## 3 Results  
### Calculation of LOD from Univariate Calibration Curve  
The currently accepted operative definition of the _LOD_ identifies this quantity as the minimum concentration of the analyte that can be detected, \"controlling the risks\" [22] of false positives (fluctuation of the blank signal mistaken for analyte signal) and false negatives (signal of the analyte mistaken for a fluctuation of the blank) (see Figure 1). \begin{table}
\begin{tabular}{c c} \hline \hline
**Sample** & **Cu (w\%)** \\ \hline S1 & 6.29 \\ \hline S2 & 0.1 \\ \hline S3 & 0.0256 \\ \hline S4 & 0.548 \\ \hline S5 & 0.0303 \\ \hline S6 & 0.2445 \\ \hline S7 & 1.231 \\ \hline S8 & 0.1276 \\ \hline S9 & 0.150 \\ \hline \hline S10 & 0.150 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Cu concentration in the samples. Figure 1: Schematic representation of the histogram of the signal from the blank (gray) and from a sample (red). The red region is the probability of false negatives (\(P_{FN}\)), the gray region is the probability of false positives (\(P_{FP}\)). Assuming a Gaussian distribution of the signal of the blank and the sample, with the same variance \(\sigma\), the signal mean intensity at the \(LOD\) corresponding to a given probability of false positives \(P_{FP}\) and false negatives \(P_{FN}\) can be written as:  
\[I_{LOD}=\Big{(}erfc^{-1}(P_{FP})+erfc^{-1}(P_{FN})\Big{)}\sigma \tag{1}\]  
where \(erfc^{-1}(x)\) is the inverse of the complementary error function (1--integral of the Gaussian function from \(-\infty\) to x). In most cases, for the calculation of the \(I_{LOD}\), the two probabilities of false positives and false negatives are assumed to be equal to a reasonably low value \(P\). In that case:  
\[I_{LOD}=2\;erfc^{-1}(P)\;\sigma \tag{2}\]  
In \(LIBS\), the calculation of the Limit of Detection of an analyte is almost universally performed, exploiting the traditional IUPAC (now obsolete) definition [13]:  
\[LOD=\frac{3\sigma}{b} \tag{3}\]  
where \(\sigma\) is the standard deviation of the \(LIBS\) signal of a sample with zero concentration of the analyte (blank) and b is the slope of the calibration curve (sensitivity). The \(LOD\) defined in Equation (3) is interpreted as the minimum concentration of the analyte that can be safely distinguished from the blank, being characterized by a signal (\(I_{LOD}=b\times LOD\)) equal to three times the standard deviation of the signal of the blank \(\sigma\). Comparing definition (3) with (1), we see that the probability of false positives (assumed equal to the probability of false negatives, as in (2)) is \(P=erfc(3/2)=7.7\%\). We have already anticipated that the definition (3) has been modified by IUPAC. The new definition for the \(LOD\) of a given element in a univariate calibration is:  
\[LOD=\frac{3.3\;\sigma_{y/x}\sqrt{1+\frac{\overline{c}^{2}}{\Sigma\left(C_{i}- \overline{c}\right)^{2}}}}{b} \tag{4}\]  
where:  
\[\sigma_{y/x}=\sum\frac{\sqrt{\left(I_{i}-\hat{I}_{i}\right)^{2}}}{N} \tag{5}\]  
where \(N\) represents the number of points in the calibration curve [22]. The term \(\sigma_{y/x}\) is an estimate of the deviation of the signals \(I_{i}\) with respect to the value predicted by the calibration curve \(\hat{I}_{i}\) at that concentration, under the hypothesis of homoscedasticity (equal variance) of the signals close to the \(LOD\). \(C_{i}\) are the concentrations of the standards and \(\overline{C}\) is the average of these concentrations. The term \(\frac{\overline{c}^{2}}{\Sigma\left(C_{i}-\overline{c}\right)^{2}}\) represent the leverage for the blank sample. The higher the leverage, the higher would be the fluctuations of the intercept of the calibration curve, which reflects the uncertainty of the signals. The new IUPAC definition differs from (4) in two aspects. Firstly, the probability of false positives/false negatives is set to 5%, instead of 7.7%. Therefore, the factor of 3 in Equation (3) becomes 3.3 in (4).