## 3 Results and Discussion
### Spectroscopic Characterization
#### 3.3.3 Electron Temperature  
The collisions of electrons with the atoms in the filamentary discharge produce the excitation and ionization of these species. The excited states with the highest excitation energy are in equilibrium with the ground state of ions. Their populations follow the Saha distribution:  
\[\frac{n(p)}{g_{p}}=\frac{n_{e}n^{+}}{2g^{+}}[\frac{h^{2}}{2\pi m_{e}k_{B}T_{e} }]^{3/2}exp\bigg{(}\frac{E_{ion}-E_{p}}{k_{B}T_{e}}\bigg{)} \tag{9}\]  
where \(n(p)\) is the population of the excited level related to \(p\), \(n_{e}\) is the electron number density, \(n^{+}\) is the density of positive ions, \(m_{e}\) is the electron mass, \(E_{ion}\) is the ionization energy of the hydrogen (13.596 eV), and \(E_{p}\) is the energy of the level [23]. The measure of intensities \(I_{exp}\) of different atomic lines allows us to obtain the relative populations of the upper levels by [24]:  
\[\frac{n(p)}{g_{p}}\propto\frac{I_{exp}\lambda_{pq}}{g_{p}.A_{pq}} \tag{10}\]  
where \(g_{p}\) is the departure level degeneracy, and \(A_{pq}\) and \(\lambda\) denote the transition probabilities and the wavelength of the lines, respectively. A Boltzmann plot is a semi-log representation of these relative populations vs. the ionization energy, which shows a linear behavior with slope equal to \(1/(k_{B}T_{e})\) when they follow the Saha distribution of Equation (9), and this allows us to estimate the electron temperature. It should be mentioned that the results of the electron temperature obtained by this procedure could give incorrect values if the populations of the excited state are far from the equilibrium of the Saha distribution. [PERSON] et al. [24] have shown that, in a pure hydrogen plasma, the separation of states corresponding to these transitions from the Saha distribution follow a \(p^{-6}\) law, with \(p\) as the principal quantum number of these levels, and they proposed a modified Boltzmann  
Figure 11: Variations of the rotational temperature with (**a**) different voltages applied on the stationary power electrode at 1 sLm gas flow rate, (**b**) rotational frequencies of the power electrode at 14 kV applied voltage and 1 sLm gas flow rate, and (**c**) gas flow rates at 14 kV applied voltage and 18,000 rpm rotation frequency of the power electrode. plot method to obtain more accurate electron temperatures in the plasma. They assumed that the atomic distribution function is given by:  
\[\frac{n(p)}{g_{p}}=p^{-6}\frac{n_{e}n^{+}}{2g^{+}}[\frac{h^{2}}{2\pi m_{e}k_{B}T_{ e}}]^{3/2}exp\bigg{(}\frac{E_{ion}-E_{p}}{k_{B}T_{e}}\bigg{)} \tag{11}\]  
The modified Boltzmann plot was formed by plotting the \(\log\Big{(}\frac{l_{ep}\cdot\lambda_{pq}}{g_{p}\cdot\lambda_{pq}\cdot p^{-6} }\Big{)}\) vs. the energy level. With this modification, the behavior was linear again and its slope allowed us to obtain the electron temperature (Figure 12). In this work, the first five lines of the hydrogen Balmer series were used in the calculation of electron temperature. Table 4 presents the parameters needed for this calculation. Figure 12 shows the modified Boltzmann plot of the emission lines of hydrogen atoms at the applied voltage of 14 kV, rotation frequency of the power electrode of 18,000 rpm, and gas flow rate of 1 sLm. By excepting the \(\alpha\)-line in the Balmer series, which can present a self-absorption in its own discharge [24], the populations due to each transition