ON THE THERMODYNAMICAL STABILITY OF THE PLASMA OF GAS DISCHARGE IN THE REAL GAS

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Submitted on 1 Jan 1979

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ON THE THERMODYNAMICAL STABILITY OF THE PLASMA OF GAS DISCHARGE IN THE REAL

GAS

S. Temko, K. Temko, S. Kuzmin

To cite this version:

S. Temko, K. Temko, S. Kuzmin. ON THE THERMODYNAMICAL STABILITY OF THE PLASMA OF GAS DISCHARGE IN THE REAL GAS. Journal de Physique Colloques, 1979, 40 (C7), pp.C7- 677-C7-678. �10.1051/jphyscol:19797328�. �jpa-00219320�

JOURNA6- DE PHYSIQUE ColZoque C7, suppZdment a u n07, Tome 40, J u i Z Z e t 1979, page C7- 677

ON THE THERMODYNAMICAL STABIUTY OF THE PLASMA ff GAS DIXHARGE IN THE REAL GAS

S.W. Ternko, K.W. ~ernko'and S.K. ~uzrnin*'.

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X X ~ c a d e m y o f ScLence U.S. S. R.

The p r e s e n t r e p o r t i s about d i s c h a r g e i n weaWy i o n i z a t e d r e a l gas. The d i s c h a r g e i s being kept up by t h e c u r r e n t being ca- used by t h e e x t e r n a l s o u r s e . The p r e s s u r e of t h e plasma i s balanced by t h e magnetic p r e s s u r e being caused by t h e own magnetic f i e l d o f t h e c u r r e n t . The d i s c h a r g e h a s t h e g e o m e t r i c a l form o f t h e e l l i p s o i d a l waordw. The plasma of t h e g a s dischange c o n s i s t s of e l e c t r o n s , p o s i t i v e i o n s and n e u t r a l s . The i n t e r a c t i o n between t h e par- t i c l e s of t h e plasma i s d e s c r i b e d by po- t e n t i a l f u i i c t i o n s forming t h e symmetric and p o s i t i v e m a t r i x . The i n t e r a c t i o n i s n o t supposed t o be weak i n comparison w i t h t h e a v e r a g e energy of c h a o t i c movement o f t h e p a r t i c l e s of t h e plasma. We t a k e i n t o c o n s i d e r a t i o n n o t o n l y t h e Coulomb i n t e - r a c t i o n , but a l s o t h e n e a r i n t e r a c t i o n t a - k i n g p l a c e when t h e d i s t a n c e between t h e p a r t i c l e s i s s h o r t . Accounting of t h e f o - r c e s of n e a r i n t e r a c t i o n i s made by t h e Lenard-Gones p o t e n t i a l s [I ]

.

l i t y of change of t h e p a r t i c l e s from one s o r t t o . a n t h e r because of chemical r e a c t i - o n s , e l a s t i c and n o n e l a s t i c i n t e r a c t i o n s . The moments of t h e random v e c t o r - d e n s i t y of t h e p a r t i c l e s a r e t h e corresponding d i - s t r i b u t i o n f u n c t i o n s of t h e p a r t i c l e s o f t h e plasma t 4 J

.

.

% e r c a l c u l a t i o n s ,we g e t :

1 ) . t h e v e c t o r - d e n s i t y of t h e p a r t i c l e s of t h e plasma;

2). t h e d i s t r i b u t i o n f u n c t i o n of t h e random v e c t o r ~ d e n s i t y of t h e p a r t i c l e s ;

3 ) . t h e f i r s t momrnt of t h e random v e c t o r - - d e n s i t y of t h e p a r t i c l e s ;

4 ).the a v e r a g e energy, t h e entropy and t h e f r e e energy of t h e plasma;

5 ) . t h e dimensions of t h e g a s d i s c h a r g e

~ l a s m a "cord", when t h e c u r r e n t through t h e plasma i s g i v e n ;

6 ) . t h e t ernaeratures of t h e e l e c t r o n s , t h e i& and t h e n e u t r a l s ;

7 ) . t h e v a l u e of t h e numerical p a r a m e t e r s of t h e Lenard-Gones p o t e n t i a l s , which ha- v e been d i s c o v e r e d by t h e method of s t a t i -

s t i c a l t e s t s ( t h e Mont e-Carlo method) ; 8). t h e average c o n c e n t r a t i o n of e l e c t r o n s , i o n s and n e u t r a l s ;

g ) . t h e corresponding d e g r e e s of i o n i z a t i - o n s , which depends on t h e average energy of t h e p a r t i c l e s of t h e plasma, on i t s geometry and t h e c h a r a c t e r of i n t e r a c t i o n of t h e p a r t i c l e s .

The f o u n d a t i o n of t h i s r e p o r t i s t h e r e - s u l t s of t h e a u t h o r s p u b l i s h e d b e f o r e [7-1 31

.

.

.

The method of s o l v i n g t h e problem. Out of t h e e q u i l i b r i u m c o n d i t i o n s and d e f i n i t i o n of t h e space l i m i t e d plasma we g e t t h e system of t h e d u a l i n t e g r a l Fredholm eq+

a t i o n s of t h e f i r s t k i n d , which a r e t o be s o l v e d i n common w i t h t h e n o r m a l i z i n g c o n d i t i o n s on t h e v e c t o r - d e n s i t y , of t h e p a r t i c l e s of t h e plasma. Having been made t h e n e c e s s a r y p r e l i m i n a r y p r e p a r a t i o n we do F o u r i e r c o n v o l u t i o n t r a n s f o r m of t h e d u a l i n t e g r a l Fredholm e q u a t i o n s of t h e f i r s t kind. S o l v i n g t h e g e n e r a l i z e d G i l - ^' b e r t problem we g e t t h e v e c t o r - d e n s i t y of t h e p a r t i c l e s of t h e plasma. Knowing t h e v e c t o r - d e n s i t y of t h e p a r t i c l e s we a r e a b l e t o examine i n d e t a i l t h e p r o p e r t i e s t h e g a s d i s c h a r g e *

he r e s u l t s . T h e r e a r e g i v e n below t h e r e - s u l t s concerning t h e weakly i o n i z a t e d g a s There a r e g i v e n on t h e p i c t u r e 1 t h e d i a - grams of t h e d e n s i t i e s of e l e c t r o n s , i o n s and n o u t r a l s . Using t h e v e c t o r - d e n s i t y of t h e articles of t h e plasma we d i s c o v e r t h e bhy s i c a l v a l u e s c h a r a c t e r i z i n g t h e

s t a t e of t h e r e a l g a s d i s c h a r g e plasma, f o r example t h e a v e r a g e p r e s s u r e of t h e plasma ( p i c t u r e 2 ) .

The c o n c l u s i o n s . The a u t h o r s have s o l v e d t h e w e l l known problem of N.N.Bogol.iubov

111 : t o c o n s t r u c t t h e s t a t i s t i c a l t h e o r y of plasma when i n t e r a c t i o n between t h e p a r t i c l e s i s d e s c r i b e d by t h e sum of the, Coulomb p o t e n t i a l and t h e Lenard-Gones p o t e n t i a l . Having been worked by t h e & u t h o r s t h e method of t h e s t a t i s t i c a l t h e o r y of f i n i t e c l u s t e r s r7-151 g i v e s i n p a r t i - c u l a r p o s s i b i l i t y t o a n a l y s e - completely t h e s t a t e of t h e r e a l g a s d i s c h a r g e

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19797328

plasma i n dependence of i t s geometry,the c h a r a c t e r of t h e i n t e r a c t i o n and t h e cur- r e n t going through t h e plasma. The c o n s i - d e r a t i o n h a s been done f o r t h e r e a l space l i m i t e d plasma being assumed t o be a ne- a r l y c l o s e d sustem. However t h e method of t h e a u t h o r s i s a b l e t o be used t o c o n s i - d e r t h e plasma a s a p h y s i c a l f i n i t e open system* ( L16] ).

P i c t u r e 1. The d e n s i t y of i o n s , e l e c - t r o n s , n e u t r a l s i n t h e p l a -

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References.

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2. S.K.Kuzmin, On t h e Minimization Prob- l e m of t h e Many-Component P o t e n t i a l Theory. Coll. llThe Modern Problems i n t h e Exact S c i e n c e s w , E d i t i o n UDN Moscow, 1977, p.17 ( i n Russian j.

3. S .K.Kuzmin, On t h e Many-Component P o t e n t i a l Theory, Dep. VINITI

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NO 2425-78, 14 j u l y 1978. ( i n R i s s l a n ) . 4. U.L. Klimantovitch, The S t a t i s t i c The+

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D i f f e r e n c i a l n i e Uravnenia, 5, 6 , 10364 1972 ( i n Russian ).

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12. S.W.Temko, S.K.Kuzmin, On t h e ~t"ili+

t y and Breaking of Plasma Bunches, 1977 IEEE, Troy, New York, USA, 1977.

13. S.W.Temko, ' ~ . ~ . ~ u & i n , On t h e D i s t r i - b u t i o n of t h e P a r t i c l e s i n t h e Weakly I o n i z a t ed G a s , 1978 IEEE, Mont e r ey ,

California,USA, 1978.

14. S.W. Temko, On' t h e General Theory of S t a b i l i t y of l a c r o s c o p i c a l Systems, Dep. V I N I T I , dep,EE 978-78, 21 M=chs 1978 ( i n Russian ).

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dep. N2 3110-78, 26 September, 1978 ( I n Russian ).

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P i c t u r e 2. The p r e s s u r e of plasma a s a f u n c t i o n of s p e c i f i c volume,