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Journal of Physics: Conference Series 717 (2016) 012074

doi:10.1088/1742-6596/717/1/012074

Stark effect modeling in the detailed opacity code SCO-RCG

J-C Pain1, F Gilleron1 and D Gilles2

1 CEA, DAM, DIF, F-91297 Arpajon, France

2 CEA, DSM, IRFU, F-91191 Gif-sur-Yvette, France

E-mail: jean-christophe .painScea. fr

Abstract. The broadening of lines by Stark effect is an important tool for inferring electron density and temperature in plasmas. Stark-effect calculations often rely on atomic data (transition rates, energy levels,...) not always exhaustive and/or valid for isolated atoms. We present a recent development in the detailed opacity code SCO-RCG for K-shell spectroscopy (hydrogen- and helium-like ions). This approach is adapted from the work of Gilles and Peyrusse. Neglecting non-diagonal terms in dipolar and collision operators, the line profile is expressed as a sum of Voigt functions associated to the Stark components. The formalism relies on the use of parabolic coordinates within SO(4) symmetry. The relativistic fine-structure of Lyman lines is included by diagonalizing the hamiltonian matrix associated to quantum states having the same principal quantum number n. The resulting code enables one to investigate plasma environment effects, the impact of the microfield distribution, the decoupling between electron and ion temperatures and the role of satellite lines (such as Li-like 1snln'l' — 1s2nl, Be-like, etc.). Comparisons with simpler and widely-used semi-empirical models are presented.

1. Introduction and outline of the method

In hot dense plasmas encountered for instance in inertial confinement fusion (ICF), the line broadening resulting from Stark effect can be used as a diagnostics of electronic temperature Te, density ne and ionic temperature Tj. This represents a challenging task, since from astrophysical dilute plasmas to ultra-dense nuclear fuel in ICF, the density varies by twenty orders of magnitude. The capability of the detailed opacity code SCO-RCG [1, 2] was recently extended to K-shell spectroscopy (hydrogen- and helium-like ions), following an approach proposed by Gilles and Peyrusse [3]. Ions and electrons are treated respectively in the quasi-static and impact approximations and the line profile reads

0(f) oc - /Re [Tr{d.X_1}j W(F)dF, (1)

n J L J

where X = 2in (v + v1) — iH(F)/h — Ac, v1 being the frequency of the lower state and H(F) = Ho — d.F the hamiltonian of the ion in the presence of an electric field F following the normalized distribution W(F). H0 is the hamiltonian without electric field while d and Ac represent respectively the dipole and collision operators. The trace (Tr) runs over the various states of the upper level. If AvD is the Doppler width and wk the weight of the kth Stark component, neglecting non-diagonal terms in dipolar and collision operators, the line profile can be written as a sum of Voigt (V) functions (parametrized as in Ref. [4]):

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Journal of Physics: Conference Series 717 (2016) 012074

doi:10.1088/1742-6596/717/1/012074

(v ) =

W (F )dF

J2wk (F )V (xk ,Vk)

Aud JO

where is the frequency of the line without external field and

_ v-vq- Cfc(F) _ (k\Ac\k)

XDH being the Debye-Hückel length.

2nAvD ' (2)

2. Hydrogen-like ions

Stark effect for hydrogenic ions can be calculated in parabolic coordinates using the basis states |nqme), where q = n1 — n2, n1 and n2 being the so-called parabolic quantum numbers, related by ni + n2 + |mi| + 1 = n, —I < mi < I being the magnetic orbital quantum number. The perturbation d is diagonal in this basis and a 2nd-order development gives

{nqme\ ~ d.F\nqme) = ~±nqF - ¿^JjL (l7n2 - 3(?2 - 9mj + 19) F2.

However, the fine-structure hamiltonian Flo is diagonal in the subset of states |nlsjmj). In order to diagonalize the total hamiltonian H in such a basis, the Stark matrix element is

(nlsjmj |— d.F Inl'sj'mj ) =

1/2 ra-1-|m£|,2

(-l)i+i'-i+3mi-ms-q-n[£,£',j,j ' ] 1/2

ms = -1/2 q=-(n- 1-|mi|) l S j

mi m g -mj

t \ / -I -I g

-mj 1 M ^ -me x(nqmil- d.Flnqmi), (5)

n— 1 n— 1

me-q me+q

l' s j' Ï

m£ mg -mj J

\ ( n— 1 n— 1

\ [ 2 2

mg-q mg+q

/ V 2 2

with s = 1/2, me + ms = mj and [x] = 2x + 1. Fig. 1 displays a comparison between our previous semi-empirical modeling (Refs. [5, 6]) and the present work in the case of Ar XVIII Lya line. Fig. 2 shows Mg XII Ly^ profile in three different conditions.

3. Helium-like ions

We consider the transitions 1snl 1P — 1s2, n > 2. For n > 5, the perturbation due to field F is much larger than the separation between terms, the levels are quasi-hydrogenic and He lines are modeled as Ly-like lines (with Z ^ Z — 1). For n < 5, singlet-triplet mixing is neglected and two-electron wavefunctions ^(1,2) of singlet states are built as [9]:

x Xa(+),s(-)

where ^f00 is the wavefunction of the fundamental state of an hydrogenic ion of charge Z and u^m,/, the wavefunction of the excited state nlml of an hydrogenic ion of charge Z — 1. Xa,s are antisymmetric (resp. symmmetric) spin functions. Such an approximation is valid

Journal of Physics: Conference Series 717 (2016) 012074

doi:10.1088/1742-6596/717/1/012074

^ 0,2 -Я

0,15 -

0,05 -

— Dimitrijevic-Konjevic + Rozsnyai _

— This work

3320 3330 3340 Energy (eV)

Figure 1. Lya line for an Ar plasma at Te=Tj=700 eV and p=3.98 g/cm3 (example chosen in Ref. [7]).

- n =7.0 1021 cn a-3, T =T=167 eV

- n =1.5 1022 cn T =T=170 eV

- n=3.0 1022 cn n-J, T =T=196 eV

1744 1746 1748 Energy (eV)

Figure 2. Ly^ line for a Mg plasma in three different conditions (example chosen in Ref. [8]).

for highly charged ions when the electron-nucleus interaction overcomes the Coulomb electron-electron repulsion. The hamiltonian H0 — e(z1 + z2)F is diagonalized in the sub-space of states 11s; n£m,f, S) with S=0 for singlet states and S=1 for triplet states. For Hea, the resonance line (1s2p 1P — 1s2) requires the energies of terms 1s2s 1S and 1s2p 1P and the intercombination line (1s2p 3P — 1s2) the energies of terms 1s2s 3S and 1s2p 3P.

Figure 3. Measured emission of aluminum "buried layers" heated by an ultra-short laser [10] (emissive volume: 400 im2 x 0.5 ¡m, duration: 3 ps) compared to SCO-RCG prediction.

4. Interpretation of a "buried-layer" experiment on aluminum

Fig. 3 shows our interpretation of the recently measured emission of aluminum micro-targets buried in plastic ("buried layers") and heated by an ultra-short laser [10]. Fig. 5 displays the He^ profile with different models for the microfield distribution function W(F) (see Fig. 4): Holtsmark model, neglecting ionic correlations and electron screening (see Ref. [11] and references therein), Mayer and "nearest neighbour" (NN) distributions, valid for strongly

Journal of Physics: Conference Series 717 (2016) 012074

doi:10.1088/1742-6596/717/1/012074

Figure 4. Different microfield distributions for an Al plasma at Te=Ti=310 eV and p=2.7 g/cm3 (units of fl = F/F0, where F0 = Z*e/r2ws).

1700 1800 1900

Energy (eV)

Figure 5. He^ line for the different microfield distributions of Fig. 4.

Model / FWHM Ly„ Ly/j Hea He/5

This work 1.80 27.03 2.46 32.84

Dimitrijevic-Konjevic 2.19 53.73 2.07 61.92

Table 1. FWHM of Ly^ and He«,^ lines for an Al plasma at Te=Ti=310 eV and p=2.7 g/cm3.

coupled plasmas, and a combination of APEX (Adjustable Parameter Exponential) method with Monte Carlo simulations proposed by Potekhin et al. [11] and parametrized by ionic coupling r = (Z*e)2 / (rwskBT) and electron degeneracy k = rws/ATF constants, rws being the Wigner-Seitz radius and Atf the Thomas-Fermi screening length. Table 1 shows the values of the full width at half maximum (FWHM) of Lya,^ and Hea,^ lines with the new and previous (Refs. [5, 6]) modelings of Stark effect in SCO-rCg.

In the future, we plan to investigate the importance of autoionizing states 1s2l2l' and 1s2l3l' of He^ (in the present work we only took into account 2121') and to include the line 1s3d 1D2 -1s2 1S0 induced by the field (mixing states 1s3d 1D2 and 1s3p 1P1) as well as the lines 1s3d 3D2 - 1s2 1So and 1s3s 3S1 - 1s2 1So. We also started to study the Stark-Zeeman splitting.

Acknowledgments

We would like to thank V. Dervieux, B. Loupias and P. Renaudin for the experimental spectrum.

References

[1] Pain J-C, Gilleron F and Blenski T 2015 Laser. Part. Beams 33 201

[2] Pain J-C and Gilleron F 2015 High Energy Density Phys. 15 30

[3] Gilles D and Peyrusse O 1995 J. Quant. Spectrosc. Radiat. Transfer 53 647

[4] Humlicek J 1979 J. Quant. Spectrosc. Radiat. Transfer 21 309

[5] Dimitrijevic M S and Konjevic N 1980 J. Quant. Spectrosc. Radiat. Transfer 24 451

[6] Rozsnyai B F 1977, J. Quant. Spectrosc. Radiat. Transfer 17 77

[7] Mancini R C, Iglesias C A, Ferri S, Calisti A and Florido R 2013 High Energy Density Phys. 9 731

[8] Nagayama T et al. 2014 Phys. Plasmas 21 056502

[9] Bethe H A and Salpeter E E 1957 Quantum Mechanics of one- and two- electron atoms (Berlin: Springer)

[10] Dervieux V et al. 2015 High Energy Density Phys. 16 12

[11] Potekhin A Y, Chabrier G and Gilles D 2002, Phys. Rev. E 65 036412