Impact of vibrationally and electronically excited H 2 on the molecular assisted recombination rate in detached plasma regimes

The vibrational redistribution of H 2 molecules via the 𝐵 1 𝛴 + 𝑢 and 𝐶 1 𝛱 𝑢 states are included in the determination of molecular assisted recombination (MAR) rates and are found to have negligible impact in detached plasma regimes. In high recycling and detached divertor plasma conditions, MAR occurs through the higher levels ( 𝑣 ≥ 4 ) of vibrationally excited H 2 which are primarily populated through direct electron impact excitation or the electronic excitation of H 2 molecules and the subsequent radiative decay to another vibrational state. We use the collisional-radiative model CRUMPET to evaluate MAR rate coefficients with electronic transitions included, which are applied to detached plasma parameters found in the linear plasma device Magnum-PSI.


Introduction
Plasma recombination in the divertor region is considered to be an explanation for the decrease of plasma particle flux to the divertor targets in detached plasma regimes [1].Plasma recombination occurs mainly from radiative and three-body recombination, which are collectively termed electron-ion recombination (EIR) [2].EIR becomes relevant only at electron temperatures of 1 eV or below.The presence of molecules contributes to the plasma recombination via molecular assisted recombination (MAR) processes, which are relevant at electron temperature higher than EIR (1 <   < 5 eV) [1,3].For hydrogen in divertor plasma conditions, MAR occurs through the higher levels ( ≥ 4) of vibrationally excited H 2 .Hence, the population distribution of the H 2 vibrational states influences plasma recombination through MAR.The vibrational states of H 2 are primarily populated through electron impact excitation of the vibrational number  to a higher number  via two channels [4][5][6]: The first reaction is a resonant mechanism involving the temporary formation of H − 2 [6,7].It is the vibrational excitation process commonly included in vibrationally resolved recombination rate coefficients [8] used in coupled fluid-kinetic edge codes such as SOLPS [9], EDGE2D-EIRENE [10,11], or B2.5-Eunomia [12].The second reaction excites the vibrational levels indirectly through electronic excitation of the molecule followed by radiative decay [5,6].To the authors knowledge, the second reaction was not commonly included in the derivation of effective recombination rate coefficients (e.g.AMJUEL [8]) used in fluid-kinetic edge codes.
This paper evaluates the effect of (2) to the vibrational population distribution of H 2 and consequently to the MAR rate for electron densities and temperatures expected in the detached plasma regime of a tokamak.The collisional-radiative model CRUMPET [3] is used to solve the population distribution of the H 2 vibrational states for certain plasma parameters and is explained in Section 2. Using the H 2 vibrational state distribution obtained from CRUMPET, the effective recombination rate coefficients of MAR processes are re-evaluated with (2) included and the results are presented in Section 3. The effect of electronic transitions of H 2 to the particle balance in detached plasma regime is evaluated using data from fluid-kinetic code simulations of plasmas in the linear plasma device Magnum-PSI and is shown in Section 4. We summarize our conclusions in Section 5.

CR model species and reactions
The population distribution of H 2 () (vibrationally excited H 2 in the electronic ground state) is calculated using the collisional-radiative model CRUMPET [3].CRUMPET allows flexibility in  number of species considered in the equation [13]: where  is the source term,  is the density of species considered, and the  ×  rate matrix .The applicability of CRUMPET is only limited by the availability of reaction rate coefficients necessary to construct  [3].The collisional processes included in  and the available cross-section or rate coefficient data are summarized in Table 1.Direct electron impact vibrational excitation ( 1) is treated as ladder-like, where only single step transitions are allowed.For electronic excitation of H 2 (2 and 3 in Table 1), all possible transitions between vibrational levels are included.The same applies for optical transitions (4 and 5 in Table 1).The H 2 () population can be depleted through dissociation, ion conversion (IC), and the dissociative attachment (DA) process (6,7 and 8 in Table 1).The reaction rates are calculated assuming   =   .The direct ionization of H 2 is not included in CRUMPET for the purpose of comparison with simulation data [14] later in Section 4, which also does not include this process.It is advised to include direct ionization in the future, especially when the electron temperature can exceed 4 eV, where it starts to compete with ion conversion and dissociation.
Two different CRUMPET inputs are created to demonstrate the effect of electronic transitions to the population distribution of H 2 ().The first input considers only the electronic ground state of H 2 , H + 2 .H − , and H.The second input considers the electronically excited states of H 2 in addition to all off the species of the first input.Hence, the rate matrix  have the size 18 × 18 and 69 × 69 for the first and second input respectively.The first input will be referred to as no electronic (no el.) in the figures presented in the paper.

Effect of H 𝟐 electronic transitions to MAR rate
With increasing   , the population distribution is shifted towards higher vibrational states of H 2 when the electronic transitions of H 2 are included in the model.This trend is reversed when electronic transitions are not included, as shown by the population distributions of H 2 () for electron temperatures   = 1, 3, and 5 eV in Fig. 1.Indeed, the reaction rates for the electronic transition to the singlet  and  states, as described in reaction number 2 and 3 in Table 1, are expected to be dominant for higher electron impact energies [4].
The shift towards higher vibrational population states can potentially increase the effective recombination rate through the MAR process, both through the IC and the DA channel.The effective rate coefficient for these two channels can be calculated as follows:

Table 2
List of subsequent processes causing the annihilation of ions H + 2 and H − .Only MAR processes result in plasma recombination.The other pathways are molecular assisted ionization (MAI) and molecular assisted dissociation (MAD) [21,22].

Nr.
Reaction formula Name Data ref.where  H 2 () = H 2 ()∕H 2 ( = 0) is the relative population fraction of H 2 ().The IC and DA processes do not necessarily result in plasma recombination and depend on the subsequent reaction of the ionic products, H + 2 for IC or H − for DA.These subsequent reactions are summarized in Table 2. Taking into account these follow-up reactions, a factor can be added to Eq. ( 4) such that an effective MAR rate can be defined as: where is the ratio between rate coefficients of reactions resulting in recombination and the total rate coefficients of all possible reaction depleting the ionic products H + 2 or H − .The effective MAR rate coefficients through the IC and DA channel are evaluated for 1000   values evenly spaced in the logarithmic scale between 0.1-1000 eV (Fig. 2).By including the electronic transitions of H 2 , the change in the population of the higher vibrational states of H 2 () alters the effective recombination rate coefficients starting from   = 3 eV.For the ion conversion channel (MAR-IC), the rate coefficient is enhanced by a factor of 3 at around   = 5 eV, and by more than an order of magnitude at around   = 15 eV.The impact of including the electronic transitions of H 2 is even more profound for effective recombination via the dissociative attachment channel (MAR-DA).This pathway becomes a relevant contributor of plasma recombination for   ≥ 2 eV, contrary to when no electronic transitions of H 2 is considered.Furthermore, the rate coefficient is larger by 2 orders of magnitude at around   = 10 eV.
The electronic transitions of H 2 starts to improve the MAR rates for   above 2-3 eV.However, in this temperature range, direct dissociation and ionization deplete the electronic ground state of H 2 as a MAR precursor [1].Direct dissociation and ionization of H 2 are still dominant for   above 2 eV even when electronic transitions can significantly enhance MAR rates as shown in Fig. 3. Hence, evaluating the overall impact of electronic transitions of H 2 to the global particle balance requires information on the H 2 ground state density which is coupled with the electron density and temperature.

Evaluation of MAR rate in detached plasma regimes
SOL plasma codes, such as SOLPS, EDGE2D-EIRENE, and B2.5-Eunomia, solve the Braginskii equations for the plasma species, and the Boltzmann equation for the neutral species using Monte Carlo methods.They provide self-consistent information of the plasma density, temperature, and neutral densities.A detachment study using B2.5-Eunomia [14] was recently conducted in the linear plasma device Magnum-PSI [23].Plasma detachment is achieved by increasing the H 2 gas pressure exclusively in a chamber housing the target of the plasma beam using gas puffing [24].The plasma was observed to be in a varying state of detachment depending on H 2 pressure that is applied within the chamber [24].We use the plasma density, temperature, and the H 2 density in the electronic and vibrational ground state from B2.5-Eunomia simulations of the several detached plasma states for H 2 pressures of 0.27-2 Pa.The plasma density, temperature, and the H 2 density associated with the different H 2 pressure values are illustrated in Fig. 4.These plasma conditions cover the temperature range where the effect of electronic transitions of H 2 to the effective recombination rate becomes noticeable (  ≥ 2 eV), based on the analysis on the effective MAR rate coefficients in Section 3.
Fig. 3.The rate coefficients of processes that deplete the electronic ground state of H 2 .The MAR precursors (IC and DA) compete with direct dissociation (black circle) and ionization (blue square) for   ≥ 2 eV even when the electronic transitions of H 2 are included.The rate coefficients are calculated for   = 10 20 m −3 .There is no ionization rate with electronic transitions as it is not included in CRUMPET.The plasma density, temperature, and the H 2 density in the electronic and vibrational ground state from B2.5-Eunomia are used to calculate the effective recombination rate   following (5): where   is the electron density,  H 2 (=0) is the hydrogen molecule density in the electronic and vibrational ground state,  is the plasma volume, and  indicates the quantities to be local in a grid cell of the finite-volume B2.5 code.The grid cells are limited to the plasma region inside the target chamber of Magnum-PSI.The global recombination rate from MAR-IC and MAR-DA pathways for each level of pressures are shown in Fig. 5.By including electronic transitions of H 2 the MAR rates are enhanced marginally in the 'attached' plasma regime for which   is around 3-4 eV.The MAR-IC pathway is enhanced by approximately 7% while the MAR-DA pathway shows a larger increase of around 16%.The impact diminishes towards detached plasma regime and becomes negligible at the highest pressure for MAR-IC and MAR-DA, at 0.7% and 2%, respectively.The same analyses are repeated for the MAD pathways (nr. 2 and 4 in Table 2).While these processes does not produce plasma recombination, they can remove electron energy from the plasma.The electron cooling rate can therefore be when electronic transitions are included.Indeed, this increase is readily observed from the increase of global MAD-IC and MAD-DA rates shown in Fig. 6.The MAD-IC and MAD-DA pathways are increased by, respectively, 18% and 59% in the attached plasma regime.Similar with MAR, the impact diminishes towards detached plasma regime to around, respectively, 1% and 3%.However, MAD is expected to be relevant at attached plasma temperatures, and so the electronic transitions should be included in determining MAD rates.

Conclusions
The impact of electronic transitions to the singlet B and C states of H 2 on the MAR rate is evaluated using the CR model CRUMPET and data of electron density, temperature and H 2 density from B2.5-Eunomia simulations of plasma detachment in the linear plasma device Magnum-PSI.The impact is highest in the attached plasma regime where the electron temperature is around 3-4 eV, as expected from the modified effective MAR rate coefficients shown in Fig. 2.However, the enhancement diminishes for further plasma detachment, and so remains insufficient to change the system significantly.On the other hand, MAD rates is enhanced substantially in the attached plasma regime where it is relevant, and thus can have major impact in the plasma energy balance.The additional electron cooling may affect plasma recombination indirectly by driving the temperature further down.While this paper uses simulation data from a linear plasma device, these effects are expected to occur in tokamak divertors with similar plasma and neutral conditions, as the effect is dependent mainly on the plasma density, temperature and neutral densities.It is important to note that reactor divertors are subject to other physics such as drifts and complex magnetic geometries which are absent ina linear plasma device.The same approach presented in this paper is readily available for tokamak simulations performed using SOLPS, EDGE2D-EIRENE or other codes, which included the aforementioned physics.These studies should provide more insight in the effect of electronic transitions of H 2 to plasma recombination, and consequently, their effects on plasma detachment.

Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Fig. 1 .
Fig.1.Population of H 2 () relative to the ground vibrational state H 2 ( = 0).The solid line indicates the population distribution when electronic states of H 2 is not considered.The dotted line indicates the population distribution when electronic states are considered.When the re-distribution of H 2 () via electronic states is included, the distribution is shifted towards higher states with increasing   .The rate coefficients are calculated for   = 10 20 m −3 .

Fig. 2 .
Fig. 2. Effective MAR rate coefficients via ion conversion (MAR-IC) (a) and via dissociative attachment (MAR-DA) (b) pathways calculated with CRUMPET.The rate coefficients are evaluated when the electronic transitions of H 2 are included (red solid line) and neglected (red dotted line).As a comparison, the effective MAR rates from AMJUEL, where electronic transitions are neglected, is shown in black.The relative difference between the CRUMPET rate coefficient and the no el.rate coefficients of AMJUEL (black) and CRUMPET (red) are shown for MAR-IC (c) and MAR-DA (d).The rate coefficients are calculated for   = 10 20 m −3 .The AMJUEL rate coefficient collapse faster than CRUMPET at very low temperatures which resulted in the very high ratio numbers shown in (c) and (d).(For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 4 .
Fig. 4.   ,   and  H 2 distributions of the Magnum-PSI plasma beam within the target chamber.The values are obtained from B2.5-Eunomia simulations of varying H 2 pressures of 0.27, 0.53, 1.0 and 2.0 Pa [14], measured near the pumping surface far from the plasma (R = 0.2 m).The target plate is located at the east boundary (Z = 0.09 m).

Fig. 5 .
Fig. 5. Global MAR-IC (a) and MAR-DA (b) rates calculated using data shown in Fig. 4. The rates are shown for each simulated pressure level representing the varying stages of plasma detachment.The rate uses ⟨⟩  with electronic transitions of H 2 included (black) and neglected (red).The impact of electronic transitions of H 2 to the MAR rate is negligible in the detached regime.(For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 6 .
Fig.6.Global MAD-IC (a) and MAD-DA (b) rates calculated using data shown in Fig.4.The rates are shown for each simulated pressure level representing the varying stages of plasma detachment.The rate uses ⟨⟩  with electronic transitions of H 2 included (black) and neglected (red).The MAD pathway does not result in plasma recombination and only removes electron energy from the plasma.The influence of electronic transition to this process can be significant at attached plasma regimes.(For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Table 1
For this paper, the species considered are: List of collisional processes included in the rate matrix  in CRUMPET.For H2VIBR,  refers to the vibrational mode number.