M Capitelli
email: mario.capitelli@ba.imip.cnr.it

R Celiberto

G Colonna

G D'ammando

O De Pascale

P Diomede

F Esposito

C Gorse

A Laricchiuta

S Longo

L D Pietanza

From Cold to Fusion Plasmas: Spectroscopy, Molecular Dynamics and Kinetic Considerations

Keywords: numbers: 34.80.Gs, 52.20.-j, 52.25.-b

Non-equilibrium effects in hydrogen plasmas have been investigated in different systems, ranging from RF plasmas to corona discharges. Existing measurements of vibrational and rotational temperatures, obtained by different spectroscopical techniques are reported, rationalized by results calculated by kinetic models. Input data of these models are discussed with particular attention on the dependence of relevant cross sections on the vibrational quantum number. Moreover the influence of vibrational excitation of the H 2 molecules in affecting the translational distribution of atoms in ground and excited levels is shown. Finally a collisional radiative model for atomic hydrogen levels, based on the coupling of the Boltzmann equation for EEDF and the excited state kinetics, is presented, emphasizing the limits of quasi-stationary approximation. In this last case, large deviations of the EEDF and atomic level distributions from the equilibrium are observed.

Introduction

Fusion plasmas are characterized by very high temperatures that, while preventing the existence of neutral species, still allows multi-charged radiating species. This radiation, coming from impurities generated by wall erosion or added by ad-hoc injection, can be used to get information about the electron temperature. In these last years however "cold plasmas" are becoming very popular in the fusion community because they determine the properties near the divertor where a moderate electron density and temperature exist. Under these conditions the vibrational manifold of the molecular species is well populated having strong consequences on the behaviour of the whole system. On the other hand, vibrationally excited molecules of D 2 and H 2 species play a paramount role in creating negative ion sources to be used as neutral heating in the ITER project [START_REF] Capitelli | Vibrational kinetics, electron dynamics and elementary processes in H 2 and D 2 plasmas for negative ion production: Modelling aspects[END_REF]. Plasma conditions in the negative ion sources (NIS) are not so far from those met in the divertor region (DR) so that spectroscopic and kinetic methods studied in NIS can be extended to understand DR properties. We refer in particular to the spectroscopic methods to measure vibrational temperature and vibrational distribution function (VDF) as well as the complex kinetic models used to rationalize these measurements [START_REF] Capitelli | Open problems in the physics of volume H -/Dsources[END_REF][START_REF] Capitelli | Non-equilibrium plasma kinetics: a state-to-state approach[END_REF]. Non equilibrium vibrational distributions are controlled not only by electron molecule impact collisions i.e. the so called e-V and E-V pumping processes [START_REF] Capitelli | Vibrational kinetics, electron dynamics and elementary processes in H 2 and D 2 plasmas for negative ion production: Modelling aspects[END_REF][START_REF] Capitelli | Open problems in the physics of volume H -/Dsources[END_REF][START_REF] Capitelli | Non-equilibrium plasma kinetics: a state-to-state approach[END_REF] but also by quenching processes involving atomic species, in particular the process H + H 2 (ν, j) ↔ H + H 2 (ν , j )

In turn the reaction rates of this process demands the knowledge of the translational temperature T g of atomic hydrogen (deuterium), which in general is much higher than the corresponding one for molecular species. This indicates the importance of the atomic neutral temperature in determining the form of VDF. Theoretical and experimental line widths of atomic H can be used to get information on T g , being the line broadening controlled by electron atom collisions [START_REF] D'ammando | Modelling spectral properties of non-equilibrium atomic hydrogen plasma[END_REF] and electron dissociativeexcitation processes [START_REF] Celiberto | Electron impact direct dissociation processes of vibrationally excited H 2 molecules to excited atomic hydrogen H*(n=1-5).i. cross sections[END_REF][START_REF] Celiberto | Electron impact direct dissociation processes of vibrationally excited H 2 molecules to excited atomic hydrogen H*(n=1-5).ii. translational energy distribution functions of dissociation products[END_REF][START_REF] Tanaka | Hydrogen molecular effects on the Hα spectrum in low-temperature plasmas[END_REF].

Atomic emission spectroscopy is still largely used to get information about the electronic distribution of excited states yielding information on the electron temperature. Also in this case one should accompany the spectroscopic measurements with suitable collisional radiative (CR) models which take into account collisions of free electrons with atoms and molecules [START_REF] Colonna | Coupled solution of a time-dependent collisionalradiative model and boltzmann equation for atomic hydrogen plasmas: possible implication with libs plasmas[END_REF][START_REF] Fujimoto | Ratio of balmer line intensities resulting from dissociative excitation of molecular hydrogen in an ionizing plasma[END_REF]. It should be noted that under many situations the electron energy distribution function (EEDF) can deviate from the Maxwell one implying the necessity of solving an appropriate Boltzmann equation for EEDF [START_REF] Capitelli | Vibrational kinetics, electron dynamics and elementary processes in H 2 and D 2 plasmas for negative ion production: Modelling aspects[END_REF][START_REF] Capitelli | Open problems in the physics of volume H -/Dsources[END_REF][START_REF] Colonna | Coupled solution of a time-dependent collisionalradiative model and boltzmann equation for atomic hydrogen plasmas: possible implication with libs plasmas[END_REF]. These considerations clearly reaffirm the strong linking between spectroscopic measurements and atomic and molecular physics, the last important for the construction of kinetic models used to deconvolute the experimental observations. The aim of this paper is indeed to present some study-cases which can emphasize this linking with particular attention on the importance of vibrationally excited molecules.

The paper is divided into different sections. Section 2 deals with the measurement of vibrational and rotational temperatures by CARS (Coherent AntiStokes Raman Spectroscopy) and by Vacuum Ultra Violet (VUV) absorption spectroscopy, as well as the kinetic models used to razionalize the experimental results. Section 3 reports emission involving Fulcher system in different plasma conditions (hollowchatode discharges and parallel plate RF reactors). Section 4 reports the influence of vibrational excitation on the translational distribution functions of atomic hydrogen and protons. Section 5 shows results of a collisional radiative model, obtained by coupling the kinetics of excited states and Boltzmann equation for EEDF.

CARS spectroscopy

CARS spectroscopy is probably the most used method for getting information on vibrational and rotational temperatures of H 2 , D 2 plasmas under different conditions including DC discharges [START_REF] Nibler | CARS measurement of vibrational temperatures in electric discharges[END_REF][START_REF] Shirley | Vibrational excitation in H 2 and D 2 electric discharges[END_REF], multipole magnetic [START_REF] Péalat | Rovibrational molecular populations, atoms, and negative ions in H 2 and D 2 magnetic multicusp discharges[END_REF] and RF plasmas [START_REF] Hassouni | Nonequilibrium vibrational excitation of H 2 in radiofrequency discharges: A theoretical approach based on coherent anti-stokes raman spectroscopy measurements[END_REF][START_REF] Shakhatov | Measurement of vibrational, gas, and rotational temperatures of H 2 (X 1 Σ + g ) in radio frequency inductive discharge plasma by multiplex coherent anti-stokes raman scattering spectroscopy technique[END_REF]. In this paper we report a sample of results recently obtained by our group under RF inductive plasma conditions [START_REF] Hassouni | Nonequilibrium vibrational excitation of H 2 in radiofrequency discharges: A theoretical approach based on coherent anti-stokes raman spectroscopy measurements[END_REF][START_REF] Shakhatov | Measurement of vibrational, gas, and rotational temperatures of H 2 (X 1 Σ + g ) in radio frequency inductive discharge plasma by multiplex coherent anti-stokes raman scattering spectroscopy technique[END_REF] as well as the theoretical model used for rationalizing the results.

The experimental set-up included the RF inductive discharge plasma reactor and vacuum chamber for CARS technique calibration, a system for gas feeding and pumping out, a pressure controlling system, the CARS system with optics to recombine and focus the pump laser beams for generating the CARS spectrum, an automatic system for data acquisition and processing [START_REF] Shakhatov | Measurement of vibrational, gas, and rotational temperatures of H 2 (X 1 Σ + g ) in radio frequency inductive discharge plasma by multiplex coherent anti-stokes raman scattering spectroscopy technique[END_REF]. The longitudinal, tube-type, RF configuration (diameter 3.2 cm and 54 cm length) was designed to achieve large concentrations of vibrationally excited H 2 . Up to 5 kW of the RF power supply operating at 27 MHz was used to inductively couple the RF power into the plasma through an impedance-matching network and a multiple-turn, water-cooled copper work coil (80 mm in length and 50 mm in inner diameter). Figure 1 shows average values of the rotational temperature, T rot , and vibrational temperature, T 1→0 , as function of the total pressure in the RF inductive discharge plasma in the molecular hydrogen for various power releases. The experimental data were evaluated from the CARS spectra processing, as illustrated in [START_REF] Shakhatov | Measurement of vibrational, gas, and rotational temperatures of H 2 (X 1 Σ + g ) in radio frequency inductive discharge plasma by multiplex coherent anti-stokes raman scattering spectroscopy technique[END_REF]. These CARS spectra were taken from the probe volume positioned on the axis and in the central zone of the RF inductive coil. Figure 1. Experimental vibrational and rotational temperatures of the H 2 ground electronic state (X 1 Σ + g ) as a function of pressure, in RF inductive discharge plasma [START_REF] Shakhatov | Measurement of vibrational, gas, and rotational temperatures of H 2 (X 1 Σ + g ) in radio frequency inductive discharge plasma by multiplex coherent anti-stokes raman scattering spectroscopy technique[END_REF].

Inspection of figure 1 shows that the average values of the vibrational temperature decreases monotonically from 4 300 K to 2 800 K by increasing the total pressure from 66.6 to 1066.6 Pa (power input from 0.5 to 2 W/cm 3 ). At the same time average values of the rotational temperature increase with the pressure from 560 to 750 K. At the low rotational levels J=0-5, the relaxation rates between the rotational and translational degree of freedom of the hydrogen molecules are very high so that the measured rotational temperature T rot , can be considered equal to the gas temperature in the investigated range of the total pressure.

T vib (γ H =10 -3 ) T vib (γ H =10 -2 ) T vib (γ H =10 -1
A detailed kinetic model describing RF plasmas has been discussed in [START_REF] Hassouni | Nonequilibrium vibrational excitation of H 2 in radiofrequency discharges: A theoretical approach based on coherent anti-stokes raman spectroscopy measurements[END_REF].

Basically the model includes:

• a non equilibrium vibrational kinetics sub-model to describe the vibrational distribution function (VDF) of H 2 molecules (ν=0-14) and the dissociation through the vibrational pumping mechanism; • a collisional radiative model describing the kinetics of electronically excited states of both atomic and molecular hydrogen; • a collisional model that describes the dissociation, attachment and ionization kinetics, ion conversion and recombination processes and mutual neutralization of negative and positive ions; • a sub-model of the electron kinetics that enables to determine the electron energy distribution function (EEDF) required to estimate the rate constants for electronheavy species collisions; • a total energy equation that yields the gas temperature that govern the rate constants of the collisions that involve heavy species; • a quasi-homogeneous plasma transport model for the estimate of species and energy losses at the plasma surrounding wall.

The model has been in particular used to reproduce the experimental data for the two following discharge conditions:

(a) pressure = 133.3 Pa; injected power density = 0.5 W cm -3 ; plasma length ∼ 27 cm; radius = 1.27 cm; wall temperature = 370 K (b) pressure = 800 Pa; injected power density = 2 W cm -3 ; plasma length ∼ 10 cm; radius = 1.27 cm; wall temperature = 550 K. In both cases, the theoretical vibrational temperatures increase with the increase of γ H , due to the corresponding decrease of the concentration of atomic hydrogen in gas phase which play a fundamental role in deactivating vibrationally excited molecules. It should be noted that a satisfactory agreement between theoretical and experimental vibrational temperatures is reached for γ H =10 -2 for case (a), while a recombination factor of the order of 10 -1 is needed to reproduce the vibrational temperature at high pressure (case (b)). The difference in the recombination coefficient values can be at least partially explained by the dependence of γ H on wall temperature. The theoretical model developed in [START_REF] Hassouni | Nonequilibrium vibrational excitation of H 2 in radiofrequency discharges: A theoretical approach based on coherent anti-stokes raman spectroscopy measurements[END_REF] is also able to describe the whole VDF and EEDF, not accessible by experiments. Figure 2 a and b report VDF and EEDF, for case (a) and for different values of γ H , respectively. Inspection of the two figures show large deviations of the two distributions from the corresponding Boltzmann and Maxwell ones, which should appear as straight lines in the two plots. The strong dependence of distributions on γ H is again due to the change in the composition of the mixture of the molecular and atomic components (see [START_REF] Hassouni | Nonequilibrium vibrational excitation of H 2 in radiofrequency discharges: A theoretical approach based on coherent anti-stokes raman spectroscopy measurements[END_REF] for details). CARS spectroscopy while being essential to measure the vibrational and rotational temperatures of H 2 and D 2 can be hardly applied for measuring high vibrational levels (ν >4), molecules which play a role of paramount importance in the formation of negative H -/D -ions through dissociative attachment. To this end many techniques have been used including multi-photon ionization [START_REF] Eenshuistra | Dissociation and vibrational excitation of H 2 molecules and wall influence on the densities in a multicusp ion source[END_REF], resonant enhanced dissociative attachment [START_REF] Hall | Vibrational excitation of hydrogen via recombinative desorption of atomic hydrogen gas on a metal surface[END_REF] and VUV (vacuum ultraviolet) absorption [START_REF] Mosbach | Population dynamics of molecular hydrogen and formation of negative hydrogen ions in a magnetically confined low temperature plasma[END_REF]. The listed techniques have been in particular used for measuring H 2 non-equilibrium vibrational distributions in magnetic multipole discharges. These discharges operate at low pressure (∼0.1 Pa) and are formed by electrons emitted by filaments and accelerated by applied voltages (∼100 V). The electrons form a plasma characterized by vibrational and electron energy distributions that markedly differ from the Boltzmann and Maxwell ones. Large deviations of EEDF from the Maxwell behavior is in fact observed as can be seen from figure 3. A quasi-bi-maxwellian EEDF is found, due to the continuous inlet of high energy electrons (90 eV) in the reported case and to the subsequent thermalization of part of this electrons due to inelastic, superelastic and elastic collisions (see [START_REF] Pagano | Modeling multicusp negative ion sources[END_REF] for details).

Moreover a large plateaux in VDF can exist in these discharges as a result of the so called E-V mechanism i.e. excitation of singlet states of H 2 , D 2 and subsequent optical decay in the ground electronic state. A lot of theoretical models have been used to rationalize the experimental VDF, essentially based on the coupled solution of the vibrational and electron kinetics, similar to the previous model [START_REF] Pagano | Modeling multicusp negative ion sources[END_REF]. The satisfactory agreement between experimental (VUV absorption) and theoretical results (see figures 4 a and b) is a further indication of the possibility of using theoretical and experimental methodologies developed for cold plasma to fusion ones, a point being emphasized by fusion researchers [START_REF] Pigarov | Collisional radiative kinetics of molecular assisted recombination in edge plasmas[END_REF][START_REF] Krasheninnikov | Molecule assisted recombination (MAR): Mechanisms and plasma conditions for effective operation[END_REF]. ] experimental results [START_REF] Mosbach | Population dynamics of molecular hydrogen and formation of negative hydrogen ions in a magnetically confined low temperature plasma[END_REF] (squares) theoretical results [START_REF] Pagano | Modeling multicusp negative ion sources[END_REF].

ε(v)-ε(v=0) [eV] (b)
3. Fulcher-α Band: a diagnostic tool

Hollow-cathode glow discharge

The Fulcher-α band spectroscopy represents a valuable tool for the indirect experimental determination of the vibrational and rotational distributions of the H 2 ground electronic state, widely used under different plasma conditions spanning from coronal discharge to the diagnostic of the divertor region in controlled fusion experiments. The emission spectrum of the excited ro-vibrational levels of the d 3 Π u state (n=3) radiatively decaying to the ro-vibrational of the a 3 Σ + g state (n=2) is, in fact, conveniently located in the visible region and is characterized by a relatively high intensity. The usable lines are actually limited to the transitions allowed according to the ro-vibronic sub-level selection rules, usually selecting the Q branch originating from d 3 Π - u term (∆J=0) due to the anomalous intensity of the spectrum for P and R branches from d 3 Π + u (∆J=±1) [START_REF] Ginsburg | Intensity measurements in the molecular spectrum of hydrogen[END_REF][START_REF] Kado | On the anomalous characteristics in the P and R branches in a hydrogen Fulcher band[END_REF][START_REF] Kovács | On the intensity anomalies in the Fulcher-α bands of the H 2 molecule[END_REF], and also considering those transitions initiated from the lowest vibrational levels (ν < 4), due the observed efficient predissociation depleting the higher levels [START_REF] Xiao | Rovibrational distribution determination of H 2 in low temperature plasmas by Fulcher-α band spectroscopy[END_REF].

The derivation of the VDF in the X 1 Σ + g state, in the frame of a "coronal model" for low-density plasmas, relies on some theoretical assumptions:

• ground state VDF is Boltzmann, characterized by the temperature T vib • the d 3 Π u state is populated by excitation of ground state hydrogen molecules in collision with electrons • the population of the considered emitting levels is negligibly affected by predissociative and collisional quenching processes

VIBRATIONAL TEMPERATURE [K] ROTATIONAL TEMPERATURE [K] PRESSURE [Pa]

Figure 5. Experimental vibrational and rotational temperatures of the H 2 ground electronic state (X 1 Σ + g ) as a function of pressure, in hollow-cathode glow plasma, obtained from the intensity of the Fulcher-α band spectra, by a coronal model [START_REF] Yamasaki | Experimental evaluation of predissociation rate in ν = 4 state usable for analysis of H 2 Fulcher-α band emission[END_REF].

In figure 5 the rotational and vibrational temperatures of the ground H 2 molecule, measured in a hollow-cathode glow discharge apparatus, are reported in a wide pressure range, thus showing the weak dependence of these quantities on the neutral density [START_REF] Yamasaki | Experimental evaluation of predissociation rate in ν = 4 state usable for analysis of H 2 Fulcher-α band emission[END_REF].

However caution should be used in the analysis of the results, obtained in the hypothesis of a VDF in the ground state perfectly mirrored by the vibrational distribution of the emitting excited state. Typical electron density values in divertor region, n e ≈ 10 14 cm -3 , are considerably higher than in technical plasmas and the theoretical model must describe the interplay of different elementary processes, thus considering the different vibrational pumping/de-activation mechanisms of excited electronic states. Attempts in this direction are reported in literature [START_REF] Fantz | Spectroscopic diagnostics of the vibrational population in the ground state of H 2 and D 2 molecules[END_REF], improving the collisional radiative model by Sawada&Fujumoto [START_REF] Sawada | Effective ionization and dissociation rate coefficients of molecular hydrogen in plasma[END_REF] for hydrogen molecules and atoms. Results [START_REF] Fantz | Spectroscopic diagnostics of the vibrational population in the ground state of H 2 and D 2 molecules[END_REF] clearly indicate that while at low electron densities the pumping mechanism of the excited emitting state is dominated by the forbidden electron-impact induced excitation of the ground state, increasing n e (> 10 12 cm -3 ) the excitation is governed by transitions involving metastable excited states (a 3 Σ + g , c 3 Π u ), with intermediate regimes where both mechanisms are important in populating the emitting state [START_REF] Fantz | Atomic and molecular emission spectroscopy in low-temperature plasmas containing hydrogen and deuterium[END_REF].

These considerations justify the efforts done in the dynamical investigation on the probability of electron-H 2 collision processes initiated from electronically excited states, with particular attention to the allowed transitions (n = 2 → n = 3) between triplet terms [START_REF] Laricchiuta | Electron-impact-induced allowed transitions between triplet states of H 2[END_REF], in the perspective of a more sophisticated molecular radiative model fully-resolved in the electronic degree of freedom. As in the case of the ground-state excitation, the role of vibration in enhancing the probability of these excited-stateexcited-state transitions should be investigated. In this context the dependence on the both initial and final vibrational quantum numbers of cross sections for the process e + H 2 (a 3 Σ + g , ν) → e + H 2 (d 3 Π u , ν ) are reported in figure 6. Results, obtained within the semi-classical impact parameter method, explore the whole domain of variability of ν, while selected ν values are considered, corresponding to the levels that, not undergoing predissociation, could contribute to the emission radiating back to the a 3 Σ + g state. The relative position of the corresponding potential energy curves strongly favour the vibrational wavefunction overlap, accounting for a dominant role of diagonal state-to-state transitions, i.e. (ν= ν ), and negligible direct dissociation.

In figure 7 the excitation cross sections for the forbidden transition X 1 Σ + g → d 3 Π u , calculated in the Born-Rudge approximation, have been displayed, for selected vibrational levels of the ground state. For the ν=0 case also the partial excitation cross section resulting from the summation over the first four vibrational levels, actually contributing to the emission, is reported. The results for ν=0 are in satisfactory agreement with the available experimental [START_REF] Möhlmann | Emission cross sections of the H 2 (3p 3 Πu→ 2s 3 Σ + g ) transition for electron impact on H 2[END_REF] and theoretical [START_REF] Chung | Dissociation of the hydrogen molecule by electron impact[END_REF] data available in literature. The vibrational excitation of the target is effective only in the near threshold region, where also the peak is located, while for higher values of collision energy the process is even less favored increasing the initial vibrational quantum number. Figure 7. Excitation cross section for electron-impact induced excitation of d 3 Πu state of H 2 molecule, as a function of collision energy for different initial vibrational levels, ν, of the X 1 Σ + g state (dotted line curve represents the integral cross section obtained summing up to ν =3). A comparison with theoretical [START_REF] Chung | Dissociation of the hydrogen molecule by electron impact[END_REF] (long-dashed line) and experimental [START_REF] Möhlmann | Emission cross sections of the H 2 (3p 3 Πu→ 2s 3 Σ + g ) transition for electron impact on H 2[END_REF] (close markers) results for ν=0 level.

Parallel plate RF discharges in H 2

In view of their importance in the context of material processing (e.g. diamond films, silicon films) and electronic manufacturing (e.g. deposition of low-k dielectrics), the electric discharge plasmas generated in hydrogen-rich gas mixtures have attracted much attention. In particular, measurements have been performed on capacitively coupled discharges in pure hydrogen and hydrogen diluted methane, reporting the emission intensity of the Balmer-alpha and Fulcher lines resolved in space and time [START_REF] Tochikubo | Study of the structure of radio frequency glow discharges in CH 4 and H 2 by spatiotemporal optical emission spectroscopy[END_REF][START_REF] Leroy | Spatiotemporal analysis of the double layer formation in hydrogen radio frequency discharges[END_REF][START_REF] Mahony | Sheath dynamics observed in a 13.56 MHzdriven plasma[END_REF][START_REF] Czarnetzki | Space and time resolved electric field measurements in helium and hydrogen RF-discharges[END_REF]. A remarkable feature of the space profile of the intensity is a double maximum observed in particular close to the electrodes at the higher values of the discharge power. In [START_REF] Leroy | Spatiotemporal analysis of the double layer formation in hydrogen radio frequency discharges[END_REF] an interpretation of this feature was provided, in terms of a double layer dynamics of the sheath field, but a theoretical model was still not possible, except for a determination of ionization rate determined by simplified fluid (macroscopic) and particle-in-cell Monte Carlo (PIC-MC, kinetic) calculations. The use of kinetic models is highly valuable for these studies since they allow to take into account the space and time dependence of the eedf and its connection to the sheath dynamics through the generation and transport and charged species. Fluid models, instead, are based on a macroscopic description of the charged particle kinetics in the discharge. These models assume for the ionization coefficient, and in general for the rate coefficients of the electron/molecule reactions, values which are at most calculated by solving the so-called quasi-isotropic Boltzmann equation assuming a uniform plasma. These are later interpolated in terms of the electron energy and applied locally depending on the calculated value of the electron temperature. Fluid models are much less computationally expensive but usually give inaccurate results at the lower pressure values. In [START_REF] Mahony | Sheath dynamics observed in a 13.56 MHzdriven plasma[END_REF] additional experimental results were provided to support an interpretation of the feature as due to collision induced ion drag leading to sheath collapse, coherently with the absence of the feature in atomic gas under similar conditions. In [START_REF] Czarnetzki | Space and time resolved electric field measurements in helium and hydrogen RF-discharges[END_REF] a connection was established between the profiles of the emission intensity and those of the electric field, providing a much convincing support to the sheath dynamic interpretation on an experimental basis. However, only in [START_REF] Diomede | Effect of discharge voltage on capacitively coupled, parallel plate RF hydrogen plasmas[END_REF], for the first time, a theoretical calculation of the excitation rates of H(n=3) and H 2 (d 3 Π u ), respectively, was reported for different discharge voltages using a hybrid model, including a PIC-MC calculation of charged species dynamics (e, H + , H + 2 , H + 3 , H -) and a reaction-diffusion calculation of the VDF of H 2 (X 1 Σ + g ). Due to the low radiative lifetimes of the emitting states it was supposed that the population rate of these last is proportional to the emission intensity for both lines. A qualitative agreement was found, with a reproduction of the double maxima of the space emission profile of both lines and a semi-quantitative agreement concerning the location of these features, which was found to depend weakly on the discharge voltage when this last was high enough to allow the feature to appear. We show for the first time results based on an updated version of the model in [START_REF] Diomede | Effect of discharge voltage on capacitively coupled, parallel plate RF hydrogen plasmas[END_REF]. The improvements concern the possibility to deliver space-time plots of the population rates and the inclusion of a more consistent set of cross sections for the Balmer-α emission via H 2 (X 1 Σ + g ,ν) dissociation to H(n=3), based on the latest results in [START_REF] Celiberto | Electron-impact excitation cross sections of vibrationally excited X 1 Σ + g H 2 and D 2 molecules to rydberg states[END_REF][START_REF] Celiberto | Cross section data for electron-impact inelastic processes of vibrationally excited molecules of hydrogen and its isotopes[END_REF].

Cross sections for the process of electron-impact induced direct dissociation through the repulsive branch of two bound states, D 1 Π u and B 1 Σ + u ), both correlating to the correlating with H(1s)+H (n=3) dissociation limit, were obtained in the frame of the semi-classical impact parameter method, with resolution on the initial vibrational quantum number, ν. The vibrational excitation of the molecular target acts enhancing the process probability, showing the dependence on ν a strongly irregular behavior that follows closely the behavior of corresponding Franck-Condon density. In figures 9 a and b we report results for the excitation rate of H 2 (d 3 Π u ) as a function of space and RF phase in the same conditions in [START_REF] Diomede | Effect of discharge voltage on capacitively coupled, parallel plate RF hydrogen plasmas[END_REF] at two different voltages. In these figures the time is measured in terms of phase with respect to the RF oscillation at 13.56 MHz: the population rate increases when the sheath dynamics heats up the free electron cloud and increases the value of the eedf beyond the process threshold. The figures show the contribution of the expanding sheath (the two maxima closer to the discharge center) as well that of the collapsing sheath (closer to the electrodes), qualitatively coherent with experimental results [START_REF] Tochikubo | Study of the structure of radio frequency glow discharges in CH 4 and H 2 by spatiotemporal optical emission spectroscopy[END_REF][START_REF] Leroy | Spatiotemporal analysis of the double layer formation in hydrogen radio frequency discharges[END_REF][START_REF] Mahony | Sheath dynamics observed in a 13.56 MHzdriven plasma[END_REF]. The effect of voltage is manifest since at lower voltage the contribution of sheath collapse to excitation rate becomes negligible. In figure 10 we show the time averaged, space dependent excitation rate of H(n=3) as results from the sum of two channels (via D 1 Π u and B 1 Σ + u ) states) showing

From Cold to Fusion Plasmas: Spectroscopy, Molecular Dynamics and Kinetic Considerations12 the detailed contribution of the vibrational states of H 2 (X 1 Σ + g ). In the figure it is possible to appreciate the space dependence of the different contributions. Note that only half of the space domain is considered because of the symmetry of the plasma conditions. As expected, since the voltage is high enough to show the double layer, a double maximum is observed for all rates in the sheath region since these processes are characterized by a high threshold. In this case, which is a typical operating condition for a low pressure discharge in H 2 , the VDF is characterized by a low vibrational temperature for the 0-1 levels (T 0→1 ) due to the low electron number density (∼10 9 cm -3 ) and with a plateau for intermediate levels in the range 4-8 due to the antagonism of e+H 2 EV processes [START_REF] Longo | Modelling of capacitively coupled RF plasmas in H 2[END_REF] and H 2 +H VT processes. These features are reflected by the rate trends with ν since the cross section increasing with ν is modest, and as a consequence most of the atoms come from ground level H 2 molecules. In different systems, like inductive discharges or high pressure discharges, where the T 0→1 is usually considerably higher we expect that the consideration of the EEDF and the vibrational state-to-state determination of the dissociation rate will be a necessity to get an accurate result. These results illustrate the power of state of the art modeling applied to these technologically important systems.

Translational energy distribution functions of atomic hydrogen and protons

The vibrational excitation of molecular target acts, in general, enhancing direct dissociation and ionization processes, thus affecting the population of electronically excited H atoms and, in turn, the intensity of atomic emission lines. Modeling of non equilibrium vibrational distributions requires state-to-state cross sections resolved on the vibrational degree of freedom for different channels [START_REF] Celiberto | Electron impact direct dissociation processes of vibrationally excited H 2 molecules to excited atomic hydrogen H*(n=1-5).i. cross sections[END_REF][START_REF] Celiberto | Electron-impact excitation cross sections of vibrationally excited X 1 Σ + g H 2 and D 2 molecules to rydberg states[END_REF][START_REF] Celiberto | Cross section data for electron-impact inelastic processes of vibrationally excited molecules of hydrogen and its isotopes[END_REF].

From the previous study cases we have underlined the role of atomic hydrogen in affecting the vibrational distribution of H 2 (D 2 ) molecules. The corresponding relaxation rates depend on the translational temperature as can be understood by inspection of the corresponding rates reported in figure 11. These rates have been obtained by intensive QCT (Quasi-Classical Trajectory) calculations [START_REF] Esposito | Detailed cross section calculations of atom-molecule energy transfer processes and dissociation for hydrogen, nitrogen and oxygen[END_REF] of the best available PES (Potential Energy Surface).

The translational temperature depends on the dissociation and ionization channels, yielding hydrogen atoms and ions by the processes

e + H 2 (X 1 Σ + g , ν) → e + H 2 → e + H(1s) + H (n) e + H 2 (X 1 Σ + g , ν) → 2e + H + 2 ( 2 Σ + g , 2 Σ + u ) → 2e + H(1s) + H +
The translational energy distribution functions of hydrogen atoms, F H (E, ε), and ions, F H + (E, ε), include the effect of the microscopic collisional dynamics, through the elementary processes cross sections, σ, weighted on the vibrational distribution of the ground state

F H (E, ε) ∝ i ν N ν N tot [σ i exc (E, ν → ε) + σ i ion (E, ν → ε)] (1)
F H + (E, ε) ∝ i ν N ν N tot [σ i ion (E, ν → ε)] (2)
The cross sections for the i-th channel, function of energy of the impinging electron E, are resolved on the initial vibrational level and on the energy of the final vibrational level, belonging to the continuum, in turn related to the translation energy of atomic fragments, through the threshold energy (ε tr =ε-ε th ). In figure 12 a pictorial view is given, showing how the VDF of the ground state is mirrored in dissociative excitation processes. Tanaka et al. [START_REF] Tanaka | Hydrogen molecular effects on the Hα spectrum in low-temperature plasmas[END_REF] showed the dependence of the kinetic energy distribution of H(1s) atoms, formed in the dissociation through the repulsive b 3 Σ + u state, the main dissociative channel, on the vibrational temperature of the ground H 2 state (see figure 13). At low values of T vib excited vibrational levels are poorly populated and the distribution is characterized by a sharp peak at ∼3 eV. Increasing the vibrational temperature, corresponding to higher excited level populations, a broadening of the distribution is observed, due to the spreading of the shading area in figure 12.

More complex is the case of dissociative excitation leading to excited atomic fragments, H (n). In [START_REF] Celiberto | Electron impact direct dissociation processes of vibrationally excited H 2 molecules to excited atomic hydrogen H*(n=1-5).ii. translational energy distribution functions of dissociation products[END_REF][START_REF] Celiberto | Electron impact direct dissociative-ionization cross sections from vibrationally excited H 2 molecules and translational energy distribution functions of protons[END_REF] the distributions of H (n) arizing from non-equilibrium ground VDFs have been investigated. In figures 15 the translational energy distribution functions (TEDFs) of atoms with principal quantum numbers n =3 and 4, for two different electron density values, corresponding to non-equilbrium VDFs [START_REF] Hiskes | Generation of negative ions in tandem high-density hydrogen discharges[END_REF] (see figure 14), are reported [START_REF] Celiberto | Electron impact direct dissociation processes of vibrationally excited H 2 molecules to excited atomic hydrogen H*(n=1-5).ii. translational energy distribution functions of dissociation products[END_REF][START_REF] Celiberto | Electron impact direct dissociative-ionization cross sections from vibrationally excited H 2 molecules and translational energy distribution functions of protons[END_REF]. Results were obtained considering the cross section vibrational dependence well described by Franck-Condon density factors. Non-equilibrium vibrational distributionas a function of the vibrational quantum number for different electron densities [START_REF] Hiskes | Generation of negative ions in tandem high-density hydrogen discharges[END_REF].

The distributions are always characterized by a bimodal shape, associated to two different groups of atoms, i.e. "cold" and "hot" atoms. The slow component is due to the dissociation occurring on the repulsive branch of bound electronic states in optically allowed transitions, in fact showing a linear increase with electron energy, while fast atoms are formed in optically forbidden excitation to purely repulsive states. Vibrational excitation, populating the VDF tail, acts reducing the translational energy separation between groups. It should be noted that some processes affecting the TEDF have been neglected in this model, such as the dissociation through the doubly excited states, leading to pairs of excited atomic fragments.

Considerations can be extended to the theoretical TEDF obtained for protons (see figure 16) [START_REF] Celiberto | Electron impact direct dissociative-ionization cross sections from vibrationally excited H 2 molecules and translational energy distribution functions of protons[END_REF] and are confirmed also by experiments [START_REF] Ajello | Kinetic energy distribution of H (2p) atoms from dissociative excitation of H 2[END_REF]. Translational energy distribution function of excited atomic hydrogen, H (n), formed in electron-impact induced direct dissociation, as a function of translational energy of atoms, at electron impact energy E=50 eV for different electron densities [START_REF] Celiberto | Electron impact direct dissociation processes of vibrationally excited H 2 molecules to excited atomic hydrogen H*(n=1-5).ii. translational energy distribution functions of dissociation products[END_REF][START_REF] Celiberto | Electron impact direct dissociative-ionization cross sections from vibrationally excited H 2 molecules and translational energy distribution functions of protons[END_REF].

n =10 11 [cm -3 ] e n =10 12 [cm -3 ] e Figure 16.
Translational energy distribution function of proton, formed in electron-impact induced ionization, as a function of translational energy of H-H + pair, at electron impact energy E=50 eV for different electron densities [START_REF] Celiberto | Electron impact direct dissociative-ionization cross sections from vibrationally excited H 2 molecules and translational energy distribution functions of protons[END_REF].

From Cold to Fusion Plasmas: Spectroscopy, Molecular Dynamics and Kinetic Considerations17

Collision Radiative model of atomic hydrogen plasma

To relate the emission properties of hydrogen plasmas to non-equilibrium distributions, the CR model is solved under the quasi-stationary state approximation [START_REF] Bates | Recombination between electrons and atomic ions. i optically thin plasmas[END_REF][START_REF] Bates | Recombination and energy balance in a decaying plasma. 2. He-He + -e plasma[END_REF] (QSS-CR). This model was applied mainly to atomic systems [START_REF] Bates | Recombination between electrons and atomic ions. i optically thin plasmas[END_REF][START_REF] Bates | Recombination and energy balance in a decaying plasma. 2. He-He + -e plasma[END_REF][START_REF] Drawin | Thermodynamic properties of the equilibrium and non-equilibrium states of plasmas[END_REF][START_REF] Biberman | Kinetics of Nonequilibrium Low-Temperature Plasmas[END_REF][START_REF] Sobelmann | Excitation of atoms and brodening of spectral lines[END_REF], but in some cases it has been extended to molecular plasmas [START_REF] Fantz | Spectroscopic diagnostics of the vibrational population in the ground state of H 2 and D 2 molecules[END_REF]. The main hypothesis of QSS-CR model is that macroscopic quantities such as electron density and temperature as well as the population of the ground state, change much slower than the population of the excited states. As a consequence, the distribution of internal states is calculated by changing parametrically the slow varying quantities [START_REF] Bates | Recombination between electrons and atomic ions. i optically thin plasmas[END_REF][START_REF] Bates | Recombination and energy balance in a decaying plasma. 2. He-He + -e plasma[END_REF][START_REF] Drawin | Thermodynamic properties of the equilibrium and non-equilibrium states of plasmas[END_REF][START_REF] Biberman | Kinetics of Nonequilibrium Low-Temperature Plasmas[END_REF][START_REF] Sobelmann | Excitation of atoms and brodening of spectral lines[END_REF].

The QSS distribution depends non-linearly on the electron density, due to three body recombination, and on electron temperature, through rate coefficients of excitation and ionization by electron impact. On the other hand, the dependence on the population of the ground state is linear. As a consequence the ratio b i between the QSS density of the i-th state, n i , and the equilibrium value obtained by Saha-Boltzmann equation [START_REF] Drawin | Thermodynamic properties of the equilibrium and non-equilibrium states of plasmas[END_REF], n Saha i , can be written as

b i = n i n Saha i = r 0 i (n e , T e ) + r 1 i n 1 n Saha 1 (n e , T e ) (3)
where r 0 i , which accounts for the contribution of electron-ion recombination, is calculated neglecting the excitation from the ground state (n 1 =0), while r 1 i , due to the excitation from the ground states, do not consider the recombination processes (n H + =0). The QSS distribution is obtained by the cooperative effects of these processes with transitions from excited states due to electronic collisions and radiative decay [START_REF] Drawin | Thermodynamic properties of the equilibrium and non-equilibrium states of plasmas[END_REF][START_REF] Fujimoto | Plasma Spectroscopy[END_REF]. A given level i-th reaches the Saha equilibrium when the b i =1. Applying this condition to 3, it is possible to calculate the threshold electron density n h e such that the levels with principal quantum number i > h are in Saha-Boltzmann partial equilibrium which has been related to electron temperature by the equation [START_REF] Fujimoto | Plasma Spectroscopy[END_REF][START_REF] Griem | Principles of Plasma Spectroscopy[END_REF]

n h e ∝ T γ e h β (4)
Different authors (see for eample [START_REF] Drawin | Thermodynamic properties of the equilibrium and non-equilibrium states of plasmas[END_REF][START_REF] Fujimoto | Plasma Spectroscopy[END_REF][START_REF] Griem | Principles of Plasma Spectroscopy[END_REF]) reports different values for the coefficients γ and β, depending on the cross sections and Einstein coefficients used.

In figure 17 the normalized distributions (f QSS = ni gin H , being g i = 2i 2 the statistical weight and n H the total density of the hydrogen atoms) calculated under QSS approximation for n 1 = 10 16 cm -3 and varying T e between 1 eV-5 eV and n e between 10 12 cm -3 -10 15 cm -3 . In figure 17a the dependence of the QSS distribution on electron density has been analyzed for T e = 1 eV. For low values of n e underpopulated tails are observed, being the plasma in ionization regime. Increasing the electron density, the QSS distribution approaches the equilibrium.

Neverthless, even if the atomic excitation from the ground state becomes more effective by increasing the electron temperature (see figure 17b for n e = 10 12 cm -3 ), the distribution does not approach the equilibrium, because also the ionization is stronger, and the equilibrium composition is obtained for higher values of the electron density. In fact, increasing the T e , the population of the first excited state increases as well, but the tail shapes slightly changes relatively to the first excited states, depending mainly on the ionization-recombination disequilibrium. This effect is confirmed by 4, being the transition to equilibrium independent of the density of the ground state.

The limits of the QSS approach can arise in systems where the external conditions rapidly changes or in presence of strong space gradients. The first approaches to this problem [START_REF] Limbaugh | Validity of the quasisteady state and collisional-radiative recombination for helium plasmas. i. pure afterglows[END_REF][START_REF] Cacciatore | Population densities and ionization coefficients of fast transient hydrogen plasmas[END_REF] consider the explicit time evolution of the level population under the assumption of Maxwell electron distributions, based on the assumption of rapidly relaxing EEDF's. Recently also the effect of non-Maxwellian distributions has been investigated [START_REF] Colonna | Coupled solution of a time-dependent collisionalradiative model and boltzmann equation for atomic hydrogen plasmas: possible implication with libs plasmas[END_REF], coupling self-consistently the level kinetics as in [START_REF] Limbaugh | Validity of the quasisteady state and collisional-radiative recombination for helium plasmas. i. pure afterglows[END_REF][START_REF] Cacciatore | Population densities and ionization coefficients of fast transient hydrogen plasmas[END_REF] with the time dependent Boltzmann equation for free electrons. The model describes at the same time also the time evolution of the electron and atom density.

As test case we have considered an hydrogen plasma with the following initial conditions: n H = 10 16 cm -3 , n e = 10 12 cm -3 , T e =5 eV, T g =1 eV, T H =0.1 eV. In these conditions the electron gas tend to cool down towards the gas temperature by elastic collisions. Due to inelastic collisions, the atoms are excited, heating the level distribution. The level(a) and electron (b) distributions are reported in figure 18. It can be observed that the quasi-stationary level distribution is reached after t=10 ns. In the meanwhile, the electron energy distribution is not constant, but due to inelastic collisions, the distribution tail for energies above the first excited state is underpopulated with respect to the Maxwell, reducing the electron temperature to 3.1 eV at t=10 ns. In this calculation, the gas composition is practically unchanged, for this reason we can compare the results with the QSS calculations considering the Maxwell distribution at T e =5 eV (the initial value) and T e =3.1 eV. Due to non-maxwellian EEDF, the QSS level distributions are more than one order of magnitude higher than the time dependent solution, showing the limit of the maxwellian approximation usually considered in QSS calculation. Moreover, when the level distribution is practically constant, especially for the tail, the high energy EEDF rapidly decreases. It should be also noted that at t =10 -7 s EEDF presents a plateau in the energy range [12-20 eV] due to second-kind collisions between cold electrons and excited atomic levels (see for example [START_REF] Colonna | Coupled solution of a time-dependent collisionalradiative model and boltzmann equation for atomic hydrogen plasmas: possible implication with libs plasmas[END_REF]).

These results have been used to model the spectra of atomic hydrogen to see the effects of non-equilibrium on the emission line intensity and shape. To calculate the spectra the following processes [START_REF] D'ammando | Modelling spectral properties of non-equilibrium atomic hydrogen plasma[END_REF] have been taken into account

• Absorption and emission in atomic lines (Bound-Bound)

H(n ) + hν nn H(n), n < n (5)
• Photoionization and radiative recombination (Bound-Free)

H(n ) + hν H + + e -( ) (6)
• Bremsstrahlung and inverse Bremsstrahlung due to ions e

-( i ) + H + + hν e -( f ) + H + (7)
Line broadening of atomic hydrogen lines is caused by Stark Broadening from electrons and ions [START_REF] Griem | Principles of Plasma Spectroscopy[END_REF][START_REF] Griem | Spectral Line Broadening by Plasmas[END_REF][START_REF] Oks | Stark Broadening of Hydrogen and Hydrogenlike Spectral Lines in Plasmas: The Physical Insight[END_REF]. Free electrons can be quite generally treated in the impact approximations while for H + ions the quasistatic approximation is adequate. At low ion and electron densities other phenomena such as doppler broadening and resonance broadening may give the most important contribution to the line half width. When many different broadening effects are active, the line shape results from convolution of the elementary profiles, and the Voigt function is generally a good approximation. This level of description is accurate enough for plasma kinetic purposes, but it is probably not adequate for detailed spectroscopic analysis. In figure 19 the calculated lines profiles in the Balmer serie are compared with experimental data of Wiese,

From Cold to Fusion Plasmas: Spectroscopy, Molecular Dynamics and Kinetic Considerations19

Kelleher and Paquette [START_REF] Wiese | Detailed study of the stark broadening of balmer lines in a high density plasma[END_REF] from an experiment at atmospheric pressure. The adopted Stark broadening model overestimates the width of H β , and the Voigt function cannot reproduce the small dip at the center of H β , due to the lack of an unshifted Stark component at line center. Better agreement would require a more detailed consideration of the Stark substructure near the line center [START_REF] Griem | Spectral Line Broadening by Plasmas[END_REF][START_REF] Lee | Plasma line shapes for selected transitions in hydrogen-, helium-and lithium-like ions[END_REF][START_REF] Oks | Stark Broadening of Hydrogen and Hydrogenlike Spectral Lines in Plasmas: The Physical Insight[END_REF][START_REF] Seaton | Atomic data for opacity calculations: Xiii. line profiles for hydrogenic ions[END_REF][START_REF] Stehlé | Line shapes in hydrogen opacities[END_REF][START_REF] Vidal | Unified Theory Calculations of Stark Broadened Hydrogen Lines Including Lower Sate Interactions[END_REF]], but in this case available theories does not lead to analytical expressions for the line profile. Simulated Balmer lines are reported in figure 20, corresponding to the level distributions of figure 18 at t=10 -8 s and the QSS results at T e =3.1 eV. Moreover, comparison is also made with the spectrum obtained using a Boltzmann distribution at T H =3.1 eV. The latter is several order of magnitude larger than the two nonequilibrium calculations because it does not take into account the underpopulation of higher excited states, due to the ionizing regime. On the other hand the difference between the time-dependent and QSS calculations reflect the already discussed effect of non-equilibrium EEDF on level populations, and thus on the emissivity. Finally, due to the low electron density, Stark broadening is not very effective, resulting in very sharp line shapes. Comparison of the results from the spectral model described in [START_REF] D'ammando | Modelling spectral properties of non-equilibrium atomic hydrogen plasma[END_REF] and the experimental data of Wiese at al. [START_REF] Wiese | Detailed study of the stark broadening of balmer lines in a high density plasma[END_REF].

From Cold to Fusion Plasmas: Spectroscopy, Molecular Dynamics and Kinetic Considerations21 Wavenumber [cm -1 ] jν [erg cm -3 s -1 Hz -1 sr -1 ] 10 -13 10 -9 10 -5 10 -1 1.4⋅10 4 1.6⋅10

Conclusions

In this paper we have presented different case studies emphasizing the linking between spectroscopical measurements, atomic and molecular physics and kinetic models. Several non-equilibrium cold plasma conditions, have been discussed, including RF inductive plasma discharges, hollow cathode corona discharges, multipole magnetic discharges, RF parallel plate reactors. Different spectroscopic techniques (CARS, VUV and emission spectroscopy) have been used to measure vibrational and rotational temperatures as well as the distribution of the whole vibrational ladder. All the results emphasize the relevance of the vibrational degree of freedom, in determining the properties of the cold plasma. A particular attention is dedicated to the characterization of the VDF and its coupling a suitable Boltzmann equation to describe EEDF, either under local approximation or including non-local effects.

Simultaneously, efforts are devoted to describe the dependence of dissociativeexcitation cross sections on the vibrational quantum number of the initial state (either the ground or excited triplets). Different observations give an indication of the primary role of atomic hydrogen in relaxing the vibrational distribution of H 2 molecules so that the knowledge of the velocity distribution of atoms, ground and excited, and protons becomes essential. On the other hand, these distributions are shown, in all the reported cases, to be strongly affected by the vibrational content of molecules, which in general spreads the translational distribution function towards higher energies.

Finally a collisional radiative model for excited atomic levels is presented and discussed with particular attention on the coupling of excited state kinetics and Boltzmann equation for EEDF. This study is in particular devoted to shed light on the non equilibrium distributions for quasi-stationary and time-dependent plasma conditions.

The presented results, even if obtained for cold plasma conditions, can be extended to fusion devices as tokamak, with direct application to divertor and edge plasma. Experimental techniques (CARS, VUV, emission spectroscopy) and theoretical methods (dynamics of elementary processes, Boltzmann and PIC models, chemical and state-to-state kinetics) can be easily shared by the two communities. In conclusion we hope that the present paper can constitute a bridge for deeply connecting cold and fusion plasma communities which only in these last years start interacting.

Figure 2 .

 2 Figure 2. VDF (a) and EEDF (b), for case (a), for different γ H values [13].

Figure 3 .

 3 Figure 3. Stationary EEDF for different discharge currents, for p=1.5 Pa and V d =100 V [18].

Figure 4 .

 4 Figure 4. H 2 VDF for different values of the discharge current ((a) I d =0.5 A (b) I d =10 A), at pressure p=1.5 Pa and discharge voltage V d =100 V, (circles)experimental results[START_REF] Mosbach | Population dynamics of molecular hydrogen and formation of negative hydrogen ions in a magnetically confined low temperature plasma[END_REF] (squares) theoretical results[START_REF] Pagano | Modeling multicusp negative ion sources[END_REF].

Figure 6 .

 6 Figure 6. State-to-state cross section for electron-impact induced excitation of selected final vibrational levels ν of d 3 Πu state of H 2 molecule, as a function of collision energy for different initial vibrational levels, ν, of the a 3 Σ + g state. (a) ν =0; (b) ν =1; (c) ν =2; (d) ν =3.

Figure 8 .

 8 Figure 8. Cross section for electron-impact induced direct dissociation of H 2 molecules through excited singlet states as a function of collision energy for different initial vibrational levels. (a) state B 1 Σ + u (b) state D 1 Πu.

Figure 9 .

 9 Figure 9. Excitation rate [cm -3 s -1 ] for the H 2 (d 3 Πu) as a function of space and RF phase for a discharge with the following parameters: p=66.6 Pa, ν RF =13.56 MHz, d=3 cm for two different RF voltages: (a) 447 V and (b) 145 V.

Figure 10 .

 10 Figure 10. Time averaged excitation rate for the H(n=3) as a function of the axial position in the discharge and the vibrational quantum number of the H 2 (X 1 Σ + g ,ν) state in the same conditions of figure 9 a.

Figure 11 .

 11 Figure 11. Vibrational deactivation rate coefficients from H 2 (X 1 Σ + g ,ν=10), as a function of final vibrational quantum number, with different translational temperatures at Trot=500 K.

Figure 12 .Figure 13 .

 1213 Figure 12. Energy state diagram of singlet electronic terms for H 2 molecule. Pictorial view of the electron-impact induced direct dissociation through the repulsive branch of bound excited states.

  Figure 14.Non-equilibrium vibrational distributionas a function of the vibrational quantum number for different electron densities[START_REF] Hiskes | Generation of negative ions in tandem high-density hydrogen discharges[END_REF].

  Figure 15.Translational energy distribution function of excited atomic hydrogen, H (n), formed in electron-impact induced direct dissociation, as a function of translational energy of atoms, at electron impact energy E=50 eV for different electron densities[START_REF] Celiberto | Electron impact direct dissociation processes of vibrationally excited H 2 molecules to excited atomic hydrogen H*(n=1-5).ii. translational energy distribution functions of dissociation products[END_REF][START_REF] Celiberto | Electron impact direct dissociative-ionization cross sections from vibrationally excited H 2 molecules and translational energy distribution functions of protons[END_REF].

Figure 17 .Figure 18 .

 1718 Figure 17. Level distribution function of hydrogen atoms as a function of level energy under QSS approximation. (a) at Te=1 eV, for different electron densities; (b) at ne=10 12 cm -3 , for different electron temperatures.

Figure 20 .

 20 Figure 20. Plasma emissivity in the Balmer lines using different models to calculate the plasma composition and excited states distribution. The lines are indexed with the principal quantum number of the upper state, (a) lines from n=3 to n=6 (b) lines from n=7 to n=11.

Table 1 .

 1 Calculated T vib for different γ H values compared with experimental values for case (a) and case (b) RF plasma conditions.

Table 1

 1

	reports a comparison between theoretical and experimental vibrational
	temperatures for cases (a) and (b).
	The theoretical values have been calculated by different γ H values (taken as a
	parameter) i.e. by different values of the heterogeneous recombination coefficient of
	the process
	2H + W ↔ H 2 + W

Acknowledgments

This work has been partially supported by MIUR PRIN 2007 (2007H9S8SW 003).

References


T Vallotton

O Peyrusse
email: peyrusse@celia.u-bordeaux1.fr

D Benredjem

Density-induced continuum resonances and quasi-bound states in the collisional-radiative equilibrium of dense plasmas

Keywords: numbers: 52, 20, Hv, 52, 25, Jm

The aim of this work is to improve the treatment of density effects in non local thermodynamic equilibrium plasmas. The density effect on atomic structure (wavefunctions and energy levels) is modeled by an ion-sphere potential. The modification of the atomic potential, continuum lowering and appearance of resonances are shown. In particular, we show that the continuum resonances are linked to the electrons in the subshells passed into the continuum. Their presence determine the existence of partially bound configurations, which must be taken into account in the collisional-radiative model. We introduce in the set of rate equations a supplementary ionization process due to the plasma environment. This process (and its inverse) enters into the balance of all of the other process. It is equivalent to tunneling ionization where an outer electron located above the ionization threshold (and trapped by the potential barrier) crosses the barrier. As an application, we studied the influence of temperature and density on the average ionization and the ionic populations of a carbon plasma. We compared these calculations with the traditional method based on the chemical picture with continuum lowering.

Introduction

Theoretical studies of atomic structure and level population distribution in hot and dense plasmas are of great interest for the understanding of many physical phenomena occuring in inertial confinement fusion plasmas, X-ray sources and astrophysical plasmas. Indeed, the knowledge of the opacity, the emissivity and the ionic charges distribution is required in plasma modelling. The density effects are usually studied in hot and dense plasmas when the local thermodynamic equilibrium (LTE) approximation is justified. However, if the density is sufficiently high to affect the atomic structure but not enough to ensure the local thermodynamic equilibrium, or if the plasma time scale is such that the bound electron distribution does not thermalize, it is necessary to model the microscopics properties of the plasma by means of a collisional-radiative model.

In the most sophisticated LTE models (average atom or superconfiguration average models), the bound and free electrons which are treated on equal footing [1,2,3,4,[START_REF] Piron | High Energ[END_REF], experience a central self consistent field. The central field model assumes that each ion in the plasma is enclosed into a sphere from which the neighboring ions are excluded. The continuum density of states (DOS) obtained in these calculations exhibits resonances. Such resonances represent formerly bound states (in the low density limit) which transit into the continuum due to the density effects.

On the other hand, non-LTE (NLTE) plasmas are usually studied in the collisionalradiative framework, where the density effects are taken into account by using ad-hoc formulae for continuum lowering [6,7]. These formulae suffer from a lack of consistency and lead to an unphysical sharp suppression of the levels. D.V. Fisher et al [8,9] have proposed a more physical method based on the introduction of collectivized states in the collisional radiative model. In this method, one take into account the nearest ions to evaluate effective statistical weights. Moreover, the calculation of the rates connecting collectivized states and connecting collectivized states and free states is very difficult.

In the present work, we will rather use an approach where the resonances are taken into account by the introduction of quasi-bound configurations. For this purpose, one has to obtain free and bound electron wavefunctions which take into account the density effects. In the present study, we use the simple ion sphere model, in which the plasma is determined by its electron density, whose potential is introduced in the self-consistent calculation of the wavefunctions. A continuum density of states (DOS) is then calculated from the free wavefunctions. We then use the obtained resonances to determine quasibound configurations which are introduced in a collisional radiative system together with a new set of processes (tunnelling ionization and its inverse) which represents the partially bound character of these configurations.

In the following, we first present the self-consistent method used to calculate bound and free wavefunctions (Section 2). Section 3 describes the density effects on atomic structure and in Section 4, the inclusion of the density effects in a collisional-radiative model is discussed. Finally, the last section presents an application to a carbon plasma.

Self-consistent calculation of wavefunctions

In order to obtain the radial wavefunctions P n i l i (r) of a bound electron i in a dense plasma, we have to solve the eigenvalue equation (in the non relativistic framework for simplicity),

- 1 2
d 2 dr 2 P n i l i (r) + l i (l i + 1) 2r 2 + V i (r) + V plasma (r) P n i l i (r) = E i P n i l i (r), ( 1
)
where the energy is in Hartree, V i (r) is the usual radial potential for the isolated ion, and V plasma (r) a radial potential due to the plasma. Concerning the isolated ion potential, we use the Hx scheme [10,[START_REF] Cowan | The theory of atomic structure and spectra[END_REF]:

V i (r) = -Z r + V D (r) + V E (r)
, where V D (r) is the direct term of the electron-electron potential and V E (r) the exchange term. According to Cowan [START_REF] Cowan | The theory of atomic structure and spectra[END_REF], we write

V D (r) = q j=1 w j ∞ 0 P n j l j (r 1 ) 2 r > dr 1 (2) V E (r) = -k x f (r) ρ (r) ρ (r) + 0.5/(n i -l i ) ρ (r) ρ(r) 24ρ(r) π 1/3 , (3)
where r > =max(r, r 1 ), w j is the number of electrons in the subshell j and ρ(r) = 1 4πr 2 i w i P 2 n i l i (r). For V plasma (r) we use (as mentioned previously) the ion sphere model, in which the free electron density is assumed uniform into a sphere of radius r 0 . We assume that up to r 0 , only bound and free electrons are present, the other ions being excluded from the sphere. Outside the ion sphere the electron and ion charge distributions compensate, i.e. the electrical neutrality is locally fulfilled for r ≥ r 0 . The radius r 0 is chosen so that for the most external bound electron, the charge of the nucleus is completely screened by the bound electrons in the lower subshells and by the free electrons into the ion sphere. Therefore (see [START_REF] Massacrier | [END_REF]),

V plasma = Z-N +1 2r 0 (3 -r 2 r 0 2 ) if r < r 0 Z-N +1 r if r > r 0 (4)
where

r 0 = 3(Z -N + 1) 4πN e 1 3 , (5)
Z denotes the atomic number of the ion, N the number of bound electrons and N e the free electron density. Note that, within the ion-sphere model, it is also possible to define an average ionization depression ∆E = -3 2 Z-N +1 r 0 [START_REF] More | Atoms in Unusual situations[END_REF] so that the simplest way of taking account of the density effect consists in eliminating all the (unperturbed) levels such that

E i > χ i -∆E, (6)
where χ i is the ionization potential of level i and E i is the energy for the isolated ion. This defines the so-called "chemical picture with continuum lowering". This simple prescription will be compared below with the present approach.

In order to solve the Schrödinger equation (Eq. 1), we use a "spectral" method where the radial wavefunctions are developed on a basis of B-splines functions [START_REF] Peyrusse | [END_REF]. This method was chosen for its robustness, promptness and accuracy. For the bound states, the radial wavefunctions expansion reads

P nl (r) = Q-1 i=2 c i B k i (r), ( 7
)
where k is the order of the B-spline, Q the total number of B-splines and c i the coefficient of P nl (r) on the i th B-spline. The knownledge of a radial wavefunction reduces to the determination of all the coefficients c i . In the previous expansion, we dropped the first and the last B-spline in order to fulfill the necessary boundary conditions, namely P nl (0) = P nl (R) = 0 with R >> r 0 . For each l value, a generalized eigensystem has to be solved. It yields as many eigenvalues and eigenfunctions as basis functions. For the bound states (and the free states satisfying the same boundary conditions) the generalized eigensystem reads

Hc = εSc, ( 8
)
where H is the matrix representing the Hamiltonian in the B-spline basis, S the overlap matrix of the B-splines, and c the vector representing the coefficients c i . We have

H ij = - 1 2 rmax r min B k i (r) d 2 dr 2 B k j (r)dr + l(l + 1) 2 rmax r min B k i (r) 1 r 2 B k j (r)dr + rmax r min B k i (r)V (r)B k j (r)dr (9)
S ij = rmax r min B k i (r)B k j (r)dr. ( 10
)
The radial wavefunctions for free electrons of arbitrary energy E are obtained by solving HP El = E P El , where P El is written as an expansion on the B-splines basis (see Eq.7), and where E is known a priori. Here, the value of the wavefunction at r = R is arbitrary so there is no need to drop the last B-spline in the expansion. Instead, we set this last coefficient so that the radial wavefunction is written

P El (r) = c N N -1 j=2 cj B k j (r) + B k N (r) , (11)
where cj = c j /c N . c N will be obtained by the necessary normalization of the continuum wavefunctions. The system of linear equations becomes

N -1 j=2 (H ij -ES ij )c j = ES iN -H iN . (12)
Finally, the total DOS is written as ρ(E) = l ρ l (E) where ρ l is the partial DOS, i.e.

ρ l (E) = 2(2l + 1) r 0 0 |P E,l (r)| 2 dr. ( 13
)
3. Density effects on electronic structure

Modification of orbital wavefunctions, formation of resonances

The plasma potential introduces a dependence on electron density N e for the orbitals and eigenvalues. Figure 1 shows the radial wavefunction 3d of the ground configuration [Ar]3d 2 of Fe VII. One can see a noticeable increasing of the radial extension at the density N e = 1.1 × 10 24 cm -3 . For excited orbitals such as 4s, the effect becomes .

noticeable for lower densities, as shown in figure 2. In all cases, the perturbation by the free electrons causes an extension of the wavefunction outside the ion cell. When the density is high enough the last occupied subshell moves into the continuum. For instance, in the case of the ion Fe VII in its ground configuration, the density threshold is 1.288 × 10 24 cm -3 . More precisely, for a density above this threshold, the self consistent solution of the eigensystem does not give a negative eigenvalue for the 3d orbital. In other words, the 3d subshell has moved into the continuum. Without wavefunction for this subshell, we cannot keep the associated configuration in the rate equation system. However, this configuration does not suddenly disappear. Indeed, if we plot the potential seen by a d electron for a density just above the threshold density (see figure 3), we observe a potential barrier. This suggests the existence of quasi-bound states with energies below this barrier. The total density of states (DOS) in the continuum of the ground configuration of Fe VII, for an electron density of 1.7 × 10 24 cm -3 , exhibits a strong resonance corresponding to these quasi-bound orbitals (see figure 4). We also plot the total and the partial DOS obtained, as well as the DOS of the free electron gas. The resonance is located at an energy of 0.9 Rydberg. At the same time, the maximum energy of the potential barrier goes up to 1.8 Rydberg (Figure 3). The free electrons states located in the resonance c orrespond to trapped electrons. The partial DOS associated with l = 2 largely exceeds all the partial DOS (as seen in figure 4) and the resonance clearly represents a d orbital. As the 3d orbital is the subshell d just passed in the continuum, it is represented by this resonance. Generally, one can say that a resonance observed in the partial DOS associated with a given l, represents the first l subshell passed into the continuum.

In order to calculate self-consistently the eigenfunctions and eigenvalues of these quasi-bound electrons, we choose to take the localized part of the free wavefunctions obtained at the maximum of the resonance, as the quasi-bound orbitals. In other words, we only consider the bound part of the wavefunction for the resonant electrons. This allows us to define partially bound configurations in the collisional-radiative system and to calculate their configuration-average energies similarly to the bound configurations. The quasi-ionized (resonant) subshells are then treated similarly to bound subshells. In figure 5, we show the free wavefunction p of the 3d4p configuration in Fe VII calculated for an energy located at the maximum of the resonance and for N e = 6.7 × 10 23 cm -3 . As expected, the radial wavefunction behaves like a bound wavefunction for small values of r, and like a free oscillating wavefunction for large r values. The bound part of the orbital is associated with the wavefunction located in the small r region. This bound part can be obtained by connecting continuously the calculated wavefunction with an exponentially decreasing function outside the ionic sphere. Such a procedure (which must also preserves the right number of nodes) is illustrated in Fig. 5 (solid line). By doing so, we are able to obtain localized wavefunctions (renormalized in the ion sphere) for the quasi-bound electrons. This will allow us to consider these electrons in configuration-average energy calculations.

Configuration-average energies and quasi-bound configurations

Following Cowan [START_REF] Cowan | The theory of atomic structure and spectra[END_REF], the configuration-average energy E ion C of a configuration (n α l α ) wα (n β l β ) w β ..., where w is the population of the nl subshells reads

E ion C = q α=1 w α E α k + E α n + 1 2 (w α -1)E αα + 1 2 q β =α w β E αβ , (14)
where the sum runs over the occupied subshells in the configuration, E α k is the kinetic energy, E α n the electron-nucleus interaction energy, and E αα the electron-electron interaction energy. The radial wavefunction involved in Eq.13 must account for the density effects. Furthermore, the interaction of the ion with the plasma gives an energy

E plasma

C

which is expressed as

E plasma C = q α=1 w α ∞ 0 P i (r)V plasma (r)P i (r)dr. ( 15
)
The configuration-average energy of a configuration in a dense plasma is then It is worth stressing that E ion C and E plasma C have to be calculated even for configurations having a quasi bound orbital which exists only as a resonance in the DOS. So, as mentioned above, we consider the wavefunction at the maximum of the resonance, and then we extract its localized part. In Fig. 6, we show the configurationaverage energy of the ground configuration of Fe VII as a function of the electron density. The solid line corresponds to a fully bound configuration while the dotted line corresponds to a quasi-bound part where the 3d orbital lies into the continuum.

E C = E ion C + E plasma C . (16)
We note that for increasing densities the configuration-average energy varies as a + b N 1/3 e i.e. similarly to the binding energy of atomic orbitals in the ion-sphere model. It is important to note that there is no discontinuity between the two parts of the configuration-average energy. Our approach is now applied to a large set of configurations in various ions stages. Figure 7 shows configuration-average energies of carbon ions for various plasma densities. The bottom curve (solid line) represents the energies of C I to C VI, with 15 configurations per ion (in each charge-state, the excited configurations are obtained by promoting sequentially one electron from the ground configuration). The energies are calculated within the isolated ion approximation. The others curves are obtained for three increasing densities.

Density modified energies and rates in the collisional radiative model

Variation of the configuration-average energies

The most excited configurations (in each ion) shift into the continuum when the density increases. Among these configurations some remain quasi-bound while others are purely deleted. The latter correspond to the dilution (in the DOS) of the resonances associated with their outer orbitals. An increasing number of electronic configurations are permanently removed from the collisional radiative model when the density increases.

Density modified rates of the collisional-radiative processes

In the central field approximation used in this study and in the configuration-average context, the collisional and radiative rates depend on the configuration-average energies and on the orbital wavefunctions. Both are affected by the density, and we have to account for the corresponding variation of the transition rates.

Concerning the radiative decay (E1) from a configuration C ((n 1 l 1 ) w 1 (n 2 l 2 ) w 2 ) to a configuration C ((n 1 l 1 ) w 1 +1 (n 2 l 2 ) w 2 -1 ), we use the following rate (see [15]):

A CC = k(E C -E C ) 3 w 1 (g 2 -w 2 ) g 1 g 2 2[l 1 , l 2 ] l 1 1 l 2 0 0 0 2 P 2 12 , (17)
where

[l 1 , l 2 ] = (2l 1 + 1)(2l 2 + 1), P 1,2 = ∞ 0 P n 1 l 1 (r)rP n 2 l 2 (r)dr, g i = 2(2l i + 1
). k = 2.677 × 10 9 , if the energy is in Rydberg and the rate A CC in s -1 . E C , E C and P 1,2 account for density effects.

Because quantum calculations of collisional rates are time consuming, we have used in this study semi-empirical formulae. For collisional excitation, we have used the Van-Regemorter formulae [16]:

K C C = N e × 3.2 × 10 -7 E H T e 3/2 f C→C e -y y G(y), (18)
where y = E C -E C kT e and E H =13.6 eV. G is an average Gaunt factor for which we have taken the generalized formulation of Mewe [18], G(y) = 0.15 + 028e y E 1 (y) where E 1 (y) is the exponential integral. The rate of the inverse process, namely the collisional deexcitation is obtained by detailed balance. For collisional ionization, the simple Lotz formulation [17] gives the following rate

S CC = 3 × 10 6 N e w nl T -3/2 e E 1 (y) y . ( 19
)
The rate of the inverse process, namely the 3-body recombination is obtained by detailed balance. Finally, the radiative recombination rate is given by the Seaton formula [19]:

R CC = 5.2 × 10 -14 N e y 1.5 e y E 1 (y)Z V ac(nl) ( 2 0 )
where V ac(nl) is the vacancy in the nl subshell. For these processes, the modifications induced by the density are taken into account only through the configuration-average energies.

Tunneling ionization and its inverse

We have to account for the partially bound nature of the quasi-bound configurations.

In the example of figure 3, the potential clearly shows that an electron in a quasibound orbital can leave an ion by tunneling through the barrier. It is then necessary to introduce a new process in the rate equation system, specific to these configurations, which account for their partially bound nature.

It is well known that a discrete state interacting with continuum states of close energies leads to a decay of the discrete state. The corresponding spectrum has a lorentzian shape whose width is related to the decay rate. Moreover, this rate, called selfionization rate or tunneling rate, is related to the width of the resonance in the partial DOS. The width of such resonances (as discussed in the previous section) provides a transition rate to be included in the collisional-radiative equations. The rate of ionization by tunneling reads:

τ t.i. = 2Γ/ (in s -1 ), ( 21
)
where Γ is the FWHM of the resonance in the partial DOS and in τ t.i. , the index t.i. stands for tunneling ionization.

The inverse process of capture of an electron inside the barrier must be also taken into account. Let us apply the microreversibility principle to the tunneling ionization process. If we consider a configuration C linked by tunneling to a configuration C with the rate τ t.i. , the rate τ capt of the inverse process is given by

P C τ t.i. = P C τ capt , (22)
where P C and P C are the respective LTE probabilities of existence of the two configurations. Having

P C = g C e -(E C -µQ C )/kTe P C g C e -(E C -µQ C )/kTe
, where g C , Q C tare the statistical weight and the number of electrons of the configuration C respectively. µ denotes the chemical potential of the free electrons. Then, we can write the rate of electron capture through the potential barrier as

τ capt = g C g C e -(E C -E C -µ)/kTe τ t.i. . (23)

Average ion charge and ionic density

Carbon plasma

In this section, we focus on a carbon plasma. The advantage of such a low Z element allows us to deal with a moderate number of configurations and to neglect some 1. Average ionization of a carbon plasma as a function of Te and for different densities. The temperature is in eV and the electron density in cm -3 .

N e = 10
processes that make the calculations more complex in heavier elements. For example, autoionisation and resonant capture which both involve doubly excited configurations, can be neglected (at least when one deals with the ionization state of the plasma). We restrict the collisional radiative system to the ground configurations and one-electron excited configurations. The configuration interaction effect plays a small role and can then be ignored. The relevant configurations considered in neutral carbon are 1s 2 2s 2 2p 2 (ground configuration), 1s 2. Average ionization of a neon plasma as a function of density for T e = 25eV . The electron density is in cm -3 .

In Table 1 we give the average ionization Z of a carbon plasma for various temperatures and densities. Here, the plasma density effects are ignored. For comparison, we give in Table 2 the Z value in a neon plasma for an electron temperature of 25 eV. These results are in agreement with other calculations ( [START_REF] Rubiano | High Energ[END_REF][START_REF] Colgan | High Energ[END_REF]22]).

The process of tunneling ionization and its inverse as well as the energies and the modified rates discussed in previous sections are now included in the rate-equation system. In figure 8, we show the average ionization of a carbon plasma, with and without plasma density effects, which can be compared with a third curve showing the average ionization obtained with the chemical picture with continuum lowering (i.e. by eliminating unperturbed levels according to Eq. 6). The comparison of the two first curves show that the pressure ionization tends to increase Z. The Z values converge for high temperatures (≥ 200 eV). At moderate temperature, many partially bound configurations are populated, and the tunneling ionization and its inverse process play an important role, the effect on Z is important. For higher temperatures, the ions are highly ionized and the tunneling ionization has a small influence on the Z value. For T e ≈ 16 eV there is a crossing of the two curves (see the inset in Fig. 8). Below this value, the density effects reduce the Z value. Comparison with the last curve (dots) shows that the simple removal of levels gives a too strong effect for T e < 25eV . Furthermore, there is no crossing with the solid curve (no density effect). The crossing is clearly an effect due to the two new processes introduced in our approach. It can be explained by the importance of the electron capture (over the barrier) with respect to the tunneling ionization in our rate equation system. The tunneling ionization alone always yields higher Z values in comparison with calculations performed without density effects (see Fig. 9). Figure 10 shows the ratio τ capt /τ t.i. for different couples of configurations, as a function of T e . For low temperatures, this ratio is larger than 1, i.e. the process of electron capture through a barrier of potential dominates the tunneling ionization process. The ratio decreases with temperature and falls below 1 for temperatures above 50 eV.

In figure 11 we show the population densities of the various ion species. We note that the plasma density effects mainly affect the two less ionized charge states. For temperatures below 20 eV, the population of neutral carbon atoms is increased if one takes into account the plasma density effects. For such a temperature range the electron capture dominates the tunneling ionization. In that sense, on can speak of over-barrier recombination. Above 20 eV, the population of the C I species decreases because the tunneling ionization dominates. For the ions C II, the temperature range where the population is increased by electron capture is much smaller. The population of C II increases very slightly only for T e < 6 eV. For other charge states these effects are less important. There is in fact less quasi-bound configurations so the tunneling ionization plays a smaller role.

In Figure 12 we present Z as a function of temperature for other densities. If the plasma is dense enough to move all the configurations of a charge state into the continuum and to strongly broaden the associated continuum resonances, the corresponding charge state disappears. We can see that for an electron density of 6.7 × 10 24 cm -3 , Z remains above 1 and is nearly constant for temperatures below 50 eV. Here again, we can see that the simple suppression of unperturbed levels gives a too strong ionization for N e > 10 21 cm -3 and at low temperature. This is because too many levels move into the continuum (for N e = 6.7 × 10 24 cm -3 all the levels of the four first charge states are lost). On the contrary, for N e = 6.7 × 10 21 cm -3 , the density is too weak to affect enough levels, the obtained ionization is then nearly the same than the ionization calculated without density effects.

Conclusion

We have obtained orbitals for bound and free electrons in a dense plasma in the framework of a typical plasma model potential, namely the ion sphere model. In this kind of central field model, high-lying states move into the continuum as resonances, which are the trace of formerly bound electrons in the low density limit. These resonances allow us to define quasi-bound orbitals which are included in the definition of quasi-bound configurations. These quasi-bound configurations can still be taken into account in the collisional-radiative system. We have added the tunneling ionization and its inverse process of capture. Our study uses a collisional radiative model which integrates these configurations. The obtained results differ from results obtained by mean of a simple suppression of levels. It is shown that the inclusion of the over barrier capture effect can lead to a significant recombination in some conditions.

The generalization to heavier elements would necessitate a larger number of configurations. The introduction of two-electron excited configurations is also necessary to account for processes such as autoionization and dielectronic capture. This would improve the treatment of the transition between the NLTE and LTE regimes at high densities. Furthermore, the results of this study may serve to realign more precisely, heuristic treatments of pressure ionization in collisional-radiative models.

Figure 1 .

 1 Figure 1. 3d radial wavefunction of the ground configuration of Fe VII at N e = 1.1 × 10 24 cm -3 . Solid line: isolated ion, dashed line: perturbed ion. r 0 is the ion sphere radius.

Figure 2 .

 2 Figure 2. 4s radial wavefunction of the excited configuration 1s 2 2s 2 3s 2 3p 6 3d4s of Fe VII at N e = 3.3 × 10 23 cm -3 . Legends as in figure 1

Figure 3 .

 3 Figure 3. Potential experienced by a 3d electron of Fe VII for N e = 1.7 × 10 24 cm -3 . The horizontal line represents the localization of the maximum of the resonance for a 3d electron.

Figure 4 .

 4 Figure 4. Total and partial DOS for the ground confiiguration of Fe VII. N e = 1.7 × 10 24 cm -3 .

Figure 5 .

 5 Figure 5. Radial wavefunction of Fe VII in the 3d4p configuration at N e = 6.7 × 10 23 cm -3 . Solid line: localized; dashed line: non localized.

Figure 6 .

 6 Figure 6. configuration-average energy of [Ar]3d 2 in Fe VII, as a function of the electron density.

Figure 7 .

 7 Figure 7. Configuration-average energy of carbon ion. Each cross represents a specific configuration

Figure 8 .

 8 Figure 8. Average ionization of a carbon plasma as a function of temperature for Ne = 6.7 × 10 22 cm -3 . Solid line: without density effects. Dashed line: with density effects (present approach). Dotted line: chemical picture with continuum lowering.

Figure 9 .

 9 Figure 9. Average ion charge of a carbon plasma as a function of temperature, for Ne = 6.7 × 10 22 cm -3 .

Figure 10 .

 10 Figure 10. Ratio τ capt /τ t.i. for different electronic configurations as a function of T e.

Figure 11 .

 11 Figure 11. Population densities of carbon ions as a function of temperature (in eV). Ne = 6.7 × 10 22 cm -3 . Dashed lines: with density effects, solid lines: without density effects.

Figure 12 .

 12 Figure 12. Z as a function of temperature and for different densities in a carbon plasma. Dashed lines: with density effects (present approach), dotted lines: chemical picture with continuum lowering, solid lines: without density effects.

Table

  13 N e = 10 15 N e = 10 17 N e = 1019

	T e	Z	T e	Z	T e	Z	T e	Z
	3 1.498 3 1.705 3 1.906 3 1.094
	5 2.130 5 2.174 5 2.805 5 2.245
	7 2.823 7 2.865 7 3.345 7 3.066
	10 3.474 10 3.519 10 3.911 10 3.813

Table

  2 2s 2 2p3s, 1s 2 2s 2 2p3p, 1s 2 2s 2 2p3d, etc..

	N e 10 14	10 15	10 16	10 17	10 18	10 19	10 20	10 21	10 22	10 23
	Z 4.954 4.962 5.006 5.210 5.694 6.002 5.8926 4.941 2.912 1.373