1. Introduction
The collisional-radiative model (CR model) [
1] provides population distributions of atoms and molecules. Using this model, the effective reaction rate coefficients of various atomic and molecular processes including the contributions of the excited states can be calculated based on the population distribution. Emission intensities of atoms and molecules are also calculated using the population distribution.
We have been developing a CR model of molecular hydrogen,
. Previously, a CR model for molecular hydrogen in which only the electronic states are considered was constructed by Sawada and Fujimoto [
2]. This model was used to calculate effective reaction rate coefficients of dissociation and ionization of the molecule in plasmas. This CR model was extended by Greenland and Reiter [
3] and Fantz and Wündrlich [
4], and the rate coefficients calculated by these models were included in the neutral transport code EIRENE [
5].
Molecular-assisted recombination (MAR) was then proposed to understand the detached recombining plasmas in fusion edge plasmas [
6]. MAR comprises two series of processes. The first starts with the production of
from the collision of
and
, and the second starts with the dissociative attachment of electrons. To evaluate the contribution of the molecular vibrational distribution to the rate coefficient of MAR, we constructed a model in which the vibrational states were included in addition to the electronic states [
7].
It has recently been recognized that the production of state-resolved cross-sections is essential for the detailed analysis of fusion divertor plasmas [
8]. In this study, we develop a CR model in which the electronic, vibrational and rotational states are considered in order to model molecular processes whose cross-sections strongly depend on the initial vibrational and rotational states, e.g., the dissociative attachment of electrons in MAR.
As for the emission spectroscopy of molecular hydrogen, a corona model [
1] for the Fulcher transition (
) that includes vibrational states was constructed to determine the vibrational temperature in the electronic ground state
[
9]. Although the model was developed to also include rotational states in order to determine both rotational and vibrational temperatures in
[
10], corona models may not be suited to high electron density fusion detached plasmas because they neglect electron impact transition among the excited states. To describe high electron density plasmas, a rovibrationally-resolved CR model is necessary. In addition to the Fulcher band, this model provides emission line intensities for many bands, which may be used to determine the electron temperature and density as in helium atom spectroscopy [
11] and to determine the rovibrational population distribution in the
without assuming the vibrational and rotational Boltzmann distributions.
The purpose of this paper is to introduce a newly-developed CR model and to provide examples of its calculation. In this study, we investigated the rovibrational population distribution in a fusion detached divertor plasma. Because the quasi-steady-state solution [
1] for rovibrational states in
cannot be applied to the rate equations in the CR model owing to the large population relaxation time, we solved the rate equations time dependently for these states.
In this paper, we first introduce our newly-developed state-resolved CR model of molecular hydrogen and then list considered energy levels and relevant data as well as spontaneous transition probabilities, cross-sections and rate coefficients used in the model. Finally, we report on the modeling of a SlimCS fusion demo detached divertor plasma [
12,
13] in which effective reaction rate coefficients including MAR are calculated using the rovibrational population distribution. An example of emission spectra is also shown.
4. Results and Discussion
We applied the
CR model to a fusion detached plasma as an example of the application of the present model using the following parameters taken from a SlimCS DEMO detached plasma simulation [
12,
13]:
,
and
. In the present calculation, background molecular hydrogen, which contributes to the rotational excitation, is limited to
, and the kinetic temperature and total density are fixed at
and
, respectively.
The present code solves the rovibrational populations of the
,
and
time dependently. The values of
from Equation (
6) for the rovibrational states of
are
. The values of
in the electronic excited states is
for a level in the
. Except for the
,
τ is smaller than
. In this case, the
, known as the metastable state, does not have a significantly larger
than the other levels owing to the large value of
. The Runge–Kutta method was used with the time step of
. The QSS was applied to the rovibrational states in the other electronic excited states.
Figure 9 shows the population of the rotational states in
as a function of time. The initial population of 300 K Boltzmann distribution is given at
s. Initial total
density was set at 1
. In the present case, the distribution may be understood to be that of a gas puffed molecule. In modeling the recycling of a molecule at a wall, a proper rovibrational distribution should be given.
Figure 10a–c shows the rovibrational population distributions at
s,
s and
s, respectively.
Figure 10b,c suggests a different populating mechanism from that in
Figure 10a.
Figure 11a–c shows the inflow to each rovibrational level at
s,
s and
s, respectively. At
s, the dominant inflow to each level of
is the electron impact transition in
. The origin of the shoulder-like profile for
in
Figure 11a is the atomic hydrogen impact transition from lower lying
states with the large population. The
J dependence of the contributions of molecular and atomic collisions can be attributed to the differences in the
J dependence of the rate coefficients or cross-sections in
Figure 5 and
Figure 6. The transition with photoemission to the
is limited to
because the upper limit of
in the electronic excited states is
in the
state. As shown in
Figure 11b,c, increasing the excited rovibrational state population leads to an increase in electron impact transition among these states.
Figure 12a–c shows the outflow from each rovibrational level at
s,
s and
s, respectively. As the highly excited rovibrational state population increases, the dissociative attachment increases. Finally, as shown in
Figure 11c and
Figure 12c, the population distribution is dominated by the large flow in the
from electron impact and quenching processes including the dissociative attachment. After
s, the total population decreases with the relative rovibrational population in
kept nearly constant.
Figure 13 shows the vibrational population distributions obtained by summing all rotational populations. The distribution is not expressed as a simple Boltzmann distribution.
Figure 14 shows the effective rate coefficients for MAR (P1–P3) and (P4–P7) in
Table 11 and other processes in which atomic hydrogen is produced (P8–P12) and (P14–P15). Here, the effective rate coefficients are defined by:
where
(P1–P12) and
(P14–P15) are the effective rate coefficients for the production of atomic hydrogen from
and
, respectively, including the contributions of their excited states,
is the population density of the ground state atom,
in Equation (
4),
is total population density of
including the ground and excited states, and
is the same as
in Equation (
5).
The effective rate coefficients are calculated using cross-section data and models in the references in
Table 11, in addition to the present
CR model. In (P3, P7, P12, P15), the atomic hydrogen CR model [
14,
15] is used to calculate population flows into the ground state from excited states. The vibrationally-resolved CR model for
[
7] that includes processes (P4–P6) and (P16) in
Table 11 is used to calculate the population distribution of
assuming the QSS. Rotational levels are not considered in the
CR model. For processes (P11) and (P12), we use dissociative excitation data derived from experimental emission cross-sections given in [
2,
71] instead of those in the references in
Table 10 in order to include the contribution from all of the excited states of the
molecule. The rovibrational dependence is not included in the present calculation owing to a lack of relevant information.
Figure 14 shows that the effective rate coefficients for (P1–P3) and (P4–P7) increase with the excited rovibrational population in
. For
s, the sum of the effective rate coefficients of the two types of MAR is larger by a factor of approximately 25 than that at
s. Furthermore, MAR cannot be neglected relative to the well-known dissociative process (P8) as a source of atomic hydrogen after time
s.
In this study, we used the rate coefficient of dissociative attachment based on the nonlocal resonance model [
69]. According to this model, it is pointed out that non-resonant scattering enhances the rate coefficient [
72] and, although a complete dataset including it is not available, this effect may enhance the effective rate coefficient of (P1–P3) by an approximate factor of two beyond the value calculated here.
The photoemission transition from triplet electronic excited states (P10) (see
Table 5) is also important for atom production. The populations of the upper states of the photoemission transitions in (P9) and (P10) increase up to
s. In a later time, these populations decrease because the relative population of lower rotational states in
, which are the sources of the populations of the upper states of the photoemission, decreases; the electron impact excitation for
is considered and the number of the rotational states of the upper state is smaller than that of the
.
In this calculation, as the quenching from higher lying rotational states, only the dissociative attachment is effectively considered, because the rotational dependence of the cross-section is not included for other quenching processes. Furthermore, the electron impact dissociation through the continuum, which may play an important role, is not included here because no relevant data are available. If the dissociative attachment process is not dominant, the population density becomes smaller, which in turn reduces the effective rate coefficient for the MAR starting from the attachment process. The quenching cross-section from the higher rotational states should be studied in detail in future research. For precise calculation of the effective rates, precise cross-section data for the various transitions in will also be indispensable.
Figure 15a–c shows an example of the emission spectra of molecular hydrogen calculated from the obtained rovibrational population distribution of the excited electronic states by multiplying the spontaneous transition coefficient and photon energy. At present, the uncertainty in the electron impact excitation cross-section between the electronic states is large. For qualitative spectroscopic diagnostic, a more reliable dataset is necessary.
Here, we would like to discuss the application of the model presented in this paper to more general cases. In this study, we addressed processes starting from the
. In future work, a model for the recombining component starting from
, (P19–P20) in
Table 11, should be constructed for lower electron temperature plasmas. Furthermore, recombination starting from atomic hydrogen, (P21) in
Table 11, may contribute to the rovibrational population of the
in high atomic density plasmas [
77]. In considering the recycling of molecules at a wall (P22), a proper initial population distribution at the wall should be given to the model. As for the present calculation, interpreting it as that for the history of a gas puffed molecule, the result is valid regardless of the negligence of the above processes.
Figure 16 shows the rate coefficients of (P6) + (P16) and (P17) in
Table 11. When the electron density is much smaller than the molecular density,
may be produced by (P17), which is a competing process of (P6) + (P16). In this case, the dissociative process from
(P18) may contribute to the population of
. Production of the rovibrationally-resolved cross-section for this process is indispensable for such plasmas.