Streamer-induced kinetics of excited states in pure N₂: I. Propagation velocity, $\textit{E}/\textit{N}$ and vibrational distributions of N₂(C$^{\text{3}}\Pi_\text{u}$) and N$^+_{\text{2}}$(B$^{\text{2}}\Sigma_\text{u}^+$) states

We carried out the present experiments using a diagnostic discharge reactor described in detail in preceding works [16–18] and illustrated in figures 1(a) and (c). The DBD consists of a point-plane electrode system placed in a stainless-steel chamber based on the KF 6-way cross. The copper tips (1 and 16 mm for the high-voltage (HV) and grounded electrode, respectively) were embedded in MACOR glass-ceramic disks (0.5 mm thick dielectric layers, relative permittivity is approximately 6 at 25 ^∘C and 1 kHz). The discharge was powered by employing an HV waveform made of two sine-waves ($f_\textrm{AC}$ = 1 kHz, peak-to-peak the amplitude of 19.4 kV) superposed with a fast positive pulse (with the amplitude of 9 kV) acting on the second positive sinusoidal half-cycle, see panel (b) of figure 1. This complex waveform was applied at a fixed repetition frequency of f$_\textrm{M}$ = 10 Hz. Within each waveform, the fast positive pulse produces a triggered streamer mono-filament that is locked with respect to the onset of the pulse (HV rise time of a few nanoseconds). Untriggered discharges regularly occur during the preceding AC phases (typically one streamer mono-filament per one AC half-cycle), leaving residual electrons that serve as initial charges for subsequent triggered events.

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Figure 2 shows the block diagram of the setup used to drive the discharge periodically by a specific HV waveform. The AC high-voltage power supply is composed of a TG1010A Function Generator (TTi), a Powertron model 250A RF amplifier, and a high-voltage step-up transformer. The pulsed power supply generates the positive HV pulse ($u_\textrm{PPS}$, peak amplitude 9 kV, 100 ns duration, the typical rising slope of 200 Vns⁻¹ , repetition rate $f_\textrm{P}$ = $f_\textrm{M}$) by discharging a 10 nF capacitor through a Behlke model HTS-81 solid state switch (8 kVdc/30 A). We used two channels of the digital pulse/delay generator (BNC Model 575) as a master trigger to synchronize the AC burst with the positive HV pulse at a suitable phase. Other BNC channels were used to synchronise the diagnostic equipment (ICCD detectors and multichannel photon counter). The discharge was fed with pure nitrogen (99.9999 % N₂ + 0.5 ppmv H₂O + 0.3 ppmv O₂ + 0.1 ppmv total hydrocarbon) through Bronkhorst HI-TEC mass flow controller (total flow of 85.3 sccm). The pressure of 200 Torr in the chamber was stabilised by using a needle valve backed by an oil-free diaphragm vacuum pump, and measured with a TPG 260 piezo gauge (Pfeiffer Vacuum). The present experimental conditions are dictated by conditions used to perform LIF/TALIF experiments [16–18]; the pressure was set as a suitable compromise, which ensures high stability (low onset jitter) of the discharge and a sufficiently large radius of the streamer filament for LIF diagnostics. A Tektronix DPO 5204B oscilloscope was used to record the discharge characteristics (the HV, current, PMT waveforms) and to control the timing of the discharge with respect to the detectors. The discharge voltage waveforms were sampled through a Tektronix P6015 high-voltage probe (1000:1$@$100 MΩ). The current pulses produced by individual micro-discharges were monitored through the Pearson probe right before the high voltage electrode.

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We used the digital pulse/delay generator (BNC Model 575) as a trigger and delay source to synchronize the HV waveforms with diagnostic equipment (ICCDs, oscilloscope). Two BNC 575 channels were used to generate the triggered discharge, while the other channels were used for synchronous electrical characteristics and spectrometric data acquisition. We simultaneously used two ICCD-based imaging spectrometers to collect spectrometric data in the 200–1100 nm spectral range. To analyze plasma-induced emission (PIE) in 200–850 nm interval, we used Jobin-Yvon iHR-320 spectrometer equipped with an Andor DH740 ICCD detector. On the other hand, the Andor Shamrock 303i spectrometer with DH340 ICCD was used to collect spectra in 600–1100 nm range. All emission spectra were calibrated only relatively since the absolute calibration has no effect on VDF analysis and $E/N$ determination. The iHR-320 was used in an imaging mode (1:1 image through a pair of UV-grade lenses). The Shamrock 303i was used in full vertical binning mode because the PIE was delivered through the 10 m long quartz fiber bundle. By directly imaging the DBD gap onto the entrance slit of the iHR-320, we could also take time-resolved images of the gap using the spectrometer's grating in the 0th-order of diffraction. Furthermore, to better follow the evolution of the discharge morphology during a single event, we used a 4-channel XX RapidFrame camera (Stanford Computer Optics, Inc), essentially a four-channel ICCD system based on UV-extended mirror-based image splitting optics and four 4-Picos ICCD cameras. The image splitter splits the arriving flux of photons equally among the four exit channels (Ch1–Ch4), each registered by one ICCD. Each ICCD detector is equipped with an 18 mm dual-stage multi-channel plate (MCP) image intensifier (S20 photocathode), allowing the acquisition of time-resolved images taken with a pre-selected MCP gating time (a minimum gate of 1 ns was used in this work). This enables the registration of four successive time-resolved images taken with a required frame rate. The four ICCD detectors' relative positions of the MCP gates were controlled through the internal delay generators of the respective 4-Picos cameras. Complementary PIE characteristics measurements were performed using fast PMTs (Hamamatsu H10721, Photek PMT210) combined with appropriate bandpass filters.

The acquired datasets were analyzed by using synthetic models for N₂ and N$_2^+$ emission spectra reported in detail in [3, 20, 21]. Based on the fact that a typical N₂/N$_2^+$ emission spectrum consists of a group of vibronic bands arranged in so-called sequences (a group of bands characterized by a constant difference between the vibrational numbers of the upper/lower states participating in the radiative transition), we use an approach based on the calculation of a set of band profiles by considering the actual experimental instrumental function of the ICCD spectrometer and assuming an equilibrium population distribution in the upper rotovibronic states (therefore characterized by a certain rotational temperature $T_\textrm{rot}$). Then the reproduction/interpretation of the experimental spectra is reduced to finding a suitable multiplicative constant for each band profile such that the superposition of all calculated band profiles (after being multiplied by the corresponding constants) minimizes the residual spectrum obtained as the difference between the experimental and calculated spectra. The set of multiplicative constants then contains information on the vibrational distribution of the upper (emitting) vibronic state.

In order to understand the discharge initiation and the charge transfer in discharge chamber, we consider an equivalent circuit which, as best as possible, represents the circuit. The equivalent electric circuit consists of lumped elements as shown in figure 3(a). The voltage source $V_\textrm{1}$ represents a device which superposes the low-frequency harmonic waveform and short (100 ns) voltage pulse (∼9 kV) with approximately exponential decay. The internal impedance of $V_\textrm{1}$ is represented by the resistor $R_\textrm{i}$ and the inductor $L_\textrm{i}$. The elements $C_\textrm{p}$ and $R_\textrm{p}$ represent loading by the Tektronix P6015 high-voltage probe. The discharge cell's inner electrodes are accessible via two short transmission lines $T_\textrm{1}$ and $T_\textrm{2}$. The anode is fed from $V_\textrm{1}$ via $T_\textrm{1}$, whereas the cathode is connected to the ground via $T_\textrm{2}$ and an inductor $L_\textrm{2}$. The capacitor $C_\textrm{1}$ represents dielectrics covering the electrodes, and $C_\textrm{2}$ defines the capacity of the gap between the dielectrics. These capacities were estimated based on the geometrical size of the dielectrics and the gap ($C = \varepsilon_0 \varepsilon_r \frac{S}{d}$, where the S denotes the area of the electrodes, d gap or dielectrics thickness and $\varepsilon_0, \varepsilon_r$ the permittivity of vacuum or the dielectrics, respectively). The streamer monofilament in the gap, and an unavoidable conductive leakage between the HV electrode and the groundings, are defined by the two variable resistors denoted filament and leakage, respectively. The meaning of the leakage resistor is that the filament current is not restricted to flow to the grounded inner walls of the discharge chamber.

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Figure 3(b) shows the experimentally measured voltage (red) and current (blue) waveforms at the $V_\textrm{1}$, when there is no discharge between the electrodes due to elevated gas pressure. The simulation on the equivalent circuit is performed in the LTspice software [22], where the measured voltage waveform is a simulation input, whereas its output is the simulated current (green). The simulated current's noisy character originates in the weak noise present in the voltage waveform as the load impedance is predominantly capacitive. The noise is not present in the measured current waveform, probably because the noise bandwidth is larger than 1 GHz, whereas the bandwidth of the Pearson probe is limited to 200 MHz. Figure 3(c) shows voltage and current waveforms at the $V_\textrm{1}$ with the discharge. The measured current waveform (blue) is obviously elevated compared to the previous case in figure 3(b). Although the discharge itself is modelled by a variable filament resistor, to achieve a similar simulated current at $V_\textrm{1}$, it is necessary to add the additional leakage resistor to the model. Without this resistor, the agreement between the measured and simulated currents is not good. Individual components of simulated current are plotted in figure 3(d), where they are labeled as follows: filament current (blue), leakage current (magenta), and the current on the dielectrics ($I_\textrm{C1}$, black). The steep increase of $I_\textrm{C1}$ is due to streamer arrival to the cathode, connected with the charging of the dielectric cathode. This increase begins approximately 14 ns after the high voltage onset. The initial charging of the dielectric cathode, defined by a relatively small capacity $C_\textrm{1}$, is after approximately 11 ns taken over by the leakage resistor. The probable explanation is that: although the streamer filament initially gaps the dielectrics, the surface ionization wave later propagates along the dielectric electrodes into remote areas of the chamber, where no insulation exists. Therefore, the leakage current flowing between conductive parts in the chamber cannot be excluded. Figure 3(e) plots the time dependence of estimated filament (blue) and leakage (magenta) resistances, which are found by the simulation to be appropriate to reproduce the measured current.

The fast time-resolved imaging using 4-channel ICCD provides information about the streamer morphology and dynamics. The most important results obtained through the 4-channel XX RapidFrame are summarized in figures 4–6. Figure 4 shows Ch1–Ch4 images acquired during a single event. Channels Ch1–Ch2 integrate the very weak emission produced during the rising slope of the second positive AC half-cycle before the onset of the superimposed HV pulse. These two ICCD images capture the very slow phase of the regeneration of the charges in the anode region after the polarity reversal of the HV electrode (temporary cathode to temporary anode). Channels Ch3 and Ch4 integrate the emission during the first 20 ns of the HV pulse and several microseconds after the pulse. Ch3 provides a characteristic image of a fully developed streamer monofilament in the gap. The image from Ch4 shows the glow disappearance after the streamer decay. Note that in order to visualize the luminosity of the formative pre-streamer phase, we had to use MCP gates with microsecond duration (5 and 2 µs for Ch1 and Ch2, respectively) compared to nanosecond gates during the streamer phase (Ch3, Ch4). Figure 5 shows a series of four consecutive frames captured during the onset of a streamer triggered by a HV pulse. Four images taken during one event in four consecutive time windows allow monitoring the dynamics of monofilament formation. They were obtained using the same MCP gate widths (1 ns) for all channels and were appropriately delayed with respect to the onset of the HV pulse and with respect to each other ($\Delta t$ = 3 ns). Events 1–3 were selected from all captured single discharge events to illustrate the basic PIE characteristics during the first tens of nanoseconds.

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Finally, figure 6 shows a kinetic series of ICCD frames created from Ch2 images and selected from a set of 500 registered events similar to those shown in figure 5 (i.e. all acquired with a 1 ns MCP gate, but with a variable delay relative to the HV pulse). Each frame in figure 6 was created by averaging five equivalent Ch2 images (taken at the same MCP delay and showing similar evolution on all four Ch1–Ch4 channels) and corrected for background, resulting in significant noise reduction. The image in the upper right corner captures the moment immediately after the impact of the streamer on the dielectric surface covering the temporary cathode, and the corresponding MCP delay is denoted as T₀. The three preceding images in the top row capture the evolution before streamer formation. The images in the middle row demonstrate the formation of the cathode spot over the next ten nanoseconds. The streamer velocity on the cathode surface can be determined by investigating the evolution of the surface streamer front in time, based on the middle row. However, this method is not very precise since figure 6 was obtained by averaging. Nevertheless, the streamer velocity on the cathode surface reaches approximately (2±1)$\times10^5$ m s⁻¹. The images in the last row show the discharge's upward expansion along the temporary anode's dielectric surface. Based on the observed features of the discharge evolution, the following mechanism of streamer initiation between the dielectric barriers can be concluded, similarly as described in detail in [23]:

• (i)  
  Townsend avalanche pre-phase (charge accumulation, $t \lt T_0$).
• (ii)  
  Avalanche to streamer transition, propagation of the cathode-directed streamer and its impact on the cathode ($t \approx T_0$).
• (iii)  
  Propagation of surface streamer on the cathode dielectric ($t \gt T_0$).
• (iv)  
  Charge redistribution: electrons, generated during the streamer impact on the cathode, are transferred to the dielectric surface on the anode ($t \gt T_0 + 16$ ns). The charge redistribution in the gap causes a decrease in the electric field.

The determination of the streamer propagation velocity in a 4 mm DBD gap is a challenging task. On the one hand, the statistical dispersion of the streamer onset time, in this case of $\sim6$ ns, makes it difficult to average the streamer propagation over several events. On the other hand, the minimum imaging gate of 2 ns of the DH740 ICCD, when it is necessary to obtain a sub-nanosecond evolution, turns the task even harder.

Herein, we propose a method to determine the streamer propagation velocity that overcomes both aforementioned problems and is described by the following steps.

• (i)  
  For different opening times of the 2 ns gate ($t^\mathrm{gate}_\mathrm{on}$), chosen to catch the streamer propagation phase, 500 ICCD images are acquired. An example of such image is represented in figure 7(a).
• (ii)  
  In parallel, for each streamer event acquired, the temporal evolution of the streamer light is collected by a PMT. This allows determining precisely the instant at which the streamer event has started ($t^\mathrm{stream}_\mathrm{start}$), apart from a constant time shift. The stochastic effect of the streamer onset is annihilated by subtracting this time from the gate opening time: $t^\mathrm{gate}_\mathrm{corr} = t^\mathrm{gate}_\mathrm{on} - t^\mathrm{stream}_\mathrm{start}$, where $t^\mathrm{gate}_\mathrm{corr}$ is a corrected time that is now given in the reference time of the streamer event.
• (iii)  
  Afterwards, each image is horizontally averaged, leaving only the light emission as a function of the vertical position. The horizontally-averaged images are plotted in figure 7(b) as a function of the corrected time $t^\mathrm{gate}_\mathrm{corr}$. Each column corresponds to one horizontally-binned image, and the colorbar quantifies the intensities of such image.
• (iv)  
  The streamer front position (SFP) is identified by finding the first vertical position, starting from the bottom, whose horizontally-averaged intensity is higher than a certain threshold. In this case, a threshold of $1.5\times10^4$ is optimal to determine the SFP. However, note that using thresholds of 10⁴ or $2\times10^4$ leads to the same trend of SFP as a function of time, with the exception of a constant time shift, which is not relevant for the velocity determination.
• (v)  
  Finally, the streamer propagation velocity is calculated by fitting the SFP($t^\mathrm{gate}_\mathrm{corr}$) with linear functions.

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From the observation of figure 7(b), we find that the streamer takes approximately 3 ns to propagate from the top (anode) to the bottom (cathode) electrode. Additionally, there are clearly two different linear trends in the streamer propagation: initially, the streamer velocity is $\sim 9.8\times10^5$ $\mathrm{m\,s}^{-1}$ and then, in the lower half of the gap, the velocity increases to $\sim 2.4\times10^6 \mathrm{m\,s}^{-1}$. After the streamer arrives at the cathode, it propagates around it, leading to the apparent deceleration pointed out by the third linear trend. Light reflections at the cathode surface may also play a role in this third stage. Note that, simultaneously with the streamer propagation over the gap, there is also propagation around the anode, as evidenced by the increased luminosity after 1 ns at positions higher than the end of the top electrode.

We should remark that the study is done with only 500 ICCD images, which corresponds to an effective measurement time of ∼1 min (considering the 10 Hz repetition frequency). A similar study done by varying the opening time of the gate $t^\mathrm{gate}_\mathrm{on}$ would require a very large number of averages at each point in order to cancel the effect of the jitter on the streamer onset. For this reason, such a study would take at least several hours. This shows the potential of the method presented in this work.

The UV part of the spectra contains a blend of the SPS of N₂ and the γ-system of nitric oxide (NO-γ), see figure 8(a). Even in high-purity nitrogen, the dissociation of O₂ impurities subsequently forms nitric oxide in the ground state ($\textrm{NO}({\mathrm{X}}^2 \Pi)$). Electronic excitation of $\textrm{NO}({\mathrm{X}}^2 \Pi)$ then occurs either through electron-impact excitation [24]:

$$\begin{equation} \textrm{e} + \textrm{NO}\left({\mathrm{X}}^2 \Pi\right) \rightarrow \textrm{e} + \textrm{NO}\left(\textrm{A}^2\Sigma^+\right) \end{equation} \tag{ 1 }$$

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or resonant energy transfer from the $\textrm{N}_2({\mathrm{A}}^3 \Sigma^+_\textrm{u})$ metastables [25–28]:

$$\begin{equation} \textrm{NO}\left({\mathrm{X}}^2 \Pi\right) + \textrm{N}_2\left({\mathrm{A}}^3 \Sigma_\textrm{u}^+, w\right) \rightarrow \textrm{NO}\left(\textrm{A}^2\Sigma^+\right) + \textrm{N}_2. \end{equation} \tag{ 2 }$$

Therefore, the NO-γ emission can serve as an important diagnostic tool about the presence of $\textrm{N}_2({\mathrm{A}}^3 \Sigma^+_\textrm{u})$ metastables. The NO-γ emission begins to occur at the time of approximately 60 ns; at these times, the vibrational levels w > 5 of $\textrm{N}_2({\mathrm{A}}^3 \Sigma_\textrm{u}^+)$ peak [16]. Therefore, we can expect that the onset emission is connected mainly with the energy transfer from the higher levels, which is later replaced by the energy transfer from the lower $\textrm{N}_2({\mathrm{A}}^3 \Sigma^+_\textrm{u})$ vibrational levels. This is also in agreement with the higher rate constants of the process (2) for the higher w-levels; see [27] and references therein.

Another interesting emission present in the observed spectra is the Goldstein–Kaplan system:

$$\begin{equation} \textrm{N}_2\left({\mathrm{C}^{^{\prime}}}^3 \Pi_\textrm{u}, v = 0\right) \rightarrow \textrm{N}_2\left({\mathrm{B}}^3 \Pi_\textrm{g}, v^{^{\prime}}\right) + h\nu, \end{equation} \tag{ 3 }$$

where the higher vibrational levels of (C$^{^{\prime} 3}\Pi_\textrm{u}$) state are strongly predissociative. The upper state (C$^{^{\prime} 3}\Pi_\textrm{u}$) of this system is populated through the pooling reaction:

$$\begin{equation} \textrm{N}_2\left({\mathrm{A}}^3 \Sigma_\textrm{u}^+\right) + \textrm{N}_2\left({\mathrm{A}}^3 \Sigma_\textrm{u}^+\right) \rightarrow \textrm{N}_2\left(\textrm{C}^{^{\prime} 3}\Pi_\textrm{u}\right) + \textrm{N}_2, \end{equation} \tag{ 4 }$$

the pooling rate constant for the process (4) is much lower (2$\cdot 10^{-12}$cm³s⁻¹) than for the pooling to the $\textrm{N}_2(\textrm{B}^3\Pi_\textrm{g}, \textrm{C}^3\Pi_\textrm{u}, \textrm{C}^{^{\prime\prime} 5}\Pi)$ [29, 30]. Moreover, the Goldstein–Kaplan emission spectra overlap with the SPS spectra, and their investigation requires careful separation of the dominant SPS emission, see figure 8(b). In the residual spectrum, the Goldstein–Kaplan bands (0,9), (0,10) and (0,11) at band head wavelengths 416.6, 443.2 and 472.8 nm, respectively, can be easily identified since they have the strongest intensity [31]. Other bands (0,2) at 286.3 nm, (0,3) at 300.5 nm, (0,4) at 315.9 nm, (0,5) at 332.6 nm, (0,6) at 350.4 nm, (0,7) at 370.7 nm, (0,8) at 392.5 nm, (0,12) at 505.9 nm, originating from the C'$^3\Pi$, v = 0 level, have much lower intensity. The observation of the Goldstein–Kaplan and NO-γ bands confirms the presence of $\textrm{N}_2({\mathrm{A}}^3 \Sigma_\textrm{u}^+)$ states at the early post-discharge phase, $t \in$ (80, 200) ns.

The investigation of the VDFs allows revealing the specific precursors of the excited N₂ species. The Franck–Condon factors (FCF) are defined using vibrational overlap integrals and serve as a benchmark when evaluating the efficiency of vertical electron-impact excitation processes [32]. Therefore, it is usually assumed that when exciting multiple vibronic levels (e.g. N₂(C$^3\Pi_\textrm{u}$, v = 0–4)) from a single ground state level ($\textrm{N}_2({\mathrm{X}}^1 \Sigma^+_\textrm{g}$, v = 0)) the population of excited states is proportional to the corresponding FCFs between $\textrm{N}_2({\mathrm{X}}^1 \Sigma^+_\textrm{g}$, v = 0) and N₂(C$^3\Pi_\textrm{u}$, v = 0–4). Under present discharge conditions, it provides important evidence justifying the use of the intensity ratio method for the determination of the reduced electric field (confirmation of N₂(C$^3\Pi_\textrm{u}$) and N$^+_2$(B$^2\Sigma_\textrm{u}^+$) state production due to direct electron-impact ionization from the $\textrm{N}_2({\mathrm{X}}^1 \Sigma^+_\textrm{g}, v = 0)$ ground state). The analysis of N₂(C$^3\Pi_\textrm{u}$) VDF is relatively simple due to the high intensity of SPS radiation and many bands/sequences that can be analyzed. Conversely, determination of the VDF of N$^+_2$(B$^2\Sigma_\textrm{u}^+$) is significantly more difficult due to the FNS lower intensity and the omnipresent overlap with SPS bands, especially under conditions of a low-to-medium range of $E/N$ (100–200 Td) [12, 33].

The SPS bands belonging to the $\Delta v$ = 0, −2, and −3 sequences were analyzed using the technique detailed in [3, 21]. When calculating synthetic SPS band profiles, we considered the experimental instrumental function given by dispersion grating and entrance slit width. Figure 9(a) shows the UV spectra from the wavelength range 385–400 nm, revealing FNS(0,0) and the SPS bands belonging to the $\Delta v$ = −3. Figure 9(b) shows the SPS(0,0), SPS(2,4) and FNS(0,0) PMT profiles as well as the band intensities of SPS(0,3), SPS(1,4) and SPS(2,5) obtained by the spectrometer using the technique of kinetic series. These measurements can be converted to the time evolution of N₂(C$^3{\Pi _{\textrm{u}}},v = 0 - {\textrm{4}})$ profiles as shown in figure 9(c). Obviously, the onset of the ICCD intensities (1–8 ns) is more gradual than the onset measured using the PMT, which is due to the 2 ns gate width of ICCD used to collect spectra. Complementary to the preceding, figure 10 shows the time evolution of the VDF within 950 ns after the discharge onset.

• (i)  
  Interval 0–8 ns, VDF shows a typical FCF-like shape, which reveals the initiation of the active discharge phase (see the population comparison in table 1 and discussion further below). During this phase, the $\textrm{N}_2({\mathrm{C}}^3 \Pi_\textrm{u}, v = 0)$ and $\textrm{N}_2({\mathrm{C}}^3 \Pi_\textrm{u}, v = 1)$ densities can be described using the following formulas. We use the shortened notation (${\mathrm{N}}_2({\mathrm{C}}_{0})$ = $\textrm{N}_2({\mathrm{C}}^3 \Pi_\textrm{u}, v = 0)$, ${\mathrm{N}}_2({\mathrm{C}}_{1})$ = $\textrm{N}_2({\mathrm{C}}^3 \Pi_\textrm{u}, v = 1)$).

  $$\begin{eqnarray} &&\frac{\textrm{d} \left[{\mathrm{N}}_2\left({\mathrm{C}}_{0}\right)\right]}{\textrm{dt}} = k_\textrm{im}^{\textrm{C}_0} \left[{\mathrm{e}}\right] \left[{\mathrm{N}}_2\right] > 0, \end{eqnarray} \tag{ 5 }$$

  $$\begin{eqnarray} &&\frac{\textrm{d} \left[{\mathrm{N}}_2\left({\mathrm{C}}_{1}\right)\right]}{\textrm{dt}} = k_\textrm{im}^{\textrm{C}_1} \left[{\mathrm{e}}\right] \left[{\mathrm{N}}_2\right] > 0, \end{eqnarray} \tag{ 6 }$$

  where the $k_\textrm{im}^{\textrm{C}_0}$ and $k_\textrm{im}^{\textrm{C}_1}$ denote rate constants of electron impact excitation of the $\textrm{N}_2({\mathrm{C}}^3 \Pi_\textrm{u}, v = 0)$ and $\textrm{N}_2({\mathrm{C}}^3 \Pi_\textrm{u}, v = 1)$ states, respectively.
• (ii)  
  Interval 12–20 ns, the N₂(C$^3{\Pi _{\textrm{u}}},v = 0 - {\textrm{4}})$ densities start to reflect quenching by molecular nitrogen and relax from the higher to the lower vibrational levels due to vibrational relaxation within the N₂(C$^3\Pi_\textrm{u})$ state:

  $$\begin{align} \begin{array}{l} {{\textrm{N}}_2}{{\textrm{C}}^3}{\Pi _{\textrm{u}}},v = 1 - {\textrm{4 + }}{{\textrm{N}}_{\textrm{2}}}\left( {{{\textrm{X}}^{\textrm{1}}}\Sigma _{\textrm{g}}^{\textrm{ + }},{{\textrm{v}}^\prime }{\textrm{ = 0}}} \right)\\ \,\,\,\,\,\,\,\,\,\,\,\,{{\textrm{N}}_{\textrm{2}}}\left( {{{\textrm{X}}^{\textrm{1}}}\Sigma _{\textrm{g}}^{\textrm{ + }},{{\textrm{v}}^{\prime \prime }}{\textrm{x2A7E0}}} \right){\textrm{ + }}{{\textrm{N}}_{\textrm{2}}}\left( {{{\textrm{C}}^{\textrm{3}}}{\Pi _{\textrm{u}}},{\textrm{w}} < {\textrm{v}}} \right). \end{array} \end{align} \tag{ 7 }$$

  The evolution of the lowest two states $\textrm{N}_2({\mathrm{C}}^3 \Pi_\textrm{u})$ can be described by the following equations:

  $$\begin{align} \frac{\textrm{d} \left[{\mathrm{N}}_2\left({\mathrm{C}}_{0}\right)\right]}{\textrm{dt}} & = k_\textrm{im}^{\textrm{C}_0} \left[{\mathrm{e}}\right] \left[{\mathrm{N}}_2\right] + k_\textrm{rel}^{\textrm{C}_{10}}\left[{\mathrm{N}}_2\left({\mathrm{C}}_{1}\right)\right]\left[{\mathrm{N}}_2\right] \nonumber\\ & \quad - \frac{\left[{\mathrm{N}}_2\left({\mathrm{C}}_{0}\right)\right]}{\tau_\textrm{eff}^{\textrm{C}_0}} \approx 0, \end{align} \tag{ 8 }$$

  $$\begin{eqnarray} &&\frac{\mathrm{d} \left[{\mathrm{N}}_2\left({\mathrm{C}}_{1}\right)\right]}{\mathrm{dt}} = k_\mathrm{im}^{\mathrm{C}_1} \left[{\mathrm{e}}\right] \left[{\mathrm{N}}_2\right] - \frac{\left[{\mathrm{N}}_2\left({\mathrm{C}}_{1}\right)\right]}{\tau_\mathrm{eff}^{\mathrm{C}_1}} \approx 0, \end{eqnarray} \tag{ 9 }$$

  where the $k_\mathrm{rel}^\mathrm{C_{10}}$ is the rate constant of the equation (7) for v = 1 and w = 0. $\tau_\mathrm{eff}^{\mathrm{C}_0}$ and $\tau_\mathrm{eff}^{\mathrm{C}_1}$ denote effective lifetimes of ${\mathrm{N}_2}({\mathrm{C}}^3 \Pi_\mathrm{u}, v = 0)$ and ${\mathrm{N}_2}({\mathrm{C}}^3 \Pi_\mathrm{u}, v = 1)$ states, respectively. Note that $\tau_\mathrm{eff}^{\mathrm{C}_1}$ includes the influence of the process (7). In the case of N₂ at 200 Torr reaching values $\tau_\mathrm{eff}^{\mathrm{C}_0}$ = (8.0 $\div$ 12) ns, $\tau_\mathrm{eff}^{\mathrm{C}_1}$ = (4.0 $\div$ 5.6) ns [34]. The change of VDF in this time interval implies that the gain of the excited states population is already balanced by the losses due to quenching by N₂, confirming that the electric field does not further increase, since $k_\mathrm{im}^{\mathrm{C}_0}$ and $k_\mathrm{im}^{\mathrm{C}_1}$ increase with increasing electric field.
• (iii)  
  Interval 40–60 ns, the VDF reaches its minimum since the electric field is already very low. Higher vibrational levels $v\unicode{x2A7E}$ 1 are relaxed to the N₂(C$^3\Pi_\mathrm{u}, v = 0$). We observe that $\frac{\sum_{v = 1}^4[{\mathrm{N}_2}({\mathrm{C}}^3 \Pi_\mathrm{u}, v)]}{[{\mathrm{N}_2}({\mathrm{C}}^3 \Pi_\mathrm{u}, v = 0)]} \lt 10 \%$, corresponding to characteristic N₂(C$^3\Pi_\mathrm{u}$) state vibrational temperature of approximately 1300 K. The equations describing the ${\mathrm{N}_2}({\mathrm{C}}^3 \Pi_\mathrm{u}, v = 0)$ and ${\mathrm{N}_2}({\mathrm{C}}^3 \Pi_\mathrm{u}, v = 1)$ densities are the following:

  $$\begin{align} \frac{\mathrm{d} \left[{\mathrm{N}}_2\left({\mathrm{C}}_{0}\right)\right]}{\mathrm{dt}} & = k_\mathrm{im}^{\mathrm{C}_0} \left[{\mathrm{e}}\right] \left[{\mathrm{N}}_2\right] + k_\mathrm{rel}^\mathrm{C_{10}}\left[{\mathrm{N}}_2\left({\mathrm{C}}_{1}\right)\right]\left[{\mathrm{N}}_2\right] \nonumber\\ & \quad - \frac{\left[{\mathrm{N}}_2\left({\mathrm{C}}_{0}\right)\right]}{\tau_\mathrm{eff}^{\mathrm{C}_0}} << 0, \end{align} \tag{ 10 }$$

  $$\begin{eqnarray} &&\frac{\mathrm{d} \left[{\mathrm{N}}_2\left({\mathrm{C}}_{1}\right)\right]}{\mathrm{dt}} = k_\mathrm{im}^{\mathrm{C}_1} \left[{\mathrm{e}}\right] \left[{\mathrm{N}}_2\right] - \frac{\left[{\mathrm{N}}_2\left({\mathrm{C}}_{1}\right)\right]}{\tau_\mathrm{eff}^{\mathrm{C}_1}} << 0. \end{eqnarray} \tag{ 11 }$$
• (iv)  
  Interval 60–1000 ns, the switching to N₂(${\mathrm{A}}^3 \Sigma^+_\mathrm{u}$)-pooling-controlled VDFs. The N₂(${\mathrm{A}}^3 \Sigma^+_\mathrm{u}$)-pooling reactions affect observed populations of the N₂(C$^3{\Pi _{\textrm{u}}},v = 3 - {\textrm{4}}$) states at first (60–80 ns) since their population by the electron impact was low. In the later times (80–1000 ns) also, the lower N₂(C$^3{\Pi _{\textrm{u}}},v = 0 - {\textrm{2}}$) levels are populated by the pooling reactions [35, 36]. The VDF still changes when comparing the (140–350 ns) and (500–950 ns) cases. The change is due to different ${\mathrm{N}_2}({\mathrm{A}}^3 \Sigma_\mathrm{u}^+, v)$ vibrational levels populating the N₂(C$^3\Pi_\mathrm{u}, w$) states by the pooling reactions. These results will be further discussed in the section 3.7. The equations describing the ${\mathrm{N}_2}({\mathrm{C}}^3 \Pi_\mathrm{u}, v = 0)$ and ${\mathrm{N}_2}({\mathrm{C}}^3 \Pi_\mathrm{u}, v = 1)$ densities are the following:

  $$\begin{align} \frac{\mathrm{d} \left[{\mathrm{N}}_2\left({\mathrm{C}}_{0}\right)\right]}{\mathrm{dt}} & = k^\mathrm{pool}\left[{\mathrm{N}}_2\left({\mathrm{A}}\right)\right]^2 + k_\mathrm{rel}^\mathrm{C_{10}}\left[{\mathrm{N}}_2\left({\mathrm{C}}_{1}\right)\right]\left[{\mathrm{N}}_2\right] \nonumber\\ & \quad - \frac{\left[{\mathrm{N}}_2\left({\mathrm{C}}_{0}\right)\right]}{\tau_\mathrm{eff}^{\mathrm{C}_0}} = 0, \end{align} \tag{ 12 }$$

  $$\begin{eqnarray} &&\frac{\mathrm{d} \left[{\mathrm{N}}_2\left({\mathrm{C}}_{1}\right)\right]}{\mathrm{dt}} = k^\mathrm{pool}\left[{\mathrm{N}}_2\left({\mathrm{A}}\right)\right]^2 - \frac{\left[{\mathrm{N}}_2\left({\mathrm{C}}_{1}\right)\right]}{\tau_\mathrm{eff}^{\mathrm{C}_1}} = 0.\end{eqnarray} \tag{ 13 }$$

  The $k^\mathrm{pool}$ denotes the pooling reaction rate constant from metastable states and $[{\mathrm{N}}_2({\mathrm{A}})]$ is the $\sum_v$ [N$_2({\mathrm{A}}^3 \Sigma^+_\mathrm{u}, v)$]. The equations can be written in the steady state, since the nanosecond timescale of quenching dominates over the microsecond timescale of pooling reactions.

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Table 1. Comparison of the experimental and FCF-like VDFs. The SPS errors were estimated using three different distributions obtained from three SPS sequences ($\Delta v$ = 0, −2 and −3).

v	N2(C$^3\Pi_\mathrm{u}$, $v =$ 0–4) experiment 0–2 ns	N2(C$^3\Pi_\mathrm{u}$, $v =$ 0–4) FCF-like [32]	N$^+_2$(B$^2\Sigma_\mathrm{u}^+$, $v =$ 0–2) experiment 0–5 ns	N$^+_2$(B$^2\Sigma_\mathrm{u}^+$, $v =$ 0–4) FCF-like [32]
0	1.0	1.0	1.0	1.0
1	0.485 ± 0.045	0.558	0.063 ± 0.015	0.129
2	0.155 ± 0.020	0.193	0.002 ± 0.001	0.00 261
3	0.042 ± 0.008	0.054	Not observed	1.59×10−5
4	0.009 ± 0.002	0.014	Not observed	4.89×10−6

The conditions for the use of the intensity ratio method are fulfilled in the interval 0–20 ns, where the population of the N₂(C$^3{\Pi _{\textrm{u}}},v = 0 - {\textrm{4}}$) states is due to the electron-impact excitation also with the contribution of the quenching and relaxation processes.

Analogous to SPS, we analyze the VDF of the N$^+_2$(B$^2\Sigma_\mathrm{u}^+$) state. We have chosen 380–480 nm interval containing $\Delta v = 0$, −1 and −2 sequences of the FNS. Figure 11 shows synthetic bands of the FNS (upper trace) and SPS (lower trace), both calculated by assuming FCF-like VDFs (using triangular instrumental function and rotational temperature of 300 K). The $\Delta v = 0$ sequence of FNS shows relatively low intensities of bands originating from $v\gt$0 vibrational levels; furthermore, this region is affected by several SPS bands. The Δv = −1 sequence of FNS shows reasonable intensities of (0,1) and (1,2) bands while $\Delta v = -$2 also predicts weak (2,4) band at 460 nm. Upon inspection of the experimental spectra obtained during the first nanoseconds of the discharge, it is clear that the FNS system cannot be analyzed without separating the FNS bands from the dominant SPS emission as illustrated in figure 12. Upper panels (a),(b) in figure 12 show experimental spectra in 405–480 nm interval. All labeled SPS bands were subtracted from (a),(b) and the resulting residuals are plotted in the lower panels (c), (d). Residuals (c) in the 420–430 nm interval clearly show the shapes of the two FNS(0,1) and FNS(1,2) bands. Residuals (d) in the interval 455–475 nm reveal the shape of the FNS(0,2) band and a small peak at the position of the FNS(2,4) band. The residual peak occurring at 465 nm is due to overlapping almost coinciding SPS(4,10) and FNS(1,3) bands. In order to estimate the VDF, we used intensities of FNS(0,1) and FNS(1,2) from (c) and FNS(0,2) and FNS(2,4) from (d). Experimental VDF and FCF-like VDF are compared in table 1. The value of VDF for N$^+_2$(B$^2\Sigma_\mathrm{u}^+$, v = 1) level is notably below the FCF-like value. The possible explanation could be that the ratio of rate coefficients for N$^+_2$(B$^2\Sigma_\mathrm{u}^+$, v = 0) and N$^+_2$(B$^2\Sigma_\mathrm{u}^+$, v = 1) levels is not simply given by a ratio of FCFs; as already reported in the case of the ${\mathrm{N}_2}({\mathrm{C}}^3 \Pi_\mathrm{u})$ state under low $E/N$ conditions [3]. The primary reason for such behavior is the non-FCF scaling of ${{\textrm{N}}_2}({{\textrm{X}}^1}\Sigma _{\textrm{g}}^ + ,v = 0){{\textrm{N}}_2}({{\textrm{C}}^3}{\Pi _{\textrm{u}}},v = 0 - {\textrm{4}})$ cross-section amplitudes in the region of the excitation threshold, which results in an increasing contribution of threshold electrons to the total rate constant with decreasing $E/N$. Similar trends should be observed in the case of FNS at comparably higher $E/N$ due to the notable difference in the ${\mathrm{N}_2}({\mathrm{X}}^1 \Sigma^+_\mathrm{g}, v = 0) \longrightarrow {\mathrm{N}_2}({\mathrm{C}}^3 \Pi_\mathrm{u})$ excitation and ${\mathrm{N}_2}({\mathrm{X}}^1 \Sigma^+_\mathrm{g}, v = 0) \longrightarrow \mathrm{N}^+_2(\mathrm{B}^2\Sigma_\mathrm{u}^+)$ ionization thresholds. However, SPS and FNS spectra obtained over a wide range of $E/N$ would be required to provide explicit experimental confirmation of this statement, which is beyond the scope of this work. On the other hand, based on the same reasoning, close to FCF-like scaling should be expected in the case of the electronic excitation process $\mathrm{e} + \mathrm{N^{+}_2}({\mathrm{X}}^2 \Sigma_\mathrm{g}^+, v = 0) \longrightarrow \mathrm{e} + \mathrm{N^{+}_2}({\mathrm{B}}^2 \Sigma_\mathrm{u}^+, v^{^{\prime}})$; which would imply a completely different VDF of the $\mathrm{N^{+}_2}({\mathrm{B}}^2 \Sigma_\mathrm{u}^+)$ state (1:0.44:0.065). Therefore, based on the previous arguments, we believe that the obtained VDF can be considered a signature of a direct $\mathrm{e} + {\mathrm{N}_2}({\mathrm{X}}^1 \Sigma^+_\mathrm{g}, v = 0) \longrightarrow \mathrm{2e} + \mathrm{N}^+_2(\mathrm{B}^2\Sigma_\mathrm{u}^+)$ process at present conditions and justifies the use of the FNS/SPS method for $E/N$ evaluation.

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Determining the reduced electric field using the FNS/SPS intensity ratio technique includes many technical details described in the following paragraph. Figure 13(a) presents band intensities obtained by the spectrometer using the technique of kinetic series with 2 nanosecond MCP gate and step. This measurement informs us about the ratio of the FNS(0,0) and SPS(2,5) intensities, also showing that the FNS signal maximum precedes the SPS maximum, which is a common phenomenon caused by the different cross sections for electron impact excitation of the FNS and SPS upper states and the much shorter effective lifetime of the FNS compared to the SPS [37]. The photomultiplier (PMT) measurements in figure 13(b) present an accurate time evolution of signals, complementary to the kinetic series measurements, showing the SPS(2,4) intensity and a blend of FNS(0,0) and SPS(2,5) intensities, which are in the range of 388–395 nm. The short response time of PMT enables measurements with a time resolution of down to 100 ps, which is important for capturing small delays between emission curves from different excited states [37]. The signals were measured using the slits of 250 µm. The instant of the FNS signal maximum in figure 13(b) was shifted to correspond with the FNS signal maximum in figure 13(a). Then, the onset of the PMT signals begin at 7 ns, and from this instant, the measured signals start to grow continuously. The maximum of the FNS/SPS blend is reached approximately at 12 ns and precedes the maximum of the SPS(2,4) signal, which is reached approximately at 13.5 ns. Both signals are scaled to 1. The FNS-narrow filter is defined by its transmission function: $T = T_0 \exp(-\sigma (\lambda-\lambda_0)^2)$ with parameters T₀, λ₀ and σ being 33.6%, 391.5 nm and 1.05, respectively. The FNS(0,0) and SPS(2,5) blend needs to be resolved into individual components. Due to the much lower effective lifetime of the FNS(0,0) compared to SPS(2,5), we can consider that the tail of the FNS/SPS blend is created only due to the pure SPS(2,5) signal. The SPS(2,5) profile scaled by a factor 0.68 to fit the tail of the FNS/SPS blend is labeled using blue dashed line. The difference of the FNS/SPS blend and the SPS(2,4) signal scaled to fit the tail of the FNS/SPS blend, then provides the pure FNS(0,0) signal labeled using red solid line.

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The dependence $R(E/N)$ relating the intensity ratio and reduced electric field [19]:

$$\begin{equation} {R}\left(E/N\right) = \frac{\displaystyle \strut \tau_\mathrm{eff}^\mathrm{\scriptscriptstyle FNS} \frac{\mathrm{d} I_\mathrm{\scriptscriptstyle FNS}}{\mathrm{d} t} + I_\mathrm{\scriptscriptstyle FNS}} {\displaystyle \strut \tau_\mathrm{eff}^\mathrm{\scriptscriptstyle SPS}\frac{\mathrm{d} I_\mathrm{\scriptscriptstyle SPS}}{\mathrm{d} t} + I_\mathrm{\scriptscriptstyle SPS}} \cdot \, \frac{g_\mathrm{\scriptscriptstyle SPS}}{g_\mathrm{\scriptscriptstyle FNS}} = a \exp \left(-b\left(E/N\right)^{-0.5}\right) , \end{equation} \tag{ 14 }$$

was used to obtain the reduced electric field. The $I_\mathrm{FNS, SPS}$ are the measured intensities of respective spectral bands and $\tau_\mathrm{eff}^\mathrm{\scriptscriptstyle FNS, SPS}$ their effective lifetimes. At the investigated pressure of 200 Torr, the values of the effective lifetimes are $\tau_\mathrm{eff}^\mathrm{B}$ = (0.15 $\div$ 0.24) ns and $\tau_\mathrm{eff}^\mathrm{C_2}$ = (3.1 $\div$ 3.5) ns [34]. The $g_\mathrm{FNS, SPS}$ are factors denoting the ratio of the bandhead intensity and the integral band intensity of the respective FNS and SPS bands, which are used to account for the different spectrometric representations of FNS(0,0) and SPS(2,5) bands. These factors depend on the $T_\mathrm{rot}$ and instrumental function parameter $\Delta_\mathrm{1/2}$. In our case $\Delta_\mathrm{1/2}$ = 0.19 nm and $T_\mathrm{rot}$ = 300 K, which, following the procedure suggested in [34], leads to the value $\frac{g_\mathrm{\scriptscriptstyle SPS}}{g_\mathrm{\scriptscriptstyle FNS}} = 0.75$. The FNS(0,0) and SPS(2,5) ratio at the time of the FNS(0,0) maximum determined using kinetic series is 0.185. Therefore, the value of 0.185 × 0.75 = 0.139 was used to scale the normed PMT signals, leading to the absolute values presented in figure 13(c). The $E/N$ with its uncertainty band was determined based on the equation (14), where we used parameters $a_\mathrm{lower} = 33.358$, $b_\mathrm{lower} = 100.1$ and $a_\mathrm{upper} = 13.0$, $b_\mathrm{upper} = 109.9$, for lower and upper boundaries of the calibration range $R(E/N)$. Note that the uncertainty springs from the uncertainty of the rate constants defining the $R(E/N)$ relation [34]. The computed $E/N$ reaches the value of 380±100 Td in its peak, and the peak precedes the FNS(0,0) peak by approximately 2 ns. Note, that the instant of $E/N$ peak does not match with the FNS(0,0) peak due to the nature of equation (14). Simply put, the increase in the electric field causes the increase of electron density; therefore, the peaks of FNS(0,0) and SPS signals follow with some delay after the peak of $E/N$. Figure 13(d) shows the contributions of the different components of the formula (14). It is apparent that the numerator of the formula (14) is defined mainly by the $I_\mathrm{\scriptscriptstyle FNS}$, since the short effective lifetime of N$^+_2$(B$^2\Sigma_\mathrm{u}^+$) state causes instant equilibrium of the FNS upper state emission and population. Thus, the FNS derivative part in the formula (14) can be neglected ($\displaystyle \strut \tau_\mathrm{eff}^\mathrm{\scriptscriptstyle FNS} \frac{\mathrm{d}I_\mathrm{\scriptscriptstyle FNS}}{\mathrm{d}t} \lt\lt I_\mathrm{\scriptscriptstyle FNS}$). In the denominator (SPS part), during the increase of the emission, the derivative component ($\displaystyle \strut \tau_\mathrm{eff}^\mathrm{\scriptscriptstyle SPS} \frac{\mathrm{d}I_\mathrm{\scriptscriptstyle SPS}}{\mathrm{d}t}$) dominates over the absolute value ($I_\mathrm{\scriptscriptstyle SPS}$). Later, about 13.5 ns after the emission onset, the ($I_\mathrm{\scriptscriptstyle SPS}$) prevails the derivative component, and the contribution of the derivative component is minor. Owing to the quite noisy nature of the $I_\mathrm{\scriptscriptstyle FNS}$ for $t \gt$ 15 ns, we restricted the reduced electric field determination just to $t \in$ (8, 15) ns.

The kinetic model used is similar to the one in detail described in [12]; moreover, the electron processes from [11] were added to describe the build-up of the electron density. The model relies on the reduced electric field obtained as a fit of the $E/N$ data determined in the previous section. The fit, labeled by the black dashed line, shown in figure 13(c) was performed using the sigmoid function suggested for streamer pulses in [11] and defined by the following formula:

$$\begin{align*} E/N\left(t\right) = \left\{ \begin{array}{ll} 0 & \mathrm{if} \, t \unicode{x2A7D} t_0 \\ \frac{A}{N}\left(g\left(t,t_\mathrm{c}, w_1, w_1\right) \times \left[1-g\left(t,t_\mathrm{c}, -w_1, w_3\right)\right]\right) & \mathrm{if} \, t > t_0 \\ \end{array} \right. \end{align*}$$

where

$$\begin{equation} g\left(x,\alpha, \beta, \gamma\right) = \left[1+\exp{-\left(x-\alpha+\beta/2\right)/\gamma}\right]^{-1}.\end{equation} \tag{ 15 }$$

The parameters $A, t_\mathrm{0}, t_\mathrm{c}, w_1, w_3$ were obtained by the fit and reached the values: 216.8 kV m⁻¹, 7.74 ns, 8.38 ns, 0.235 ns and 4.09 ns, respectively. The non-zero initial densities ${\mathrm{N}_2}({\mathrm{A}}^3 \Sigma^+_\mathrm{u}, v = 0)$, ${\mathrm{N}_2}({\mathrm{A}}^3 \Sigma^+_\mathrm{u}, v = 1)$ and N(⁴S) are set according to the experimental results of [16, 18] to be 10¹³, 10¹³ and 7·10¹⁴ cm⁻³, respectively, since, a certain density of these species is already built up within two breakdowns preceding the third breakdown during which measurements were performed [16, 18]. The initial electron density was defined to achieve a maximum electron density between 10¹³ and 10¹⁴ cm⁻³. The electron density is related to the electric field pulse, increasing exponentially at the onset of the streamer pulse due to electron-impact ionization of N₂. The loss of electrons in the model is due to electron-ion recombination, which is more efficient at lower $E/N$. Note that the 0D modeling cannot capture all of the experimentally observed features since some are also related to spatial characteristics during streamer propagation, where both electric field and electron density are strongly spatially dependent.

Figure 14 shows the comparison of the ${\mathrm{N}_2}({\mathrm{A}}^3 \Sigma^+_\mathrm{u}, w)$ density profiles obtained from the experiment [16] and model, using $E/N(t)$ from figure 13(c) as input. Despite all the approximations involved in the calculations, a good agreement on the microsecond timescale is found. The non-monotonic behavior observed in figure 14 for N₂(A$^3 \Sigma^+_\mathrm{u}$, $v =$ 5–9) has already been reported in [16] and is probably the result of two opposing processes: quenching due to vibrational relaxation and excitation by pooling process (16) followed by radiative quenching via SPS. Simply put, once the population of levels v = 0 and 1 of the N₂(A$^3 \Sigma^+_\mathrm{u}$) metastable state reaches certain values due to vibrational relaxation, the pooling reaction refills the N₂(B$^3\Pi_\mathrm{g}$) and N₂(C$^3\Pi_\mathrm{u}$) states and their relaxation (collision-radiative) causes a local increase in the population of $v =$ 5–9 levels of the N₂(A$^3 \Sigma^+_\mathrm{u}$) metastable.

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Also, the time of the N₂(C$^3\Pi_\mathrm{u}$,$v =$ 0–4) peaks obtained by the model (solid lines in figure 15(a)) show good agreement with the experimental data (points in figure 9(b)). The decay of the N₂(C$^3\Pi_\mathrm{u}$,$v =$ 0–4) densities for $t \in$ (10, 100) ns is much faster than in the experiment, which reports that the excited state source term ($k^\mathrm{C}_\mathrm{im} n_\mathrm{e} n_{\mathrm{N}_2}$) in the model diminishes faster than in the experiment. However, to account for this effect, it would require the development of a high-dimensional fluid model, as in [38, 39]. For t > 100 ns, the 0D model predicts the increase of N₂(C$^3\Pi_\mathrm{u}$, $v =$ 0–4) densities and their peak at $t \approx$ 6 µs due to the pooling from the ${\mathrm{N}_2}({\mathrm{A}}^3 \Sigma^+_\mathrm{u}, v = 0)$ and ${\mathrm{N}_2}({\mathrm{A}}^3 \Sigma^+_\mathrm{u}, v = 1)$ metastable states, see solid lines in figure 15(a). The disagreement of the model and experiment for $t \in$ (60, 1000) ns is substantial since the experiment predicts continuous decay of the N₂(C$^3\Pi_\mathrm{u}$, $v =$ 0–4) densities, which did not change even for $t \gt$ 1000 ns, where we obtain spectra at very low signal-to-noise ratio. This is a sign of the possible role of the pooling reactions from the higher vibrational levels of the metastable state ${\mathrm{N}_2}({\mathrm{A}}^3 \Sigma^+_\mathrm{u})$. The pooling reactions can be written generally, for pooling from v and w vibrational levels of ${\mathrm{N}_2}({\mathrm{A}}^3 \Sigma^+_\mathrm{u})$, as:

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$$\begin{align} {{\textrm{N}}_2}\left( {{{\textrm{A}}^3}\Sigma _{\textrm{u}}^ + ,v} \right) + {{\textrm{N}}_2}\left( {{{\textrm{A}}^3}\Sigma _{\textrm{u}}^ + ,w} \right) \to {{\textrm{N}}_2}{\textrm{(}}{{\textrm{C}}^3}{\Pi _{\textrm{u}}},{v^\prime } = 0 - {\textrm{4 + }}{{\textrm{N}}_{\textrm{2}}}. \end{align} \tag{ 16 }$$

However, only rate constants for (v, w) combinations (0,0); (1,0) and (1,1) were determined [35, 40] and are the only pooling processes included in the 0D kinetic model. The remaining (v, w) combinations were never inspected since it is very challenging to match the metastable levels participating in the pooling reactions with the specific N₂(C$^3\Pi_\mathrm{u}$, vʹ = 0–4) products. Moreover, the procedures based on reactants' and products' energy balance also fail; thus, it is difficult to estimate these rate constants [35]. In our model, where we consider 22 vibrational levels of ${\mathrm{N}_2}({\mathrm{A}}^3 \Sigma^+_\mathrm{u})$, we could potentially arrive at $\frac{n(n+1)}{2}$ = 253 combinations of (v, w) for n = 22. This can be simplified if we realize that the source term of pooling reactions is proportional to the $[{\mathrm{N}_2}({\mathrm{A}}^3 \Sigma^+_\mathrm{u}, v)]\cdot[{\mathrm{N}_2}({\mathrm{A}}^3 \Sigma^+_\mathrm{u}, w)]$ product, which can be analyzed for the different (v, w) combinations in our specific case of streamer afterglow, supposing similar pooling rate constant magnitude for all combinations. Figure 15(b) shows the twelve most important $[{\mathrm{N}_2}({\mathrm{A}}^3 \Sigma^+_\mathrm{u}, v)]\cdot[{\mathrm{N}_2}({\mathrm{A}}^3 \Sigma^+_\mathrm{u}, w)]$ products, having the maximum higher than 3·10$^{27}\,$cm⁻⁶. It is apparent that the (v, w) combinations (1,1); (0,0) and (1,0) dominate for $t \gt$ 2 µs, while the combinations (3,3); (3,2); (4,3) and (5,3) for $t \in$ (0, 2) µs. To illustrate the potential influence of the pooling from the ($v\gt$1, $w\gt$1) level combinations, we have added to the model the important pooling process within (0, 2) µs time interval, (v, w) = (3,3):

$$\begin{align} \begin{array}{l} {{\textrm{N}}_2}\left( {{{\textrm{A}}^3}\Sigma _{\textrm{u}}^ + ,v = 3} \right) + {{\textrm{N}}_2}\left( {{{\textrm{A}}^3}\Sigma _{\textrm{u}}^ + ,w = 3} \right)\\ \,\,\,\,\,\,\,\,\,\, \to {{\textrm{N}}_2}{{\textrm{C}}^3}{\Pi _{\textrm{u}}},{v^\prime } = 0 - {\textrm{4 + }}{{\textrm{N}}_{\textrm{2}}}, \end{array} \end{align} \tag{ 17 }$$

its pooling reaction rates $k^{v^{^{\prime}}}_\mathrm{p}$ reach values (1.0, 2.0, 1.0, 0.7, 0.6) × 10⁻¹¹ cm³s⁻¹ for respective ${{\textrm{N}}_2}({{\textrm{C}}^3}{\Pi _{\textrm{u}}},{v^\prime } = 0 - {\textrm{4}})$ vibrational levels, chosen to be the same as the value for (v, w) = (1,1) combinations suggested by Piper [35]. Figure 15(a) now compares the N₂(C$^3\Pi_\mathrm{u}$, vʹ = 0–4) densities from the model, where we do not consider (solid line) and consider (dashed line) the process (17). The agreement of the model with the experiment improves, which gives us the confirmation that the inclusion of the pooling from ($v\gt$1, $w\gt$1) level combinations is important for $t \in$ (0, 2) µs in our case of N₂ at 200 Torr and could be important even for longer times for reduced N₂ pressures.

We obtained fundamental information about the onset of luminosity and subsequent Townsend-to-streamer transition. Thanks to the 4-channel ICCD framing camera, we were able to observe µs formation of diffuse phase covering the anode dielectric surface from which a single cathode directed streamer is born. The propagation of the streamer according to the velocity measurements takes approximately 3 ns and it is in agreement with the $E/N$ peak, which has a rise time shorter than 2 ns according to the FNS/SPS intensity ratio measurements. After the streamer passes through the gap, a surface streamer develops on the dielectric cathode. Moreover, the voltage and current measurements performed in the section 3.1 inform us about the charge leak created by the discharge through the flow of the current to the conductive parts of the discharge chamber. The 14 ns delay between the onset of the current flowing to dielectrics ($I_{C1}$) and current flowing to conductive parts of the chamber ($I_\mathrm{leakage}$) is consistent with the results observed by the 4-channel ICCD in section 3.2, since the bottom row of the figure 6 for $t \gt$ T₀ + 14 ns shows the light emission also from distant parts of the discharge chamber. This can be a sign of the charge transfer to the grounded chamber walls through the surface streamer developing in the vicinity of both electrodes.

Our experimental results can be compared with the modeling works of Yurgelenas and Wagner [38] and Papageorghiou et al [39]. Yurgelenas and Wagner [38] developed a model of DBD in air at atmospheric pressure in the plane-to-plane geometry with the gas gap of 1.2 mm. In their model, they found two different modes during the discharge initiation: (i) the diffuse discharge and (ii) the ionization wave (streamer). Moreover, they observed a different velocities in these two modes. The streamer mode has approximately twice the speed of the diffuse discharge which is similar as in our case, where we determined velocities 9.8×10⁵ m s⁻¹ and 2.4×10⁶ m s⁻¹ in the anode and the cathode parts of the discharge gap, respectively. According to Yurgelenas and Wagner [38] the inclusion of the ion–electron emission from the cathode dielectric is sufficient to explain the formation of this type of the discharge and the change of the initial conditions in their model (residual surface charges, secondary emission coefficient from cathode) affects the prevalence of the diffuse or ionization wave character and shifts the boundary between these two modes.

In the work of Papageorghiou et al [39] the interaction of a volume streamer with a dielectric cathode is studied in N₂ at atmospheric pressure using a two-dimensional model. They consider plane-to-plane geometry with the gas gap of 1 mm and observe that the electric field increases from 120 kV cm⁻¹ to 300 kV cm⁻¹ during the streamer arrival on the cathode. These electric fields correspond to the values of 450–1100 Td, higher than the values determined in the present work (380 ± 100 Td). Papageorghiou et al [39] model the discharge just by incorporating the photoionization and photoemission from the cathode and they show that the photoionization is a key process also affecting the propagation of surface streamer on the cathode. They explain in detail several effects also observed in our case. The observed glow near the cathode is caused by the volume streamer impact and subsequent creation of surface streamer on the cathode dielectric. They determined the surface streamer velocity to be 4.5$\times10^5$ m s⁻¹ and the velocity was found to be decreasing with radial distance from the cathode spot. This value is close to the surface streamer velocity determined in our case, which is (2±1)$\times10^5$ m s⁻¹. After the volume streamer contact with the cathode, a local minimum of the axial $E/N$ magnitude is predicted behind the surface streamer layer. This phenomenon is also present in our experiment since we observe the emission extinction at the boundary of the volume and the surface streamers in figures 4 and 6. The volume streamer velocity determined in [39] is ∼7×10⁵ m s⁻¹, close to the value determined in section 3.3. Papageorghiou et al [39] also observe that after the contact of the streamer with the cathode, the electron density in the middle of the gap still doubles. Such behavior is not present in our 0D model, but this could actually explain the slower relaxation of N₂(C$^3\Pi_\mathrm{u}$) observed by the experiment compared to results of 0D model.