## 2 Avalance Photodiodes (APDs)  
The performance of the X-ray polarimeter strongly depends on the performance of the photodetectors used. Thus, the characterization of photodetectors is fundamental for the right operativity of the final device. The measurement of the detector parameters is an interesting challenge from the instrumentation and measurement point of view since it involves the measurement of a large number of different physical quantities that must be accurately controlled in the space parameters of interest for the operative scenario. This section provides a brief introduction to avalanche photodiodes (APDs), focusing on the physical quantities that this work aims to characterize. APDs are semiconductor devices used in the detection of low-level radiation signals. Their principle of operation is the same as standard photodiodes: when a photon with energy higher than the energy band gap arrives in the sensitive region, an electron-hole pair can be generated. The ratio between the number of generated pairs to the number of impinging photons is called the quantum efficiency (QE) of the device [24; 26]. Apart from the photocurrent-generating mechanism, APDs differ from photodiodes due to their ability to multiply the photocurrent. The high electric field applied to the PN junction accelerates the carriers to a drift speed that is useful for ionizing the atoms in the crystal lattice during the collisions. This creates a chain reaction called avalanche multiplication. The entity of this multiplicative effect is described by the APD gain \(G_{APD}\)[25; 26; 27]. As long as the APD leakage current can be neglected, \(G_{APD}\) can be defined as:  
\[G_{APD}\approx I_{D}/I_{dg}\, \tag{1}\]  
where \(I_{D}\) is the total APD dark current and \(I_{dg}\) the dark current generated inside the APD substrate. The multiplicative effect, i.e., \(G_{APD}\), depends on the drift velocity of the carriers. Thus, \(G_{APD}\) is expected to be dependent on the externally applied voltage \(V_{bias}\). In addition to this, the temperature \(T\) also plays an important role: as the temperature increases the lattice vibration of the APD semiconducting material, the possibility of collision between the accelerated carriers and the lattice itself also increases, even before the carriers reach a sufficiently high speed to generate ionization, thus bringing a reduction in \(G_{APD}\)[25; 28]. One of the most used models describes all these phenomena with the following exponential law [29]:  
\[G_{APD}(T,V_{bias})=a+b\mathrm{e}^{(eV_{bias}+dT)}\, \tag{2}\]  
where \(a\), \(b\), \(c\), and \(d\) are device-dependent parameters. Other models [29; 30] are also used in the literature to describe the dependence of \(G_{APD}\) on \(T\) and \(V_{bias}\) in restricted parameter spaces. The different models are detailed and further discussed in Section 5. It is worth noting that the so-defined \(G_{APD}\) is a measure of the multiplication of charge carriers, and thus it does not include the QE of the device. This means that \(G_{APD}\) does not depend on the wavelength of the incident photons or other extrinsic phenomena. Due to the \(G_{APD}\) dependence on \(T\), applications involving APDs in environments characterized by \(T\) variations impose the temperature characterization of their \(G_{APD}\). In this work, a measurement system able to measure \(G_{APD}\) over the temperature range expected in orbiting CubeSats is designed. In particular, the bi-dimensional function given by (2) will be experimentally determined in order to exploit the possibility of controlling \(V_{bias}\) to compensate the effects of \(T\) variations on \(G_{APD}\). Another important physical property of APDs that can affect the final performance of a polarimeter is energy resolution [25; 31]. This refers to the ability of the device to distinguish between photons of different energies.