Streamer-induced kinetics of excited states in pure N₂: II. Formation of N₂(B$^3\Pi_\mathrm{g}, v = 0\textrm{-}21$) through analysis of emission produced by the first positive system

The experimental setup was described in detail in part I [19] of this work, therefore, we recall only the most important facts. A mono-filamentary streamer discharge is produced in a 4 mm gap in a DBD point-plane electrode geometry, as shown in figure 1(a). A specific high-voltage (HV) waveform based on periodic (10 Hz) bursts composed of two consecutive HV AC waveforms (1 kHz) and a nanosecond HV pulse was used to power the discharge. The HV pulse was superimposed with an appropriate phase shift during the second positive AC half-cycle. Within each burst, the superimposed HV pulse produces a triggered streamer mono-filament that is locked with respect to the pulse onset (typical rising slope is 0.2 kVns⁻¹) and captured in figure 1(b). 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 events. All ICCD spectra were acquired from the centre of the gap simultaneously using iHR-320 and Shamrock 303i spectrometers; see figure 1(c).

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The acquired vis–NIR emission spectra (ICCD kinetic series) were corrected for the spectral response of the respective ICCD spectrometers. Corrected spectra were then analysed using the synthetic FPS emission model [1, 20, 21]. As in the case of the Second Positive System (SPS) and First Negative System (FNS) analysis performed in part I [19], the approach is based on the calculation of a set of band profiles by considering the actual experimental instrumental function of the ICCD spectrometer and assuming a Boltzmann temperature ($T_\mathrm{rot}$ = 300 K) to calculate populations of the upper rotovibronic levels of the B$^3\Pi_\mathrm{g}$ state. Compared to SPS/FNS, the FPS analysis is much more difficult because the FPS involves more upper vibrational levels, and, as a result, FPS band sequences cover larger spectral intervals with significant overlap between neighbouring sequences. Another problem is related to the fact that the lowest N₂(B$^3\Pi_\mathrm{g}, v = 0\textrm{-}2$) vibrational levels produce bands in the NIR range; for example, the FPS(0,0) band emits at 1050 nm [1], while the higher N₂(B$^3\Pi_\mathrm{g}, v\gt 9$) levels emit the strongest bands in the visible range (below 650 nm). Figures 2 and 3 show two sets of synthetic FPS band profiles used to analyse the FPS emission acquired with low spectral resolution (using 150 and 300 G mm⁻¹ gratings) and medium spectral resolution (using 1200 G mm⁻¹ grating), respectively. All plotted FPS bands were synthesized for equilibrium rotational temperature $T_\mathrm{rot}$ = 300 K and given instrumental function of the ICCD spectrometer. The relative intensity of the FPS bands corresponds to the same (unit) population of all vibronic levels; therefore, each FPS band contributes to the resulting complete FPS spectrum through a weight given by the respective multiplication factor (given by the VDF). It can also be seen that for the analysis of higher ($v \unicode{x2A7E} 9$) vibrational levels, the best choice is the spectral range between 500 and 650 nm, as shown in figures 2(a) and 3, while in the case of the lowest (v < 3) levels, the interval 850–1100 nm is a must, see figure 2(b). This implies the need for (and justifies the use of) two different ICCD spectrometers. Each spectrometer is optimized for a different spectral range, aiming to obtain the FPS($v, v-\Delta v$) emission ($\Delta v$ = 0, +1,...,+5 sequences) in the interval 500–1100 nm and subsequently, to complete VDF of the N₂(B$^3\Pi_\mathrm{g}$) state. To determine the tail of the VDF, the spectral interval between 500–580 nm containing many FPS bands originating from high vibrational levels (v > 10) is particularly important. However, this spectral range is affected by interfering bands originating from other emission systems as discussed further in the text. Therefore, a higher spectral resolution, such as demonstrated in figure 3, is needed to correctly distinguish the FPS band structures from other N₂ and N$_2^+$ bands, especially during the first few nanoseconds.

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When obtaining FPS with high spectral resolution (sufficient to partially resolve the structure of individual bands) and a good signal-to-noise ratio, as developed in [20], a simplified approach based on successive subtraction of synthetic FPS bands within a given sequence from the experimental spectra works quite well, and it allows obtaining VDF with good accuracy. Under the current experimental conditions, i.e. low-resolution spectra acquired at high temporal (ns) resolution (often with very low signal-to-noise ratio $\lt 1$), such a simplified analytical approach would not work (the error associated with determining the initial part of the VDF would propagate and gradually accumulate toward the end of the VDF). Therefore, we developed and used a more sophisticated approach where the set of populations is fitted using a least-squares technique. The expression to fit the experimental spectrum is the following:

$$\begin{align} \mathcal{S}_\mathrm{FPS}\left(\lambda\right) = \mathcal{F} \sum_{v = 0}^{v_\mathrm{max}} p_v\cdot \mathrm{FPS}_v^\mathrm{synt}\left(\lambda\right)\ , \end{align} \tag{ 1 }$$

where $\mathcal{F}$ is a normalization constant; $p_v = [\mathrm{N_2(B^3} \Pi_\mathrm{g},\textit{v}\mathrm{)]/}$ $\mathrm{\sum_{\textit{v} = 0}^{\textit{v}_\mathrm{max}}[N_2(B^3} {\Pi_\mathrm{g}}, \mathrm{\textit{v})}]$ is the relative population of the vibrational level v; $\mathrm{FPS}_v^\mathrm{synt}(\lambda)$ is the synthetic spectrum of the vibrational level v that contains the summed contributions of the $\Delta v$ = 0, +1,...,+5 sequences, calculated for a unit vibrational population [1, 20]; and $v_\mathrm{max}$ is the maximum vibrational level containing bands in the fitted region. $\mathcal{F}$ and the various values of p_v are fitted such that the distance between the experimental spectrum ($\mathcal{S}^\mathrm{exp}_\mathrm{FPS}(\lambda)$) and the synthetic spectrum ($\mathcal{S}^\mathrm{fit}_\mathrm{FPS}(\lambda)$) is minimal. It is important to note that the proper use of this approach requires the exclusion of spectral intervals with potential overlaps with other emissions, such as atomic lines, SPS bands, HIR bands, etc., from the fitting algorithm. These emissions may be hidden under the FPS spectrum and interfere during specific discharge phases. First, the $\mathcal{S}^\mathrm{exp}_\mathrm{FPS}(\lambda)$ is fitted through a simpler approach where the populations p_v are assumed to follow a Boltzmann distribution at a certain temperature T_v . Then, the results of the preliminary fit are used as initial parameters for the fit where the vibrational populations can have an arbitrary distribution. This procedure allows convergence for all the spectra analysed in this work.

The kinetic model used to describe the population and VDFs of N₂(B$^3\Pi_\mathrm{g}, v = 0\textrm{-}21$) levels is based on the reaction scheme listed in [22]. In comparison with the kinetic scheme presented in [22], we added some reactions and species potentially important for the N₂(B$^3\Pi_\mathrm{g}$) state:

• N₂(B$^{{\prime} 3}\Sigma_\mathrm{u}^-$) state, which levels are energetically close to N₂(B$^3\Pi_\mathrm{g}, v\unicode{x2A7E}4$) levels,
• radiative processes connected with the emission of Wu–Benesch system (N₂(W$^3\Delta_\mathrm{u}$) $\longrightarrow$ N₂(B$^3\Pi_\mathrm{g}$)) and infrared afterglow (N₂(B$^{{\prime} 3}\Sigma_\mathrm{u}^-$) $\longrightarrow$ N₂(B$^3\Pi_\mathrm{g}$)) [1],
• predissociation of N₂(B$^3\Pi_\mathrm{g}, v = 13\textrm{-}18$) levels based on [16].
• We modified some of the quenching reactions, the products of N₂(C$^3\Pi_\mathrm{u}$) quenching by molecular nitrogen are now considered to be transferred to higher levels of N₂(W$^3\Delta_\mathrm{u}$) and N₂(B'$^3\Sigma_\mathrm{u}^-$) states, as suggested in [23].

The kinetic model contains 160 species participating in 7312 reactions. The species are listed in table 1, and the reaction scheme is presented in the supplementary material.

Table 1. Species considered in the full kinetic model($^*$).

Charged species	e, N$_2^+$, N$_4^+$
Neutral species	N2(X$^1\Sigma_\mathrm{g}^+$,v = 0–58), N2(A$^3\Sigma_\mathrm{u}^+$,v = 0–21), N2(B$^3\Pi_\mathrm{g}$,v = 0–21),
	N2(B$^{{\prime} 3}\Sigma_\mathrm{u}^-$,v = 0–21), N2(W$^3\Delta_\mathrm{u}$,v = 0–21), N2(C$^3\Pi_\mathrm{u}$,v = 0–4),
	N2(C$^{{\prime} ^{\prime} 5}{\Pi}_\mathrm{u}$,v = 2–3), N(4S), N(2D), N(2P)

($^*$) conditions considered: p = 200 Torr, $T_\mathrm{gas}$ = 300 K

The initial value problem for the reaction-resulting ordinary differential equations is integrated in time using an implicit solver, RADAU5 [24], with the reduced electric field ${\frac{E}{N}}(t)$ as a parameter. The ${\frac{E}{N}}(t)$ profile was obtained using the FNS/SPS intensity ratio method, as described in part I [19]. 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 [25, 26] 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) [25, 26]. The initial electron density was defined to achieve a maximum electron density between 10¹³ and 10¹⁴ cm⁻³ and to obtain the best agreement with the experimental data of [25]. In the model at the current conditions, the excited states are produced by electron impact excitation from the ${\mathrm{N}_2}(\mathrm{X}^1 \Sigma^+_\mathrm{g}, v = 0)$, and we consider that the specific vibrational level populations correspond to FCF-like distribution. The excited states can be converted to other states in intramolecular and intermolecular energy transfers. During the intramolecular energy transfer, the collisional partner (in this work purely ${\mathrm{N}_2}(\mathrm{X}^1 \Sigma^+_\mathrm{g}, v = 0)$) of the excited electronic state N₂(Z) does not change its vibrational level:

$$\begin{align} & \mathrm{N}_2\left({Z}, v^{{\prime}}\right) + \mathrm{N}_2\left(\mathrm{X}^1 \Sigma^+_\mathrm{g},v = 0\right) \rightarrow \mathrm{N}_2\left({Y}, v^{^{\prime\prime}}\right) \notag\\ & \quad + \mathrm{N}_2\left(\mathrm{X}^1 \Sigma^+_\mathrm{g},v = 0\right), \end{align} \tag{ 2 }$$

and therefore, intramolecular energy transfers can be probable in the case of the low energy difference between the reactant (N₂(Z, vʹ)) and the product (N₂(Y, vʹʹ)) vibrational levels. The considered intramolecular energy transfers are between the N₂(B$^3\Pi_\mathrm{g}$) and N₂(A$^3\Sigma_\mathrm{u}^+$), N₂(W$^3\Delta_\mathrm{u}$), N₂(B$^{{\prime} 3}\Sigma_\mathrm{u}^-$), N₂(C$^3\Pi_\mathrm{u}$) states, because the selection rules in the process allow just u$\leftrightarrow$g transitions for homonuclear molecules, and we assume that the allowed transitions are favoured in the collisional transitions.

In the intermolecular energy transfer, the energy lost by the quenching of the excited state transfers to the higher vibrational levels of the collisional partner (in this case to $\mathrm{N}_2(\mathrm{X}^1 \Sigma^+_\mathrm{g},w^{^{\prime\prime}}\unicode{x2A7E}0)$):

$$\begin{align} & \mathrm{N}_2\left({Z}, v^{{\prime}}\right) + \mathrm{N}_2\left(\mathrm{X}^1 \Sigma^+_\mathrm{g},w^{{\prime}} = 0\right) \rightarrow \mathrm{N}_2\left(\mathrm{X}^1 \Sigma^+_\mathrm{g},w^{^{\prime\prime}}\unicode{x2A7E}0\right) \notag\\ & \quad + \mathrm{N}_2\left({Y}, v^{^{\prime\prime}}\right). \end{align} \tag{ 3 }$$

The experimental distinction between the intramolecular and intermolecular processes was made just in one rare study [27], where the authors found out that the intramolecular and intermolecular components defining the transfer of N₂(A$^3\Sigma_\mathrm{u}^+$) and N₂(W$^3\Delta_\mathrm{u}$) states into the N₂(B$^3\Pi_\mathrm{g}$) state are comparable. The theoretical differentiation between the intramolecular and intermolecular processes applying the Landau–Zener and Rosen–Zener approximations was performed by Kirillov [28–30] and similar calculations were also done to obtain the rate constants for intramolecular and intermolecular processes to construct the kinetic scheme for the presented model.

The Rosen–Zener approximation is used for {N₂(A$^3\Sigma_\mathrm{u}^+$), N₂(W$^3\Delta_\mathrm{u}$), N₂(B$^{{\prime} 3}\Sigma_\mathrm{u}^-$)} $\rightarrow$ N₂(B$^3\Pi_\mathrm{g}$) endothermic and N₂(B$^3\Pi_\mathrm{g}$) $\rightarrow$ {N₂(A$^3\Sigma_\mathrm{u}^+$), N₂(W$^3\Delta_\mathrm{u}$), N₂(B$^{{\prime} 3}\Sigma_\mathrm{u}^-$)} exothermic transitions [28]. Furthermore, the Rosen–Zener approximation is used for intermolecular transitions defined by the equation (3) for all combinations of $Z~\rightarrow~Y$, where Z, $Y \in \{$N₂(A$^3\Sigma_\mathrm{u}^+$), N₂(W$^3\Delta_\mathrm{u}$), N₂(B$^{{\prime} 3}\Sigma_\mathrm{u}^-$), N₂(C$^3\Pi_\mathrm{u}$)} and also for N₂(B$^3\Pi_\mathrm{g}$) $\rightarrow$ N₂(B$^3\Pi_\mathrm{g}$) transition. The Rosen–Zener approximation is defined as:

$$\begin{align} k_\mathrm{R}\left(\Delta E, T\right) = g_{Y} f k_0 \sqrt{\frac{T}{T_0}} \exp{\left(-\frac{|\Delta E|}{\gamma \sqrt{T/T_0}} + \frac{\Delta E}{2 k_\mathrm{B} T} \right)} \end{align} \tag{ 4 }$$

where $k_0 = \frac{1}{6}\times10.2\times$10⁻¹⁰ cm³s⁻¹, γ = 105 cm⁻¹ ($k_0 = \frac{1}{6}\times2.63\times$10⁻⁹ cm³s⁻¹, γ = 200 cm⁻¹ for collisions involving N₂(C$^3\Pi_\mathrm{u}$)). Note that the k₀ and γ used in this work are those recommended by [23, 29, 31] based on comparison with experimental data. The g_Y factors are statistical weights denoting the degeneracy of given electronic states, $g_{Y} = 6$ for $Y \in$ {N₂(B$^3\Pi_\mathrm{g}$), N₂(W$^3\Delta_\mathrm{u}$), N₂(C$^3\Pi_\mathrm{u}$)} and $g_{Y} = 3$ for $Y \in$ {N₂(A$^3\Sigma_\mathrm{u}^+$), N₂(B$^{{\prime} 3}\Sigma_\mathrm{u}^-$)}. The temperature T stands here to express the dependence of the rate coefficient on the neutral gas temperature, and T₀ is the room temperature, but in this work, we set $T = T_0 = 300$ K. The f represents the Franck–Condon factors used during the transition, where $f = q_{v^{{\prime}}v^{^{\prime\prime}}}$ and $f = q_{v^{{\prime}}w^{^{\prime\prime}}} q_{v^{^{\prime\prime}}w^{{\prime}}}$ for intramolecular and intermolecular processes, respectively. The used Franck–Condon factors are from [32]. The energy differences were calculated using the following formulas: $\Delta E = \varepsilon_{Z, v^{{\prime}}} - \varepsilon_{Y, v^{^{\prime\prime}}}$ and $\Delta E = \varepsilon_{Z, v^{{\prime}}} - \varepsilon_{Y, v^{^{\prime\prime}}} - \varepsilon_{\mathrm{N}_2(\mathrm{X}^1 \Sigma^+_\mathrm{g},w^{^{\prime\prime}}\unicode{x2A7E}0)}$ for intramolecular and intermolecular processes, respectively. The energy levels of the investigated electronic states were calculated using formulas and vibrational constants from [33]. The vibrational levels of N₂(B$^3\Pi_\mathrm{g}$) and N₂(A$^3\Sigma_\mathrm{u}^+, v\gt 7$), N₂(W$^3\Delta_\mathrm{u}$), N₂(B$^{{\prime} 3}\Sigma_\mathrm{u}^-$), N₂(C$^3\Pi_\mathrm{u}$) states considered in the model are shown in figure 4 and labelled with solid lines (higher levels marked by dashed lines were not implemented because of missing FCFs). The Landau–Zener approximation is applied to {N₂(A$^3\Sigma_\mathrm{u}^+$), N₂(W$^3\Delta_\mathrm{u}$), N₂(B$^{{\prime} 3}\Sigma_\mathrm{u}^-$)} $\rightarrow$ N₂(B$^3\Pi_\mathrm{g}$) exothermic transitions and it is the same for endothermic transitions N₂(B$^3\Pi_\mathrm{g}$) $\rightarrow$ {N₂(A$^3\Sigma_\mathrm{u}^+$), N₂(W$^3\Delta_\mathrm{u}$), N₂(B$^{{\prime} 3}\Sigma_\mathrm{u}$)} [28] as:

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$$\begin{align} & k_\mathrm{L}\left(\Delta E, T\right) \notag\\ & \quad = g_{Y} f k_0 \sqrt{\frac{T}{T_0}} \left(1 + \frac{|\Delta E|}{\beta T}\right) \exp{\left(-\frac{|\Delta E|}{\beta T} + \frac{\Delta E}{2 k_\mathrm{B} T} \right)}. \end{align} \tag{ 5 }$$

where β = 0.466 cm⁻¹K⁻¹. Note that the definition of rate constants as described using equations (4) and (5) also satisfies the principle of detailed balance for reverse rate constants applied for intramolecular processes:

$$\begin{align} \frac{k_{YZ}}{k_{ZY}} = \frac{g_{Z}}{g_{Y}} \exp \left(-\frac{\Delta E}{k_\mathrm{B} T}\right). \end{align} \tag{ 6 }$$

The computation of the processes included in the scheme was performed by the testing the $k_\mathrm{R}$ and $k_\mathrm{L}$ values for all $(Z, v^{{\prime}})$, $(Y, v^{^{\prime\prime}})$ and $(Z, v^{{\prime}})$, $(Y, v^{^{\prime\prime}})$, $\mathrm{N}_2(\mathrm{X}^1 \Sigma^+_\mathrm{g},w^{^{\prime\prime}}\unicode{x2A7E}0)$ combinations for intramolecular and intermolecular processes, respectively. The processes with rate constants higher than 10⁻¹³cm³s⁻¹ were then incorporated to the kinetic scheme. The rate constants calculated in the model depend on the $\gamma, \beta$ and k₀ parameters, which were set based on the comparison of rate constants with the experimental data of [6, 35], see [29].

Finally, we would like to underline the limits of the developed model. The model does not take into account the rotational structure of the vibrational levels since $\Delta E$ and f are computed considering rotational number J = 0. This might influence the results since in general at $T_\mathrm{g}$ = 300 K the transition does not necessarily occur only between $\mathrm{N}_2({Z}, v^{{\prime}}, J = 0)$ and $\mathrm{N}_2({Y}, v^{^{\prime\prime}}, J = 0)$. However, the development of the presented model appears to be the first logical choice for an approach to complex kinetics of the N₂(B$^3\Pi_\mathrm{g}$) state. Moreover, the number of the vibrational levels $v\unicode{x2A7D}21$ used in the model corresponds with the available Franck–Condon factors in [32]. Up to now, the most detailed N₂ kinetic scheme was developed by [30] to study N₂(B$^3\Pi_\mathrm{g}$) state distribution in ionosphere.

One problem to overcome is that other spectral systems can potentially interfere with the FPS emission in the investigated spectral range. This is clearly shown in figure 6(a), which presents three spectra for comparison: one from the beginning of the discharge and two spectra from different post-discharge phases. The upper curve (2 ns MCP gate delayed by 2 ns after discharge onset) shows the FPS spectrum strongly disturbed by SPS emission in the 500–600 nm interval. The overlapping of the SPS and the FPS in this interval provides a valuable link to the SPS emission in the UV region, in particular, the vibrational distribution determined for the N₂(C$^3\Pi_\mathrm{u})$ state (cf. figure 10 in [19]) can be used to separate SPS from FPS since this interval is important to determine the N₂(B$^3\Pi_\mathrm{g}, v\unicode{x2A7E}9$) levels. The middle curve in figure 6(a) illustrates the post-discharge phase (40–60 ns) with significantly reduced SPS emission, in which the entire spectral interval is practically dominated by FPS, with a small exception given by the weak but still observable SPS(0,6). The shape of emission spectra between 510 and 560 nm shows characteristics of the FPS bands originating from high vibrational levels (v > 13). The third, lower curve in figure 6(a) reveals a very different picture when N₂(A$^3\Sigma_\mathrm{u}^+, v = 0\textrm{-}1$) metastables come into the game. The PIE starts to be controlled by the N₂(A$^3\Sigma_\mathrm{u}^+$)+N₂(A$^3\Sigma_\mathrm{u}^+$) energy pooling; consequently, two other spectral systems emerge and overlap with FPS, namely HIR system (${\mathrm{N}_2}(\mathrm{C^{^{\prime\prime}}}^5 \Pi_\mathrm{u}, v) \longrightarrow \mathrm{N}_2(\mathrm{A^{{\prime}}}^5 \Sigma^+_\mathrm{g},v-\Delta v)$ with $\Delta v = 2, 3$) and GHG system (N₂(H$^3\Phi_\mathrm{u}$) $\longrightarrow$ N₂(G$^3\Delta_\mathrm{g}$)). Note that the GHG system was observed in high-purity nitrogen [1, 36]. Intense FPS(2,0) band is clearly distinguishable from the FPS due to the high pooling rate for production of N₂(B$^3\Pi_\mathrm{g}, v = 2$) as suggested in [10]. As a result, the FPS analysis becomes very difficult for longer post-discharge times (hundreds of nanoseconds) unless appropriate subtractions of the dominant HIR and GHG bands are made.

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Figure 6(b) zooms in the 550–675 nm spectral region where the most intense FPS bands originating from the higher vibrational levels are found. For a proper VDF analysis of the B$^3\Pi_\mathrm{g}$ state, it is clear that the overlap of the SPS bands with the FPS (e.g. the SPS(1,9) band overlaps the FPS(12,8) at 575 nm) can be neglected for delays greater than 20 ns (compare the upper spectrum with the middle and the lower spectrum in figure 6(b)).

Figure 7(a) shows ICCD spectra obtained using a Shamrock 303i spectrometer in the 620–1080 nm. The upper curve (integrating the first five nanoseconds after the start of the discharge) shows the FPS spectrum, which is characterized by a relatively weak intensity of the FPS(0,0) band at 1050 nm as compared to FPS bands around and below 750 nm and originating from higher v-levels ($v\unicode{x2A7E} 2$). After only 20 nanoseconds (see the middle curve in figure 7(a)), there is a clear trend characterized by an increase in the intensity of FPS(0,0) relative to all other bands. This indicates continued relaxation in the N₂(B$^3\Pi_\mathrm{g}$) manifold, which is consistent with the relaxation also observed in the N₂(C$^3\Pi_\mathrm{u})$ state (cf Part I [19]). Finally, after another 20 nanoseconds, the FPS spectrum is dominated by bands originating from the lowest three N₂(B$^3\Pi_\mathrm{g}$) state levels ($v = 0\textrm{-}2$).

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Complementary evidence of the rapid intensity evolution of the FPS(0,0) band relative to all other bands is given in figure 7(b), which shows spectra obtained with better resolution (300 G mm⁻¹ grating). Furthermore, the spectral range 950–1080 nm seems to be particularly suitable for studying the evolution of the $v = 0\textrm{-}3$ levels, since even after the onset of HIR emission (cf figure 7(b)) due to energy pooling processes, there is no apparent interference of the FPS bands with the HIR bands.

The FPS spectra acquired by both spectrometers were fitted using a synthetic model for the FPS emission, as described in subsection 2.2. In order to correctly determine the VDF of $\mathrm{N_2(B^3} \Pi_\mathrm{{g}})$, we had to split the analysis into three wavelength ranges:

• (i)  
  610–900 nm to determine relative populations of v = 1–12 by employing 300 and 150 G mm⁻¹ gratings (iHR320 spectrometer). The lower bound of the wavelength interval was chosen to avoid the influence of SPS emission in the initial times ($t \lt$10 ns). For $t \gt$10 ns, we can shift the lower bound even to 560 nm.
• (ii)  
  830–1080 nm to determine relative populations of the lowest levels v = 0–3 by employing 300 and 150 G mm⁻¹ gratings (Shamrock 303i spectrometer) through intensities of the FPS(0,0), FPS(1,1), FPS(2,2) and FPS(3,3) bands. Note that the sensitivity of the ICCD decreases drastically for wavelengths higher than 1060 nm, as a result FPS(0,0) is the only band originating from v = 0 level available for the current analysis.
• (iii)  
  510–585 nm to determine relative populations of v = 11–21 by employing the 1200 G mm⁻¹ grating and 100 µm entrance slit (iHR320 spectrometer) in the latter times ($t \gt$10 ns).

Figure 8(a) shows a comparison of the experimental spectrum obtained with a 300 G mm⁻¹ grating and 50 µm entrance slit (averaged over a time interval of 20 ns) with two synthetic FPS band profiles simulated at $T_\mathrm{rot}$ = 300 K using the actual instrumental function (determined using a Hg–Ar pencil lamp). It is clear that the band profiles generated by the simulation program [1, 20] reproduce very well the partially resolved band-head structure of the FPS(2,0) and FPS(1,0) bands. We have analysed the FPS(2,0) and FPS(1,0) emission during first 200 nanoseconds. In this time interval we do not see any significant modification of the shape of FPS(2,0) and FPS(1,0) partially-resolved bandprofiles. This means that the rotational temperature of the $\mathrm{N_2(B^3} \Pi_\mathrm{{g}},v = 1\textrm{-}2)$ levels does not change significantly and $T_\mathrm{rot}$ = 300 K is a good approximation for using synthetic FPS bands for vibrational analysis in the entire time interval considered in this work.

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Figures 8(b) and (c) shows examples of the fitted experimental FPS spectra, acquired during the 2–4 ns and 10–15 ns by iHR320 and Shamrock 303i spectrometers, respectively. The agreement between the experiment and the fit is overall very good. Note that this interval includes an overlap of the FPS($v^{{\prime}}\gt 11$, $v^{{\prime}} - \Delta v$) bands belonging to the $\Delta v$ = +3 sequence and FPS($v^{{\prime}}\gt 3$, $v^{{\prime}} - \Delta v$) bands belonging to the $\Delta v$ = +4 sequence. For example, the level $v^{{\prime}} = 13$ contributes to the detected emission close to 610 nm through the FPS(13,10) band and overlaps with the FPS(5,1) whose radiative probability is roughly one half of FPS(13,10); however population ratio between $v^{{\prime}} = 5$ and $v^{{\prime}} = 13$ is about 20–25. Therefore, FPS(13,10) band intensity is much lower compared with FPS(5,1), consequently, FPS(13,10) cannot be simply resolved using 300 G mm⁻¹ grating. To study $v^{{\prime}} = 13$, it is necessary to observe the FPS(13,9) at 570 nm, where there is less significant overlap and higher transition probability (about 6 times higher) compared with FPS(13,10). In addition, we used better resolution (1200 G mm⁻¹ grating) to get clearer evidence about the $v^{{\prime}}\gt 12$ populations.

In the region 685–700 nm (see figure 8(b)) several bands overlap, namely FPS(14,13), FPS(18,18) with the HIR(4,2) band and therefore this interval was not used to fit experimental spectra. The visible discrepancy observed in figure 8(c) in 930–950 nm is caused by a low signal-to-noise ratio induced by a local drop in the photocathode efficiency in this wavelength region containing the FPS(4,4) band (which was excluded from the VDF analysis).

Figure 9(a) illustrates the difference between the shape of the FPS spectrum simulated for the complete vibrational distribution ($v\unicode{x2A7D}21$) and the shape of the FPS spectrum produced by all FPS($v^{{\prime}}\unicode{x2A7D}12$, vʹʹ) transitions. This difference is caused by a significant variation between the transition probabilities of the FPS($v^{{\prime}}\gt 12$, vʹʹ) and FPS($v^{{\prime}} = 5\textrm{-}12$, vʹʹ) bands, which more than compensates for the general trend of a decrease in the population of vibrational levels with increasing vibrational quantum number. It is also worth noting that the FPS($v^{{\prime}}\gt 12$, $v^{{\prime}} - \Delta v$) bands belonging to the $\Delta v$ = +2 sequence and the overlapping $\Delta v$ = +3 FPS bands between 610 and 660 nm should be taken into account, so the FPS spectrum is correctly reproduced in this spectral interval.

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Figure 9(b) compares the synthetic FPS spectrum with the experimental one obtained with much better spectral resolution (1200 G mm⁻¹ grating and 100 µm entrance slit). The blue and red solid curves show the synthetic FPS spectra considering full ($v^{{\prime}}\unicode{x2A7D} 21$) and partial ($v^{{\prime}}\unicode{x2A7D} 12$) VDFs, respectively. While the FPS spectrum generated for the partial VDF (red curve) shows a clear separation of the $\Delta v$ = +4 and +5 sequences, the FPS spectrum for the full VDF proves that the region of 500–570 nm is completely dominated by bands ($v^{{\prime}}\gt 12$) belonging to the $\Delta v$ = +4 sequence. The agreement is evident when comparing the synthetic FPS spectrum (blue curve) with the experimental data (dash-dot black curve).

The populations and VDFs obtained for v = 1–21 are based on the measurements performed by iHR320 spectrometer, as defined in (i)+(iii). These data are presented in table 2. The VDFs were acquired using MCP gate/step of 2 ns for $t \unicode{x2A7D} 40$ ns and the MCP gate/step of 20 ns for t > 30 ns. The error at the initial time 0–2 ns was estimated based on the comparison between various fitting bands belonging to different sequences. The critical for the VDF determination seems to be the incorporation of the 610 - 700 nm region since the bands of the FPS $\Delta v = 3$ sequence are well separated and do not overlap with other bands (except $v \unicode{x2A7E} 12$) and, therefore, provide the most trustworthy information concerning a large part of the VDF ($v = 3\textrm{-}11$). We cannot estimate the error in the fraction of v = 1 since the investigated region includes just FPS(1,0) at 880 nm, but the relative error is probably comparable to the other levels. The VDFs for v = 0–3 were obtained using the Shamrock 303i spectrometer, as defined in (ii) and are listed in table 3. We used the MCP gate/step of 2 and 5 ns for $t \unicode{x2A7D} 25$ ns and the MCP gate/step of 20 ns for t > 10 ns. The appropriate time shift between spectrometers was determined by comparing (i) and (ii) 610 - 900 nm spectra at different times. In both tables 2 and 3, populations are normalized to the $\mathrm{N}_2(\mathrm{B}^3\Pi_\mathrm{g},v = 2)$; thus, defined as ${[\mathrm{N}_2(\mathrm{B}^3\Pi_\mathrm{g},v)]}/{[\mathrm{N}_2(\mathrm{B}^3\Pi_\mathrm{g},v = 2)]}$. The relative populations obtained by the Shamrock 303i and iHR320 spectrometers were stitched together with respect to $\mathrm{N}_2(\mathrm{B}^3\Pi_\mathrm{g},v = 2)$ in order to relate v = 0 population to the data obtained by the iHR320 spectrometer. An interpolation was performed to impose measurements of (ii) with 2/5 ns gates to the measurements of (i) with 2 ns gates. Figure 10 shows a comparison between experimental and model populations of N₂(B$^3\Pi_\mathrm{g}, v = 0\textrm{-}12)$ levels. The experimental intensities obtained from the kinetic series were scaled in such a way that the population at the maximum of the experimental curve corresponds to the model value for the given vibrational level (the maxima of the vibrational levels are reached, except for v = 0, between 10 and 20 ns, where the model and experimental VDFs are in reasonable agreement). This allows a simple comparison of the post-discharge evolution for all plotted levels. Figure 10(a) illustrates the evolution of the level populations up to v = 7, revealing a distinctly different trend for the $v = 0\textrm{-}2$ and $v = 4\textrm{-}7$ groups. However, in all cases the experimentally observed evolution in early post-discharge (${\gt}20$ ns) is faster compared to the model predictions. The higher levels ($v = 7\textrm{-}12$) are shown in figure 10(b). The experimental data indicate that the higher vibrational levels decay at very similar rates, which is quite well reproduced by the model results, except for the v = 7 and v = 11 levels.

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Table 2. Comparison of the experimental and FCF-like VDFs. The experimental VDFs were obtained using the iHR320 spectrometer and averaged over the indicated time interval. The experimental data from 0–2 ns are compared with the FCFs of [32].

v	FCF-like	0–2 ns	2–4 ns	10–12 ns	20–22 ns	30–32 ns	30–50 ns	50–70 ns
1	$7.54\times 10^{-01}$	0.52±0.05	$4.53\times 10^{-01}$	$5.49\times 10^{-01}$	$7.68\times 10^{-01}$	$9.83\times 10^{-01}$	$1.12\times 10^{+00}$	$1.56\times 10^{+00}$
2	$1.00\times 10^{+00}$	1	$1.00\times 10^{+00}$	$1.00\times 10^{+00}$	$1.00\times 10^{+00}$	$1.00\times 10^{+00}$	$1.00\times 10^{+00}$	$1.00\times 10^{+00}$
3	$9.74\times 10^{-01}$	0.95±0.05	$9.00\times 10^{-01}$	$6.76\times 10^{-01}$	$3.46\times 10^{-01}$	$2.75\times 10^{-01}$	$2.89\times 10^{-01}$	$2.27\times 10^{-01}$
4	$7.74\times 10^{-01}$	0.80±0.04	$7.50\times 10^{-01}$	$6.04\times 10^{-01}$	$2.46\times 10^{-01}$	$1.87\times 10^{-01}$	$1.84\times 10^{-01}$	$1.15\times 10^{-01}$
5	$5.38\times 10^{-01}$	0.53±0.03	$5.51\times 10^{-01}$	$4.18\times 10^{-01}$	$2.15\times 10^{-01}$	$1.43\times 10^{-01}$	$1.35\times 10^{-01}$	$9.36\times 10^{-02}$
6	$3.41\times 10^{-01}$	0.36±0.01	$3.78\times 10^{-01}$	$3.23\times 10^{-01}$	$2.23\times 10^{-01}$	$1.47\times 10^{-01}$	$1.40\times 10^{-01}$	$1.00\times 10^{-01}$
7	$2.00\times 10^{-01}$	0.22±0.01	$1.99\times 10^{-01}$	$1.48\times 10^{-01}$	$8.45\times 10^{-02}$	$5.41\times 10^{-02}$	$5.47\times 10^{-02}$	$3.91\times 10^{-02}$
8	$1.10\times 10^{-01}$	0.12±0.01	$1.17\times 10^{-01}$	$7.14\times 10^{-02}$	$3.99\times 10^{-02}$	$3.44\times 10^{-02}$	$2.79\times 10^{-02}$	$2.28\times 10^{-02}$
9	$5.90\times 10^{-02}$	0.044±0.003	$6.26\times 10^{-02}$	$4.40\times 10^{-02}$	$2.84\times 10^{-02}$	$2.36\times 10^{-02}$	$2.68\times 10^{-02}$	$2.11\times 10^{-02}$
10	$3.04\times 10^{-02}$	0.033±0.006	$6.01\times 10^{-02}$	$3.57\times 10^{-02}$	$3.15\times 10^{-02}$	$2.36\times 10^{-02}$	$3.24\times 10^{-02}$	$2.09\times 10^{-02}$
11	$1.53\times 10^{-02}$	0.022±0.008	$3.24\times 10^{-02}$	$3.30\times 10^{-02}$	$2.46\times 10^{-02}$	$1.48\times 10^{-02}$	$2.01\times 10^{-02}$	$1.56\times 10^{-02}$
12	$7.59\times 10^{-03}$	0.010±0.005	$4.42\times 10^{-02}$	$2.20\times 10^{-02}$	$1.69\times 10^{-02}$	$9.83\times 10^{-03}$	$1.23\times 10^{-02}$	$9.38\times 10^{-03}$
13	$3.72\times 10^{-03}$	—	-	$1.10\times 10^{-02}$	$6.53\times 10^{-03}$	$6.39\times 10^{-03}$	$8.38\times 10^{-03}$	$5.47\times 10^{-03}$
14	$1.82\times 10^{-03}$	—	-	$6.59\times 10^{-03}$	$9.22\times 10^{-03}$	$6.39\times 10^{-03}$	$1.01\times 10^{-02}$	$8.59\times 10^{-03}$
15	$8.82\times 10^{-04}$	—	-	$1.10\times 10^{-02}$	$1.69\times 10^{-02}$	$9.83\times 10^{-03}$	$1.51\times 10^{-02}$	$1.20\times 10^{-02}$
16	$4.30\times 10^{-04}$	—	-	$1.37\times 10^{-02}$	$2.54\times 10^{-02}$	$1.67\times 10^{-02}$	$2.12\times 10^{-02}$	$1.48\times 10^{-02}$
17	$2.11\times 10^{-04}$	—	-	$8.24\times 10^{-03}$	$2.38\times 10^{-02}$	$2.36\times 10^{-02}$	$2.07\times 10^{-02}$	$1.64\times 10^{-02}$
18	$1.05\times 10^{-04}$	—	-	$7.69\times 10^{-03}$	$2.15\times 10^{-02}$	$2.75\times 10^{-02}$	$1.68\times 10^{-02}$	$1.72\times 10^{-02}$
19	$5.23\times 10^{-05}$	—	-	$6.59\times 10^{-03}$	$1.69\times 10^{-02}$	$2.16\times 10^{-02}$	$1.23\times 10^{-02}$	$1.09\times 10^{-02}$
20	$2.65\times 10^{-05}$	—	-	$1.92\times 10^{-03}$	$1.15\times 10^{-02}$	$1.18\times 10^{-02}$	$8.94\times 10^{-03}$	$7.81\times 10^{-03}$
21	$1.37\times 10^{-05}$	—	-	$4.40\times 10^{-04}$	$6.14\times 10^{-03}$	$5.90\times 10^{-03}$	$5.59\times 10^{-03}$	$4.69\times 10^{-03}$
$\mathcal{F}$	—	$2.29\times 10^{+03}$	$5.95\times 10^{+03}$	$3.15\times 10^{+05}$	$2.11\times 10^{+05}$	$1.13\times 10^{+05}$	$1.03\times 10^{+05}$	$8.92\times 10^{+04}$

The experimental data show that the population of the higher levels N₂(B$^3\Pi_\mathrm{g}, v = 3\textrm{-}12)$ decay immediately after reaching their peak at approximately 12 ns. The populations of lower levels N₂(B$^3\Pi_\mathrm{g}, v = 0\textrm{-}2)$ in the range 12–80 ns grow or do not decay so rapidly as the higher levels. However, this behaviour is not well reproduced in the kinetic model. The most apparent is this behaviour when comparing N₂(B$^3\Pi_\mathrm{g}, v = 3\textrm{-}4)$ and N₂(B$^3\Pi_\mathrm{g}, v = 0)$. Obviously the N₂(B$^3\Pi_\mathrm{g}, v = 0)$ should be better populated at the expense of depopulation of N₂(B$^3\Pi_\mathrm{g}, v = 2\textrm{-}4)$. This suggests that pathways (and corresponding rate constants) connecting these states are likely underestimated in the model, or the rate of quenching of individual N₂(B$^3\Pi_\mathrm{g}, v)$ levels out from triplet states needs to be reevaluated.

The VDFs of $\mathrm{N}_2(\mathrm{B}^3\Pi_\mathrm{g},v = 0\textrm{-}21)$ as shown in figure 11(a) present the combination of (i), (ii) and (iii) measurements for times 0–70 ns. During the first interval (0–2 ns), we observe a very good agreement with the theoretical Franck–Condon factors [32]. The agreement is also good for the 10–12 ns time interval, except v > 8. Similarly, as in [19] for the N₂(C$^3\Pi_\mathrm{u}$) state case, the agreement proves the dominant electron-impact excitation from the ground state $\mathrm{N_2(X^1} {\Sigma}_\mathrm{g}^+, v = 0)$ to the excited state N₂(B$^3\Pi_\mathrm{g}$). During the later times (t > 20 ns), the relaxation from higher to lower vibrational levels is evident. At the initial times (0–2 ns, 10–12 ns), the v = 2 has the highest population; however, for the higher times (30–70 ns), the dominance of v = 2 is replaced by the $v = 0\textrm{-}1$ vibrational levels. We also observe a local maximum for v = 6 in the VDF. The figure 11(b) shows VDFs of N₂(B$^3\Pi_\mathrm{g}, v = 0\textrm{-}21)$ computed using the kinetic model for the same time intervals as in the experiment. The model predicts the local maximum for v = 6 as well as the increase of the VDF for v > 13, which is caused due to the $\mathrm{N}_2(\mathrm{C}^3\Pi_\mathrm{u},v = 0\textrm{-}4)$ and $\mathrm{N}_2(\mathrm{B}^3\Pi_\mathrm{g},v \gt 8)$ collisional coupling through N₂(W$^3\Delta_\mathrm{u}$) and N₂(B'$^3\Sigma_\mathrm{u}$) intermediate states. The local minimum observed in the VDF for $v = 12\textrm{-}15$ is due to the coupling of these vibrational levels to the ${\mathrm{N}_2}(\mathrm{A^{{\prime}}}^5 \Sigma^+_\mathrm{g},v^{{\prime}}\gt 5)$ levels, which are above the dissociation limit [15, 16]. Geisen et al [16] showed that the predissociation rate constants for $\mathrm{N}_2(\mathrm{B}^3\Pi_\mathrm{g}, v = 13\textrm{-}18)$ depend on v and reach the maximum values 3.08×10⁸, 1.71×10⁸, 1.13×10⁸ s⁻¹ for $v = 13, 14, 15$, respectively. On the other hand, it is obvious that the presented kinetic model is unable to reproduce the growth of the $v = 0\textrm{-}1$ with respect to the v = 2 as observed in the experiment. We observe also a significant difference in the relative populations of $\mathrm{N}_2(\mathrm{B}^3\Pi_\mathrm{g}, v\gt 9)$ levels, which is most apparent for t = 10–12 ns. The corresponding VDF fraction ${\sum_{v = 9}^{21}[\mathrm{N}_2(\mathrm{B}^3\Pi_\mathrm{g},v)]}/{\sum_{v = 0}^{21}[\mathrm{N}_2(\mathrm{B}^3\Pi_\mathrm{g},v)]}$ is below 0.05 in the experiment while 0.16 in the model. This discrepancy could be due to the missing N₂(A$^3\Sigma_\mathrm{u}^+, v \gt 21$) and N₂(A$^{{\prime} 5}\Sigma_\mathrm{g}^+$) states, which are energetically close to the $\mathrm{N}_2(\mathrm{B}^3\Pi_\mathrm{g}, v = 9\textrm{-}13)$ levels. The presence of these vibrational levels would lead to the population transfer to the additional states and would decrease the $\mathrm{N}_2(\mathrm{B}^3\Pi_\mathrm{g}, v\gt 9)$ populations. More details on the model results will be presented in the following section.

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We were able to obtain the VDFs of the $\mathrm{N_2(B^3} \Pi_\mathrm{{g}})$ state in a time interval covering both the formation/propagation phase of the streamer front and the later relaxation phase of the streamer channel. During the initial phase (t < 10 ns), the accuracy of the VDF determination is strongly limited by the low signal intensity ($v\unicode{x2A7D}1$) and the overlap of the FPS bands with the SPS bands ($v\unicode{x2A7E}10$). During the relaxation phase (t > 100 ns), a complete VDF analysis is almost impossible because many FPS bands overlap with the significantly more intense bands of the HIR/GHG systems produced by $\mathrm{N}_2(\mathrm{A}^3\Sigma_\mathrm{u}^+) + \mathrm{N}_2(\mathrm{A}^3\Sigma_\mathrm{u}^+)$ pooling. On the other hand, the fact that the population of both $\mathrm{N_2(B^3} \Pi_\mathrm{{g}})$ and $\mathrm{N_2(C^3}\Pi_\mathrm{u})$ states can be monitored even during the relaxation phase driven by $\mathrm{N}_2(\mathrm{A}^3\Sigma_\mathrm{u}^+)$ metastables [19] offers a unique chance not only to obtain the VDFs of both electronic states but in principle also their absolute densities, as proposed in [37] (out of the scope of present work).

The comparison of experimentally determined N₂(B$^3\Pi_\mathrm{g},$ $v = 0\textrm{-}21)$ evolution with the kinetic model revealed that the pathways connecting N₂(B$^3\Pi_\mathrm{g}, v = 3\textrm{-}4)$ with N₂(B$^3\Pi_\mathrm{g}, v = 0\textrm{-}1)$ states are probably underestimated in the presented kinetic model. According to the model, existing pathways are not only vibrational relaxations within the N₂(B$^3\Pi_\mathrm{g})$ state, for example:

$$\begin{align} & \mathrm{N}_2\left(\mathrm{B}^3\Pi_\mathrm{g}, v = 4\right) + \mathrm{N}_2\left(\mathrm{X}^1 \Sigma^+_\mathrm{g},v = 0\right) \rightarrow \mathrm{N}_2\left(\mathrm{X}^1 \Sigma^+_\mathrm{g},v = 1\right) \notag\\ & \quad + \mathrm{N}_2\left(\mathrm{B}^3\Pi_\mathrm{g}, v = 1\right), \end{align} \tag{ 7 }$$

but also pathways leading through the intermediate $\mathrm{N}_2(\mathrm{A}^3\Sigma_\mathrm{u}^+)$ and $\mathrm{N}_2(\mathrm{W}^3\Delta_\mathrm{u})$ states:

$$\begin{align} & \mathrm{N}_2\left(\mathrm{B}^3\Pi_\mathrm{g}, v = 4\right) + \mathrm{N}_2\left(\mathrm{X}^1 \Sigma^+_\mathrm{g},v = 0\right) \rightarrow \mathrm{N}_2\left(\mathrm{X}^1 \Sigma^+_\mathrm{g},v = 0\right)\notag\\ & \quad + \mathrm{N}_2\left(\mathrm{A}^3\Sigma_\mathrm{u}^+, v = 13\right), \end{align} \tag{ 8 }$$

$$\begin{align} \notag\\[-6pt] & \mathrm{N}_2\left(\mathrm{A}^3\Sigma_\mathrm{u}^+, v = 13\right) \!+\! \mathrm{N}_2\left(\mathrm{X}^1 \Sigma^+_\mathrm{g},v = 0\right) \rightarrow \mathrm{N}_2\left(\mathrm{X}^1 \Sigma^+_\mathrm{g},v = 3\right)\notag\\ & \quad + \mathrm{N}_2\left(\mathrm{B}^3\Pi_\mathrm{g}, v = 0\right) \end{align} \tag{ 9 }$$

and

$$\begin{align} \notag\\[-6pt] & \mathrm{N}_2\left(\mathrm{B}^3\Pi_\mathrm{g}, v = 3\right) + \mathrm{N}_2\left(\mathrm{X}^1 \Sigma^+_\mathrm{g},v = 0\right) \rightarrow \mathrm{N}_2\left(\mathrm{W}^3\Delta_\mathrm{u}, v = 3\right) \notag\\ & \quad + \mathrm{N}_2\left(\mathrm{X}^1 \Sigma^+_\mathrm{g},v = 0\right), \end{align} \tag{ 10 }$$

$$\begin{align} \notag\\[-6pt] & \mathrm{N}_2\left(\mathrm{W}^3\Delta_\mathrm{u}, v = 3\right) + \mathrm{N}_2\left(\mathrm{X}^1 \Sigma^+_\mathrm{g},v = 0\right) \rightarrow \mathrm{N}_2\left(\mathrm{X}^1 \Sigma^+_\mathrm{g},v = 1\right)\notag\\ & \quad + \mathrm{N}_2\left(\mathrm{B}^3\Pi_\mathrm{g}, v = 1\right). \end{align} \tag{ 11 }$$

Note that processes (7)-(11) are those having rate coefficients higher than $1.0\times10^{-12}\mathrm{cm}^{3}\mathrm{s}^{-1}$. In the literature, the effective rate coefficients for the $\mathrm{N_2(B^3} \Pi_\mathrm{g}, v \unicode{x2A7D} 12)$ quenching by N₂ can be found in many works [9, 10, 38–41]. Some of these works [10, 38, 39, 41] refer that the effective quenching rate coefficient for N₂(B$^3\Pi_\mathrm{g}, v = 3)$ is at least twice as large as for N₂(B$^3\Pi_\mathrm{g}, v = 2)$ state, thus, in the agreement with our experimental results. However, the use of the effective quenching rate coefficient is in general misleading since it ignores the structure of the energetically adjacent vibrational levels of different electronic states. The concept based on effective rate coefficients is valid just after creating equilibrium with the energetically closest levels. Therefore, some authors [9] distinguish two quenching rates: fast and slow, where the first corresponds to the population transfer to the energetically closest levels and the second to the quenching after finding equilibrium with the closest levels.

Moreover, the use of the effective rate coefficients in the kinetic models, as it was done, for example, in [18], is not able to predict energy transfers from the higher to lower vibrational levels under highly transient discharge conditions. Therefore, the development of the state-to-state kinetic model, as done within this work, seems to be the only possible way to describe the kinetics of N₂(B$^3\Pi_\mathrm{g})$ state correctly.

To the best of our knowledge, no other study determined the complete VDF of $\mathrm{N_2(B^3}\Pi_\mathrm{g}, v = 0\textrm{-}21)$ under the streamer discharge conditions. A list of works on $\mathrm{N_2(B^3}\Pi_\mathrm{g})$ VDF at different discharge conditions can be found in [42], and in general, we observe similar effects as reported in those works. The VDF observed for t > 20 ns tends to relax towards the lower vibrational levels. We also observe a local VDF maximum for v = 6. This phenomenon is usually connected with the intramolecular coupling of $\mathrm{N_2(B^3}\Pi_\mathrm{g})$ and N₂(W$^3\Delta_\mathrm{u}$) states [43] since the $\mathrm{N_2(B^3}\Pi_\mathrm{g}, v = 6)$ and N₂(W$^3\Delta_\mathrm{u}, w = 7$) states have low energy difference, which implies the following process:

$$\begin{align} & \mathrm{N}_2\left(\mathrm{W}^3\Delta_\mathrm{u}, v = 7\right) + \mathrm{N}_2\left(\mathrm{X}^1 \Sigma^+_\mathrm{g},v = 0\right) \leftrightarrow \mathrm{N}_2\left(\mathrm{B}^3\Pi_\mathrm{g}, v = 6\right)\notag\\ & \quad + \mathrm{N}_2\left(\mathrm{X}^1 \Sigma^+_\mathrm{g},v = 0\right), \end{align} \tag{ 12 }$$

with energy difference of 132 cm⁻¹ [32]. The N₂(W$^3\Delta_\mathrm{u}, w$ = 7) state is populated by electron impact with a much higher rate constant than the $\mathrm{N_2(B^3}\Pi_\mathrm{g}, v = 6)$ state, therefore during first tens of nanoseconds the N₂(W$^3\Delta_\mathrm{u}, w$ = 7) population is converted to energetically close $\mathrm{N_2(B^3}\Pi_\mathrm{g}, v = 6)$ state by the process (12). Later, both close states find equilibrium and evolve as a single state.

The FPS radiation observed in the range 510–585 nm, belonging to the ($v^{{\prime}}\gt 12,v^{^{\prime\prime}}$) bands, is also quite important for understanding underlying microscopic processes. This radiation is a characteristic of the so-called Lewis-Rayleigh afterglow, which is regularly observed in the flowing afterglow rich in atomic species produced by a continuous discharge in low-pressure nitrogen [44, 45]. In our case of a transient streamer monofilament at high pressure, the origin of radiation from higher ($v\unicode{x2A7E}13$) levels is different. The observed overpopulation of the higher levels $\mathrm{N}_2(\mathrm{B}^3\Pi_\mathrm{g}, v = 13\textrm{-}21)$ can be roughly reproduced by introducing the intermolecular coupling of the $\mathrm{N}_2(\mathrm{C}^3\Pi_\mathrm{u})$ state to the higher levels of N₂(W$^3\Delta_\mathrm{u}$) and N₂(B'$^3\Sigma_\mathrm{u}^-$) states in the kinetic model, as proposed in [23]:

$$\begin{align} & \mathrm{N}_2\left(\mathrm{C}^3\Pi_\mathrm{u}, v^{{\prime}}\right) + \mathrm{N}_2\left(\mathrm{X}^1 \Sigma^+_\mathrm{g},v = 0\right) \rightarrow \mathrm{N}_2\left(\mathrm{X}^1 \Sigma^+_\mathrm{g},v\unicode{x2A7E}0\right) \notag\\ & \quad + \mathrm{N}_2\left(\mathrm{W}^3\Delta_\mathrm{u}, w\right) \end{align} \tag{ 13 }$$

and

$$\begin{align} & \mathrm{N}_2\left(\mathrm{C}^3\Pi_\mathrm{u}, v^{{\prime}}\right) + \mathrm{N}_2\left(\mathrm{X}^1 \Sigma^+_\mathrm{g},v = 0\right) \rightarrow \mathrm{N}_2\left(\mathrm{X}^1 \Sigma^+_\mathrm{g},v\unicode{x2A7E}0\right)\notag\\ & \quad + \mathrm{N}_2\left(\mathrm{B}^{{\prime} 3}\Sigma_\mathrm{u}^-, v^{^{\prime\prime}}\right). \end{align} \tag{ 14 }$$

The N₂(W$^3\Delta_\mathrm{u}$) and N₂(B$^{{\prime} 3}\Sigma_\mathrm{u}^-$) state levels are efficiently coupled to N₂(B$^3\Pi_\mathrm{g}, v \gt$12) and mediate the transfer of $\mathrm{N}_2(\mathrm{C}^3\Pi_\mathrm{u})$ state population to $\mathrm{N}_2(\mathrm{B}^3\Pi_\mathrm{g})$ state. According to the developed model [23, 46], the most dominant collisional transfers from $\mathrm{N}_2(\mathrm{C}^3\Pi_\mathrm{u}, v^{{\prime}} = 0)$ are to $\mathrm{N}_2(\mathrm{W}^3\Delta_\mathrm{u}, w = 20\textrm{-}21)$ and $\mathrm{N}_2(\mathrm{B}^{{\prime} 3}\Sigma_\mathrm{u}^-, v^{^{\prime\prime}} = 14\textrm{-}18)$. Similarly, $\mathrm{N}_2(\mathrm{W}^3\Delta_\mathrm{u}, w = 22)$ and $\mathrm{N}_2(\mathrm{B}^{{\prime} 3}\Sigma_\mathrm{u}^-, v^{^{\prime\prime}} = 13\textrm{-}15)$ for collisional quenching of $\mathrm{N}_2(\mathrm{C}^3\Pi_\mathrm{u}, v^{{\prime}} = 1)$. Note that, based on results of part I of this work [19], the populations of $\mathrm{N}_2(\mathrm{C}^3\Pi_\mathrm{u}, v^{{\prime}} = 0\textrm{-}1)$ contain approximately 90 % of the total $\mathrm{N}_2(\mathrm{C}^3\Pi_\mathrm{u})$ state population. The energy of the $\mathrm{N}_2(\mathrm{W}^3\Delta_\mathrm{u}, w = 20\textrm{-}22)$ and $\mathrm{N}_2(\mathrm{B}^{{\prime} 3}\Sigma_\mathrm{u}^-, v^{^{\prime\prime}} = 13\textrm{-}18)$ levels coincides with the energy of N₂(B$^3\Pi_\mathrm{g}, v = 16\textrm{-}21$) levels, see figure 4, therefore allowing the intramolecular energy transfers:

$$\begin{align} & \mathrm{N}_2\left(\mathrm{W}^3\Delta_\mathrm{u}\right)/\mathrm{N}_2\left(\mathrm{B}^{{\prime} 3}\Sigma_\mathrm{u}^-\right) + \mathrm{N}_2\left(\mathrm{X}^1 \Sigma^+_\mathrm{g},v = 0\right) \rightarrow \mathrm{N}_2\left(\mathrm{B}^3\Pi_\mathrm{g}\right) \notag\\ & \quad + \mathrm{N}_2\left(\mathrm{X}^1 \Sigma^+_\mathrm{g},v = 0\right), \end{align} \tag{ 15 }$$

which populate the $\mathrm{N}_2(\mathrm{B}^3\Pi_\mathrm{g})$ state during early post-discharge phase.

Since the kinetic model is very complex, it is useful to test the contribution of specific processes on the resulting VDF. Table 4 and figure 12 demonstrate the potential of the selected processes for VDF shaping. The inclusion of the specific processes in the model is denoted using check marks. Insets of figure 12 show the evolution of $S = \sum_{v = 13}^{21}[\mathrm{N}_2(\mathrm{B}^3\Pi_\mathrm{g},v)]$ in each test case. First four columns allow insight at pressure used in the present experiments (200 Torr). Last two columns with corresponding plots show predictions at pressure of 20 Torr, where the $E/N$ and $n_\mathrm{e}$ profiles are set the same as in the case of 200 Torr.

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Table 4. Test of the influence of selected processes on the shape of the VDF of N₂(B$^3\Pi_\mathrm{g}$) state. Each letter in the first row corresponds to a specific test case shown in the corresponding panel of figure 12.

	(A)	(B)	(C)	(D)	(E)	(F)
N2(C$^3\Pi_\mathrm{u}$) $\rightarrow$ N2(W$^3\Delta_\mathrm{u}$)/N2(B'$^3\Sigma_\mathrm{u}^-$)	✓	—	✓	✓	✓	—
${\mathrm{N}_2}(\mathrm{A}^3 \Sigma_\mathrm{u}^+)$ pooling	✓	✓	—	✓	✓	✓
$\mathrm{N}_2(\mathrm{B}^3\Pi_\mathrm{g}, v = 13\textrm{-}18)$ predissociation	✓	✓	✓	—	✓	✓
pressure [Torr]	200	200	200	200	20	20

The participation of the $\mathrm{N}_2(\mathrm{C}^3\Pi_\mathrm{u})$ to N₂(W$^3\Delta_\mathrm{u}$) and N₂(B'$^3\Sigma_\mathrm{u}^-$) coupling on the $\mathrm{N}_2(\mathrm{B}^3\Pi_\mathrm{g}, v = 13\textrm{-}21)$ population is apparent when comparing figures 12(A) and (B). In the test case (B), the population of $\mathrm{N}_2(\mathrm{C}^3\Pi_\mathrm{u})$ state contributes only by radiative quenching to the population of state $\mathrm{N}_2(\mathrm{B}^3\Pi_\mathrm{g})$ (low vibrational levels). Obviously, the electron-impact excitation of $\mathrm{N}_2(\mathrm{W}^3\Delta_\mathrm{u}, v = 15\textrm{-}21)$, $\mathrm{N}_2(\mathrm{B}^{{\prime} 3}\Sigma_\mathrm{u}^-, v = 9\textrm{-}21)$ and $\mathrm{N}_2(\mathrm{B}^3\Pi_\mathrm{g}, v = 13\textrm{-}21)$ states; which are the levels with energies higher than the energy of $\mathrm{N}_2(\mathrm{B}^3\Pi_\mathrm{g}, v = 12)$ level brings just 20% of $\mathrm{N}_2(\mathrm{B}^3\Pi_\mathrm{g}, v = 13\textrm{-}21)$ population. Therefore, the electron impact excitation is dominant few nanoseconds after the streamer onset, compare the insets of figures 12(A) and (B). Moreover, the electron impact excitation alone is not able to explain the well-defined VDF maximum for $v = 16\textrm{-}18$. There is no difference between figures 12(C) and 12(A), because pooling reactions have just a minor effect on the VDF for t < 70 ns due to low density of ${\mathrm{N}_2}(\mathrm{A}^3 \Sigma_\mathrm{u}^+, v = 0,1)$:

$$\begin{align} & {\mathrm{N}_2}\left(\mathrm{A}^3 \Sigma_\mathrm{u}^+, v\right) + {\mathrm{N}_2}\left(\mathrm{A}^3 \Sigma_\mathrm{u}^+, w\right)\notag\\ & \quad \rightarrow \mathrm{N}_2\left(\mathrm{B}^3\Pi_\mathrm{g}, v = 13\textrm{-}21\right) + {\mathrm{N}_2}. \end{align} \tag{ 16 }$$

However, we cannot definitely exclude the population of $\mathrm{N}_2(\mathrm{B}^3\Pi_\mathrm{g}, v = 13\textrm{-}21)$ levels from ${\mathrm{N}_2}(\mathrm{A}^3 \Sigma_\mathrm{u}^+,~v\gt~5)$ and ${\mathrm{N}_2}(\mathrm{A}^3 \Sigma_\mathrm{u}^+,~w\gt~5)$ combinations, which are already present tens of nanoseconds after the discharge onset [19].

Figure 12(D) shows that the predissociation of $\mathrm{N}_2(\mathrm{B}^3\Pi_\mathrm{g}, v = 13\textrm{-}18)$ levels is responsible for the local minimum of the VDF. From the comparison of insets of figures 12(A) and (D), it is apparent that the contribution of the predissociation to the atomic nitrogen production approximately $4 \times 10^{13}$cm⁻³, seems to be just a minor source of the atomic nitrogen [26]. Further insight into the role of collisional quenching of $\mathrm{N}_2(\mathrm{C}^3\Pi_\mathrm{u})$ on the VDF shape of $\mathrm{N}_2(\mathrm{B}^3\Pi_\mathrm{g})$ state might be obtained by comparing distributions formed at high pressures ($\unicode{x2A7E}$200 Torr) with those produced at lower pressures (20 Torr), see figures 12(E) and (F). The fraction of the $\mathrm{N}_2(\mathrm{C}^3\Pi_\mathrm{u}, v = 0)$ which participates in the collisional quenching can be determined as ${k_\mathrm{q} [\mathrm{N}_2]}\cdot{(1/\tau_0 + k_\mathrm{q} [\mathrm{N}_2])^{-1}}$, where $k_\mathrm{q} = 1.1\times10^{-11}$cm³s⁻¹ is the quenching rate constant for $\mathrm{N}_2(\mathrm{C}^3\Pi_\mathrm{u}, v = 0)$ quenching by N₂ and $\tau_0 = 36$ ns is the lifetime of spontaneous emission of the $\mathrm{N}_2(\mathrm{C}^3\Pi_\mathrm{u}, v = 0)$ state. For the $[\mathrm{N}_2] = 6.4\times10^{18}$cm⁻³ density corresponding to the pressure of 200 Torr, we find that approximately 70$\%$ of the $\mathrm{N}_2(\mathrm{C}^3\Pi_\mathrm{u}, v = 0)$ population is quenched collisionally. At the pressure of 20 Torr it is 20$\%$. Consequently, higher pressure leads to populations of higher vibrational levels of $\mathrm{N}_2(\mathrm{B}^3\Pi_\mathrm{g})$ state, contrary to lower pressure which tends to support population of lower levels through the radiative quenching (SPS).

In figure 13 the evolution of $\sum_{v = 13}^{21}[\mathrm{N}_2(\mathrm{B}^3\Pi_\mathrm{g},v)]$ and $\sum_{v=0}^{4}\lbrack{\mathrm N}_2({\mathrm C}^3\Pi_{\mathrm u},v)\rbrack$ determined by the model is compared with the SPS($w = 0\textrm{-}4,w^{{\prime}}$) and FPS($v = 13\textrm{-}21,v^{{\prime}}$) intensity profiles, both normed to maximum. The SPS($w = 0\textrm{-}4,w^{{\prime}}$) profile was obtained in [19], and the FPS($v = 13\textrm{-}21,v^{{\prime}}$) profile was obtained by integrating the intensity observed in the interval 500–570 nm. We observe that the time delay between the peaks is approximately 10 ns, which corresponds with ${\mathrm N}_2({\mathrm C}^3\Pi_{\mathrm u})$ effective lifetime, 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 [47]. The right axis of figure 13(a) presents the evolution of the fraction ${\sum_{v = 13}^{21}[\mathrm{N}_2(\mathrm{B}^3\Pi_\mathrm{g},v)]}/{\sum_{v = 0}^{21}[\mathrm{N}_2(\mathrm{B}^3\Pi_\mathrm{g},v)]}$ obtained by model and experiment. The maximum of this fraction corresponds to the peak of the FPS($v = 13\textrm{-}21,v^{{\prime}}$) intensity profile, indicating that the main population of the $\mathrm{N}_2(\mathrm{B}^3\Pi_\mathrm{g}, v = 13\textrm{-}21)$ occurs during 10–30 ns.

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We should discuss also other possible channels to be considered for production of $\mathrm{N}_2(\mathrm{B}^3\Pi_\mathrm{g}, v\gt 12)$ levels. Such channels could be associated with the atomic species $\mathrm{N}(^4\mathrm{S},^2\mathrm{D},^2\mathrm{P})$ and three-body recombination [48]:

$$\begin{align} & \mathrm{N}\left(^4\mathrm{S}\right) + \mathrm{N}\left(^2\mathrm{D}\right)/\mathrm{N}\left(^2\mathrm{P}\right) + \mathrm{N}_2\left(\mathrm{X}^1 \Sigma^+_\mathrm{g},v\unicode{x2A7E}0\right) \rightarrow \mathrm{N}^*_2 \notag\\ & \quad + \mathrm{N}_2\left(\mathrm{X}^1 \Sigma^+_\mathrm{g},v\unicode{x2A7E}0\right), \end{align} \tag{ 17 }$$

where $\mathrm{N}^*_2$ denotes higher vibrational levels of $\mathrm{N}_2(\mathrm{W}^3\Delta_\mathrm{u})$, $\mathrm{N}_2(\mathrm{B}^{{\prime} 3}\Sigma_\mathrm{u}^-)$ and $\mathrm{N}_2(\mathrm{B}^3\Pi_\mathrm{g})$ states, which can be converted to $\mathrm{N}_2(\mathrm{B}^3\Pi_\mathrm{g}, v = 13\textrm{-}21)$ levels. Nevertheless, the results of the kinetic model show that the densities of $\mathrm{N}(^4\mathrm{S}), \mathrm{N}(^2\mathrm{D}), \mathrm{N}(^2\mathrm{P})$ atoms are rather constant in the time $t = 10\textrm{-}1000$ ns, therefore the radiation produced in the $\mathrm{N}_2(\mathrm{B}^3\Pi_\mathrm{g}, v = 13\textrm{-}21)$ levels should not decay rapidly after reaching the maximum population between 20 and 30 ns.

Further channel could be the following pooling mechanism studied by Piper [49]:

$$\begin{align} & {\mathrm{N}_2}\left(\mathrm{A}^3 \Sigma_\mathrm{u}^+, v\right) + {\mathrm{N}_2}\left(\mathrm{X}^1 \Sigma_\mathrm{g}^+, v \unicode{x2A7E} 5\right)\notag\\ & \quad \rightarrow \mathrm{N}_2\left(\mathrm{B}^3\Pi_\mathrm{g}, v = 0\textrm{-}12\right) + {\mathrm{N}_2}\left(\mathrm{X}^1 \Sigma_\mathrm{g}^+, w\unicode{x2A7E}0\right), \end{align} \tag{ 18 }$$

but he did not observe $\mathrm{N}_2(\mathrm{B}^3\Pi_\mathrm{g}, v\gt 12)$ levels and used a low pressure N₂ afterglow at high vibrational temperatures (1800 and 3700 K). We can estimate the magnitude of the source term of (18) in our case: $P = k\cdot[{\mathrm{N}_2}(\mathrm{A}^3 \Sigma_\mathrm{u}^+)]\cdot[{\mathrm{N}_2}(\mathrm{X}^1 \Sigma_\mathrm{g}^+, v \unicode{x2A7E} 5)]$, the rate constant $k = (3.0\pm1.5)\times10^{-11}$cm³s⁻¹, $\sum_{v\unicode{x2A7E}0}[{\mathrm{N}_2}(\mathrm{A}^3 \Sigma_\mathrm{u}^+, v)] \approx 10^{15}$cm⁻³ and the $\sum_{v\unicode{x2A7E}5}[{\mathrm{N}_2}(\mathrm{X}^1 \Sigma_\mathrm{g}^+, v)] \approx 10^{14}$cm⁻³ based on the kinetic model. Such values lead to $P \approx 3.0\times10^{18}$cm⁻³s⁻¹, which requires approximately 0.1 ms to build up $\mathrm{N}_2(\mathrm{B}^3\Pi_\mathrm{g})$ densities 10¹⁴cm⁻³ as predicted by our model. Therefore, this process should not be significant in our case.