# Laser-Induced Breakdown Spectroscopy for Determination of Spectral Fundamental Parameters
## 2 Plasma Models  
The LIBS plasmas are difficult to study because of the multitude of complex phenomena and interactions that characterize them. Sticking momentarily to LIBS plasma generated on solid samples, one should consider that one of the peculiar characteristics of LIBS is the key role of the laser in both sampling (through laser ablation, in solids) and excitations of the ablated material. While this characteristic is at the basis of all the practical advantages of LIBS (simplicity, possibility of remote analysis, no sample treatment, etc.) the intertwining of laser-sample, laser-plasma, plasma-environment, laser-environment, interactions is extremely complex and difficult to model. Changing the main parameters of the laser beam (pulse length and shape, energy, focusing conditions) and of the environment (composition and density of the environmental gas) may result in large changes in the radiation emitted by the plasma. Moreover, the LIBS plasmas evolve in time, expanding in the surrounding environment. Consequently, spatial gradients of the plasma parameters (atomic and electron number density, electron temperature) may occur, which must be considered in the analysis of the LIBS spectra. Different emission regimes may take place at different times after the onset of the plasma, and the emitted radiation would interact with the plasma itself. An approximation which is often used for modeling the plasma and its emission is the local thermal equilibrium (LTE) approximation [39]. In the framework of LTE approximation, it is assumed that all the processes in the plasma (ionization-recombination, excitation-deexcitation, etc.) would be at the equilibrium at the same temperature T (eV), except for the radiation, in which the absorption process would not balance the emission. The equilibrium between ionization and recombination is described by the Saha-Eggert equation:  
\[\frac{n^{II}}{n^{I}}=\frac{2}{n_{e}}\Lambda^{-3}\frac{U^{II}(T)}{U^{I}(T)}e^{- E_{ion}/k_{B}T} \tag{1}\]  
\(E_{ion}\) is the (first) ionization energy of the element, \(n^{I}\) represents the number density of the neutral species, \(n^{II}\) is the number density of the ionic species, \(n_{e}\) is the electron number density and \(k_{B}\)is the Boltzmann constant. \(\Lambda\) (cm) is the average thermal De Broglie wavelength of the electrons in the plasma, defined as:  
\[\Lambda=\sqrt{\frac{2\pi\hbar^{2}}{m_{e}k_{B}T}} \tag{2}\]  
where \(m_{e}\) is the electron mass and \(\hbar\) is the reduced Planck constant and \(U^{I}(T)\) and \(U^{II}(T)\) are the partition functions of the neutral/ionic species. In LIBS, ionization stages higher than the first are seldom observed, therefore one can safely assume that \(n^{I}+n^{II}=n_{el}\). While the concentration \(n_{el}\) of the element can be considered as constant (excluding the possible formation of molecular species in the plasma), the concentration of the species varies during the evolution of the plasma, following the variation of the plasma temperature and electron number density. The excitation-deexcitation equilibrium is described by Equation (3), which links the number of photons emitted per second in a given transition between two levels of energy \(E_{k}\) and \(E_{i}\), respectively (central wavelength \(\lambda_{0}=hc/(E_{k}-E_{i})\)) to the number density of the corresponding species \(n^{a}\), at the same temperature \(T\):  
\[I_{0}=Fn^{a}g_{k}A_{ki}\frac{e^{\frac{E_{k}}{k_{B}T}}}{U^{a}(T)}\qquad\quad a= \left\{ \begin{array}{ll}I&\quad\text{for neutral lines}\\ II&\quad\text{for ionized lines}\end{array} \right. \tag{3}\]  
\(g_{k}\) is the degeneracy of the upper level and \(A_{ki}\) (s\({}^{-1}\)) is the transition probability. \(I_{0}\) (photons per second) is thus proportional, through a factor F depending on the acquisition geometry, to the population of the upper level \(E_{\text{k}}\) (eV) of the transition, given by the Boltzmann equation:  
\[n^{a}(E_{k})=n^{a}g_{k}\frac{e^{-\frac{E_{k}}{k_{B}T}}}{U^{a}(T)} \tag{4}\]  
times the transition probability \(A_{ki}\).