# Laser-Induced Breakdown Spectroscopy for Determination of Spectral Fundamental Parameters
## 2 Plasma Models
When \(\tau(\lambda)\) is \(<<1\), it's easy to check that Equation (8) coincides with Equation (3), but, since the effect of self-absorption is higher near the maximum of the emission, with its increase the line shape becomes progressively distorted, as shown in Figure 1.  
Figure 1: Effect of self-absorption on the line shape (numeric calculation). For resonant lines (\(E_{i}=0\)) the lower level of the transition is generally very populated and, depending on the value of \(n^{a}\), a strong self-absorption can be observed. Strong self-absorption would also mean a nonlinear relation between observed intensity and the number density of the corresponding species (see Figure 1). In the limit of very strong self-absorption, the line profile flattens out at the Black-body limit, and the intensity becomes independent on the species concentration. If the plasma is not homogeneous (one can realistically consider a situation where the temperature in the plasma periphery is lower than in the center, for example) the emitted radiation can suffer from a further self-absorption, which affects only the central part of the line. This phenomenon is called self-reversal because it produces a dip in the peak of the line (see Figure 2). When the plasma is not homogeneous, moreover, even Equation (6) should be reconsidered, since we may have plasma conditions satisfying the McWhirter criterion, and at the same time not being in LTE conditions. The same may occur if the plasma is expanding too fast for the equilibrium to be established [39]. From the above discussion, it is clear the 'optimum' conditions for measuring spectral fundamental parameters by LIBS are the ones where Equations (1) and (3) hold. One should thus prefer experimental conditions leading to the creation of a homogeneous, stationary, optically thin plasma, in LTE conditions. In addition, the spectral signal/background ratio and its reproducibility should be high enough to guarantee a good quality of the results. However, as we will see, sometimes it is not possible to fulfil all the above requirements. For example, the extent of the self-absorption effect in LIBS plasma is critical because, on one hand, Equation (1) holds only when the self-absorption of the plasma is negligible. On the other hand, self-absorption contributes to the establishment of the LTE conditions, therefore a balance between these two opposite requirements has to be found. Moreover, the expansion of the LIBS plasma depends on the type and density of the surrounding atmosphere, as well as on the laser pulse length, energy and focusing conditions on the target. The hypothesis of stationary and homogeneous plasma could be fulfilled only in a limited range of the LIBS plasma lifetime, or not being fulfilled at all. Similarly,  
Figure 2: Self-reversal in a silver plasma. The different spectra were acquired at different delays after the onset of the plasma, with a gate of 500 ns. At later times, the plasma homogenizes, and the self-reversal disappears. the strategy of using samples at low concentration of the element under study, to reduce the effect of self-absorption, may result in weak emission signals. In the following, we discuss the experimental strategies used by LIBS researchers for determining fundamental spectral parameters (namely, line broadening parameters and transition probabilities) from the analysis of LIBS spectra.