A collisional-radiative (CR) model that extracts the electron temperature, *T*_{e}, of hydrogen plasmas from Balmer-line-ratio measurements is examined for the plasma electron density, *n*_{e}, and *T*_{e} ranges of 10^{10}–10^{15} cm^{−3} and 5–500 eV, respectively. The CR code, developed and implemented in Python, has a forward component that computes the densities of excited states up to *n* = 15 as functions of *T*_{e}, *n*_{e}, and the molecular-to-atomic neutral ratio *r*(H_{2}/H). The backward component provides *n*_{e} and *r*(H_{2}/H) as functions of the Balmer ratios to predict the *T*_{e}. The model assumes Maxwellian electrons. The density profiles of the electrons and of the molecular and atomic hydrogen neutrals are shown to be of great importance, as is the accuracy of the line-ratio measurement method.

## I. INTRODUCTION

Collisional-radiative (CR) models^{1,2} are utilized for studying the population distribution of atoms and molecules over their excited vibronic and ionized states, giving information about the relative importance of the different populating and depopulating processes. CR models are at the basis of many plasma diagnostics in the contexts of plasma processing and fusion research. Key parameters^{3} are electron density (*n*_{e}), velocity distribution,^{1,4} and molecular and atomic densities.

CR models for inert gases are relatively straightforward, there being only one species of neutrals. Sasaki *et al.*^{5} measured He–I line ratios in helium plasma to investigate the electron temperature (*T*_{e}) and *n*_{e} in NAGDIS-I (Nagoya University Divertor Simulator). Bogaerts and Gijbels^{6} used a CR model for argon glow discharges, employing a Monte Carlo model to calculate the electron energy distribution. Balmer-line series are also used to obtain particle and power balances and *T*_{e} and *n*_{e} measurements in the divertors of fusion research devices.^{7,8}

Hydrogen, being a molecule, is more complex. Johnson and Hinnov^{9} developed a CR model for hydrogen but only considered the atomic species. Since then, other CR models for the hydrogen atom have been presented, e.g., Ref. 10, Guzman *et al.*^{11} updated the ADAS^{12} tools for CR modeling to include hydrogen molecules.

The CR model that we have developed^{13} can provide *T*_{e} for a mixture of molecular ($nH2$) and atomic neutrals. This paper shows that knowledge of *n*_{e}, *n*_{H}, and $nH2$ profiles is essential to extract *T*_{e}. Only Maxwellian distributions are considered herein. Importantly, the measured H^{o} density^{14} in our fusion research device, the Princeton Field Reversed Configuration (PFRC-2), is of order 10^{11} cm^{−3} and constant in radius while that of H_{2} is $\u223c20\xd7$ higher at the plasma edge and about 10× lower in the plasma core.

## II. CR MODEL

In CR models, the densities of the various excited states of a specific atom or ion are expressed as functions of a number of relevant parameters, usually the ground-state density, *T*_{e}, and *n*_{e}. We constructed in Python a forward component CR model for hydrogen plasma based on Eq. (1). The first and the third terms represent collisional excitation or de-excitation into and out of state *i*, respectively; the second and fourth terms represent the radiative decay into and out of state *i*, respectively; the fifth term represents the ionization of state *i*; the sixth term gives the ground-state atomic-hydrogen excitation to state i; and the last term is used for population-of-state based on molecular dissociation. Negligible recombination occurs in the warm (*T*_{e} > 5 eV), relatively tenuous (*n*_{e} < 10^{14} cm^{−3}), high heating-power density PFRC-2 plasmas that are the central focus of our study,

where *n*(*i*) is the density of the excited state *i*, ⟨*σv*⟩’s are the Maxwellian-averaged electron-impact reaction rate coefficients, and *A*’s are the Einstein coefficients. The electron-impact rate coefficients for hydrogen are incorporated from Johnson,^{15} Vriens and Smeet,^{16} and Sawada *et al.*^{17} (Ion-neutral collisions proceed at far slower rates hence are not included.)

The emission-line intensity ratios of the Balmer lines are generated for the *n*_{e} range 10^{10}–10^{15} cm^{−3} and *T*_{e} from 5 to 500 eV. The Balmer-line ratios depend strongly on the ratio of molecular to atomic hydrogen densities; see Fig. 1. (The H_{α} line brightness produced by electron-impact dissociative excitation of H_{2} is $\u223c5\xd7$ less than that caused by electron-impact excitation of H.) The densities of up to *n* = 15 excited states of hydrogen atoms, molecules, or molecular ions are expressed as a function of *T*_{e}, *n*_{e}, and *r*(H_{2}/H).^{2} The H_{β}/H_{α} intensity ratio vs *n*_{e} for various *T*_{e} are shown in Fig. 1(a) for both pure H and pure H_{2} neutrals and those for H_{γ}/H_{β} are in Fig. 1(b).

Because the Balmer transitions are of low energy, $\u223c2$ eV, the change in line ratios becomes increasingly smaller as *T*_{e} rises, as seen in Fig. 1.

### A. CR model tends to coronal equilibrium (CE) model

A coronal equilibrium (CE) model is appropriate at low density where collisional de-excitation is unimportant. Figure 1 indicates that this occurs for *n*_{e} ≲ 10^{11} cm^{−3}. In CE, the fundamental approximations are that all upward atomic transitions are (electron) collisional and all downward transitions are radiative by spontaneous emission.

The upper limit on *n*_{e} for a CE model to be applicable can found by balancing collisional de-excitation and radiative decay. At low density, Eq. (1) becomes

For the *n* = 5 state, and similarly for the *n* = 3 and 4 states, the left-side term in Eq. (3) is the collisional de-excitation rate and the right-side term is the radiative decay rate

where

## III. SENSITIVITY OF LINE RATIO

### A. Sensitivity to *T*_{e}

Comparing the Balmer-line ratio for H_{2} neutrals at *T*_{e} = 100 eV to that at 120 eV (see Fig. 1) shows very small differences: the changes in the ratios are Δ(H_{β}/H_{α}) = 0.7% and Δ(H_{γ}/H_{β}) = 0.5% for the *n*_{e} range 7 × 10^{11–12} cm^{−3}. The Balmer-line ratios for H_{β}/H_{α} and H_{γ}/H_{β} vs *T*_{e} are shown in Fig. 3 at *n*_{e} = 10^{12} cm^{−3}. For Δ*T*_{e}/*T*_{e} = ±10% at *T*_{e} = 100 eV, the required accuracy for a plasma with only atomic neutrals is Δ(H_{β}/H_{α}) = 0.9%. Below *T*_{e} = 25 eV, about 10× less accuracy is needed in the measured H_{β}/H_{α}.

At somewhat higher density, *n*_{e} ∼ 10^{13} cm^{−3}, an accuracy of ±1% in H_{β}/H_{α}, and an accuracy of ±10% in H_{γ}/H_{β} are required for a ±10% accuracy when *T*_{e} near 100 eV.

### B. Sensitivity to *n*_{e}

The sensitivity of the Balmer-line ratio to *n*_{e} was evaluated at 100 eV (see Fig. 4. The difference in the Balmer ratio for H_{2} neutrals in the range *n*_{e} = (1–10) × 10^{12} cm^{−3} shows that Δ(H_{β}/H_{α}) = −42% while for H neutrals, Δ(H_{β}/H_{α}) = −50%.

### C. Sensitivity to neutral population densities

The Balmer-line ratio is highly sensitive to the ratio of the molecular to the sum of all neutral densities (Fig. 5). The Balmer-line ratio difference at *r*(H_{2}/(H + H_{2})) of 0.99 and 0.9 was Δ(H_{β}/H_{α}) ∼ Δ(H_{γ}/H_{β}) ∼ 14%. This density-ratio range is representative of that in the PFRC-2 edge plasma.

The effect of the neutral density ratio on the extracted *T*_{e} is shown in Fig. 6 for H_{β}/H_{α} = 0.13. At a 1% concentration of H_{2}, the inferred *T*_{e} is 6 eV, while at 90%, it is 110 eV. Knowledge of the neutral densities is clearly essential to accurately determine *T*_{e}. Two-photon absorption laser induced flourescence (TALIF)^{14} can provide the atomic neutral density profile. At present, the H_{2} profile is calculated for the PFRC-2 using the DEGAS neutral transport code. It shows the H_{2}/H ratio changes from 1:10 in the PFRC-2 plasma’s core to 10:1 at its edge. Use of this data also requires knowledge of the plasma density profile because the Balmer line ratios are line averages. Spectrometer observations through chords having different tangency radii will reduce this uncertainty.

## IV. DETERMINATION OF LINE RATIO

This section describes the methods that we have evaluated for obtaining the strength of a spectral line when computing Balmer line ratios for PFRC-2.

Pulsed, odd-parity rotating magnetic field (RMF) antennas create a hydrogen plasma in the center cell with the goal of inducing a field-reversal configuration (FRC). Hydrogen line intensities are measured using a Model 9590212 Ocean FX-Spectrometer, set at 150 *µ*s integration time. The PFRC-2 was operated at vacuum central magnetic field of 140 G, absorbed power $\u223c16$ kW, initial fill pressure of 0.3 mTorr, and pulse width of 7 ms. Several methods were compared for finding the Balmer-line ratios. Prior to these tests, the spectrometer sensitivity vs wavelength was measured with two integrating spheres, sources of nominally black-body-like radiation.

### A. Counts under the peak

In this method, the line intensity is determined by summing all the counts “under” the peak, as shown in Fig. 7. These data were produced in less than three minutes of clock time by accumulating photons over 159 shots at *t* = 1.08 ± 0.075 ms of the 7 ms width pulses. Near the H_{β} wavelength, the spectrometer dispersion was about 6 pixels/nm. 17 pixels to either side of the maximum have been included; outside this range, the counts per pixel are lesser than 1/500 of the peak value and are the region used for background subtraction.

### B. Gaussian fit

In this approach, a Gaussian fit is applied to the data obtained from the spectrometer (see Fig. 8). The area under the Gaussian is used to obtain the line intensity. The Gaussian is chosen to match the peak height and half-width; however, the fit is poorer on the line’s wings. In this wavelength range, the line of a laser also shows similar asymmetry, leading us to attribute the non-Gaussian shape to an instrumental effect of the Ocean FX-spectrometer.

### C. Peak height

In this approach, the pixel with the greatest number of counts near the line of interest provides the peak height from which the intensity is computed. The peak-height method differs from the Gaussian method because the peak height of a narrow line depends on where its central wavelength falls relative to the pixel boundaries in the spectrometer’s detector. A narrow line centered on one pixel will have one peak height, but centered between two pixels, and the same line would have a smaller peak height.

### D. Comparison

The Balmer-line ratios vs time for the mentioned experimental condition are shown in Fig. 9. Each method yields a different line ratio; thus, different values for *T*_{e} when those line ratios are used with the CR model. The Gaussian and area-under-peak methods differ by about 5%, while the peak-amplitude method is lower by 25%. Consideration of the physics issues leads us to conclude that the most accurate method for determining the line ratios is to sum the counts under the peak, as it accounts for factors, particularly the spectrometer dispersion, that affect the distribution of counts vs wavelength.

## V. AN ELECTRON TEMPERATURE MEASUREMENT

In consideration of the three aforementioned factors—the measured line ratios, the % of H_{2}, defined as $Pc\u2261100nH2/(nH2+nH)$, and *n*_{e}—we extract estimates of *T*_{e}(*t*). The line ratio statistics provide the error bar, the shaded region in Fig. 10.

The *n*_{e} measurement is a line average. For the same profile shape, *n*_{e} depends inversely on the separatrix radius. Based on probe data and fast camera observations, we use a separatrix radius of 5 cm. We note, in passing, that, if *n*_{e} is known, using two line ratios, e.g., H_{β}/H_{α} and H_{γ}/H_{β}, can reduce the uncertainty in the value of *P*_{c} used to calculate *T*_{e}.

The *P*_{c} value in the core depends on the accuracy of the TALIF measurements and the DEGAS simulations; the latter depends on *T*_{e}. The *T*_{e} considered in DEGAS span the range 20–100 eV, predicting 5 < *P*_{c} < 25%. Based on pressure-gauge and TALIF measurements, *P*_{c} at the edge exceeds 10. However, the plasma edge *n*_{e} is lower, hence the emission dimmer. We do not include the high *P*_{c} at the edge in the data analysis, hence provide a conservative estimate of *T*_{e}.

Figure 10 shows the extracted *T*_{e}(*t*) for two values of *P*_{c}, 3% and 5%. For these PFRC-2 experimental conditions, if *P*_{c} is above 10%, this CR model predicts that *T*_{e} exceeds 500 eV. The shaded region indicates the ±*σ* uncertainty due to the Balmer-line ratio statistics. Near the measured plasma line-average density, *T*_{e} is approximately proportional to 4/r (cm).

The increasing line ratio during the discharge (see Fig. 9) causes the extracted *T*_{e} to increase. Figure 10 displays the important feature first noted in Fig. 1: the neutral density ratio, *P*_{c}, is critical in extracting *T*_{e}. Conversely, if *T*_{e} is known via another diagnostic, *P*_{c} can be determined. We note that x-ray diagnostics on the PFRC-2, which measure the Bremsstrahlung spectrum above 500 eV, have shown *T*_{e} in the range of 100–400 eV. The number of plasma discharges to acquire the x-ray data with the same time resolution was about 30 greater than the Balmer-line-ratio technique described herein.

## VI. SUMMARY

A CR model developed for the hydrogen plasma was evaluated. It converts to CE model when *n*_{e} is less than 10^{11} cm^{−3}. The CR method is highly sensitive to *n*_{e} and its radial profile, and the ratio of molecular to atomic neutrals and their radial profiles. Accurate determination of line intensities, that is, spectrometer calibration, is also crucial for use of the model to extract accurate *T*_{e}. First tests of the CR model on the PFRC-2 have been made, including comparisons with other diagnostics. These comparisons are informative as different diagnostics sample different parts of the electron energy distribution function.

## ACKNOWLEDGMENTS

This work was supported, in part, by the U.S. Department of Energy under Contract No. DE-AC02-09CH11466, by ARPA-E Award No. DE-AR0001099 (Princeton Fusion Systems), and by the Princeton Program in Plasma Science and Technology, Princeton University. We are grateful to M. Paluszek and S. J. Thomas for their continued support and B. Berlinger and C. Brunkhorst for excellent technical work.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**S. P. Vinoth**: Formal analysis (lead); Software (equal). **E. S. Evans**: Software (equal). **C. P. S. Swanson**: Formal analysis (equal). **E. Palmerduca**: Software (equal). **S. A. Cohen**: Supervision (equal).

## DATA AVAILABILITY

The data that support the findings of this study are openly available at https://dataspace.princeton.edu/handle/88435/dsp01x920g025r.^{18}