Spectral line shapes modeling in laboratory and astrophysical plasmas

New Astronomy Reviews 53 (2009) 272–276

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Spectral line shapes modeling in laboratory and astrophysical plasmas R. Stamm a,*, D. Boland a, M. Christova a,b, L. Godbert-Mouret a, M. Koubiti a, Y. Marandet a, A. Mekkaoui a, J. Rosato a a b

Physique des Interactions Ioniques et Moléculaires, UMR6633, Université de Provence et CNRS, Marseille, France Department of Applied Physics, Technical University-Sofia, BG-1000 Sofia, Bulgaria

a r t i c l e

i n f o

Article history: Available online 6 August 2009 Keywords: Stark broadening Zeeman effect Doppler broadening Turbulence

a b s t r a c t An overview of several spectral line shapes studies of common interest in astrophysical and laboratory plasmas is presented. For lines dominated by Stark broadening, approaches taking into account the dynamics of numerous perturbers are sometimes required. We briefly recall ab initio simulation techniques and model microfield methods used for such conditions. Together with the impact approximation, such models may also be used for studying the effects on a line profile of a magnetic field of the order of the tesla, allowing the diagnostic of stellar objects or magnetic fusion devices. The problem of the apparent spectral line emitted in a plasma affected by strong fluctuations of the plasma parameters is discussed for the case of optically thin plasmas. Ó 2009 Elsevier B.V. All rights reserved.

1. Introduction Astrophysicists share with laboratory plasma physicists an interest for the details of line shapes emitted in plasmas. Many studies have demonstrated that an exhaustive analysis of spectral line shapes provides invaluable information on the emitter environment (Popovic and Dimitrijevic, 2007; Stehlé, 1995). Ongoing progresses in observations drive a parallel effort in the modeling, since there can be no accurate diagnostic without a deep understanding of the emitter interaction with the plasma. Emphasizing on hydrogen emitters, we discuss in this paper the line shapes emitted by atoms perturbed by charged particles. These studies are of a particular interest for plasma diagnostic, and also provide input data for investigating plasmas with the help of numerical models. Astrophysical applications are numerous, ranging from diagnostic issues such as the knowledge of effective stellar temperature, and the surface gravity. Line shapes enter in the modeling of different regions of a star, like the stellar atmosphere and its opacity properties, stellar winds, various kinds of stellar interiors, and of many extragalactic objects. A very similar use of line shapes is found in laboratory or fusion plasmas. For instance, the modeling of edge plasmas in magnetic fusion devices requires detailed profiles for an accurate diagnostic and complete understanding of the plasma properties. In the following, we illustrate with several examples some of the tools used in our laboratory for modeling the line shapes emitted in weakly coupled plasmas. An old problem concerns the many body dynamic interactions with the emitter which occur for a large * Corresponding author. Tel.: +33 491288621; fax: +33 491670222. E-mail address: [email protected] (R. Stamm). 1387-6473/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.newar.2009.07.011

range of plasma conditions. We will present some of the approaches which are useful, on the one hand for an accurate and ab initio description of the line shape, and on the other hand for obtaining a large amount of profiles by a computer efficient model for a real time diagnostic. Interesting application conditions for these models concern magnetized plasmas found in astrophysics or magnetic fusion plasmas. For magnetic fields of several teslas, there can be a complex interplay of Stark and Zeeman effect, and line shape models in plasma conditions of interest in astrophysics and magnetic fusion will be discussed with the help of an impact approximation. A more recently studied problem of general interest is the modeling of spectra observed in a plasma with spatially and temporally fluctuating parameters, a common situation in astrophysical conditions. Magnetic fusion plasmas and some laboratory experiments also experience strong fluctuations for instance due to drift wave turbulence. We have identified plasma conditions for which a simple statistical model can be applied for a calculation of the apparent line shape in such plasmas. Results for our model will be shown for Doppler dominated profiles.

2. Stark effect Stark broadening may affect line shapes as the electronic density becomes significant. Hydrogen or helium lines are usually dominantly affected, but lines of all elements may be concerned in dense plasmas. In dense laboratory or astrophysical plasmas, a standard Stark broadening approach consists in a binary impact approximation for the electron perturbers, together with a static approximation for the ionic perturbers (Griem, 1964). For many laboratory or astrophysical plasma conditions, ion dynamic effects

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have to be retained since the time of interest of the transition considered (defined as the inverse of the line width) may be comparable to the ion collision time (defined for the ions as their average interparticle distance r0 over their thermal velocity v0). For hydrogen lines emitted from levels of a low principal quantum number n, with a plasma in the eV range, this may occur for densities as high as Ne = 1017 cm3. On the other hand, the impact approximation for the electrons becomes more and more questionable as lines with large principal quantum numbers are considered (n > 10–12). In such cases the electron broadening mechanism is similar to the ion dynamics problem, since it involves many body dynamic interactions between charged particles and the emitter. Several approaches exist for a realistic description of line shapes formed during an interaction with many moving perturbers. As an external magnetic field can be present, one has in general to distinguish between the different directions of polarization for the line shape. In units of seconds/radians, the line shape I^e ðxÞ for radiation polarized along the unit vector ^e, is given by a one sided Fourier transform:

I^e ðx; ~ BÞ ¼

1

p

Re

Z

1

C ^e ðt; ~ BÞeixt dt;

ð1Þ

0

where the dipole autocorrelation function C ^e ðt; ~ BÞ is defined by the following trace over the atomic states:

C ^e ðt; ~ BÞ ¼ Trh^e  ~ Dð0Þ^e  ~ DðtÞqi:

ð2Þ

In this expression ~ D is the dipole operator, the angle brackets denote an average over all charged perturbers, and q is the density matrix for the atom only. Using Eqs. (1) and (2), and a kinetic model for obtaining the elements of the density matrix, non local thermodynamic equilibrium conditions may be taken into account. This point will not be considered below, where we will focus on conditions such that all the substates of the atomic levels are equally populated and both the upper and lower levels of the radiative transition are in local thermodynamic equilibrium. For a plane wave propagating in a direction given by a vector ~ k (direction of observation), the general state of polarization can be described by combining two independent linearly polarized plane k (Branswaves with polarization vectors ^ ek ðk ¼ 1; 2Þ orthogonal to ~ den and Joachain, 1983). This leads to the general expression of the Stark–Zeeman line shape for a single atom in the case of an observation with an angle h0 with the magnetic field: 2

IðxÞ ¼ cos2 h0 Ill ðxÞ þ sin h0 I? ðxÞ;

ð3Þ

where the difference between the line shape observed in directions parallel (Ill) and perpendicular (I\) to the magnetic field are due to the radiation polarization properties resulting from Eq. (2) (Nguyen-Hoe et al., 1967). The dipole operator at time t may be expressed with the time evolution operator U(t) of the emitter:

~ DðtÞ ¼ U þ ðtÞ~ Dð0ÞUðtÞ:

273

dinger equation for U(t). This procedure is repeated a large number of times in order to perform the statistical average present in Eq. (2). With three decades of development, such computer simulations have now been used many times to benchmark studies on the effect of the emitter–perturbers dynamic on a line shape (Stamm and Voslamber, 1979; Stamm et al., 1986; Calisti et al., 1987; Gigosos and Cardenoso, 1987; Stambulchik et al., 2007). With the help of molecular dynamic techniques several other physical phenomena have been studied, like the correlation effects (Calisti et al., 2008), or the various contributions to asymmetric broadening (Wujec et al., 2002). An example of another study concerns the effect of time ordering in the evaluation of the evolution operator (Rosato et al., 2008b). In Fig. 1 we show the effect on the Lyman a dipole correlation function due to a single collision with an ion. The calculation has been performed with and without the time ordering effect, revealing the large error which would result by omitting this dynamic quantum effect. Retaining time ordering in a collision operator for the electrons, and in a numerical simulation for the ion leads to line shapes which are in a close agreement with experimental observations for this line. In Fig. 2 we have plotted the central part of Lyman a obtained from a calculation for a density Ne = 21017 cm3, and a temperature of 15,500 K, in an argon plasma containing traces of hydrogen (Geisler et al., 1981; Stamm et al., 1984). It can also be seen in Fig. 2 that a static ion approximation leads to a profile narrower by more than a factor 2, demonstrating on this line the role of ion dynamics even for such high densities. Other approaches are used today for retaining the dynamic effects on the profiles due to multiple collisions, and give accurate line profiles with much less computer intensive calculations. Among those, the so called model microfield approaches have proved to provide a realistic description of the line shapes by using statistical properties of the microfield, together with a procedure for the field time evolution. The Model Microfield Method, first proposed by Brissaud and Frisch (1971), has been developed and applied to a large range of astrophysical conditions by Stehlé and Hutcheon (1999). Using an efficient procedure for mixing Stark components, the Frequency Fluctuation Model, another fast approach developed in our laboratory (Talin et al., 1995), is available as a line shape code able to compute an arbitrary atomic or ionic line. 3. Effects of the magnetic field The presence of a static magnetic field is common for many types of plasmas. Laboratory plasma experiments use a magnetic

ð4Þ

The time evolution operator U(t) obeys to the following Schrödinger equation:

ih

dUðtÞ ¼ HðtÞUðtÞ; dt

ð5Þ

where H(t) = H0 + V(t), with H0 being the time independent atomic Hamiltonian (possibly including a static magnetic term) and VðtÞ ¼ ~ D ~ EðtÞ the interaction potential between the emitter and the electric field of all surrounding charged particles. In order to go beyond the standard approximation of Stark broadening, numerical simulations are suitable. Basically, an ab initio technique consists in a computer simulation of the motion of a large number of charged particles, followed by a numerical solution of the Schrö-

Fig. 1. Lyman alpha dipole correlation function calculated for a single collision with (solid line) and without (dashed line) time ordering effects.

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Fig. 2. Profile of Lyman alpha line for Ne = 21017 cm3, and a temperature of 15,500 K, in an argon plasma containing traces of hydrogen. The simulation calculation retaining ion dynamic (solid line) is compared to an experiment (dotted line, Geisler et al., 1981), and to a calculation using static ions (dashed line).

field for confining the plasma and controlling its interaction with the wall. In magnetic confinement fusion devices operated with intense magnetic fields the radiative properties are strongly dependent of the magnetic field. With the decision of building the ITER tokamak, which is designed for high density steady state discharges, there is a new interest for modeling the line shapes affected simultaneously by Zeeman and Stark effects, requiring a specific approach (Günter and Könies, 1999; Godbert-Mouret et al., 2001; Adams et al., 2002). In astrophysics, magnetic fields have been detected in several types of main sequence stars, in white dwarfs and neutron stars. Whereas the typical field strength on the surface of some main sequence stars is about as strong as the magnetic field of a good permanent horseshoe magnet (0.1 T), fields of the order of 10–105 T are measured in white dwarfs, and can be another factor 104 larger for some neutron stars. Also very weak field strengths are measured today with the help of Zeeman splitting, implying that quite different analysis are required for these different objects. In the following we will discuss about spectra observed in the tesla range, since those are of present interest in magnetic fusion. In such cases, the interest for detailed line shapes is motivated by the need of accurate diagnostics and the modeling of photon transport with numerical codes (Reiter et al., 2002). For the opacity calculations, the dominant contribution is provided by Lyman alpha photons, motivating the need of ultra fast, but accurate line shape model for this optically thick line. Since the final convolution with the simulated velocity distribution function is performed by a transport code, we present a Lyman alpha line shape without Doppler effect, but retaining Zeeman and Stark effect. For magnetic fields of the order or larger than a tesla, the Lyman alpha fine structure splitting is smaller than the magnetic interaction energy. Let us thus discuss Zeeman–Stark effect in the spherical n, l, m states, thus using the strong magnetic field limit. In a perpendicular observation, the spectrum consists of a central component (p component) connecting the l = 1, m = 0 substate to the groundstate (n = 1, l = 0, m = 0), and two symmetric (r components) connecting the m = ±1 substates to the groundstate (the three components forming the so-called Lorentz triplet). Now, adding an electric field interaction will broaden differently the central and lateral component for conditions where the magnetic term is larger than the electric one. The central component will be dominantly broadened by the linear Stark effect resulting from the coupling of the l = 0, m = 0 and l = 1, m = 0 substates, whereas the lateral components are mainly broadened by quadratic Stark effect

resulting from the coupling of Zeeman shifted l = 1, m = ±1 substates with the other substates. Our first calculations have been performed analytically, by using an impact approximation (Griem et al., 1959) for both ions and electrons. An efficient way of taking into account the effect of the magnetic field consists in using the impact formalism set up for non degenerate atoms by Griem et al. (1962). The Zeeman splitting may indeed be retained with what we call a magnetic cutoff in the a(z) and b(z) functions defined by Griem et al. (1962), resulting in a narrowing of the lateral components (Rosato et al., 2009). In Fig. 3 it can be seen that compared to a standard impact theory (Griem et al., 1959) our model predicts a strong narrowing and a significant additional shift on the lateral component of Lyman alpha for Ne = 1013 cm3, T = 1 eV, in a magnetic field of 5 T. This effect has been confirmed using the numerical simulation approach described in Section 2, and studied in conditions of higher densities for which the impact approximation for the ions is no longer valid (Rosato et al., 2009). 4. Role of fluctuating plasma parameters Many kinds of plasmas are affected by the turbulent fluctuations of their particle density, temperature and velocity. Turbulence is indeed ubiquitous in the interstellar medium (Elmegreen and Scalo, 2004), and is studied since it is strongly involved in the process of star formation (McKee and Ostriker, 2007). Surveys of the local interstellar medium using different emitters allow to discriminate between thermal and turbulent broadening by using the line widths of both light and heavy emitters (Redfield and Linsky, 2004). Observations of atomic and molecular spectra suggest that the interstellar gas is self-organized, may have a fractal structure, and exhibit intermittency properties (Lequeux, 2002). Turbulence effects are also important in the modeling of cool dwarf stars (Barklem et al., 2002), because the wings of the hydrogen lines are believed to form in deep layers which may be affected by convection. In the following, we again focus on hydrogen emission which often plays an important role in the search of a spectroscopic signature of turbulent ionized gases (Westmoquette et al., 2007). Our study is fitted to the plasmas of tokamaks which are affected by low frequency turbulence, but other kind of plasmas may be concerned. This turbulence affects the atomic hydrogen emission through populations created by charge exchange with the protons, and therefore the emitted Doppler line shapes reflects the ion velocity distribution function. The ion turbulence is produced by small scale instabilities, induced by gradients. It may be described by fluid models (Horton, 1999), which predict fluctu-

Fig. 3. Profile of a Lyman alpha lateral component for Ne = 1013 cm3, a temperature of 1 eV, and a magnetic field of 5 T. A standard impact calculation (dashed line) is compared to our impact calculation including a magnetic cutoff (solid line).

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ations rates reaching several tens of percent. For tokamak edge plasmas, their characteristic fluctuation time is of the order of a few 10 ls, and their typical spatial length is about 10 Larmor radius (The Larmor radius is the ion gyroradius in the magnetic field, mp v? , where v\ is defined for a proton of mass mp and charge e as eB the ion velocity component transverse to the magnetic field B). The characteristic turbulent time is usually much larger than the typical time scale of the atomic processes, such as for instance the line shape time of interest. On the other hand, the acquisition time sm for a line shape measure is generally much larger than this turbulent fluctuation time. In the following, we restrict ourselves to the case of optically thin plasmas, with the photon mean free path large compared to the typical fluctuation length. A line shape measurement thus performs a spatial and time average over fluid cells having each a specific set of values for the density, velocity and temperature:

Imes ðxÞ ¼

1

sm

Z sm Z L 1 I0 ðx; z; tÞdzdt; L 0 0

ð6Þ

Fig. 4a. Profile of the Deuterium Balmer alpha line calculated with (solid line) and without (dashed line) a temperature fluctuation rate of 50%, for an average temperature of 1 eV.

where I0 is the local profile at location z and time t, and L is the length of the line of sight. Using a statistical approach, it is convenient to introduce a set of sample variables Y associated with the fluid variables. It has been shown (Marandet et al., 2005, in press) that the measured profile may be expressed as an integral over Y of the local profile I0(x, Y), multiplied by the local brightness B(Y), and weighed by the probability density function (PDF) of the fluctuating hydrodynamic variables W(Y):

Imes ðxÞ ¼

Z

dYWðYÞBðYÞI0 ðx; YÞ:

ð7Þ

The PDF of only one hydrodynamic variable has been studied theoretically (Marandet et al., 2006), and has been measured in a few devices (Graves et al., 2005). Since this quantity is of great interest for characterizing the plasma turbulence, the link made with the line shape may lead to a useful analysis of turbulence by spectroscopy. A recent application of our model concerns the study of temperature fluctuations on a Doppler profile. Using for the local profile I0(x, T) a thermal Gaussian function, the apparent (measured) profile is given by:

Iapp ðxÞ ¼

Z

1

dTWðTÞBðTÞI0 ðx; TÞ:

ð8Þ

0

For densities lower than 5  1013 cm3, the temperature of electrons and ions are weakly correlated, and it is possible to show (Marandet et al., 2006) that the wings of a spectral line are governed by the asymptotic decay of the ion temperature PDF. In the high density regime the large collision rate ensures that electron, ion and neutral temperatures are equal (Rosato et al., 2008a). At this stage, we need to choose an explicit temperature PDF W(T). Experimental and theoretical studies of the plasma parameter fluctuations reported in the literature often use two different PDF, the gamma and log-normal functions (Rosato et al., 2008a). We shall use the gamma function for the calculations which follow, but note that using the log-normal function would not significantly change our conclusions. The gamma PDF is defined by:

WðTÞ ¼

1 T l1 expðT=qÞ; ql CðlÞ

ð9Þ

where C is the gamma function, and q and l are related to the mean temperature and variance by hTi = lq and hDT2i = lq2. The fluctuation p rate is given by 1/ l and in the following it will be expressed in percentage. The brightness B(T) has been obtained from a collisionalradiative model (SOPHIA code) for the Balmer alpha line (Rosmej et al., 2006), and its main characteristic is to be strongly peaked

Fig. 4b. Same as Fig. 4a for an average temperature of 20 eV.

at about 1.5 eV. Ignoring Zeeman effect, we have calculated the apparent profile of the Deuterium Da Balmer line for average temperatures of 1 and 20 eV, and compared between profiles obtained with and without fluctuations. In Fig. 4a, the Da apparent profile obtained for a temperature of 1 eV is broadened by a fluctuation rate of 50%. For a temperature of 20 eV, it is remarkable that the same fluctuation rate results in a strong narrowing of the line (Fig. 4b). It can be shown that the effect of fluctuations switches from a broadening to a narrowing as the average temperature increases to a temperature where the brightness function is maximum. This behavior may be understood by analyzing the contribution of each function entering in the integral in Eq. (8) (Rosato et al., 2008a; Marandet et al., in press).

5. Conclusion Several line shape problems are of a common interest in laboratory and astrophysical plasmas. An accurate description of Stark profiles requires the modeling of the emission by atoms perturbed by a large number of moving charged particles. An ab initio simulation provides an accurate result, and helps in the understanding of specific physical situations. Less computer intensive calculations are provided by the Model Microfield Method or the Frequency Fluctuation Model. The combined Zeeman–Stark effect may be calculated by the same approaches but also, in the low density limit, by a suitably

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modified impact theory. We have shown that the Zeeman splitting can strongly reduce the Stark broadening of shifted components. Another domain of common interest is the possible role of hydrodynamic turbulence on the line shapes. Using a statistical approach, our study shows that a link exists between the apparent line shape and the PDF of the fluctuating variables. Our study of the Deuterium Balmer alpha line in a plasma affected by strong temperature fluctuations demonstrates that if a broadening is observed for averaged temperature smaller than 1.5 eV, an actual narrowing of the line occurs for larger temperatures. Acknowledgements This work is part of a collaboration (LRC DSM 99-14) between the Laboratory of Physique des Interactions Ioniques et Moléculaires and the Fédération de recherche en fusion par confinement magnétique. It has also been supported by the Agence Nationale de la Recherche (ANR-07-BLAN-0187-01), Project PHOTONITER. References Adams, M.L., Lee, R.W., Scott, H.A., Chung, H.K., Klein, L., 2002. Phys. Rev. E 66, 066413. Barklem, P.S., Stempels, H.C., Allende Prieto, C., Kochukhov, O.P., Piskunov, N., O’Mara, B.J., 2002. A&A 385, 951. Bransden, B., Joachain, C., 1983. Physics of Atoms and Molecules. Longman Scientific and Technical, Essex. Brissaud, A., Frisch, U., 1971. J. Quant. Spectr. Rad. Transfer 11, 1767. Calisti, A., Stamm, R., Talin, B., 1987. Europhys. Lett. 4, 1003. Calisti, A., Talin, B., Ferri, S., Mossé, C., Lisitsa, V., Bureyeva, L., Gigosos, M.A., Gonzalez, M.A., del Ro Gaztelurrutia, T., Dufty, J.W., 2008. In: Spectral Line Shapes, vol. 15, AIP, vol. 1058, p. 27. Elmegreen, B.G., Scalo, J., 2004. Annu. Rev. Astron. Astrophys. 42, 211. Geisler, M., Grützmacher, K., Wende, B., 1981. In: Spectral Line Shapes, vol. 1. de Gruyter, p. 27.

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