# H-β Line in a Corona Helium Plasma: A Multi-Code Line Shape Comparison

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## Abstract

**:**

## 1. Introduction

^{+}ions of helium plasma. The second part is focused on comparisons with experimental spectra of the same line measured in a corona discharge in helium. The aim in the first part was to compare the pure Stark-broadened profiles of this line calculated by the contributing codes for a selected set of plasma parameters. The latter, assumed to correspond to conditions of a corona discharge in helium, are the following: two values for the electron density n

_{e}= 10

^{15}and 10

^{16}cm

^{−3}, three values for the electron and ion temperatures assumed to be equal: T

_{e}= T

_{i}= 0.1, 0.2, and 0.4 eV. The choice to retain only the Stark broadening is aimed to highlight the differences between the participating codes and is justified by the fact that other broadening mechanisms are less important even though they are not completely negligible. Indeed, as we will see in the section devoted to the data fitting, Doppler and van der Waals broadenings have to be included as well as the instrumental function. The van der Waals broadening, which is due to the interactions of the hydrogen atoms with neutral helium atoms present in the partially ionized helium plasma, is roughly comparable to the Doppler broadening, unlike resonance broadening, which is due to the interactions of neutral emitters with neutrals of the same species.

## 2. Brief Description of the Codes

_{e}= 10

^{15}cm

^{−3}and n

_{e}= 10

^{16}cm

^{−3}.

## 3. Code Comparison through Profiles and Line Widths

#### 3.1. H-β Line Profiles for the Lower Density

_{e}= 10

^{15}cm

^{−3}) are respectively illustrated for equal ion and electron temperatures of 0.1, 0.2, and 0.4 eV in Figure 1, Figure 2 and Figure 3 corresponding respectively to subcases 1, 2, and 3. It can be seen that all the codes except PPP are in an overall agreement in terms of widths and line shapes with some differences in the central line dip. The profiles calculated by PPP are broader than all the others, especially at the lowest temperature of 0.1 eV. This was excepted because this version of the PPP line shape code used for the treatment of the electron broadening the GBK (Griem-Blaha-Kepple) [14] collision operator does not depend on the frequency. This operator is known to overestimate the broadening in comparison to frequency-dependent collision operators in the framework of the impact approximation used for the treatment of the emitter-electron interactions. Note that in ZEST calculations, the electron broadening was treated using the impact limit of the model reported in Reference [10]. Such an operator is different from the GBK one used in PPP, but does not account for frequency-dependence either. Note that the small oscillations of the LSNS profiles highlighted by the use of a semi-logarithmic scale are a feature of the numerical Fourier Transform, which has no physical origin. They can be eliminated by computing the dipole autocorrelation function over longer times. This simulation tool has been applied to the spectrum corresponding to the lowest pressure in a separate study [15]. The calculations of the PPP code tend towards the other code calculations as the temperature increases from 0.1 to 0.4 eV. At 0.4 eV, all the profiles give close widths even though they differ in the filling of the line center dip. It is worth noting that PPP allows the use of a frequency-dependent collision operator for the electron broadening, but at the expense of computing time or through a post-processing treatment. This possibility was not used at the time of the SLSP 4 workshop, hence, we only report on the PPP calculations based on the GBK collision operator model.

#### 3.2. H-β Line Profiles for the Highest Density

_{e}= 10

^{16}cm

^{−3}) and the same temperatures as in the previous subsection are respectively illustrated in Figure 4, Figure 5 and Figure 6 which refer to subcases 4, 5, and 6, respectively. For this density, the expected discrepancy between the calculations carried out by the PPP code using the GBK collision operator and those of the other codes are more pronounced, confirming the non-validity of this form of this operator at high densities and low temperatures even though it tends to disappear with increasing temperatures. One can see that for the remaining calculations, SimU agrees well with QC_FFM on the one side and ZEST agrees well with LSNS on the other side. Note that the profiles calculated by LSNS and ZEST are a bit broader than those calculated by SimU and QC_FFM. These agreements concern the overall profiles, the line widths, and wings, but not the line center dip for which there are some discrepancies between the various code calculations.

#### 3.3. Comparison of the FWHM of the H-β Line

_{e}= 10

^{15}and 10

^{16}cm

^{−3}. This figure shows the relative dispersion of the FWHM as deduced from the codes when those of PPP are higher, especially at the highest density and lowest temperature. More details are shown in Table 1.

_{e}, T

_{e}, are presented as well as the ratio of the highest value of the FWHM to each FWHM value. In addition, the mean value of the FWHM calculated using all the values but the highest is also shown. The factor in the last column of the table represents the ratio of the FWHM obtained by the PPP to the mean value. For the lowest density, the mean value <FWHM> varies between 6.71 cm

^{−1}for 0.1 eV to 7.71 cm

^{−1}for 0.4 eV. The FWHM values of the profiles calculated by PPP are roughly about 1.2–1.5 higher than the average values. For the highest density, the mean value of the FWHM (excluding the highest value) is in the range 27–36 cm

^{−1}, with the lower value (27 cm

^{−1}) corresponding to the lowest temperature, that is, 0.1 eV. As explained previously, the FWHMs calculated by PPP are higher than those of the other codes. It is about 2.8 times the mean value for 0.1 eV and decreases down to be 1.7 times higher at 0.2 eV to eventually become equal to the mean value at temperatures of 0.4 eV or higher, leading to a very good agreement with the other codes. This means that the calculations done by PPP using the non-frequency depending collision operator can be safely used for comparisons with the experimental data provided that the temperatures are not too low and the density too high.

#### 3.4. Reconsidering the PPP Calculations of the H-β Line

#### 3.5. Electric Microfield Distributions

_{e}= 10

^{15}cm

^{−3}. One can see no difference between the distributions calculated by LSNS, PPP, and ZEST but that which was calculated by SimU is different. This is not surprising since SimU, by simulating both electrons and ions, provides the total electric field corresponding to the doubled charge density. Its distribution is broader and shifted towards higher field values. For the other codes, the field distributions shown here are for ions only.

## 4. Comparison with Experimental Spectra: Line Shape Fitting

#### 4.1. Introducing the Experimental Spectra

#### 4.2. Broadening Mechanisms of H-β Line in a Helium Plasma

#### 4.3. Comparison with Experimental Data: Spectral Fitting

#### 4.3.1. Cases of Low Pressure

#### 4.3.2. Cases of Intermediate Pressures

#### 4.3.3. Cases of High Pressures

#### 4.4. Interpretation of the Fitting Parameters

_{a}) and ion (T

_{i}) temperatures (T

_{a}= T

_{i}= 300 K), electron temperature (1 eV = 11,604 K). The only free parameter was the electron density. Profiles calculated with PPP to fit the spectra were based on the following: the Stark effect including ion dynamics, no instrumental function, the inclusion of van der Waals broadening without (PPP n°1), and with Doppler broadening (PPP n°2). In both cases, the temperatures were fixed and only the electron density was allowed to vary. The fixed temperatures were as follows: T

_{a}= T

_{i}= T

_{e}= 300 K for PPP n°1 calculations and T

_{a}= T

_{i}= 300 K and T

_{e}= 1 eV for PPP n°2 calculations. In the profiles computed with PPP_GC, the following effects were retained: the Stark broadening with the ion dynamics included as well as the Doppler and the van der Waals broadenings. The instrumental function was ignored. The ion and neutral temperatures were assumed to be equal and fixed to 300 K while allowing both the electron density and the temperature to vary in the prescribed domains. In QC_FFM calculations, in addition to the Stark broadening, the instrumental function and the Doppler broadening were added but not the van der Waals broadening. The temperatures were fixed at T

_{i}= T

_{a}= 0.1 eV; T

_{e}= 1 eV. ZEST calculations are based on the Stark broadening without the inclusion of ion dynamics nor van der Waals broadening. The temperatures were T

_{i}= T

_{e}= 0.1 eV.

## 5. Discussion and Conclusions

^{14}–1.2 × 10

^{15}cm

^{−3}. If one looks in more detail at the results, he/she can see that the PPP n°1 and PPP_GC values are very close even though the Doppler broadening in PPP n°1 was not included and the electron temperature was only 300 K while in PPP_GC, the Doppler effect was retained and the electron temperature was found to be ~1000 K. The PPP n°2 calculations retaining the Stark, Doppler, and van der Waals broadening for an electron temperature of 1 eV gave an electron density of the same order of magnitude, that is, 2.6 × 10

^{14}cm

^{−3}. For the same temperatures as those of the PPP n°2 calculations and by retaining only the Stark and van der Waals broadenings, the LSNS calculations gave a density about 2 times higher, that is, 5 × 10

^{14}cm

^{−3}. The electron densities obtained by QC_FFM and ZEST were respectively 8 × 10

^{14}and 1.2 × 10

^{15}cm

^{−3}for an electron temperature of 1 eV and 0.1 eV, respectively. For the 1.5 bars case, similar results were obtained where the electron density values cover the range 3 × 10

^{14}–2.7 × 10

^{15}cm

^{−3}. The lowest densities were obtained by the PPP and PPP_QC followed by LSNS. Again, the highest densities were obtained by QC_FFM and ZEST. For the P = 2 bars case, the PPP n°1 calculations gave the lowest density value, followed by the PPP n°2 and PPP_QC calculations with comparable values; QC_FFM gave an electron density about 5 times higher. For cases 4, 5, and 6 corresponding to pressures P = 3, 4, and 5 bars, PPP again gave electron densities about 5 times lower than those given by QC_FFM. It is clear that these comparisons do not allow for the determination of the electron density with accuracy as its determination depends on many factors such as the retained broadening mechanisms and the temperatures of the electrons in the addition of those of the ions and neutrals. It is clear that for low-temperature dense plasmas, accounting for neutral broadening is essential and any code which ignores it would inevitably overestimate the electron density to compensate for the missing van der Waals broadening in a way in which the higher the pressure, the larger the mistake. These are not the only reasons explaining the large dispersion of the deduced electron densities since, as we have seen, even the pure Stark calculations differ for some unfavorable conditions like high electron densities and low temperatures. Further, we concluded to the necessity of carrying out further analyses consisting in preferably fitting/analyzing both the H-β line and He i 492 nm lines simultaneously or at least fitting them separately but in a consistent manner. Some papers submitted to this issue have started in this way to improve this spectroscopic diagnostic.

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## References

- Stambulchik, E. Review of the 1st spectral line shapes in plasmas code comparison workshop. High Energy Density Phys.
**2013**, 9, 528. [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.; Pain, J.C.; Stambulchik, E. Broadening of the neutral helium 492 nm line in a corona discharge: Code comparison and data fitting. Atoms
**2018**, 6, 19. [Google Scholar] [CrossRef] - Rosato, J.; Bufferand, H.; Koubiti, M.; Marandet, Y.; Stamm, R. A table of Balmer γ line shapes for the diagnostic of magnetic fusion plasmas. J. Quant. Spectrosc. Radiat. Transfer
**2015**, 165, 102–107. [Google Scholar] [CrossRef] - Stambulchik, E.; Maron, Y. A study of ion-dynamics and correlation effects for spectral line broadening in plasma: K-shell lines. J. Quant. Spectrosc. Radiat. Transf.
**2006**, 99, 730–749. [Google Scholar] [CrossRef] - Stambulchik, E.; Alexiou, S.; Griem, H.R.; Kepple, P.C. Stark broadening of high principal quantum number hydrogen Balmer lines in low-density laboratory plasmas. Phys. Rev. E
**2007**, 75, 016401. [Google Scholar] [CrossRef] [PubMed] - 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] - Stambulchik, E.; Maron, Y. Quasicontiguous frequency-fluctuation model for calculation of hydrogen and hydrogenlike Stark-broadened line shapes in plasmas. Phys. Rev. E
**2013**, 87, 053108. [Google Scholar] [CrossRef] [PubMed] - Gilleron, F.; Pain, J.C. ZEST: A fast code for simulating Zeeman-Stark line-shape functions. Atoms
**2018**, 6, 11. [Google Scholar] [CrossRef] - Lee, R.W. Plasma line shapes for selected transitions in Hydrogen-, Helium- and Lithium-like ions. J. Quant. Spectrosc. Radiat. Transf.
**1988**, 40, 561–568. [Google Scholar] [CrossRef] - Potekhin, A.; Chabrier, G.; Gilles, D. Electric microfield distributions in electron-ion plasmas. Phys. Rev. E
**2002**, 65, 036412. [Google Scholar] [CrossRef] [PubMed] - Gilles, D.; Peyrusse, O. Fast and accurate line shape modeling of the H- and He-like Lyman series for radiative transfer calculations in plasmas. J. Quant. Spectrosc. Radiat. Transf.
**1995**, 53, 647–661. [Google Scholar] [CrossRef] - Calisti, A.; Mossé, C.; Ferri, S.; Talin, B.; Rosmej, F.; Bureyeeva, L.A.; Lisitsa, V.S. Dynamic Stark broadening as the Dicke narrowing effect. Phys. Rev. E
**2010**, 81, 016406. [Google Scholar] [CrossRef] [PubMed] - Griem, H.R.; Blaha, M.; Kepple, P.C. Stark-profile calculations for Lyman-series lines of one-electron ions in dense plasmas. Phys. Rev. A
**1979**, 19, 2421. [Google Scholar] [CrossRef] - 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] - Li, Z.-L.; Bonifaci, N.; Aitken, F.; Denat, A.; von Haeften, K.; Atrazhev, V.M.; Shakhatov, V.A. Spectroscopic investigation of liquid helium excited by a corona discharge: Evidence for bubbles and “red satellites”. Eur. Phys. J. Appl. Phys.
**2009**, 47, 2821. [Google Scholar] [CrossRef] - 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] - Ali, A.W.; Griem, H.R. Theory of resonance broadening of spectral lines by atom-atom impacts. Phys. Rev.
**1965**, 4A, A1044. [Google Scholar] [CrossRef] - Laux, C.O.; Spence, T.G.; Kruger, C.H.; Zare, R.N. Optical diagnostics of atmospheric pressure air plasmas. Plasma Sources Sci. Technol.
**2003**, 12, 125–138. [Google Scholar] [CrossRef] - Nikiforov, A.Y.; Leys, C.; Gonzalez, M.A.; Walsh, J.L. Electron density measurement in atmospheric pressure plasma jets: Stark broadening of hydrogenated and non-hydrogenated lines. Plasma Sources Sci. Technol.
**2015**, 24, 034001. [Google Scholar] [CrossRef] - Yubero, C.; Dimitrijevic, M.S.; García, M.C.; Calzada, M.D. Using the van der Waals broadening of the spectral atomic lines to measure the gas temperature of an argon microwave plasma at atmospheric pressure. Spectrochim. Acta B
**2007**, 62, 169–176. [Google Scholar] [CrossRef] - Allard, N.; Kielkopf, J. The effect of neutral non resonant collisions on atomic spectral lines. Rev. Mod. Phys.
**1982**, 54, 1103. [Google Scholar] [CrossRef] - Muñoz, J.; Dimitrijevic, M.S.; Yubero, C.; Calzada, M.D. Using the van der Waals broadening of spectral atomic lines to measure the gas temperature of an argon-helium microwave plasma at atmospheric pressure. Spectrochim. Acta B
**2009**, 64, 167–172. [Google Scholar] [CrossRef] - 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 PPP. Atoms
**2018**. submitted. [Google Scholar]

**Figure 1.**The theoretical Stark profiles of the H-β line calculated for hydrogen atoms in a helium plasma with an electron density of 10

^{15}cm

^{−3}and a temperature of 0.1 eV for both plasma He

^{+}ions and electrons. The profiles are centered at 486.1 nm and the wavenumber units are used; (

**a**) linear scale. (

**b**) semi-logarithmic scale.

**Figure 4.**The theoretical Stark profiles of the H-β line calculated for hydrogen atoms in a helium plasma with an electron density of 10

^{16}cm

^{−3}and a temperature of 0.1 eV for both plasma He

^{+}ions and electrons. The profiles are centered at 486.1 nm and wavenumber units are used. (

**a**) linear scale; (

**b**) semi-logarithmic scale.

**Figure 7.**The H-β line FWHM versus the electron temperature as deduced from the profiles calculated by the codes. (

**a**) Lower electron density (n

_{e}= 10

^{15}cm

^{−3}); (

**b**) Higher electron density (n

_{e}= 10

^{16}cm

^{−3}).

**Figure 8.**The theoretical Stark profiles of the H-β line of Figure 1 and Figure 2 with the inclusion of the modified PPP calculations (PPPm: thick red solid line) where the electron density was scaled down by the mean corresponding factor explained at the end of Section 3.3. The same factor of 1.5 was used for both subcases (

**a**) T

_{e}= T

_{i}= 0.1 eV (

**b**) T

_{e}= T

_{i}= 0.2 eV.

**Figure 9.**The theoretical Stark profiles of the H-β line of Figure 3 (T

_{e}= T

_{i}= 0.4 eV) with the inclusion of modified PPP calculations (PPPm: thick red solid line) where the electron density was scaled down by the mean corresponding factor explained at the end of Section 3.3. The factor used here is 1.2.

**Figure 10.**The electric field distributions as calculated by the different line shape codes for the lowest electron density and temperature n

_{e}= 10

^{15}cm

^{−3}and T

_{e}= T

_{i}= 0.1 eV. These concern ions only for LSNS, PPP, and ZEST codes. The SimU distribution represents that of the total electric field, that is, the ions and electrons.

**Figure 11.**The superposition of the available experimental spectra of the H-β 486.1 nm (left peak) and the He i 492 nm (right peak) lines measured at room temperature for pressures in the range of 1–5 bars.

**Figure 12.**The comparison/fitting of the low-pressure H-β spectra taking into account the Stark effect. Other broadening mechanisms like Doppler and van der Waals were not necessarily taken into account by all the codes. (

**a**) P = 1 bar; (

**b**) P = 1.5 bars.

**Figure 13.**The comparison/fitting of the intermediate pressure H-β spectra. (

**a**) P = 2 bars; (

**b**) P = 3 bars. The contributions are similar to those of Figure 12.

**Table 1.**FWHM in units of cm

^{−1}as deduced from the profiles calculated by each of the five codes LSNS, PPP, C_FFM, SimU, and ZEST. The FWHM ratio stands for the ratio of the highest value of the FWHM to each FWHM. The highest value corresponds to the PPP calculations except for n

_{e}= 10

^{16}cm

^{−3}and T

_{e}= 0.4 eV where it corresponds to the LSNS calculations.

Plasma Parameters | Codes | ||||||
---|---|---|---|---|---|---|---|

(n_{e}, T_{e} = T_{i}) | LSNS | PPP | QC_FFM | SimU | ZEST | Mean | Factor |

(10^{15} cm^{−3}, 0.1 eV) | |||||||

FWHM (cm^{−1}) | 7.71 | 10.26 | 5.75 | 6.29 | 7.09 | 6.71 | - |

FWHM ratio | 1.3 | 1.0 | 1.8 | 1.6 | 1.4 | 1.53 | 1.53 |

(10^{15} cm^{−3}, 0.2 eV) | |||||||

FWHM (cm^{−1}) | 8.03 | 9.61 | 6.62 | 7.15 | 7.78 | 7.39 | - |

FWHM ratio | 1.2 | 1.0 | 1.4 | 1.3 | 1.2 | 1.30 | 1.30 |

(10^{15} cm^{−3}, 0.4 eV) | |||||||

FWHM (cm^{−1}) | 7.74 | 9.19 | 7.17 | 7.85 | 8.10 | 7.71 | - |

FWHM ratio | 1.2 | 1.0 | 1.3 | 1.2 | 1.1 | 1.19 | 1.19 |

(10^{16} cm^{−3}, 0.1 eV) | |||||||

FWHM (cm^{−1}) | 32.24 | 73.00 | 21.27 | 24.40 | 30.00 | 26.97 | - |

FWHM ratio | 2.3 | 1.0 | 3.4 | 3.0 | 2.4 | 2.7 | 2.7 |

(10^{16} cm^{−3}, 0.2 eV) | |||||||

FWHM (cm^{−1}) | 38.59 | 54.67 | 27.56 | 29.98 | 34.60 | 32.68 | - |

FWHM ratio | 1.4 | 1.0 | 2.0 | 1.8 | 1.6 | 1.67 | 1.67 |

(10^{16} cm^{−3}, 04 eV) | |||||||

FWHM (cm^{−1}) | 40.18 | 36.00 | 32.34 | 34.73 | 36.82 | 36.01 | - |

FWHM ratio | 1.0 | 1.1 | 1.2 | 1.2 | 1.1 | 1.11 | 1.00 |

**Table 2.**The summary of the FWHM of the van der Waals broadening compared with the experimental FWHM of the six experimental spectra of the H-β line in a helium plasma considered in this paper.

Pressure (bars) | $\Delta {\mathsf{\lambda}}_{VdW}$ (nm) | $\Delta {\mathsf{\lambda}}_{exp}$ (nm) | $\Delta {\mathsf{\lambda}}_{VdW}/\Delta {\mathsf{\lambda}}_{exp}$(%) |
---|---|---|---|

1 | 0.051 | 0.143 | 35 |

1.5 | 0.076 | 0.206 | 36 |

2 | 0.101 | 0.335 | 30 |

3 | 0.152 | 0.510 | 30 |

4 | 0.202 | 0.604 | 31 |

5 | 0.253 | 0.819 | 31 |

**Table 3.**The electron densities (in units of 10

^{15}cm

^{−3}) as inferred from the fit of the experimental H-β spectra by the contributing codes. P is the pressure in units of bars.

Case n° | P | LSNS | PPP (n°1) | PPP (n°2) | PPP_GC | QC_FFM | ZEST |
---|---|---|---|---|---|---|---|

1 | 1 | 0.5 | 0.15 | 0.26 | 0.18 | 0.8 | 1.2 |

2 | 1.5 | 1.1 | 0.3 | 0.58 | 0.38 | 2.2 | 2.7 |

3 | 2 | - | 0.55 | 1.0 | 1.3 | 4.7 | - |

4 | 3 | - | 0.9 | 2.0 | - | 10.0 | - |

5 | 4 | - | 1.3 | 2.8 | - | 15.0 | - |

6 | 5 | - | 1.9 | 3.8 | - | 27.0 | - |

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## Share and Cite

**MDPI and ACS Style**

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.
https://doi.org/10.3390/atoms6020029

**AMA Style**

Sheeba RR, 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(2):29.
https://doi.org/10.3390/atoms6020029

**Chicago/Turabian Style**

Sheeba, Roshin Raj, Mohammed Koubiti, Nelly Bonifaci, Franck Gilleron, Caroline Mossé, Jean-Christophe Pain, Joël Rosato, and Evgeny Stambulchik.
2018. "H-β Line in a Corona Helium Plasma: A Multi-Code Line Shape Comparison" *Atoms* 6, no. 2: 29.
https://doi.org/10.3390/atoms6020029