## 2 Materials and Methods
### Semi-Quantitative Analysis of Different Elements in LIB Spectra of Gemstones
#### 3.2.3. Local Thermal Equilibrium  
The local thermal equilibrium (LTE) is one of the most critical parameters to be verified in the laser-induced plasma. For LTE, some necessary and sufficient conditions must be met.  
Necessary condition: The electron number density should be higher than the minimum limit given by the McWhirter relation. The relation is shown below:  
\[\text{N}_{\text{e}}>1.6\times 10^{12}\,[\text{T}]^{1/2}\,[\Delta\text{E}]^{3}, \tag{1}\]  
where N\({}_{\text{e}}\) is the electron density, \(\Delta\)E is the maximum energy difference between two states of an atom in eV, and T is the plasma temperature in Kelvin [38]. To obtain the plasma temperature, the Boltzmann relation, given in Equation (2), can be used:  
\[I_{\lambda}^{ki}=C_{s}\ F\ A_{ki}\ g_{k}\frac{e^{-(E_{k}/k_{B}T)}}{U(T)}, \tag{2}\]  
where \(A_{ki}\) (s\({}^{-1}\)) is the transition probability, \(g_{k}\) is the statistical weight, \(F\) is the experimental factor, \(C_{s}\) is the concentration of species of elements present in the sample, \(k_{B}\) is theBoltzmann constant, and \(U(T)\) is the partition function [39; 40]. Taking the logarithm of Equation (2) and rearranging it yields  
\[ln\left[\frac{I_{\lambda}^{ki}}{A_{ki}*g_{k}}\right]=-\frac{E_{k}}{k_{B}T}+ln \left[\frac{C_{s}F}{U(T)}\right]. \tag{3}\]  
The partition function for the emitting species is defined as \(U(T)=\Sigma  
The electron density can be calculated by measuring the full width at half maximum (FWHM) (\(\Delta\lambda\)) of the Stark broadened line given by the expression:  
\[\text{N}_{\text{e}}\approx 10^{16}\;\Delta\lambda/2\text{w}. \tag{4}\]  
Here, w is the electron impact parameter whose value is 9.25 \(\times\) 10\({}^{-4}\) nm for the Ca (422.6 nm) spectral line obtained from the reference [41], and \(\Delta\lambda\) is obtained using the relation \(\Delta\lambda_{\text{True}}=(\Delta\lambda^{2}_{\text{Observed}}-\Delta \lambda^{2}_{\text{Spectrometer}})^{1/2}\). The parameter \(\Delta\lambda_{\text{Observed}}\) is FWHM of the Stark broadened line, and \(\Delta\lambda_{\text{Spectrometer}}\) is the instrumental line width. The FWHM calculated for the spectral line of CaI at 422.6 nm is 0.13 nm. It was determined from a Lorentzian fit of the line, as illustrated in Figure 5. A similar procedure has been followed for the remaining samples. The lower value of Ne for McWhirter criteria using Equation (1) and the value of Ne has been calculated using Equation (4) and is listed in Table 5. It is clear from Table 5 that for all the samples, the electron number density is greater than the lower limit. Thus, the necessary condition is satisfied.  
Sufficient condition: The ionization temperature must be within 15% of the excitation temperature. The Saha-Eggert [38] equation given by Equation (5) is used to obtain ionization temperature:  
\[\ln\frac{I_{mn}^{III}\;A_{ki}^{I}\delta_{A}^{II}\lambda^{II}}{I_{ki}^{I}A_{m}^{ III}\delta_{m}^{I}\lambda^{I}}=\ln\left(\frac{2(2\pi kTm_{e})^{\frac{2}{3}}}{N_{e} \hbar^{3}}\right)-\frac{(V^{+}+E^{II}+E^{I})}{kT^{ion}}, \tag{5}\]  
where \(m_{e}\) is the electron mass, \(h\) is [PERSON]'s constant, \(V^{+}\) is the first ionization potential of the element, \(E^{II}\) and \(E^{I}\) are the upper energy levels of the ionic and atomic species of the element having transition probabilities \(A_{nm}^{II}\) and \(A_{ki}^{I}\) and statistical weights \(g_{m}^{II}\) and \(g_{k}^{I}\)[38; 39]. We have plotted the Saha-Boltzmann curve for the element Si of all the samples and calculated the ionization temperature. Figure 6 shows the plot for sample S_1. The calculated ionization and excitation temperatures for all the samples are summarized in Table 4. It shows that for all the samples, both calculated temperatures are in good agreement and remain within a range of 15%. Thus, the sufficient condition is also verified.  
Figure 5: Lorentzian plot of the atomic line of Ca present in the LIB spectra of sample S_1.