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ANGULAR DISTRIBUTION AND POLARIZATION OF THE CONTINUUM EMISSION IN PLASMAS
WITH TOROIDAL SYMMETRY
M. Lamoureux, J. Jacquet, R. Pratt
To cite this version:
M. Lamoureux, J. Jacquet, R. Pratt. ANGULAR DISTRIBUTION AND POLARIZATION OF THE
CONTINUUM EMISSION IN PLASMAS WITH TOROIDAL SYMMETRY. Journal de Physique
Colloques, 1988, 49 (C1), pp.C1-223-C1-225. �10.1051/jphyscol:1988145�. �jpa-00227465�
JOURNAL D E PHYSIQUE
Colloque C1, Supplbment au n03, Tome 49, Mars 1988
ANGULAR DISTRIBUTION AND POLARIZATION OF THE CONTINUUM EMISSION IN PLASMAS WITH TOROIDAL SYMMETRY
M. LAMOUREUX* * * *
,
J. J A C Q U E T * * * ( ~ ' and R.H. PRATT*"'* ~ a b o r a t o i r e de Spectroscopie Atomique et Ionique, Universite Paris-Sud, F-91405 Orsay, France
"'DRFC, CEN F-92260 ~ o n t e n a y - a u x - ~ o s e s , France
* * * Department of Physics and Astronomy, University of
Pittsburgh, Pittsburgh, PA 15260, U.S.A.
Resume. Les 6lectrons suprathermiques dans les plasmas sont habi- tuellement fortement directionnels, et cela entrafne une dspendance en angle et e n polarisation pour le rayonnement continu de rayons X. Dans ce travail, nous donnons les relations entre les fonctions de distribution anisotropiques f(v,e) et le degr6 d e polarisation de l f 6 m i s s i o n de recombinaison radiative directe et du rayonnement de freinage. Nous les appliquons ensuite 2 un plasma de tokamak aprss avoir d6terminP les f(v,B) par une simulation d e Monte Carlo.
Abstract. Superthermal electrons i n plasmas are usually strongly directional, and this confers angular dependence and polarization to the X-ray continuum radiation emitted. Here, we give the relations between the anisotropic distribution functions f(v,B) and the degree of polarization of the emission due to direct radiative recombination and bremsstrahlung. A n application i s then made to a tokamak plasma whose f(v,B) we obtained from a Monte Carlo code.
I n most plasmas for which the free electron distributions exhibit a superthermal tail, these highly energetic electrons are strongly anisotropic with regard to an axis of symmetry. Their distribution functions f(v,B)-v being the scalar velocity and @ the pitch angle (angle between the axis of symmetry and the direction, of the incident free electron)- can be evaluated from Monte Carlo simulations. Thus, in tokamak plasmas /I/, the motion of the test particle in the simulation i s determined as due to a toroidal electric force, to a n average drag and to an average randomly directed acceleration, and also if present to a perpendicular acceleration due to Electron Cyclotron Radiofrequency Heating. Fig. 1 shows results which we qbjtaine-% for a typical TFR plasma with electron density of 1 . 5 x 1 0 c m a n d "cold" t e m p e r a t u r e of a b o u t 1 keV. T h e superthermal electrons correspond to energies higher than about
1 5 keV, which was indeed confirmed by experimental observations 111.
T h e figure also illustrates the anisotropy of these superthermal electrons. T h e ensuing polarization of line emission has already been studied, especially in relation to astrophysics 121. We deal here with the resulting polarization of the continuum emission due to direct radiative recomkination (DRR) and to bremsstrahlung (Br). T o obtain simple estimates we restrict ourselves to dipole approximation, which is reasonably accurate for the situation considered here.
("present address : CEA Limeil, F-94190 Villeneuve-Saint-Georges. France
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1988145
C1-224 JOURNAL DE PHYSIQUE
Ln f ( v , % ) X distribution fore= 0'
distribution fore= 36'
I
in\:
---Maxwellian distribution for T = 1 keV XX X X x x
\ u X Y
. '
\
\
\
I \ I I I I I
10 20 30 40 50 60 2
mv / 2 in keV Fig. 1 : Free electron distribution in a TFR tomplasm
versus electron energy for the pitch angles = 0 and 36'.
Degree of polarization in %
RO,
Photon energy in keVFig. 2 : w e e of p l a r i z a t i o n f o r the
total
emission ( DRR+
B r )due t o the ~ i ~ * + impurity in a TFR tokamak plasna vs. photon energy.
The emissivity coefficients for the continuum radiation are calculated from the atomic cross sections of the elementary radiative process by integrating over the solid angles of the incoming and scattered electrons and over the incident electron velocities v (these last two integrations only for the bremsstrahlung process). By starting from analytical expressions for cross sections, we succeeded i n carrying o n all integrations analytically except those over the pitch angle and the velocity v, which involve data for f(v,%)- which is given numerically anyway. When using DRR cross sections i n the dipole length approximation, as Milchberg and Weisheit did / 3 / , we find that the polarization corresponding to the emission coming from a cone of pitch angle is :
2 2 0.75 b sin
Y
(3cos 8 - 1 )pDRR(h)) = 2 2
2
-
b+
1.5 b(1-cos2@ +0.75 b sin y(3cos &-1)'
where y i s the angle between the direction of observation and the axis of symmetry, and b is the asymmetry parameter for the shell into which the electron is recombining. T o obtain similar relatively simple expressions for the bremsstrahlung process, we have used a s our starting point two types of analytical cross sections /4//5/, after having checked their validity by comparison with exact values 161.
B o t h of t h e s e e x p r e s s i o n s l e a d to n e a r l y i d e n t i c a l p r e d i c t e d polarizations i n the present case. Making use of Born Elwert cross sections 4 , the degree of polarization for the pitch angle
e
(already summed over the polar angle of the incoming electron, and the solid angle of the scattered electrons) is :
2 2 -v 2
+
3v'P ~1 = sin
w
(1-~COSe) r
-3+ ~n-1 v + v l I
y
being as before the angle between the direction of observation and the axis of symmetry and v' the velocity of the scattered electron (h3= m/2 (vL-
v")).Using the emissivity coefficients of the tokamak plasma ta- k e n as a n example here, we obtained the polarization shown in fig. 2 for the total emission (DRR
+
Br) radiated by the impurity niZS+ at= 90". Because of the strong anisotropy of the superthermal electrons, the degree of polarization of the X-Ray continuum is large, even greater than 5 0 % for h v larger than 3 0 keV.
Because of the strong polarization of the continuum X-Ray spectrum induced by the anisotropy of the free electrons, the observation of the continuum emission (provided its polarization i s also measured) i s a good way not only to diagnose the superthermal temperature but also the directionality of the superthermal electrons.
Acknowledgments to A. Bekkali, A. Pain, and M. A. Dubois. This work has been made possible through a grant CNRS-IRF-MFT
"
L e confinement magngtique des plasmas d e fusion".
(1) L: Jacquet and M. Pain of TFR group. Proceedings of the 13 th European Conference o n Controlled Fusion and Plasma Heating, Schliersee, April 1986, p. 86.
(2) M.K. Inal and J. Dubau, J. Phys. B , 20, 4221 (1987) (3) H.M. Milchberg and J.C. Weisheit, Phys. Rev. A 26,
p. 1023 (1982)
(4) H. A. Bethe and E . E. Salpeter,Quantum mechanics of one -and two- electron atoms,Plenum Publishing Corporation,= (1977).
(5) P. Kirkpatrick and L. Wiedmann, Phys. Rev. A, p 321 (1945)
(6) M. Lamoureux, R. H. Pratt and L. Jacquet, proceedings of the 14 th International Conference on X-Ray and Inner-Shell Processes
(Paris, September 14-18), to be published at J. de Physique.
