1 Introduction

Molecular hydrogen (H2) occurs in a variety of plasmas, e.g. astrophysical plasmas, low-temperature plasmas for technological applications, plasmas in sources for negative hydrogen ions and fusion divertor plasmas. Population models enable not only in combination with emission spectroscopy characterization of these plasmas by their parameters, but also to evaluate effective reaction rates. These can for instance be applied to molecular-assisted recombination (MAR), a mechanism that may contribute to the detachment process and consequently to a drastic reduction of the plasma pressure in tokamak divertors [1]. The vibrational excitation of the hydrogen molecule is especially important for the plasma kinetics in the divertor region, as it enhances MAR [2]. This is highly relevant for fusion research, as (partially) detached operation is a key element in ensuring a high lifetime of the divertor in future tokamaks (e.g. ITER) [3].

Population models balance the probabilities of populating and depopulating processes for the excited states of a particle (e.g. atom or molecule) in the form of coupled rate equations (in nonequilibrium plasmas). In hydrogen plasmas, if the radiation field, the ionization degree of the plasma and the electron density are low (e.g. ne ⪅ 1017 m−3) and if both the collision reactions between excited states and absorption of photons are negligible, corona models are applicable. Such models solely balance excitation by collision from the ground state and spontaneous emission. Collisional-radiative (CR) models follow a more general approach and consider a much larger number of reaction channels including in addition collision processes between excited states and for instance recombination processes. Consequently, the excited state densities can depend not only on the electron temperature and density, but also on the properties of other species. CR models inherently comprise corona models. For very high electron densities (ne > 1022 m−3), the population densities of the excited states calculated by CR models (should) approach the (local) thermodynamic equilibrium distribution, provided that no influential processes (e.g. the radiation transport) are neglected.

Population models need several quantities as input: the plasma parameters (e.g. density, temperature and energy distribution function (EDF) of each quasi-constant species (see below)), the photon characteristics (e.g. opacity) and the reaction probabilities between states. The latter one can either be in the form of Einstein coefficients, rate coefficients or cross sections. Rate coefficients for arbitrary temperatures of the colliding particles can be evaluated by integration of the energy-dependent cross sections over a suitable EDF. The application of cross sections features the advantage that individually (non-Maxwellian) EDF (e.g. two-temperature or Druyvesteyn distributions) is applicable and that processes are better comparable in this form (e.g. directly the energy dependence).

Since the accuracy of the modelling results correlates with the accuracy of the reaction probabilities that are entered into the models, the need of profound cross sections, rate coefficients or Einstein coefficients is evident. Therefore, the respective coupling data must be critically evaluated from a physical perspective before its inclusion in models. However, especially for molecular hydrogen, these data are often not available. Hence, the completeness of the models determines their accuracy too. In any case, benchmarking the models against measurements is of utmost importance.

(Optical) emission spectroscopy (OES) is due to its noninvasive character; one of the most established plasma diagnostic tools [4] and population densities deduced from measured emission spectra can be utilized for benchmarking compared to the results of CR models if plasma parameters like the electron temperature Te (provided that the electron EDF is Maxwellian and thus Te is defined) and electron density ne are known (e.g. from other diagnostics like Langmuir probes). Furthermore, with the help of plasma spectroscopy and the backward application of CR models, it is also possible to determine the plasma parameters.

This work focuses on molecular hydrogen plasmas. H2 is a diatomic, homonuclear molecule that can store energy in the vibrational and rotational motion of the nuclei against each other. Depending on whether the alignment of the individual spins in the H2 molecule is parallel or antiparallel, the molecular states are divided into a singlet and a triplet system. The ground state is part of the singlet system. Processes coupling the two systems would need to involve a change in spin orientation and are optically forbidden. Thus, for low-pressure plasmas with a low collisionality usually CR models are applied that individually describe the two systems. Within the adiabatic approximation, the total wave function of molecular hydrogen can be separated in an electronic, a vibrational and a rotational part. The first CR models for H2 solely considered electronic states [5, 6]. However, advances in scattering theory and other fields of molecular data production allow also the determination of (ro-)vibrationally resolved coupling data. Such coupling data enable the composition of (ro-)vibrationally resolved population models involving (ro-)vibrational levels.

In the pioneering works [7, 8], collisional-radiative modelling focused on atomic hydrogen. The method was generalized in [9] and applied to molecular hydrogen in [6]. Nowadays, there are numerous works extending the original approach with different emphases: in several works, the initial procedure is improved by considering possible non-Maxwellian energy distribution functions for free electrons by means of appropriate Boltzmann equations for atomic and molecular hydrogen plasmas (see e.g. [10,11,12,13,14,15]). For such investigations, level kinetics general purpose codes are also available (see e.g. [16]). Furthermore, neutral gas transport codes for hydrogen (e.g. the Eirene code [17]) are available, which also comprise of CR models. Multiple works explore the coronal approximation for the simulation of the Fulcher transition (see e.g. [18, 18]). Other works try to focus on the plasma chemistry (especially regarding the vibrational kinetics of the electronic ground state) and calculate effective MAR rate coefficients in detached divertor plasmas (see e.g. [20, 21]). The issue of the necessity of accurate collisional data is specifically addressed also in several works. To give some examples: sets of approximate formulae for evaluating collisional-radiative transition rates for atomic hydrogen were already presented in [22], a large list of chemical process data in hydrogen plasmas is included in [22] and even the assessment of the collision data included in the aforementioned, widely used EIRENE code is an ongoing topic [24].

The aim of this paper is to give an overview of the status of the population models based on the Yacora solver [25] and to highlight the relevance of molecular data in CR modelling of molecular hydrogen plasmas. Firstly, the basics of CR modelling are deepened, and the Yacora code is discussed. Secondly, relevant physical effects are presented. Finally, the data needs for collisional-radiative modelling are demonstrated by means of three models as examples comprising different detail levels for different fields of application.

2 Collisional-radiative modelling

2.1 Principles of CR models

In a general case, the temporal variation of the population density of an excited state k can be described by a rate equation

$$\begin{aligned}\frac{{dn}_{k}}{dt}&= \sum_{i,j}{X}_{ijk}\left(T\right){n}_{i}{n}_{j}+\sum_{i}{A}_{ik}{n}_{i}- {n}_{k}\\ & \quad \times \left[\sum_{i,j}{X}_{ikj}\left(T\right) {n}_{i}+\sum_{i}{A}_{ki}\right],\end{aligned}$$

where Xijk(T) is the rate coefficient for populating the state k from collisions between species i and j at temperature T, and Aik is the Einstein coefficient for spontaneous emission from state i towards state k. When modelling H2 plasmas, Xijk(T) or Xikj(T) commonly represent electron collisions as (de-)excitation, ionization and dissociative electron attachment. Nevertheless, other processes, for instance heavy particle collisions, can be considered as well. The equation must be extended when collisions involving more than two particles (e.g. three-body recombination populating atomic states) are to be included. The respective rate equations for all states of interest included in the CR model form a set of ordinary differential equations. If the relaxation times for the excited states are much shorter than the relaxation time for the ground level, the quasi steady state solution is usually applied, e.g. the left side of the equation system is set equal to zero [7]. Consequently, for a set of linear equations, the population density of an excited state k can be expressed as

$${n}_{k}= {R}_{0k}\left({n}_{e},{T}_{e}\right){n}_{e}{n}_{0},$$

with the population coefficient R0k, the electron density ne and the ground state density n0. If other processes involving collision partners that also have a quasi—constant density (e.g. ions) are to be considered, the population coefficients for each of these species must be added in the equation above as separate terms. In [26, 27], the time-scale criterion is universalized to criteria depending on the properties of the coupling matrix underlying the system of equations.

2.2 The flexible solver Yacora: overview and features

Yacora is a flexible, zero-dimensional code solving systems of rate equations by integration. Thus, it is applicable not only for CR models, but also for (nonlinear) dissociation and ionization models. For collision processes, it is possible to implement both cross sections and rate coefficients in the model as reaction probabilities. The code independently calculates rate coefficients from the cross sections using an EDF defined by the user of the code. For the results of the models presented in this work, Maxwellian EDF is applied. In a plasma, the occurring processes show a large variability of relaxation times. For example, photon-emitting processes happen typically much faster than collisional processes (e.g. for Te = 2 eV and ne = 1017 m−3 the reaction rate for spontaneous emission C1 → X1 in the hydrogen molecule is by a factor more than 106 larger than the rate for electron impact de-excitation). Consequently, the equation system at hand is stiff and ordinary solution techniques (e.g. Runge–Kutta method) are computationally heavy. Therefore, Yacora utilizes the CVODE package including backward differentiation formulas in fixed-leading coefficient form for solving initial value problems for stiff ordinary differential equation systems [28]. CVODE is part of the SUNDIALS framework developed at the Lawrence Livermore National Laboratory.

Numerous population models based on the Yacora solver are relevant for application in fusion plasmas. A CR model for atomic hydrogen comprises all excited states with principal quantum number n ≤ 40 and includes as reaction channels for instance H+ three-body and radiative recombination, H− mutual neutralization with H+ and H2+, H2+ and H3+ dissociative recombination and H2 dissociation [25, 29, 30]. For the hydrogen molecule, there are several CR models comprising different detail levels for different fields of application (based on previously available cross sections) [30, 31]:

Figure 1 depicts the energy level diagram of the singlet and triplet system in the hydrogen molecule. An available model for the electronic states of the triplet system of the hydrogen molecule [31] is based on the fully quantal electron impact cross sections calculated by the molecular convergent close—coupling (MCCC) method in the adiabatic—nuclei formulation [32]. The MCCC method is an ab initio approach for electronic scattering problems with the ability to provide accurate cross sections for all incident energies and allows in principle accurate calculation of electronically, vibrationally and ro-vibrationally resolved cross sections [33]. In [31], it is demonstrated that the model agrees very well with measurements, which were conducted in the electron temperature and density range of Te = 2.5–10 eV and ne = 1.8–3.3 × 1016 m−3.

Fig. 1
figure 1

Electronic energy level diagram of the hydrogen molecule delimiting the population models discussed in the following. The states of the CR singlet model are marked in blue. The vibrationally resolved CR model focuses on the vibrational levels of the ground state X1. The corona model (magenta coloured) couples ro-vibrationally resolved the ground state X1 with the a3 and d3 states of the triplet system (exemplified by the arrows). In the interest of clarity, the figure depicts for these states solely the vibrational, but not the rotational levels

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A CR model from 2011 [34] based on different, previously available cross section sets for the electronic states of the singlet system (states marked in blue in Fig. 1) is discussed in more detail later. It is compared to an improved singlet model applying MCCC cross sections likewise as the respective triplet model.

To address the reduced energy threshold for instance for electron impact excitation due to excitation of the vibrational levels of H2, there is a vibrationally resolved CR model from 2016 [29]. In this model, the states X1, B1, C1, EF1, GK1, I1, c3, a3, e3 and d3 are vibrationally resolved. A vibrationally resolved model developed in the scope of this work, which focuses on the vibrational levels of the ground state X1, is discussed later.

In (low-pressure) optical plasma diagnostics, the Fulcher-α emission lines (d3 → a3 for Δv = 0) are often addressed, as they are in the visible wavelength range (~ 600–640 nm), are easily distinguishable and hardly overlap with other emission band lines [35]. Due to the lack of consistent ro-vibrationally resolved reaction data, there is (instead of a fully resolved CR model) a ro-vibrationally resolved corona model for the Fulcher-α band (d3 → a3) from 2016 [29]. A version of this model for D2 exists also. An improved ro-vibrationally resolved corona model for the H2-Fulcher-α band applying MCCC cross sections is presented in Sect. 4.1. In Fig. 1, the vibrational levels of the X1, d3 and a3 states that constitute this ro-vibrationally resolved model are marked in magenta. Further, ro-vibrationally resolved corona models are also available for the Lyman (B1 → X1) and Werner (C1 → X1) band [29].

The Yacora solver is also used for CR models for helium, argon, caesium, hydrocarbons, the nitrogen molecule (N2) and the carbon molecule (C2) (see for example [34]).

Electronically resolved CR models for atomic and molecular hydrogen as well as atomic helium are available as web application “Yacora on the Web” [30].

3 On the physics background of low-temperature plasmas

3.1 Recombining and ionizing plasmas

As mentioned above, the population densities of exited atomic or molecular states can be connected, depending on the plasma parameters, to the respective ground state and to different other heavy particle species as atomic or molecular positive or negative ions.

A prominent example is the hydrogen atom where in ionizing plasmas (Te⪆10 eV) mainly direct excitation from the atomic ground state plays a role. Depending on the dissociation degree also dissociative excitation of the hydrogen molecule can be a relevant population channel enhancing Balmer (Lyman) line emission. In recombining plasmas (Te⪅1.5 eV), three-body and radiative recombination of H+ are the main excitation channels. Depending on the respective particle densities also molecules may play a role via dissociative recombination (mainly of H2+ but also of H3+) as well as negative hydrogen ions via mutual neutralization with H+ or H2+. The situation is more complex in plasmas that are in-between the ionizing and recombining regimes: depending on the plasma parameters, any combination of the named channels coupling the excited states H* to the different heavy particle species can be present.

As a consequence, regarding the data needs for CR modelling besides an accurate description of the collisional and radiative processes within the hydrogen atom also accurate reaction probabilities for the named coupling processes are needed. For the respective reaction probabilities either the recent literature sources or (in case of no-existing data) approximations are used [29].

An example [29] is the magnetized plasma expansion described in [36, 37]. In this experiment, the plasma is generated by a cascaded arc, leaves a nozzle and expands into a low-pressure surrounding (p≈10 Pa) where an axial magnetic field confines it. Within the first few centimetres of the discharge, an electron current is driven between anode and cathode, which heats the plasma by means of ohmic heating. The current decreases with increasing distance from the nozzle. At a certain axial position (at z ≈20 cm), the ohmic heating becomes inefficient: a sudden drop in electron temperature (from ≈1.2 eV to ≈0.1 eV) and electron density (from ≈2 × 1019 m−3 to ≈1017 m−3) is observed. The drop of the plasma parameters is accompanied with a change of the plasma emission: for smaller distances from the nozzle, the plasma’s colour is red, and for larger distances, it appears to be blue.

A double Langmuir probe has been used to measure the electron temperature and density. Line-of-sight averaged densities of the atomic ground state have been determined [38] by two-photon absorption laser-induced fluorescence (TALIF, [39]), population densities of the excited state n = 2 by tunable diode laser absorption spectroscopy (TDLAS, [40]) and the population densities of all other exited states by OES. Axially and radially resolved population density profiles have been obtained by de-convoluting the results of the optical measurements by means of Abel inversion [41]. As a benchmark case, the Yacora CR model for atomic hydrogen was applied, in connection with the measured profiles of the electron temperature and density.

Figure 2 shows the population densities simulated by Yacora (divided by the statistical weight of the states) and the relative contribution of the different coupling channels together with the measured results. The results point out that the colour change of the plasma results from a shift in the relative relevance of the different excitation channels: for smaller distances from the nozzle recombination of the molecular ion H2+ is the most relevant channel for exciting the upper states (n ≥ 3) of the Balmer emission lines, see (Fig. 2a). The population density of these states decreases with the quantum number, making Hα (n’ = 3 → n’’ = 2) the strongest emission line in the visible range, explaining the red colour of the plasma. For larger distances, mutual neutralization of H− with atomic and molecular ions results in an overpopulation of the excited states n = 4…7, see (Fig. 2b), which explains the observed blue colour of the light emitted from the plasma.

Fig. 2
figure 2

Comparison of population densities due to different reaction channels (H: effective excitation, H+: recombination, H2+: dissociative recombination, H3+: dissociative recombination, H2: dissociative excitation, H−: mutual neutralization with H+ and H2+) in the hydrogen atom (divided by the statistical weight) calculated by the CR model and measured in the plasma of a magnetized plasma expansion: a red part of the plasma where dissociative recombination of H2+ is the most relevant excitation channel for H*; b blue part of the plasma with a high relevance of mutual neutralization of H−

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The agreement between measurement and the model is in both cases good, demonstrating that by including the involved coupling channels, CR models for the hydrogen atom can describe well the transition between different plasma regimes. Such models thus can be used for assessing the relative relevance of the different excitation channels and the densities of the involved neutral or ionic species. (Some of these particle densities are difficult to measure using different diagnostic techniques.)

3.2 Isotope effect for molecular hydrogen and deuterium

The application of hydrogen in plasmas is not restricted to the isotope H2. A prominent example is fusion experiments where a mixture of D2 and T2 can be present [42, 43]. Dissociation of the molecules followed by reformation into molecules, mostly at the walls of the experiment, will result in the formation of the isotopologue DT [44, 45]. Consequently, for the description of the plasma by means of population modelling also models for D2, T2 and DT are desirable.

Cross sections and Einstein coefficients can differ for the same reaction between different isotopes of hydrogen [46, 47]. The main physical reason being the different position of the vibrational and rotational energy levels within the molecule [48]. Consequently, population models for hydrogen including purely atomic processes are applicable directly to atomic deuterium and tritium—as soon as molecules are involved (even only as an intermediate state during a reaction process)—the impact of the isotope mass on the reaction probability needs to be checked. As a consequence, the data need for population models increases drastically for molecular deuterium or tritium, as the availability of data is very much limited here.

As the atomic Yacora model includes the coupling of excited states to different molecular species (see Sect. 3.1), the reaction data for the respective processes need to be exchanged for creating a model for isotopologues of H. This process is not straightforward as the available data base is partially scarce even for H. Prominent examples for data non-available for H, and its isotopes are cross sections for excitation of the individual excited states H* by means of dissociative excitation of H2 and by means of recombination of H+. A critical review of the currently available H data, including the possibility to generate new reaction probabilities also for isotopologues of H is currently ongoing.

The situation becomes better for molecular CR models as the recent MCCC calculations can be performed with reasonable effort also for other isotopologues: cross sections for electron impact excitation and ionization, resolved for the initial (and final) vibrational quantum numbers have been for instance recently published for deuterium and tritium [49]. A Yacora CR model for D2 is currently under development.

3.3 Optical thickness

A crucial part in constructing CR models is the treatment of the radiation transport: photons emitted within the plasma need to be transported through the plasma before being able to reach an observer (e.g. a detector). If the plasma is optically thick, the number of photons emitted by an atom or molecule by means of spontaneous emission is effectively reduced by self-absorption during their transport through the plasma. Consequently, the population density of the upper state is locally increased which in turn can affect the population densities of other excited states. The relevance of optical thickness depends on the gas temperature, the oscillator strength of the transition, the reduced mass of the emitting particle, the density of the lower state of the transition and the plasma length [50].

As the emission of a photon and its reabsorption do not necessarily take place at exactly the same location, in principle optical thickness cannot be expressed by a zero-dimensional set of ordinary differential equations as used by CR models. Usually, it is approximated by means of so-called escape factors: population escape factors are multiplied with the Einstein coefficient of the respective transition in order to reduce the transition probability and thus mimic the effect of optical thickness on the excited state population densities.

The population escape factors are proportional to the population density of the lower state of the transition. Thus, optical thickness typically affects resonant atomic emission lines only. In low-temperature plasmas, it is of particular relevance for a high dissociation degree and consequently a high atomic density as well as in fusion plasmas in regions where high atomic densities are achieved. Typically, the escape factors for non-resonant emission lines (Balmer, Paschen, … series in the case of the hydrogen atom) are equal to one, which means that self-absorption does not play a role. As shown in Fig. 3a are population escape factors [50] versus the density of the atomic hydrogen ground state for the resonant emission lines Lα to Lδ, determined for the geometry and for typical parameters of a small-scale ICP plasma reactor. The relevance of self-absorption increases with the wavelength of the transition, e.g. for the Lα line, it is highest, and it decreases with the upper state of the resonant transition. The grey bar illustrates the atomic ground-state density used in the CR model during calculating the result, as shown in Fig. 3b. If the threshold density for optical thickness of a specific transition is reached, its effect strongly increases with increasing density of the lower state density.

Fig. 3
figure 3

a Population escape factors for the first four Lyman emission lines of atomic hydrogen for the geometry and typical parameters of an exemplary low-pressure, low-temperature experiment. The grey bar illustrates the typical ground state density present in this experiment. b Impact of optical thickness on excited state population densities predicted by a CR model for atomic hydrogen [51]

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Figure 3b shows (again for typical parameters inside an ICP reactor) how the application of the population escape factors is shown in Fig. 3a affects population densities from the CR model. The strong increase of the population densities for low principal quantum number is a direct effect of the small population escape factors of the Lyman emission lines as shown in Fig. 3a.

Optical thickness affects not only the population density of excited states, it also reduces the line emission measured by techniques like emission spectroscopy. More strictly speaking, the emission from the plasma is reduced along its transport towards the detector and has to be multiplied by the reciprocal line escape factor [51]. Because both the population escape factors and the line escape factors can reach very small values (see Figs. 3a) and 7 in [52]), precise knowledge of both factors is mandatory to keep the error bar of diagnostic results reasonably small.

4 Towards a ro-vibrationally resolved CR model for molecular hydrogen

4.1 Ro-vibrationally resolved corona model for the Fulcher-α band

The Yacora corona models for the Fulcher-α band determine the population of the ro-vibrational levels of the electronic d3 state. Spectra can be calculated from these results by multiplication with the respective Einstein coefficients and by convolution with the apparatus profile.

For the included states X1, d3 and a3 ro-vibrational splitting is considered. In order to accurately describe radiative transitions, the energetically degenerate d3 levels are treated separately according to the different orientation of the projection of the electronic orbital angular momentum onto the internuclear axis (Λ) and the different parity. In total, the model consists of 990 different ro-vibrationally resolved (sub-)levels resulting from the states X1, d3 and a3.

In a first step, the model calculates the rotational and vibrational population of the ground state n(X1,v,J) based on rotational and vibrational temperatures defined as input parameters by the user of the model. In each case, it is possible to implement two-temperature distributions [53].

$$ n\left( {X^{1} ,v^{\prime } ,J^{\prime } ~} \right) \propto \left( {1 - \beta } \right)\tilde{n}_{{rot}} \left( {v^{\prime } ,\;J^{\prime } ,~T_{{rot\;1}} } \right)\tilde{n}_{{vib}} \left( {v^{\prime } ,\;T_{{vib\;1}} } \right)~ + ~\beta\, \tilde{n}_{{rot}} \left( {v^{\prime } ,\;J^{\prime } ,~T_{{rot~2}} } \right)\tilde{n}_{{vib}} \left( {v^{\prime } ,\;T_{{vib~2}} } \right).$$

This is required since the distribution of the lower vibrational levels ñvib(v, Tvib 1) is usually characterized by a cold temperature Tvib 1 resulting from plasma processes. However, reactions between different species (e.g. surface recombination of H to H2) can lead to a deviation from this distribution [54] and can be described by Tvib 2. Similarly, the population of the lowest rotational levels ñrot(v, J, Trot 1) can be described by a rotational temperature Trot 1 that has been found to reflect the gas temperature [55]. The higher rotational levels in turn are often influenced by surface processes that increase their population and can be described by Trot 2. Since both the hot rotational and vibrational populations result from recombination at the surface, they are usually equated (e.g. Trot 2 = Tvib 2 = Trot,vib 2). The respective two distributions are weighted by the weighting factor β. In the second step, the ground-state levels are coupled to the d3 state via electron impact excitation considering the optical selection rules for the quantum number J of the total angular momentum. While the corona model from 2016 applied vibrationally resolved cross sections calculated according to the semiclassical Gryzinski method [56, 57], an improved corona model developed within the scope of this work applies vibrationally resolved MCCC electron impact excitation cross sections stemming from [32]. The d3 state is linked to the a3 state via spontaneous emission (indicated by the arrows in Fig. 1). In both the previous and improved corona models, the transition probabilities for spontaneous emission are calculated using the LEVEL code [58]. This programme solves the radial Schrödinger equation and calculates eigenvalues, eigenfunctions, Franck–Condon factors and off-diagonal matrix elements for diatomic molecules.

Figure 4 compares the H2 band spectra for v′ = 0 → v′′ = 0 and v’ = 1 → v′′ = 1 in the wavelength range 600–618 nm calculated using the improved corona model with results from intensity calibrated [51] OES at the plasma grid of the negative ion source prototype for ITER neutral beam injection (PRF = 7.5 kW and pfill = 0.3 Pa). The values for the electron temperature and density, the vibrational temperature and the rotational temperature (Te = 4.9 eV, ne = 5.4 × 1015 m−3, Tvib 1 = 3000 K, Trot 1 = 500 K, Trot, vib 2 = 4400 K, β = 0.21) are derived from the experiment and inserted into the model as input quantities. The ro-vibrational band structure of the calculation and the measurement are in good agreement. However, the absolute emission of the calculation is steadily overestimated.

Fig. 4
figure 4

Fulcher-α emission lines (v′ = 0 → v′′ = 0 and v′ = 1 → v′′ = 1) in the wavelength range 600–618 nm calculated using the ro-vibrationally resolved Yacora corona model compared with results from OES measurements in an ionizing plasma

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One possible reason for the observed deviations can be understood by Fig. 5, showing the relative density of the electronic d3 state (as the sum of all ro-vibrational levels) versus the electron density from the ro-vibrationally resolved corona model and the CR model for the electronic triplet states of H2. Results are shown for different electron temperatures (Te = 1 eV and 10 eV). For the corona model, different vibrational temperatures Tvib = 1000 K and 10,000 K are used. Comparing the slope of the relative densities (n(d3)/n(X1)) as a function of ne from the corona and the electronic triplet model, completely different dependencies are found. This can be attributed to the (de-) population mechanisms neglected in the corona model but included in the CR model, which become more relevant with increasing electron density due to a higher collisionality. This includes not only additional population mechanisms due to stepwise excitation and cascades, but also the depopulating mechanisms of most importantly dissociative electron attachment and electron impact de-excitation, as well as ionization and proton charge transfer. Furthermore, the relative densities from the corona model show a dependency on the vibrational temperature, which is more pronounced at low Te. However, the influence of this dependency is much smaller than the difference between the corona and CR models.

Fig. 5
figure 5

Comparison of the relative density of the d3 state in dependence on the electron density for the corona and the electronic triplet model shown for Te = 1 and 10 eV. While the circles represent the results from the electronic triplet model, the triangles (Tvib = 1000 K) and squares (Tvib = 10,000 K) are the summed results from the ro-vibrationally resolved corona model for different Tvib input parameters

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These observations demonstrate that neither a ro-vibrationally resolved corona model, nor a purely electronic CR model are suited to provide a full picture of the (de-)population dynamics in the plasmas of interest. This is due to the lack of the consideration of (de-)populating mechanisms in the corona model and due to the disregard of ro-vibrational effects in the electronic model. Consequently, a fully ro-vibrationally resolved CR model for the hydrogen molecule is required. The first step towards such a model is described in Sect. 4.3.

4.2 Electronic CR model of the H2 singlet system

In the Yacora model for the singlet system of the hydrogen molecule from 2011 [34] electronically excited states up to n = 10 (united atom approximation) are included. For the n ≤ 3 states splitting due to Λ is considered and the energies assigned to the states correspond to the energies of the state’s potential curve minima. As mentioned above, in this model different input data sets have been used: The model could be switched between two sets of cross sections for electron impact excitation, namely the semiempirical cross sections from Miles [59] or a combination of the recommendations from Janev [60] (fits based on several measurements and calculations) together with the calculations from Celiberto [61] (impact parameter method). Since the cross sections from [61] were the only vibrationally resolved ones, they were used to evaluate effective cross sections for Tvib = 5000 K that were implemented in the model. In previous comparisons of measurements with the results of the model based on the two different sets of cross sections no general conclusion could be reached as to which model describes more accurately the population dynamics in low-pressure plasmas.

For this work, an improved singlet model has been constructed applying (in the same manner as the electronically resolved triplet model for H2 [31]) for electron impact (de-)excitation between Λ-resolved states MCCC cross sections taken from [32]. Figure 6 compares the electron impact excitation cross sections applied in the previous and improved singlet models (Celiberto/Janev, Miles and MCCC) for two different transitions (X1 → C1 and X1 → GK1). While for excitation into C1 the MCCC cross section lies for a broad energy range in-between the previously used cross sections and approaches for high energies the cross section from Celiberto, for excitation into GK1 the MCCC cross section shows a sharper threshold and exhibits beginning from Ee≈20 eV values in-between the other data sets.

Fig. 6
figure 6

Comparison of electron impact excitation cross sections for excitation from the ground state to the states C1 (a) and GK1 (b) in the hydrogen molecule stemming from Celiberto ([61])/Janev ([60]), Miles ([59]) and [32] (MCCC method). The cross sections from Celiberto and Janev are used combined as one set of cross sections in the model from 2011

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The improved singlet model contains the same electronically resolved states as the previous Yacora model, but the energies assigned to the n ≤ 3 states correspond to the respective vibrational ground state (v = 0) energies. For the higher n states, the energies are approximated by \({E}_{n}={Ry}^{*}\left(\frac{1}{{0.94}^{2}}-\frac{1}{{n}^{2}}\right)\) with the ionization energy of the H-atom Ry*. Table 1 gives an overview of the molecular processes interconnecting the states considered in the improved singlet model as well as the respective references of the data sources.

Table 1 Processes included in the collisional-radiative model for the singlet system of the hydrogen molecule and their respective input data sources
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For exciting electron collisions with an initial state of n = 1 or n = 2 towards the states n = 4–10, approximations inspired by the recommended scaling in [60] (depending on the dipole oscillator strength for transitions X1 → n) are applied to the MCCC cross sections. For electron impact excitation from initial states n = 3–9 to final states n = 4–10 rate coefficients stemming from the CR model described in [6] are used. These rate coefficients follow the recommendation from [62] and average the atomic hydrogen data of [22] and [63]. For future models electron impact excitation cross sections evaluated for the excited states of the hydrogen molecule are highly desirable, as they are to the knowledge of the authors currently not available.

Electron impact de-excitation cross sections for incidental n ≥ 3 states are calculated from their backward process cross section applying the detailed balance principle using the electron temperature. Regarding electron impact induced non-dissociative ionization also MCCC cross sections are used for the n = 2 states. For the states with n = 3–10, the applied rate coefficients originate from the model described in [6].

Dissociative electron attachment cross sections and rate coefficients for excited H2 states are scarce in the literature. Therefore, this process is described applying the rate constants from [64] for the n = 2 and from [65] for the n ≥ 3 states and illustrates the ongoing need for state-resolved cross sections or rate coefficients for H2.

With the exception of proton charge transfer other collisions between hydrogen molecules and heavy particles are neglected. The cross sections for this process stem from [60] and are estimated by the classical over-barrier transition model adapted to molecules. Due to the lack of alternative cross sections or scaling laws, these proton charge transfer cross sections are also used for incidental states n < 4, even though the cross sections are only explicitly recommended therein for states n ≥ 4. The proton density is calculated under assumption of quasi neutrality neglecting other positive ion species.

For spontaneous emission transition between Λ-resolved states effective Einstein coefficients evaluated for the temperature of 0 K [47] are applied. For all other spontaneous emission transitions approximations stemming from [22] and private communication with K. Sawada in 2000 are used.

In order to experimentally benchmark the improved singlet model, experimental results [66] (absolute population densities of the electronically excited H2 states) from emission spectroscopy in a low-pressure, low-temperature plasma reactor [67] are employed. Special features of the experiment are the constant gas temperature (Tgas≈450 K) and constant electron density (ne≈1017 m−3).

The examined plasmas consist of mixtures of hydrogen and helium. (The latter is added for diagnostic purposes to determine the electron temperature from the absolute emission.) Measurements are conducted with two intensity calibrated spectrometers (VIS, VUV) [51, 68] using the same radial sight of line capturing in combination the wavelength range of 116–900 nm.

For benchmarking the singlet CR model, the total population densities of the electronic states are determined from the measured total radiation of an entire electronic state by dividing by the respective Einstein coefficients. To obtain the measured total radiation of an entire electronic state, the individual rotational lines must be summed (or the measured lines must be scaled up) to obtain the radiation of the vibrational bands, which in turn must be summed [69].

Figure 7 compares the relative densities of the C1 a) and the GK1 b) state (from measurements of the Werner band and the transition GK1 → B1) normalized over the ground-state density in dependence on the electron temperature determined by the models based on the different available sets of cross sections with the measurements. For both investigated excited states, the model based on the MCCC cross sections features an improved agreement with the measurements in comparison to the previous model. Analogous comparisons for the states B1 and I1 (not shown here) demonstrate good agreement between the measurements and the results of the model applying MCCC cross sections too. However, the measured population density of the state B1 shows a stronger increase with increasing electron temperature than the modelling results. Since this trend increases with the electron temperature, which during the performed measurements was turned by changing the pressure, a pressure-dependent effect may explain the differences.

Fig. 7
figure 7

Comparison of measured [66], relative population densities of the C1 (a) and GK1 (b) state with modelling results of the different Yacora models for the singlet system of molecular hydrogen

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4.3 Vibrationally resolved CR model of the H2 ground state X1

In the previous Yacora model considering vibrationally resolved states of the hydrogen molecule from 2016 [29] mainly electron collision processes influencing the vibrational distribution are considered. During the time of developing this model, the availability of molecular data has been scarcer than now; hence, the model neglected, e.g. heavy particle collisions (except for proton charge transfer) and consequently the resulting vibrational distribution was not generally valid. Instead Tvib was introduced as a quasi-constant input parameter. For a future fully vibrationally resolved model, a self-consistent treatment of the population density distribution of the vibrational levels is desirable. As a first step towards this fully vibrationally resolved model for H2, a vibrationally resolved model for its ground state X1 is developed. Below the dissociation limit, 15 vibrational eigenstates are situated in the potential curve of the electronic ground state X1 (also depicted in the energy level diagram in Fig. 1) and considered in this model.

Table 2 compares the processes influencing the vibrational population in the previous vibrationally resolved model for H2 from 2016 with the improved vibrational model for the ground state X1. If not otherwise indicated, H2(v) refers to the electronic ground state H2(X1,v). With respect to proton collision reactions, besides the proton charge transfer (already considered in the model from 2016), proton impact (de-)excitation and proton impact dissociation are now implemented. Collisions among H2 molecules themselves also lead to a redistribution of the vibrational population distribution and are implemented via the conversion of kinetic energy into vibrational excitation (or the contrary, VT) and vibrational excitation of one molecule at the energetic cost of another one (VV). While hydrogen atom impact dissociation leads to depopulation of the excited states, H− associative detachment and H3+ dissociative recombination act as populating mechanisms. For hydrogen atom impact dissociation and hydrogen molecule impact (de-)excitation (VT) only rate coefficients and for the processes of H3+ dissociative recombination only rough estimate cross sections are available. Nevertheless, the rates estimated with these data sets for typical parameters of negative ion source plasmas are not neglectable small. Consequently, sets of vibrationally resolved cross sections are desirable especially for these processes.

Table 2 Comparison of the considered processes influencing the vibrational state population between the model 2016 and the vibrationally resolved model of the ground state X1
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The vibrational levels of the first two electronically excited states B1 and C1 are coupled to the improved CR model for the vibrational levels of X1 via electron impact (de-)excitation and spontaneous emission.

Table 2 does not claim to list all of the reactions occurring in the plasmas intended to be described by this model but is rather limited by the availability of the respective input data. Since negative ion source plasmas are characterized by a high atomic hydrogen density (nH≈1018 m−3, derived with the ideal gas law from the parameters in [70]), it would be for instance also of interest to include the process of hydrogen atom impact (de-)excitation (H + H2(v′) → H + H2(v′′)). (For such a plasma, the hydrogen atom impact excitation rate can have the same magnitude as the rate for electron impact excitation) However, for this process to the knowledge of the authors solely collision data is available describing only the excitation of the hydrogen molecule from the ground to excited vibrational states and not excitation of excited states among each other. Hence, this demonstrates again the ongoing H2 data needs for CR modelling.

An example for the critical evaluation of coupling data before its inclusion in a model is given by means of the dissociative electron attachment. The process occurs when the incoming electron and the neutral hydrogen molecule form an intermediate resonant anion state, which then decays by auto-detaching the electron (e + H2(v) → H2− → H + H−) [71]. In particular the process can proceed from the vibrational ground state via the resonances X2Σu+, B2Σg+ and higher excited electronic (Rydberg-) resonance states of H2−.

Figure 8a compares the dissociative electron attachment cross sections for initial state H2(X1, v = 0) from several literature sources for molecular hydrogen. While the semiempirical cross sections from Bardsley ([72]) are applied in the 2016 model, the calculations from Horáček ([73]) stem from a nonlocal resonance model and the cross sections from Laporta ([74]) are calculated within a local complex potential approach considering (as the only one shown) besides the X2Σu+ also the B2Σg+ channel. While [73] gives a set of initial v–resolved H2 dissociative electron attachment cross sections for v = 0–13, [72] provides them for v = 0–9. In [74] cross sections for H2 are only presented for v = 0 and 10. Since the cross sections from [72] and [74] are in good agreement up to Ee ~ 6 eV, e.g. for energies below the threshold for the B2Σg+ channel, the influence of this channel on the reaction rate can be determined qualitatively by comparison of the respective rate coefficients calculated using a Maxwellian electron EDF, as depicted in Fig. 8b. For a temperature of 2 eV the rate coefficient evaluated based on the cross sections from [74] is a factor more than seven times higher than the respective one for [72]. This demonstrates that the consideration of the B2Σg+ channel can be influential and a separate evaluation for each initial vibrational state is necessary. Hence, analyses for all v-states for H2 with data derived with the method from [74] would be of high interest but are currently not possible as only cross sections for initial v = 0,10 are given therein. Therefore, even though the influence of the B2Σg+ channel for H2 is recognized, it cannot be accounted for in the improved model due to the lack of molecular process data. In [73], the authors demonstrate that their cross section shows higher values in comparison to other theoretical calculations. To avoid the underestimation following the disregard of the B2Σg+ channel, the data set from [73] is implemented in the vibrationally resolved model for the H2 ground state X1. Nevertheless, calculations considering dissociative electron attachment via several resonances for each vibrational state of H2 are highly desirable for future CR modelling and from comparisons of the absolute rates for different processes for the different v levels, the highest influence may be expected for the states v = 4–10.

Fig. 8
figure 8

Comparison of dissociative electron attachment cross sections (a) and rate coefficients (b) for different data sources regarding collisions with initial state H2(X1,v = 0). While Bardsley ([72]) and Horáček ([73]) solely consider the X2Σu+ channel, Laporta ([74]) includes transitions via the B2Σg+ resonance as well

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5 Conclusions

The status of CR modelling for molecular hydrogen plasmas has been introduced by three examples.

Great advances in population modelling were made owing to the MCCC cross sections. It was demonstrated for the singlet system of H2 (like previously shown for the triplet system) that the application of MCCC cross sections in low-temperature plasmas (Te≈1.5–4 eV, ne≈1017 m−3) shows better agreement with the measurements than the preceding model based on previous cross sections. Future work is planned to extend the benchmark to a broader range of plasma parameters (e.g. measurements stemming from negative ion source plasmas and fusion divertor plasmas).

By comparing results of the electronic CR model for the triplet system and a (ro-)vibrationally resolved corona model for the Fulcher-α band as example case it was demonstrated that both models are incapable of providing a complete picture of the (de-)populating dynamics. While only electron impact excitation from X1 to d3 and spontaneous emission from d3 to a3 are considered in the corona model, other influential (de-)population mechanisms remain untreated. In contrast, the electronic model considers the coupling of excited states and other influential (de-)population mechanisms but lacks the consideration of the dependence on the vibrational temperature.

A fully ro-vibrationally resolved CR model for the hydrogen molecule is required. The development of such a Yacora model is currently ongoing. One of the first steps on the way to this model is the coupling of the electronic singlet and triplet models in order to be able to account also for (optically forbidden) spin-mixing processes. Further steps towards the fully (ro-)vibrationally resolved Yacora model include the development of a vibrationally resolved model for the electronic ground state X1 of the hydrogen molecule. This model is intended to manage a self-consistent calculation of the vibrational temperature by accounting for additional heavy particle collisions (and in a future step for surface processes). The ability to account for influential heavy particle collisions is limited by the availability of accurate molecular coupling data. Concrete needs of cross sections (de-)populating molecular hydrogen states have been pointed out in the discussion of the presented models.

For fusion research molecular deuterium and tritium are of high relevance too. However, cross sections and rate coefficients for the analogues processes covered in the hydrogen models of this work are even scarcer (with the exception of electron impact excitation and ionization cross sections calculated with the MCCC method). Therefore, advances in molecular data generation go hand in hand with advances in population modelling and are highly desirable. Advances in population modelling, in turn, enable improvements in their application in various fields of plasma physics (e.g. evaluating spectroscopic measurements) and fusion research.