COMPARISON OF MONTE CARLO AND BOLTZMANN CALCULATIONS OF ELECTRON DIFFUSION TO AN ANODE

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COMPARISON OF MONTE CARLO AND BOLTZMANN CALCULATIONS OF ELECTRON

DIFFUSION TO AN ANODE

G. Braglia, J. Lowke

To cite this version:

G. Braglia, J. Lowke. COMPARISON OF MONTE CARLO AND BOLTZMANN CALCULATIONS

OF ELECTRON DIFFUSION TO AN ANODE. Journal de Physique Colloques, 1979, 40 (C7), pp.C7-

17-C7-18. �10.1051/jphyscol:1979708�. �jpa-00219067�

JOURNAL DE PHYSIQUE CoZZoque C7, suppl&ment a u n07, Tome 40, J u i Z Z e t 1979, page C7, 17

COMPARISON OF MONTE CARL0 AND BOLTZMANN CALCULATlONS OF ELECTRON DIFFUSION TO A N ANODE

G.L. i3ragli6 J.J. ~ o w k e r * X X U n i v e r s i t y P a m a , I t a l y . X

U n i v e r s i t y o f Sydney, A u s t r a l i a .

I n t r o d u c t i o n : Our c o n c e p t i o n of t h e d i f f u s i o n of Boltzmann C a l c u l a t i o n s : The Boltemann e q u a t i o n ; e l e c t r o n s i n a g a s i n which t h e r e i s a uniform e l e e

t r i c f i e l d h a s changed markedly i n r e c e n t y e a r s . 2 $ [ N C ~ ~ ( e + k ~ $)+ _6mN0 g ( e ~ _a€ + _az

E x p e r i m e n t a l [I] and t h e o r e t i c a l [2,3] i n v e s t i g a t -

i o n s of t h e d i f f u s i o n of e l e c t r o n p u l s e s changed t h e a

view t h a t d i f f u s i o n was i s o t r o p i c and i n s t e a d d i -

f f u s i o n l o n g i t u d i n a l and t r a n s v e r s e t o t h e e l e c t r i c was solved [51 f o r the distribution function docE,

f i e l d h a s been r e p r e s e n t e d by two s e p a r a t e d i f f u s i o n _{z )} f o r the above cross-section, field and pressure;

c o e f f i c i e n t s D _L and D _T [41 . _{m i s} t h e e l e c t r o n mass, M t h e atom mass, N t h e g a s number d e n s i t y , e t h e e l e c t r o n i c c h a r g e , E t h e R e c e n t l y , however, [ 5 ] , s o l u t i o n s of t h e Boltzmann e l e c t r i c f i e l d , k Boltemanns c o n s t a n t and T t h e e q u a t i o n t o g i v e t h e e l e c t r o n d i s t r i b u t i o n i n e n e r g y t e m p e r a t u r e . Boundary c o n d i t i o n s were (1) a t t h e

&

and p o s i t i o n z f o r a c o n t i n u o u s s t r e a m of e l e c - anode, do ^{= 0,} ( 2 ) a t E = 1 eV, do - ^{0 , ( 3 ) a t E =}

t r o n s d i f f u s i n g i n a u n i f o r m e l e c t r i c f i e l d t o a n 0. eE a d 0 / a E = - a d O / a z and ( 4 ) a t a d i s t a n c e f a r anode, showed t h a t n e i t h e r D T o r D L was a p p r o p r i a t e f r o n the anode, e.g., cm, do equaLs the equilib- t o g i v e t h e e l e c t r o n d e n s i t y d i s t r i b u t i o n n ( z ) . The

rium energy distribution appropriate t o a60/az = o.

p r e s e n t paper r e p e a t s t h i s c a l c u l a t i o n u s i n g Monte

C a r l o methods t o t e s t t h e v a l i d i t y of t h e p r e v i o u s R e s u l t s : a r e summarised i n F i g s . 1-4 and i t i s Boltzmann s o l u t i o n , which used a s s u m p t i o n s s u c h a s

s e e n t h a t t h e r e i s g e n e r a l l y good agreement between a n e x p a n s i o n i n two t e r m s i n s p h e r i c a l harmonics of t h e Boltzmann and Monte C a r l o solutions.

t h e d i s t r i b u t i o n f u n c t i o n .

Monte C a r l o C a l c u l a t i o n s : Monte C a r l o c a l c u l a t i o n s u s i n g t h e methods o f [6] were made f o r a c o n t i n u o u s s t r e a m o f e l e c t r o n s d i f f u s i n g t o a n anode i n a g a s of a t o m i c weight 4 a t a p r e s s u r e o f 2.775 t o r r a t

2 9 3 ' ~ i n a u n i f o r m f i e l d of 1 V/cm. E l a s t i c c o l l - i s i o n s o n l y were c o n s i d e r e d u s i n g a momentum t r a n s - f e r c r o s s - s e c t i o n Q = 6 10-~~(&/0.1)'crn* where ^E i s e l e c t r o n e n e r g y i n e l e c t r o n v o l t s . For Q a c a l - c u l a t i o n s a r e s i m p l i f i e d [ 6 ] , b u t even s o w i t h 16,000 e l e c t r o n s , r e l e a s e d 4 a t a t i m e , a t r e g u l a r i n t e r v a l s of 1 0 - ~ / 4 , 0 0 0 S o v e r a n i n t e r v a l of S, - 1 0 c o l l i s i o n s need t o be c o n s i d e r e d . 8 Elec-

F i g . 1 i n d i c a t e s t h a t t h e Boltemann-Monte C a r l o re- s u l t s f o r t h e e l e c t r o n d e n s i t y n d i f f e r s u b s t a n - t i a l l y f r ~ m t h e c l a s s i c a l s o l u t i o n s o f t h e . e o n t i n - u i t y e q u a t i o n

- - ( ~ W - D % ) = O a

a z a z

where t h e v a l u e s of t h e d r i f t v e l o c i t y W and e l e c - t r o n d i f f u s i o n c o e f f i c i e n t D a r e t a k e n a s c o n s t a n t s independent of p o s i t i o n a p p r o p r i a t e t o t h e v a l u e of E / N . S o l u t i o n s u s i n g D = D o r D = D u n d e r e s t i -

L T

mate and o v e r e s t i m a t e , r e s p e c t i v e l y , t h e d i f f u s i o n . The e l e c t r o n d e n s i t y g r a d i e n t s p e r t u r b do s o t h a t W and D a r e r e a l l y complex f u n c t i o n s o f z . t r o n s were r e l e a s e d i s o t r o p i c a l l y a t a p o i n t s o u r c e ,

F i g . 2 i n d i c a t e s t h a t t h e a v e r a g e e l e c t r o n e n e r g y 1 cm from t h e anode, w i t h t h e e q u i l i b r i u m e n e r g y

n e a r t h e anode i s a l m o s t a f a c t o r of two l a r g e r d i s t r i b u t i o n . Any e l e c t r o n s c r o s s i n g t h e anode

t h a n t h e e q u i l i b r i u m a v e r a g e e n e r g y a p p r o p r i a t e t o p l a n e were removed from t h e system. The s p e e d s and

E/N. T h i s i n c r e a s e i s a r e s u l t of t h e a b s e n c e of r a d i a l and a x i a l p o s i t i o n s a t t h e end o f s

were u s e d t o compile t h e accompanying f i g u r e s . e l e c t r o n s back s c a t t e r e d from p o s i t i o n s i n f r o n t of t h e e l e c t r o d e which would have a low e n e r g y b a c a u s e o f t h e i r d r i f t a g a i n s t t h e e l e c t r i c f i e l d . S i m i l a r

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1979708

e f f e c t s a c t t o r e d u c e t h e a v e r a g e e n e r g y i n t h e Monte C a r l o s o l u t i o n a t t h e s o u r c e of e l e c t r o n s .

I n F i g . 3 t h e e f f e c t i v e d r i f t v e l o c i t y W n e a r t h e anode i s shown t o be markedly p e r t u r b e d by t h e e f f e c t of t h e anode.

To o b t a i n a c c u r a t e p l o t s of d O ( s , z ) from Monte C a r l o c a l c u l a t i o n s would r e q u i r e s t i l l more c o l l - i s i o n s t h a n we have c o n s i d e r e d . I n s t e a d we show

skgodr a s a f u n c t i o n of z i n F i g . 4 a s a n i n d i - c a t i o n of do. F i g . 5 shows p r e d i c t e d c u r r e n t r a t i o s f o r t h e Townsend-Huxley e x p e r i m e n t . When a b s o r p t i o n a t t h e c a t h o d e i s i n c l u d e d , Monte C a r l o r a t i o s a r e between p r e d i c t i o n s from t h e c o n t i n u i t v e q u a t i o n u s i n g (1) i s o t r o p i c d i f f u s i o n , and ( 2 ) c a l c u l a t i o n s u s i n g DL and D _T [2] .

Conclusion: E l e c t r o n d r i f t and d i f f u s i o n i n a u n i - form e l e c t r i c f i e l d c a n n o t be r e p r e s e n t e d by d r i f t and d i f f u s i o n c o e f f i c i e n t s t h a t a r e i n d e p e n d e n t o f p o s i t i o n . For a c o n t i n u o u s s t r e a m of e l e c t r o n s ab- s o r b e d a t e l e c t r o d e s t h e e f f e c t i v e l o n g i t u d i n a l d i - f f u s i o n c o e f f i c i e n t d i f f e r s from t h a t a p p r o p r i a t e t o d i f f u s i n g p u l s e s .

R e f e r e n c e s :

[ l ] Wagner, E.B., Davis, F. J. and H u r s t , G. S.

J.Chem.Phys. 47, 3138 (1967).

[ A P a r k e r , J . H . , J r . and Lowke, J.J. Phys.Rev.

181, 290 (1969).

[3] S k u l l e r u d , H.R. J.Phys.B. 2 , 696 (1969).

[41 Huxley, L.G.H. and Crompton, R.W. The Di-

f f u s i o n and D r i f t o f E l e c t r o n s i n Gases, Wiley 1974.

[51 Lowke, J . J . , P a r k e r , J . H . , Jr. and H a l l , C . A . Phys.Rev. A , 1 5 , 1237 (1977).

[61 B r a g l i a , G.L. P h y s i c a 92C, 91 (1977).

Cork-

O

1.2 -.

4

sourea

hod.

F i g . 2 Average E l e c t r o n Energy

1 .8 .6 .4 a 2 0

Distonce from Anode

cm

F i g . 3 E f f e c t i v e D r i f t V e l o c i t y

Energy : ev

F i g . 4 Energy I n t e g r a l s n e a r Anode

'

7)

1

0 With Cathode Anode Only

Classical / ^iuxley

Isotropic With DL

Source Dirtonce fmm Anode : A d o

F i g . 1 Normalised E l e c t r o n D e n s i t y

Radius Inner Disk : cm

F i g . 5 C u r r e n t R a t i o s f o r Townsend Huxley experiment