# A New Procedure to Determine the Plasma Parameters from a Genetic Algorithm Coupled with the Spectral Line-Shape Code PPP

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. A Genetic Algorithm Coupled with a Stark Broadened Line-Shape Model

#### 2.1. Genetic Algorithm

_{e}and the electron temperature T

_{e}. Both of these plasma parameters are related to the ionic and electronic Stark broadening models. In addition, the Lorentzian width related to van der Waals broadening of spectral lines and the Gaussian width which corresponds to instrumental broadening have been investigated. Specific hypotheses which could affect the obtained parameters have been adapted in the proposed spectral model. For instance, the plasma is considered to have uniform temperature and density for which both lines of interest are optically thin. However, there could be spatial temperature and density gradients, and the emission lines may be emitted in different physical locations for various conditions. Such variations have not been considered in the spectra modelling.

#### 2.2. Spectral Line Shape Calculations

_{e}= 1 × 10

^{15}cm

^{−3}and T

_{e}= 1.16 × 10

^{4}K (T

_{i}= 300 K). Moreover, in this particular case, this line profile is also affected by van der Waals broadening which results from the dipolar interaction between excited atoms and induced dipoles from the neutral perturbers [21]. In Figure 2, in addition to Stark broadening, both the van der Waals and instrumental broadening mechanisms are included in this analysis to show their influence on the line profile. It appears that these broadening effects cannot be ignored in density diagnostic methods. Detailed analysis of the van der Waals broadening effects is provided in Section 3.2. The Balmer-β line profile has also been calculated by the PPP code for a fixed temperature, T

_{e}= T

_{i}= 10,000 K, and for different densities in the range between 10

^{14}and 10

^{16}cm

^{−3}for the specific case of reduced mass of 0.8. The line widths have been compared with those tabulated in [22] and are found to be in good agreement within 5%.

^{1}D – 1s2p

^{1}P

^{0}line at 492.2 nm with He

^{+}perturbers for different electronic densities: n

_{e}= 3 × 10

^{14}cm

^{−3}, n

_{e}= 2 × 10

^{15}cm

^{−3}, n

_{e}= 10

^{16}cm

^{−3}. Electronic and ionic temperatures were fixed at 1.16 × 10

^{4}K and 300 K, respectively. No additional broadening was taken into account in this calculation. When the electronic density increases, the forbidden components (the transitions 1s4p 1P

^{0}– 1s2p

^{1}P

^{0}and 1s4f

^{1}F

^{0}– 1s2p

^{1}P

^{0}) become more intense, and their position is shifted towards the blue wing of the allowed components. The distance between the peaks of the allowed and forbidden components can be used for an estimation of the electronic density [23]. The He I 492.2 nm line has then been synthesized using the PPP code for various densities, from ~10

^{14}cm

^{−3}to 10

^{16}cm

^{−3}. The dependency of the distance between the allowed and forbidden component peaks versus the electronic density is shown in Figure 4. According to these results and by comparison with the experimental data (blue points in Figure 4), the estimate electronic densities for pressures of 0.1, 0.2, and 0.3 MPa have been found to be n

_{e}= 10

^{15}cm

^{−3}, n

_{e}= 5.2 × 10

^{15}cm

^{−3}, and n

_{e}= 9 × 10

^{15}cm

^{−3}, respectively.

## 3. Analysis of the Hydrogen Balmer-β and He I 492.2 nm Lines in Corona Discharge Plasma

#### 3.1. Corona Discharges Excited in Gaseous Helium: Experimental Setup and Methodology

^{−4}Pa and filled with ultra-pure helium of grade N 60 (l’Air Liquide) with nominal concentration of impurities of <0.1 ppm of oxygen. The gas was directed through a series of traps filled with a mixture of molecular sieves (3–10 $\mathrm{\AA}$) and charcoal, activated under vacuum typically at 350 °C for three days and then immersed in liquid N

_{2}.

_{ins}~ 0.09 nm for the 1800 grooves/mm grating.

#### 3.2. Analysis of the Observed H-β and He I 492 nm Lines

^{−3}nm for the H-β line and 7 × 10

^{−4}nm for the He I 492 nm line, leading to negligible effects on these line profiles compared to overall broadenings. Thus, the combination of these different broadening mechanisms results in complex spectral line profiles showing an inhomogeneous structure, i.e., a set of Stark components due to ionic Stark effect, where each component is homogeneously broadened by electronic collisions, van der Waals effects, and instrumental artifacts.

_{he}= P/kT, where k is the Boltzmann constant, and the hydrogen density by 0.1 × 10

^{−6}n

_{he}, it is possible to estimate the line center opacity assuming a spatially uniform plasma of given geometrical path length, ℓ, at local thermodynamic equilibrium. For a geometrical path length ℓ = 100 μm, the line center opacities are of the order of a few 10

^{−6}for the helium line and of a few 10

^{−8}for the hydrogen line (for these three pressure values). Thus, in the following analysis, it is assumed that both lines are optically thin.

_{e}and T

_{e}are determined with the GA procedure, together with the potential additional broadening of spectral lines corresponding to the full (Lorentzian) width at half maximum (FWHM) for van der Waals broadening, and the full (Gaussian) width at half maximum (FWHM) for instrumental broadening. The latter has been measured in the range between 0.08 nm and 0.09 nm which has been set in the GA procedure in order to determine the Gaussian width.

_{e}, the electron temperature T

_{e}, the van der Waals width $\Delta {\mathsf{\lambda}}_{\mathrm{VDW}}$, and the additional Gaussian width Δλ

_{ins}, are given in Table 5 for the H-β line and in Table 6 for the He I 492 nm line.

^{15}cm

^{−3}and ~10

^{15}cm

^{−3}have been obtained for the He I and for the H-β lines, respectively. In these preliminary works, the authors do not consider either van der Waals or instrumental broadening. Thus, it is difficult to compare the results of these papers [25,26] with the results obtained in the present paper, as both broadening mechanisms are not negligible. However, there is no contradiction between these papers [25,26] and the present study, as in articles [25,26] the authors obtained higher electronic densities to compensate the omission of additional broadening mechanisms.

^{14}, 1.19 × 10

^{15}, and 2.5 × 10

^{15}cm

^{−3}for 0.1, 0.2, and 0.3 MPa, respectively. However, these values are still below the density values obtained by analyzing the helium line.

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Griem, H.R. Spectral Line Broadening by Plasma; Academic Press: New York, NY, USA, 1974. [Google Scholar]
- Golovkin, I.E.; Mancini, R.C.; Louis, S.J.; Lee, R.W.; Klein, L. Analysis of X-ray spectral data with genetic algorithms. JQSRT
**2002**, 75, 625–636. [Google Scholar] [CrossRef] - Chung, H.K.; Cohen, D.H.; MacFarlane, J.J.; Bailey, J.E.; Wang, P.; Moses, G.A. Statistical fitting analysis of Stark-broadened optically thick Ar II spectra measured in ion beam transport experiments. JQSRT
**2000**, 65, 135–149. [Google Scholar] [CrossRef] - Marandet, Y.; Genesio, P.; Godbert-Mouret, L.; Koubiti, M.; Stamm, R.; Capes, H. Determination of edge plasma parameters by a genetic algorithm analysis of spectral line shapes. Contrib. Plasmas Phys.
**2004**, 44, 289–293. [Google Scholar] [CrossRef] - 4th SLSP Workshop. Available online: http://plasma-gate.weizmann.ac.il/projects/slsp/slsp4/ (accessed on 22 February 2018).
- Sheeba, R.R.; Koubiti, M.; Bonifaci, N.; Gilleron, F.; Mossé, C.; Pain, J.-C.; Rosato, J.; Stambulchik, E. H-β Line in a Corona Helium Plasma: A Multi-Code Line Shape Comparison. Atoms
**2018**, 6, 29. [Google Scholar] [CrossRef] - Sheeba, R.R.; Koubiti, M.; Bonifaci, N.; Gilleron, F.; Pain, J.-C.; Stambulchik, E. Broadening of the Neutral Helium 492 nm Line in a Corona Discharge: Code Comparisons and Data Fitting. Atoms
**2018**, 6, 19. [Google Scholar] [CrossRef] - Calisti, A.; Khelfaoui, F.; Stamm, R.; Talin, B.; Lee, R.W. Model for the line shapes of complex ions in hot and dense plasmas. Phys. Rev. A
**1990**, 42, 5433–5440. [Google Scholar] [CrossRef] [PubMed] - Talin, B.; Calisti, A.; Godbert, L.; Stamm, R.; Lee, R.W.; Klein, L. Frequency-fluctuation model for line-shape calculations in plasma spectroscopy. Phys. Rev. A
**1995**, 51, 1918–1928. [Google Scholar] [CrossRef] [PubMed] - Charbonneau, P. Genetic algorithms in astronomy and astrophysics. Astrophys. J. Suppl. Ser.
**1995**, 101, 309. [Google Scholar] [CrossRef] - Ferri, S.; Calisti, A.; Mossé, C.; Rosato, J.; Talin, B.; Alexiou, S.; Gigosos, M.A.; González, M.A.; González-Herrero, D.; Lara, N.; et al. Ion Dynamics Effect on Stark-Broadened Line Shapes: A Cross-Comparison of Various Models. Atoms
**2014**, 2, 299–318. [Google Scholar] [CrossRef] [Green Version] - Godbert-Mouret, L.; Meftah, T.; Calisti, A.; Stamm, R.; Talin, B.; Gigosos, M.; Cardenoso, V.; Alexiou, S.; Lee, R.W.; Klein, L. Accuracy of stark broadening calculations for ionic emitters. Phys. Rev. Lett.
**1998**, 81, 5568. [Google Scholar] [CrossRef] - Bonifaci, N.; Li, Z.; Eloranta, J.; Fiedler, S.L. Interaction of Helium Rydberg State Molecules with Dense Helium. J. Phys. Chem. A
**2016**, 120, 9019–9027. [Google Scholar] [CrossRef] [PubMed] - Galtier, E.; Rosmej, F.B.; Calisti, A.; Talin, B.; Mossé, C.; Ferri, S.; Lisitsa, V.S. Interference effects and Stark broadening in XUV intrashell transitions in aluminum under conditions of intense XUV free-electron-laser irradiation. Phys. Rev. A
**2013**, 87, 033424. [Google Scholar] [CrossRef] - Mancini, R.C.; Iglesias, C.A.; Ferri, S.; Calisti, A.; Florido, R. The effect of improved satellite line shapes on the argon Heβ spectral feature. High Energy Density Phys.
**2013**, 9, 731–736. [Google Scholar] [CrossRef] - Iglesias, C.A.; DeWitt, H.E.; Lebowitz, J.L.; MacGowan, D.; Hubbard, W.B. Low-frequency electric microfield distributions in plasmas. Phys. Rev. A
**1985**, 31, 1698. [Google Scholar] [CrossRef] - Iglesias, C.A.; Rogers, F.J.; Shepherd, R.; Bar-Shalom, A.; Murillo, M.S.; Kilcrease, D.P.; Calisti, A.; Lee, R.W. Fast electric microfield distribution calculations in extreme matter conditions. J. Quant. Spectrosc. Radiat. Transf.
**2000**, 65, 303–315. [Google Scholar] [CrossRef] - Hooper, C.F., Jr. Electric microfield distributions in plasmas. Phys. Rev.
**1966**, 149, 77. [Google Scholar] [CrossRef] - Hooper, C.F., Jr. Low-frequency component electric microfield distributions in plasmas. Phys. Rev.
**1968**, 165, 215. [Google Scholar] [CrossRef] - Calisti, A.; Mossé, C.; Ferri, S.; Talin, B.; Rosmej, F.; Bureyeva, L.A.; Lisitsa, V.S. Dynamic Stark broadening as the Dicke narrowing effect. Phys. Rev. E
**2010**, 81, 016406. [Google Scholar] [CrossRef] [PubMed] - Drake, G.W.F. Atomic, Molecular, and Optical Physics Handbook; AIP Press: Woodbury, NY, USA, 1996. [Google Scholar]
- Gigosos, M.A.; Cardenoso, V. New plasma diagnosis tables of hydrogen Stark broadening including ion dynamics. J. Phys. B Atomic Mol. Opt. Phys.
**1996**, 29, 4795. [Google Scholar] [CrossRef] - Lara, N.; González, M.A.; Gigosos, M.A. Stark broadening tables for the helium I 492.2 line-Application to weakly coupled plasma diagnostics. Astron. Astrophys.
**2012**, 542, A75. [Google Scholar] [CrossRef] - Nguyen, T.H.V. Spectroscopie d’émission d’un plasma crée par des décharges couronne dans l’hélium. Ph.D. Thesis, Université de Grenoble, Grenoble, France, 2015. [Google Scholar]
- Rosato, J.; Bonifaci, N.; Li, Z.; Stamm, R. A spectroscopic diagnostic of the electron density in a corona discharge. J. Phys. Conf. Ser.
**2017**, 810, 012057. [Google Scholar] [CrossRef] [Green Version] - Rosato, J.; Bonifaci, N.; Li, Z.; Stamm, R. Line shape modeling for the diagnostic of the electron density in a corona discharge. Atoms
**2017**, 5, 35. [Google Scholar] [CrossRef]

**Figure 1.**A schematic representation of the operation of the genetic algorithm (GA) coupled with PPP.

**Figure 2.**Comparisons between different broadening effects on the H-β line profile in He plasma at n

_{e}= 1 × 10

^{15}cm

^{−3}, T

_{e}= 1.16 × 10

^{4}K, and T

_{i}= 300 K. Dotted black line: electronic impact broadening; Red line: total Stark broadening with quasi-static approximation; Blue line: total Stark broadening taking into account ion dynamics effect; Black line: total Stark profile taking into account relevant, van der Waals, and instrumental broadening.

**Figure 3.**Synthetized profiles of spectral line He I 492.2 nm for T

_{e}= 1.16 × 10

^{4}K and T

_{i}= 300 K and for three densities, i.e., n

_{e}= 3 × 10

^{14}cm

^{−3}(green line), n

_{e}= 2 × 10

^{15}cm

^{−3}(blue line), n

_{e}= 1 × 10

^{16}cm

^{−3}(red line).

**Figure 4.**Distance between allowed and forbidden component peaks versus electronic density for He I 492.2 nm line synthesized at T

_{e}= 1.16 × 10

^{4}K and T

_{i}= 300 K.

**Figure 6.**Examples of fitting of the experimental spectral lines. (

**a**) H-β line profile (

**b**) He I 492 nm line profile for pressure value (1) 0.1 MPa (2) 0.2 MPa (3) 0.3 Mpa.

**Table 1.**van der Waals width formulas for the three radiative transitions of the H-β line in helium plasma.

Transition | Δλ^{th} _{vdw} |
---|---|

4s – 2p | 2.20 × 10^{−5}PT^{−7/10} |

4p – 2s | 2.12 × 10^{−5}PT^{−7/10} |

4d – 2p | 1.98 × 10^{−5}PT^{−7/10} |

Wavelength (nm) | Transition | Δλ^{th} _{vdw} |
---|---|---|

491.0747 | ^{1}P_{o} – ^{1}P_{0} | 1.6906 × 10^{−5}PT^{−7/10} |

492.0612 | ^{1}P_{o} – ^{1}F_{0} | 1.3342 × 10^{−5}PT^{−7/10} |

492.1931 | ^{1}P_{0} – ^{1}D_{1} | 1.5424 × 10^{−5}PT^{−7/10} |

Pressure (10^{5} Pa) | $\Delta {\mathbf{\lambda}}_{\mathbf{V}\mathbf{D}\mathbf{W}}\text{}\left(\mathbf{nm}\right)$ |
---|---|

1 | 3.65 × 10^{−2} |

2 | 7.31 × 10^{−2} |

3 | 10.96 × 10^{−2} |

Pressure (10^{5} Pa) | $\Delta {\mathbf{\lambda}}_{\mathbf{V}\mathbf{D}\mathbf{W}}\text{}\left(\mathbf{nm}\right)$ |
---|---|

1 | 2.85 × 10^{−2} |

2 | 5.69 × 10^{−2} |

3 | 8.54 × 10^{−2} |

**Table 5.**Results of the fitting GA analysis of the H-β line. n

_{e}: electron density; T

_{e}: electron temperature; $\Delta {\mathsf{\lambda}}_{\mathrm{VDW}}$: van der Waals width; Δλ

_{ins}: Gaussian width.

Pressure (bar) | n_{e} (cm^{−3}) | T_{e} (10^{4} K) | $\Delta {\mathbf{\lambda}}_{\mathbf{V}\mathbf{D}\mathbf{W}}\text{}\left(\mathbf{nm}\right)$ | Δλ_{ins} (nm) |
---|---|---|---|---|

1 | 10^{14} | 1.23 | 7.2 × 10^{−2} | 8.0 × 10^{−2} |

2 | 8 × 10^{14} | 1.17 | 15.2 × 10^{−2} | 8.0 × 10^{−2} |

3 | 1.85 × 10^{15} | 1.21 | 24.2 × 10^{−2} | 8.0 × 10^{−2} |

Pressure (bar) | n_{e} (cm^{−3}) | T_{e} (10^{4} K) | $\Delta {\mathbf{\lambda}}_{\mathbf{V}\mathbf{D}\mathbf{W}}\text{}\left(\mathbf{nm}\right)$ | Δλ_{ins} (nm) |
---|---|---|---|---|

1 | 10^{15} | 1.21 | 2.93 × 10^{−2} | 8.0 × 10^{−2} |

2 | 3.96 × 10^{15} | 1.16 | 5.82 × 10^{−2} | 8.0 × 10^{−2} |

3 | 8 × 10^{15} | 1.16 | 9.7 × 10^{−3} | 8.0 × 10^{−2} |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Mossé, C.; Génésio, P.; Bonifaci, N.; Calisti, A.
A New Procedure to Determine the Plasma Parameters from a Genetic Algorithm Coupled with the Spectral Line-Shape Code PPP. *Atoms* **2018**, *6*, 55.
https://doi.org/10.3390/atoms6040055

**AMA Style**

Mossé C, Génésio P, Bonifaci N, Calisti A.
A New Procedure to Determine the Plasma Parameters from a Genetic Algorithm Coupled with the Spectral Line-Shape Code PPP. *Atoms*. 2018; 6(4):55.
https://doi.org/10.3390/atoms6040055

**Chicago/Turabian Style**

Mossé, Caroline, Paul Génésio, Nelly Bonifaci, and Annette Calisti.
2018. "A New Procedure to Determine the Plasma Parameters from a Genetic Algorithm Coupled with the Spectral Line-Shape Code PPP" *Atoms* 6, no. 4: 55.
https://doi.org/10.3390/atoms6040055