On electron temperature rise in divertor relevant recombining plasma along magnetic field line

Volumetric recombination, the inverse process of ionization, is one of the most fundamental processes in plasmas and observed in various fields, for instance, in a supernova remnant,¹ in an electric double layer like plasma,² and in an arc plasma jet.³ Volumetric recombination is also of great interest in fusion research. When heat and particles in confined plasma region move across the separatrix, they enter the scrape-off layer and eventually terminate at divertor plates. In terms of divertor protection, heat removal capability is a critical issue that determines the output of fusion reactors.^4,5 Divertor plasma detachment, which is caused by increased impurity radiation loss and volumetric recombination, is considered to be a promising solution that allows the divertor to withstand the heat flux and sputtering.^6–8 The problem is that the modeling and code development on divertor detachment are premature although numerous simulations on divertor detachment have been carried out to find possible operation scenarios.⁹ 

The reaction rate of volumetric recombination is determined by electron temperature (⁠$Te$⁠) and electron density (⁠$ne$⁠): for example, that of three body recombination is proportional to $neTe−9/2$⁠.¹⁰ This means plasma modeling that allows reliable predictions on spatial structure of detached divertor, especially in terms of $Te$⁠, is necessary for fusion reactor design. The region where volumetric recombination strongly occurs is referred to as “recombination front” and formed in the vicinity of divertor plates, which is far away from core plasma. With respect to the spatial structure, $Te$ and $ne$ in detached or recombining plasma are considered to decrease monotonically along the magnetic field line at downstream of the recombination front,¹¹ and some experiments support this understanding.^12–15 However, there is a sign of temperature increase at downstream of the recombination front region even though there is no additional electron heating source, albeit not claimed in these reports. For example, in the ULS device, although it is within the range of error bar, $Te$ derived from the Boltzmann plot method increases from less than 0.1 eV to approximately 0.2 eV toward downstream.¹³ The double probe measurement conducted in the NADGIS-II device shows similar spatial structure.¹⁵ This phenomenon is likely common in laboratory plasmas regardless of the device, discharge gas species, and diagnostic method. Recently, a numerical study showed that spatial structure of $Te$ in detached plasma is modified when the heating effect due to three body recombination is taken into consideration.¹⁶ This work indicated that the heating effect has an important role in determination of spatial structure of detached plasma. However, monotonic decrease in $Te$ is still maintained. Therefore, it suggests that a different mechanism remains to be found to account for the temperature rise phenomenon described above.

In the present work, $Te$ and $ne$ in recombination front region and its downstream region were investigated. Temperature rise from below 0.5 to above 1 eV along the magnetic field line was clearly observed. This phenomenon was analyzed, and a plausible mechanism that causes temperature rise is provided for the first time.

Experiments were conducted using an RF plasma device DT-ALPHA.¹⁷ Figure 1(a) represents a schematic of the device. DT-ALPHA consists of a quartz pipe and a stainless steel chamber, and the total length is approximately 2 m. Inner diameter of the quartz pipe and vacuum chamber are 36 and 63 mm, respectively. An RF antenna is wound around the quartz pipe and connected with an RF power supply through a matching circuit. Plasma is produced by 13.56 MHz RF discharge. In the present experiment, RF heating power was kept at approximately 250 W. Helium working gas and secondary gas were supplied into the device near upstream edge and the downstream region. In order to optimize spatial distribution of neutral pressure, three orifice units are installed. By controlling the amount of secondary gas, helium recombining plasma can be produced at slightly upstream of the secondary gas feeding port.^18,19 Hereafter, recombining plasma region and secondary gas feeding position are referred to as the “midstream region” and the “downstream region,” respectively. When recombining plasma is formed, the midstream region corresponds to the recombination front region. Helium neutral pressure (p) was measured at the downstream region. Since no orifice unit is installed between the two regions, p at the midstream region is considered to be almost the same with that at the downstream region. Magnetic field strength at the two regions was maintained at approximately 0.2 T. Plasma diagnostics were conducted at the midstream and the downstream regions using Langmuir probes (LPs) and a spectroscope. Langmuir probes were utilized as a single probe. Figure 1(b) represents a cross section at the midstream region. A viewing chord was kept at almost central position of a cylindrical plasma but moved vertically when radial profiles were observed, as illustrated in Fig. 1(b).

Typical wavelength spectrum observed at the midstream, and the downstream regions are shown in Fig. 2. As described later, when the neutral pressure was increased above 6.5 Pa, recombining plasma was formed at the midstream region. Therefore, line spectrum attributed to three body recombination and continuum spectrum attributed to radiative recombination were clearly observed around $λ=350$ and $λ<345 nm$⁠, respectively. On the other hand, no clear indicator of volumetric recombination was seen at the downstream region as shown with dashed line. It is well known that use of a Langmuir probe, especially a single probe, becomes difficult for diagnosing detached or recombining plasmas because it significantly overestimates electron temperature.^20,21 Instead, $Te$ and $ne$ in recombining plasma can be derived from both high-n line emissions (the Boltzmann plot method) and continuum emissions.^21,22 Here, n represents principal quantum number. In the present work, continuum emission was utilized for the derivation because electron energy distributions (EEDs) in an RF plasma sometimes deviate from the Maxwellian distribution, and the use of the Boltzmann plot method leads to the underestimation of $Te$ and $ne$ when electrons form bi-Maxwellian one.^23,24 In Fig. 2, a typical fitting of continuum emission is also plotted with a solid line, and this line gives $Te$ and $ne$⁠. To derive $ne$ from line-integrated emission, one needs to determine the characteristic length of recombining plasma. Figure 3 shows radial profile of high-n line emission intensity at $p=7.2 Pa$⁠. Here, emission intensities are line-integrated value. Circles, squares, and diamonds correspond to $n=9, 10, and 11$⁠, respectively. As shown in Fig. 3, line emissions showed center-peaked profile, and the full width at half maximum was approximately 7 mm. Therefore, the characteristic length was assumed to be 7 mm. Figure 4 represents line-integrated emission intensity due to $23P$–$n3D$ transitions, electron temperature, and electron density as a function of helium neutral pressure. Optical emissions were collected at the midstream region, and a viewing chord was adjusted to almost plasma center. $Te$ and $ne$ were obtained at both midstream and downstream regions, respectively. As illustrated in Fig. 4(a), high-n line emissions showed rapid increase when the neutral pressure was increased above 6.5 Pa, and then, at $p>7.8 Pa$⁠, it decreased to the same level with that at $p<6.5 Pa$⁠. This trend indicates that helium recombining plasma was formed at the midstream region when $6.5<p<7.8 Pa$⁠. In contrast, it is considered that ionizing plasma was formed at $p>7.8 Pa$⁠. One likely cause of this trend is the influence of secondary gas backflow on plasma production. Here, it should be noted that measurement at the downstream region can be affected if position of the recombination front shifts as amount of secondary gas changes. However, the recombination front stayed at almost the same position during this experiment. $Te$ and $ne$ at the midstream and downstream regions are plotted in Figs. 4(b) and 4(c), respectively. Circles and squares were obtained at the midstream and the downstream regions, respectively. Open and filled symbols correspond to the Langmuir probe and spectroscopic measurements, respectively. At the midstream region, Langmuir probe can yield reliable $Te$ and $ne$ below 6.5 and above 7.8 Pa, respectively, because ionizing plasma was formed at these pressure regions, as Fig. 4(a) indicates. On the other hand, the spectroscopic method should be utilized within $6.5<p<7.8 Pa$ because recombining plasma was formed. Although a Langmuir probe is not available at this pressure, $Te$ and $ne$ obtained by the probe are plotted just for reference. As Fig. 2 indicates, ionizing plasma was formed at the downstream region even when recombining plasma was formed at the midstream region. Therefore, a Langmuir probe was utilized at the downstream region for all neutral pressure cases. As shown in Figs. 4(b) and 4(c), when recombining plasma was not formed, little difference was seen in $Te$ and $ne$ at two regions. $Te$ at this pressure range was roughly 2–3 eV. Mean free path of the electron impact ionization by such low temperature electrons is much longer than the length between the two regions (0.15 m), which means change of $Te$ and $ne$ due to ionization is negligible. Therefore, showing almost constant $Te$ and $ne$ is reasonable. When neutral pressure was increased above 6.5 Pa, ionization strongly proceeded, and thereby $ne$ increased up to approximately $5×1018 m−3$ and $Te$ decreased below 0.5 eV. Obviously, when recombining plasma was formed, $Te$ at the downstream region was much greater than that at the midstream region, even though there was no electron heating source between these two regions.

Rise in electron temperature of approximately 1–2 eV was observed at downstream of recombining plasma. This result should be discussed carefully because, as Fig. 4(b) indicates, a Langmuir probe overestimates electron temperature to similar extent when that is utilized for measurement of recombining plasma. If electron temperature at the downstream region was too low to use a Langmuir probe, the result should be interpreted simply as the overestimation. Therefore, validity of the measurement is to be confirmed first. After that, the mechanism of the temperature rise is discussed from the viewpoint of electron energy distribution.

Figure 5(a) represents wavelength spectrum observed at the midstream region. Neutral pressure was 7.1 Pa, so this spectrum is the same one with that of Fig. 2. As described in Sec. III, both continuum and high-n line spectrum associated with volumetric recombination are clearly observed at this pressure. $Te$ and $ne$ derived from the continuum spectrum are 0.47 eV and $5.2×1018 m−3$⁠, respectively. Circles represent line emission intensities correspond to $23P−n3D (n=8–11)$ transitions calculated by a collisional-radiative model (CR model)²⁵ with above plasma parameters. As shown in Fig. 5(a), relative relation of high-n line emissions can be reproduced very well by the CR model when appropriate parameters are utilized. Here, experimentally observed spectrum and calculated one show slightly different relative relation. One probable reason for this difference is the line-integral effect because that is not taken into consideration in the calculation.

Similarly, experiment and calculation were compared for the measurement at the downstream region and summarized in Fig. 5(b). Neutral pressure was also 7.1 Pa, so this spectrum corresponds to the dashed line in Fig. 2. No clear high-n emissions are seen in this spectrum, which indicates that ionizing plasma was formed at the downstream region. It can be also confirmed that calculation with the aforementioned $Te$ and $ne$ cannot reproduce relative relation of high-n line emissions (⁠$23P–n3D$⁠), at all. In contrast, line emissions due to low-n transitions (⁠$21S–n1P; n=4–6$⁠) are reproduced very well when $Te=1.4 eV$ and $ne=1.2×1018 m−3$ are utilized, as shown with squares. This temperature and density were that determined by the Langmuir probe at the downstream region. The comparison confirms that the Langmuir probe can obtain valid $Te$ without causing overestimation, highlighting the fact that the temperature rise is not originated to use of the Langmuir probe.

Interpretation of the temperature rise was attempted from the viewpoint of electron energy distribution. Electrons at the recombination front region have various energy according to the Maxwellian distribution, and low energy electrons recombine preferentially, whereas recombination of high energy electrons is less frequent enough to reach the downstream region. The effective electron temperature $Teeff$⁠, average energy of un-recombined electrons $⟨E⟩$⁠, and effective electron density $neeff$ are expressed as

where $Eth$ and f(E) represent the energy threshold and electron energy distribution function, respectively. Hereafter, experimental results are discussed with the assumption that $Teeff$ or $⟨E⟩$ and $neeff$ were measured as electron temperature and density at the downstream region. In this analysis, f(E) was assumed to be Maxwellian distribution, and $Te$ at the midstream region derived by spectroscopy was utilized to determine the shape of f(E). The value of f(E) was normalized to unity. To obtain $Teeff$ and $⟨E⟩$ from Eq. (1), one needs to evaluate the value of $Eth$⁠. In the present work, $Eth$ was determined focusing on balance between ionization and recombination rates. When one considers collision processes among many excited levels, use of a collisional-radiative model (CR model) is very effective. Therefore, the CR model for helium atom²⁵ was utilized to determine $Eth$⁠. Assuming quasi-steady-state approximation (QSS approximation) for all excited states except for the ground state, time derivative of the electron density is expressed as

where $n(11S), SCR$⁠, and $αCR$ represent density of ground state helium atom, CR ionization, and recombination rate coefficients, respectively. Usually, particle transport is not taken into consideration when QSS approximation is assumed because extinction of excited atoms caused by electron impact excitation is much larger than transport loss. However, it has been reported that transport of $23S$ metastable atoms is not negligible in low electron temperature recombining plasma.²⁶ In Ref. 26, characteristic time of neutral transport and extinction time of $23S$ atoms are compared, and when $ne∼1019 m−3$⁠, these two values become almost same when $Te$ decreased to approximately 0.2 eV. Although $Te$ in the present experiment is higher than 0.2 eV, validation of the QSS approximation without particle transport, especially for two metastable states, should be confirmed before $Eth$ is evaluated. Outflow from an excited state p due to electron impact excitation is described as $∑q>pC(p,q)nen(p)$⁠. Here, $q, C(p,q)$⁠, and n(p) represent another excited state, the rate coefficient for electron impact excitation, and population density in excited state p. C(p, q) and n(p) can be calculated by the CR model, and $ne$ can be obtained experimentally. For example, $∑q>pC(p,q)ne$ at $p=7.1 Pa$ is calculated as follows. As shown in Fig. 4, $Te$ and $ne$ at 7.1 Pa were $Te=0.47 eV$ and $ne=5.2×1018 m−3$⁠. With these parameters, $∑q>pC(p,q)ne$ becomes approximately $3.5×106 s−1$ for the $21S$ state and $2.5×105 s−1$ for the $23S$ state. The transport term can be described as $∇·Γ$⁠, where $Γ$ represents the particle flux. When one assumes diffusive transport, $Γ$ is written as $−D∇n(p)$ with a diffusion coefficient D. Introducing the characteristic length of neutral density gradient d, $∇·Γ$ can be simplified as $D/d2×n(p)$⁠. In this manner, transport terms are also proportional to n(p) with the coefficient $D/d2$⁠. Although the determination of D and d is difficult, when we assume D = 1–10 $m2s−1$ and $d=10−2 m$⁠, $D/d2$ becomes 10⁴–$105 s−1$⁠. This is much smaller than $∑q>pC(p,q)ne$⁠. As well as at 7.1 Pa, $D/d2$ is always smaller than $∑q>pC(p,q)ne$ for all neutral pressure cases. Therefore, particle transport can be ignored in the QSS approximation within the present experimental condition.

Typical curves of the first and second terms of Eq. (3) are plotted in Fig. 6 as a function of electron temperature. Electron density and neutral pressure utilized for the calculation are $ne=5.2×1018 m−3$ and $p=7.4 Pa$⁠, respectively, which corresponds to the condition at which recombining plasma was formed (see Fig. 4). Production and extinction rates of electrons balance at $Te=1.3 eV$⁠. For convenience, let $Eth$ be the electron temperature where these two terms cross. $Eth$ for other pressure cases was determined in the same manner.

Calculation results of $Teeff, ⟨E⟩$⁠, and $neeff$ are shown in Fig. 7. In Fig. 7(a), $Teeff$ and $⟨E⟩$ are compared with experimentally obtained $Te$ at the downstream region. Red open and red filled squares represent $Teeff$ and $⟨E⟩$⁠, respectively. Except for 6.8 Pa, the value of $Te$ is between $Teeff$ and $⟨E⟩$⁠. This result indicates that the observed $Te$ is reflecting $Teeff$ or $⟨E⟩$ of un-recombined electrons. With respect to $neeff$⁠, difference is relatively large especially at $p=7.1 Pa$⁠. However, that is at most approximately two times of magnitude, even though $neeff$ strongly depends on uncertainties in $Eth$ and $ne$⁠. Therefore, it can be concluded that electron density at the downstream region would reflect the number of un-recombined electrons. Finally, a possible explanation for increase in electron temperature along magnetic field line is given as follows. At the recombination front region, low energy electrons, such as $E<Eth$⁠, are preferentially consumed by recombination, while high energy electrons reach the downstream region and then measured by a Langmuir probe there. As described above, the apparent increase in $Te$ at the downstream of the recombination front region is likely common in detached and recombining plasmas. Not only in such low temperature plasmas, the deformation of EED is also observed when the electric double layer exists²⁷ or electrons are collisional with neutral particles,²⁸ but the mechanism provided in this report is disparate from those because deformation is caused by low energy electron depletion associated with electron-ion recombination.

Here, it should be noted that $Te$ and $ne$ at the downstream region were obtained by the same manner with usual single probe analysis because I-V curves at this region showed similar shape to that of Maxwellian distribution plasma. However, this does not necessarily mean that the depletion did not deform EED because energy relaxation among remaining electrons could also determine final shape of EED as well as the depletion of low energy electrons. In addition, how I–V curves reflect deformed EED is not also clearly understood. Investigation of these problems will deepen the understanding of the experimental results. This is a future work.

In summary, the rise of electron temperature along the magnetic field line in helium recombining plasma was observed and analyzed. When ionizing plasma was formed, $Te$ and $ne$ obtained at the midstream and the downstream regions were in good agreement. On the other hand, when recombining plasma was formed, $Te$ at the midstream region was approximately 0.5 eV, whereas it increased up to above 1 eV at the downstream region. Introducing energy threshold $Eth$⁠, effective electron temperature $Teeff$ and average electron energy $⟨E⟩$ were calculated. As for $Eth$⁠, electron temperature where ionization and recombination rates balance was utilized. Through comparison between calculation and experiment, it was found that both $Teeff$ and $⟨E⟩$ were in good agreement with $Te$ at the downstream region. With respect to $ne$⁠, although the difference between $ne$ and effective electron density $neeff$ was relatively large, it was within two times of magnitude. It is concluded that relatively high energy un-recombined electrons remained by depletion of low energy electrons are responsible for the apparent temperature rise along magnetic field line. This conclusion suggests that the evolution of EED should be taken into consideration when volumetric recombination strongly occurs because Maxwellian distribution is no longer conserved due the depletion of low energy electrons.

At present, numerical simulation codes such as SONIC^29,30 and SOLPS³¹ are being developed to predict edge plasma and impurity transport in fusion reactors. In these codes, plasma transport is solved with fluid equations. This treatment is based on the assumption that the Maxwellian distribution is conserved along magnetic field line, and thereby electron temperature can be defined everywhere. Contrary to the present understanding, this work suggests that the validity of the Maxwellian assumption needs to be carefully examined in the simulation study.

The authors would like to thank Professor Hoshino of Keio University and Dr. Homma of National Institutes for Quantum Science and Technology for their helpful discussion of the results. This work was supported by Japan Society for the Promotion of Science JSPS KAKENHI under Grant Nos. JP20H01883 and JP19H01869.

The authors have no conflicts to disclose.

The data that support the findings of this study are available from the corresponding author upon reasonable request.