## 5 Results
### Energy Calibration and Dead Layer Estimation
The cluster-level calibration is obtained by plotting the mean values of the cluster ToT (obtained by fitting the curves in Figure 10) as a function of the corresponding deposited energies in the active volume of the sensor (i.e., \(E_{depeff}\) in the table), as illustrated in Figure 13. As saturation effects at high energy are clearly visible in the figure, the linear calibration fit based on Equation (1) is performed using only the four lowest energy data points (i.e., those with beam energy up to 1 MeV) obtaining the following result:  
\[ToT_{reg}(E_{dep})= E_{dep}\cdot\Big{[}993\pm 51(\sigma_{ToT})\pm 30(\sigma_{E})\pm 12( \sigma_{fit})\Big{]}\Big{[}\frac{25\ \mathrm{ns}}{\mathrm{MeV}}\Big{]}\] \[+ N\cdot\Big{[}23.10\pm 1.18(\sigma_{ToT})\pm 0.69(\sigma_{E})\pm 0.27(\sigma_{fit})\Big{]}\Big{[}\frac{25\ \mathrm{ns}}{\mathrm{pixel}}\Big{]} \tag{8}\]  
where the variable cluster size \(N\) is considered. The uncertainties are as follows: \(\sigma_{ToT}\) is the averaged \(1-\sigma\) of the measured ToT values per cluster for all the beam energies, \(\sigma_{E}\) is the beam energy uncertainty, evaluated at an average of 3% over all beam energies as communicated by the facility, and \(\sigma_{fit}\) is the error on the least squares fit of the calibration curve. The interpolated cluster ToT volume as expected from Equation (8) is included in Figure 13 for all energies, confirming the manifestation of saturation effects at high energies. The saturation is visible for all beam particles above 2 MeV, and more prominent for alphas than for protons, which is explored in the next section. To further explore the origin of the saturation, the distribution of the ToT per pixel is plotted in Figure 14 for all beam energies and particle types. For 2 MeV protons, which represent the lowest-energy beam for which a mild saturation is visible in Figure 13, the pixels with the highest ToT are  
Figure 12: Values of the silicon dead layer thickness \(T_{DL}\), and of the calibration parameters \(a\) and \(b\) from Equation (1), at each step of the iterative data analysis process employed for their evaluation. measuring between 550 and 600 ToT [25 ns] units. The evidence of saturation effects with this beam indicates that such pixels are presumably already beyond the linear regime of the calibration (i.e., beyond Region B of Figure 6), and likely in the first energy range where saturation is exhibited (Region C). According to the measured calibration (Equation (8)), a ToT range between 550 and 600 ToT [25 ns] units corresponds to a deposited energy of the order of 600 keV per pixel, which is rather consistent with the expectations, as the characteristic pixel-level saturation threshold found in the literature is of 850 keV [24]. Moreover, Figure 14 shows that a few pixels occasionally record high ToT values, beyond the edge of the main distributions, occasionally reaching the maximum digitally measurable ToT (1024 DAC units, given by the 10-bit registry), while these cases are very rare, they could indicate that such pixels have entered the parabolic calibration region (Region D in Figure 6). No evidence of pixels entering Region E and F of the same figure is observed, coherently with the absence of volcano effects in the cluster shapes in Figure 9.  
Figure 14: Measured distribution of pixel ToT at different beam energies, for (**Left**) protons and (**Right**) alphas. The maximum ToT is given by the 10-bit registry as \(2^{10}=1024\) ToT units. \[E_{dep}\ [\mathrm{keV}]=(598\pm 68)\ [\mathrm{keV}]-(51.5\pm 5.8)\frac{1}{ \cos\theta}\ [\mathrm{keV}], \tag{6}\]0 Energy calibration per cluster, with the cluster ToT volume plotted against the FLUKA simulated energy deposition in the active sensor, for protons (blue points) and alphas (red points).