# Laser-Induced Breakdown Spectroscopy for Determination of Spectral Fundamental Parameters
## 4 Transition Probabilities
The integral over the line profile described by Equation (8) in not analytical; however, it has been numerically solved by [PERSON] et al. [54], leading to a very simple relation with the \(SA\) parameter:  
\[\beta_{ji}=\frac{A_{ji}}{\sum_{k}A_{jk}}=\frac{I_{ji}}{\sum_{k}I_{jk}} \tag{21}\]2  
In the same work, the authors also determined the dependence of the line FWHM on \(SA\), in the form:  
\[\beta_{ji}=\frac{A_{ji}}{\sum_{k}A_{jk}}=\frac{I_{ji}}{\sum_{k}I_{jk}} \tag{21}\]3  
These two relations can be merged to obtain a link between measured intensity and FWHM of the emission line with the corresponding values in the optically thin limit [41]:  
\[\beta_{ji}=\frac{A_{ji}}{\sum_{k}A_{jk}}=\frac{I_{ji}}{\sum_{k}I_{jk}} \tag{21}\]4  
Equation (26) is extremely useful for evaluating (and compensating) the effects of self-absorption. Using the above relation, Equation (3) can be rewritten in terms of the measured intensities as:  
\[\beta_{ji}=\frac{A_{ji}}{\sum_{k}A_{jk}}=\frac{I_{ji}}{\sum_{k}I_{jk}} \tag{21}\]5  
To link the measured intensities to the \(A_{ki}\) spectral parameters, the optically thin FWHM of the lines (\(\Delta\lambda_{0}\)) must be known. This link shows that in optically thick plasmas, the knowledge of the Stark coefficient of the emission lines of interest is essential for the determination of the \(A_{ki}\), using either Equation (20) or the Boltzmann plot methods, unless the \(SA\) parameter can be evaluated with other independent methods. The application of the [PERSON] et al. relation requires the knowledge of the electron number density of the plasma, which can be obtained measuring the Stark broadening of a such as the hydrogen Balmer alpha line or another non-self-absorbed line whose Stark coefficient is known. The incidence of self-absorption on the measured line intensity can be evaluated (and compensated) also using the dependence of the line intensity on the concentration of the corresponding emitting species. The \(SA\) parameter depends on the optical thickness of the plasma for the transition under study, and the optical thickness depends linearly on the concentration of the species \(n^{a}\), according to Equations (9) and (10), as:  
\[\beta_{ji}=\frac{A_{ji}}{\sum_{k}A_{jk}}=\frac{I_{ji}}{\sum_{k}I_{jk}} \tag{21}\]6  
If the electron temperature and electron number density of the plasma would not vary appreciably with the concentration of the element, the analysis of the LIBS line intensity vs. the element's concentration would provide information about the \(A_{ki}\) or the \(\Delta\lambda_{0}\) associated with the transition. This idea is at the basis of the curve-of-growth method [55; 56; 57] and its variations, as the C-sigma method, introduced by [PERSON] and [PERSON] in 2014 [58; 59] and recently improved by [PERSON] et al. [60]. According to the C-sigma method, the LIBS intensity of several lines and different species can be plotted on a universal curve, as a function of \(\tau(\lambda_{0})\). In the [PERSON] et al. treatment (extended C-sigma), this universal curve is linear. [PERSON] et al. [61] further improved the method, by introducing a Time-independent extended C-Sigma curve (TIECS) which does not depend on the time at which the LIBS spectrum is acquired. The equation expressing the TIECS curve is [61]:  
\[\beta_{ji}=\frac{A_{ji}}{\sum_{k}A_{jk}}=\frac{I_{ji}}{\sum_{k}I_{jk}} \tag{21}\]7  
where \(F\) is a constant, which will be determined by the best fit of the experimental data. The RHS of Equation (30) is called C-sigma.