Investigation of N(⁴S) kinetics during the transients of a strongly emissive pulsed ECR plasma using ns-TALIF

The microwave (MW)-assisted plasma was generated using a commercial ECR source (Sairem Aura-Wave) detailed in [33]. The magnet is placed on the plasma source such that the magnetic field ensuring the ECR regime is directed toward the center of the reactor chamber, thus minimizing the electron losses at the walls. The magnetic field has an intensity of 0.0875 T and the resulting plasma has a toroidal shape (figure 1). The MW power is supplied through a coaxial feed from a solid-state MW generator (Sairem GMS 200) that delivers a maximum power of 200 W. The operating MW frequency was tuned between 2.4 and 2.5 GHz in order to achieve optimal impedance matching between the source and the discharge system. The plasma may be operated in both continuous and pulsed modes. The pulsed mode regime makes use of a square waveform power pulse and is characterized by the pulse period, the peak power and the duty cycle which represents the ratio of the high power phase duration to the pulse period. The plasma source was pulsed at the laser frequency of 10 Hz and the pressure was fixed at 20 Pa in all the experiments of this work.

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The source was embedded in a 100 mm diameter stainless-steel six-way cross and was placed on a specifically designed flange at the top of the cross. Three flanges were dedicated to optical viewports. The laser beam enters and exits the plasma reactor through two fused silica viewports (90% transmission at 205 nm). The fluorescence was collected perpendicularly to the laser beam through a borosilicate viewport. A metallic grid was placed at each viewport in order to prevent MW leakage outside the cavity. The vacuum was obtained using a set of turbo-molecular and rotary pumps (Edwards EXT75DX and Pfeiffer Duo 6M), allowing a residual pressure of 10⁻⁵ Pa to be achieved. The working pressure in the chamber was monitored using a capacitive gauge (Pfeiffer CCR363) working in the range 0.133–1333 Pa. The gases were injected in the reactor using mass flow controllers (Bronkhorst EL flow), the flowrate being set at 5 sccm.

The characterization of the pulsed MW plasma by means of a nanosecond-two-photon absorption laser-induced fluorescence (ns-TALIF) diagnostic was carried out using a dye laser pumped by a pulsed Nd:YAG laser, the TALIF signal being acquired with a gated photomultiplier tube. In the following a detailed description of each component of this setup is given (figure 1).

2.2.1. Laser system

2.2.2. Fluorescence collection system

We have used the TALIF scheme proposed by Bengtsson et al [10] that shows much higher sensitivity [11] than the very first scheme proposed by Bischel et al [8]. In this scheme, a N-atom in the (2p³)⁴ ${S}_{3/2}^{\text{o}}$ ground state, denoted as N in the following, absorbs two photons produced by a laser pulse centred around 206.67 nm, and undergoes an electronic transition to the (3p)⁴ ${S}_{3/2}^{\text{o}}$ excited state hereafter denoted N^*. Assuming that the laser–atom interaction is dominated by the two-photon absorption, the density of the produced excited state is directly related to the offset frequency δν_N = 2ν_l − ν_N where ν_l and ν_N are the laser central frequency and the atomic transition frequency of the absorption line, respectively. After the laser pulse, N^* undergoes a first-order de-excitation through radiative decay and collisional quenching. The radiative de-excitation to the (3s)⁴P_1/2,3/2,5/2 degenerate state produces three fluorescence lines at 742.364, 744.230 and 746.831 nm. Thus, during the de-excitation phase just after the laser pulse, the N^* density follows an exponential decay, with a time constant ${\tau }_{\text{f,N}}=\frac{1}{{\sum }_{i}{A}_{i,3\text{p}\to 3\text{s}}+Q}$, A_i,3p→3s being the Einstein coefficient of the ith radiative de-excitation process [34] and Q being the rate of collisional quenching of N^*. The resulting incremental time-variation, ${\mathrm{d}}^{2}{n}_{{\mathrm{N}}^{{\ast}}}$, during dt, of the instantaneous density of the excited state, $\mathrm{d}{n}_{{\mathrm{N}}^{{\ast}}}$(ν_l ± d(ν_l), t), resulting from a laser pulse centred at a laser frequency ν_l ± d(ν_l), corresponding to an offset frequency δν_N ± d(δν_N), can be expressed as:

$$\begin{align*}{\mathrm{d}}^{2}{n}_{{\mathrm{N}}^{{\ast}}}& ={G}^{\left(2\right)}{\sigma }_{\text{N}}^{\left(2\right)}{n}_{\text{N}}{\left(\frac{{I}_{\text{l}}T\left(t\right)}{h{\nu }_{\text{l}}}\right)}^{2}g\left({\delta}{\nu }_{\text{N}}\right)\mathrm{d}\left({\delta}{\nu }_{\text{N}}\right)\mathrm{d}\mathrm{t}\\ & \quad -\frac{\mathrm{d}{n}_{{\mathrm{N}}^{{\ast}}}({\nu }_{\text{l}}{\pm}\mathrm{d}({\nu }_{\text{l}}),\mathrm{t})}{{\tau }_{\text{f,N}}}\mathrm{d}t,\end{align*} \tag{ 1 }$$

where ${G}^{\left(2\right)}$ and ${\sigma }_{\text{N}}^{\left(2\right)}$are the two-photon statistical factor and the two-photon excitation cross-section (m⁴) for the 2p → 3p transition at a frequency ν_N, respectively. I_l and $T\left(t\right)$ are the radiant exposure (J m⁻²) and the normalized time profile (s⁻¹) of the laser pulse. $g\left({\delta}{\nu }_{\text{N}}\right)$ is the statistical overlap factor between the nitrogen absorption line and the normalized laser line such that ${\int }_{-\infty }^{+\infty }g\left({\delta}{\nu }_{\text{N}}\right)\mathrm{d}({\delta}{\nu }_{\text{N}})=1$. $g\left({\delta}{\nu }_{\text{N}}\right)\mathrm{d}\left({\delta}{\nu }_{\text{N}}\right)$ also represents the laser energy fraction that is actually used to excite the atoms.

The measured fluorescence signal ${\mathrm{d}}^{2}{s}_{\text{f,N}}{\left({\nu }_{\text{l}},t\right)}_{\text{N}}$ during a time interval dt is proportional to the instantaneous density of the excited state $\mathrm{d}{n}_{{\mathrm{N}}^{{\ast}}}({\nu }_{\text{l}}{\pm}\mathrm{d}({\nu }_{\text{l}}),t)$ as follows:

$$\begin{equation}{\mathrm{d}}^{2}{s}_{\text{f,N}}\left({\nu }_{\text{l}},t\right)={K}_{\text{g}}{\eta }_{\text{N}}\sum\limits _{i}\;{T}_{i,3\text{p}\to 3\text{s}}{A}_{i,3\text{p}\to 3\text{s}}\mathrm{d}{n}_{{\mathrm{N}}^{{\ast}}}\mathrm{d}t\end{equation} \tag{ 2 }$$

with T_i,3p→3s the transmittance of the optics used for collecting the fluorescence signal and η_N the sensitivity of the sensor over the spectral width of the fluorescence signal. K_g is a geometrical factor that depends on several optical parameters that are difficult to estimate, e.g. the probed volume and the solid angle of the fluorescence volume subtended on the sensor. This factor shows the same value for N and Kr TALIF systems.

Substituting $\mathrm{d}{n}_{{\mathrm{N}}^{{\ast}}}\mathrm{d}t$ by its value as a function of ${\mathrm{d}}^{2}{s}_{\text{f,N}}\left({\nu }_{\text{l}},t\right)$ given by equation (2) in equation (1), and integrating over a long time period, one gets:

$$\begin{align*}\frac{{\int }_{t=0}^{\infty }{d}^{2}{s}_{\text{f,N}}\left({\nu }_{\text{l}},t\right)}{{\left({E}_{\text{l}}/h{\nu }_{\text{l}}\right)}^{2}}& =\frac{{K}_{\text{g}}}{{S}^{2}}{\eta }_{\text{N}}\sum\limits _{i}\;{T}_{i,3\text{p}\to 3\text{s}}{A}_{i,3\text{p}\to 3\text{s}}{G}^{\left(2\right)}\\ & \quad {\times}\;{\sigma }_{\text{N}}^{\left(2\right)}{n}_{\text{N}}{\tau }_{\text{f,N}}g\left({\delta}{\nu }_{\text{N}}\right)\mathrm{d}\left({\delta}{\nu }_{\text{N}}\right)\\ & \quad {\times}{\int }_{0}^{\infty }{T}^{2}\left(t\right)\mathrm{d}t,\end{align*} \tag{ 3 }$$

where E_l is the laser pulse energy recorded by the photodiode such that I_l = E_l/S, S being the cross-section of the laser beam. The factor $\mathrm{d}{{\Psi}}_{\text{N}}=\frac{{\int }_{t=0}^{\infty }{\mathrm{d}}^{2}{s}_{\text{f,N}}\left({\nu }_{\text{l}},t\right)}{{\left({E}_{\text{l}}/h{\nu }_{\text{l}}\right)}^{2}}$ is the ratio of the number of detected fluorescence photons during dt to the square of the number of laser photons for a laser central frequency in the range ν_l ± d(ν_l).

In order to eliminate the unknown geometrical factors and to determine the N-atom density, Kr is used as calibrating gas as described by Niemi et al [13]. A factor dΨ_Kr may also be defined for Kr and expressed as a function of the Kr-atom density using an equation similar to (3):

$$\begin{align*}\mathrm{d}{{\Psi}}_{\text{Kr}}& =\frac{{K}_{g}}{{S}^{2}}{\eta }_{\text{Kr}}{T}_{\text{Kr}}{A}_{\text{Kr}5\text{p}\to 5\text{s}}{G}^{\left(2\right)}{\sigma }_{\text{Kr}}^{\left(2\right)}{n}_{\text{Kr}}{\tau }_{\text{f,Kr}}g\left({\delta}{\nu }_{\text{Kr}}\right)\\ & \quad {\times}\;\mathrm{d}\left({\delta}{\nu }_{\text{Kr}}\right){\int }_{0}^{\infty }{T}^{2}\left(t\right)\mathrm{d}t,\end{align*} \tag{ 4 }$$

where T_Kr and η_Kr are the transmittance and the sensitivity of the photomultiplier tube for the spectral range used for the detection of the Kr-atom fluorescence. A_Kr5p→5s is the radiative decay rate for the Kr 5p → 5s transition observed [35].

Taking the ratio of equations (3) and (4) one may express the ground state N-atom density as a function of the known Kr-atom density:

$$\begin{align*}{n}_{\text{N}}& =\frac{{\eta }_{\text{Kr}}}{{\eta }_{\text{N}}}\frac{{T}_{\text{Kr}}{A}_{\text{Kr}5\text{p}\to 5\text{s}}}{{\sum }_{i}{T}_{i,3\text{p}\to 3\text{s}}{A}_{i,3\text{p}\to 3\text{s}}}\frac{{\sigma }_{\text{Kr}}^{\left(2\right)}}{{\sigma }_{\text{N}}^{\left(2\right)}}\frac{{\tau }_{\text{f,Kr}}}{{\tau }_{\text{f,N}}}\\ & \quad {\times}\;\frac{\mathrm{d}{{\Psi}}_{\mathrm{N}}}{\mathrm{d}{{\Psi}}_{\mathrm{K}\mathrm{r}}}\frac{g\left({\delta}{\nu }_{\text{Kr}}\right)}{g\left({\delta}{\nu }_{\text{N}}\right)}{n}_{\text{Kr}}.\end{align*} \tag{ 5 }$$

The ratio of the generalized two-photon excitation cross-sections ${\sigma }_{\text{Kr}}^{\left(2\right)}$ and ${\sigma }_{\text{N}}^{\left(2\right)}$, for Kr (4p⁶)¹S₀ → 5p'[3/2]₂ and N (2p³)⁴ ${S}_{3/2}^{\text{o}}$ → (3p)⁴ ${S}_{3/2}^{\text{o}}$, respectively, is taken from [13].

In equation (5), all the quantities except the ratio $\frac{g\left({\delta}{\nu }_{\text{Kr}}\right)}{g\left({\delta}{\nu }_{\text{N}}\right)}$ are readily available after capturing the fluorescence generated by one laser pulse. The estimation of this ratio requires knowledge of the profiles of the laser beam and the absorption lines of the N-atom and the Kr-atom. In particular, the absorption line profiles depend on the local plasma conditions and are not always easy to determine from the experiments.

This difficulty is generally circumvented by capturing the fluorescence signals generated by several laser pulses with different central frequency values ν_l around 0.5ν_A (A = N or Kr) and integrating dΨ_A over the whole laser central frequency range. As a matter of fact, if we consider the case of N-atom fluorescence, integrating equation (3) over the laser frequency domain yields the following relationship:

$$\begin{align*}{\int }_{{\delta}\nu =-\infty }^{\infty }\mathrm{d}{{\Psi}}_{\text{N}}& =\frac{{K}_{\text{g}}}{{S}^{2}}{\eta }_{\text{N}}\sum\limits _{i}\;{T}_{i,3\text{p}\to 3\text{s}}{A}_{i,3\text{p}\to 3\text{s}}{G}^{\left(2\right)}\\ & \quad {\times}\;{\sigma }_{\text{N}}^{\left(2\right)}{n}_{\text{N}}{\tau }_{\text{f,N}}{\int }_{0}^{\infty }{T}^{2}\left(t\right)\mathrm{d}t.\end{align*} \tag{ 6 }$$

A similar equation can be derived for the Kr-atom which is:

$$\begin{align*}{\int }_{{\delta}\nu =-\infty }^{\infty }\mathrm{d}{{\Psi}}_{\mathrm{K}\mathrm{r}}& =\frac{{K}_{\text{g}}}{{S}^{2}}{\eta }_{\text{Kr}}{T}_{\text{Kr}}{A}_{\text{Kr}5\text{p}\to 5\text{s}}{G}^{\left(2\right)}\\ & \quad {\times}\;{\sigma }_{\text{Kr}}^{\left(2\right)}{n}_{\text{Kr}}{\tau }_{\text{f,Kr}}{\int }_{0}^{\infty }{T}^{2}\left(t\right)\mathrm{d}t.\end{align*} \tag{ 7 }$$

Taking the ratios of equations (6) and (7), yields the following expression for the N-atom density:

$$\begin{equation}{n}_{\text{N}}=\frac{{\eta }_{\text{Kr}}}{{\eta }_{\text{N}}}\frac{{T}_{\text{Kr}}{A}_{\text{Kr}5\text{p}\to 5\text{s}}}{{\sum }_{i}\;{T}_{i,3\text{p}\to 3\text{s}}{A}_{i,3\text{p}\to 3\text{s}}}\frac{{\sigma }_{\text{Kr}}^{\left(2\right)}}{{\sigma }_{\text{N}}^{\left(2\right)}}\frac{{\tau }_{\text{f,Kr}}}{{\tau }_{\text{f,N}}}\frac{{\int }_{-\infty }^{\infty }\mathrm{d}{{\Psi}}_{\mathrm{N}}}{{\int }_{-\infty }^{\infty }\mathrm{d}{{\Psi}}_{\mathrm{K}\mathrm{r}}}{n}_{\text{Kr}}.\end{equation} \tag{ 8 }$$

Equation (8), referred to as the FEM, is widely used in the literature to determine N-atom densities by TALIF experiments. It implicitly captures the overlap factor without any assumption. However, its use requires the determination of the time- and frequency-integrated fluorescence signals for each density value measurement. In our experiments, this typically would require the acquistion of 70 spectral points, taking approximately 20 minutes. This is obviously not suitable for the time-resolved measurements required in the context of the present work on pulsed plasma.

To circumvent this issue, we first make use of the specific discharge conditions investigated here in order to estimate $g\left({\delta}{\nu }_{\text{A}}\right)\mathrm{d}\left({\delta}{\nu }_{\text{A}}\right)$, with A = N or Kr, and subsequently use equation (5) to readily determine the N-atom density using a single laser shot at the central frequency of 0.5ν_A. The procedure used for this purpose is discussed in the following.

From the experimental point of view, $g\left({\delta}{\nu }_{\text{A}}\right)\mathrm{d}\left({\delta}{\nu }_{\text{A}}\right)$ may be determined from the full excitation spectrum; by taking the ratio of equations (3) and (6), one obtains for an atom A = N or Kr:

$$\begin{equation}g\left({\delta}{\nu }_{\text{A}}\right)\mathrm{d}({\delta}{\nu }_{\text{A}})=\frac{\mathrm{d}{{\Psi}}_{\text{A}}}{{\int }_{{\delta}\nu =-\infty }^{\infty }\mathrm{d}{{\Psi}}_{\text{A}}}.\end{equation} \tag{ 9 }$$

Actually, $g\left({\delta}{\nu }_{\text{A}}\right)\mathrm{d}({\delta}{\nu }_{\text{A}})$ depends on the spectral profiles of the laser and the absorption lines. The spectral profile of the laser in this study is Gaussian and this is also the case for the absorption lines under our conditions, since they are dominated by Doppler broadening for both the N-atom and the Kr-atom. In this case, the overlap factors may be easily expressed as:

$$\begin{equation}g\left({\delta}{\nu }_{\text{A}}\right)\mathrm{d}({\delta}{\nu }_{\text{A}})=\frac{\sqrt{4\enspace \text{ln}\enspace 2/\pi }}{{\Delta}{\nu }_{\text{G,A}}^{\text{eff}}}\enspace {\text{e}}^{-4\enspace \text{ln}\enspace 2{\left(\frac{{\delta}{\nu }_{\text{A}}}{{\Delta}{\nu }_{\text{G,A}}^{\text{eff}}}\right)}^{2}}\mathrm{d}({\delta}{\nu }_{\text{A}}),\end{equation} \tag{ 10 }$$

where ${\Delta}{\nu }_{\text{G,A}}^{\text{eff}}=\sqrt{2{\Delta}{{\nu }_{\text{G,l}}}^{2}+{\Delta}{{\nu }_{\text{G,A}}}^{2}}$ is the effective Gaussian full-width at half maximum (FWHM) due to convolution of the laser (Δν_G,l) and the transition profiles (Δν_G,A). ${\Delta}{\nu }_{\text{G,A}}^{\text{eff}}$ is experimentally obtained from the full excitation spectrum.

The laser broadening Δν_G,l was determined from the TALIF signal of Kr at 300 K (without plasma). In such a condition, the Doppler broadening of the Kr absorption line is estimated to be 0.13 cm⁻¹. The measured broadening of the Kr full excitation spectrum line is 1.59 cm⁻¹ which is much greater than the Doppler broadening of Kr at 300 K. The resulting value of the laser line broadening is 1.10 cm⁻¹. It is worth mentioning that this value is significantly larger than those usually reported in the literature, e.g. 0.25 and 0.36 cm⁻¹ in [14] and [21], respectively.

Determination of the absorption line broadening, Δν_G,N, for the N-atom in the plasma requires knowledge of the gas temperature. This was estimated from the rotational temperature of the N₂(C³Π_u) electronically excited state, using OES applied to the transition N₂(C³Π_u →B³Π_g, Δv = −2) and for stationary operation of the MW plasma. The assumption of rotational equilibrium with the gas temperature (T_rot,_N2(C) = T_g) may be questionable for the N₂(C) system in some cases. However, under the considered discharge conditions, the N₂(C) is predominantly produced by electron impact excitation of N₂ ground states and lost by radiative de-excitation. The Franck–Condon principle makes the rotational distribution of the ground (X) and excited (C) states identical and in thermal equilibrium with the gas temperature [36]. Therefore, we can use the rotational temperature of the N₂(C) system as a measure of the gas temperature. The rotational temperature inferred from the rotational structure of the emission band varied between 500 and 600 K in the investigated range of MW power. The resulting Doppler broadening for nitrogen is almost constant at a value of approximately 0.46 cm⁻¹, which is two times smaller than the laser broadening. Consequently, Δν_G,N and $g\left({\delta}{\nu }_{\text{N}}\right)$ should remain almost constant for the considered discharge parameters. This was confirmed by measuring the FWHM of the excitation spectrum ${\Delta}{\nu }_{\text{G,N}}^{\text{eff}}$ over a wide range of plasma conditions as shown in figure 2.

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This finding resulted in a significant simplification of the measurement procedure. As a matter of fact, the experiments to estimate the full integral ${\int }_{{\delta}\nu =-\infty }^{\infty }\mathrm{d}{\Psi}$ have to be performed only once in order to obtain $g\left({\delta}{\nu }_{\text{A}}\right)$. The N-atom density can then be readily determined through equation (5) using a single laser shot at a central frequency of 0.5ν_A, which significantly simplifies the measurement procedure. The use of the peak value at 0.5ν_A achieves the maximum overlap between the laser and the absorption lines, thus limiting the uncertainties of the measured N-atom density.

The temporal integral ${\int }_{t=0}^{\infty }{\mathrm{d}}^{2}{s}_{\text{f,N}}\left({\nu }_{\text{l}},t\right)$ and τ_f are directly determined from the analysis of the time-resolved TALIF signal. The raw photomultipliertube signal for nitrogen was composed of two parts: (a) the time varying N^*atom fluorescence and (b) a plasma background emission identified as the N₂ FPS, i.e. N₂(B³Π_g → A³ ${{\Sigma}}_{\mathrm{u}}^{+}$, Δv = −2), that was collected in the 728–750 nm wavelength range (denoted as 728–750 nm FPS emission). The overlap between these two components resulted in very poor signal-to-noise ratios, varying between 8 for the best case and 1.6 for most cases.

In order to capture the spectral profile of the laserexcitation spectrum, a two-step time- and spectral-fitting/filtering procedure was used, as described below. Firstly, the photomultiplier signal was post-processed using a non-linear least squares fit prior to performing the temporal integration. Figure 3 shows a typical raw photomultipliertube signal along with its non-linear least squares fit. As indicated in figure 3, one may distinguish three different sections of the signal: (I) the constant plasma background region before the initiation of the laser pulse at a time t₀, (II) the fast increase of the fluorescence signal during the laser pulse and (III) the exponential decay with time constant τ_f after time t₁.

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The fitting function is given by:

$$\begin{equation}{\mathrm{d}}^{2}{s}_{\text{f,N}}\left({\nu }_{\text{l}},t\right)=\begin{cases}t{< }{t}_{0}:\enspace & {A}_{0}\\ {t}_{0}{\leqslant}t{< }{t}_{1}:\enspace & {A}_{0}+{A}_{1}\left(t-{t}_{0}\right)\\ t{\geqslant}{t}_{1}:\enspace & {A}_{0} + {A}_{1}\left({t}_{1} - {t}_{0}\right)\mathrm{exp}\left(-\frac{t-{t}_{1}}{{\tau }_{f}}\right).\end{cases}\end{equation} \tag{ 11 }$$

All the parameters of the fit (τ_f, A₀, A₁, t₁ and t₀) are obtained simultaneously. Thus, the time-integrated signal (filtered) for the N-atom can be written as follows:

$$\begin{equation}{\int }_{t=0}^{\infty }{\mathrm{d}}^{2}{s}_{\text{f,N}}\left({\nu }_{\text{l}},t\right)=-{\tau }_{\text{f}}{A}_{1}\left({t}_{1}-{t}_{0}\right)-\frac{1}{2}{A}_{1}{\left({t}_{1}-{t}_{0}\right)}^{2}.\end{equation} \tag{ 12 }$$

${\int }_{{\delta}{\nu }_{\text{N}}=-\infty }^{\infty }\mathrm{d}{{\Psi}}_{\text{N}}$ was determined from the laser excitation spectral profile of dΨ_N(δν_N). For this purpose, the fitting procedure was applied to a set of 70 raw photomultiplier tube signals (figure 3) resulting from the fluorescence generated by 70 laser pulses with central frequencies varying between 96 743 cm⁻¹ and 96 754 cm⁻¹ (103.35–103.37 nm). The as determined $\mathrm{d}{{\Psi}}_{\text{N}}\left({\delta}{\nu }_{\text{N}}\right)$ laser excitation spectral profile (red-filled circle) along with its Gaussian fit is shown in figure 4. A similar procedure was applied to determine the LIF laser excitation spectrum of I₂ used for laser wavelength calibration. This is also depicted in figure 4 showing that the spectrum broadening extends over several I₂ fluorescence maxima, which indicates a fairly large broadening of the laser line, typically 2 cm⁻¹.

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The determination of the N-atom density (equation (8) or (10)) requires determining the ratio of the fluorescence decay times of Kr and N. The decay time for the Kr fluorescence line intensity at 587.09 nm was measured at 10 Pa and a value of 31.0 ± 1.2 ns was found. In fact under the low-pressure conditions considered, the quenching is negligible and the fluorescence decay time is almost equal to the 5p'[3/2]₂ radiative lifetime. Using the quenching rate constant values given in [13] we estimated a value of 31.8 ± 1.3 ns for this radiative lifetime. This value is slightly lower than those previously reported in reference [13], i.e. 34.1 ns, and [37], i.e. 35.4 ns, but larger than the value of 26.9 ns given in reference [38].

The fluorescence decay time for N(⁴S)-atoms was measured at 20 Pa and was found to be 26.0 ± 1.4 ns for all the investigated plasma conditions (figure 5). Using the rate constant values given in references [11, 13] for the quenching of N(4S) by N₂ molecules, we found a value of 26.2 ± 2 ns for the N(3p⁴S⁰) radiative lifetime in agreement with the literature [10, 11, 15]. As expected, the collisional quenching by N₂ molecules is negligible and barely reduces the fluorescence decay time.

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The uncertainty of the measured N-atom densities was calculated using the propagation of uncertainties of the independent variables [39]. The uncertainties of the TALIF integrals (dΨ and ${\int }_{{\delta}\nu =-\infty }^{\infty }\mathrm{d}{\Psi}$), ${\Delta}{\nu }_{\text{G,A}}^{\text{eff}}$ and τ_f were obtained directly from the non-linear fitting procedures [40]. The other independent variables were either obtained from the literature (two-photon absorption cross-sections) or datasheets from the manufacturers (transmission of filters and photomultiplier sensitivity). Figure 6 shows the contributions of the different independent variables to the uncertainty of the N-atom density. The largest contribution to the relative error on N-atom density comes by far from the uncertainty of the ratio of the two-photon absorption cross-sections for nitrogen and krypton, 50% [13]. The second error source comes from the uncertainty of the Kr radiative lifetime, 20% [35]. The uncertainties of the time-integrated peak fluorescence intensity, dΨ, determined using the procedure described above contributes 10%–15% to the global relative error on the density. The contributions of the other error sources are below 5%.

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The different standard procedures used to determine atomic densities from TALIF measurements are generally valid only if the laser instantaneous and local power density remains low enough to avoid additional laser–atom interaction phenomena such as field coupling, photoionization, photo-dissociation and stimulated emission [29, 41]. The analysis of the TALIF signal intensity dependence on the laser energy, i.e. the so-called power dependence or PD factor [42], makes it possible to check if the experimental conditions ensure a laser–atom interaction dominated by the processes involved in the TALIF scheme. A value of 2 for the PD factor, i.e. a quadratic regime of excitation, indicates that the fluorescence dominates the laser–atom interaction processes.

From the experimental point of view several methods may be used to vary the laser energy in order to determine the PD value. Unfortunately, there is no clear conclusion on the best one. Therefore, in this work, we tested three approaches. In the first one the energy was changed using a half-wave plate and a Glan–Taylor polarizer. In the second one, the energy was varied by modifying the angle of the second BBO crystal in the dye laser. The third method was based on the use of a set of neutral density (ND) filters.

The method based on Glan–Taylor polarizer is very sensitive to the beam polarization, which resulted in repeatability issues since it was very difficult to inject the laser beam at the Brewster angle in the Pellin–Broca prism. As for the second method, the laser shape changed significantly with the energy, which prevented using equation (1) that assumes a constant beam shape. The last method was in fact the most reliable and straightforward. It yielded a power dependence of ∼2 for both nitrogen TALIF at 20 Pa and krypton TALIF at 1 Pa over the laser energy range 100–300 μJ (figure 7).

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Figures 8(a) and (b) show the variation of N-atom densities measured using the FEM and the PEM at different times and as a function of the power coupled to the plasma. The considered discharge conditions correspond to a duty cycle of 0.4. The results obtained by the two approaches are in a very good agreement. Hence, we subsequently employed the straightforward PEM to determine the N-atom density.

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Figure 9 shows the variations of the stationary values reached by the N-atom density and the 728–750 nm FPS emission intensity during the high-power phase as a function of the MW power deposited during this phase for a pulse period and a duty cycle of 100 ms and 0.5, respectively. Both are seen to increase similarly and linearly with the MW power.

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In fact, for the considered values of duty cycle and pulsed period, the plasma reaches a stationary state during the pulse (see section 4.3 and figure 10). The electron temperature is therefore mainly determined by the balance between the electron production and loss terms and does not depend on the deposited power [43, 44]. The increase in the 728–750 nm FPS emission and the N-atom density with the power can only be explained by the increase in the electron density which is proportional to the power density if the plasma composition does not change significantly [43, 44]. Thus, for the current experiments, we have a constant electron temperature, and the N-atom density and the FPS emission intensity would be proportional to the electron density and therefore to the power density. Since the measurements reported in figure 9 show that the N-atom density and the FPS emission intensity are proportional to the power, we can conclude that the power density is proportional to the absorbed power, which means that the plasma volume remains almost constant. We can also conclude that, despite the very emissive nature of the plasma, the methodology developed in this work enables prediction of consistent trends of N-atom densities with MW power and FPS emission.

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The time evolution of the N-atom density, the 728–750 nm FPS emission intensity and the MW power are shown in figure 10. In this figure, the time origin, t = 0 s, is chosen at the start of the power increase (a) or decrease (b) at the transition between the high- and low-power phases. The variation of the 728–750 nm FPS emission intensity is indicative of the change in the population of the N₂(B³Π_g) electronic state (denoted as N₂(B) in the following).

Also shown in the figure are the time variation fits using first-order exponential growth and decay functions that are used to estimate the characteristic rise and decay times. The rise and decay times of the absorbed MW power are approximately 0.45 ms and 0.07 ms, respectively. These are much smaller than the rise and decay times of the N₂(B) emission intensity and the N(⁴S)-atom density. Also, the measured plasma characteristics reach steady-state values during both high- and low-power phases. This indicates that the characteristic times for the governing processes of N(⁴S)-atom and N₂(B) kinetics are much smaller than the 100 ms pulse period.

The dynamics of the electron heating in pulsed plasmas is very fast and the electron temperature usually closely follows the timescale of the absorbed MW power [45] (see also the discussion and figures at the end of this section). The sudden increase in the MW power should result in substantial electron heating that leads to enhanced excitation, ionization and dissociation kinetics. Figure 10(a) shows that N(⁴S)-atom density and N₂(B) emission exhibit similar characteristic rise times, i.e. around 1 ms. Assuming average values of 2 eV and 5 × 10¹⁰ cm⁻³ for the electron temperature and density, respectively, during the ignition phase, we estimate a time constant of 300 μs for the ionization kinetics. This means that the electron density is constantly evolving during almost 1 ms after the plasma ignition. Therefore, the excitation and dissociation rates are also evolving with time and it is indeed the ionization kinetics that drives the growth of the FPS and N-atom during the discharge ignition phase.

Since the gas temperatures is fairly low, i.e. T_g = 500–600 K, the excitation is governed by electron impact processes (R1) while the dissociation involves both an electron impact process on the ground state (R2) and the well-known vibrational excitation/relaxation route (R3) which also requires vibrational excitation by electron impact. Thus the time variation of the N(⁴S)-atom density and the N₂(B) emission intensity are also indicative of electron density buildup and ionization timescale.

$$\begin{equation}\mathrm{e}+{\mathrm{N}}_{2}\left({\mathrm{X}}^{1}{{\Sigma}}_{\mathrm{g}}^{+}\right)\to \mathrm{e}+{\mathrm{N}}_{2}\left({\mathrm{B}}^{3}{{\Pi}}_{\mathrm{g}},v\right)\end{equation} \tag{ R1 }$$

$$\begin{equation}\mathrm{e}+{\mathrm{N}}_{2}\left({\mathrm{X}}^{1}{{\Sigma}}_{\mathrm{g}}^{+}\right)\to \mathrm{e}+\mathrm{N}\left({}^{4}\mathrm{S}\right)+\mathrm{N}\left({}^{4}\mathrm{S}\enspace \enspace \mathrm{o}\mathrm{r}\enspace {\enspace }^{2}\mathrm{D}\right)\end{equation} \tag{ R2 }$$

$$\begin{align*}\mathrm{e}+{\mathrm{N}}_{2}\left(v\right)\to {\mathrm{N}}_{2}\left(\mathrm{w}{ >}v\right)& +\mathrm{N}\enspace \enspace \mathrm{o}\mathrm{r}\enspace \enspace {\mathrm{N}}_{2}\to \mathrm{N}\left({}^{4}\mathrm{S}\right)\\ & \quad +\mathrm{N}\left({}^{4}\mathrm{S}\right)+\mathrm{N}\enspace \enspace \mathrm{o}\mathrm{r}\enspace \enspace {\mathrm{N}}_{2}.\end{align*} \tag{ R3 }$$

As for the decay phase, since the dynamics of the electron heating is very fast, one would expect the electron temperature drop to closely follow the MW power variation. Therefore the high-energy threshold electron impact excitation, ionization and dissociation processes are very likely to be rapidly quenched during the power decay phase (see also the discussion at the end of this section).

If the loss process of N₂(B) was mainly through radiative de-excitation, which has a characteristic lifetime of 10 μs [46], the decay of the N₂(B) emission would have closely followed the MW power decay. This is obviously not the case in this experiment which showed a much longer decay time of 600 μs for the FPS emission. Actually, N₂(B) kinetics probably follows a complex mechanism similar to that observed in a short-lived afterglow [47–49]. In that case, the kinetics of N₂(A), N₂(B) and N₂(C) are strongly coupled in a collisional-radiative loop involving several processes that are reported in reference [50]. The major exit channels from this loop are the processes:

$$\begin{equation}{\mathrm{N}}_{2}\left(\mathrm{A}\right)+\mathrm{w}\mathrm{a}\mathrm{l}\mathrm{l}\to {\mathrm{N}}_{2}\end{equation} \tag{ R4 }$$

$$\begin{equation}{\mathrm{N}}_{2}\left(\mathrm{A}\right)+{\mathrm{N}}_{\mathrm{2}}\left({\mathrm{a}}^{\prime }\right)\to {\mathrm{N}}_{4}^{+}+\mathrm{e}\quad k=1{0}^{-11}\enspace {\mathrm{c}\mathrm{m}}^{3}\enspace {\mathrm{s}}^{-1}\enspace \left[51\right]\end{equation} \tag{ R5 }$$

$$\begin{align*}{\mathrm{N}}_{2}\left(\mathrm{A}\right)& +{\mathrm{N}}_{2}\left(\mathrm{A}\right)\to {\mathrm{N}}_{2}\left(\mathrm{X},\enspace v=0\right)+{\mathrm{N}}_{2}\left(\mathrm{B}\enspace \enspace \mathrm{o}\mathrm{r}\enspace \enspace \mathrm{C}\right)\\ & \quad k=7.7{\times}1{0}^{-11}\enspace {\mathrm{c}\mathrm{m}}^{3}\enspace {\mathrm{s}}^{-1}\enspace \left[51\right]\end{align*} \tag{ R6 }$$

$$\begin{align*}{\mathrm{N}}_{\mathrm{2}}\left(\mathrm{A}\right)& +\mathrm{N}\left({}^{4}\mathrm{S}\right)\to {\mathrm{N}}_{2}\left(\mathrm{X},\enspace v=\mathrm{6}\text{--}9\right)+\mathrm{N}\left({}^{2}\mathrm{P}\right)\qquad \\ & \quad {k}_{4}=4{\times}1{0}^{-11}\enspace {\mathrm{c}\mathrm{m}}^{3}\;{\mathrm{s}}^{-1}\enspace \left[50\right]\end{align*} \tag{ R7 }$$

$$\begin{align*}{\mathrm{N}}_{\mathrm{2}}\left(\mathrm{B}\right)& +{\mathrm{N}}_{2}\left(\mathrm{X}\right)\to {\mathrm{N}}_{2}\left(\mathrm{X},\enspace v=0\right)+{\mathrm{N}}_{2}\left(\mathrm{X}\right)\quad \\ & \quad {k}_{3}=0.15{\times}1{0}^{-11}\enspace {\mathrm{c}\mathrm{m}}^{3}\;{\mathrm{s}}^{-1}\enspace \left[50\right].\end{align*} \tag{ R8 }$$

If we assume a large de-excitation probability of N₂(A) at the wall, the diffusion/wall de-excitation of this metastable should be diffusion limited. In this case, the N₂(A) de-excitation characteristic time, i.e. reaction (R4), would correspond to the diffusion characteristic time of N₂(A) that is given by Λ²/D where Λ is the plasma characteristic length and D the diffusion coefficient. Assuming the same value for the diffusion coefficients of all N₂ electronic states and a Lenard-Jones N₂–N₂ interaction potential, a value of 0.03 m² s⁻¹ was estimated for D at a pressure of 20 Pa and a temperature of 600 K. This yields a 5 ms diffusion characteristic time for the 2 cm length scale estimated from the volume-to-surface ratio in our system. This diffusion timescale is much longer than the observed decay time and we can conclude that reaction (R4) does not dictate the loss mechanism of electronic excited states of N₂. The estimation of the characteristic times of reactions (R5) and (R6) is more difficult since it requires knowledge of the N₂(A) and N₂(a') densities. Lower limits for these timescales however may be estimated by noticing that the relative densities of these excited states with respect to N₂ ground states are always below 10⁻³ [47, 48]. Using this value of relative density and the rate constants given in [51] we can estimate lower limits of 42 ms and 5 ms for the characteristic times of reactions (R5) and (R6), respectively. These values are also much larger than the observed sub-millisecond decrease of the FPS emission intensity. These processes do not result in a depletion of the excited triplet state that is consistent with the one observed for the FPS emission during the early stage of the low-power phase. Using the N-atom density measured in the present study, i.e. n_N ≈ 10¹⁹ m⁻³, a characteristic time of 2.5 ms was estimated for reaction (R7) that is also not fast enough to explain the time decrease of the FPS just after the transition to the low-power phase. Eventually, a value of 275 μs was estimated for reaction (R). It appears therefore that reaction (R8) is the major loss mechanism of excited electronic states in our discharge conditions. The characteristic time inferred from this reaction rate constant is not very far from the one measured experimentally, i.e. 600 μs. Taking into account the qualitative nature of our reaction timescale analysis as well as the uncertainty of the rate constant value for (R8) [50] one can even consider that the agreement is satisfactory.

The characteristic time of N-atom density decay is longer than that of the plasma emission. It was estimated to be 2.5 ms. Over such a long timescale, it is usually assumed that the consumption of N-atoms takes place through surface recombination processes at the reactor walls:

$$\begin{equation}\mathrm{N}\left({}^{4}\mathrm{S}\right)+\mathrm{w}\mathrm{a}\mathrm{l}\mathrm{l}\to 0.5{\mathrm{N}}_{2}.\end{equation} \tag{ R9 }$$

Surface recombination is generally a first-order reaction which is controlled by the balance between the diffusion and surface recombination at the wall:

$$\begin{equation}\frac{D}{{\Lambda}}\left({x}_{\text{g}}-{x}_{\text{w}}\right)=-\frac{\gamma }{4}{v}_{\text{s}}^{{\ast}}{x}_{\text{w}},\end{equation} \tag{ 13 }$$

where D is the diffusion coefficient, Λ is the plasma characteristic length, γ is the sticking coefficient and ${v}_{\text{s}}^{{\ast}}$ is the mean thermal velocity. Further, x_g and x_w are the N-atom mole-fractions in the gas phase and at the reactor wall, respectively. Assuming a Lennard-Jones N–N₂ interaction potential, a value of 0.15 m² s−1 was estimated for the diffusion coefficient of N-atoms at 20 Pa and 600 K.

This yields a value of 3.4 ms for the diffusion characteristic time, which is very close to the decay time seen in our experiments. This shows that the measured depletion kinetics of N(⁴S)-atoms agrees with a diffusion-limited recombination process. In fact if we assume that the recombination process is limited by the surface reaction, the value of γ determined from the measured decay time would be approximately 10⁻² which is significantly larger than the value of 10⁻³ reported in reference [27] for stainless steel. Therefore, we cannot comment on the surface recombination. The only thing we can conclude is that the surface recombination timescales are either of the same order of magnitude or shorter than the diffusion timescales. In both cases, the measured characteristic time would be consistent with the diffusion-limited recombination. Of course the surface recombination timescales may have similar values to diffusion. In any case, the agreement between the measured decay time and estimated diffusion time would tend to validate our N(⁴S)-density measurements.

Actually, the N(⁴S) loss kinetics involves much more complex kinetics than a simple diffusion/recombination process. Looking at figure 10(b), one can notice a surge in the N-atom density just at the transition between the high- and low-power phases despite the poor time resolution in this figure. This reproducible surge is similar to the one observed in reference [17] where the authors showed an unexpected increase of N(⁴S) TALIF signal at the start of the plasma-off phase in a pulsed microwave plasma. The methodology used in the present work provides TALIF measurements with a time resolution as low as 5 μs and straightforward signal processing which enables investigating short timescale kinetics effects for N(⁴S). This has been used to confirm and further study the existence of this fast increase in N-atom TALIF signal that is expected to occur within 500 μs during the early stage of the low-power phase. Figure 11 shows time-resolved measurements with a much better resolution, i.e. 10 μs resolution, of the N(⁴S)-atom density and the background plasma emission near the transition between the high- and low-power phases. This figure confirms the existence of a significant increase in the N-atom density (more than 10¹⁸ m⁻³, which represents more than 10% relative variation) just after the transition to the low-power phase. In fact, a high N(⁴S) density level is maintained during almost 1 ms after the power transition. Note that such behavior is not observed for the plasma emission intensity that decreases sharply just after the transition.

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The increase in the N(⁴S) density can be only due to the change in the kinetics of very short characteristic time processes. This may involve either a decrease in the rates of N-atom consumption processes or an increase in the rates of N-atom production processes at the transition between the high- and low-power phases. Therefore, beside reactions (R2), (R7) and (R9), we considered the following gas-phase processes that are reported in the literature [50–52] as far as N(⁴S) kinetics is concerned:

$$\begin{equation}\mathrm{N}\left({}^{2}\mathrm{D}\right)+{\mathrm{N}}_{2}\to \mathrm{N}\left({}^{4}\mathrm{S}\right)+{\mathrm{N}}_{2}\end{equation} \tag{ R10 }$$

$$\begin{equation}\mathrm{N}\left({}^{2}\mathrm{D}\enspace \enspace \mathrm{o}\mathrm{r}\enspace \enspace {}^{2}\mathrm{P}\right)+\mathrm{w}\mathrm{a}\mathrm{l}\mathrm{l}\to \mathrm{N}\left({}^{4}\mathrm{S}\right)\end{equation} \tag{ R11 }$$

$$\begin{equation}\mathrm{e}+{\mathrm{N}}_{2}^{+}\to \mathrm{N}+\mathrm{N}\end{equation} \tag{ R12 }$$

$$\begin{equation}\mathrm{e}+\mathrm{N}\left({}^{2}\mathrm{P}\enspace \enspace \mathrm{o}\mathrm{r}\enspace \enspace {}^{2}\mathrm{D}\right)\to \mathrm{e}+\mathrm{N}\left({}^{4}\mathrm{S}\right)\end{equation} \tag{ R13 }$$

$$\begin{equation}\mathrm{e}+\mathrm{N}\left({}^{4}\mathrm{S}\right)\to \mathrm{e}+\mathrm{N}\left({}^{2}\mathrm{P}\enspace \enspace \mathrm{o}\mathrm{r}\enspace \enspace {}^{2}\mathrm{D}\right).\end{equation} \tag{ R14 }$$

As mentioned previously (see also the discussion at the end of this section), the rapid transition from the high - to the low-power phases will first result in a decrease of the electron temperature, which immediately affects the rates of the electron impact processes, and more particularly the ionization and dissociation processes. Then, the time evolution of N(⁴S) is governed by the interplay of all the process mentioned previously, i.e. (R2), (R3), (R7) and (R9)–(R14).

To evaluate the relative predominance of these channels in the evolution of the N(⁴S)-atom density, we made use of a self-consistent N₂ plasma model that solves for the two-term expansion electron Boltzmann equation [45] coupled to the balance equations of the vibrational states of the N₂ ground state (46 levels), and the N₂ and N electronically excited states that result from collisional-radiative models of N₂ (18 states) and N (26 states), as well as the total energy equation that yields the gas temperature. The detailed description of this model, that makes use of the methodology described in reference [45] and a collisional-radiative model similar to those described in references [50–52], is out of the scope of this paper. Here, we limit ourselves to the use of this model for the analysis of the time variation of the electron density, the electron temperature, the N₂ vibrational distribution and the N-atom densities in order to evaluate the relative predominance of the different processes involved in the N(⁴S) kinetics.

Figure 12 shows the timevariation of the calculated electrontemperature (T_e) and electron density (n_e) at the transition between the high- and low-power phases. When the power is suddenly decreased at the end of the high-power phase, the electron recombination being much slower than the electron energy relaxation, the electrons experience very strong cooling which result in the observed strong and very fast decrease of T_e from 17000 K to 7000 K. During this rapid decrease phase, the temperature adjusts so as to fulfil the power balance and the time evolution of T_e is actually driven by the recombination kinetics (electron density decrease). The T_e value required to fulfil the power balance when the recombination is under progress, is much lower than the value of 15 000 K required for sustaining the low-power phase plasma. Eventually, under a given level of n_e, T_e starts increasing again so as to fulfil the power and density balances of charged species. It ends up reaching a 15000 K steady-state value for the low-power phase. Therefore, it appears that the calculated n_e and T_e are coupled. They are governed by the electron kinetics and more precisely by the balance between the ionization and electron loss processes. These later are dominated by wall recombination.

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Figure 13 shows the calculated timevariation of the vibrational distribution functions (VDF) at different stages of the transient between high- and low-power phases. It is worth mentioning that the y-axis scale is extended for the low population value in order to emphasize the relevant population changes during the transition. The relative density of high-lying vibrational states (v > 34) strongly decreases during the early low-power phase. A detailed rate analysis shows that this decrease benefits the low-lying vibrational states through a vibrational de-excitation process, the contribution towards vibrational dissociation during the 500 μs being negligible (see figure 14). Note that the total variation of relative density of the high vibrational levels is below 10⁻⁵ which is much below the observed N(⁴S) relative density increase that is around 10⁻³.

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The time variation of the rates of the processes involved in the kinetics of N(⁴S) are shown in figure 14 confirming that the contribution of process (R3), i.e. v–d mechanism, to N(⁴S) kinetics is very weak. Actually, during the high-power phase, the kinetics of N(⁴S) is essentially governed by the balance between (a) the production through electron impact dissociation of N₂ (reaction (R2)) and the de-excitation of N(²D/²P) at the wall (reaction (R11)) and, to a lesser extent, by super-elastic collisions (reaction (R13)), on one hand and (b) the loss through electron impact excitation of N(⁴S) to N(²D) and N(²P) states (reaction (R14)) and wall recombination (reaction (R9)) on the other hand. When the power is decreased, the electron cooling affects this balance through a strong decrease in the electron impact dissociation of N₂, the N(⁴S)-atom excitation and, to a lesser extent, the N(²D/²P) de-excitation. As a result, the kinetics of the N(⁴S)-atom becomes dominated by the wall de-excitation of N(²D/²P), as a source, and the wall recombination, as a loss term, during the transient. This results in a net production of N(⁴S) during 200 μs, which explains the observed short timescale increase in N(⁴S) density in both the experiment and simulation (not shown). Actually, the surface recombination rate increases by approximately 10% during the early low-power phase (which can be easily seen in the insert in figure 14). This indicates a slight increase in the N(⁴S) density over the short timescale considered in figure 14, which is in agreement with the TALIF measurements (figure 11).

This finding leads to a slightly different interpretation from the one given in reference [17] where the observed increase in N(⁴S)-atom density during the early post-discharge phase of a pulsed plasma was attributed to super-elastic collisions on N(²P/²D). Here, we showed that this increase is most probably due to the wall de-excitation of the metastable state N(²D/²P)-atoms that results in a net production of N(⁴S), and therefore an increased N(⁴S) density, during 200 μs, which is consistent with the results obtained from the TALIF measurements. Note that this short timescale kinetic effect may be more or less important depending on the discharge conditions. It seemed in particular much more pronounced under the experimental conditions considered in reference [17].

In any case, the contribution of N(²D) and N(²P) metastable states to N(⁴S) kinetics should be considered if the N-atom density in a strongly emissive plasma is to be extrapolated from post-discharge density TALIF measurements. In fact, the increase in N(⁴S) in the early low-power discharge phase is directly related to the densities of the N(²D/²P)-atoms during the high-power phase. Our results tend to show that the metastable density represents approximately 10% of the N(⁴S) density during the high-power phase.