# Laser-Induced Breakdown Spectroscopy for Determination of Spectral Fundamental Parameters
## 3 Spectral Broadening and Shift Parameters  
The main broadening effect in a LIBS plasma is the Stark effect [37]. It is due to the perturbation induced by the electric field of the electrons (and, at lesser extent, of the ions) on the energy levels of a given transition. The Stark broadening depends on the number density of the charged species (mainly the electron number density) and, at the same electron number density, it is in general larger for ions than for the neutral atoms. This general model of Stark effect in a plasma can be summarized in the equation [43]:  
\[ \begin{array}{c}\Delta\lambda_{S}=2\;w(T,n_{e})\Big{(}\frac{n_{e}}{10^{16}} \Big{)}+3.5\;A\;\Big{(}\frac{n_{e}}{10^{16}}\Big{)}^{1/4}\Big{[}1-BN_{D}^{-1/3} \Big{]}w(T,n_{e})\Big{(}\frac{n_{e}}{10^{16}}\Big{)}\\ N_{D}=1.72\cdot 10^{9}\;T^{\frac{3}{2}}/n_{e}^{1/2}\end{array}  \tag{11}\]  
where  
\[B=\left\{ \begin{array}{c}0.75\;\text{for neutral lines}\\ 1.2\;\text{for ionic lines}\end{array} \right. \tag{12}\]  
In Equation (11) \(w(T,n_{e})\) is the Stark broadening coefficient, depending weakly on the electron temperature and number density, A is the ion broadening parameter and \(N_{D}\) represents the number of particles in the Debye sphere. The Stark effect is the main broadening phenomenon in LIBS plasmas (Doppler effect is typically negligible with respect to the Stark broadening). Moreover, In LIBS plasmas the ion contribution to Stark broadening can also be considered as negligible. Therefore, the FWHM of the Lorentzian line shape \(L(\lambda)\) can be linked to the Stark broadening as:  
\[\Delta\lambda_{0}=2\;w(T,n_{e})\Big{(}\frac{n_{e}}{10^{16}}\Big{)} \tag{13}\]  
For the ions, a shift in the position of the peak of the line can also be observed, depending on the plasma temperature and electron number density. The simple relation between line width and electron number density is valid for all the elements, except hydrogen and hydrogenoid atoms. For this class of atoms, the Stark broadening is much more pronounced than in other elements; however, the dependence on the electron number density is also more complex [37; 44]. An empirical formula linking the FWHM of the hydrogen Balmer alpha line (656.28 nm) to the electron number density has been derived in [45]:  
\[\Delta\lambda_{S}=\gamma_{\alpha}(T,n_{e})\;n_{e}^{2/3} \tag{14}\]  
Similarly, [PERSON] [44] proposed a formula for the FWHM of the hydrogen Balmer beta line (486.135 nm), as:  
\[\Delta\lambda_{S}=\gamma_{\beta}(T,n_{e})\;n_{e}^{0.68116} \tag{15}\]  
\(\gamma_{\alpha}(T,n_{e})\) and \(\gamma_{\beta}(T,n_{e})\) depend weakly on the plasma temperature and electron number density. The description of the Stark broadening effect does not rely on the existence of LTE conditions in the plasma. The measurement of the Stark broadening coefficient of a spectral line is, apparently, simple. Once the electron number density of the plasma is determined (from another line with known Starkbroadening coefficient--or from one of the hydrogen lines just mentioned [46], acquired at the same time and from the same spatial region) the Stark broadening coefficient could be determined as:  
\[w(T,n_{e})=\frac{\Delta\lambda_{0}}{2\left(\frac{n_{e}}{10^{16}}\right)} \tag{16}\]  
The electron temperature can be determined, assuming LTE conditions in the plasma, using the usual Boltzmann [47] or Saha-Boltzmann [48] plot methods. If the electron number density is determined from the Stark broadening \(\Delta\lambda_{\text{0 ref}}\) of another reference line whose Stark broadening coefficient \(w_{ref}\) is known(not necessarily of the same species, since the electron number density is a global property of the plasma), the unknown Stark broadening coefficient can be simply calculated as:  
\[w=w_{ref}\frac{\Delta\lambda_{0}}{\Delta\lambda_{0 ref}} \tag{17}\]  
without even measuring the electron number density. In this case, however, the dependence of the coefficient on the plasma temperature and electron number density cannot be made explicit.