Electric field determination in air plasmas from intensity ratio of nitrogen spectral bands: I. Sensitivity analysis and uncertainty quantification of dominant processes

A zero-dimensional chemical kinetic model for dry air (N₂–O₂, 4:1) at pressure of 10⁵ Pa and temperature 300 K, is used to determine the population ratio of ${{\rm{N}}}_{2}^{+}{({{\rm{B}}}^{2}{{\rm{\Sigma }}}_{{\rm{u}}}^{+})}_{v=0}$ and ${{\rm{N}}}_{2}{({{\rm{C}}}^{3}{{\rm{\Pi }}}_{{\rm{u}}})}_{v=0}$ vibrational levels of electronically excited states. To shorten the notation, we further refer to those states simply as ${{\rm{N}}}_{2}^{+}({{\rm{B}}}^{2})$ and ${{\rm{N}}}_{2}({{\rm{C}}}^{3})$. We base our chemical kinetic model on a list of plasmachemical processes from [25–28] compiled in ZDPlasKin [29] library, input file for N₂–O₂ mixture (version 1.03). Note that some simplifications have been introduced in the scheme. For example, we discarded vibrational manifolds of N₂ and O₂ ground states. On the other hand, the ZDPlasKin input file does not include processes for ${{\rm{N}}}_{2}^{+}({{\rm{B}}}^{2})$ and these processes were, therefore, added to the kinetic scheme. The resulting kinetic model used in this work includes a set of 617 chemical reactions listed in the appendix involving 42 species listed in table 1. In order to compare with other experimental and theoretical works dealing with steady state conditions, we seek for a steady state value of the ratio of ${{\rm{N}}}_{2}^{+}({{\rm{B}}}^{2})$ and ${{\rm{N}}}_{2}({{\rm{C}}}^{3})$ densities. The initial value problem for the resulting ordinary differential equations is integrated in time with the reduced electric field E/N and electron density n_e as fixed parameters. We use an implicit solver radau5 to integrate the stiff reaction system [30]. The complete list of reactions is listed in the table A1. The dependence of the rate constants for electron-impact processes on the reduced electric field E/N is found by solving the Boltzmann equation in the two-term approximation with the use of the BOLSIG+ solver [31] with cross-section sets [32–38]. With 42 species, the kinetic model is fully described by a set of 42 ordinary differential equations. Within this framework the densities of ${{\rm{N}}}_{2}({{\rm{C}}}^{3})$ and ${{\rm{N}}}_{2}^{+}({{\rm{B}}}^{2})$ obey the following two:

$$\begin{eqnarray}\displaystyle \frac{{\rm{d}}[{{\rm{N}}}_{2}({{\rm{C}}}^{3})]}{{\rm{d}}{t}} & = & [{\rm{e}}]({k}_{010}[{{\rm{N}}}_{2}]+{k}_{041}[{{\rm{N}}}_{4}^{+}])\\ & & -{k}_{105}[{{\rm{N}}}_{2}({{\rm{C}}}^{3})]\\ & & +{k}_{124}[{{\rm{N}}}_{2}({{\rm{A}}}^{3})][{{\rm{N}}}_{2}({{\rm{A}}}^{3})]\\ & & -[{{\rm{N}}}_{2}({{\rm{C}}}^{3})]({k}_{129}[{{\rm{N}}}_{2}]+{k}_{130}[{{\rm{O}}}_{2}]),\end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray}\displaystyle \frac{{\rm{d}}[{{\rm{N}}}_{2}^{+}({{\rm{B}}}^{2})]}{{\rm{d}}{t}} & = & [{\rm{e}}]([{{\rm{N}}}_{2}]{k}_{028}+[{{\rm{N}}}_{2}^{+}]{k}_{050})\\ & & -[{{\rm{N}}}_{2}^{+}({{\rm{B}}}^{2})]{k}_{106}\\ & & -[{{\rm{N}}}_{2}^{+}({{\rm{B}}}^{2})][{{\rm{N}}}_{2}]([{{\rm{N}}}_{2}]{k}_{366}+[{\rm{N}}]{k}_{367})\\ & & -[{{\rm{N}}}_{2}^{+}({{\rm{B}}}^{2})]\left(\displaystyle \sum _{n\in { \mathcal S }}{k}_{n}[{\rm{Y}}]-\displaystyle \sum _{m\in { \mathcal T }}{k}_{m}[{\rm{Z}}][{\rm{M}}]\right),\end{eqnarray} \tag{ 4 }$$

where [·] denotes species number density, k_x the rate coefficient for process Px as listed in table A1, and the summations goes over the set of second order processes ${ \mathcal S }:= a$ {324–332, 410, 418, 426, 434, 442, 450, 458, 466, 474, 484, 494, 504, 514, 524, 534, 544, 556}, and the set of third order processes ${ \mathcal T }:= a$ {562, 568, 574, 585, 593, 601, 609, 617}.

Table 1. Species considered in the full kinetic model: 12 positive species, 9 negative species, 21 neutral species.

Positive species	N+, ${{\rm{N}}}_{2}^{+}$, ${{\rm{N}}}_{3}^{+}$, ${{\rm{N}}}_{4}^{+}$, O+, O2 +, ${{\rm{O}}}_{4}^{+}$, NO+, N2O+, NO2 +, ${{\rm{O}}}_{2}^{+}$ N2, ${{\rm{N}}}_{2}^{+}({{\rm{B}}}^{2})$
Negative species	e, O−, ${{\rm{O}}}_{2}^{-}$, ${{\rm{O}}}_{3}^{-}$, ${{\rm{O}}}_{4}^{-}$, NO−, N2O−, NO2 −, NO3 −
Neutral species	N2, N2(A3), N2(B3), N2(a'1), ${{\rm{N}}}_{2}({{\rm{C}}}^{3})$ , N, N(2D), N(2P), O2, O2(a1), O2(b1), O2(4.5 eV), O, O(1D), O(1S), O3, NO, N2O, NO2, NO3, N2O5

The relation between the band emission intensity and the density of the upper excited state ${n}_{v^{\prime} }^{* }$ is given as

$$\begin{eqnarray}&&{I}_{v^{\prime} v^{\prime\prime} }=L\displaystyle \frac{{hc}}{{\lambda }_{v^{\prime} v^{\prime\prime} }}{n}_{v^{\prime} }^{* }{A}_{v^{\prime} v^{\prime\prime} },\end{eqnarray} \tag{ 5 }$$

where $v^{\prime} v^{\prime\prime}$ defines the vibronic transition, ${A}_{v^{\prime} v^{\prime\prime} }$ is the rate of spontaneous emission (band Einstein coefficient) (s⁻¹)and ${\lambda }_{v^{\prime} v^{\prime\prime} }$ is the characteristic wavelength of emitted radiation during transition between upper ($v^{\prime}$) and lower (v'') vibronic states. The constant L corresponds to the thickness of the emitting medium, as long as the medium is uniform. This intensity is integrated spectrally over the band as well as over the solid angle and it has the unit of (W m⁻²).

Finally, the FNS/SPS ratio can be expressed as

$$\begin{eqnarray}&&R(E/N)=\displaystyle \frac{{I}_{\mathrm{FNS}}}{{I}_{\mathrm{SPS}}}=\displaystyle \frac{{\lambda }_{\mathrm{SPS}}[{{\rm{N}}}_{2}^{+}({B}^{2})]{A}_{\mathrm{FNS}}}{{\lambda }_{\mathrm{FNS}}[{{\rm{N}}}_{2}({C}^{3})]{A}_{\mathrm{SPS}}}.\end{eqnarray} \tag{ 6 }$$

Since the densities depend on the reduced electric field through the electron-impact processes, the formula (6) gives the relation between steady state FNS/SPS intensity ratio and the reduced electric field.

The EEM according to Morris [24] belongs to the family of screening methods. As such, it is capable of identifying inputs, to which a numerical model is insensitive. The main advantage of the method is the number of necessary model evaluations, which is substantially lower compared to variance-based sensitivity analysis methods, e.g. the Sobol method [39]. In contrast to partial-derivative based sensitivity analysis methods [40, 41] used e.g. in [42, 43], EEM scales better with a large number of model inputs. At the same time, the method is capable of detecting nonlinear interactions between the model inputs (e.g. ith input is influential only if jth input is sufficiently large). As the trade-off for the low number of evaluations, the EEM provides only approximate ranking of the importance of the individual model inputs. This is, however, sufficient for reduction of a kinetic scheme, as previously illustrated in [44]. Since the EEM requires that the model inputs be distributed on a unit k-dimensional hypercube, with k being the number of model inputs, we define a vector of factors ${\bf{x}}=({x}_{1},\,\ldots ,\,{x}_{k})$, with each x_j is restricted to a finite number of possible choices on ${x}_{j}\in [0,1]$.

The key quantity for the EEM is then the so-called elementary effect. For ith model input, the elementary effect for model output Y(x) is obtained as

$$\begin{eqnarray}&&{d}_{i}({\bf{x}})=\displaystyle \frac{Y({x}_{1},\,\ldots ,\,{x}_{i}+{\rm{\Delta }},\,\ldots ,\,{x}_{k})-Y({x}_{1},\,\ldots ,\,{x}_{k})}{{\rm{\Delta }}},\end{eqnarray} \tag{ 7 }$$

where Δ is a constant value by which ith input factor is modified. The Morris' EEM method is based on generating so-called trajectories through the configuration space. Each step of Morris trajectory corresponds to a single elementary effect evaluation, proceeding randomly through elements of x, though, there is exactly one evaluation for each factor per trajectory. To generate trajectories we used basic sampling scheme by Morris [24]. In this way n = 10¹–10² elementary effects is required to be generated for each model input to provide sufficient statistics, this corresponds to n trajectories through the hypercube, resulting to a set of n elementary effects ${{\bf{d}}}_{i}=\{{d}_{i}^{1},\,\ldots ,\,{d}_{i}^{n}\}$, for each input i. In this work, we used n = 100. Some further improvements for sampling strategy and method efficiency have been proposed in later works, see [45].

Within our work, we apply the EEM to the 'full kinetic model' of nitrogen–oxygen plasma in 0D, as described in section 2.1, looking for the FNS/SPS intensity ratio R(E/N) as the output of the model, i.e. the R(E/N) is the observable for the EEM. We aim to determine the sensitivity of R(E/N) to all, k, rate constants which are included in the full kinetic model. The vector ${\bf{k}}=\left({k}_{1}\left(\tfrac{E}{N}\right),\,\ldots ,\,{k}_{k}\left(\displaystyle \frac{E}{N}\right)\right)$ denotes the vector of reaction rates, which have to be varied in order to assess the model sensitivity to these reactions. To link k with vector of factors x in the EEM we create a mapping so that

$$\begin{eqnarray}&&{k}_{i}\left(\displaystyle \frac{E}{N}\right)=\left({x}_{i}+\displaystyle \frac{1}{2}\right)\cdot {k}_{0,i}\left(\displaystyle \frac{E}{N}\right),\end{eqnarray} \tag{ 8 }$$

where ${k}_{0,i}\left(\displaystyle \frac{E}{N}\right)$ is the unperturbed rate coefficient for ith process used in the full kinetic model. This mapping ensures the rate coefficients are varied from $0.5{k}_{0,i}$ to $1.5{k}_{0,i}$ and that the model can be expressed consistently with the EEM.

In order to encode output of the model into a single number to be used in (7), we construct L1-error of the time-evolution of the FNS/SPS intensity ratio comparing solutions F and with perturbed and unperturbed rate constants respectively:

$$\begin{eqnarray}&&Y({\bf{k}})={\int }_{0}^{T}| F({\bf{k}},t)-F({{\bf{k}}}_{0},t)| \,{\rm{d}}t,\end{eqnarray} \tag{ 9 }$$

where k is given by (8) and ${{\bf{k}}}_{0}$ is the unperturbed rate coefficient used in the full kinetic model.

The quantities which reveal whether the model is sensitive to ith input are then the mean value μ_i over the set of n elementary effects d_i :

$$\begin{eqnarray}&&{\mu }_{i}=\displaystyle \frac{1}{n}\displaystyle \sum _{j=1}^{n}{d}_{i}^{j},\end{eqnarray} \tag{ 10 }$$

and the standard deviation of the mean, denoted σ_i :

$$\begin{eqnarray}&&{\sigma }_{i}=\sqrt{\displaystyle \frac{1}{n}\displaystyle \sum _{j=1}^{n}{({d}_{i}^{j}-{\mu }_{i})}^{2}}.\end{eqnarray} \tag{ 11 }$$

To be able to conclude that the model is insensitive to ith input, both σ_i , and μ_i have to be sufficiently low. A high value of μ_i shows strong linear interaction of the model with the input while a high value of σ_i suggests that there is a nonlinear interaction between the output and the input. The choice for the criterion for μ_i and σ_i which distinguishes between influential and non-influential reactions is model-specific and is discussed, for the presented case, in section 4.1.

The origin of the uncertainty in calculated R(E/N) is the spread of values in kinetic data, i.e. quenching rates, reaction rates, cross-sections, available in literature. These data were obtained experimentally and the spread is a consequence of both aleatoric (statistical) uncertainties, i.e. the accuracy and noise level of the instrument, and epistemic (systematic) uncertainties, e.g. non-selectivity of the method, different assumptions in data processing. The aleatoric uncertainties can be, in part, eliminated by sufficient statistics, although subtle differences in experimental setups always lead to different values when an experiment is reproduced by different author. The epistemic uncertainties are a much bigger challenge, since their elimination requires deep insight into the methods that have been used for the determination of the kinetic data and assessment of their reliability.

Within this work, we use Monte Carlo-based uncertainty quantification in order to determine the confidence band for the R(E/N). Specifically, we perform 5000 runs of the tested model for each E/N. In each of these runs, the kinetic data for influential processes are randomly varied with uniform distribution within a certain range. The width of the confidence band of the FNS/SPS ratio is then obtained as the maximum and minimum value of the FNS/SPS ratio from the 5000 runs. The range, in which the kinetic data are varied is derived from literature.

In this paragraph we find a reduced kinetic model which, for given electron density n_e and reduced electric field E/N, predicts the same R(E/N) ratio (within the chosen tolerance) as the full kinetic model. The reduction of the kinetic scheme is crucial for the subsequent uncertainty quantification and detailed revision of the kinetic data. In order to identify processes that are influential for R(E/N), we perform sensitivity analysis using EEM on the full kinetic model of 617 reactions. Reduced electric field E/N is varied in the range from 100 to 3000 Td. The range for n_e was chosen to represent conditions similar to experimental works [5, 46]. These experiments were performed in a Townsend-like discharge, where the electric field distortion E_s by space charge was checked to be below 1% of the external electric field magnitude E. Considering the experimental setup in [5, 46], we can estimate the maximum value of n_e, for which the discharge still operates in the Townsend mode, by considering an infinite planar plasma slab with thickness of d = 0.2 mm filled with uniformly distributed charge of density ${q}_{0}{n}_{{\rm{e}}}$. For the electric field distortion E_s/E = 0.01, the reduced electric field E/N = 200 Td, and air number density N = 2.5 × 10¹⁹ cm⁻³, the estimated threshold electron density is approximately:

$$\begin{eqnarray*}&&{n}_{{\rm{e}}}=\displaystyle \frac{2{\varepsilon }_{0}}{{{dq}}_{0}}\displaystyle \frac{{E}_{{\rm{s}}}}{E}\displaystyle \frac{E}{N}N\approx 2.8\times {10}^{10}\ {\mathrm{cm}}^{-3}.\end{eqnarray*}$$

To cover safely the experimental conditions, we therefore choose n_e to vary from 10⁸ to 10¹² cm⁻³. Further decrease of n_e would not affect the results in terms of the number of influential processes. For the analysis in this section, we use cross-sections from [32, 33] to calculate electron-impact reaction rates. Note that choice of cross-section set is not crucial here, because resulting reaction rates are anyway artificially perturbed on the random walk through the Morris trajectories.

Figure 1 shows the mean value μ_i and standard deviation σ_i of the elementary effects for the most important processes at reduced electric field 2000 Td and electron density n_e = 10¹⁰ cm⁻³, and at reduced electric field 200 Td and electron density n_e = 10¹² cm⁻³. Note that μ_i and σ_i where normalized by ${\delta }_{\max }=\max ({\delta }_{i})$, where ${\delta }_{i}=\sqrt{{{\mu }_{i}}^{2}+{{\sigma }_{i}}^{2}}$ is the distance of each point from zero in σ_i –μ_i plot. Processes with values of both μ_i and σ_i close to zero can be considered non-influential for the model and vice versa. This provides relative ranking for each process considered in the full kinetic model.

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Figure 2 shows the relative importance of the 12 most influential processes in terms of δ_i /δ_max, as a function of E/N for n_e = 10¹⁰ cm⁻³ and n_e = 10¹² cm⁻³. The choice of the minimum value of δ_i /δ_max for which the processes are still considered influential, is strongly model-specific, and requires heuristic analysis. We have found that using the criterion δ_i /δ_max ≥ 0.1, the maximum relative difference between values of R(E/N) obtained from the full model and the reduced model will be no more than 10⁻². The 12 most influential processes are listed in table 2 in the order of importance. There are, in total, 9 processes that meet the δ/δ_max ≥ 0.1 criterion. Note that this set of processes includes the three-body quenching of ${{\rm{N}}}_{2}^{+}({{\rm{B}}}^{2})$, i.e. the process P366, which was a point of discussion in [23, 47]. It is clear from figure 2 that this process is only important at low values of E/N. Furthermore, in figure 2, we see that electron-impact processes P050 and P023 are getting more influential as the electron density increases and might need to be considered a part of the reduced kinetic model when calculating the FNS/SPS intensity ratio at electron densities higher than 10¹² cm⁻³.

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Table 2. The most important processes for the FNS/SPS intensity ratio listed in order of importance. The R1–R9 set defines the reduced model. Processes P023 and P050 may be influential for electron densities higher than ${10}^{12}\,$ cm⁻³. Full spectroscopic notation of species is used in this table: ${{\rm{N}}}_{2}{({{\rm{X}}}^{1}{{\rm{\Sigma }}}_{{\rm{g}}}^{+})}_{v=0}$, ${{\rm{N}}}_{2}^{+}{({{\rm{X}}}^{2}{{\rm{\Sigma }}}_{{\rm{g}}}^{+})}_{v=0}$, ${{\rm{N}}}_{2}^{+}{({{\rm{B}}}^{2}{{\rm{\Sigma }}}_{{\rm{u}}}^{+})}_{v=0}$ and ${{\rm{N}}}_{2}{({{\rm{C}}}^{3}{{\rm{\Pi }}}_{{\rm{u}}})}_{v^{\prime} =0}$ is represented by short notation ${{\rm{N}}}_{2}$, ${{\rm{N}}}_{2}^{+}$, ${{\rm{N}}}_{2}^{+}({{\rm{B}}}^{2})$ and ${{\rm{N}}}_{2}({{\rm{C}}}^{3})$, respectively in table A1.

Reaction no.	Process no.	Reaction
R1	P028	${\rm{e}}+{{\rm{N}}}_{2}{({{\rm{X}}}^{1}{{\rm{\Sigma }}}_{{\rm{g}}}^{+})}_{v=0}\longrightarrow {{\rm{N}}}_{2}^{+}{({{\rm{B}}}^{2}{{\rm{\Sigma }}}_{{\rm{u}}}^{+})}_{v^{\prime} =0}+2{\rm{e}}$
R2	P010	${\rm{e}}+{{\rm{N}}}_{2}{({{\rm{X}}}^{1}{{\rm{\Sigma }}}_{{\rm{g}}}^{+})}_{v=0}\longrightarrow {{\rm{N}}}_{2}{({{\rm{C}}}^{3}{{\rm{\Pi }}}_{{\rm{u}}})}_{v^{\prime} =0}+{\rm{e}}$
R3	P106	${{\rm{N}}}_{2}^{+}{({{\rm{B}}}^{2}{{\rm{\Sigma }}}_{{\rm{u}}}^{+})}_{v=0}\longrightarrow {{\rm{N}}}_{2}^{+}{({{\rm{X}}}^{2}{{\rm{\Sigma }}}_{{\rm{g}}}^{+})}_{v^{\prime\prime} =0}+h\nu$
R4	P105	${{\rm{N}}}_{2}{({{\rm{C}}}^{3}{{\rm{\Pi }}}_{{\rm{u}}})}_{v^{\prime} =0}\longrightarrow {{\rm{N}}}_{2}{({{\rm{B}}}^{3}{{\rm{\Pi }}}_{{\rm{g}}})}_{v^{\prime\prime} =0}+h\nu$
R5	P331	${{\rm{N}}}_{2}^{+}{({{\rm{B}}}^{2}{{\rm{\Sigma }}}_{{\rm{u}}}^{+})}_{v=0}+{{\rm{N}}}_{2}{({{\rm{X}}}^{1}{{\rm{\Sigma }}}_{{\rm{g}}}^{+})}_{v=0}\longrightarrow {{\rm{N}}}_{2}^{+}{({{\rm{X}}}^{2}{{\rm{\Sigma }}}_{{\rm{g}}}^{+})}_{v=0}+{{\rm{N}}}_{2}{({{\rm{X}}}^{1}{{\rm{\Sigma }}}_{{\rm{g}}}^{+})}_{v=0}$
R6	P332	${{\rm{N}}}_{2}^{+}{({{\rm{B}}}^{2}{{\rm{\Sigma }}}_{{\rm{u}}}^{+})}_{v=0}+{{\rm{O}}}_{2}\longrightarrow {{\rm{N}}}_{2}^{+}{({{\rm{X}}}^{2}{{\rm{\Sigma }}}_{{\rm{g}}}^{+})}_{v=0}+{\rm{O}}+{\rm{O}}{(}^{1}{\rm{S}})$
R7	P129	${{\rm{N}}}_{2}{({{\rm{C}}}^{3}{{\rm{\Pi }}}_{{\rm{u}}})}_{v^{\prime} =0}+{{\rm{N}}}_{2}{({{\rm{X}}}^{1}{{\rm{\Sigma }}}_{{\rm{g}}}^{+})}_{v=0}\longrightarrow {{\rm{N}}}_{2}({\rm{a}}^{\prime} 1)+{{\rm{N}}}_{2}{({{\rm{X}}}^{1}{{\rm{\Sigma }}}_{{\rm{g}}}^{+})}_{v=0}$
R8	P130	${{\rm{N}}}_{2}{({{\rm{C}}}^{3}{{\rm{\Pi }}}_{{\rm{u}}})}_{v^{\prime} =0}+{{\rm{O}}}_{2}\longrightarrow {{\rm{N}}}_{2}{({{\rm{X}}}^{1}{{\rm{\Sigma }}}_{{\rm{g}}}^{+})}_{v=0}+{\rm{O}}+{\rm{O}}{(}^{1}{\rm{S}})$
R9	P366	${{\rm{N}}}_{2}^{+}{({{\rm{B}}}^{2}{{\rm{\Sigma }}}_{{\rm{u}}}^{+})}_{v^{\prime} =0}+{{\rm{N}}}_{2}{({{\rm{X}}}^{1}{{\rm{\Sigma }}}_{{\rm{g}}}^{+})}_{v=0}+{\rm{M}}\longrightarrow {{\rm{N}}}_{4}^{+}+{\rm{M}}$
	P324	${{\rm{N}}}_{2}^{+}{({{\rm{B}}}^{2}{{\rm{\Sigma }}}_{{\rm{u}}}^{+})}_{v=0}+{{\rm{O}}}_{2}\longrightarrow {{\rm{O}}}_{2}^{+}+{{\rm{N}}}_{2}$
	P023	${\rm{e}}+{{\rm{N}}}_{2}{({{\rm{X}}}^{1}{{\rm{\Sigma }}}_{{\rm{g}}}^{+})}_{v=0}\longrightarrow {{\rm{N}}}_{2}^{+}{({{\rm{X}}}^{2}{{\rm{\Sigma }}}_{{\rm{g}}}^{+})}_{v^{\prime\prime} =0}+2{\rm{e}}$
	P050	${\rm{e}}+{{\rm{N}}}_{2}^{+}{({{\rm{X}}}^{2}{{\rm{\Sigma }}}_{{\rm{g}}}^{+})}_{v^{\prime\prime} =0}\longrightarrow {\rm{e}}+{{\rm{N}}}_{2}^{+}{({{\rm{B}}}^{2}{{\rm{\Sigma }}}_{{\rm{u}}}^{+})}_{v=0}$

To conclude this section, using the EEM of sensitivity analysis, we were able to reduce the full model of 617 reactions to a minimum model of only 9 reactions, maintaining less than 1% deviation for $R(E/N)$ compared to the full model. Note that this minimal model proofs the model generally used, for instance in [3, 48]. This work mathematically shows that the generally used minimal model does not lack any of the known reactions. And that the three-body quenching which has been disputed in the community [23, 47] might be of importance, but only at a very low value of the reduced electric field.

For the sake of brevity, the table 2 introduces simplified numbering of the reactions in the reduced model: R1–R9. This numbering is used further on in this work, where we focus on identification and quantification of uncertainties for these reactions.

Using the sensitivity analysis, we obtained a reduced model comprising of only 9 processes. Since there are numerous kinetic data available in literature for these processes, the uncertainty of model should be quantified. To get a deeper insight, we divide the sources of uncertainty into the following four components.

• (i)  
  uncertainty in EEDF,
• (ii)  
  uncertainty in electron-impact cross-sections (R1, R2),
• (iii)  
  uncertainty in radiative de-excitaions (R3, R4),
• (iv)  
  uncertainty in quenching rate constants (R5–R9).

The full-range model represents the most straightforward approach to the uncertainty quantification wherein all the kinetic data available in literature are considered in the uncertainty quantification (apart from obvious outliers). The references that have been used for the kinetic data are listed in table 3. The table 3 also includes the exact intervals, in which the individual kinetic parameters were varied.

Table 3. Rate constants of the most important processes and relevant interval of experimentally measured data. While there is a good agreement between different references for the quenching rate constants of $({{\rm{C}}}^{3}{{\rm{\Pi }}}_{{\rm{u}}})$ state (processes R7 and R8). There are much larger discrepancies in quenching rates that are available in literature for $({{\rm{B}}}^{2}{{\rm{\Sigma }}}_{{\rm{u}}}^{+})$ state (R5 and R6).

No.	Rate constant	Range of kinetic data	References for UQ
R1	${k}_{\mathrm{im}}^{{\rm{B}}}=\sqrt{\tfrac{2{q}_{{\rm{e}}}}{{m}_{{\rm{e}}}}}{\int }_{0}^{\infty }\epsilon f(\epsilon )\sigma (\epsilon ){\rm{d}}\epsilon$	EEDF × CSS	[49–56]
R2	${k}_{\mathrm{im}}^{{\rm{C}}}=\sqrt{\tfrac{2{q}_{{\rm{e}}}}{{m}_{{\rm{e}}}}}{\int }_{0}^{\infty }\epsilon f(\epsilon )\sigma (\epsilon ){\rm{d}}\epsilon$	EEDF × CSS	[1, 55, 57–66]
R3	${A}_{0{\rm{B}}}$	(1.11 $\div$ 2.43) × 107 (s−1)	[67–93]
R4	${A}_{0{\rm{C}}}$	(1.78 $\div$ 3.57) × 107 (s−1)	[67, 70, 72–75, 78, 82, 86, 89, 91, 92, 94–108]
R5	${k}_{q,{{\rm{N}}}_{2}}^{{\rm{B}}}$	(1.37 $\div$ 9.80) × 10−10 (cm3 s−1)	[75, 78, 82, 84, 85, 91–93, 109–118]
R6	${k}_{q,{{\rm{O}}}_{2}}^{{\rm{B}}}$	(4.6 $\div$ 11.0) × 10−10 (cm3 s−1)	[91, 93, 109, 112, 113, 115, 117, 118]
R7	${k}_{q,{{\rm{N}}}_{2}}^{{\rm{C}}}$	(0.40 $\div$ 1.70) × 10−11 (cm3s−1)	[82, 91, 94, 96, 98, 100, 108–110, 112, 117–124]
R8	${k}_{q,{{\rm{O}}}_{2}}^{{\rm{C}}}$	(1.90 $\div$ 3.30) × 10−10 (cm3 s−1)	[91, 98, 109, 112, 117–120, 124]
R9	${k}_{{\rm{q}},3{\rm{b}}}^{{\rm{B}}}={k}_{0}{\left(\tfrac{{T}_{\mathrm{gas}}}{{T}_{\mathrm{eff}}}\right)}^{1.6}$	${k}_{0}\,\in \,$(5.0 $\div$ 8.5) × 10−29 (cm6 s−1)	[125–132]

In the sections below, we briefly discuss the impact of each of the class of processes on the uncertainty in the FNS/SPS ratio. In part II to this publication, we further provide a thorough analysis of these references, deeming some of the values less reliable than others, thereby reducing the model uncertainty.

4.2.1. Uncertainty in EEDF

Both the full and the reduced models utilize BOLSIG+ [133] for calculating the EEDF. The input settings of BOLSIG+ were ionization degree 10⁻⁴, plasma density 10¹² cm⁻³ and temporal model for growth. Four BOLSIG+ compliant cross-sectional databases are available: Triniti [36] , IST Lisbon [37, 38], Biagi [32, 33] and Phelps [34, 35]. Since the cross-sections available in each of these databases differ slightly, the calculated rates of R1 and R2 are influenced and the FNS/SPS ratio will also vary.

If we fix the processes R3–R9 from table 3 on the mean value from interval of their rate coefficients and for processes R1, R2 we use rate coefficients combining electron-impact cross-sections of [56, 64] with different EEDFs obtained from the cross-section libraries (Biagi, Phelps, Triniti, IST Lisbon), we can read the uncertainty of the model from the blue band in figure 3. Specifically, the model gives the electric field ratio anywhere in the range of 440 and 500 Td for the FNS/SPS ratio of 10⁻². Compared to the other sources of uncertainty, the cross-sectional databases present only a minor source.

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4.2.2. Uncertainty in electron-impact cross-sections of R1 and R2

4.2.3. Uncertainty in radiative de-excitation lifetimes

4.2.4. Uncertainty in quenching rate constants

In figure 4, we plot the total uncertainty band of the model together with theoretical R(E/N) curves obtained by other authors and one experimental. It is apparent that all the curves available in literature lie within the confidence band, as they should since the authors constructing these models worked with equivalent literature. This illustrates that the theoretical R(E/N) curve has very large uncertainty that has to be considered in data processing using this method. Considering that the electric field measured by the intensity ratio method might be further utilized for calculating the plasma kinetics and that electron-impact reactions often show exponential dependence on E/N, we can assert that reducing the R(E/N) uncertainty is very desirable.

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Summarizing the results above, the differences among considered cross-section datasets present only a minor source of uncertainty of the theoretical R(E/N) curves.

On the other hand, the ionization and excitation cross-sections for the ${{\rm{N}}}_{2}^{+}({{\rm{B}}}^{2})$ and ${{\rm{N}}}_{2}({{\rm{C}}}^{3})$ states, as well as their quenching rates and radiative lifetimes all present substantial sources of uncertainty. Taking all of them into account, we find that the total uncertainty of the R(E/N) curves is very significant. The spread of R(E/N) curves available in literature confirms that there is no agreement in the community as to which kinetic data should be used for obtaining R(E/N).

Figure 5 shows the synthetic spectra of SPS and FNS generated using models detailed in [1]. It should be underlined that in the case of FNS most of the emission originating from the v = 0 vibrational level goes indeed through the $v^{\prime} =0\to v^{\prime\prime} =0$ transition, therefore the (0, 0) band of the FNS is the obvious choice for detection. However, in the case of the SPS emission, there are other options because several bands originating from the v = 0 level of the ${{\rm{N}}}_{2}({{\rm{C}}}^{3}{{\rm{\Pi }}}_{{\rm{u}}})$ state occur with sufficient intensities, e.g. (0, 1), (0, 2) or (0, 3) bands indicated by arrows in the figure 5 at 357.7, 380.5 and 405.9 nm, respectively. Considering that the detection efficiency often increases from UV to visible spectral range, the combination of FNS(0, 0) and SPS(0, 2), or FNS(0, 0) and SPS(0, 3) might be better choice compared with the FNS(0, 0) and SPS(0, 0) case. This is certainly true in the case of remote spectrometric observation of transient luminous events because of heavy attenuation of the UV part of emission spectra, including the SPS(0, 0) band intensity. It is worth noting that Paris et al [5] also quantified the electric field dependence for the ratios of FNS(0, 0)/SPS(2, 5) and SPS(2, 5)/SPS(0, 0). The variation of the SPS(2, 5)/SPS(0, 0) intensity ratio in the low field region <300 Td is consistent with predicted E/N dependence of the rate constant for electron-impact excitation of individual vibration levels of the ${{\rm{C}}}^{3}{{\rm{\Pi }}}_{{\rm{u}}}$ state [1]. A similar variation with the E/N should be obtained for other SPS bands, the (3, 6) and (1, 4) bands, occurring next or close to the FNS(0, 0) band (see figure 5). The FNS(0, 0)/SPS(3, 6) and FNS(0, 0)/SPS(1, 4) might provide alternative R(E/N) curves to be used simultaneously with R(E/N) curves derived from FNS(0, 0)/SPS(0, 3) and FNS(0, 0)/SPS(2, 5).

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Each of these ratios dependences represents a diagnostic tool for the determination of the reduced electric field. Each of these dependencies would require additional uncertainty quantification, which would also consider the uncertainty in the respective vibrational transitions and Frank–Condon factors. However, the 9 influential processes identified in this work are also valid for other FNS/SPS bands.