## 5 Results  
The designed system is finally used to characterize the APD behavior in the whole \(T\)-\(V_{bias}\) space of interest. In this section, the \(G_{APD}\) measurements are shown and discussed. In particular, the experimentally obtained \(G_{APD}(V_{bias},T)\) surface is used to obtain the parametric curve \(V_{bias}(T)\) useful to stabilize \(G_{APD}\) against the \(T\) variation in the operative conditions using a \(G_{APD}\) control loop based on \(T\) measurements and \(V_{bias}\) regulation. The APD gain \(G_{APD}\) is measured as detailed in the previous Section, and the results are shown in Figure 10.  
As expected from the theory, \(G_{APD}\) increases when \(V_{bias}\) is increased and \(T\) is decreased, whereas for what concerns the APD energy resolution measurements, the curves show a slight upward curvature with minima centered around \(V_{bias}=380\) V. This is qualitatively expected due to the trade-off between the signal-to-noise ratio and noise dependence on \(V_{bias}\). The dependence of \(G_{APD}\) on \(V_{bias}\) was fitted using different empirical models from the literature and with a standard exponential growth model \(G_{APD}=a+b\mathrm{e}^{(GV_{bias})}\)[29]. The results are shown in the \(-20\)\({}^{\circ}\)C measurement in Figure 11. The residuals and determination coefficients obtained with the three models show that the standard exponential growth model is sufficient for the fit of the experimental data. Figure 10: APD gain \(G_{APD}\) measured for different biasing voltages \(V_{bias}\) and temperatures \(T\). Measurements are obtained by placing the \({}^{55}\)Fe source in direct contact with the APD under test. Figure 9: The position of the measured centroids \(Ch_{pk}\) with respect to the energy \(E\) of the incident X-ray photons. Data are measured using the \({}^{109}\)Cd, \({}^{241}\)Am, \({}^{57}\)Co samples with GAGG. Uncertainty bars are contained in the dimension of the symbols. The same data points reported in Figure 10 are shown in the 3D plot in Figure 12 and fitted with the bi-dimensional function described by Equation (2). The best fit is obtained with the Levenberg-Marquardt method: \(a=16.2\), \(b=6.0\times 10^{-5}\), \(c=3.67\times 10^{-2}\) V\({}^{-1}\), and \(d=-2.65\times 10^{-2}\)\({}^{\circ}\)C\({}^{-1}\). The parameter covariance matrix \(\mathbf{U_{p}}\) was determined through a Monte Carlo simulation [38] assuming the measured \(T\) and \(V_{bias}\) as determination of stochastic variables with uniform distributions centered on the measured points and large \(\pm 1\)\({}^{\circ}\)C for \(T\) and \(\pm 2\) V and \(\pm 0.02\%\) of the readings for \(V_{bias}\). The width of the distribution is chosen accordingly with the experimentally measured maximum discrepancy between the AD590 sensors after the calibration process and the maximum error on the output voltage of the DT1471 HET CAEN HV generator. In particular, for what concerns the distribution of the \(V_{bias}\) points, the \(\pm 2\) V is considered a strongly correlated systematic effect affecting all the points the same way, while the \(\pm 0.02\%\) part is propagated as a random uncorrelated contribution. The simulation gives the following covariance matrix \(\mathbf{U_{p}}\):  
\[\left( \begin{array}{cccc}0.04&-1.54\times 10^{-6}&6.44\times 10^{-5}\text{ V}^{-1}&-3.09\times 10^{-5}\text{ }^{\circ}\text{C}^{-1}\\ -1.54\times 10^{-6}&7.28\times 10^{-11}&-2.76\times 10^{-9}\text{ V}^{-1}&1.01\times 10^{-9}\text{ }^{\circ}\text{C}^{-1}\\ 6.44\times 10^{-5}\text{ V}^{-1}&-2.76\times 10^{-9}\text{ V}^{-1}&1.16\times 10^{-7}\text{ V}^{-2}&-4.30\times 10^{-8}\text{ V}^{-1}\text{ }^{\circ}\text{C}^{-1}\\ -3.09\times 10^{-5}\text{ }^{\circ}\text{C}^{-1}&1.01\times 10^{-9}\text{ }^{\circ}\text{C}^{-1}&-4.30\times 10^{-8} \text{ V}^{-1}\text{ }^{\circ}\text{C}^{-1}&4.62\times 10^{-8}\text{ }^{\circ}\text{C}^{-2}\\ \end{array} \right)\,, \tag{7}\]  
which is used for the evaluation of the uncertainties in the following data elaboration.