## 2 Theoretical Approach
### Stark Broadening  
Besides the natural (due to quantum mechanical energy uncertainty) and Doppler broadening (due to thermal velocity of the emitting particle), several further broadening mechanisms of spectral lines exist. Responsible for the broadening are frequent interactionswith neutral or charged particles, which are too weak to produce the excitation responsible for a radiation itself but still strong enough to modify the lifetime of the excited electron. These interactions are in general of a electromagnetic nature, with the coupling of any charge \(q\) to the electric field being much stronger than to the magnetic field (assuming that the velocity of the charged particle is much lower than the speed of light, which is usually the case). If an interaction with neutral particles is considered, the effect is referred to as pressure broadening. For charged particles, it is referred to as Stark broadening or the Stark effect [44,45]. The Stark effect occurs due to the interaction of the atom's bound electrons with the plasma's free electrons and can be mainly described by one of the two following approximations:  
1. Quasistatic approximation: The electron distribution within the atom is not spherically symmetric (such as in the hydrogen atom) and the atom already displays a permanent electric dipole. In this case, the plasma's electric free charges can be assumed to act collectively, direct, as a \"cloud\", creating the external electric field \(\vec{E}_{0}\), which interacts with the atom's dipole moment. This approximation is also called the linear Stark effect.
2. Impact approximation: The electron distribution within the atom is spherically symmetric (such as in a noble gas atom) and there is no permanent electric dipole. In this case, the interaction is mediated by two electrons: the first one, which after approaching the atom polarizes the latter for a very short time interval, and the second one, which during such time interval interacts with the induced, short-lived, electric dipole moment. This effect is referred to as the quadratic Stark effect. Since a consecutive collision with two electrons is seldom enough, the quadratic Stark effect is much weaker than the linear one.  
In the case of non-equilibrium plasmas, most commonly hydrogen lines are used to determine the electron number density, for which the quasistatic approximation applies [49, 50, 51, 52, 53, 54, 55]. Typically, the widths of the hydrogen's alpha line \(H_{\alpha}\) (\(\lambda_{if}\) = 656.28 nm) for the transition \(i=3\to f=2\) and hydrogen's beta line \(H_{\beta}\) (\(\lambda_{if}\) = 486.13 nm) for the transition \(i=4\to f=2\) are used for the estimation according to the model proposed originally by [PERSON] et al. [50] and discussed by [PERSON] et al. [51]. According to the authors, as well as to [PERSON] et al. [52] and [PERSON] et al. [53], the \(H_{\beta}\) line represents a good compromise between an acceptable signal-to-noise ratio and self-absorption, with the latter occurring if the lines are too strong. It is also less dependent on the electron temperature and thus more suitable for electron densities in the range of \(10^{21}\) m\({}^{-3}\), which are expected for the plasma studied in this work at atmospheric pressure [56]. The weak temperature dependence can actually be nearly completely removed if, instead of the full width at half maximum, the full width at half area (area contained inside the broadened spectral line, short FWHA) of the hydrogen line is considered [50]. Therefore, in this work the FWHA of the \(H_{\beta}\) line is used to estimate the electron density according to the correlation  
\[\text{FWHA}\ [nm]=1.666\left(\frac{n_{e}\ [nm^{-3}]}{10^{23}}\right)^{0.68777}, \tag{1}\]  
given by [PERSON] et al. in [50].