Role of the excited electronic states in the ionization of ambient air by a nanosecond discharge N. Minesi 1 , P. Mariotto 2 , G. D. Stancu 3 and C. O. Laux 4 EM2C, CNRS, CentraleSupélec, Université Paris Saclay, 3 Rue Joliot-Curie, 91190 Gif-sur-Yvette, France

In this paper, the mechanism of air ionization by a single nanosecond discharge under atmospheric conditions is studied by numerical simulations. The plasma kinetics is solved with ZDPlasKin and the electron energy distribution function is calculated with BOLSIG+. The originality of the model is to consider not only the excited electronic states of N2, but also the excited electronic states of O and N. These states are shown to have a primary importance in the ionization of the plasma for ne > 10 17 cm -3 . It is shown that a non-equilibrium plasma (Te > Tgas) at ne = 10 17 cm -3 can reach full ionization and thermalization (Te = Tgas ≈ 3 eV, ne ≈ 10 19 cm -3 ) in less than half a nanosecond under a field usually encountered in nanosecond discharges.

I. Nomenclature

II. Introduction

Nanosecond Repetitively Pulsed (NRP) discharges are currently studied for their potential application such as plasma assisted combustion [START_REF] Ju | Plasma Assisted Combustion: Dynamics and Chemistry[END_REF][START_REF] Pilla | Stabilization of a Turbulent Premixed Flame Using a Nanosecond Repetitively Pulsed Plasma[END_REF][START_REF] Pilla | Etude Expérimentale de La Stabilisation de Flammes Propane-Air de Prémélange Par Décharges Nanosecondes Impulsionnelles Répétitives[END_REF][START_REF] Barbosa | Influence of Nanosecond Repetitively Pulsed Discharges on the Stability of a Swirled Propane/Air Burner Representative of an Aeronautical Combustor[END_REF], control of thermo-acoustic instabilities [START_REF] Lacoste | Effect of Nanosecond Repetitively Pulsed Discharges on the Dynamics of a Swirl-Stabilized Lean Premixed Flame[END_REF] and ignition [START_REF] Pancheshnyi | Ignition of Propane-Air Mixtures by a Repetitively Pulsed Nanosecond Discharge[END_REF][START_REF] Boumehdi | Ignition of Methane-and n-Butane-Containing Mixtures at High Pressures by Pulsed Nanosecond Discharge[END_REF]. The knowledge of plasma properties prior to their introduction in industrial devices is important to predict their performances. NRP discharges have been known for being weakly ionized and out of equilibrium in preheated air [START_REF] Pai | Nanosecond Repetitively Pulsed Discharges in Air at Atmospheric Pressure -The Glow Regime[END_REF][START_REF] Pai | Transitions between Corona, Glow, and Spark Regimes of Nanosecond Repetitively Pulsed Discharges in Air at Atmospheric Pressure[END_REF][START_REF] Pai | Nanosecond Repetitively Pulsed Discharges in Air at Atmospheric Pressure-the Spark Regime[END_REF][START_REF] Rusterholtz | Ultrafast Heating and Oxygen Dissociation in Atmospheric Pressure Air by Nanosecond Repetitively Pulsed Discharges[END_REF]. The different states of non-equilibrium NRP plasmas (corona, glow and spark) have been described in [START_REF] Pai | Transitions between Corona, Glow, and Spark Regimes of Nanosecond Repetitively Pulsed Discharges in Air at Atmospheric Pressure[END_REF]. Recently, the study of these discharges at ambient conditions has shown that NRP discharges can also be fully ionized. In [START_REF] Parkevich | Investigation of a Near-Electrode Plasma Formed in the Atmospheric Discharge with Employment of Picosecond Laser Probing[END_REF][START_REF] Parkevich | Anode Plasma Formation at the Initial Stage of a Nanosecond Air Discharge[END_REF], Parkevitch et al. showed by laser interferometry that approximately one nanosecond after a 25-kV voltage rise, a fully-ionized plasma is formed near the cathode surface in a pin-to-plane configuration. Moreover, they observed the formation of "clots" of dense plasma with electron number densities greater than 10 20 cm -3 . The typical dimension of these clots was around 1 μm. It was supposed that these clots arise from the explosion of the electrode material. In Ref. [START_REF] Orrière | Ionization and Recombination in Nanosecond Repetitively Pulsed Microplasmas in Air at Atmospheric Pressure[END_REF], Orriere et al. studied the formation of a plasma during a 2.5 kV pulse of 10 ns in ambient air. The distance between the electrodes was 200 μm, and the maximum reduced field E/N was 600 Td (1 Td being equal to a field E = 2.5 × 10 25 V/m at P = 1 atm and T = 300 K). The authors measured the electron number density by Stark broadening of H, O and N atomic lines and measured a maximum density of 10 19 cm -3 . They also showed that due to the extremely high temperature (Te > 30,000 K) of the plasma after the discharge, the electrons recombine with O + and N + (instead of N2 + or NO + as is the case in non-equilibrium plasmas [START_REF] Laux | Ionization Mechanisms in Two-Temperature Air Plasmas[END_REF]).

Stark broadening was also used for electron number density measurement in several works in pin-to-pin or pinto-plane configuration [START_REF] Lo | Streamer-to-Spark Transition Initiated by a Nanosecond Overvoltage Pulsed Discharge in Air[END_REF][START_REF] Minesi | On the Arc Transition Mechanism in Nanosecond Air Discharges[END_REF][START_REF] Sainct | Etude de La Réactivité de Décharges Électriques Nanoseconde à La Pression Atmosphérique Dans La Vapeur d'eau[END_REF][START_REF] Houpt | Charge Transfer in Constricted Form of Surface Barrier Discharge at Atmospheric Pressure[END_REF] and all authors reported electron number densities higher than 10 18 cm -3 . Such a high electron number density was also reported in SDBD discharges [START_REF] Shcherbanev | Optical Emission Spectrum of Filamentary Nanosecond Surface Dielectric Barrier Discharge[END_REF][START_REF] Shcherbanev | Filamentary Nanosecond Surface Dielectric Barrier Discharge. Plasma Properties in the Filaments[END_REF][START_REF] Stepanyan | A Nanosecond Surface Dielectric Barrier Discharge in Air at High Pressures and Different Polarities of Applied Pulses: Transition to Filamentary Mode[END_REF]. The authors demonstrated that the increase of pressure can be responsible for the transition from a weakly ionized plasma (called "streamer") to a fully-ionized plasma (called "filament").

Simulations of partially ionized plasmas have already been performed [START_REF] Macheret | Modeling of Air Plasma Generation by Repetitive High-Voltage Nanosecond Pulses[END_REF] with the kinetic mechanism of Kossyi et al. [START_REF] Kossyi | Kinetic Scheme of the Non-Equilibrium Discharge in Nitrogen-Oxygen Mixtures[END_REF] or Capitelli et al. [START_REF] Capitelli | Plasma Kinetics in Atmospheric Gases[END_REF] which are well suited for simulations of non-equilibrium plasmas. However, there is yet no chemical model describing the mechanism of full ionization by nanosecond discharges in air. Recombination rate coefficient (cm 6 s -1

)

T e (K)

III. General view of the chemical mechanism

Numerical simulations of the plasma kinetics under a given reduced field are performed using ZDPlasKin [START_REF] Pancheshnyi | Computer Code ZDPlasKin[END_REF].

The kinetic model is based on a ready-to-use mechanism published by Flitti and Pancheshnyi [START_REF] Flitti | Gas Heating in Fast Pulsed Discharges in N 2 -O 2 Mixtures[END_REF] based on the work of Capitelli et al. [START_REF] Capitelli | Plasma Kinetics in Atmospheric Gases[END_REF]. This mechanism is intended to simulate nanosecond discharges out of equilibrium (i.e. Tgas ≪ Te). The Electron Energy Distribution Function (EEDF) is calculated by the Boltzmann equation solver BOLSIG+ [START_REF] Hagelaar | Solving the Boltzmann Equation to Obtain Electron Transport Coefficients and Rate Coefficients for Fluid Models[END_REF]. The cross-sections of electron reactions are used to calculate the corresponding reaction rates and are taken mostly from the LXCat database [33][34][35][36]. Because excited states of molecular nitrogen can play a role in the ionization process [START_REF] Shcherbanev | Filamentary Nanosecond Surface Dielectric Barrier Discharge. Plasma Properties in the Filaments[END_REF], the ionization cross-sections of N 2 * are added, based on the work of Bacri and Medani [START_REF] Bacri | Electron Diatomic Molecule Weighted Total Cross Section Calculation. III. Main Inelastic Processes for N2 and N+ 2[END_REF]. The aim of this work being to simulate a fully ionized plasma, the ionization of atomic species and their reverse process must be included.

𝐴 + 𝑒 ⇌ 𝐴 + 𝑒

Reac. 1

For electronic transitions between the states i and j of species A, Reac. 1, the BOLSIG+ solver calculates forward and backward rate coefficients given the excitation cross-section and the ratio of the state degeneracies. This calculation is based on the detailed balance principle. In the case of ionization, Reac. 2, the backward rate coefficient is obtained by Eq. 1.

𝐴 + 𝑒 ⇌ 𝐴 + 𝑒 + 𝑒 Reac. 2

𝑘 𝑘 = 𝑍 𝑍 𝑍 𝑉 exp - I 𝑘 𝑇
Eq. 1

According to the law of mass action, the ratio of the forward and backward reaction rate coefficients of Reac. 2 is equal to the ratio of the partition functions divided by the volume, Eq. 1. It is assumed that the EEDF is Maxwellian (use of Te) and that excited states of species A follow a Boltzmann distribution. Three-body recombination rates computed in various conditions are available in Ref. [START_REF] Biberman | Kinetics of Impact-Radiation Ionization and Recombination[END_REF][START_REF] Bourdon | Three-Body Recombination Rate of Atomic Nitrogen in Low-Pressure Plasma Flows[END_REF][START_REF] Park | Comparison of Electron and Electronic Temperatures in Recombining Nozzle Flow of Ionized Nitrogen-Hydrogen Mixture. Part 2. Experiment[END_REF][START_REF] Annaloro | Global Rate Coefficients for Ionization and Recombination of Carbon, Nitrogen, Oxygen, and Argon[END_REF]. A recombination rate coefficient valid for any ion is calculated by Bibermann et al. in Ref. [START_REF] Biberman | Kinetics of Nonequilibrium Low-Temperature Plasmas[END_REF]. This coefficient is proportional to 𝑇 . and said to be valid if Te is smaller than the energy of the first excited state of the considered atom. This rate coefficient is later reported by Capitelli et al. and its validity is limited to 2000 K [START_REF] Capitelli | Plasma Kinetics in Atmospheric Gases[END_REF]. Its evolution is plotted in Fig. 1. Bibermann et al. also calculated the N + recombination rate coefficients at several temperatures (reported in Fig. 1). The deviation between the 𝑇 . law and the N + recombination rate coefficient begins at 𝑇 ~2000 𝐾 which justifies the more restrictive limit of Capitelli et al. A review of other N + three-body recombination rates can be found in [START_REF] Annaloro | Global Rate Coefficients for Ionization and Recombination of Carbon, Nitrogen, Oxygen, and Argon[END_REF].

The use of these recombination rates implies that N + always recombines into the ground state of N. However, recombination into excited electronic states should be considered. It is also shown in Fig. 1 that recombination rates of different sources can differ by two orders of magnitude. Therefore, the model will be electronic statespecific, and the recombination rates will be computed by detailed balance using state-specific ionization crosssection or using state-specific reaction rate coefficients (detailed in next section).

In all simulations performed in this work, the initial gas temperature is equal to 300 K and the initial electron temperature to 1000 K. The initial total gas density (79% of N2 and 21% of O2) is equal to the Loschmidt number at 300 K, 𝑛 = 2.45 × 10 cm . The initial electron density is equal to 𝑛 = 10 cm -3 . The integration step is below 10 ps. The nanosecond pulse is modeled with two phases:

1. For t = 0 -1 ns, the applied field increases linearly with time up to its maximum value. 2. For t = 1 -15 ns, the applied field is constant. 3. At t = 15 ns, the simulation is stopped. This is summarized in Eq. 2 where E0 is the applied field.

E [V. cm ] = 𝐸 × 𝑡 1 ns , 𝑡 ≤ 1 ns 𝐸 , 1 ns ≤ 𝑡 ≤ 15 ns Eq. 2 𝐸 𝑁 [Td] = 𝐸[V. cm ] 𝑁[cm -3 ] × 10 Eq. 3
If not specified, 𝐸 = 56 kV.cm -1 , corresponding to (𝐸/𝑁) = 230 Td in ambient conditions. This generic pulse is representative of a 6-kV pulse applied across a 1-mm gap. Because BOLSIG+ is adapted for partially ionized plasma, the reduced field E/N is usually equal to be the ratio of the electric field to the neutral particle density. In this work, the neutral particle density can drop to negligible values and the reduced field tends toward infinity. Therefore, we set N as the heavy particle density, i.e. equal to the density of neutral and ions. At ne < 10 18 cm -3 , this assumption induces negligible variations in the results. At ne > 10 18 cm -3 , the simulations are marked by a dotted line (instead of a full line) to indicate these results are subject to the previous assumption. Because this simulation is 0-D, field screening effects such as the electrode fall are not considered.

IV. Determination of ionization and recombination rates

In this section, the procedure to determine the recombination rate coefficients is detailed. If each electronic level of the species is considered, only the EEDF has to be Maxwellian to apply detailed balancing. The partition function of each level is simply its degeneracy multiplied by the exponential factor. In this work, we only consider the ground state of the ions, N + ( 3 P) and O + ( 4 S).

𝐴 + 𝑒 ⇌ 𝐴 + 𝑒 + 𝑒 Reac.

3

𝑑[𝐴 ] 𝑑𝑡 = - 𝑑[𝐴 ] 𝑑𝑡 = 𝑘 , [𝐴 ]𝑛 Eq. 4 𝑑[𝐴 ] 𝑑𝑡 = - 𝑑[𝐴 ] 𝑑𝑡 = 𝑘 , [𝐴 ]𝑛 Eq. 5
The electronic state-specific forward and backward rate coefficients of Reac. 3 are defined in Eq. 4 and Eq. 5. The forward rate coefficients are calculated in BOLSIG+ based on the cross-sections of the reactions. According to the detailed balance principle, it is possible to express 𝑘 , in Eq. 6.

𝑘 , = 1 2 2𝜋𝑚 𝑘 𝑇 ℎ 𝑔 𝑔 exp I -𝐸 𝑘 𝑇 𝑘 , Eq. 6
This procedure has been applied to the first 8 electronic levels of atomic nitrogen and first 3 electronic levels of atomic oxygen. These levels are listed in Table 1 and are all included in the chemical mechanism if not specified.

The excitation cross-sections by electron impact between states of atomic oxygen and nitrogen are taken from Refs. [START_REF] Wang | B-Spline R-Matrix-with-Pseudostates Calculations for Electron-Impact Excitation and Ionization of Nitrogen[END_REF] and [START_REF] Tayal | B -Spline R -Matrix-with-Pseudostates Approach for Excitation and Ionization of Atomic Oxygen by Electron Collisions[END_REF]. The ionization cross-sections by electron impact of the first three levels of oxygen and nitrogen are also calculated in Refs. [START_REF] Wang | B-Spline R-Matrix-with-Pseudostates Calculations for Electron-Impact Excitation and Ionization of Nitrogen[END_REF] and [START_REF] Tayal | B -Spline R -Matrix-with-Pseudostates Approach for Excitation and Ionization of Atomic Oxygen by Electron Collisions[END_REF]. The ionization rates of nitrogen levels above 10 eV are calculated using the reaction rate coefficients calculated from BEB cross-sections in Ref. [START_REF] Ciccarino | Electron-Impact Ionization of Atomic Nitrogen[END_REF] for Te up to 50,000 K. These rate coefficients are function of Te and the EEDF is therefore assumed to be Maxwellian when these rates are dominant. Net Ionization Rate (cm -3

.s -1

) Time (ns)

Fig. 4 Net rates of ionization (backward rates subtracted). For the sake of clarity, the net ionization rates of the two metastable states of N at 2.38 and 3.57 eV, red curve, are grouped. The next 5 levels are gathered in two groups: blue curve (10.33, 10.68, 10.92 eV) and green curve (11.60 and 11.75 eV).

V. Ionization mechanism

The reduced field, Eq. 3, is expected to follow the variation of E. However, at the onset of N2 and O2 dissociation, the total gas density increases, implying a decrease of E/N. This effect is shown in Fig. 2 where a generic field of 237 Td is applied. At t = 12 ns, the reduced field begins to decay due to molecular dissociation. Shortly after, the atoms are ionized, and E/N is constant.

In the chemical mechanism, no heating is associated with the exothermic (or endothermic) reactions. However, the heat exchange due to electron-neutral and electron-ion elastic collisions is computed by BOLSIG+. In nonequilibrium discharges, the electron number density is lower (𝑛 = 10 -10 cm ) and the temperature increases (Δ𝑇 ≈ 1000 K) mainly due to the ultrafast heating mechanism [START_REF] Rusterholtz | Ultrafast Heating and Oxygen Dissociation in Atmospheric Pressure Air by Nanosecond Repetitively Pulsed Discharges[END_REF]. Models to determine the temperature evolution of non-equilibrium discharges are already available in the literature [START_REF] Popov | Investigation of the Mechanism for Rapid Heating of Nitrogen and Air in Gas Discharges[END_REF]. In the case of a thermal spark i.e. a fully ionized nanosecond spark, the heating induced by electron collisions is dominant. The temperature evolution is demonstrated in Fig. 3. While the plasma is partially ionized, the electron temperature is constant around Te = 45,000 K. The gas (heavy) temperature remains equal to Tgas = 300 K. The heavy particles and the electrons start their thermalization at t = 10.5 ns, when the electron number density starts to increase. The decrease of the reduced field (due to molecules dissociation) induces a decrease of Te from 40,000 K to 35,000 K. At t = 11.25 ns, Tgas being close to Te, less energy is transferred from the electron to the heavy particles and Te increases by 7000 K. Soon after, at t = 12 ns, the electrons finally thermalize with the heavy particles at Tgas = Te = 42,000 K.

The electron production is first dominated by the ionization of N2 and O2, see Fig. 4. In the appendix, the evolution of the density of N2 excited states is plotted and the importance N2(a') ionization is discussed. The ionization rates of N2(A, B, C) are lower by an order of magnitude compared to the ground state. Fig. 4 distinguishes the net contribution to the production of electrons from the different states of N. For t in the range 10.75 -11.25 ns, the electrons are mainly produced by electron-impact ionization of highly excited states of N. At t = 11.25 ns, the gas is ionized at ~83 %. The plasma parameters at the transition from a non-equilibrium to a thermalized plasma are summarized in Table 2. It is shown that the electron number density gains two orders of magnitude in less than 0.5 ns. At the end of the pulse the electron number density saturates to 5×10 19 cm -3 . The plasma recombination phase has not been studied and is part of future work.

VI. EEDF deviation from the Maxwellian distribution

The EEDF is computed by BOLSIG+ considering the total cross-section of the plasma mixture. Departure from a Maxwellian distribution is expected in non-equilibrium plasmas generated by nanosecond discharges. The transition between t = 10.75 ns and t = 11.25 ns marks the thermalization of the electron and heavy temperatures and a set of EEDF calculated in this interval are shown in Fig. 5. The deviation from the Maxwellian distribution is quantified in Eq. 7, where 𝑓(𝜖) is the EEDF and 𝑓 (𝜖) is the corresponding Maxwellian distribution with the same averaged energy. Compared to the equivalent Maxwellian distribution, the EEDF is first overpopulated at approximately 10 eV. From t = 10.9 ns to t = 11.25 ns, the peak is shifted to higher energies and flattens. This shift can be due to the peak of atomic nitrogen effective cross-section at 10 eV. At t = 11.5 ns, the EEDF is Maxwellian (𝛿(𝜖) = 1 for 𝜖 < 20 eV) due to the electron-electron collisions.

𝛿(𝜖) = 𝑓 (𝜖) 𝑓 (𝜖)

Eq. 7

The recombination rates are calculated by detailed balancing, assuming the EEDF to be Maxwellian. However, the recombination rates are not significant during the ionization phase. Therefore, the approximation of the latter assumption is sufficient for results below t < 10.25 ns. In Fig. 4, the net ionization rates of the nitrogen states are compared. The rates of N( 4 S), N( 2 D) and N( 2 P) ionization, computed by BOLSIG+, are accurate given the precision of the BSR cross-section. The recombination in these three states becomes significant at t = 11.25 ns, as the EEDF is getting close to a Maxwellian. It is therefore valid to use detailed balancing to calculate the recombination rates.

The rates of the higher excited states ionization are Arrhenius laws based on BEB cross-sections integrated assuming a Maxwellian EEDF [START_REF] Ciccarino | Electron-Impact Ionization of Atomic Nitrogen[END_REF]. This assumption could have an impact on the results if the EEDF strongly deviates from the Maxwellian. During the interval 10.75 -11.25 ns, the ionization of N*(Eup > 10 eV) is dominant, but the EEDF is quickly becoming Maxwellian. The approximation is therefore assumed to be valid on this time interval. More accurate calculations would require the direct integration of the BEB cross-section instead of the Arrhenius rates. This possibility is currently explored in our group for future work. After t = 11.5 ns, the EEDF is Maxwellian and the application of the Arrhenius rates is therefore accurate.

VII. Influence of the number of atomic states in the model

The nitrogen excited electronic states have shown a real influence on the electron production. This was already seen when the excited states of molecular nitrogen were included in Ref. [START_REF] Shcherbanev | Filamentary Nanosecond Surface Dielectric Barrier Discharge. Plasma Properties in the Filaments[END_REF]. New simulations are performed changing the number of nitrogen and oxygen excited states included in the chemical mechanism. The applied reduced field is equal to (E/N)0 = 230 Td in ambient conditions. The evolution of the electron number density is showed in Fig. 6, for 𝑡 > 9 ns. The 8-levels (full) model will be the reference of comparison. If only six excited states of nitrogen (Eup < 11.61 eV in Table 1) are included, a similar ionization that described in previous sections is encountered. However, if only the three first levels of nitrogen are included, the total ionization of the gas is delayed by 1 ns. An intermediate result is obtained if four or five levels are included. If only the ground states of nitrogen and oxygen are included in the simulation, the maximal (final) electron number density is decreased by two orders of magnitude compared to the reference. These results show again the dynamics of the thermal spark formation is strongly influence by the presence of atomic excited states in the simulation. The deviation between the models starts at t = 10 -10.5 ns, corresponding to ne = 10 16 -10 17 cm -3 .

Following, the previous procedure, several simulations are performed for different (E/N)0 varying the number of excited states included in the simulation. The maximal electron number density (reached at the end of the pulse) is shown in Fig. 7. As before, the two excited levels of atomic oxygen, O( 1 D, 1 S), are not included if only the nitrogen ground state is used. For E/N < 213 Td, the deviation between all the models is relatively small. The 8levels model shows full ionization is reached for (E/N)0 > 216 Td. The deviation with the reference of the 6-model is small on the entire studied range. If only three excited states of nitrogen are included, the full ionization is reached for (E/N)0 > 224 Td. If only the ground states of nitrogen and oxygen are included, the electron number density reaches 10 19 cm -3 for (E/N)0 > 260 Td (not shown in Fig. 7). In term of electron number density, the deviation between the different models is marked at a value of ne ~ 10 17 cm -3 . The same value is encountered in Fig. 6 when the models do not agree with each other's at t = 10 -10.5 ns. We therefore conclude that the correct description of the plasma dynamics must include at least the first six electronic states of atomic nitrogen, if ne > 10 17 cm -3 .

Conclusion

The mechanism of ionization in ambient air by a generic 15-ns pulse of 56 kV.cm -1 is studied with 0-D simulations using ZDPlaskin and BOLSIG+. The electron production phase is first dominated, as usual, by electron-impact ionization of N2 and O2. Then, molecules dissociate, and the density of atomic species increases. Excited electronic states of atoms and molecules are populated by electron-impact excitation. We show that the ionization from N2(a'), N and N* soon becomes more important than ionization from the ground states of N2 and O2. We also show that ionization from the excited N states with energy greater than 10 eV quickly dominates. The electron number density increases by two orders of magnitude (from 10 17 to more than 10 19 cm -3 ) in less than half a nanosecond. Simultaneously, the gas temperature increases from 300 to 42,000 K and equilibrates with the electron temperature. The EEDF is found to be Maxwellian by the time the plasma is fully ionized. This mechanism explains in our opinion the fast transition (< 1 ns) from a non-equilibrium to a thermal plasma as experimentally observed with nanosecond discharges in Refs. [START_REF] Minesi | On the Arc Transition Mechanism in Nanosecond Air Discharges[END_REF][START_REF] Shcherbanev | Filamentary Nanosecond Surface Dielectric Barrier Discharge. Plasma Properties in the Filaments[END_REF][START_REF] Parkevich | Fast Fine-Scale Spark Filamentation and Its Effect on the Spark Resistance[END_REF].

Ai = generic species

  in the state i BEB = Binary-Encounter-Bethe BSR = B-spline R-matrix 𝛿 = deviation of the EEDF from the Maxwellian distribution E = electric field E0 = maximal applied electric field (E/N)0 = reduced electric field at ambient conditions Eup = energy of the excited state 𝜖 = electron energy 𝑓 = Electron Energy Distribution Function (EEDF) 𝑓 = Maxwellian EEDF h = Planck constant IA = ionization energy of the generic species A kb = Boltzmann constant kf = forward reaction rate coefficient kr = backward reaction rate coefficient me = electron mass N = heavy particles density, equal to the neutral and ion density ne = electron number density Te = electron temperature Tgas = heavy particle temperature Z = partition function

Fig. 2 Fig. 3

 23 Fig.2Variation of the electron number density during the 15-ns pulse (black line). The decrease of the reduced electric field (red line) is due to the increase of heavy particles' density during dissociation. The region of fast ionization is delimited by the gray lines between 10.75 and 11.25 ns.

29 N( 4 PN

 294 , 2 P, 4 P) + e « N + + 2e N( 2 S 0 , 4 D 0 ) + e « N + + 2e

Fig. 5 3 )Fig. 6 Fig. 7

 5367 Fig. 5 Deviation of the EEDF (δ) from a Maxwellian EEDF of the same average energy

9.25 9 .Fig. 8 Fig. 9

 989 Fig. 8 Evolution of the species density during the generic 15-ns pulse of 𝑬 𝟎 = 56 kV.cm -1 according to the 8-levels model. The region of fast ionization is delimited by the gray lines between 10.75 and 11.25 ns.

  Fig.10Ionization rates of N2(X, A, B, C, a'). The net ionization rates (i.e. with recombination rate subtracted) is shown in Fig.4(purple solid curve).

Table 1

 1 Excited levels of N and O included in the present work. The excitation rate coefficients between states is computed with BOLSIG+ based on the BSR cross-sections calculated in Ref.[START_REF] Wang | B-Spline R-Matrix-with-Pseudostates Calculations for Electron-Impact Excitation and Ionization of Nitrogen[END_REF][START_REF] Tayal | B -Spline R -Matrix-with-Pseudostates Approach for Excitation and Ionization of Atomic Oxygen by Electron Collisions[END_REF]. When available, the ionization BSR cross-sections are used, otherwise, the reaction rates computed in Ref.[START_REF] Ciccarino | Electron-Impact Ionization of Atomic Nitrogen[END_REF] on the basis of BEB cross-sections are used. All reverse rates are calculated by detailed balancing.

		Configuration	Term	g	Eup (eV)	Electronic transitions rate	Ionization rate
		2s 2 2p 3	4 S 0	4	0		Integral of BSR
		2s 2 2p 3	2 D 0	10	2.38		cross-section
		2s 2 2p 3	2 P 0	6	3.575		[39]
	N	2s 2 2p 2 ( 3 P)3s 2s 2 2p 2 ( 3 P)3s	4 P 2 P	12 6	10.33 10.68	Integral of BSR cross-section [39]
		2s 2 2p 4 2s 2 2p 2 ( 3 P)3p	4 P 2 S 0	12 2	10.92 11.60		Arrhenius rate [41]
		2s 2 2p 2 ( 3 P)3p	4 D 0	20	11.75
		2s 2 2p 4	3 P	9	0
	O	2s 2 2p 4	1 D	5	1.97	Integral of BSR cross-section [40]
		2s 2 2p 4	1 S	1	4.19

Table 2

 2 Plasma parameters before (t = 10.75 ns) and after (t = 11.25 ns) the fast ionization phase. At t = 11.25 ns, 36% of atomic nitrogen is in the ground state (2.7×10 18 cm -3 ).

	t (ns)	E/N (Td)	Te (K)	Tgas (K)	N2 (cm -3 )	N (cm -3 )	ne (cm -3 )
	10.75	207.0	39600	1400	1.6×10 19	4.7×10 18	1.6×10 17
	11.25	116.9	41300	38000	7.8×10 15	7.4×10 18	3.3×10 19

Appendix

We show in Fig. 8 the evolution of the main species involved in the ionization mechanism. The transition between a non-equilibrium to a thermal plasma is marked by two grey lines in Fig. 8 at 10.75 and 11.25 ns. During this transition, N2 is completely dissociated. The excited states of N2 are also completely dissociated. The evolution of N2(a') is however surprisingly peaked compared to the other excited states. This is due to the differences in the dissociation cross-sections showed in Fig. 9. As the energy of the electrons increases, the dissociation of N2 excited states is expected before their ionization. However, the dissociation cross-section of N2(a') in Fig. 9 is low compared to the other excited states. N2(a') is therefore less affected by electron-impact dissociation and is efficiently ionized compared to N2 other states. This is shown in Fig. 10 where the ionization rates are plotted against time. For t > 10 ns, the ionization rate of N2(a') is higher than the ionization rate N2(X).