## 2 Materials and Methods
### Plasma Diagnostics  
A closed plasma system with constant temperature is in thermodynamic equilibrium if the radiation field is given by [PERSON]'s law, the energy distribution of each type of particle in the system corresponds to the classical Maxwell-Boltzmann distribution, the ratio of the population densities of the two energy states is given by the Boltzmann distribution and the principle of detailed equilibrium is applied. Complete thermodynamic equilibrium cannot be achieved in laboratory plasmas, and deviations from equilibrium occur. Radiation losses are always present. Electrons react more strongly to an electric field than ions due to their much smaller mass. Therefore, electrons gain more energy per unit of time than ions. The temperature of the ions remains lower than the temperature of the electrons. The energy difference between the ground  
Figure 3: High-voltage pulse generator. state and the first excited state is usually large, while the energy difference between the excited states is small, especially in noble gases. However, at high plasma densities, collisions occur frequently enough to maintain the steady-state population density according to the Boltzmann relation and the Saha equation, which indicates the degree of ionisation of the plasma system. Therefore, we can use the concept of local thermodynamic equilibrium (LTE). Electron collisions are much faster and more frequent than ion collisions and therefore provide equilibrium [53]. Radiative emission from atoms and ions occurs when electrons in the observed system transition from one energy level to another and is characterised by spectral lines. Spectroscopic methods are used to observe the radiation emitted by the plasma. In emission spectroscopy, the intensity of the spectral lines can provide various data about the plasma. The radiation intensity \(I_{ij}\) is expressed as follows:  
\[I_{ij}=E_{ij}N_{i}A_{ij} \tag{1}\]  
where \(E_{ij}\) is the difference between the energy levels \(E_{i}\) and \(E_{j}\), and \(E_{i}>E_{j}\), \(A_{ij}\) is the transition probability and \(N_{i}\) is the population of the upper energy state of \(i\). In most laboratory experiments, measuring the absolute intensity is difficult and sometimes impossible, so the intensity ratio of the spectral lines of the plasma is used:  
\[\frac{I_{ij}}{I_{mm}}=\frac{\lambda_{mm}N_{i}A_{ij}}{\lambda_{ij}N_{m}A_{mm}} \tag{2}\]  
where \(\lambda_{ij}\) is the wavelength of the spectral line corresponding to the transition from energy state \(i\) to energy state \(j\). The population of atomic states \(N\) generally depends on the temperature and density of the plasma. The simplest approach to determine the electron temperature from the emission of spectral lines is possible if the population densities of the upper levels of the two lines are in partial local thermodynamic equilibrium (PLTE), i.e., if they are connected by the Boltzmann factor, \(e^{-E_{i}/k_{B}T_{e}}\), i.e., the probability that a certain number of atoms of energy \(E_{i}\) are present in a system of temperature \(T_{e}\). Figure 4: Optical emission spectrometer (OES). A more precise estimate of the temperature of the generated plasma can be obtained using the Boltzmann plot, which is a straight line with the slope \(k_{B}T_{c}\):  
\[\ln\left(\frac{I_{ij}\lambda_{ij}}{g_{i}A_{ij}}\right)=-\frac{1}{k_{B}T_{e}}E_{ i}+\ln\left(\frac{4\pi Z}{hcN_{i}}\right) \tag{3}\]  
where the above quantities (\(I_{ij}\),\(A_{ij}\), \(A_{ij}\), \(N_{i}\)) are also refer to \(g_{i}\) as the statistical weight of the energy level, \(E_{i}\) as the energy level of the upper state for the emission, \(T_{e}\) the electron temperature, \(Z\) the partition function, \(k_{B}\) the Boltzmann constant, \(h\) the Planck constant and \(c\) the speed of light [53]. Different mechanisms of spectral line broadening (Doppler, strong, natural, instrumental) influence the spectral profile of the emitted light [53].