# Compositional Analysis of Chalcopyrite Using Calibration-Free Laser-Induced Breakdown Spectroscopy
## 3 Results
### Quantitative Analysis
#### 3.2.6 Calibration-Free Analysis  
For quantification of the chalcopyrite sample, a few emission lines of the elements identified in the sample were selected. Line selection was made on the basis of criteria given in [19]. The chosen lines of Cu-I, Fe-I, and Ca-I, along with their relevant spectroscopic parameters, are listed in Table 2. No suitable line of Na was found for quantification. The following expression has been used for the calculation of atomic number density of neutral atoms of different elements in plasma:  
\[N_{x}^{i}=\frac{I\lambda U_{x}^{i}(T)}{A_{iki}g_{k}}\Big{(}4\times 10^{14}\Big{)}e ^{(\frac{E_{k}}{g_{B}T})} \tag{5}\]  
where \(U_{x}^{i}(T)\) is the partition function of the neutral atoms, and \(E_{k}\) is the energy of upper level, and other parameters are the same as already mentioned. Number density of singly ionized atoms was estimated using Saha equation [40]:  
\[N_{x}^{ii}= 2(N_{x}^{i})(N_{e}^{-1})\Big{(}6.04\times 10^{21}\Big{)}(T_{e}^{ \frac{3}{2}})\Bigg{(}\frac{U_{x}^{ii}(T)}{U_{x}^{i}(T)}\Bigg{)}e^{(\frac{-x_{i }}{g_{B}T})} \tag{6}\] \begin{table}
\begin{tabular}{c c c c c c c} \hline \multirow{2}{*}{**Element**} & **Wavelength** & **Transition** & **Upper level** & **Lower level** & **Upper Level** & **Lower level** \\  & **\(\lambda\) (nm)** & **Probability A** & **Statistical** & **Statistical** & **Energy E\({}^{\textbf{k}}\)** & **Energy E\({}^{\textbf{i}}\)** \\  & **(s\({}^{-1}\))** & **Weight g\({}^{\textbf{k}}\)** & **Weight g\({}^{\textbf{i}}\)** & **Weight g\({}^{\textbf{i}}\)** & **(cm\({}^{-1}\))** & **(cm\({}^{-1}\))** \\ \hline Ca-I & 643.58 & \(5.3\times 10^{7}\) & 9 & 7 & 35,896.889 & 20,371.00 \\ \hline Ca-I & 646.30 & \(4.7\times 10^{7}\) & 7 & 5 & 35,818.317 & 20,349.260 \\ \hline Ca-I & 649.61 & \(4.4\times 10^{7}\) & 5 & 3 & 35,730.454 & 20,335.360 \\ \hline Cu-I & 510.82 & \(2.0\times 10^{6}\) & 4 & 6 & 30,783.696 & 11,202.618 \\ \hline Cu-I & 515.30 & \(6.0\times 10^{7}\) & 2 & 4 & 49,935.200 & 30,535.324 \\ \hline Cu-I & 521.80 & \(7.5\times 10^{7}\) & 6 & 4 & 49,942.051 & 30,783.686 \\ \hline Cu-I & 578.19 & \(1.65\times 10^{6}\) & 2 & 4 & 30,535.324 & 13,245.443 \\ \hline Fe-I & 357.15 & \(6.76\times 10^{7}\) & 9 & 11 & 35,379.208 & 22,845.869 \\ \hline Fe-I & 404.82 & \(8.6\times 10^{7}\) & 9 & 9 & 36,686.174 & 11,976.238 \\ \hline Fe-I & 438.58 & \(5.0\times 10^{7}\) & 11 & 9 & 34,782.419 & 11,976.238 \\ \hline Fe-I & 487.41 & \(2.54\times 10^{7}\) & 3 & 3 & 43,763.980 & 23,244.838 \\ \hline Fe-I & 489.35 & \(3.01\times 10^{7}\) & 9 & 7 & 43,434.627 & 22,996.674 \\ \hline Fe-I & 492.25 & \(3.58\times 10^{7}\) & 11 & 9 & 43,163.326 & 22,845.869 \\ \hline \end{tabular}
\end{table} 
Table 2: Optical parameters of chosen emission lines of different compositional elements of Chalcopyrite sample used for calibration free analysis. where, \(N_{x}^{i}\), \(U_{x}^{ii}(T)\), and \(\chi_{i}\) are atomic number density, partition functions of singly ionized atoms at calculated \(T_{e}\), and ionization potential of elements, respectively. Total relative atomic number density of elements was obtained by adding values of \(N_{x}^{ii}\) as calculated by Equation (6) in values of \(N_{x}^{i}\) as calculated by Equation (5). The relative weight of each element was estimated by multiplying the relative atomic number density of each element with the atomic weight of the corresponding element, i.e., \((WA_{x}\times N_{x})\). The elemental weights so obtained were added together, and then the corresponding sum, \((\sum_{x}WA_{x}\times N_{x})\), was used to determine the weight percentage of compositional elements by using the following formula:  
\[W\%N_{x}=\frac{WA_{x}\times N_{x}}{\sum_{x}WA_{x}\times N_{x}}\times 100 \tag{7}\]  
Theoretical calculations revealed that relative concentrations of Cu, Fe, and Ca in the chalcopyrite sample were 58.9%, 40.2%, and 0.9% by weight respectively. To validate the quantitative results as obtained using CF-LIBS analysis, X-ray fluorescence (XRF) analysis was performed on the same sample.