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

Abstract

In this study, eight emission lines in the visible wavelength range of neutral helium were used to diagnose the electron density and temperature of the Large Helical Device (LHD) helium plasma instead of the conventional three-line method. The collisional-radiative (CR) model for low-pressure helium plasma was revised to include the optical escape factors for spontaneous transition from the n¹P 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

Currently, optical emission spectroscopy (OES) measurement is generally used to acquire plasma parameters. Because it does not require direct contact with the plasma, this method is relatively more effective and safer than other applicable methods and is capable of measuring the plasma parameters under different conditions, such as low-pressure microwave-induced plasma [1], atmospheric pressure non-equilibrium plasma [2], and Large Helical Device (LHD) plasma [3]. OES diagnoses employing the collisional-radiative (CR) model are used to diagnose the electron density and temperature of different types of gases [4,5].

Emission lines in the visible wavelength range are widely used for plasma diagnosis. Three-line diagnosis is commonly used to diagnose the electron density and temperature of helium plasmas. This method is relatively safe and effective because it requires the measurement of only three lines in the visible wavelength range; however, the results are sometimes inconsistent with other diagnostic results. Therefore, the conventional method requires further improvements to enhance its reliability.

For plasmas with lower electron temperatures, the performance of the line intensity ratio (LIR) method and laser Thomson scattering (LTS) method has been investigated for recombining detached plasmas [6]. Additionally, the optical escape factor in cylindrical geometries has been calculated and applied to weakly ionized helium using the CR model [7,8]. However, the escape factor for helium plasmas with high electron temperatures in LHD has not yet been studied.

In this study, the CR model developed by Goto [9] and Fujimoto [10] was revised. The model considers 65 discrete energy levels of neutral helium, including one ground state (1¹S) 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].

This study demonstrates a new algorithm for determining the electron density and temperature of helium plasma. Section 2 demonstrates that the conventional method, which uses only three emission lines, exhibits inconsistencies seen in the results. Section 3 illustrates the revision of the CR model and the development of the algorithm. Section 4 and Section 5 present relevant experiments, discussions, and conclusions, respectively.

2. Experiment

The measurements were performed for the LHD, which is a heliotron-type experimental device for magnetically confined fusion plasmas. We measured the spectra for 10 discharges with helium gas, for which the magnetic axis radius was fixed at 3.6 m, while the magnetic field strength was changed to 0.75, 1.5, and 2.75 T. The line-of-sight was roughly in the radial direction, passing through the central plasma region on a horizontally elongated poloidal cross-section, as shown in Figure 1.

The field view collimated by the lens had an approximately 30 mm wide cylindrical profile. The light introduced into an optical fiber with a core diameter of 100 μm was guided to a Czerny-Turner type spectrometer with a focal length of 0.5 m. The spectrometer was equipped with a grating of 100 grooves/mm, and the reciprocal linear dispersion was measured to be 19.976 nm/mm. We used a charge-coupled device (CCD) consisting of 1024 pixels × 255 pixels, with each pixel measuring 26 μm × 26 μm, as the detector. Consequently, a wavelength of approximately 520 nm could be measured simultaneously. The central wavelength was adjusted to cover all six transitions from n = 3 to n = 2, i.e., from 388.9 nm to 728.1 nm. The absolute sensitivity of the entire system was calibrated using a standard light source, which consisted of a halogen lamp and an integrated sphere (Labsphere USS-600C). The radiance (W·m⁻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.

An example of the measured spectra is shown in Figure 2.

Eight emission lines corresponding to neutral helium were identified. In LHD, it is known that neutral helium line emissions are localized in a thin layer at the plasma edge region [15]. Therefore, the local T_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:

$$n_p=\frac{{\lambda }_{p,q}}{hc}\frac{{\Phi }_{p,q}}{A_{p,q}}$$

(1)

where h is Planck’s constant, c is the speed of light, λ_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:

$${\Phi }_{p,q}=4\pi {\int }_{\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{e}}^{}L\left(\lambda \right)d\lambda .$$

(2)

Table 1. Helium lines used in the line spectrum analysis.

Wavelength λp,q (nm)	Transition (n2S+1L→n′2S′+1L′)	Ap,q (s−1)
728.135	$3^1\mathrm{S}\to 2^1\mathrm{P}$	1.8291 × 107
706.525	$3^3\mathrm{S}\to 2^3\mathrm{P}$	2.7849 × 107
501.568	$3^1\mathrm{P}\to 2^1\mathrm{S}$	1.3368 × 107
388.864	$3^3\mathrm{P}\to 2^3\mathrm{S}$	0.9472 × 107
667.815	$3^1\mathrm{D}\to 2^1\mathrm{P}$	6.3676 × 107
587.566	$3^3\mathrm{D}\to 2^3\mathrm{P}$	7.0693 × 107
492.193	$4^1\mathrm{D}\to 2^1\mathrm{P}$	1.9855 × 107
447.150	$4^3\mathrm{D}\to 2^3\mathrm{P}$	2.4574 × 107

lists the parameters of the emission lines examined. A conventional three-line diagnosis for deriving T_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.

The parameters T_e and n_e were determined by minimizing the function as follows:

$$f\left(T_{\mathrm{e}},n_{\mathrm{e}}\right)=\sum _p{\left(\frac{n_p^{}-n_p^{\prime }}{n_p^{}}\right)}^2$$

(3)

where $n_p^{}$ is the measured number density, and $n_p^{\prime }$ was calculated using the CR model with the given T_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.

We examined the results obtained using emission lines that were not used for T_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 3¹P–1¹S, 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

At an earlier time, Kajita et al. investigated radiation trapping by additionally measuring one or two specific lines. The escape factors for the spontaneous transition from 2¹P, 3¹P, 4¹P, and 5¹P 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 (1¹S–4¹P), 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:

$${\Lambda }_{p,q}={\int }_0^{1}T\left(\frac{\kappa D}{2x}\right)\mathrm{d}x$$

(4)

where x is the cosine of the inclination of the z-axis in Ref. [20], $\kappa$ is the absorption coefficient [16] which can be obtained as follows:

$$\kappa =\frac{e^2}{4{\epsilon }_0{m}_{\mathrm{e}}c}{n}_q{f}_{q,p}P\left(\nu \right)$$

(5)

where $P\left(\nu \right)$, ${\epsilon }_0$, $m_{\mathrm{e}}$, $e$, and $f_{q,p}$ are the line profile, vacuum permittivity, mass of the electron, elementary charge, and oscillator strength, respectively. The Doppler profile [21,22] is used for calculating the line profile $P\left(\nu \right)$. It is expressed as follows:

$$P(\nu )\mathrm{d}\nu =\frac{1}{\sqrt{\pi }{\mathsf{\Delta }\nu }_{\mathrm{D}}}\mathrm{e}\mathrm{x}\mathrm{p}\left[-{\left(\frac{\nu -{\nu }_0}{{\mathsf{\Delta }\nu }_{\mathrm{D}}}\right)}^2\right]\mathrm{d}\nu$$

(6)

with the Doppler width

$${\mathsf{\Delta }\nu }_{\mathrm{D}}=\frac{{\nu }_0}{c}\sqrt{\frac{2k_{\mathrm{B}}{T}_{\mathrm{g}}}{m_{\mathrm{H}\mathrm{e}}}}$$

(7a)

and the central frequency

$${\nu }_0=\frac{{ϵ}_p^{\mathrm{e}\mathrm{x}}-{ϵ}_q^{\mathrm{e}\mathrm{x}}}{h}$$

(7b)

where $k_{\mathrm{B}}$, $T_{\mathrm{g}}$, and $m_{\mathrm{H}\mathrm{e}}$ are the Boltzmann’s constant, gas temperature, and mass of the helium atom, respectively. ${ϵ}_p^{\mathrm{e}\mathrm{x}}$ and ${ϵ}_q^{\mathrm{e}\mathrm{x}}$ are the excitation energies of levels p and q, respectively. $T\left({\tau }_0\right)$ is the average probability that a photon within the line profile propagates ${\tau }_0$:

$$T({\tau }_0)={\int }_{-\infty }^{\infty }\mathrm{e}\mathrm{x}\mathrm{p}\left[-{\tau }_0\frac{P\left(\nu \right)}{P\left(0\right)}\right]P(\nu )d\nu$$

(8)

Figure 4 shows the calculated optical escape factors for the levels n¹P as a function of the ground-state density.

Level 2¹P exhibited the lowest optical escape factor among the six n¹P levels. This indicates that lower levels have a relatively stronger absorption effect. The absorption effect of n¹P (n = 5, 6, and 7) can be neglected when the ground-state density is lower than 10¹³ 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:

$${\Gamma }_{p,q}^{\mathrm{r}\mathrm{a}\mathrm{d}}={\Lambda }_{p,q}{A}_{p,q}{n}_p$$

(9)

where ${\Lambda }_{p,q}^{}$ is set to 1 for optically thin states. This is the same technique employed in Ref. [8].

3.2. Bias–Variance Analysis

An algorithm for diagnosing the electron density and temperature was developed in this study. We evaluated the escape factors with a slab structure; the fitting was conducted with the restriction that the derived escape factors should not differ significantly from the evaluated escape factors. This restriction can be realized using a regularization term in the object function

$$f_{\mathrm{o}\mathrm{b}\mathrm{j}}=\sum _p{\left(\frac{n_p^{}-n_p^{\prime }}{n_p^{\mathrm{s}\mathrm{m}\mathrm{a}\mathrm{l}\mathrm{l}}}\right)}^2+\mu {\left(\frac{{\Lambda }_{n^1\mathrm{P},1^1\mathrm{S}}^{\mathrm{c}\mathrm{a}\mathrm{l}}-{\Lambda }_{n^1\mathrm{P},1^1\mathrm{S}}^{\prime }}{{\Lambda }_{n^1\mathrm{P},1^1\mathrm{S}}^{\mathrm{c}\mathrm{a}\mathrm{l}}}\right)}^2$$

(10)

where ${\Lambda }_{n^{1}P,1^{1}S}^{\prime }$ is the escape factor generated by the optimizing algorithm. $n_p^{s\mathrm{m}\mathrm{a}\mathrm{l}\mathrm{l}}$ denotes the smaller one among $n_p^{}$ and $n_p^{\prime }$. ${\Lambda }_{n^1\mathrm{P},1^1\mathrm{S}}^{\mathrm{c}\mathrm{a}\mathrm{l}}$ is the optical escape factor calculated using Equation (4), which is a function of $n_1^{\prime }$ when T_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.

The algorithm employs the number of densities corresponding to the eight measured emission lines as inputs. The interior point method [23,24] is applied to optimize the object function. The optimizing algorithm generates $n_{\mathrm{e}}^{\prime }$, $T_{\mathrm{e}}^{\prime }$, $n_{1^1\mathrm{S}}^{\prime }$, and ${\Lambda }_{p,q}^{\prime }$ with different values and determines the global minimum of the object function. The outputs are $n_{\mathrm{e}}^{\prime }$ and $T_{\mathrm{e}}^{\prime }$, which minimize the object function. An optimum value of μ is determined via bias–variance analysis [25,26,27]. The analysis helps us understand the trade-off between the new model’s ability to fit the measured line intensity well (low bias) and its ability to generalize accurate electron density, electron temperature, ground-state density, and escape factors (low variance). By analyzing the bias and variance of the new model, we made informed decisions about parameter selection and complexity to achieve better performance. In the bias–variance analysis, additional measurements have been taken. The helium plasma is generated under stable conditions. In total, 40 spectra are obtained from the OES measurements every 0.01 s. It is assumed that the electron density and temperature are constant during the 0.4 s. The acquired spectra are used for analysis. The bias is obtained as follows:

$${\left(bias\right)}^2=\frac{1}{8}\sum _p{\left[\frac{{\mathrm{l}\mathrm{o}\mathrm{g}}_{10}\left(\left|{\overline{n}}_p^{\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{s}}-{\overline{n}}_p^{\mathrm{f}\mathrm{i}\mathrm{t}}\right|\right)}{{\mathrm{l}\mathrm{o}\mathrm{g}}_{10}\left({\overline{n}}_p^{\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{s}}\right)}\right]}^2$$

(11)

and the variance “var” is obtained by the following:

$$var=\frac{1}{8}\sum _p\left\{\frac{1}{K}\sum _{k=1}^K{\left[\frac{{\mathrm{l}\mathrm{o}\mathrm{g}}_{10}\left(\left|{\overline{n}}_p^{\mathrm{f}\mathrm{i}\mathrm{t}}-n_p^{\left(k\right)\mathrm{f}\mathrm{i}\mathrm{t}}\right|\right)}{{\mathrm{l}\mathrm{o}\mathrm{g}}_{10}\left({\overline{n}}_p^{\mathrm{f}\mathrm{i}\mathrm{t}}\right)}\right]}^2\right\},$$

(12)

where ${\overline{n}}_p^{\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{s}}$ is the mean value of the measured number density of level p, ${\overline{n}}_p^{\mathrm{f}\mathrm{i}\mathrm{t}}$ is the mean value of the number density obtained via the fitting algorithm, and $n_p^{\left(k\right)\mathrm{f}\mathrm{i}\mathrm{t}}$ is the fitted number density of level p in the k-th measured spectrum. This bias expresses the discrepancy between the fitted results and measurements. The variance corresponds to the degree of variation in the fitted results under approximately identical plasma conditions. The bias and variance with different μ, in the range of 0.01–100, are shown in Figure 6.

The bias increases with an increase in the regularization parameter; it is relatively steady when μ is higher than 3. The variance also decreases rapidly when μ is lower than 3 and is steady when μ is higher than 3. This indicates that the algorithm can be overfitted when μ < 3. It is difficult to determine the global minimum of the total error; however, μ = 5 is considered reasonable in the present case. It restricts the optical escape factors generated by the algorithm within a reasonable range but is not completely a function of the ground-state density.

The present model could still suffer from an overfitting problem, and we attempted to optimize the number of escape factors considered in the model. We conducted a bias–variance analysis by increasing the number of escape factors considered in the model. The used fitting parameters and results are shown in

Table 2. Fitting parameters used in bias–variance analysis.

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	$n_{\mathrm{e}}^{}$, $T_{\mathrm{e}}^{}$, $n_{1^1\mathrm{S}}^{}$, ${\Lambda }_{2^1\mathrm{P},1^1\mathrm{S}}^{}$, ${\Lambda }_{3^1\mathrm{P},1^1\mathrm{S}}^{}$ ${\Lambda }_{4^1\mathrm{P},1^1\mathrm{S}}^{}$, ${\Lambda }_{5^1\mathrm{P},1^1\mathrm{S}}^{}$, ${\Lambda }_{6^1\mathrm{P},1^1\mathrm{S}}^{}$

and Figure 7, respectively.

The variance increases steadily with an increase in the number of fitting parameters. The bias decreases rapidly when the number of fitting parameters is less than six. The minimum total error was recorded when the number of fitting parameters was six. Therefore, six parameters ($n_{\mathrm{e}}^{}$, $T_{\mathrm{e}}^{}$, $n_1^{}$, ${\Lambda }_{2^1\mathrm{P},1^1\mathrm{S}}^{}$, ${\Lambda }_{3^1\mathrm{P},1^1\mathrm{S}}^{}$, and ${\Lambda }_{4^1\mathrm{P},1^1\mathrm{S}}^{}$) were selected as the fitting parameters for the algorithm.

4. Results and Discussion

We conducted fitting using the model described in Section 2 for the line intensity data shown in Figure 2. The electron densities and temperatures diagnosed are shown in Figure 8.

Both the electron density and temperature obtained using the novel method showed increasing and decreasing tendencies, respectively, with an increase in the line-averaged electron density, which was similar to the results obtained using the three-line method, as shown in Figure 3b. When the line-averaged electron density was low, the electron temperatures obtained using the two methods were similar. The electron temperature diagnosed using the new model decreased faster when the line-averaged electron density increased. The electron density obtained using the proposed model was generally slightly lower than that obtained using the three-line method.

A comparison of the normalized line intensities is shown in Figure 9. An example of line spectra reproduced by the new model and three-line analysis is shown in Figure 10. Compared to the three-line method, the difference between the fitted and measured results for the 706.5 and 728.1 nm lines increased slightly. A relatively large difference appeared when the line-averaged electron density was higher than 2 × 10¹³ cm⁻³. The fitting of the 501.6 and 587.6 nm lines improved when the line-averaged electron density was between 10¹² and 2 × 10¹³ cm⁻³. When the line-averaged electron density was higher than 2 × 10¹³ 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.

Figure 11 shows the escape factors obtained by fitting. The escape factors for all three states exhibited similar trends, with an increase in the line-averaged electron density. State 2¹P had the smallest escape factor, which decreased from approximately 0.6 to 0.005 when the line-averaged electron density was lower than 6 × 10¹³ cm⁻³. Then, it increased to approximately 0.02 at 10¹⁴ cm⁻³. State 4¹P had the largest escape factor in the range of 0.07–0.9. The escape factor for state 3¹P was between those for states 2¹P and 4¹P (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 N₁. Because N₁ 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

In this study, the helium CR model was modified by including an optical escape factor. An algorithm was developed to use $n_{\mathrm{e}}^{}$, $T_{\mathrm{e}}^{}$, $n_1^{}$, ${\Lambda }_{2^1\mathrm{P},1^1\mathrm{S}}^{}$, ${\Lambda }_{3^1\mathrm{P},1^1\mathrm{S}}^{}$, and ${\Lambda }_{4^1\mathrm{P},1^1\mathrm{S}}^{}$ to fit the eight emission lines in the visible wavelength range. According to the results, the algorithm can precisely diagnose the electron density and temperature of the LHD helium plasma. The disagreement of the line at 501.6 nm in the conventional three-line diagnosis can be solved by including the optical escape factor in the CR model and increasing the number of input lines from three to eight.

However, in the developed algorithm, the differences between the measured and fitted results of states 3¹S, 3³S, and 3¹D 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.

The algorithm provides another option to diagnose the electron density and temperature of low-pressure helium plasma using the OES method, and it can be applied to the plasma under various conditions. The validity of the algorithm for other types of helium plasmas will be investigated in the future.