# Collisional line profiles of sodium perturbed by H2 from the optical to the infrared spectral range

Nicole F. Allard, J. Kielkopf, Fernand Spiegelman, G. Tinetti, J.P. Beaulieu

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## To cite this version:

Nicole F. Allard, J. Kielkopf, Fernand Spiegelman, G. Tinetti, J.P. Beaulieu. Collisional line profiles of sodium perturbed by H2 from the optical to the infrared spectral range. EAS Publications Series, 2012, 58, pp.239-242. 10.1051/eas/1258038. hal-00862175

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Distributed under a Creative Commons Attribution 4.0 International License
COLLISIONAL LINE PROFILES OF SODIUM PERTURBED
BY H₂ FROM THE OPTICAL TO THE INFRARED
SPECTRAL RANGE

N.F. Allard<fnref n="1" />,<fnref n="2" />, J.F. Kielkopf<fnref n="3" />, F. Spiegelman<fnref n="4" />, G. Tinetti<fnref n="5" />
and J.P. Beaulieu<fnref n="2" />,<fnref n="4" />

**Abstract.** Our aim is to present new unified line profiles of sodium perturbed by H₂ from the optical to the infrared spectral range. We compare the specific results obtained for the 3s-3p transition to laboratory spectra.

**1 Introduction**

In Allard et al. (2003) and Tinetti et al. (2007) we presented the first applications of semi-classical profiles of sodium and potassium perturbed by molecular hydrogen to the modeling of brown dwarfs and extra-solar planets. The resonance line profiles were calculated using the Rossi & Pascale (1985) pseudo-potentials (hereafter labeled RP85). We now extend this work to excited states. Na-H₂ *ab initio* potentials (hereafter labeled S11) were recently computed allowing a more accurate determination of the far wings. Preliminary results for the 3s-3p transition are reported and compared to laboratory spectra.

**2 Theoretical spectra**

A unified treatment of the shape of a pressure-broadened alkali absorption line from near resonance to the far wing is obtained using autocorrelation formalism. Complete details and the derivation of the theory are given by Allard et al. (1999).

<footnote n="1">¹ GEPI, Observatoire de Paris, 61 avenue de l'Observatoire, 75014 Paris, France</footnote>

<footnote n="2">² Institut d'Astrophysique de Paris, UMR 7095, CNRS, Université Paris VI, 75014 Paris, France</footnote>

<footnote n="3">³ Department of Physics and Astronomy, University of Louisville, Louisville, KY 40292, USA</footnote>

<footnote n="4">⁴ Laboratoire de Physique Quantique, UMR 5626, CNRS, Université Paul Sabatier, 118 route de Narbonne, 31400 Toulouse, France</footnote>

<footnote n="5">⁵ Department of Physics and Astronomy, University College London, Gower Street, WC1E6BT London, UK</footnote>
<table><caption>**Table 1.** Half-width ($10^{-20}$ cm$^{-1}$/cm$^{-3}$) of Na 3s-3p and 3p-4s lines.</caption><thead><tr><th>Transitions</th><th>500 K</th><th>650 K</th><th>1000 K</th><th>1500 K</th><th>2000 K</th><th>3000 K</th></tr></thead><tbody><tr><td>3s <sup>2</sup>S<sub>1/2</sub>-3p <sup>2</sup>P<sub>1/2</sub></td><td>1.46</td><td>1.6</td><td>1.86</td><td>2.16</td><td>2.42</td><td>2.8</td></tr><tr><td>3s <sup>2</sup>S<sub>1/2</sub>-3p <sup>2</sup>P<sub>3/2</sub></td><td>1.96</td><td>2.16</td><td>2.49</td><td>2.94</td><td>3.24</td><td>3.87</td></tr><tr><td>3p <sup>2</sup>P<sub>1/2</sub>-4s <sup>2</sup>S<sub>1/2</sub></td><td>4.43</td><td>5.07</td><td>5.71</td><td>6.67</td><td>7.87</td><td>8.89</td></tr><tr><td>3p <sup>2</sup>P<sub>3/2</sub>-4s <sup>2</sup>S<sub>1/2</sub></td><td>4.34</td><td>5.43</td><td>5.58</td><td>7.17</td><td>7.6</td><td>9.35</td></tr></tbody></table>

<figure><img src="image_3.png" /><figcaption>**Fig. 1.** Absorption cross sections of the 3s-3p D1 line component of NaH$_2$, taking into account the two symmetries $C_{2v}$ and $C_{\infty v}$ using RP85 (full line) compared to S11 (dotted lines) compared to the corresponding Lorentzian profile. The density of perturbers is $n_{H_2} = 10^{20}$ cm$^{-3}$. The temperature is 1000 K.</figcaption></figure>

This treatment includes the finite duration of collision. It is well known that the impact approximation, which assumes that collisions occur instantaneously, causes the Lorentz theory to fail not too far from the line center. Consequently, it is not correct to use Lorentzian profiles in the far wings, as is the standard practice for most stellar atmosphere work (Désert et al. 2008). Figures 1 and 2 emphasize the non-Lorentzian behavior of the two components of the Na-H$_2$ resonance line. Figure 9 of Allard et al. (2007) presents the potential energy surfaces of Na-H$_2$ with spin-orbit coupling in the $C_{2v}$ and $C_{\infty v}$ symmetries. We present in Figure 3 the absorption cross section of the resonance line of Na compared to those of the first excited states. The Lorentzian profiles are calculated using the line widths presented in Table 1. Calculations have been done for the D1 and D2 lines
<figure><img src="image_4.png" /><figcaption>**Fig. 2.** Absorption cross sections of the 3s-3p D2 line component of NaH$_2$, taking into account the two symmetries C$_{2v}$ and C$_{∞v}$ using RP85 (full line) compared to S11 (dotted lines) compared to the corresponding Lorentzian profile. The density of perturbers is n$_{H_2}$ = 10$^{20}$ cm$^{-3}$, the temperature is 1000 K.</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>**Fig. 3.** Absorption cross section of the 3s-3p, 3p-3d, 3p-4s, 4s-4p lines (from the left to the right) Dashed curves represent the corresponding Lorentzians. ($T$ = 1000 K, n$_{H_2}$ = 1 × 10$^{19}$ cm$^{-3}$).</figcaption></figure>

from $T_{\text{eff}}$ = 500 K to 3000 K with a step size of 500 K. Opacity tables of alkalies perturbed by He and H$_2$ can be obtained from an on-line database<fnref n="1" />.

<footnote n="1">¹http://mygepi.obspm.fr/~allard/</footnote>
### 3 Far wing and quasi-molecular satellite of the 3s-3p transition

The line shape theory requires accurate atomic potentials and transition moments. Blue satellite bands in alkali-He/H₂ profiles can be predicted from the maxima in the difference potentials ΔV for the *B*-X transition. In calculations of Rossi & Pascale (1985), the alkali-H₂ complex was considered as a system with a single active electron. In Allard et al. (2012) we describe the new S11 calculations of the Na-H₂ potential energy surfaces in which the two electrons of H₂ and the valence electron of Na are explicitly treated. The major difference is that RP85 potentials are systematically less repulsive than those in S11. This difference affects the blue satellite position shown in Figure 2. The experimental absorption coefficients in Figure 2 have been scaled for comparison to the theoretical cross sections. On the blue side, the observed NaH₂ wing matches the profile from the RP85 potential. In this region the unified theory and experiment are approximately 100× a Lorentzian extrapolation of the line core.

### References

* Allard, N.F., Allard, F., Hauschildt, P.H., Kielkopf, J.F., & Machin, L., 2003, A&A, 411, 473

* Allard, N.F., Kielkopf, J.F., & Allard, F., 2007, EPJ D, 44, 507

* Allard, N.F., Royer, A., Kielkopf, J.F., & Feautrier, N., 1999, Phys. Rev. A, 60, 1021

* Allard, N.F., Spiegelman, F., Kielkopf, J.F., Tinetti, G., & Beaulieu, J.P., 2012, A&A, 543, A159

* Désert, J.-M., Vidal-Madjar, A., Lecavelier Des Etangs, A., et al., 2008, A&A, 492, 585

* Rossi, F., & Pascale, J., 1985, Phys. Rev. A, 32, 2657

* Tinetti, G., Vidal-Madjar, A., Liang, M.-C., et al., 2007, Nature, 448, 169