Characterization of a transient spark micro-discharge in nitrogen using simultaneous two-wavelength diagnostics

The emission spectrum of nitrogen under elevated pressure conditions consists of two intensive emission systems N₂(C-B) and ${\text{N}_{2}^{+}}$(B-X). As was mentioned before, the upper states of this emission bands exhibit considerably different energy resulting in a very sensitive intensity behavior for variable electron energy distribution function. The cross sections for electron impact excitation of excited states of nitrogen are well known [19–21]. Thus, nitrogen is often applied as a test gas for OES characterizations of low and atmospheric pressure plasmas [16, 18, 22].

The scheme of energetic levels and electron impact excitation transitions, which are applicable for this OES diagnostic in an active nitrogen plasma, are presented in figure 1. Under elevated pressure conditions, the molecular emission N₂(C-B) and ${\text{N}_{2}^{+}}$(B-X) can be excited by direct electron impact of the nitrogen ground state N₂(X) or stepwise [2] via the nitrogen metastable N₂(A) and the ground state of nitrogen ions ${\text{N}_{2}^{+}}$(X). The latter process is conditionally stepwise excitation, only in respect to neutral nitrogen, and used here on a purpose of commonality, because both above mentioned stepwise excitations (N₂(X)→N₂(A)→N₂(C) and N₂(X)→${\text{N}_{2}^{+}}$(X)→${\text{N}_{2}^{+}}$(B)) are relevant under similar conditions, namely at high electron density and low electric field. In general, the N₂(C) state can also be excited in an 'energy pooling reaction' equation (1) with a rate constant of k_pooling = 3 × 10⁻¹⁰ cm³ s⁻¹ [23]:

$$\begin{equation}{{\text{N}}_2}\left( {\text{A}} \right) + {{\text{N}}_2}\left( {\text{A}} \right) \to {{\text{N}}_2}\left( {\text{C}} \right) + {{\text{N}}_2}\left( {\text{X}} \right).\end{equation} \tag{ 1 }$$

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In contrast to the afterglow phase [24], the intensity of N₂(C-B), excited by reaction (1) in the active plasma volume, is usually much lower than the emission excited by electron impact excitation because of the relatively low density of metastable nitrogen. For this reason, we neglect this excitation process. Moreover, the vibrational distribution of N₂(C) state produced in the pooling reaction differs significantly from the vibrational distribution which is characteristic of electron impact excitation, and can be easily distinguished [25].

The densities of excited molecular states which are presented in following equations in square brackets and the intensities ${I_{{N_2}\left( C \right)}}$ and ${I_{N_2^ + \left( B \right)}}$ (in photon   cm^− 3 s⁻¹) of the N₂(C-B,0–0) and ${\text{N}_{2}^{+}}$(B-X,0–0) emissions at steady-state plasma conditions can be calculated using equations (2)–(7) in the frame of the electron impact excitation model presented graphically in figure 1. The steady-state density of metastable state N₂(A) (cf. equation (2)) is determined applying the respective rate equation.

$$\begin{equation}\left[ {{N_2}\left( A \right)} \right] = \left[ {{N_2}} \right]\frac{{k_{{N_2}\left( A \right)}^{{N_2}\left( X \right)}}}{{k_{{N_2}\left( B \right)}^{{N_2}\left( A \right)} + k_{{N_2}\left( C \right)}^{{N_2}\left( A \right)} + k_{ion}^{{N_2}\left( A \right)} + k_{diss}^{{N_2}\left( A \right)}}}.\end{equation} \tag{ 2 }$$

Top and sub-indices of the rate constants in equations (2)–(7) represent the molecular states before and after electron impact excitation, respectively. Under elevated plasma conditions, collisional quenching of the excited states influences the intensities of photoemission. The quenching factors, ${Q_{{N_2}(C)}}$ and ${Q_{N_2^ + (B)}}$, are calculated in equations (3), (4) using the known Einstein coefficients $({A_{{N_2}(C)}}$, ${A_{N_2^ + (B)}})$ of the respective emission transitions, N₂(C-B) and ${\text{N}_{2}^{+}}$(B-X), and the rate constants of the collisional quenching by nitrogen molecules, $k_q^{{N_2}(C)}$ and $k_q^{N_2^ + (B)}$, from [26]. The influence of the elevated gas temperature in our spark micro-discharge on the efficiency of the quenching process is considered by substituting the rate constants for the collisional induced deactivation of electronically excited nitrogen states in Arrhenius form:

$$\begin{equation}{Q_{{N_2}\left( C \right)}} = \frac{{{A_{{N_2}\left( C \right)}}}}{{{A_{{N_2}\left( C \right)}} + k_q^{{N_2}\left( C \right)} \cdot \left[ {{N_2}} \right]}}\end{equation} \tag{ 3 }$$

$$\begin{equation}{\text{ }}{Q_{N_2^ + \left( B \right)}} = \frac{{{A_{N_2^ + \left( B \right)}}}}{{{A_{{N_2}\left( C \right)}} + k_q^{N_2^ + \left( B \right)} \cdot \left[ {{N_2}} \right]}}.\end{equation} \tag{ 4 }$$

Both direct and stepwise electron impact excitation are considered separately by the calculation of the nitrogen molecular emissions (cf. equation (5)):

$$\begin{equation}{I_{{N_2}\left( C \right)}} = {Q_{{N_2}\left( C \right)}} \cdot {n_e}\left( {\left[ {{N_2}} \right] \cdot k_{{N_2}\left( C \right)}^{{N_2}\left( X \right)} + \left[ {{N_2}\left( A \right)} \right] \cdot k_{{N_2}\left( C \right)}^{{N_2}\left( A \right)} \cdot B1 \cdot B2} \right).\end{equation} \tag{ 5 }$$

Here, n_e is the electron density, and factor B1 = 0.5 represents the relative population of the vibrational level v' = 0 of N₂(C) by electron impact excitation of the N₂(A) state, which was estimated in [12]. Factor B2 = 0.5 [27] is the branching fraction of the Einstein coefficient of the N₂(C-B,0–0) transition with respect to the total probability of photoemission from level N₂(C,v' = 0). The rate constants, which are used for the calculation of the intensities of the molecular nitrogen emission (cf. equations (6) and (7)), are determined using the known cross sections of electron impact excitations (σ_exc) [19–21] and the electron velocity distribution function evaluated by the procedure discussed below:

$$\begin{align} &{I_{{N_2}\left( C \right)}} = {Q_{{N_2}\left( C \right)}} \cdot \left[ {{N_2}} \right] \nonumber\\ & \cdot {n_e}\left( {k_{{N_2}\left( C \right)}^{{N_2}\left( X \right)} + k_{{N_2}\left( C \right)}^{{N_2}\left( A \right)} \cdot \frac{{B1 \cdot B2 \cdot k_{{N_2}\left( A \right)}^{{N_2}\left( X \right)}}}{{k_{{N_2}\left( B \right)}^{{N_2}\left( A \right)} + k_{{N_2}\left( C \right)}^{{N_2}\left( A \right)} + k_{ion}^{{N_2}\left( A \right)} + k_{diss}^{{N_2}\left( A \right)}}}} \right) \end{align} \tag{ 6 }$$

$$\begin{equation}{I_{N_2^ + \left( B \right)}} = {Q_{N_2^ + \left( B \right)}} \cdot \left( {\left[ {{N_2}} \right] \cdot {n_e} \cdot k_{N_2^ + \left( B \right)}^{{N_2}\left( X \right)} + \left[ {N_2^ + \left( X \right)} \right] \cdot {n_e} \cdot k_{N_2^ + \left( B \right)}^{N_2^ + \left( X \right)}} \right).\end{equation} \tag{ 7 }$$

Under elevated pressure conditions, the electron distribution functions differ strongly from Maxwellian ones and a reliable electron temperature cannot be determined (see chapter 4). We determine the electron distribution functions by solving the Boltzmann equation numerically for nitrogen plasmas and for different reduced electric field values. For this purpose, the program code 'EEDF', developed in the group of Prof. A. Napartovich [28], is applied. The electron velocity distribution function (EVDF) in kinetic energy scale (f_v in eV^−3/2) and the electron drift velocity (in cm/s) are calculated. Moreover, electron-electron collisions are neglected. A possible influence of Coulomb collisions on the reliability of the determined plasma parameters will be discussed below. The EVDF is normalized to fulfill equation (8) and the rate constants for electron impact excitations are calculated by equation (9):

$$\begin{equation}4\pi \sqrt 2 \mathop \int \limits_0^\infty {f_v}\left( {{E_{kin}}} \right) \cdot \sqrt {{E_{kin}}} d{E_{kin}} = 1\end{equation} \tag{ 8 }$$

$$\begin{equation}{k_{exc}} = 4\pi \sqrt 2 \mathop \int \limits_0^\infty {f_v}\left( {{E_{kin}}} \right) \cdot \sqrt {\frac{{2C}}{{{m_e}}}} \cdot {E_{kin}} \cdot {\sigma _{exc}}\left( {{E_{kin}}} \right)d{E_{kin}}\end{equation} \tag{ 9 }$$

where E_kin is the kinetic energy of electrons, m_e the electron mass and the coefficient C = 1.602 × 10⁻¹² g · cm² · s⁻² · eV⁻¹. Equation (7) contains the unknown density of diatomic ions ([${\text{N}_{2}^{+}}$(X)]). Positive molecular ions in a nitrogen plasma under elevated pressure plasma conditions are mostly two (${\text{N}_{2}^{+}}$) and four atomic (${\text{N}_{4}^{+}}$) [29]. Atomic nitrogen ions are excluded from the consideration because the density of N⁺ is relatively small due to the reduced density of neutral nitrogen N and the small cross section of electron impact ionization [30]. In assumption of plasma quasi-neutrality, the electron density is equal to the sum of molecular ion densities:

$$\begin{equation}{n_e} \approx \left[ {N_2^ + \left] + \right[N_4^ + } \right].\end{equation} \tag{ 10 }$$

${\text{N}_{4}^{+}}$ ions are produced in a three particles reaction of ${\text{N}_{2}^{+}}$ (cf. equation (11)) [29]. The lifetime of four-atomic ions at elevated pressure depends on the electron density (cf. equation (12)):

$$\begin{equation}{\text{N}}_2^ + + {{\text{N}}_2} + {{\text{N}}_2} \to {\text{N}}_4^ + + {{\text{N}}_2}\end{equation} \tag{ 11 }$$

$$\begin{equation}{\text{N}}_4^ + + {\text{e}} \to {\text{products}}.\end{equation} \tag{ 12 }$$

The rate constants of the reactions in equations (11) and (12) are k₈ = 5 × 10⁻²⁹${\left( {\frac{{300K}}{{{T_g}}}} \right)^{1.67}}$ cm⁶ s⁻¹ [29] and k₉ = 2.0 × 10⁻⁶${\left( {\frac{{300K}}{{{T_e}}}} \right)^{0.5}}$ cm³ s⁻¹ [31]. Equation (13) represents the rate equation for the density of ${\text{N}_{4}^{+}}$ under steady-state conditions:

$$\begin{equation}\left[ {N_2^ + \left] { \cdot {{\left[ {{N_2}} \right]}^2} \cdot {k_8} = } \right[N_4^ + } \right] \cdot {n_e} \cdot {k_9}.\end{equation} \tag{ 13 }$$

In assumption of quasi-neutrality (equation (14)), the density of diatomic nitrogen ions is proportional to the electron density in first approximation (equation (15)):

$$\begin{equation}{n_e} \approx \left[ {N_2^ + } \right] + \left[ {N_4^ + } \right] = \left[ {N_2^ + } \right] + \frac{{\left[ {N_2^ + } \right] \cdot {{\left[ {{N_2}} \right]}^2} \cdot {k_8}}}{{{n_e} \cdot {k_9}}}\end{equation} \tag{ 14 }$$

$$\begin{equation}\left[ {N_2^ + } \right] = \frac{{{n_e}}}{{1 + \frac{{{{\left[ {{N_2}} \right]}^2} \cdot {k_8}}}{{{n_e} \cdot {k_9}}}}} = \frac{{{n_e}}}{{1 + K\left( {{n_e},{T_{g}},{T_e}} \right)}}.\end{equation} \tag{ 15 }$$

The coefficient K(n_e,T_g,T_e) depends on the electron density, the gas temperature and on the electron temperature. Under plasma conditions with both high gas temperature and high electron density, the term K(n_e,T_g,T_e) is much lower than unity. Thus, in a first approximation, K(n_e,T_g,T_e) is neglected in equation (7) and only taken into account in a second approach. Therefore, we start to characterize the plasma in a simplified model, with ${n_e} \approx \left[ {N_2^ + } \right]$, which is represented by equations (4) and (16):

$$\begin{equation}{I_{N_2^ + \left( B \right)}} = {Q_{N_2^ + \left( B \right)}} \cdot \left( {{n_e} \cdot \left[ {{N_2}\left( X \right)} \right] \cdot k_{N_2^ + \left( B \right)}^{{N_2}\left( X \right)} + n_e^2 \cdot k_{N_2^ + \left( B \right)}^{N_2^ + \left( X \right)}} \right).\end{equation} \tag{ 16 }$$

Equations (6) and (16) form a system of non-linear equations with regards to n_e. This system has two solutions, which can be interpreted in the following way: the observed nitrogen emission spectrum can be caused by excitation processes under two different conditions, namely by a low electric field and a high electron density or by a high electric field and a low electron density. As was shown in [11, 12, 16], this ambiguous problem can be solved using the rotational distribution in the emission spectra of neutral and ionized nitrogen molecules. Under elevated pressure conditions, the rotational relaxation is very effective and the population of rotational levels of neutral nitrogen molecules in ground state (N₂(X)) corresponds to the gas temperature, even in a short transient discharge. The kinetic energy of molecular ions ${\text{N}_{2}^{+}}$(X) is higher than the kinetic energy of neutrals N₂(X) because the ions are formed by electron impact of thermalized neutral molecules and then are accelerated in the electric field between elastic collisions with the surrounding gas species [32]. The lifetime of ${\text{N}_{2}^{+}}$ ions, which is determined by the electron-ion neutralization and by the formation of ${\text{N}_{4}^{+}}$ ions, amounts to about 0.1–1 ms at low pressure conditions (cf. [16]) and 0.1–1 μs under elevated pressure conditions (cf. [11, 12]). The frequency of elastic collisions with the surrounding gas species is in both cases about two orders of magnitude higher than the respective decay. At these conditions, an overflow of kinetic energy of diatomic ions is uniformly distributed between three translational and two rotational degrees of freedom. Due to the selection rules ΔJ = 0, ± 1, the rotational distributions of molecular states, excited by electron impact, are similar to those before the collision with electrons. The rotational relaxation of the ${\text{N}_{2}^{+}}$(B) state at elevated pressure in nitrogen plasmas is ineffective because each collision with a nitrogen molecule causes quenching of this excited state. In this case, the rotational distribution of the emission spectrum ${\text{N}_{2}^{+}}$(B-X) corresponds to the rotational population of the lower molecular state of which ${\text{N}_{2}^{+}}$(B) was excited by electron impact excitation. Therefore, if both nitrogen molecular emissions (neutrals and ions) are excited from ground state of neutral molecules (N₂(X)), the rotational temperature in the emission spectrum of ions (${\text{N}_{2}^{+}}$(B-X)) is equal to that for the emission of neutrals N₂(C-B). These conditions correspond to 'direct' electron impact excitation, namely to a high electric field and a low electron density. If the ${\text{N}_{2}^{+}}$(B) state is excited by electron impact from the ion ground state ${\text{N}_{2}^{+}}$(X), the rotational temperature in the emission spectrum of ions is higher than the rotational temperature of neutrals. Such plasma conditions correspond to 'stepwise' excitation, namely to a low electric field and a high electron density. The rotational structure of the nitrogen emission spectrum was successfully used to solve the ambiguous problem under low [16] and elevated pressures in [17, 33, 34]. Unfortunately, the rotational distribution in the emission spectra of nitrogen molecular species is not always measurable with sufficient accuracy if emission bands overlap [12, 35]. Moreover, under plasma conditions with a low electric field (< 30 Td [36]) ions with similar mass can be almost thermalized because of frequent elastic collisions and the low energy gain in the low electric field. In that case, a comparison of the measured and the calculated electric current can be applied for the solution of the ambiguous problem. The electric discharge current density of the spark micro-discharge can be calculated using the measured plasma parameters (cf. equation (17)), namely electron density, drift velocity v_d of the electrons (calculated using the Boltzmann solver 'EEDF' [28]):

$$\begin{equation}j = {n_e} \cdot {v_d} \cdot e\end{equation} \tag{ 17 }$$

and the elementary charge (e). The measured plasma parameters for both possible excitation models are determined and evaluated. This procedure allows the solution of the ambiguity problem [12].

Plasma conditions with direct or stepwise excitation of the first negative system are very different and one excitation mechanisms can be neglected. Therefore, the intensity of ${\text{N}_{2}^{+}}$(B-X) emission (equation (16)) can be represented in form of equation (18) for direct electron impact excitation from ground state N₂(X) or in form of equation (19) for electron impact excitation of ${\text{N}_{2}^{+}}$(X):

$$\begin{equation}{I_{N_2^ + \left( B \right)}} = {Q_{N_2^ + \left( B \right)}} \cdot {n_e} \cdot \left[ {{N_2}\left( X \right)} \right] \cdot k_{N_2^ + \left( B \right)}^{{N_2}\left( X \right)}\end{equation} \tag{ 18 }$$

$$\begin{equation}{I_{N_2^ + \left( B \right)}} = {Q_{N_2^ + \left( B \right)}} \cdot n_e^2 \cdot k_{N_2^ + \left( B \right)}^{N_2^ + \left( X \right)}.\end{equation} \tag{ 19 }$$

The reduced electric field (E/N in Td) can be determined under both conditions by combining equation (6) with equation (18) or equation (19) yielding equations (20) and (21), respectively:

$$\begin{equation} \frac{{{I_{{N_2}\left( C \right)}} \cdot {Q_{N_2^ + \left( B \right)}}}}{{{I_{N_2^ + \left( B \right)}} \cdot {Q_{{N_2}\left( C \right)}}}} = \frac{{k_{{N_2}\left( C \right)}^{{N_2}\left( X \right)} + k_{{N_2}\left( A \right)}^{{N_2}\left( X \right)} \cdot \frac{{B1 \cdot B2 \cdot k_{{N_2}\left( C \right)}^{{N_2}\left( A \right)}}}{{k_{{N_2}\left( B \right)}^{{N_2}\left( A \right)} + k_{{N_2}\left( C \right)}^{{N_2}\left( A \right)} + k_{ion}^{{N_2}\left( A \right)} + k_{diss}^{{N_2}\left( A \right)}}}}}{{k_{N_2^ + \left( B \right)}^{{N_2}\left( X \right)}}} = \frac{{k_{{N_{2\left( C \right)}}}^{exc}}}{{k_{N_2^ + \left( B \right)}^{{N_2}\left( X \right)}}} = {F_1}\left( {\frac{E}{N}} \right)\end{equation} \tag{ 20 }$$

$$\begin{align} \frac{{{{\left( {{I_{{N_2}\left( C \right)}}} \right)}^2} \cdot {Q_{N_2^ + \left( B \right)}}}}{{{I_{N_2^ + \left( B \right)}} \cdot Q_{{N_2}\left( C \right)}^2 \cdot {{\left[ {{N_2}} \right]}^2}}} = \frac{{{{{\left( {k_{{N_2}\left( C \right)}^{{N_2}\left( X \right)} + k_{{N_2}\left( A \right)}^{{N_2}\left( X \right)} \cdot \frac{{B1 \cdot B2 \cdot k_{{N_2}\left( C \right)}^{{N_2}\left( A \right)}}}{{k_{{N_2}\left( B \right)}^{{N_2}\left( A \right)} + k_{{N_2}\left( C \right)}^{{N_2}\left( A \right)} + k_{ion}^{{N_2}\left( A \right)} + k_{diss}^{{N_2}\left( A \right)}}}} \right)}^2}}}}{{{k_{N_2^ + \left( B \right)}^{N_2^ + \left( X \right)}}}} = \frac{{{{\left( {k_{{N_{2\left( C \right)}}}^{exc}} \right)}^2}}}{{k_{N_2^ + \left( B \right)}^{N_2^ + \left( X \right)}}} = {F_2}\left( {\frac{E}{N}} \right). \end{align} \tag{ 21 }$$

The electron density is calculated using equation (22) for both electron impact excitation mechanisms:

$$\begin{equation}{n_e} = \frac{{{I_{{N_2}\left( C \right)}}}}{{{Q_{{N_2}\left( C \right)}} \cdot \left[ {{N_2}} \right] \cdot k_{{N_{2\left( C \right)}}}^{exc}}}.\end{equation} \tag{ 22 }$$

The above presented collisional-radiative model (or variations of it) was applied for the characterization of different low and atmospheric pressure nitrogen containing plasmas [cf. 11, 12, 17, 18]. However, the characterization of a transient spark discharge has peculiar properties because of the low electric field in the determined discharge phase and the spatial inhomogeneity of the small plasma channel.

The transient spark micro-discharge to be characterized is time variant and not stationary. The discharge is ignited between two sharpened stainless steel electrodes with 0.7 mm distance, which are placed inside a discharge vessel to establish a controlled atmosphere (cf. figure 2). A nitrogen flow (Alphagaz 1 5.0) of 250 sccm and a pumping system are applied to sustain a constant gas pressure of 0.5 bar in the discharge vessel. A high voltage power supply provides a negative voltage pulse with a maximum of up to 45 kV. The spark discharge is ignited while the positive streamer reaches the cathode and a conductive channel is established between the electrodes. The very fast ionization wave in the spark channel causes a strong increase of the electron density and gas temperature. Because of the very low resistivity of the plasma, the voltage across the discharge gap drops and a diffuse glow phase of the discharge starts after several microseconds (cf. figure 3). We characterize the plasma conditions in this discharge phase using a delayed trigger of 100 µs to the first light emission (measured with photodiode Femto, HCA-S-400M-SI-SMA) for the spectrometer and the intensified CCD camera. The exposure times for taking a camera picture and measure a spectrum amount to 1 ms. Because the propagation of the streamer is a stochastic process, the shape of the plasma channel varies. Thus, the consecutive application of the ICCD camera with two optical filters for the first negative and the second positive systems of nitrogen photoemission will cause a significant error when determining plasma parameters with spatial resolution.

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To overcome this issue and characterize the transient discharge, an optical arrangement that consists of two achromatic lenses, two optical beam splitters, two bandpass filters for the respective nitrogen emission bands, two deflecting prisms and an ICCD camera as an image collector (cf. figure 2), is used. The application of the optical arrangement yields two discharge images, optimized for N₂(C-B) and ${\text{N}_{2}^{+}}$(B-X) molecular emissions, respectively, displayed simultaneously but separated on the CCD chip of the camera (cf. figure 4). A fast ICCD camera (PCO DiCam pro, PCO, Germany) is used with an adjustable exposure time of 3 ns to 1000 s. The exposure time chosen in our experiment is 1 ms delayed to 100 µs to the first light emission of the spark discharge using a pulse delay generator (Stanford Research Systems, Inc, Model: DG 535, four channel digital pulse delay generator). To obtain the spatial distribution of N₂(C-B) and ${\text{N}_{2}^{+}}$(B-X) emissions, a bandpass filter with a transmittance at either 380 ± 5 nm or 390 ± 5 nm (Thorlabs GmbH, Germany) is aligned with two achromatic lenses to achieve a high resolution, respectively. Under this condition, the spatial resolution of the ICCD camera is about 13 μm (two pixel-length).

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The transmission profiles of the applied narrowband optical filters are measured using a broadband light source (DH-Mini, OceanOptics), a grating imaging spectrometer (QE65000, OceanOptics, spectral resolution: 1.3 nm), and two collimators to produce a parallel light beam. The broadband light source is connected to the first collimator via an optical fiber. After passing the narrowband filter, the light beam is collected by the second collimator and guided by an optical fiber to the spectrometer.

The current-voltage characteristics are measured using a current probe (AM 503B, Tektronix, USA) and a voltage probe (P6015A, Tektronix, USA) connected to an oscilloscope (DPO 4104, Tektronix, USA), respectively.

2.3.1. Calibration of the echelle spectrometer.

To determine the gas temperature in the studied micro-discharge and to calibrate the ICCD camera in combination with the simultaneous two-wavelength arrangement, an absolutely calibrated broadband echelle spectrometer (ESA4000, LLA Instruments, Germany) is used. A brief overview of the calibration procedure is given in the following while further details can be found in [37]. The ESA4000 spectrometer is capable of measuring 90 spectral orders (30th to 120th) simultaneously without any moving parts due to the echelle spectrometer arrangement with two dispersive elements (prism and echelle grating, cf. figure 5) and an ICCD camera. By separating the spectral orders of the echelle grating with the prism and imaging them on the ICCD camera, a total spectrum with a spectral range of 200 nm < λ < 800 nm, composed of the separated spectral orders, is obtained with a spectral resolution of R = λ/Δλ ≈ 13 333 (Δλ = 0.015 nm at 200 nm to Δλ = 0.060 nm at 800 nm). However, this compact spectrometer configuration results in a complex spectral efficiency structure as the spectral sensitivity decreases at the edges of each spectral order, yielding regions with high spectral gradients. Therefore, a continuum reference light source from 200 nm to 800 nm is necessary to illuminate the echelle spectrometer and determine the spectral efficiency. The calibration procedure yields the spectral efficiency (ε_echelle(λ)) in counts · photons⁻¹ of the spectrometer including its optical fiber. The measured spectra are automatically corrected by the ESA4000 control software via the efficiency function. A relatively calibrated deuterium lamp from 200 nm to 400 nm (X2D2 L9841, Hamamatsu Japan), and a tungsten ribbon lamp, calibrated in absolute units from 350 nm to 2500 nm (WI 17/G, OSRAM, Germany), are used for the calibration procedure.

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The relative calibration in the UV range from 200 nm to 400 nm is realized using a stabilized UV/VUV D₂ lamp L9841 (peak–peak fluctuation 0.005%) with a MgF₂ window. The L9841 was calibrated at the electron-storage ring BESSY II of the Physikalisch-Technische Bundesanstalt (PTB) from 116 nm to 400 nm. If this lamp is driven in air, ozone is produced, due to the absorption in the VUV spectral range (λ < 200 nm). Ozone itself causes a broadband absorption in the UV spectral range with a maximum at λ_max ≈ 253 nm. This leads to a systematical error of the calibration procedure. To prevent the ozone formation, the calibration of the ESA4000 is performed with the lamp and the optical fiber adapted to the vacuum chamber. For the relative calibration of the spectrometer, the optical fiber is adjusted in line-of-sight to the whole discharge area of the D₂ lamp. As the total flux is not of interest in the relative calibration, no focusing optics are needed.

To calibrate the spectrometer in absolute numbers, a stable and reliable secondary standard must be used. In our case a tungsten ribbon lamp is applied which is calibrated in W · cm⁻² · sr⁻¹ · nm⁻¹ at the center of the tungsten ribbon. As the temperature of the tungsten is not constant over the whole ribbon, the center of the ribbon needs to be focused onto the optical fiber entrance to calculate a reliable photon flux entering the spectrometer. The tungsten ribbon is projected onto the optical fiber by a concave mirror (Sigma Koki TCA-30C05-500, Japan) placed at 2 f distance to the ribbon and the fiber entrance. The mirror is Al-MgF₂ coated with a radius of curvature of r = 500 mm, a focal point f = 250 mm, and a diameter of d = 10 mm. In the spectral range of the spectrometer from 350 nm to 800 nm, a reflectivity of 90% is assumed. The resulting solid angle of the mirror is calculated to Ω = 3.14 × 10⁻⁴ sr. The diameter of the optical fiber is 600 µm and yields an irradiated area of A_fiber = 2.83 × 10⁻³ cm². With these values, the spectral flux at the optical fiber entrance can be determined in W · nm⁻¹ or in photons · nm⁻¹ · s⁻¹ (cf. figure 6).

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2.3.2. Calibration of the ICCD camera.

A point-like microwave plasma source is used for absolute calibration of the ICCD camera with the simultaneous two-wavelength arrangement. This plasma source is ignited in a flow of 2 slm nitrogen at atmospheric pressure (cf. figure 7). The MW plasma source is positioned in the same location as the studied spark micro-discharge (cf. figure 2). The image of this point-like plasma, recorded for the calibration, is presented in figure 7. The absolute intensities of N₂(C-B,0–0) and ${\text{N}_{2}^{+}}$(B-X,0–0) are measured for this point-like source using the echelle spectrometer (cf. figure 8), while taking the geometrical factor into account, as the optical fiber collects only a fraction of the emitted photons.

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To determine spatially resolved emissions from N₂(C,0) and ${\text{N}_{2}^{+}}$(B,0), the bandpass filters 380 ± 5 nm and 390 ± 5 nm are used. Emission bands, namely N₂(C-B,0–2) at 380.5 nm and ${\text{N}_{2}^{+}}$(B-X,0–0) at 391.4 nm, are selected. These wavelengths are chosen according to the intensity distribution in the emission spectrum of nitrogen, the transmission of the applied optical elements and the sensitivity of the ICCD camera. As discussed in chapter 2.1, for the plasma characterization absolute intensities of the emission bands of N₂(C-B,0–0) at 337.1 nm and ${\text{N}_{2}^{+}}$(B-X,0–0) at 391.4 nm are required. The relation between the intensities of the N₂(C-B,0–0) and N₂(C-B,0–2) bands corresponds to the Franck–Condon factors of these transitions, which are independent from plasma conditions and probabilities of excitation and de-excitation processes for the N₂(C,0) level. This relation is automatically included in the calibration factor.

In the work presented here, the rotational distribution of molecular nitrogen is of great importance for the plasma characterization. First, the gas temperature is evaluated at steady-state conditions and used to obtain the gas density with the ideal gas law. Second, the rate constants for collisional processes of atomic and molecular species are corrected using the gas temperature. Third, as discussed, is the choice of the proper excitation mechanism for the molecular nitrogen emission. The determination of the gas temperature from rotational bands using neutral and ionized nitrogen molecules is a well-known technique for many decades (e.g. [2, 35, 39]). However, depending on the discharge conditions, the analysis has to be adapted to the excitation and relaxation mechanisms. In an ideal case, which is often the case at pressures of a few Pa in, for example, RF discharges, the ground state molecules achieve thermal equilibrium through collisions before being dissociated or leaving the reactor. Then, their rotational distribution ${n_J}$ is proportional to the Boltzmann relation (equation (23)):

$$\begin{equation}{n_J} \propto \left( {2J + 1} \right){\text{exp}}\left( { - \frac{{({B_e}{\text{ }}J\left( {J + 1} \right)}}{{k{T_{rot}}}}} \right)\end{equation} \tag{ 23 }$$

with the rotational quantum number J, the rotational constant of the molecular state B_e, and the rotational temperature T_rot. Because of the selection rule ΔJ = 0, ± 1 for electron impact excitation, the rotational distribution is projected to excited states and therefore can be observed as long as the excited state is depopulated by spontaneous emission before colliding with the background gas. In this scenario, the observed rotational distribution is described by the rotational constant from the ground state, although being emitted from an excited state. Therefore, to calculate the rotational and the gas temperature of the studied plasma, the knowledge of excitation-deactivation mechanism is essential. In the presented discharge, the gas temperature is determined using the N₂(C-B) band. As shown before, the dominant excitation mechanism of N₂(C) under the given discharge conditions (0.5 bar) is direct excitation from the ground state (N₂(X)) by electron impact [2] and the ground state rotational distribution is transferred into the excited state. The effective lifetime of N₂(C) (equation (24)), due to spontaneous emission and quenching by the background gas, yields ${\tau _{{N_2}\left( C \right)}} = 13.9{\text{ ns}}$:

$$\begin{equation}{\tau _{{N_2}\left( C \right)}} = \frac{1}{{{A_{{N_2}\left( C \right)}} + k_q^{{N_2}\left( C \right)} \cdot \left[ {{N_2}} \right]}}.\end{equation} \tag{ 24 }$$

The rotational-translational relaxation time of the N₂(C) rotational states is in the range of 0.5 ns at 1 bar [2]. Thus, it can be assumed that the projected rotational distribution from N₂(X) to N₂(C) is thermalized in the N₂(C) state before spontaneous emission takes place. Thus, the rotational distribution of the N₂(C-B) transition is described by the rotational constant of the N₂(C) state [40].

The rotational distributions of neutral and ionized nitrogen molecular emissions, the measured spectra of N₂(C-B,0–0) and ${\text{N}_{2}^{+}}$(B-X,0–0) vibrational bands are compared to simulated spectra calculated using a program code developed by Prof. K. Behringer. The program determines the intensities of the emission lines proportional to Hönl-London factors for respective transitions and populations of rotational levels. A Boltzmann distribution of the rotational levels of the electronically excited states N₂(C) and ${\text{N}_{2}^{+}}$(B) is assumed. The simulated intensities of the emission spectra are convoluted with the instrumental function of the applied spectrometer, which is approximated by a Gaussian profile. The bandwidth (FWHM) of Δλ = 0.025 nm is used in the simulation of N₂(C-B,0–0) in the spectral range of 320 nm to 338 nm (spectral orders of 75–71) and Δλ = 0.029 nm in the simulation of ${\text{N}_{2}^{+}}$(B-X,0–0) in the spectral range of 380 nm to 392 nm (spectral orders of 63–61). The most important sources of error when determining the rotational temperature of molecular nitrogen are statistical errors in the measurement of a low signal, the influence of continuous and discrete emission spectra of other species (besides nitrogen) excited in the studied discharge, and inaccuracies of spectroscopic constants applied in the simulation. Both simulated and measured spectra are normalized at the band heads and compared in the spectral range, where at our spectral resolution the single rotational lines overlap and produce a broadband structure.

The OES plasma characterization with spatial resolution is only possible through a consecutive determination of the electric field (equations (20) and (21)) with a subsequent calculation of the electron density (equation (22)). However, first, the mechanism of ${\text{N}_{2}^{+}}$(B-X,0–0) excitation (equation (20) or equation (21)) must be determined because the plasma conditions of both excitation schemes are very different and one of these mechanisms can always be neglected. A robust method (in most cases) to evaluate the dominate process is the comparison of the rotational temperature of neutral and ionized molecular photoemission of nitrogen. However, this method is not always applicable, because of the low intensity in the measured spectra of ions or too low electric field values at the studied plasma conditions (cf. capture 2.1). Therefore, we now discuss a general case, namely the a priori absence of any reliable information of the excitation mechanism of molecular nitrogen emission. Both direct and stepwise excitation must be considered separately under these plasma conditions. To determine the reduced electric field in our experiment, we simulate the EVDF in a nitrogen plasma using the Boltzmann solver (cf. section 2.1) for different electric field values yielding the rate constants for electron impact processes (equation (9)). To enable an automated investigation of the camera images for a plasma characterization with spatial resolution, the rate constants are determined at different reduced electric field values E/N and are fitted with polynomials afterwards over a broad E/N range yielding the functions F₃, F₄, and F₅ (equations (25)–(27)). Here, F₃ in equation (25) describes the invers of the intensity ratio of ${\text{N}_{2}^{+}}$(B-X) to N₂(C-B) in relation to the reduced electric field, when the ${\text{N}_{2}^{+}}$(B-X) emission is excited only by electron impact of the ground state of the neutral molecular nitrogen:

$$\begin{equation}\frac{E}{N} = {F_3}\left( {\frac{{k_{N_2^ + \left( B \right)}^{{N_2}\left( X \right)}}}{{k_{{N_{2\left( C \right)}}}^{exc}}}} \right) = {F_3}\left( {\frac{{{I_{N_2^ + \left( B \right)}} \cdot {Q_{{N_2}\left( C \right)}}}}{{{I_{{N_2}\left( C \right)}} \cdot {Q_{N_2^ + \left( B \right)}}}}} \right).\end{equation} \tag{ 25 }$$

F₄ in equation (26) describes the same ratio but with excitation of ${\text{N}_{2}^{+}}$(B-X) only via ${\text{N}_{2}^{+}}$(X):

$$\begin{equation}\frac{E}{N} = {F_4}\left( {\frac{{{{\left( {k_{{N_{2\left( C \right)}}}^{exc}} \right)}^2}}}{{k_{N_2^ + \left( B \right)}^{N_2^ + \left( X \right)}}}} \right) = {F_4}\left( {\frac{{{{\left( {{I_{{N_2}\left( C \right)}}} \right)}^2} \cdot {Q_{N_2^ + \left( B \right)}}}}{{{I_{N_2^ + \left( B \right)}} \cdot Q_{{N_2}\left( C \right)}^2 \cdot {{\left[ {{N_2}} \right]}^2}}}} \right).\end{equation} \tag{ 26 }$$

At last, F₅ in equation (27) is the rate coefficient for excitation to N₂(C) via N₂(X) and N₂(A) as a function of E/N:

$$\begin{align} k_{{N_{2\left( C \right)}}}^{exc} & = \left( {k_{{N_2}\left( C \right)}^{{N_2}\left( X \right)} + k_{{N_2}\left( A \right)}^{{N_2}\left( X \right)} \cdot \frac{{B1 \cdot B2 \cdot k_{{N_2}\left( C \right)}^{{N_2}\left( A \right)}}}{{k_{{N_2}\left( B \right)}^{{N_2}\left( A \right)} + k_{{N_2}\left( C \right)}^{{N_2}\left( A \right)} + k_{ion}^{{N_2}\left( A \right)} + k_{diss}^{{N_2}\left( A \right)}}}} \right)\nonumber \\ & = {F_5}\left( {\frac{E}{N}} \right). \end{align} \tag{ 27 }$$

The same procedure is also applied for the drift velocity for low and high electric fields (cf. figures 9 and 10). Electric field ranges by direct or stepwise electron impact excitation are very different. A fitting of the polynomial in very broad variation of electric field values is complicated and the fitted function has in wide field regions a large difference to the calculated points. Therefore, a first step is the determination of possible electric field variation regions for both excited molecules and after that the electric field and excitation rate constants functions can be fitted with polynomials.

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The measured intensities of the N₂(C-B,0–0) and the ${\text{N}_{2}^{+}}$(B-X,0–0) emission (averaged or spatially resolved), corrected for the respective collisional quenching ${Q_{{N_2}\left( C \right)}}$ and ${Q_{N_2^ + \left( B \right)}}$, are applied for the determination of the electric field (averaged or spatially resolved). In assumption of direct or stepwise excitation of the ${\text{N}_{2}^{+}}$(B) state, equation (25) or equation (26) is used respectively.

After the calibration of the ICCD camera using the echelle spectrometer and the point-like MW plasma source operated in nitrogen flow, the radial emissivity distribution of N₂(C-B,0–0) and ${\text{N}_{2}^{+}}$(B-X,0–0) (in photon · s⁻¹ · cm⁻³) is determined using the inverse Abel transformation [41]. The volumetric pixel amounts to 6.73 µm × 6.73 µm × 6.73 µm. In order to choose the electron impact excitation mechanism, we use one full cross section of the plasma filament with a thickness of 6.73 µm, determine the averaged plasma parameters, reduced electric field and electron density separately in assumption of direct or stepwise excitation and calculate the resulting electric current through the whole cross section. By comparing the measured current (cf. figure 3) to the calculated currents from the OES model (direct or stepwise excitation), a determination of the excitation mechanism is possible.