#
Evaluation of State-Resolved Reaction Probabilities and Their Application in Population Models for He, H, and H_{2}

^{1}

^{1}

^{*}

## Abstract

**:**

_{2}, in contrast, significant deviations exist between reaction probabilities taken from different literature sources. The reason for this is the more complex internal structure of molecules compared to atoms. Vibrationally resolved models are applied to demonstrate how these deviations affect the model results. Steps towards a consistent input data set are presented: vibrationally resolved Franck–Condon factors, transition probabilities, and ionization cross-sections have been calculated and are available now. Additionally, ro-vibrational models for selected transitions are applied successfully to low-density, low-temperature plasmas. For further improving the accuracy of population models for H

_{2}, however, it is necessary to establish a comprehensive data set for ro-vibrationally resolved excitation cross-sections based on the most recent calculation techniques.

## 1. Introduction

_{e}, electron density n

_{e}, and the quasi-constant densities of the ground states of one or more particle species. While in equilibrium plasmas the (local) thermodynamic equilibrium is fulfilled, for nonequilibrium plasmas, corona or collisional radiative (CR) models have to be applied.

_{e}and n

_{e}): the plasma parameters used as input to the model are varied until the simulated population densities best match the measured population densities of one or more excited states in the respective atom or molecule [1,5]. Forward calculations allow for known plasma parameters predicting the population densities of excited states. The latter information can be useful; for example, for predicting the photon emission of atomic lines as well as molecular bands, and the impact of this radiation on surfaces [6].

## 2. Population Models

#### 2.1. Theory of Population Modeling

_{e}≥ a few eV and/or low densities of ionic species) and collision reactions connecting different excited states are negligible (n

_{e}< 10

^{17}m

^{−3}), corona models can be applied. These models balance electron collision excitation from the atomic or molecular ground state with spontaneous emission. The density of the ground state is assumed to be quasi-constant and is used as an input parameter.

_{e}(>10

^{22}m

^{−3}) the results of population models should approach the local thermodynamic equilibrium or even the thermodynamic equilibrium. The latter is, however, not always the case for CR models applied to low-pressure plasmas since these models often neglect or strongly simplify the radiation transport.

#### 2.2. The Flexible Sover Yacora

## 3. Atomic Population Models

#### 3.1. Helium

#### 3.1.1. Properties of the Helium Atom

^{1}S in the singlet system and 2

^{3}S in the triplet system. For high electron densities, the dominant depopulating process for these states is excitation and de-excitation by electron collisions. In plasmas with low n

_{e}transport of particles in the metastable states can take over. If population models for helium are applied to such plasmas, the relevant loss processes for the metastable states have to be included either self-consistently or by using fixed transport coefficients as input.

^{1}S and 2

^{3}S can also be of relevance.

#### 3.1.2. The CR Model for Helium and Results

_{0}) together with line-of-sight averaged population densities measured for a pressure of 10 Pa by absorption spectroscopy (p = 2) and OES (p = 3) in a microwave electron cyclotron resonance (ECR) discharge (f = 2.45 GHz) in which the magnetic field is created by permanent magnets attached to one of the outer walls of the vacuum vessel (diameter d = 15 cm, height h = 56 cm). The plasma experiment is described in detail in [15]. For deducing the population densities of the excited states with p = 3, measured intensities of emission lines originating from these states have been divided by the respective Einstein coefficients.

_{e}= 3.8 eV from a Langmuir probe, n

_{e}= 7 × 10

^{16}m

^{−3}from microwave interferometry) and the calculations have been performed using a Maxwell EEDF. The agreement between model and measurement is excellent (deviations well below 35%), indicating a high accuracy of the experimental results (population densities and plasma parameters) but in particular also of the reaction probabilities used in the CR model.

#### 3.2. Atomic Hydrogen

#### 3.2.1. Properties of the Hydrogen Atom

_{e}> a few eV) plasmas direct excitation from the ground state H(1) and dissociative excitation from H

_{2}take place predominately (Figure 3b). In recombining (typically T

_{e}≤ 1 eV) plasmas, (dissociative) recombination of positive ions and mutual neutralization of negative ions, H

^{−}and positive ions can dominate (Figure 3c). Models for so-called partially recombining plasmas have to include all six particle species and reaction channels shown in Figure 3.

#### 3.2.2. The CR Model for Atomic Hydrogen

^{+}[18], dissociative excitation of H

_{2}[18], dissociative recombination of H

_{2}

^{+}[17], dissociative recombination of H

_{3}

^{+}[19,20], mutual neutralization of H

^{−}with positive ions [17,21].

_{3}

^{+}, two different reaction channels are possible: producing either three atoms in the ground state or one molecule in its ground state and an excited atom. While the total cross-section for this recombination process and the branching ratio for the two reaction channels are well known, mainly from storage ring experiments [20], not much is known about the quantum state distribution of the excited atom produced by the second reaction. It is stated in [19] that for low-collision energies, predominately atoms in p = 2 are produced. Thus, in the CR model for atomic hydrogen, p = 2 is implemented as the only product for dissociative recombination of H

_{3}

^{+}.

^{−}with the atomic positive ion H

^{+}produces an atom in its ground state and a second atom in p = 2 or p = 3 (depending on the collision energy), disagreement exists regarding the reaction channels for mutual neutralization with H

_{2}

^{+}: while in [21] it is stated that this reaction ends in a hydrogen molecule in its ground state and an excited atom, according to [17] the reaction products are an atom in the ground state and an excited molecule. In order to enable investigations on this topic, both reaction channels have been implemented to the CR model. By changing the respective branching ratio, the relative relevance of the two channels can be varied.

#### 3.2.3. Application of the CR Model for H to an Ionizing Plasma

_{2}.

_{e}and n

_{e}in ionizing plasmas is to compare measured line ratios H

_{α}/H

_{β}and H

_{β}/H

_{γ}with results of an atomic CR model, as described in detail in [1,25]. If, additionally, the molecular emission is taken into account, also the ratio n(H)/n(H

_{2}) can be determined and additionally the uncertainty of the results can be reduced. The aim of the present work was to implement and benchmark an automated version of this technique, based on a fitting procedure.

_{RF}of the RF power coupled into the plasma and of the filling pressure p

_{fill}. Measured are the Balmer lines H

_{α}… H

_{δ}and the Q lines of the first four diagonal vibrational bands (0→0 … 3→3) of the molecular transition d

^{3}→a

^{3}(see Section 4.1). By assigning rotational temperatures to the vibrational bands of this band and appropriate scaling based on the CR model for H

_{2}(see Section 4.2) the total emission of d

^{3}→a

^{3}is deduced.

_{RF}= 70 kW, p

_{fill}= 0.8 Pa, and a broad range of T

_{e}and n

_{e}. The residual is defined as the absolute value of the logarithm of the deviation between calculated (based on a Maxwell EEDF) and measured emission, summed for H

_{α}, H

_{β}, H

_{γ}, H

_{δ}, and d

^{3}→a

^{3}. The blue band depicts the parameter space with the smallest residuals. The band shows a horizontal structure and n

_{e}can be determined with an acceptable error bar to 1.25 ± 0.75 × 10

^{18}m

^{−3}.

_{e}with good accuracy. Thus, T

_{e}is taken from the absolute minimum of the residuals (12.1 eV).

^{3}→a

^{3}as well as (in red) the model result for T

_{e}= 12.1 eV and n

_{e}= 1.25 × 10

^{18}m

^{−3}. The agreement between measurement and the model is excellent (the deviations are below 12%). Additionally shown are the fractions of the Balmer radiation that can be attributed to direct (“H”) and dissociative (“H

_{2}”) excitation.

_{e}, the result can have a high uncertainty.

#### 3.2.4. Application of the CR Model for H to a Recombining Plasma

^{19}m

^{−3}to ≈10

^{17}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 is red (partially recombining plasma), for larger distances it is blue (fully recombining plasma).

_{3}

^{+}are predominately in the state p = 2 (Section 3.2.2); secondly, to check the existence and relevance of the two different reaction channels for mutual neutralization of H

^{−}with H

_{2}

^{+}(production of either excited atoms or excited molecules) suggested by [17,21] (Section 3.2.2).

_{e}and n

_{e}, a fitting procedure was performed in order to adapt the measured excited state population densities to results of the CR model for H, calculated using a Maxwell EEDF. Variable parameters in this fit are the unknown particle densities (n(H

_{2}), n(H

^{+}), n(H

_{2}

^{+}), n(H

_{3}

^{+}), and n(H

^{−})), considering the plasma quasi-neutrality. An additional free parameter is the branching ratio of the two reaction channels for mutual neutralization of H

^{−}with H

_{3}

^{+}. Since in recombining plasmas the amount of relevant excitation channels can be higher than in ionizing plasmas (as shown in Figure 3), the fitting procedure was performed manually.

_{3}

^{+}are predominately in the state p = 2 is correct. Secondly, in the plasma under investigation both proposed reaction channels for mutual neutralization of H

^{−}with H

_{2}

^{+}take place. The branching ratio between the two channels was determined to be approximately 0.16:0.84 over the complete volume of the plasma expansion, i.e., 16% of such reactions create an excited atom and the other 84% an excited molecule.

## 4. Population Models for Molecular Hydrogen and Deuterium

#### 4.1. The Hydrogen Molecule

^{1}.

^{1}are indicated in the figure. The rotational levels are not shown since the energy difference between two consecutive rotational levels is significantly smaller than the one between the vibrational levels.

^{1}, even in in low-density, low-temperature plasmas the population of these states can (partially) thermalize [30]. Typically, the population of the lowest rotational levels can be described by a rotational temperature that is identical to the gas temperature [30]. For higher rotational quantum numbers, a significantly increased population can occur—caused most likely by surface recombination of H atoms to H

_{2}—resulting in a so-called hockey-stick structure of the rotational population distribution [31]. Typical vibrational temperatures are much higher than the rotational temperature [32]. The rotational and vibrational population distributions in X

^{1}are correlated to the respective distributions in the electronically excited states by an excitation–deactivation balance [30].

^{3}, an energy interval is indicated because the potential energy curve (the electronic eigenvalues of the total wave function vs the internuclear distance) for this state is repulsive, i.e., it shows no minimum. The internuclear distance of a hydrogen molecule in b

^{3}will increase until dissociation into two atoms takes place. Radiative transition into b

^{3}does not result in a ro-vibrational emission band structure but in continuum radiation [35].

^{3}in the triplet system by spontaneous emission can take place only via electric quadrupol or magnetic dipole radiation with very low transition probability, and the radiative lifetime is around 1 ms [17]. Other reactions will take over the role as relevant depopulation mechanisms. One of these reactions can be electron collision transfer into the state a

^{3}. These two states are energetically very close and depending on the involved vibrational and rotational sublevels, the cross-section for electron collision can reach high values (up to 10

^{−16}m

^{2}[17]). Secondly, quenching (de-excitation by heavy particle collisions) can be dominant for high molecular densities. Cross-sections of 7.5 × 10

^{−16}m

^{2}have been measured for gas temperatures of 300 K [36]. Implementing such processes as accurately as possible in population models for H

_{2}is of high relevance since, as shown in [37], stepwise excitation via the c

^{3}state can play an important role for populating energetically higher levels.

_{2}couples with the molecular ion or with the continuum of another bound state, such a transition will be equivalent to an ionizing reaction (autoionization) or dissociation (predissociation), respectively.

#### 4.2. Characteristics of the Models for H_{2} and D_{2}

^{1}, I

^{1}, e

^{3}, and d

^{3}in p = 3 (the states indicated in red in Figure 6) are resolved for their vibrational levels. The population density of the vibrational levels in the ground state is treated as quasi-constant and thus T

_{vib}is an input parameter for the CR model.

^{1}→B

^{1}, X

^{1}→C

^{1}) vibrationally resolved cross-sections from [40,41] are available; these data are implemented instead of the cross-sections from [17] or [38]. The total cross-sections for the transition X

^{1}→C

^{1}from [17,38,40,41] are compared in Figure 8a. The cross-sections from [40,41] have been used while generating the data from [17] also, and thus these two curves show small deviations only (with the exception of the region close to the threshold energy).

^{1}→X

^{1}(Lyman band), C

^{1}→X

^{1}(Werner band), and d

^{3}→a

^{3}(Fulcher band) in H

_{2}have been set up. Additionally, a ro-vibrationally resolved corona model for d

^{3}→a

^{3}in D

_{2}was constructed. Due to the quasi-constant character of the sublevels in the ground state T

_{vib}and T

_{rot}of X

^{1}are input parameters for the model.

^{1}are denominated by v and J, the ones in the electronically excited states by v’ and J’). In the corona model for constant v and v’ the identical cross-section is used for all combinations of rotational substates J and J’. As a result of this approach the rotational distribution in the ground state X

^{1}is not mapped correctly to the rotational sublevels in the upper electronic state of the modeled transition. Thus, it is necessary to introduce an artificial thermalization process for the rotational sublevels, taking into account the rotational constants of the ground state and the upper electronic state.

#### 4.3. Franck–Condon Factors and Einstein Coefficients

_{2}and its isotopomeres (D

_{2}, T

_{2}, HD, DT) [42]. Additionally available are FCF for coupling of the neutral molecule H

_{2}with its positive ion H

_{2}

^{+}[47]. Both datasets are accessible online [48,49]. Recently, ro-vibrationally resolved Einstein coefficients for some selected emission bands (B

^{1}→X

^{1}, C

^{1}→X

^{1}and d

^{3}→a

^{3}) in H

_{2}(and d

^{3}→a

^{3}in D

_{2}) have been calculated.

^{3}from the ground state X

^{1}and de-excitation of d

^{3}via spontaneous transition to a

^{3}. Figure 7a shows for v and v’ < 10 vibrationally resolved FCF for X

^{1}(v)→d

^{3}(v’). The highest values of the FCF do not follow the diagonal defined by v = v’. The reason is that the minima of the potential curves of X

^{1}and d

^{3}are located at different internuclear distances (0.74 Å for X

^{1}compared to 1.1 Å for d

^{3}). As a consequence, the vibrational population in X

^{1}and d

^{3}differs and techniques like scaling with the FCF [32] have to be applied in order to deduce the vibrational population in d

^{3}from the one in X

^{1}.

^{3}and a

^{3}is very similar, the Einstein coefficients for the emission band d

^{3}(v’)→a

^{3}(v’’) follow—as can be seen in Figure 7b for v’ and v’’ < 10—the diagonal defined by v’ = v’’. The diagonal transitions v’ = v’’ = 0 … 3, in the wavelength range 600–640 nm are the strongest parts of this system.

#### 4.4. Electron Collision Excitation Cross-Sections

^{1}to different electronically excited states exist: the one from [38] was created by semiempiric methods based on experimental information and phenomenological extensions of the Born approximation into the low-energy region. The data given in [17] represents a summary of recent measurements and calculations. Within the process of validating and benchmarking the models, a critical check of these cross-sections has been performed.

^{1}to the excited state C

^{1}(optically allowed transition), Figure 8b shows cross-sections for excitation of c

^{3}(spin-exchange process). The difference in the type of excitation process results in a distinctively different shape of the cross-sections.

^{1}and [50,53,54,55,56] for excitation of c

^{3}; some of these data have been used for compiling the cross-sections given in [17]). All these cross-sections—with the exception of the vibrationally resolved data from [40,41]—are resolved only for the electronic levels. To do this comparison, the vibrationally resolved cross-sections for excitation X

^{1}→C

^{1}were summed over the vibrational levels v’ in C

^{1}.

_{thr}= 12.3 eV) of the excitation X

^{1}→C

^{1}the discrepancies between the cross-sections from the different data sources reach factors of larger than 10, for X

^{1}→c

^{3}they reach a factor of about 5 at E

_{thr}= 11.8 eV. Thus, also the uncertainty of plasma parameters determined using the current population models for H

_{2}will be quite high for low T

_{e}(see Section 4.6).

#### 4.5. Electron Collision Ionization Cross-Sections

_{2}, vibrationally resolved electron collision ionization cross-sections for the ground state X

^{1}and the first five electronically excited states (EF

^{1}, B

^{1}, C

^{1}, a

^{3}, and c

^{3}) have been calculated using the Gryzinski method [39] together with the Franck–Condon theory.

^{1}(v = 0) that for ionization of H

_{2}—one electron is completely removed from the molecule—the Gryzinski method produces surprisingly accurate results [57].

_{2}can take place via two different reactions: non-dissociative ionization, ending in a molecular ion H

_{2}

^{+}, and dissociative ionization, producing an atom H in its ground state and a positive atomic ion H

^{+}:

_{2}+ e

^{−}→ H

_{2}

^{+}+ 2e

^{−}

_{2}+ e

^{−}→ H + H

^{+}+ 2e

^{−}

_{2}into the vibrational continuum of the H

_{2}

^{+}ground state

^{2}Σ

_{g}

^{+}(reaction 2a) and excitation into the repulsive state

^{2}Σ

_{u}

^{+}(reaction 2b). As a prerequisite for determining cross-sections for Reactions 1 and 2, the respective Franck–Condon densities have been calculated [57].

^{1}(v = 0) with cross-sections available in the literature. For non–dissociative ionization an excellent agreement was found between the present data and experimentally determined data [58,59,60,61] as well as theoretical [62] cross-sections.

_{2}.

_{2}.

#### 4.6. Application of the Models

_{2}on the input data, measurements have been performed at two different low-pressure, low-temperature laboratory experiments: first, the emissivity of the molecular bands GK

^{1}→B

^{1}, I

^{1}→B

^{1}, e

^{3}→a

^{3}, and d

^{3}→a

^{3}(see Figure 6) have been measured at the uniform ECR plasma experiment described in Section 3.2.2 both in hydrogen and deuterium. And second, the emission of all emission bands indicated by the blue arrows in Figure 6 has been determined for a hydrogen plasma in the microwave experiment described in Section 3.1.2.

^{1}(i.e., an optically allowed excitation mechanism) and d

^{3}(spin-exchange excitation) vs the electron temperature. Additionally shown in Figure 10 (in green and red lines) are the CR model results (summed over the vibrational substates) based on the input data sets by [17,38]. T

_{e}used as input for the calculations was determined by evaluating the Balmer line emission, as described in Section 3.2.3, and n

_{e}(≈10

^{17}m

^{−3}) by means of microwave interferometry and double probe measurements. Different values of the electron temperature were achieved by varying the pressure. The ground state density was deduced from the ideal gas law, taking into account in an iterative way the dissociation of H

_{2}into hydrogen atoms.

^{3}were performed for different probabilities for quenching of the c

^{3}state: no quenching and the quenching cross-section taken from [36]. Thus, for the calculated population density of d

^{3}, shaded areas are shown. This strong influence of the quenching probability (more than a factor of 2) demonstrates the high relevance of stepwise excitation via the c

^{3}state for the population of the triplet states in H

_{2}.

^{1}are more or less on top of each other, the results for d

^{3}based on the data from [17] are higher by a factor of 3–4 than the results calculated using the data from [38]. This strong correlation of model results to input data is valid for all investigated emission bands.

^{1}, C

^{1}, GK

^{1}and I

^{1}), no clear statement can be made which of the two cross-section sets describes the measurement best. In the triplet system (a

^{3}, b

^{3}, d

^{3}), however, generally the data from [38] seems to yield better results—as can be seen exemplarily for the state d

^{3}in Figure 10b. This result is quite surprising since the cross-sections from [17] are a summary of measurements and calculations that are based on much more sophisticated techniques than the data from [38]. Either processes relevant for the population densities of triplet states missing in the current status of the CR model or discrepancies in the underlying cross-sections result in high error bars of the data suggested in [17]. In order to check the second explanation—and since most available cross-sections are not vibrationally or rotationally resolved—calculations aiming at a comprehensive electron collision excitation cross-section database for both multiplet systems are highly desirable.

^{3}→a

^{3}in the visible wavelength range (600–640 nm) can easily be distinguished from each other and the overlap with lines originating from other emission bands is negligible. Thus, this emission band is frequently used for plasma diagnostics [32,64,65,66]. Figure 11a,b show spectra of this band for v’ = v’’ = 0 and v’ = v’’ = 1 in hydrogen (between 600 nm and 618 nm) and deuterium (between 598 nm and 612 nm), respectively. The spectra in the upper part of the figures have been calculated using the ro-vibrationally resolved corona models (T

_{e}= 10 eV, n

_{e}= 10

^{18}m

^{−3}) whereas the ones in the lower part are measurements (taken in the plasma generation region of the negative ion source prototype for ITER NBI, P

_{RF}= 70 kW and p

_{fill}= 0.6 Pa). The calculations have been performed using the excitation cross-sections from [38] since these data, as discussed above, yield better results for the triplet states. The theoretical position of the most intense emission lines (the lines of the Q branch) is symbolized in Figure 11 by the blue stripes.

_{e}, n

_{e}, the vibrational temperature T

_{vib}, and the rotational temperature T

_{rot}used as input for the model are taken from the experiment. For H

_{2}, besides the nominal T

_{vib}= 5000 K, calculations for 1000 K and 10,000 K were performed in order to demonstrate the sensitivity of the band structure on T

_{vib}. The resulting spectra are shown in red and green in the upper part of Figure 11a (in order to increase the visibility of these spectra, the wavelength axes have been shifted slightly toward smaller and larger values, respectively).

_{vib}or T

_{rot}. This result can be explained on the one hand by the fact that the model does not calculate the rotational population distribution in d

^{3}self-consistently (due to the lack of ro-vibrationally excitation cross-sections, an artificial thermalization process was introduced, see Section 4.2). On the other hand, corona models do not take into account stepwise excitation (e.g., via the c

^{3}state) and population via cascades from energetically higher states. For the future it is planned to construct an extended corona model including such processes.

^{3}→a

^{3}band, the lines in each of the two emission bands B

^{1}→X

^{1}and C

^{1}→X

^{1}are much closer together and—depending on the apparatus profile of the used spectroscopic system—a significant overlap of lines can occur. Additionally, as can be seen in Figure 12, the two bands themselves overlap. The figure shows for the wavelength range between 110 nm and 175 nm in the upper part a spectrum calculated using the corona models for B

^{1}→X

^{1}and C

^{1}→X

^{1}(the radiation emitted by the two bands is shown in green and red, respectively) and in the lower part a spectrum measured for a pressure of 3 Pa (inductively coupled plasma (ICP) discharge, f = 13.56 MHz, d = 10 cm, h = 40 cm). While the measured spectrum comprises also the Lyman line L

_{α}, this line was omitted by purpose in the calculation. In the model the cross-sections for excitation from the ground state by [40,41] have been used. T

_{e}and n

_{e}used as input for the calculations (T

_{e}= 2.7 eV, n

_{e}= 1.3 × 10

^{17}m

^{−3}) have been determined by evaluating the Balmer line emission, as described in Section 3.2.3.

^{1}→X

^{1}(at 160 nm) the modeled spectrum, however, shows a smaller number of photons emitted in the wavelength range between 135 nm and 155 nm. This smaller number of photons is most probably due to cascading processes (e.g., EF

^{1}→B

^{1}) that will be implemented to extended corona models for B

^{1}and C

^{1}in a next step.

^{3}→a

^{3}band—not determined by scaling the emission of a few measured lines. Instead, it is possible to fit a simulated spectrum (free parameters: T

_{e}, n

_{e}, ground state density, T

_{vib}, and T

_{rot}) to the measured one. A similar procedure is described in [67] for the emission band C

^{3}→B

^{3}of molecular nitrogen. However, identifying proper values for all free parameters during performing such a fitting procedure can be quite elaborate.

^{1}→X

^{1}and 117–130 nm for C

^{1}→X

^{1}). The integrated radiation in these intervals can be scaled to the total band emission by multiplication with a scaling factor. The following scaling factors have been derived from the simulated of the molecular bands for the parameters of the used ICP discharge: 2.0 for B

^{1}→X

^{1}and 2.9 for C

^{1}→X

^{1}.

_{vib}have been performed: with decreasing T

_{vib}from 4500 K to 3000 K the relative changes in the scaling factors are below 6%. This indicates that a rough knowledge of the plasma parameters is sufficient for determining the scaling factors with sufficient accuracy. Preparing a set of scaling factors for the typical range of plasma parameters in a specific plasma discharge and scaling to the full band emission using these factors can significantly speed up the evaluation process compared to the fitting procedure mentioned above.

## 5. Conclusions

_{3}

^{+}and mutual neutralization of H

^{−}with positive ions) cross-sections or branching ratios are known only with large error bars or are missing completely.

## Abbreviations

OES | Optical emission spectroscopy |

TALIF | Two-Photon Excited laser Induced Fluorescence |

TDLAS | Tunable Diode Laser Absorption Spectroscopy |

CR model | Collisional radiative model |

EEDF | Electron energy distribution function |

ECR | Electron cyclotron resonance |

RF | Radio frequency |

ITER | The internuclear thermonuclear experimental reactor (or latin for “the way”) |

NBI | Neutral beam injection |

FCF | Franck-Condon factor |

ICP | Inductively coupled plasma |

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**Figure 2.**Line-of-sight averaged population densities n

_{p}for electronically excited states of He measured by optical absorption (p = 2) and emission (p = 3) spectroscopy and results of calculations using the Yacora collisional radiative (CR) model for He. The population densities are divided by the statistical weight of the states and normalized to the ground state density n

_{0}of He.

**Figure 3.**Excitation channels for atomic hydrogen included to the Yacora CR model for atomic hydrogen: (

**a**) all channels; (

**b**) channels relevant in ionizing plasmas; (

**c**) channels relevant in recombining plasmas.

**Figure 4.**Comparison of Yacora CR model results with the Balmer line emission and the emission of the molecular band d

^{3}→a

^{3}measured in an ionizing plasma: (

**a**) 2D matrix illustrating a parameter space in T

_{e}and n

_{e}the best possible agreement between measurement and model; (

**b**) comparison of the measured emission vs model results for T

_{e}= 12.1 eV and n

_{e}= 1.25 × 10

^{18}m

^{−3}(depicted by the orange ellipse in Figure 4a). Additionally, shown are the calculated fractions of emission that can be attributed to direct excitation of H and from dissociative excitation of H

_{2}.

**Figure 5.**Comparison of population densities 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**) partially recombining red part of the plasma; (

**b**) fully recombining blue part of the plasma.

**Figure 7.**(

**a**) Franck–Condon factors (FCF) for excitation from the ground state X

^{1}to the excited state d

^{3}in the triplet system; (

**b**) A

_{ik}for spontaneous emission from d

^{3}to a

^{3}.

**Figure 8.**Electron collision excitation cross-sections from the literature for molecular hydrogen: (

**a**) excitation from the ground state X

^{1}to the excited state C

^{1}in the singlet system; (

**b**) excitation from the ground state X

^{1}to the excited state c

^{3}in the triplet system.

**Figure 9.**Non-dissociative and dissociative ionization cross-sections for the vibrational level v = 0 in the electronic ground state X

^{1}of the hydrogen molecule.

**Figure 10.**Comparison of population densities (normalized to the molecular ground state density) calculated by the Yacora corona model for H

_{2}with measurements taken in an ionizing plasma: (

**a**) excited state I

^{1}; (

**b**) excited state d

^{3}.

**Figure 11.**Spectra of the emission band d

^{3}→a

^{3}calculated by the Yacora corona model for molecular hydrogen and measured in an ionizing plasma: (

**a**) hydrogen; (

**b**) deuterium.

**Figure 12.**VUV/UV spectra for molecular hydrogen: (

**a**) calculated by the corona model; (

**b**) measured in an inductively coupled plasma (ICP) discharge.

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Wünderlich, D.; Fantz, U.
Evaluation of State-Resolved Reaction Probabilities and Their Application in Population Models for He, H, and H_{2}. *Atoms* **2016**, *4*, 26.
https://doi.org/10.3390/atoms4040026

**AMA Style**

Wünderlich D, Fantz U.
Evaluation of State-Resolved Reaction Probabilities and Their Application in Population Models for He, H, and H_{2}. *Atoms*. 2016; 4(4):26.
https://doi.org/10.3390/atoms4040026

**Chicago/Turabian Style**

Wünderlich, Dirk, and Ursel Fantz.
2016. "Evaluation of State-Resolved Reaction Probabilities and Their Application in Population Models for He, H, and H_{2}" *Atoms* 4, no. 4: 26.
https://doi.org/10.3390/atoms4040026