X-RAY BREMSSTRAHLUNG EMISSION DUE TO PLASMA SUPERTHERMAL ELECTRONS

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X-RAY BREMSSTRAHLUNG EMISSION DUE TO PLASMA SUPERTHERMAL ELECTRONS

M. Lamoureux, R. Pratt, L. Jacquet

To cite this version:

M. Lamoureux, R. Pratt, L. Jacquet. X-RAY BREMSSTRAHLUNG EMISSION DUE TO PLASMA SUPERTHERMAL ELECTRONS. Journal de Physique Colloques, 1987, 48 (C9), pp.C9-355-C9-358.

�10.1051/jphyscol:1987961�. �jpa-00227379�

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Colloque C9, supplement au n012, Tome 48, decembre 1987

X-RAY BREMSSTRAHLUNG EMISSION DUE TO PLASMA SUPERTHERMAL ELECTRONS

M. LAMOUREUX*,~', R.H. PRATT*" and L. JACQUET****(~'

" ~ a b o r a t o i r e d e Spectroscopie Atomique et Ionique, Universitk Paris-Sud, F-91405 Orsay Cedex, France

" " ~ e p a r t m e n t of Physics and Astronomy, University of Pittsburgh, Pittsburgh, PA 15260, U.S.A.

* * *

DRFC, CEN Fontenay-dux-Roses, BP 6 , F-92260 Fontenay-aux-Roses, France

R C s u m C . N o u s C t u d i o n s l a p o l a r i s a t i o n l i n 6 a i r e d u rayonnement d e freinage d D 2 la diffusion d'dlectrons d e quelques dizaines d e keV par des a t o m e s trSs ionisCs. A cause d e leur interet pour les plasmas d e fusion, n o u s e x a m i n o n s l a v a l i d i t 6 d'i5valuations n o n r e l a t i v i s t e s analytiques par comparaison avec d e s calculs relativistes exacts. N o u s prenons l'exemple des ions d e nickel qui s o n t prdsents & lti5tat d'impureti5.s dans les plasmas d e tokamak.

Abstract. We are studying the linear polarization of the bremsstrahlung emission resulting from the scattering of electrons of a f e w 10's of Kev by highly ionized atoms. I n view of i t s interest for f u s i o n plasmas, we investigate the appropriateness of non-relativistic analytical evaluations versus

"

^exact

"

relativistic numerical calculations. Nickel ions have been chosen f o r c a s e s t u d i e s because of their presence a s impurities i n tokamak plasmas.

T h e d e t a i l e d p o l a r i z a t i o n c h a r a c t e r i s t i c s of bremsstrahlung have been determined in the theoretical model d e v e l o p e d by T s e n g a n d P r a t t 1 T h e r a d i a t i o n i s determined i n the static screened potential of the ion, the transition matrices being calculated f r o m relativistic wave functions and including multipole effects. Besides the description of the model, the paper by T s e n g and Pratt / I / presents results for carbon and aluminium neutral atoms at incident electron energies E = 5 to 50 keV. H e r e , we apply this elaborate method of calculation to determine the polarization of radiation f r o m the completely stripped ion

~ i ~and f r o m the highly ionized atom ~i'" ~ +

.

T h e f i g u r e s show results when 9 5 % of the incident electron kinetic energy i s radiated, i. e. h 3 / E = 0 . 9 5 . T h e incident electron energies considered, E = 1 0 , 5 0 k e V , f a l l in the typical energy range of superthermal electrons in most L a b o r a t o r y p l a s m a s . T h e d e g r e e o f p o l a r i z a t i o n P = I - 1 / I I ) i s plotted versus the electron scattering angle 8

.

R e s u l t s obtained w i t h - t h i s detailed calculation correspond to the solid l i n e s of the figures ( t h e l i n e a r p o l a r i z a t i o n d e g r e e i s t h e t r a d i t i o n n a l y so-called C O 3 coefficient ) F i g u r e 1 s h o w s the case

~i~'+ f o r these two energies. W h e n the incident electron e n e r g y d e c r e a s e s f r o m E = 5 0 to E = 1 0 k e V t h e p o l a r i - zation curve i s lowered, and it tends to be more

(')present address : CEA Limeil, BP 27, F-94190 Villeneuve-Saint-Georges, France

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1987961

C9-356 JOURNAL DE PHYSIQUE

s y m m e t r i c a b o u t 9 0 '

,

t h u s l o s i n g i t s r e l a t i v i s t i c features. Fig. 2 compares results obtained in the Coulomb potential of NiZ8+ and in the partly screened potential of

~i'$+

,

for the 10 keV case. The effect of screening is relatively small at these energies, because they are larger than the typical energies of the remaining bound electrons.

Since the polarization features of bremsstrahlung are indeed predominantly Coulombic for these highly ionized atomic systems at these energies, we now investigate the validity of some analytical formulae which have been proposed for the Coulomb case and which would be simpler to use in the various applications to plasma physics. The simplest analytical formula corresponds to non-relativistic

Fig- 1. Degree of polarization of the bremsstrahlung emission f o r NiZ8+, at the two incident electron energies of E = 10 and 5 0 keV, vs. the scattering angle

.

Proportion of energy radiated h 3 / ~ = 9 5 %.

(-) : Full numerical results

( - - - - 1 : Values obtained from the Kirkpatrick and Wiedmann

formula / 3 / .

(--. -) : Non-relativistic Born approximation.

Born Approximation ( S e e for example ref. 2 for the expression of the cross section. W e obtained the results shown i n f i g s 1 and 2 by integrating it analytically over t h e d i r e c t i o n s of t h e s c a t t e r e d e l e c t r o n .

.

^{T h i s}

approximation leads to d e g r e e s of polarization which are independent of the degree of ionization Z* and depend only o n the ratio h 3 / ~ . As expected and illustrated i n Fig. 1 , the Born approach i s more appropriate at larger E , though i t still overestimates the degree of polarization by a large amount at large angles ( L e t u s mention -though w e did not u s e i t h e r e - that the relativistic Born approximation would i m p r o v e this aspect ). T h e o t h e r analytical formula w h i c h w e considered w a s proposed by Kirkpatrick and W i e d m a n n 131 f o r the intensity of radiation. T h i s model, w h i c h i s a n approximation to the full non-relativistic dipole Sommerfeld

Fig. 2. A s i n Fig. _{1 ,} but f o r the two ions Ni 2 8 + and N i ^{1 8 +} and a t the one energy E = 10 keV. ( F o r ~ i " +

,

the results corresponding to Kirkpatrick and Wiedmann are shown in the Coulomb potential for 2" = 1 8 . )

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theory, goes beyond Born approximation. T h e degree of polarization i s dependent on

z*,

E and h V / E and its values are much closer to the exact o n e s than in B o r n approximation ( S e e F i g s 1 and 2 ) . However, since the treatment i s non-relativistic, i t fails to reproduce the asymmetry around

8 =90 " w h i c h s e t s i n for the higher incident electron

energies. T h i s leads to a significant overestimate of the polarization a t large scattering angles, a s illustrated i n Fig.1 a t E = 5 0 keV. Fortunately, this shortcoming i s of little importance : the d i r e c t i o n of observation i s usually perpendicular to the a x i s of symmetry of the current anisotropic plasmas and the superthermal electrons are strongly directional a l o n g that axis. G e n e r a l l y , Figs. 1 and 2 s h o w a good agreement between the f u l l numerical results and the predictions f r o m Eq. 8-9 of Kirkpatrick and W i e d m a n n 131. We e v e n notice good agreement in Fig. 2 between the exact v a l u e s obtained from the screened potential of ~i''+ and the analytical results corresponding to the Coulomb potential 1 8 1 ~ . We c o n c l u d e that the analytical formula proposed i n ref. 3 i s of very satisfying validity f o r the present cases.

T h e availability of relevant analytical cross sections ( resolved in angle and polarization ) i s welcome w h e n studying the bremsstrahlung emission i n axi-symmetrical plasmas. A s a result, t h e integration o v e r a l l angles except f o r the pitch angle c a n be carried out analytically. As an example, w e have evaluated the bremsstrahlung polarization r e s u l t i n g f r o m a t o k a m a k p l a s m a o f s p e c i f i e d c h a r a c t e r i s t i c s , w i t h t h e c o r r e s p o n d i n g e l e c t r o n distributions determined from a simulation code 141. T h e h i g h l y a n i s o t r o p i c s u p e r t h e r m a l e l e c t r o n s p r o d u c e a bremsstrahlung emission w h i c h i s strongly polarized ( degree of polarization of 7 0 % and u p f o r a perpendicular d i r e c t i o n of observation). T h e study presented here enables us to take into account the anisotropy of the superthermal electrons and the polarization of the bremsstrahlung emission i n a relatively easy way.

(1) H. K. T s e n g and R. H . P r a t t , Phys. Rev. A 7, p 1502 (1973)

(2) H . A. B e t h e and E. E. Salpeter,Quantum mechanics of one-

and two-eiectron atoms,Plenum P u b l i s h i n g Corporation,= (1977).

(3) P. Kirkpatrick and L. W i e d m a n n , Phys. Rev. A , p 321 (1945)

(4) L. J a c q u e t and M. P a i n of TFR group. P r o c e e d i n g s of the 13 t h E u r o p e a n Conference o n Controlled F u s i o n and Plasma Heating, Schliersee, April 1986, p. 86.

Acknowledgments to A. B e k k a l i , A. P a i n , and M. A. Dubois.

T h i s w o r k h a s b e e n m a d e p o s s i b l e t h r o u g h a g r a n t CNRS-IRF-MPT

"

L e confinement magnCtique d e s plasmas d e f u s i o n

".