# Laser-Induced Breakdown Spectroscopy for Determination of Spectral Fundamental Parameters
## 2 Plasma Models
As discussed before, in LIBS the process of emission-absorption of the radiation is not at thermal equilibrium (and it should not be, otherwise we would have only the continuous black-body radiation). However, a certain degree of absorption of the plasma radiation by the plasma itself (self-absorption) is physically unavoidable. In fact, according to the [PERSON] theory, the transition probability \(A_{ki}\) is linked to the absorption coefficient B by the relation:  
\[A_{ki}=B\ \frac{8\pi h}{\lambda_{0}^{3}} \tag{5}\]  
The critical parameter for the validity of Equations (1) and (3) is the ratio between the characteristic times of the radiative processes (\(A_{ki}\)-1) and the time needed for reaching the equilibrium in the ionization-recombination and excitation-deexcitation processes. To be in LTE conditions, the former should be much longer than the latter. The ionization-recombination and excitation-deexcitation equilibrium occur because of the collisional processes between the electrons (light and fast) and the atoms/ions (heavy and slow). The electron number density is thus a key factor for a plasma to be or not in LTE conditions. When the plasma can be considered as homogeneous and stationary, for being in LTE condition is sufficient to have:  
\[n_{e}>1.6\cdot 10^{12}\ T^{\frac{1}{2}}\ (\Delta E_{max})^{3} \tag{6}\]  
where \(\Delta E_{max}\) is the largest energy gap between consecutive levels. The numerical coefficient is calculated assuming the electron number density is expressed in cm\({}^{-3}\), the temperature in K and \(\Delta E_{max}\) in eV. Equation (6) is known as the McWhirter condition [40], and is in general calculated considering for \(\Delta E_{max}\) the gap between the ground state and the first excited level, since the higher levels are more closely spaced. Although in the general case, the McWhirter condition is only necessary, but not sufficient, in the case of homogeneous and stationary plasma, it is also sufficient. One should note again that the LTE conditions could be incompatible with the validity of Equation (3), because of the self-absorption in the plasma. In this case, when self-absorption is not negligible (optically thick plasma), Equation (3) should be rewritten considering the escape of the radiation from an absorbing (and emitting) plasma [41]. In LIBS, the emission line profile is usually well described by a Lorentzian function \(L(\lambda)\), with the maximum at \(\lambda_{0}\) and the full-width at half-maximum (FWHM) equal to \(\Delta\lambda_{0}\), represented by the formula:  
\[L(\lambda)=I_{0}/\pi\frac{\Delta\lambda_{0}/2}{\left(\lambda-\lambda_{0} \right)^{2}+\left(\frac{\Delta\lambda_{0}}{2}\right)^{2}} \tag{7}\]  
In Equation (7), \(I_{0}\) is defined as in Equation (3), representing the integral of the Lorentzian function. Equation (7) describes the emission of a plasma in which the photons are not substantially reabsorbed by the emitting atoms, a situation that is likely to occur when the number concentration of the emitters in the plasma is very low. In most of the cases, however, the propagation of the radiation through the plasma would produce a deformation of the line shape with respect to the original Lorentzian profile, characterized by a reduction of the peak intensity and a broadening of the observed line shape. The observed line shape of the emission line, after traveling a length \(L\) in the plasma, can be expressed by the relation:  
\[I(\lambda)=S(\lambda)\left(1-e^{-k(\lambda)I}\right) \tag{8}\]  
where, neglecting the stimulated emission,  
\[k(\lambda)=\frac{\lambda_{0}^{4}}{8\pi c}A_{ki}g_{k}\frac{n_{i}^{a}}{g_{i}}L( \lambda) \tag{9}\]  
\(n_{i}^{a}\) is the population of the lower level of the transition, given by the Boltzmann equation as:  
\[n_{i}^{a}=n^{a}\frac{e^{-\frac{E_{i}}{g_{i}}T}}{U^{a}(T)} \tag{10}\]  
In Equation (8) S(\(\lambda_{0}\)) represents the Planck black-body radiation function. The product \(\pi(\lambda)=k(\lambda)I\) is called the optical depth (or optical thickness) of the transition [42].