# Laser-Induced Breakdown Spectroscopy for Determination of Spectral Fundamental Parameters
## 3 Spectral Broadening and Shift Parameters
There are, however, several obstacles to the direct application of the simple procedure envisaged in Equation (15) or Equation (16). For the calculation of the Stark coefficient from the broadening the emission lines, one has to assume that:  
1. The plasma is homogeneous;
2. The acquisition of the spectral signal is done in an interval short enough to maintain a reasonably constant electron number density;
3. The Stark coefficient of the reference line is known with reasonable accuracy;
4. The measured line widths (of the reference and unknown lines) are given only by the Stark effect. While points (a), (b) and (c) are important, point (d) is evidently the key issue for the determination of the Stark coefficients of emission lines by LIBS. When factors other than Stark effect contribute to the measured line broadening, the Stark coefficient evaluated from Equation (15) would not be correct. One obvious effect on the measured line broadening would be given by the instrumental broadening. When the spectral resolution of the spectrometer is comparable to the measured line broadening, its contribution could not be neglected. Moreover, the observed line shape would also change, from a pure Lorentzian to a Voigt profile, given by the convolution of the Lorentzian line with FWHM = \(\Delta\lambda_{0}\) and a Gaussian with \(\sigma\) = spectral resolution of the spectrometer:  
\[V(\lambda)=\frac{1}{2\pi\sigma}\int_{-\infty}^{+\infty}\exp{(-\frac{(x- \lambda)^{2}}{2\sigma^{2}})}\ \frac{\Delta\lambda_{0}}{(\lambda-\lambda_{0})^{2}+\left(\frac{\Delta \lambda_{0}}{2}\right)^{2}}dx \tag{18}\]  
The two line widths (Gaussian and Lorentzian) could be, in principle, deconvolved by fitting the experimental line profile with a Voigt line shape. However, since the resolution of the spectrometer can be obtained with good precision from the information provided by the manufacturer or determined by measuring the broadening of a narrow emission line, a more effective strategy would be directly measuring the FWHM of the experimental line profile with a Voigt function. After that, the Lorentzian component can be obtained by subtracting the experimental broadening from this value. The subtraction should be performed correctly considering how the Gaussian and Lorentzian line widths contribute to the FWHM of the Voigt profile. [PERSON] et al. [49] proposed a formula for the Voigt FWHM that is correct within \(\pm 0.01\%\), as:\[FWHM_{V}=FWHM_{L}\left(0.5346+0.4654\sqrt{1+\left(\frac{FWHM_{G}}{0.4654~{} FWHM_{L}}\right)^{2}}\right) \tag{19}\]  
This relation can be inverted numerically for deriving the Lorentzian FWHM (the one related to the Stark effect) once the Voigt and the Gaussian FWHM are known. The second problem, much more difficult to solve, is related to the self-absorption of the line, which introduces an apparent broadening of the line (see Figure 1) which increases with the increase of self-absorption. Dealing with the self-absorption is made complicated by the fact that the line shape of a self-absorbed emission line is not described by a Lorentzian or Voigt profile anymore. Moreover, the presence of self-absorption sometimes is not easy to spot. One should consider, however, that even when the effect of self-absorption is not evident, the apparent FWHM of the line can be substantially larger than the Stark broadening (for \(\tau(\lambda_{0})=1\), the apparent FWHM is about 30% larger than the unperturbed Stark width) [50]. The effects of self-absorption should thus be checked carefully, for avoiding errors in the determination of the Stark coefficients. A (conceptually) simple method for determining experimentally the degree of self-absorption in LIBS plasma is the duplicating mirror method (see Figure 3) [51; 52]. The method assumes that the radiation emitted by the plasma in the direction opposite to the detector is sent back through the plasma, using a spherical mirror.