## 2 Theoretical Approach
### Molecular Spectroscopy  
In case of a molecule, apart from the internal electronic energy and kinetic energy due to the movement of the whole molecule (i.e., its center of mass) in three directions, the relative, vibrational motion of the atoms along the internuclear axis and the relative, rotational motion about two axes perpendicular to the internuclear axis contribute to molecule's energy. Since both the vibrational and rotational energy states are quantized, each transition described by the quantum number \(\
u\) and \(J\), respectively, emits radiation at a specific frequency \(\omega_{if}\), resulting in a much more complex spectrum. The radiation emittedby each transition forms spectral bands that reflect the rotational and vibrational states, as exemplified in Figure 1.  
For a known instrumental profile, the molecular spectrum of a particular transition can be calculated if several selection rules are followed, as described in [46; 47; 48]. Thereafter, the experimentally registered spectral intensity within the infinitesimal spectral window between \(\omega\) and \(\omega+d\omega\) can be written as  
\[I(\omega_{if})d\omega=K\,S_{F}\Big{(}\omega-\omega_{if}\Big{)}\,n_{N_{2}}\, \Big{(}\frac{\omega_{if}}{2\pi c_{0}}\Big{)}^{4}S_{j,if}\,\mathrm{e}^{-\frac{ \big{(}E_{rotj}-E_{rotj}\big{)}}{k_{B}\,T_{rot}}}\,S_{v,if}\,\mathrm{e}^{- \frac{\big{(}E_{rotj}-E_{v,if}\big{)}}{k_{B}\,T_{ub}}}\,\mathrm{e}^{-\frac{ \big{(}E_{v,j}-E_{v,if}\big{)}}{k_{B}\,T_{ub}}}\,d\omega \tag{2}\]  
with \(n_{N_{2}}\) being the number density of the emitting molecules and \(c_{0}\) the speed of light. The probability of each rotational transition is represented by the Honl-London factor \(S_{j,if}\) as well as by the tabulated Franck-Condon coefficient \(S_{v,if}\) for vibrational transitions. Furthermore, each transition is described by the according energy difference and temperature, with \(T_{rot}\) being the rotational, \(T_{vib}\) the vibrational and \(T_{e}\) the electron temperature. The transfer function of the experimental setup, denoted with \(S_{F}\) above, can be determined before measurement with a spectral calibration lamp, whereas the detection sensitivity \(K\) is also determined beforehand with a calibrated tungsten ribbon lamp.  
As can be deduced from Equation (2), several parameters need to be adjusted iteratively for an experimentally measured spectrum to be fitted with a synthetic one. The ionization and dissociation reactions are governed by the excitation temperature, which can be assumed to be equal to \(T_{e}\) or \(T_{h}\) depending on the plasma state [57; 58]. According to [PERSON] et al. [57], for electron densities below \(10^{21}\) m\({}^{-3}\), inelastic atomic and molecular collisions are mainly responsible for the ionization of nitrogen molecules, since the heavy particles collide more frequently with each other under such conditions and, due to a similar mass, more energy is transferred in each collision than in a collision with an electron. Thereafter, if the rotational energy levels are assumed to be populated by heavy particle collisions, the computations can be simplified by assuming that \(T_{rot}=T_{h}\), with \(T_{h}\) being the temperature of heavy particles (i.e., gas temperature) [59]. A detailed discussion of the temperatures governing the dissociation and ionization reactions can also be found in [58; 60].