Impact of Nonuniform Spatial Distribution of Ground-State Atoms on He I Line Intensity Ratio Analysis

Abstract

This study developed a model to calculate the optical escape factor in consideration of the spatial distribution of ground-state (1¹S) helium atoms. Based on the improved model, the impact of spatially nonuniform distribution on the helium line intensity ratio (LIR) method was investigated. The LIR method reproduced the input n_e and T_e values with good agreement when the spatial distribution was considered appropriately, whereas n_e and T_e evaluated without considering the distribution exhibited considerable deviations. These results suggest that the spatial distribution of the 1¹S atoms can be a significant source of uncertainty in the helium LIR method.

The He I line intensity ratio (LIR) method is widely used to diagnose electron density (n_e) and electron temperature (T_e). However, in the LIR method, radiation trapping becomes a significant source of uncertainties under optically thick plasma conditions. A convenient way to compensate for radiation trapping is by introducing the optical escape factor (OEF) for 1¹S-n¹P resonant transitions, where n denotes the principal quantum number. The effect of radiation trapping is more pronounced for resonant levels (n¹P); thus, several OEF calculation models have been proposed [1, 2] for various spatial distributions of the upper level. In most cases, these models assume that the spatial distribution of the helium atoms in the ground state (1¹S) is uniform. However, in realistic plasmas, this distribution can be nonuniform, primarily due to neutral pumping [3], which is caused by ion drag and gas heating. Discrepancies between the assumed and actual spatial distributions lead to increased uncertainties in the LIR method; however, their impact has remained unclear. Thus, in this study, we developed the OEF calculation model proposed by Iida [2] to account for the spatial distribution of the 1¹S atoms.

OEF (Λ) at a certain normalized position (R_N) in cylindrical coordinates is calculated based on Eqs. (1) and (2). Here, q and p denote the upper and lower levels, specifically corresponding to the n¹P and 1¹S levels, respectively. L_q denotes the characteristic length of the spatial profile of the population density of the upper level (n_q). Note that atoms excited in n¹P levels are strongly coupled with those in the ground state; thus, the same value can be utilized for temperatures T_q and T_p [4]. In this study, 300 K was assumed. The definitions of other parameters are the same as reported in the literature [2]. k′_pq0 is the absorption coefficient at the spectral center. The effect of spatially nonuniform 1¹S atoms was incorporated by spatially integrating the term in square brackets in Eq. (1) and considering the spatial profiles of n_p(R_N) in Eq. (2). In this study, the OEF was calculated up to n = 7.

  

$$\begin{array}{cc}\hfill {\Lambda }_{qp}\left(R_{\mathrm{N}}\right)=& 1-\frac{2}{\pi \sqrt{\pi }}\frac{k_{pq0}^{\prime }(R_{\mathrm{N}})L_q}{n_q(R_{\mathrm{N}})}{\int }_0^{+\infty }dx{\int }_0^{\pi }d\phi {\int }_0^{\frac{\pi }{2}}d\theta \exp [-(1+\frac{T_q}{T_p})x^2]\hfill \\ & \times {\int }_0^{r_{Nj}}d{r}_{\mathrm{N}}\times \left\{n_q(R_{\mathrm{N}}\prime (r_{\mathrm{N}}))\exp [-{\int }_0^{r_{\mathrm{N}}}\frac{{k^{\prime }}_{pq0}(r_{\mathrm{N}}^{\prime })}{\sin \theta }\exp (-\frac{T_q}{T_p}{x}^2)d{r}_{\mathrm{N}}^{\prime }]\right\},\hfill \end{array}$$(1)

  

$$k_{pq0}^{\prime }(R_{\mathrm{N}})=\frac{e^2}{4m_{\mathrm{e}}c{\epsilon }_0}{f}_{pq}{\lambda }_{qp}\sqrt{\frac{m_{\mathrm{He}}}{2\pi {\kappa }_{\mathrm{B}}{T}_p}}{n}_p(R_{\mathrm{N}}).$$(2)

The impact of the spatially nonuniform 1¹S atoms on the helium LIR method was investigated as follows based on the developed OEF model. (1) First, the population densities excited into the 3¹S, 3¹D, and 3³S levels were calculated using a collisional-radiative model for the He atom [5], assuming the spatial distributions of n_e, T_e, and 1¹S atoms. Simultaneously, the OEFs for 1¹S-n¹P transitions were calculated. (2) The LIRs of 2¹P-3¹D/2¹P-3¹S (667.8/728.1 nm) and 2¹P-3¹S/2³P-3³S (728.1/706.5 nm) were obtained based on the calculated population densities. This combination is suitable for investigating the impact of the 1¹S atom distribution because it is the simplest one in LIR analysis for helium plasma. (3) The above ratios were calculated again for various combinations of n_e and T_e using the OEFs calculated assuming spatially uniform and nonuniform 1¹S atoms. (4) Finally, n_e and T_e were obtained by minimizing the difference between the line ratios obtained in Steps (2) and (3).

As shown in Fig. 1(a), we assumed a center-peaked distribution for n_e and a spatially uniform distribution for T_e because such distributions are commonly observed in laboratory plasmas. Here, the horizontal axis is the plasma radius normalized by L_q (the e-folding length of n_e). In this study, L_q was set to 0.01 m. At the plasma core, neutral atoms in a cylindrical plasma are sometimes depleted by neutral pumping [6]. To consider this effect, four spatial distributions with different depletion rates were employed in this study (Fig. 1(b)). In contrast, although the temperature profile of the 1¹S atoms can also be nonuniform, a spatially uniform profile was employed to investigate the effect of the density profile.

Fig. 1.  Assumed spatial distributions of (a) n_e and T_e, and (b) 1¹S atoms.

Prior to investigating the impact of the spatial distribution of the 1¹S atoms, we confirmed that the assumed n_e and T_e could be appropriately reproduced when the assumed distributions of the 1¹S atoms were utilized in Step (3). The results are summarized in Fig. 2. As can be seen, both n_e and T_e were reproduced with good agreement regardless of the assumed depletion rates. Note that n_e and T_e at R_N ≥ 1.2 are not plotted because negative OEFs were obtained at this region, making effective interpretation of the results difficult [2].

Fig. 2.  (a) n_e and (b) T_e obtained by LIR analysis using spatially varying 1¹S atom distribution in Step (3).

In contrast, as shown in Fig. 3, when the 1¹S population was assumed to be spatially uniform in Step (3), the LIR method yielded different n_e and T_e values depending on the depletion rates. For example, at depletion rates of 0% and 10%, the LIR method obtained values that were consistent with the inputs. However, with a large depletion rate (90%), the reproducibility deteriorated considerably at R_N ≤ 0.5 and R_N ≥ 1.0. At R_N ≤ 0.4, n_e and T_e are not plotted due to negative OEF values. At R_N ≥ 1.0, the OEFs were positive; however, n_e and T_e were overestimated compared with the input values. These results suggest that the spatial distribution of 1¹S atoms can be a significant source of uncertainty in helium LIR analysis.

Fig. 3.  (a) n_e and (b) T_e obtained by LIR analysis using spatially uniform 1¹S atom distribution in Step (3).

Using the improved OEF model, we have demonstrated that performing helium LIR analysis without considering appropriate spatial distribution of the 1¹S atoms leads to uncertainties in the evaluation of n_e and T_e.

This work was supported by JSPS KAKENHI Grant Number JP24K00607.

References

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