## 3 Design of the Measurement System
### Design of the System for Gain and Energy Resolution Measurement
Due to the random (Poisson) statistics of the APD emitted charge pulses (once one of the X-ray sources is placed in contact), it is not possible to exclude the possibility that two pulses are emitted so close that the \(C_{f}\) is not completely uncharged before the arrival of the second pulse. This would bring a pile-up of the pulses, providing incorrect amplitude information. To avoid this phenomenon, pulse shaping stages are usually used. The pulse-shaping stage aims to provide an output of well-separated pulses, avoiding their overlap, and without losing/distorting the information of the preamplifier signal [35]: pulse amplitude and time. The easiest way to accomplish this is by using high-pass RC filters and differentiation circuits; the output of these circuits enhances the fast variations of the voltage \(v(t)\) signal corresponding to the arrival of a new photon, reducing the effect, if properly designed, of the long tails of the preamplifier output signal. Alternatively, Gaussian filters are more commonly used. These are filters whose impulse response is an approximation of the Gaussian function. This kind of filter is generally integrated with spectroscopy amplifiers. In this case, the Silena 7611 amplifier is used. Its pulse-shaping stage is based on a Gaussian filter with variable shaping time \(0.25<t_{s}/(\upmu\mathrm{s})<6\), and it is followed by a voltage amplifier whose gain can be continuously regulated by the user from 2.5 to 3000. The set-up of the system will require a fine-tuning of these parameters, i.e., \(t_{s}\) and the gain of the Silena 7611 amplifier. The signal elaboration chain from the APD to the amplifier output is summarized in Figure 3.  
Figure 2: Charge preamplifier electrical circuit [34]. * _Readout system_. Once the signal is properly conditioned, it can be measured. Since the information about the quantity of charge generated is preserved in the amplitude of the voltage pulse output from the spectroscopy amplifier, a system capable of classifying voltage pulses based on their amplitude is required. A multi-channel analyzer (MCA), operating in pulse height analysis (PHA) mode, can be used for this aim. MCAs are based on fast ADC used to digitize the incoming pulses and collect information about these. In the PHA mode, pulses are classified based on their peak voltage. Thus, the output is a histogram where each class (channel) contains the number of pulses counted during the measurement time with an amplitude corresponding to the channel number. Thus, the MCA resolution depends on the number of channels in this operation mode. Since scintillators in spectroscopy experiments are often used to convert the energy of the incident in a proportional quantity of charge, photons with different energies correspond to voltage pulses with different amplitudes. Thus, once the MCA channels are calibrated in energy, then the output of the MCA, after a certain measurement time, directly reports the spectra of the radiation incident on the detector. In this work, we will mainly exploit the MCA channel not to discriminate photons with a different energy but to evaluate how \(G_{APD}\) changes when different \(V_{bias}\) are applied at different \(T\). In this work, the Amptek MCA8000D is used.