# Laser-Induced Breakdown Spectroscopy for Determination of Spectral Fundamental Parameters
## 4 Transition Probabilities  
Assuming a homogeneous and optically thin plasma, the spontaneous transition probability associated with the emission line could be, in principle, calculated using the branching ratio method, as:  
\[A_{ji}=\frac{\beta_{ji}}{\tau_{j}} \tag{20}\]  
where  
\[\beta_{ji}=\frac{A_{ji}}{\sum_{k}A_{jk}}=\frac{I_{ji}}{\sum_{k}I_{jk}} \tag{21}\]  
and \(\tau_{j}\) is radiative lifetime of the upper level which can be measured or calculated theoretically. Figure 3: Duplicating mirror scheme. When a reference emission line of the same species (with known transition probability) is available and the plasma is in LTE conditions, the unknown \(A_{ki}\) coefficient can be determined using the relation:  
\[A_{ki}=A_{kirf}\frac{g_{kref}}{g_{k}}\frac{\text{I}_{0}}{\text{I}_{0 ref}}\;e^{- \frac{E_{kref}-E_{k}}{g_{k}T}} \tag{22}\]  
Equation (21) is independent on the concentration of the species and, apparently, it provides a very simple and straightforward way for determining the unknown transition probabilities of the lines on interest. The determination of the unknown transition probability can be made more precisely, exploiting the information that can be derived from the Boltzmann or Saha-Boltzmann plot for the corresponding species/element, as schematically represented in Figure 4.  
Using a calibration-free LIBS approach [53], the unknown \(A_{ki}\) can be determined even when another emission line of the same species is not available (or its \(A_{ki}\) is not known). In this case, the plasma electron temperature is determined using the lines of another species (the temperature is a global parameter of the plasma and, in LTE conditions, is the same for all the species in the plasma). The knowledge of the composition of the sample is then used to determine the intercept of the curve, with slope \(-1/k_{B}T\) for the species of interest (see Figure 5):  
From the discussion on the determination of the Stark coefficients, it should be evident the limit of the above-discussed approaches. For the application of these simple methods, the plasma must be, during the time of the measurements:  
1. Homogeneous;
2. Stationary;
3. In LTE conditions; and, most of all
4. Optically thin (at least for the lines considered). Equation (3), in fact, is valid in its form only when the self-absorption effects can be neglected. When this is not possible, the measured line intensity will be always lower than the corresponding \(I_{0}\) in Equation (3), as described in Equation (8) for a line with Lorentzian line shape. Figure 4: Determination of the unknown \(A_{ki}\) of a spectral line from the Boltzmann plot of the same species. The measured peak intensity of this line, after traveling a distance \(l\) in the plasma, would be:  
\[I(\lambda_{0})=S(\lambda_{o})\Big{(}1-e^{-k(\lambda_{0})l}\Big{)} \tag{23}\]  
or equivalently:  
\[I(\lambda_{0})=SA\;I_{0}(\lambda_{o}) \tag{24}\]  
where \(I_{0}(\lambda_{0})\) is the optically thin limit of the peak intensity, and the self-absorption parameter SA is defined as:  
\[SA=\frac{1-e^{-\tau(\lambda_{0})}}{\tau(\lambda_{0})} \tag{25}\]  
\(SA\) is comprised between 1 (\(\tau(\lambda_{0})<<1\), negligible self-absorption) and 0 (infinite self-absorption). Note that, in the limit of infinite self-absorption, the peak line intensity is equal to the black-body intensity \(S(\lambda_{0})\) at the peak wavelength and temperature \(T\) of the plasma, as shown in Figure 6.  
\[\beta_{ji}=\frac{A_{ji}}{\sum_{k}A_{jk}}=\frac{I_{ji}}{\sum_{k}I_{jk}} \tag{21}\]0 Determination of the unknown \(A_{ki}\) of a spectral line from the knowledge of the composition of the sample. \[\beta_{ji}=\frac{A_{ji}}{\sum_{k}A_{jk}}=\frac{I_{ji}}{\sum_{k}I_{jk}} \tag{21}\]1 Dependence of \(SA\) (black) and normalized line peak intensity (red) on \(\tau(\lambda_{0})\). The line peak intensity thus depends strongly on the plasma opacity. However, the Boltzmann equation (Equation (3)) is written in terms of the integral intensity of the emission line.