o n f u n c t i o n of t h e e l e c t r o n s i n t h e H2/H-mixture.This r e q u i r e d t h e s o l u - t i o n o f t h e corresponding BOLTZIvIANI? equa- tion.Taking i n t o account a l l important e l a s t i c and i n e l a s t i c c o l l i s i o n s between e l e c t r o n s and H2 and H r e s p e c t i v e l y ( f o r d e t a i l s s e e / l / / 2 / ) t h e d i s t r i b u t i o n func- t i o n f(xD,E/po)was c a l c u l a t e d i n t h e frame of u s u a l approximations f o r d i f f e r e n t de- g r e e s of d i s s o c i a t i o n xD=O
...
1 and d i f f e r - e n t reduced e l e c t r i c a l f i e l d s t r e n g t h s E/po=7.. .I GO V/cm Torr.By t h i s t h e t r a n s - p o r t c o e f f i c i e n t s and v a r i o u s c o l l i s i o n r a t e s of t h e e l e c t r o n s could be d e t e r - inined, t o o .C o n t r a r y t o / I / i n t h e k i n e t i c model of t h e p o s i t i v e column now we have a l s o con- s i d e r e d t h e i o n H; b e s i d e s t h e i o n s H+,H;
and H;.~his expanded t h e p r o c e s s scheme a l t o g e t h e r t o t h e follw+ving 14 r e a c t i o n
H
+
e-+H++ 2e ; z 1 = z 1 ( ~ / p , , x p ) Hz+ e+$+ 2e ; z 2 = z 2 ( E / p o , 5 )g+
HL+H2+ H++ H ; z =a (E/p,)HI+ 2HZ+HL+ H; ; ~ ~ = i - l & ~ ~ ~ ~ ~ / s
+ +
H,+ H,*H,+ H ; z5=z5(E/p,)
%+
e +Hz+ H ; z ~ = z ~ ( E / P ~ , % )%+
e42H,+ H ; z7=z7(E/p,, H i + ~ K L J H ~ HL;+
H C H 2HL; ~ ~ = 2 * 1 0 - * ~ ~ ~ ' / ~ ; ( 8 0 K ) 3+
%+
e -+H2(bE.)
-+2H+
e ; $ O = ~ , O ( E / ~ o , x3)~ , + e + ~ ~ ( a ~ ~ ~ + c ' T,)-+2~+e; w z i ( ~ / p , ,
xd
- 1 , 8 * 1 0 - ~
H1+ 2 H t H L + HZ;z4&- cm /s; (80K) 3 H 4 H 2 + H ; z43=3,5~l~-32cm6/s ; (80K) H
+
H(wal1) -Jc Hz; zI,+= @, accomodat i o n coef.
The system of b a l a n c e e q u a t i o n s f o r t h e d i f f e r e n t p a r t i c l e s
q,
H;, ,:H$
and H (in-c l u s i v e of some c o n d i t i o n s f o r n e u t r a l i t y , c u r r e n t d e n s i t y and p a r t i c l e number) r e a d s i n d i f f e r e n t i a l form :
2
+
divrj: =n n z
+
nln+lzJ- n2n, Z+e 4 4
d iv r
jl=
%n.z2- z3- + g z2.
+
divrj: =nLniz&%n, z+tnznzz9-n,n:z,-nLn3z8 d i v r j $ = n x z8- nen:z,- nLn;z9
w i t h j:=-bi(~=/b=) (</ne) (dne/dr) ; k= I , 2 , 3 nI+n'+n++n+=ne ; ( n e u t r a l i t y condi: i o n )
L 3 5
2pnL(z,o+%r )+nLn~(z3+z4)+ngn~~zb+~cLn~z++
+(j:+j>jz) d ( r / r , - I )/r,=nBnlz4+2n,n,zA~
+ 2 n 3 ~ + n @ ~ d ( r / r , - l )/4r, I 4
n4+n2_=ponp; (open tube, l a r g e dead s p a c e ) n ~ / 2 + n z = p o , n ~ ; ( c l o s e d tu b e , no dead s p a c e ) j3=n&E; ( a x i a l e l e c t r i c c u r r e n t d e n s i t y ) ( j : c u r r e n t d e n s i t y ; z : c o l l i s i o n r a t e ; n : p a r t i c l e c o n c e n t r a t i o n ; r, ro : r a d i u s coor- d i n a t e , t u b e radius;b:mobility;D:diffusion c o e f f i c i e n t ;ng=3, 54- 10' 6cm-3~orr-1 ; po:re- duced p r e s s u r e ( 0 0 C) ; p o s : i n i t i a l p r e s s u r e ; 5:therrnal v e l o c i t y ) .
I n t h e balance o f t h e H atoms t h e terms 6 w i t h t h e d e l t a f u n c t i o n r e p r e s e n t t h e t u b e w a l l l o c a l i z e d p r o d u c t i o n and l o s s process- e s by recombination.Radia1 changes of t h e c h a n n e l s :
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1979765
H and
H2
c o n c e n t r a t i o n a r e n e g l e c t e d . A f t e r r a d i a l a v e r a g i n g of t h e b a s i c equa- t i o n s f o r d i f f e r e n t pressures(0,3...5Torr) and d i f f e r e n t accomodation c o e f f i c i e n t s (low7.. . 1 0 - ~ ) t h e a x i a l e l e c t r i c f i e l d s t r e n g t h E, t h e r e l a t i v e p o r t i o n of t h e-+ -
vari-ous ionsatk=nk/ne and t h e degree of d i s s o c i a t i o n xD=nl/(nl+n2) were c a l c u l a t e d i n dependence on t h e d i s c h a r g e current.At t h i s c a l c u l a t i o n t h e g a s temperature was s e t constant(80K)and we used f o r t h e rad- i a l d i s t r i b u t i o n of t h e charge c a r r i e r s a BESSEL p r o f i l e .
RESULTS:Figs.l and 2 compare t h e import- ance of t h e v a r i o u s k i n d s of i o n s a t d i f - f e r e n t prassures.The H; i o n s a r e e n t i r e l y i n s i g n i f i c a n t a t low p r e s s u r e s but g a i n c o n s i d e r a b l e importance a t h i g h e r p r e s s - ures.0n t h e o t h e r hand t h e molecular i o n H
: always p l a y s only a s u b o r d i n a t e r o l e i n t h e i o n i c budget .Fig.3 shows t h e i n - c r e a s e of t h e degree of d i s s o c i a t i o n w i t h i n c r e a s i n g current.At low p r e s s u r e s s m a l l c u r r e n t s ,can a l r e a d y produce n e a r l y com- p l e t e d i s s o c i a t i o n . A t h i g h e r p r e s s u r e s xD d e c r e a s e s but t h e a b s o l u t e c o n c e n t r a t i o n of H atoms i s r e l a t i v e independent on t h e p r e s s u r e . E s p e c i a l l y a t low c u r r e n t s and r e l a t i v e low p r e s s u r e s t h e accomodation c o e f f i c i e n t
#
i n f l u e n c e s c o n s i d e r a b l y t h e degree of dissociation(Fig.4).Small val- u e s o f f r e l a t e t o low t e m p e r a t u r e s a t c l e a ~ t u b e wall.Also a s p e c i a l c o a t i n g of t h e w a l l ( f o r i n s t a n c e w i t h p h o s p h i t e ) re- d u c e s v . ~ h e r e g i o n ofTshown i n Big.4 cor- responds t o t h e r e a l l y occuring v a l u e s un- d e r d i f f e r e n t c o n d i t i o n s .P i n a l l y Figs.5 and 6 show examples o f t h e c a l c u l a t e d e l e c t r i c c h a r a c t e r i s t i c of t h e p o s i t i v e column i n hydrogen.Contrary t o a l l t h e o t h e r f i g u r e s number 6 a l s o con- t a i n s r e s u l t s f o r t h e c a s e of an open d i s - charge t u b e ( l a r g e dead space ) ,The n e g a t i v e s l o p e of t h e c h a r a c t e r i s t i c i s caused by t h e t r a n s f e r of t h e p o s i t i v e column from a p r e v a i l i n g molecular(H2)dischqrge a t
-
low c u r r e n t s t o s t a t e s w i t h a c o n s i d e r a b l e atomic component a t h i g h c u r r e n t s . I n con- sequence of t h e s m a l l degree of d i s s o c i a - t i o n a t h i g h g a s p r e s s u r e t h e c h a r a c t e -
r i s t i c remains f l a t i n t h i s case.
REFERENCES :
/ 1 /NICHEL,P.
,
PFAU, S. RUTSCHER, A.,
7IIhTLER, Re Contr.Pap.13th ICPIG,Berlin 1977,p.245 /2/!&ICHEL, P., PrlkU, S., GIRKLER, R., i n prepar.Fig. I Pig. 2
a-
ElPo parameter pOs [Torrl '30
20-
I [A1
-
10 1 1 I l l 1 1 I l l 1 I I l l I I 1 1
10-L 10-3 10-2 10-1 100
Pig. 5
[VlcmTorrl
pos-085 Torr (closed tube 1 p, 4 8 5 Torr lopen tube )
IS
I [Al
-
I 1 1 1 1 1 1 1 1 1 1 1 1 1 1 I 1 1 1 1 1 1 1 1 l 1 1 1 1
10.5 10-4 103 10-I lo0 10'
Fig. 6
