ELECTRON EXCITATION RATES IN NON-MAXWELLIAN PLASMAS

HAL Id: jpa-00225851

https://hal.archives-ouvertes.fr/jpa-00225851

Submitted on 1 Jan 1986

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

ELECTRON EXCITATION RATES IN NON-MAXWELLIAN PLASMAS

M. Lamoureux, P. Alaterre, J. Matte

To cite this version:

M. Lamoureux, P. Alaterre, J. Matte. ELECTRON EXCITATION RATES IN NON- MAXWELLIAN PLASMAS. Journal de Physique Colloques, 1986, 47 (C6), pp.C6-57-C6-61.

�10.1051/jphyscol:1986608�. �jpa-00225851�

JOURNAL DE PHYSIQUE,

Colloque C6, supplQment au n o 10, Tome 47, octobre 1986

ELECTRON EXCITATION RATES IN NON-MAXWELLIAN PLASMAS

M. LAMOUREUX, P. ALATERRE* and J.P. M A T T E *

Laboratoire de Spectroscopie Atomique et Ionique, Bdt. 350, Universite Paris-Sud, F-91405 Orsay Cedex, France

' ~ n s t i t u t National de l a Recherche Scientifique-Energie.

Universite du Quebec, C.P. 1020, Varennes, Quebec, JOL 2P0, Canada

RdsumQ. Nous 6tudions l'excitation collisionnelle relative 2 des transitions permise et interdites d'ions h6liumoi'des dans deux types de plasmas non-llax- welliens : plasmas presentant une population suprathermique et plasmas ayant une distribution d'blectrons en ~ X ~ - ( V / V ~ ) ~

.

(z:

Abstract. Electron excitation rates relative to allowed and forbidden transi- tions of He-like ions are evaluated for two current types of non-bTaxwellian plasmas : plasmas witha suprathermal population and plasmas with electron dis- tribution functions of the form e ~ ~ - ( v / v ~ ) ~

.

1. Introduction.

In many natural and laboratory plasmas, the free electrons do not follow a Maxwellian distribution with kinetic temperature Te. First, in an extended group of plasmas, a tail of b(axwel1ian suprathermal electrons has to be affixed to the cold Maxwellian distribution. This situation is found in astrophysical plasmas [I], Z pinch plasmas [2] , tokamak plasmas with runaway electrons as well as in laser plasmas [ 3 , 4 , 5 ] subject to resonance absorption heating and/or parametric instabi- lities. Second, in another group of plasmas, the distribution is genuinely non-Max- wellian over the whole velocity range and takes the form e~~-(v/v,)~ with 2 < m < 5.

This is the case in plasmas perturbed by ion acoustic turbulences [63 and in laser plasmas where " classical absorption " is the dominant heating mechanism [7,8]. Last situation occurs when using short wavelength lasers, as is usually done to curb the generation of suprathermal electrons deleterious from the viewpoint of inertial fusion. Whichever the cause and the nature of the non-Faxwellian character of the plasma, it will affect the properties which predominantly depend on electrons of energies for which the distribution is far from Yaxwellian, that is on electrons energetic in comparison to the average kinetic energy kT

e' 2. Calculation of the excited rates for He-Fike ions.

We consider electron collisional excitations in He-like ions, one Is electron beinp: brought to a 2s or 2p subshell. We study the allowed transitions ls2 I S +]P

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1986608

JOURNAL DE PHYSIQUE

and t h e t h r e e f o r b i d d e n t r a n s i t i o n s + l ~ , 3 ~ , 3 s f o r t h e a t o m i c numbers Z= 3 , 6 , 8 , and 14. F o r e a c h e x c i t a t i o n c o n s i d e r e d , we u s e t h e c o l l i s i o n s t r e n g t h o b t a i n e d from a c l o s e c o u p l i n g model (1) [93 and e x p r e s s e d i n a p a r a m e t r i c way. I n t h e f o l l o w i n g ,

Fig. 1 E l e c t r o n d i s t r i b u t i o n f u n c t i o n s f o r two t y p e s o f non-Maxwellian ~ l a s m a s a t t h e ( c o l d ) k i n e t i c t e m p e r a t u r e kTe- S o l i d u p p e r c u r v e : plasma w i t h a s u p r a t h e r m a l ~ o ~ u l a t i o n , 5 % of t h e e l e c t r o n s c o r r e s ~ o n d i n g t o t h e h o t t e m p e r a t u r e 8 kT

.

X i s t h e e x c i t a t i o n t h r e s h o l d e n e r g y , E= 112 M , V ~ t h e i n c i d e n t e l e c t r o n e n e r g y and x = E/X t h e reduced e n e r g y . The r a t e s amount t o

- i ^{i = 5}

<TV)= ( 1 )

(Z - ^{i = O}

where a. i s t h e Rohr r a d i u s and f ( E ) ( v i z . f ( v ) ) t h e i s o t r o p i c d i s t r i b u t i o n f u n c t i o n n o r m a l i z e d by _$ffjv) v 2 dv = l . B e s i d e s b e i n g c o n v e n i e n t f o r n u m e r i c a l e v a l u a t i o n s , t h e f a c t t h a t t h e c o l l i s i o n s t r e n n t h i s w r i t t e n a s a sum o f terms i n r e l a t i o n ( 1 ) i n v i t e s f o i p h y s i c a l comments. Thus, we p o i n t o u t t h a t t h e i = 5 term i s p r e s e n t p n l y f o r t h e 1 s S + 'P e x c i t a t i o n w h i l e t h e i = 0 term i s p r e s e n t o n l v f o r t h e -+ S , 'P t r a n s i t i o n s . F o r t h e h y d r o g e n i c i o n s , t h e i o n i z a t i o n r a t e i s

p r o p o r t i o n a l t o < c r v h , p s t a n d i n g f o r t h e i o n i z a t i o n p o t e n t i a l . F o r t h e comdetelv S t r i p p e d i o n s , t h e b r e m s s t r a h l u n g e m i s s i v i t y c o e f f i c i e n t i s p r o p o r t i o n a l t o < v v ) o i n Kramers approximation,;l: c o r r e s p o n d i n g now t o t h e photon energy.

R e s u l t s f o r t h e s e f o u r c o l l i s i o n a l e x c i t a t i o n s a r e g i v e n below f o r v a l u e s of

JILT, r a n g i n g from 0.5 t o 10.

( 1 ) It would b e a r d u o u s and h a r d l y a f f e c t o u r r e s u l t s o r c o n c l u s i o n s t o i n c l u d e t h e a t o m i c r e s o n a n c e s n e g l e c t e d i n [9],and t r e a t e d and d i s c u s s e d i n [ l o ] .

3. E x c i t a t i o n r a t e s i n plasmas w i t h a s u p r a t h e r m a l ~ o ~ u l a t i o r ~

The r a t e i s a l i n e a r c o m h i n a t i o n of t h e Fraxwellian r a t e s r e l a t i v e t o t h e c o l d and h o t t e m a e r a t u r e s . F i q u r e 2 shows t h e e v o l u t i o n o f t h e " e l e m e n t n r v r a t e s "

<bv> i d e f i n e d i n ( 1 ) v e r s u s t e m p e r a t u r e i n t h e mono-Maxwellian s i t u a t i o n ; i t c a n e a s i l y be a c c o u n t e d f o r by l o o k i n a hack a t t h e x dependence i n ( 1 ) and a t t h e shane of t h e Maxwellian f d e x n - ( x $ / k T e ) . Conseauences on t h e f o u r e x c i t a t i o n s c o n s i d e r e d a r e shown i n f i g u r e 3 i n t h e example of O V I I f o r t h e hi-Maxwellian plasma whose d i s t r i b u t i o n i s i n d i c a t e d i n f i g u r e 1. A s e x o e c t e d , t h e r a t i o s i n c r e a s e g r e a t l y w i t h t h e l a r g e r v a l u e s of x / k T e , p a r t i c u l a r l y f o r t h e allowed t r a n s i t i o n .

F i g . 2 Elementary r a t e s ( 0 - v > ~ a s d e f i n e d i n r e l a t i o n ( 1 ) v e r s u s ~ T ~ I X (2 b e i n a t h e e x c i t a t i o n t h r e s h o l d e n e r g y ) i n Maxwellian plasmas a t t e m e r a t u r e Te.

The d r a m a t i c s e n s i t i v i t v of t h e r a t e s t o t h e p e r c e n t a g e of h o t e l e c t r o n s h a s a l r e a d y been emphasized, m o s t l y f o r c o l l i s i o n a l i o n i z a t i o n ; t h e r e s u l t i n a i o n i z a t i o n e q u i l i b r i u m i s t h e r e b y m o d i f i e d [ I ,11,12,133. It was a l r e a d y shown i n a few c a s e s t h a t t h i s f e a t u r e s h o u l d b e t a k e n i n t o a c c o u n t t o j u d i c i o u s l v i n t e r p r e t e x p e r i m e n t a l l i n e i n t e n s i t i e s L5,ll-j. As c l e a r l y p o i n t e d o u t h e r e ( F i g u r e 3 ), t h e f a c t t h a t t h e allowed and f o r b i d d e n t r a n s i t i o n s respond d i f f e r e n t l y t o t h e s u p r a t h e r m a l t a i l s h o u l d make one c a r e f u l when comparing i n t e n s i t i e s of l i n e s whose upper s t a t e s a r e p o p u l a t e d by d i f f e r e n t wavs. Thus, i n t h e p r e s e n c e of s u p r a t h e r m a l e l e c t r o n s , t h e t e m p e r a t u r e d i a g n o s t i c c u r r e n t l y used i n tokamaks [14] and c o n s i s t i n g i n c o m ~ a r i n p t h e i n t e n s i t i e s of t h e r e s o n a n t and f o r b i d d e n l i n e s of A L X I I would be i n v a l i d

,

c o u l d t h e n be changed i n t o a d i a g n o s t i c on t h e s u p r a t h e r m a l e l e c t r o n s . F i n a l l v t h e i n c r e a s e of t h e r a t e s w i t h t h e amount qf h o t e l e c t r o n s c a n be t a k e n a d v a n t a p e of i n working o u t X-Ray l a s e r schemes. H u ~ e a m p l i f i c a t i o n s o f t h e g a i n were t h u s f o r s e e n [12,13] by u s i n a s u p r a t h e r m a l e l e c t r o n s ( m o n o e n e r g e t i c f o r t h a t p u r p o s e ) which match t h e e x c i t a t i o n t h r e s h o l d e n e r g y o f t h e d e s i r e d u p p e r l a s i n g s t a t e .

JOURNAL DE PHYSIQUE

Ratios of " o n - ~ a x w . over M a x w . quantities

Fig. 3 R a t i o s of t h e non-Maxwellian r a t e s q v e r t h e i r Maxwellian

c o u n t e r p a r t s f o r t h e t r a n s i t i o n s I s ( f o r b i d d e n ) v e r s u s J / kTe,,Z b e i n g t h e t h r e s h o l d e n e r g y of t h e t r a n s i t i o n I S + 'P ( a l l o w e d ) , and

-

c o n s i d e r e d and Te t h e k i n e t i c ( c o l d ) t e m p e r a t u r e . Example of O V I I , u s i n g t h e d i s t r i b u t i o n s of f i g u r e I. The u p p e r art o f t h e f i g u r e c o r r e s ~ o n d s t o t h e d i s t r i b u t i o n w i t h a s u p r a t h e r m a l t a i l . The lower p a r t c o r r e s p o n d s t o t h e d i s t r i b u t i o n w i t h a d e p l e t e d t a i l ; a l s o shown a r e t h e r a t i o s fm=5 /fm,2 a t t h r e s h o l d and t h e r a t i o s of t h e < T V > ~ .

4. E x c i t a t i o n r a t e s i n plasmas w i t h e l e c t r o n d i s t r i b u t i o n s exp-(v/yn) m

.

Concerning t h e u n d e r d e n s e r e g i o n of a l a s e r plasma h e a t e d by i n v e r s e

b r e m s s t r a h l u n g , t h e r e l a t i o n o f 2 < m < 5 t o t h e ~ l a s m a p a r a m e t e r s i s g i v e n e l s e w h e r e [7,8]. It h a s been o b t a i n e d by c o n s i d e r i n g . a s e r i e s of l a s e r plasmas s t u d i e d w i t h Fokker P l a n c k s i m u l a t i o n s [15). Above 1 / 2 M , V ~ = 3kT, t h e d i s t r i b u t i o n D r e s e n t s a c h a r a c t e r i s t i c a l l y d e p l e t e d t a i l , a s s e e n i n f i g u r e 1 , and t h e r e f o r e l e a d s t o d r a s t i c a l l y reduced r a t e s , c o n t r a r y t o t h e above c a s e w i t h s u p r a t h e r m a l e l e c t r o n s . F i g u r e 3 shows t h e r a t i o of t h e non-Maxwellian r a t e s o v e r t h e i r Maxwellian

c o u n t e r p a r t s f o r t h e maximum v a l u e of m=5, t h a t i s when 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 c a n b e n e g l e c t e d [lb]. A t ; Y l k ~ ~ =5, t h e r a t e s a r e reduced by around two o r d e r s o f magnitude ( f o r m =3.5 t h e r e d u c t i o n i s of c o u r s e s m a l l e r b u t s t i l l i m p o r t a n t : one o r d e r of maanitude ). A s i l l u s t r a t e d i n F i g u r e 3 , t h e r a t i o fm=5 / fm=2 a t t h e t h r e s h o l d E=,X ( x = l ) r e p r e s e n t s a rough and o v e r e s t i m a t e d v a l u e o f t h e r a t i o of t h e r a t e s t h e m s e l v e s , b e c a u s e of t h e s t e e p d e c r e a s e of f w i t h E. On t h e o t h e r h a n d ,

t h e f a c t o r of r e d u c t i o n c o r r e s p o n d i n g t o 4 6 ~ 7 5 g i v e s an u n d e r e s t i m a t e o f them.

According t o what was s a i d i n p a r a g r a p h 2 , t h e f a c t o r s of r e d u c t i o n f o r ( b v ) 5 and (6~)- a r e i n d e e d what we found f o r c o l l i s i o n a l i o n i z a t i o n [83 and bremmstrahluna [17]. N o t i c e t h a t t h e f o u r e x c i t a t i o n r a t e s of t h e Re-like s y s t e m s , whether allowed o r n o t , a r e reduced by a n e a r l y i d e n t i c a l f a c t o r ( v e r v c l o s e t o t h e one

c o r r e s p o n d i n g t o < b v > - ) , a l m o s t i n d e p e n d e n t from Z w i t h t h i s a b s c i s s a s c a l i n g

*.

5. Conclusion.

T h i s s t u d y of c o l l i s i o n a l e x c i t a t i o n i n He-like i o n s shows i n which wav t h e c o l l i s i o n a l r a t e s a r e e x t r e m e l y s e n s i t i v e t o t h e f e a t u r e s o f t h e t a i l of t h e e l e c t r o n d i s t r i b u t i o n f o r h i g h v a l u e s of x / k ~ ~ ( : e x c i t a t i o n t h r e s h o l d e n e r g y ).

I f t h e t a i l i s i n f l a t e d , a s when s u o r a t h e r m a l a r e g e n e r a t e d , t h e r a t e s a r e i n c r e a s e d by amounts which depend on t h e e x c i t a t i o n c o n s i d e r e d and a r e h i a h e r f o r t h e allowed ones. T h i s p l a y s i n f a v o r of t h e h i a h l y e x c i t e d s t a t e s and s h o u l d b e k e p t i n mind when i n t e r p r e t i n g d i a g n o s t i c s based on l i n e i n t e n s i t i e s . On t h e c o n t r a r y , i f t h e t a i l i s d e p l e t e d , a s o f t e n i n t h e u n d e r d e n s e r e g i o n of l a s e r p l a s m a s , t h e r a t e s a r e d r a s t i c a l l y reduced i n a s i m i l a r f a s h i o n f o r a l l t r a n s i t i o n s , t h u s n e a r l y p r o h i - b i t i h g t h e c o l l i s i o n a l e x c i t a t i o n s . H e r e , we r e s t r i c t e d o u r s e l v e s t o i s o t r o p i c d i s t r i b u t i o n s , which i s v a l i d i n t h e l a t t e r p h y s i c a l s i t u a t i o n where p o l a r i z a t i o n e f f e c t s happen t o be n e g l i g i b l e 1 7 1 . However, when t h e hot' e l e c t r o n s a r e c r e a t e d a n i s o t r o p i c [ l ] o r d r i f t i n m a g n e t i c f i e l d s ( 1 8 1 , t h i s a s p e c t s h o u l d a l s o b e c o n s i d e r e d i n a complete i n v e s t i g a t i o n o f t h e r a d i a t i v e e m i s s i o n [l8] and o f t h e v a r i o u s r a t e s . When d e a l i n g w i t h l a s e r plasmas d u r i n g t h e i r r a d a t i o n o f t h e t a r e e t , we g e n e r a l l y have t o f a c e a non-Maxwellian plasma o f one k i n d o r t h e o t h e r . T h i s i n e v i t a b l e f a c t s h o u l d b e t a k e n c a r e of when m o d e l l i n g X-UV a m p l i f i c a t i o n i n s u c h mediums. It makes t h e c a l c u l a t i o n s l e n g t h i e r b u t b r i n g s w i t h i t t h e o v ~ o r t u n i t y t o f a v o r t h e a m p l i f i c a t i o n by p l a y i n g on t h e s e non-Maxwellian e f f e c t s , a s i t h a s a c t u a l l y been once s u g g e s t e d [12,13].

R e f e r e n c e s

1. ROUSSEL-DUPRE R., S o l a r P h y s i c s 68 (1980) 242 and 265

2. HAMMEL Bruce A. and JONES L a r r y A., Appl. Phys. Let. 7 (1984) 667.

3. KECK R. L.

,

11

4. YOUNG F. C., HERBST M. J., MANKA C. K., ORENSCHAIN S. P., and GARDER J. H., Phys. Rev. L e t t . 2 (1985) 2509.

5. EPSTEIN R., SKUPSKY S. and DELETTREZ J.,

J. Ouant. S o e c t r o s c . R a d i a t . T r a n s f e r 2 (1986) 131.

6. DUM C. T., Phys. F l u i d s 21 ( 1 9 7 8 ) 945 and 956.

7. LAMOUREUX M.. MATTE J. P.. MOLLER C. and Y I N R. Y..

P r o c e e d i n g s bf t h e 3 r d ~ k t e r n a t i o n a l Conference on R a d i a t i v e P r o p e r t i e s of Hot Dense M a t t e r , W i l l i a m s b u r ~ VA, U.S.A., 14-18 Oct. 1985, t o b e p u b l i s h e d a t World S c i e n t i f i c P u b l i s h i n g Co., S i n g a p o r e .

8. ALATERRE P., MATTE J. P. and LAMOUREUX M., t o b e p u b l i s h e d a t Phys. Rev. A.

9. WYNGAARDEN W. L., BHADRA K. and HENRY Donald J. W., Phys. Rev:= (1979) 1409.

10. PRADHAN A. K., NORCROSS D. W. and HUMMER D. G., Phvs. Rev. A23 (1981) 619.

-

11. JAYAKUMAR R. and FLEISCHMANN H. H.,

J. Quant. S p e c t r o s c . R a d i a t . T r a n s f e r , 2 (1985) 177.

12. APRUZESE J. P. and DAVIS J.,

P r o c e e d i n g s of t h e 2 d I n t e r n a t i o n a l Conference on R a d i a t i v e P r o p e r t i e s of Hot Dense M a t t e r , S a r a s o t a , F1. U.S.A.,Oct 31- Nov 4 1983, ed. by Davis J., Hoover C., Lee R.,. Merts A. and Rozsnyai R., p u b l i s h e d by World S c i e n t i f i c P u b l i s h i n g Co., S i n g a p o r e , ( 1 9 8 5 ) , p 221.

13. APRUZESE J. P. and DAVIS J., Phys. Rev.

A28

14. KLAPISCH M., p r i v a t e communication.(l986)

15. MATTE J. P., JOHNSTON T. W., DELETTREZ J. and McCRORY, Phys. Rev. L e t t . 53, (1984) 1461.

16. LANGDON A. B., Phys. Rev. L e t t . 46, (1980) 575.

17. LAMOUREUX M. MOLLER C. and JAEGLE P., Phys. Rev. (1984) 429.

18. MILCHBERG H. M. and WEISHEIT J. C., Phys. Rev. (1982) 1023.