## 4 System Set-Up and Measurement Procedure
The grey points are the experimental data and the continuous lines are the fitted Gaussian curves. Data are obtained by placing the \({}^{55}\)Fe sample directly in contact with the APD, at room temperature. The arrow indicates the direction in which the centroid shifts when \(V_{bias}\) is increased. covariance matrix given by the least-square fitting method. These are then propagated [37] to \(\delta E/E\):  
\[u^{2}\bigg{(}\frac{\delta E}{E}\bigg{)}=\bigg{(}\frac{\delta E}{E}\bigg{)}^{2} \Bigg{[}\Bigg{(}\frac{u(Ch_{pk})}{Ch_{pk}}\bigg{)}^{2}+\bigg{(}\frac{u(\sigma_{ pk})}{\sigma_{pk}}\bigg{)}^{2}+2\rho\frac{u(\sigma_{pk})}{\sigma_{pk}}\frac{u(Ch_{ pk})}{Ch_{pk}}\Bigg{]}. \tag{5}\]  
This procedure is performed for all the following measurements. The tuning of the Silena 7611 amplifier shaping time \(t_{s}\) is then performed through \(\delta E/E\). In Figure 8, \(\delta E/E\) is shown as a function of \(t_{s}\) when the \({}^{55}\)Fe is placed at room temperature in direct contact with the APD under test and it is biased with \(V_{bias}=380\) V. The data in Figure 8 show that for higher \(t_{s}\), \(\delta E/E\) increases, and thus the system characteristic time masks the real resolution of the APD. Since below \(t_{s}\approx 3\) us, \(\delta E/E\) saturates, we conclude that for these \(t_{s}\), the effect of the \(t_{s}\) setting is negligible on \(\delta E/E\). Thus, all the following measurements are performed setting the Silena 7611 gain at 210 and \(t_{s}=1\) us. Then, the system losses/gain must be properly calibrated in order to obtain \(G_{APD}\). As discussed in Section 2, this would require the calibration of the gains and losses of each part of the system, but due to the difficulties and large uncertainties associated with this method, the use of a reference APD is preferred. The APD used in this work as a reference is known to have \(G_{APD,ref}=G_{APD}(382)=50\pm 1\) at 25 \({}^{\circ}\)C. In these conditions of \(T\) and \(V_{bias}\), the MCA gives a peak centered at \(Ch_{pk}=1051\pm 1\). Due to the linearity between the channel number and voltage of the MCA, this point can be used to obtain the calibration curve:  
\[G_{APD}=\frac{Ch_{MCA}G_{APD,ref}}{Ch_{ref}}=\frac{Ch_{MCA}(50\pm 1)}{1052\pm 1}. \tag{6}\]  
Finally, the final operative condition of the APD coupled with the GAGG crystal is preliminarily tested to verify the linearity of the system. The light yield by GAGG is in the optical spectrum, with a peak at \(\sim\)520 nm and intensity \(\sim\)50 photons/keV. This characteristic is exploited in the polarimeter configuration and can be used here to check the linearity of the MCA channel and of the system itself. The results are obtained by placing the GAGG crystal between the different radioisotope samples (listed in Table 1) and the APD. The \({}^{55}\)Fe sample is not used in this case due to the lower energy of the emitted photons which is out of the X-ray band of interest. The result is shown in Figure 9. Data are fitted with the linear model \(Ch_{pk}=mE\), with \(m=10.9(8)\) keV\({}^{-1}\), where the number in parentheses is the numerical value of the standard uncertainty expressed in the unit of the quoted result. The good linear fit experimentally demonstrates the linearity of the whole system. Figure 8: Energy resolution \(\delta E/E\) measured at room temperature by placing the \({}^{55}\)Fe sample directly in contact with the APD, for different shaping times \(t_{s}\).