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REMARKS ON THE X-RAY CONTINUUM SPECTRUM EMITTED BY A HOT PLASMA
C. Möller, R. Yin, M. Lamoureux
To cite this version:
C. Möller, R. Yin, M. Lamoureux. REMARKS ON THE X-RAY CONTINUUM SPECTRUM EMITTED BY A HOT PLASMA. Journal de Physique Colloques, 1987, 48 (C9), pp.C9-383-C9-386.
�10.1051/jphyscol:1987968�. �jpa-00227386�
REMARKS ON THE X-RAY CONTINUUM SPECTRUM EMITTED BY A HOT PLASMA
C. MOLLER, R.Y. YIN* and M. LAMOUREUX
Laboratoire de Spectroscopie Atomique et Ionique, Bat. 350, Universite Paris-Sud, F-91405 Orsay Cedex, France
'~epartment of Physics and Astronomy, University of Pittsburgh, Pittsburgh, PA 15260, U.S.A.
Resum5.- Nous calculons l'emission continue d'un plasma fortement ionis6, due au bremsstrahlung (br) et 5 la recombinaison radiative directe (DRR). Sur l'exemple de l'aluminium nous indiquons la precision atteinte lorsque l'on utilise des formules analytigues au lieu aes sectlors efficaces numeriques exactes our calculer le coefficient dt6missivit6 de recombinaison J ou lorsque ?'on n6glige la recombinaison sur les niveaux de n
>
2. AlorgR6ue pour un plasm maxwelllen le rap ort J est pres ue constant 5 l'intgrieur de chaque zone du spectre ( d 6 f i m i t 6 ~ ~ ~ ~ f e s seulls !e recombinaison successifs), il depend de 176nergie de photon cons~derCe pour mes plasmas dont la fonction de distribution Blectronique est de la forme exp-(v/v ) Le spectre total a alors un corn ortement qui est loin d'Ptre rnaxwellien, ma& iorsque l'on somme sur les dnergies Ke photons, la puissance totale 6mise est trPs proche de la valeur maxwellienne.Abstract.- We calculate the continuum emission due to bremsstrahlung (br) and direct radiative recombinaison (DRR) in a highly ionized plasma. Taking the example of Z = 13, we indicate the precision achieved when using analytical formulas for the DRR cross sections instead of exact numerical ones or when neglecting the recombination into shells of higher n. Whereas the' ratio of the emissivity coefficients J /J is nearly constant over ~ h o t o n energy for each spectrum zone (delimited byD%ccksive recombination thres olds) in a Maxwellian f lasma ,wiik depends on the ghoton energy considered if we deal wi&h non-Maxwellian p asmas electron distri ution functions of the form exp-(v/v )
.
The cumulative spectrum has a behaviour which is far from Maxwellian. However, wRen summed over photon energies, the resulting power loss is very close to the Maxwellian value.I
-
INTRODUCTION.
We have previously given general expressions for the bremsstrahlung (br) and direct radiative recombination (DRR) emissivity coefficients for the type of non-Maxwellian plasmas found in laser-irradiated targets below critical density /I/. We have then dealt only with bare emitting ions and used only Kramers cross sections for both atomic radiative processes. We shall now consider a plasma of a more general composition. Taking the example of an aluminium plasma, we will consi- der the ion mixture predicted by the hydrodynamical code FILM /2/ for current experimental conditions at the Nd laser of the GRECO "Interaction Laser-Matiire" at Palaiseau. For electron density and temperature of the order of Ne = 1021 cm-3 and Te = 1 keV, the ionic composition is approximately 25 % All3+, 50 % All2+ and 25 %
~1l'+.
I1
-
INFLUENCE OF THE TYPE OF DRR CROSS SECTION.
The total DRR emissivity coefficient JR tot is calculated from the so-called modified Kramers cross sections /3/ for the above mixture of the three
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1987968
C9-384 JOURNAL DE PHYSIQUE
ions and for recombination into shells of quantum numbers n= 1 to 7. These cross sections go beyond Kramers approximation in that, for the partially stripped ions, they involve the relevant quantity (Z
+
2*)/2,
Z being the atomic number and Z* the charge of the ion. At photon energies above the n = 1 thresholds of ~ 1 and ~1'~+, ~ ~ + the DRR emissivity strongly dominates the bremsstrahlung emissitivity (calculated in the Born-Elwert approximation) as can be seen on fig. 1. This comes from strong recombination into the n=l and n=2 shells and leads us to investigate the precision of the DRR cross sections. We therefore calculated the recombination into all n = 1 and n = 2 levels of the three ions considered with monoelectronic relativistic photoionization cross sections 141. To enable a rigorous comparison we have also calculated JR tot with the modified Kramers cross sections for recombination into n = 1 and n = 2 levels only. Figure 2, showing both brensstrahlung alone and total(br
+
DRR) emissivity JtOt, sums up our conclusions.Fig. 1 : Bremsstrahlung (br) and Direct Radiative Recombination (DRR) emissivity coefficienrns versus photon energy flo for electron distribution functions of the form
;;{-(v/v -mand ) with m
+.
= 2 (Maxwellzan) br... ,
DRR--- and. ,
and n = 3.5 br- . - ,
Above the n = 1 thresholds of ~1'~' and ~1'" the use of the exact numerical cross sections significantly lowers the emissivity (up to 20 % near threshold), whereas the disregard of DRR into the higher n shells leads to a hardly noticeable under-estimation of Jtot (3 % at most). This is due to the fact that recombination into the n = 1 levels is the main radiative process in that energy
and to BRR are of the same order of magnitude (best seen on fig. 1). But here, neglecting recombination into the upper levels leads to a larger error than making use of the modified Kramers cross sections and taking all levels up to n = 7 into consideration. Additional discrepancies arise from the fact that the simpler DRR evaluations do not differentiate between the various levels of the same n value. The emissivities obtained with modified Kramers cross sections are slightly over-estimated with respect to those calculated with the exact cross sections, but they are obtained at much smaller computational costs. At smaller photon energies, that is below the n = 3 thresholds, bremsstrahlung becomes even more important.
Pig. 2 : Bremsstrahlung (...) and total (br
+
DRR) emissivity coefticients versus ghyton eneTgy, calculated with a Maxwellian electron distri1.1ition function, DRRelng obtalned from :
-
exact numerical cross sections for recombination into levels n = 1 and 2.---
modified Kramers cross sections for recombination into levels n = 1 and 2.-.-
modified Kramers cross sections for recombination into levels n = 1 to 7.C9-386 JOURNAL DE PHYSIQUE
Bremsstrahlung emissivity coefficients are then more than an order of magnitude larger than DRK emissivity coefficients, and the consequence on the total emissivity of the choice of the type of DRR cross sections becomes completely negligible.
111
-
INFLUENCE OF THE ELECTRON DISTRIBUTION FUNCTION.
Up to now we have considered a plasma with a ElaxwelLian electron distribution function. If it is not the case, the preceding remarks remain valid but further conclvsions can be drawn. Bremsstrahlung !Jbr) and DRR (JK) enissivities have also been calculated with a distribution function of the form e ~ p - ( v / v ~ ) ~ with m = 3.5 as found in the underdense region of laser plasmas.
We can see in fig. 1 that for results obtained with a Maxwellian distribution function, the ratio Jbr/JK remains nearly constant vithin each of the spectrum segments delimited by the successive ionization thresholds (as long as Kramers cross sections are reasonable, that is not at low photon energies where the bremsstrahlung curve In Jbr is far from being a linear function of 5 4 , ) . This is due to the fact that, when using Krarners cross sections, the ratio J ~ ~ is a function / J ~ ~ of the ionization potential In and independent of 5 w for recombination into each n level of each ion.
The curves for the m = 3.5 distribution function show no such proportionality. The ratio J ~ ~are not constant when energy varies and J ~ " / J ~ ~ being a function of ( 5 w
-
In) the proportionality constant is not the same for all ternis of the sum leading to Jtot. This difference in the emissivity coefficients seriously affects the traditional temperature diagnostic based on the slope of InJ with respect to 5 w /5/. However, when summing over photon energies, the total emitted power, calculated with the m = 3.5 distribution function ends up, because of cancellation effects, to be very close to the Haxwellian value. It is only about 5 % less. This last remark is of interest for plasza simulation codes : after subtraction of the continuum radiative losses the amount of energy available for other processes (ionization...) is hardly affected by the non-Viaxwellian character of this type of plasxa.Acknowledgments
.-
We thank Dr R. H. Pratt for having enabled us to carry out the photoionization study and Dr C. Chenais-Popovics for discussions on the thermodynamical characteristics.References
/1/ M. LAbIDUREUX, C. M ~ L L E R and P. JAEGLE, Phys. Rev. A
30
(19R4), 429./2/ J.C GAUTHIER J.P. GEINIIRE, N. GKANUJOUAN and J. VIKMONT, J. Phys. i) : Appl.
Phys. (1983), 121.
/3/ YOUNG SOON KIM and R.H. PRATT, Phys. Rev. A
27
(1983), 2913./4/ I.B. GOLDBERG, University of Pittsburgh, Internal Report No YITT-291 (1982) unpublished.
/5/ W. LOCHTE-lIOLTGREVEN, in "Plasma Diagnostics", edited by W. Lochte-Holtgreven (North-Holland, Amsterdam, 1968), p.183.
