# Improved Line Intensity Analysis of Neutral Helium by Incorporating the Reabsorption Processes in a Helium Collisional-Radiative Model

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

^{1}P states to the ground state so that the influence of the absorption effect under optically thick conditions could be considered. The developed algorithm was based on fitting the number densities of eight excited states obtained using optical emission spectroscopy (OES). The electron density, electron temperature, ground-state density, and optical escape factors were selected as the fitting parameters. The objective function was set as the summation of the residual errors between the number densities measured in the experiment and those calculated using the revised model. A regularization term was introduced for the optical escape factor and optimized through bias and variance analyses. The results show that the agreement between the number density calculated by the algorithm and its counterpart measured in the experiment was generally improved compared to the method using three lines.

## 1. Introduction

^{1}S) and 64 excited states (2≤ n ≤ 26). Each L (orbital angular momentum quantum number) level with n (principal quantum number) less than or equal to seven is considered independently, except for levels with L greater than or equal to three. These specific levels with L ≥ 3 are combined into a single level. For levels with n between 8 and 10, the different L levels are grouped together. Levels with n greater than or equal to 11 are approximated using hydrogenic levels that possess statistical weights twice as those of hydrogen. It has all the fundamental electron collisions and radiation processes for low-pressure helium plasmas. Particularly, it considers electron collision excitation and de-excitation, electron collision ionization, electron three-body recombination, spontaneous transitions, and dielectronic and radiative recombination. The corresponding rate coefficients were obtained using the equations presented in Refs. [11,12,13,14].

## 2. Experiment

^{−2}·nm

^{−1}·sr

^{−1}) at the aperture of the integrated sphere was known, and we derived the radiance of the plasma by comparing the signal counts at the actual measurement and at the calibration measurement. We collected the data for 12 discharges under various plasma conditions. Each discharge typically lasted for 2 s, and spectra were recorded every 5 min.

_{e}and n

_{e}values at the line emission location could be diagnosed through an analysis of the spectra. In this study, the selected lines for diagnosis were not optically thick and were not subjected to the reabsorption effect. The line-integrated number density of the upper level of the corresponding spontaneous transition is determined as follows:

_{p,q}is the photon wavelength of the corresponding transition, A

_{p,q}is Einstein’s A coefficient, and Φ

_{p,q}(W·m

^{−2}) is the line-integrated spectral flux density, which was obtained by integrating the spectral radiance L(λ) (W·m

^{−2}·nm

^{−1}·sr

^{−1}) over the corresponding emission line:

_{e}and n

_{e}was first attempted using three emission lines, i.e., 667.8 nm, 706.5, and 728.1 nm [15]. The results are shown in Figure 3. The red and blue symbols in Figure 3a represent the intensity ratios of the specific lines for the diagnosis plotted against the line-averaged electron density. The line-averaged electron density was calculated by dividing the line-integrated density measured by an interferometer by the plasma length. This measurement provided information about the entire plasma, including the core region. On the other hand, helium line emissions were localized at the plasma edge. As a result, it is generally expected that the electron density derived from these emissions will be lower than the line-averaged electron density.

_{e}and n

_{e}were determined by minimizing the function as follows:

_{e}and n

_{e}. The corresponding electron densities and temperatures are indicated by the red and blue symbols in Figure 3b, respectively. The diagnosed electron density increased with the increase in the line-averaged electron density ${\overline{n}}_{\mathrm{e}}$, and the electron temperature decreased. The changes in electron density and temperature relative to the line-averaged electron density showed similar trends to the changes in the 667.8/728.1 nm and 728.1/706.5 nm line ratios relative to the line-averaged electron density, respectively. This three-line method for determining the electron density and temperature was first suggested by Schweer et al. [16]. The results were fitted with polynomial functions, and they are shown as dashed lines. The synthetic intensity ratios (dashed lines) obtained using the fitted T

_{e}, n

_{e}, and CR model [17] are shown in Figure 3a with the dashed lines. The synthetic results for the intensity ratios of the three lines agreed well with their original values, as shown in Figure 3a.

_{e}and n

_{e}determinations. The intensities of the three lines (from n = 3 to n = 2, normalized by the 667.8 nm line intensity) obtained via the OES measurement were compared with the synthetic results shown in Figure 3c. Some disagreements can be observed, i.e., the synthetic results of the lines at 388.9 and 587.6 nm have tendencies similar to the corresponding measured results; however, constant differences in the results (lines and symbols) exist. In addition, the measured results of the line at 501.6 nm show a trend opposite to that of the corresponding synthetic result. We suspect that the reabsorption effects of the transition 3

^{1}P–1

^{1}S, which has the same upper level as the line at 501.6 nm, could be causing this. Thus, it can be considered that using three lines to diagnose the electron density and temperature with the OES measurement and CR model cannot perfectly fit the measured results. Conventional diagnoses can be further improved.

## 3. Model Extension

#### 3.1. Optical Escape Factor

^{1}P, 3

^{1}P, 4

^{1}P, and 5

^{1}P states to the ground state have been evaluated in general cases [18]. In this study, the reabsorption effect in LHD was incorporated into the model as an escape factor. An accurate evaluation of the escape factor is generally difficult. In a previous study, the escape factor was considered the fitting parameter [17]. However, the escape factor was introduced only for a single resonance line (1

^{1}S–4

^{1}P), and the validity of the obtained value was not examined. Thus, we developed a method to determine the fitting conditions for the escape factors. The escape factors can be evaluated by assuming that the plasma has a slab or a cylindrical structure. Iida suggested a complete analytic expression for the optical escape factor in a cylindrical geometry [19]. For LHD helium plasma, in a previous study, we found that the line emissions of neutral helium are localized within a layer of thickness in the order of 1 cm at the plasma boundary, whereas the minor radius of the plasma is in the order of 1 m [15]. Because the line-of-sight of the present measurement is almost perpendicular to the emission layer, we believe that the situation can be approximated by a slab model. In addition, the precise plasma geometry is not well understood, and it is difficult to accurately calculate the theoretical escape factor. Instead, we can use it as a fitting parameter and use the calculated theoretical value as a constraint. Thus, an infinite plane-parallel slab model [20] with a thickness of D = 0.01 m was applied. The optical escape factor at the center of the slab is expressed as follows:

^{1}P as a function of the ground-state density.

^{1}P exhibited the lowest optical escape factor among the six n

^{1}P levels. This indicates that lower levels have a relatively stronger absorption effect. The absorption effect of n

^{1}P (n = 5, 6, and 7) can be neglected when the ground-state density is lower than 10

^{13}cm

^{−3}. In the revised CR model, we used the following equation to calculate the depopulating flux contributed by a spontaneous transition in the rate equation:

#### 3.2. Bias–Variance Analysis

_{g}and D are fixed. The hyperparameter μ controls the weight of restriction of the escape factors. A schematic of the fitting algorithm is shown in Figure 5.

## 4. Results and Discussion

^{13}cm

^{−3}. The fitting of the 501.6 and 587.6 nm lines improved when the line-averaged electron density was between 10

^{12}and 2 × 10

^{13}cm

^{−3}. When the line-averaged electron density was higher than 2 × 10

^{13}cm

^{−3}, the 501.6 nm line had a relatively better fitting. For the 447.2 and 492.2 nm lines, the difference between the measured and fitted results improved. In general, the results fitted with the new model exhibited relatively better performance.

^{1}P had the smallest escape factor, which decreased from approximately 0.6 to 0.005 when the line-averaged electron density was lower than 6 × 10

^{13}cm

^{−3}. Then, it increased to approximately 0.02 at 10

^{14}cm

^{−3}. State 4

^{1}P had the largest escape factor in the range of 0.07–0.9. The escape factor for state 3

^{1}P was between those for states 2

^{1}P and 4

^{1}P (0.03–0.8). This is a reasonable result because the reabsorption rate is proportional to the Einstein A coefficient, which decreases with an increase in the principal quantum number of the upper level. In the low line-averaged electron density region, the fitted escape factors are restricted by the calculated ones, which are functions of N

_{1}. Because N

_{1}increased with increasing line-averaged electron density, the reabsorption rate increased and the escape factors showed decreasing trends. The increase in the escape factor in the high line-averaged electron density region could be due to a decrease in atom densities. The intensities of all measured lines decreased in the high line-averaged electron density region, which is consistent with the behavior of the optical escape factor. The reason for the decrease in the line intensity in the high line-averaged density region is not yet clear and requires further study.

## 5. Conclusions

^{1}S, 3

^{3}S, and 3

^{1}D increased slightly compared to the conventional method. This can be improved by including the statistical weights of the object functions. In general, the algorithm performs well in determining the electron density and temperature.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Lin, K.; Nezu, A.; Akatsuka, H. Optical emission spectroscopy diagnosis of low-pressure microwave discharge helium plasma based on collisional-radiative model. Jpn. J. Appl. Phys.
**2022**, 61, 116001. [Google Scholar] [CrossRef] - Yuji, T.; Fujii, S.; Mungkung, N.; Akatsuka, H. Optical emission characteristics of atmospheric-pressure nonequilibrium microwave discharge and high-frequency DC pulse discharge plasma jets. IEEE Trans. Plasma Sci.
**2009**, 37, 839–845. [Google Scholar] [CrossRef] - Goto, M.; Morita, S. Determination of the hydrogen and helium ion densities in the initial and final stages of a plasma in the Large Helical Device by optical spectroscopy. Phys. Plasmas
**2003**, 10, 1402. [Google Scholar] [CrossRef] [Green Version] - Onishi, H.; Yamazaki, F.; Hakozaki, Y.; Takemura, M.; Nezu, A.; Akatsuka, H. Measurement of electron temperature and density of atmospheric-pressure non-equilibrium argon plasma examined with optical emission spectroscopy. Jpn. J. Appl. Phys.
**2021**, 60, 026002. [Google Scholar] [CrossRef] - Sawada, K.; Yamada, Y.; Miyachika, T.; Ezumi, N.; Iwamae, A.; Goto, M. Collisional-radiative model for spectroscopic diagnostic of optically thick helium plasma. Plasma Fusion Res.
**2010**, 5, 001. [Google Scholar] [CrossRef] [Green Version] - Kajita, S.; Suzuki, K.; Tanaka, H.; Ohno, N. Helium line emission spectroscopy in recombining detached plasmas. Phys. Plasmas
**2018**, 25, 063303. [Google Scholar] [CrossRef] - Lee, W.; Oh, C. Optical diagnostics of helium recombining plasmas with collisional radiative model. Phys. Plasmas
**2018**, 25, 113504. [Google Scholar] [CrossRef] - Nishijima, D.; Hollmann, E.M. Determination of the optical escape factor in the He I line intensity ratio technique applied for weakly ionized plasmas. Plasma Phys. Control. Fusion
**2007**, 49, 791. [Google Scholar] [CrossRef] - Goto, M. Collisional-radiative model for neutral helium in plasma revisited. J. Quant. Spectrosc. Radiat. Transf.
**2003**, 76, 331–344. [Google Scholar] [CrossRef] - Fujimoto, T. A collisional-radiative model for helium and its application to a discharge plasma. J. Quant. Spectrosc. Radiat. Transfer
**1979**, 21, 439–455. [Google Scholar] [CrossRef] - Ralchnko, Y.V.; Janev, R.K.; Kato, T.; Fursa, D.V.; Bray, I.; Heer, F.J.D. Cross Section Database for Collisional Processes of Helium Atom with Charged Particles. I. Electron Impact Processes; Rep. NIFS-DATA-59 2000; NISF: Nagoya, Japan, 2000. [Google Scholar]
- Shah, M.B.; Elliott, D.S.; McCallion, P.; Gilbody, H.B. Single and double ionisation of helium by electron impact. J. Phys. B
**1988**, 21, 2751–2761. [Google Scholar] [CrossRef] - Fujimoto, T. Semi-Empirical Cross Section and Rate Coefficients for Excitation and Ionization by Electron Collision and Photoionization of Helium; Rep. IPP-AM-8 1978; Institute of Nagoya Physics, Nagoya University: Nagoya, Japan, 1978. [Google Scholar]
- Goto, M.; Fujimoto, T. Collisional-Radiative Model for Neutral Helium in Plasma: Excitation Cross Section and Singlet-Triplet Wave-Function Mixing; Rep. NIFS-DATA-43 1997; NISF: Nagoya, Japan, 1997. [Google Scholar]
- Goto, M.; Morita, S. Determination of the line emission locations in a large helical device on the basis of the Zeeman effect. Phys. Rev. E
**2002**, 65, 026401. [Google Scholar] [CrossRef] - Schweer, B.; Mank, G.; Pospieszczyk, A.; Brosda, B.; Pohlmeyer, B. Electron temperature and electron density profiles measured with a thermal He-beam in the plasma boundary of TEXTOR. J. Nucl. Mater.
**1992**, 196–198, 174–178. [Google Scholar] [CrossRef] - Goto, M.; Sawada, K. Determination of electron temperature and density at plasma edge in the Large Helical Device with opacity-incorporated helium collisional-radiative model. J. Quant. Spectrosc. Radiat. Transf.
**2014**, 137, 23–28. [Google Scholar] [CrossRef] - Kajita, S.; Ohno, N. Practical selection of emission lines of He I to determine the photon absorption rate. Rev. Sci. Instrum.
**2011**, 82, 023501. [Google Scholar] [CrossRef] [PubMed] - Iida, Y.; Kado, S.; Tanaka, S. Calculation of spatial distribution of optical escape factor and its application to He I collisional-radiative model. Phys. Plasmas
**2010**, 17, 123301. [Google Scholar] - Irons, F.E. The escape factor in plasma spectroscopy-I. the escape factor defined and evaluated. J. Quant. Spectrosc. Radiat. Transf.
**1979**, 22, 1–20. [Google Scholar] [CrossRef] - Fujimoto, T. Plasma Spectroscopy; Clarendon Press: Oxford, UK, 2004. [Google Scholar]
- Akatsuka, H.; Suzuki, M. Numerical study on population inversion and lasing conditions in an optically thick recombining helium plasma. Contrib. Plasma Phys.
**1994**, 34, 539–561. [Google Scholar] [CrossRef] - Kim, S.; Koh, K.; Lustig, M.; Boyd, S.; Gorinevsky, D. An interior-point method for large-scale l
_{1}-regularized least squares. IEEE J. Sel. Top. Signal Process.**2007**, 1, 606–617. [Google Scholar] [CrossRef] - Zhang, Y. Solving large-scale linear programs by interior-point methods under the Matlab Environment. Optim. Methods Softw.
**1998**, 10, 1–31. [Google Scholar] [CrossRef] - Rajasekhar Reddy, M.; Nithish Kumar, B.; Madhusudana Rao, N.; Karthikeyan, B. A new approach for bias–variance analysis using regularized linear regression. In Advances in Bioinformatics, Multimedia, and Electronics Circuits and Signals; Springer: Singapore, 2020. [Google Scholar]
- Wilson, R.C.; Hancock, E. Bias-variance analysis for controlling adaptive surface meshes. Comput. Vis. Image Underst.
**2000**, 77, 25–47. [Google Scholar] [CrossRef] - Doroudi, S. The bias-variance tradeoff: How data science can inform educational debates. AERA Open
**2020**, 6, 1–18. [Google Scholar] [CrossRef]

**Figure 2.**Spectra of the LHD helium plasma in the visible wavelength range. The designations indicate the upper levels of the transitions, while the corresponding lower levels are shown in Table 1.

**Figure 3.**Results obtained with three emission lines: (

**a**) intensity ratio of three lines for diagnosis, (

**b**) electron density and temperature diagnosed via the three-line diagnosis, (

**c**) comparison between the normalized intensities fitted by three-line diagnosis and their counterparts measured via OES method directly. The triangles represent the data obtained using the OES method.

**Figure 7.**Calculated bias and variance of the algorithm with different numbers of the fitting parameters.

**Figure 8.**Electron density and temperature obtained using the novel fitting algorithms (symbols and dashed lines represent results obtained using the new model and three-line analysis, respectively).

**Figure 9.**Comparison of the normalized intensity (colored dots and circles represent the measured line intensity and line intensity obtained using the novel model, respectively, and the crosses represent the line intensity obtained using the three-line method). (

**a**) Comparison of the 706.5 nm and 728.1 nm lines. (

**b**) Comparison of the 388.9 nm, 501.6 nm, and 587.6 nm lines. (

**c**) Comparison of the 492.2 nm line. (

**d**) Comparison of the 447.2 nm line.

**Figure 10.**Comparison of line spectra obtained in experiments with counterparts reproduced by the new model and three-line analysis (line-averaged electron density: ${\overline{n}}_{\mathrm{e}}=2\times {10}^{13}{\mathrm{c}\mathrm{m}}^{-3}$).

**Figure 11.**Escape factor fitted with the novel model (blue, orange, and yellow symbols represent the escape factor for transition from states 2

^{1}P, 3

^{1}P, and 4

^{1}P to ground state 1

^{1}S, respectively).

Wavelength λ_{p}_{,q} (nm) | Transition (n^{2S+1}L→n′^{2S′+1}L′) | A_{p,q} (s^{−1}) |
---|---|---|

728.135 | ${3}^{1}\mathrm{S}\to {2}^{1}\mathrm{P}$ | 1.8291 × 10^{7} |

706.525 | ${3}^{3}\mathrm{S}\to {2}^{3}\mathrm{P}$ | 2.7849 × 10^{7} |

501.568 | ${3}^{1}\mathrm{P}\to {2}^{1}\mathrm{S}$ | 1.3368 × 10^{7} |

388.864 | ${3}^{3}\mathrm{P}\to {2}^{3}\mathrm{S}$ | 0.9472 × 10^{7} |

667.815 | ${3}^{1}\mathrm{D}\to {2}^{1}\mathrm{P}$ | 6.3676 × 10^{7} |

587.566 | ${3}^{3}\mathrm{D}\to {2}^{3}\mathrm{P}$ | 7.0693 × 10^{7} |

492.193 | ${4}^{1}\mathrm{D}\to {2}^{1}\mathrm{P}$ | 1.9855 × 10^{7} |

447.150 | ${4}^{3}\mathrm{D}\to {2}^{3}\mathrm{P}$ | 2.4574 × 10^{7} |

Number of Fitting Parameters | Fitting Parameter |
---|---|

3 | ${n}_{\mathrm{e}}^{}$, ${T}_{\mathrm{e}}^{}$, ${n}_{{1}^{1}\mathrm{S}}^{}$ |

4 | ${n}_{\mathrm{e}}^{}$, ${T}_{\mathrm{e}}^{}$, ${n}_{{1}^{1}\mathrm{S}}^{}$, ${\Lambda}_{{2}^{1}\mathrm{P},{1}^{1}\mathrm{S}}^{}$ |

5 | ${n}_{\mathrm{e}}^{}$, ${T}_{\mathrm{e}}^{}$, ${n}_{{1}^{1}\mathrm{S}}^{}$, ${\Lambda}_{{2}^{1}\mathrm{P},{1}^{1}\mathrm{S}}^{}$, ${\Lambda}_{{3}^{1}\mathrm{P},{1}^{1}\mathrm{S}}^{}$ |

6 | ${n}_{\mathrm{e}}^{}$, ${T}_{\mathrm{e}}^{}$, ${n}_{{1}^{1}\mathrm{S}}^{}$, ${\Lambda}_{{2}^{1}\mathrm{P},{1}^{1}\mathrm{S}}^{}$, ${\Lambda}_{{3}^{1}\mathrm{P},{1}^{1}\mathrm{S}}^{}$, ${\Lambda}_{{4}^{1}\mathrm{P},{1}^{1}\mathrm{S}}^{}$ |

7 | ${n}_{\mathrm{e}}^{}$, ${T}_{\mathrm{e}}^{}$, ${n}_{{1}^{1}\mathrm{S}}^{}$, ${\Lambda}_{{2}^{1}\mathrm{P},{1}^{1}\mathrm{S}}^{}$, ${\Lambda}_{{3}^{1}\mathrm{P},{1}^{1}\mathrm{S}}^{}$, ${\Lambda}_{{4}^{1}\mathrm{P},{1}^{1}\mathrm{S}}^{}$, ${\Lambda}_{{5}^{1}\mathrm{P},{1}^{1}\mathrm{S}}^{}$ |

8 | ${n}_{\mathrm{e}}^{}$, ${T}_{\mathrm{e}}^{}$, ${n}_{{1}^{1}\mathrm{S}}^{}$, ${\Lambda}_{{2}^{1}\mathrm{P},{1}^{1}\mathrm{S}}^{}$, ${\Lambda}_{{3}^{1}\mathrm{P},{1}^{1}\mathrm{S}}^{}$ ${\Lambda}_{{4}^{1}\mathrm{P},{1}^{1}\mathrm{S}}^{}$, ${\Lambda}_{{5}^{1}\mathrm{P},{1}^{1}\mathrm{S}}^{}$, ${\Lambda}_{{6}^{1}\mathrm{P},{1}^{1}\mathrm{S}}^{}$ |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 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 (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Lin, K.; Goto, M.; Akatsuka, H.
Improved Line Intensity Analysis of Neutral Helium by Incorporating the Reabsorption Processes in a Helium Collisional-Radiative Model. *Atoms* **2023**, *11*, 94.
https://doi.org/10.3390/atoms11060094

**AMA Style**

Lin K, Goto M, Akatsuka H.
Improved Line Intensity Analysis of Neutral Helium by Incorporating the Reabsorption Processes in a Helium Collisional-Radiative Model. *Atoms*. 2023; 11(6):94.
https://doi.org/10.3390/atoms11060094

**Chicago/Turabian Style**

Lin, Keren, Motoshi Goto, and Hiroshi Akatsuka.
2023. "Improved Line Intensity Analysis of Neutral Helium by Incorporating the Reabsorption Processes in a Helium Collisional-Radiative Model" *Atoms* 11, no. 6: 94.
https://doi.org/10.3390/atoms11060094