## 4 System Set-Up and Measurement Procedure  
Before starting to use the system for the APD calibration, an accurate tuning of the parameters acting on it and its calibration are necessary. To accomplish this, a reference APD can be used. In fact, the APD gain measurement would be possible with the system described here only if all the parts of the system itself, including cables and connectors, were properly calibrated. In order to visualize the effect of different cables on the signal sent as input to the MCA, in Figure 6 the shaped pulses, visualized on an analog oscilloscope, are shown when LEMO cables of 150 cm or 15 cm are used to connect the APD to the preamplifier. Apart from the increased noise level, the peak heights are also different, highlighting the attenuation effect of the cables. Since the quantitative evaluation of the losses introduced by each part of the system would not only be tricky but also unlikely to yield very accurate results, in this work, a pre-calibrated APD is used as a reference to calibrate the entire system. In this work, the Si S8664-55 Hamamatsu photodiode, serial number AA4400, was used as a reference. For this diode, the calibration certificate states that its breakdown voltage is  
Figure 5: Picture of the measurement setup loaded into the climate chamber. Figure 6: Effect of different wiring in the system on the voltage pulses generated by the spectroscopy amplifier. (**a**) The pulses when a 150 cm long LEMO cable is used. (**b**) When the cable used in (**a**) is substituted with one 15 cm long. In both pictures, the vertical scale is set to 200 mV/div. \(V_{b}\approx 433\) V at 25 \({}^{\circ}\)C, and, at the same temperature, with \(V_{bias}=(382.3\pm 0.1)\) V, the gain \(G_{APD,ref}=50\pm 1\). The number following the \(\pm\) symbol is to be read as the numerical value of an expanded uncertainty with 100% probability level based on a uniform distribution. This APD will be used to calibrate the whole measurement system. The two main parameters of the system that can affect its metrological performance are the amplification gain and shaping time. So, by placing a \({}^{55}\)Fe sample on the APD, the shaped pulses were collected by the MCA and the amplification gain was set to exploit the MCA dynamic range properly, and the shaping time \(t_{s}\) was set at a value sufficently low so that the effect of this parameter on the measured energy resolution was negligible; see also below. To this aim, the Silena 7611 gain is set to 210 and several peaks are measured for different \(t_{s}\) values to evaluate its effect. Figure 7 shows an example of the histograms formed by the MCA when the \({}^{55}\)Fe source is placed in direct contact with the APD and the bias voltage \(V_{bias}\) is changed from 260 V to 410 V. The effect of \(V_{bias}\) on \(G_{APD}\) is evident: for higher \(V_{bias}\) values, the histogram peaks center shift at higher MCA channels, meaning that in the input to the MCA, the voltage peaks arrive with a larger amplitude, i.e., \(G_{APD}\) is increased. The height of the histogram peaks depends on the integration time, number of events, and energy resolution. The histograms of the peak heights are then fitted with the Gaussian function to extract the centroid \(Ch_{pk}\) and standard deviation \(\sigma_{pk}\). The fit is performed by trimming the experimental histogram to a width that maximizes the \(\chi^{2}\) test value with a Gaussian distribution. Taking into account the Gaussian function, the energy resolution \(\delta E/E\) is obtained from \(\sigma_{pk}\) and \(Ch_{pk}\) as follows:  
\[\frac{\delta E}{E}=\frac{\text{FWHM}}{Ch_{pk}}=2.35\frac{\sigma_{pk}}{Ch_{pk}}\, \tag{4}\]  
with the FWHM, Full Width Half Maximum, of the Gaussian curve. The standard uncertainties \(u(Ch_{pk})\) and \(u(\sigma_{pk})\) and the Pearson correlation index \(\rho\) are obtained by the  
Figure 7: Histograms of the heights of the voltage peaks in input to the MCA and for different \(V_{bias}\) values.