## 3 Results
### LOD of Cu in Cast Iron from Univariate Calibration  
For determining the _LOD_ of copper in cast iron using a univariate approach, we firstly built the corresponding calibration curve. We focused on the narrow spectral interval comprised between 324.4 nm and 328 nm, where the two most intense Cu I emission lines (peaks at 324.75 and 327.40 nm) are visible (see Figure 2). For building a univariate calibration curve for Cu, we defined the _LIBS_ signal as the sum of the areas of the two copper lines (red zones in Figure 3). The background (corresponding to zero analyte signal) was estimated from the gray area in Figure 3.  
Plotting the _LIBS_ signal versus the concentration of the samples, we obtain the calibration curve shown in Figure 4.  
It should be noted that in _LIBS_, the signal should be ideally proportional to the number concentration of the analyte, which is different from the weight concentration [9]. However, in our case, the evident non-linearity of the calibration curve must be attributed to the effect of the self-absorption of the Cu line emission [27]. For the determination of the _LOD_ of Cu in our experimental conditions, we will, thus, limit the range of the calibration curve to the linear region comprised between zero (background signal) and 0.55 w%, thus eliminating the two samples at concentration higher than 1 w% (see Figure 5). Figure 2: Detail of the _LIBS_ spectrum of the ten cast iron samples, showing the two most intense Cu lines (average spectra). The spectra are shifted on the vertical axis for making easier the comparison. Figure 4: Calibration curve for copper. Figure 3: Spectral region considered for the determination of Cu in cast iron. The red areas represent the Cu signal used for univariate analysis; the gray one is the region used for estimating the background (zero) signal. In this range, the relation between Cu signal and concentration in weight is substantially linear and it is, therefore, possible to define the _LOD_ of the analyte. Note that the calibration curve does not reach zero at zero copper concentration. This is also evident from Figure 1, where the Cu line at 324.75 nm seems to be observable even in the spectra of the S3 and S5 samples, which have very low copper concentrations (0.02% and 0.03%, respectively). This is probably due to the interference of some (weak) iron line emitted at a close wavelength, but does not represent a problem for the calculation of the _LOD_, as soon as the uncertainty on the value of the offset is taken into account. By applying the formula reported in Equation (4), in our experimental conditions, we obtain an _LOD_ for Cu in cast iron equal to about 0.35 w% (0.2 w% according to the old IUPAC definition, Equation (4)). However, some caution must be taken when interpreting this result; observing directly the intensity distribution of the blank (estimated as the background signal of sample with the lower Cu concentration in the calibration set) and of sample S9, whose Cu concentration is about one half of the estimated _LOD_, we observe that the Gaussian fittings of the two intensity distributions overlap well above the 5% confidence limit, but the two distributions of the blank and of the signal do not overlap at all (and thus, Cu at this concentration can be discriminated from the blank with 100% probability, see Figure 6). This is because the distribution of the _LIBS_ signal of the sample is strongly asymmetric towards the higher intensities, and thus, the approximation of having a Gaussian distribution does not apply (the standard deviation of the signal is also larger than the one of the background/blank). This effect has been studied a few years ago by [PERSON] et al., which demonstrated that the statistical distribution of the _LIBS_ line intensities in some cases is better described by a General Extreme Value Distribution (GEVD) curve, rather than by a Gaussian curve [33]. The asymmetry of the _LIBS_ intensity distribution can be also due to the spectral selection procedure used for eliminating the outliers in the measurement.