# Geant4 Simulations of a Scintillator Cosmic-Ray Detector
## 3 Estimation of Sensitivity
### Calculation of Signal's Amplitude  
Photomultipliers produce signals when they are hit by photons of relevant wavelengths. They have their own internal sensitivity to photons of different wavelengths, called Photon Detection Efficiency (PDE). It is a function of overvoltage which is the difference between the applied voltage, \(V_{SiPM}\), and the minimum voltage required for an SiPM to work as a light sensor. It is called breakdown voltage, \(V_{Br}(T)\), which is device-specific and slightly depends on temperature, \(T\). The PDE (\(\lambda\)) for the SiPM analysed in this work is presented in Figure 3. \begin{table}
\begin{tabular}{l l l l l l} \hline
**Parameter** & **Scintillator** & **Al** & **Black Tape** & **SiPM** & **Vasculine** \\  & & **Foil** & & & \\ \hline
**Density [g cm\({}^{-3}\)]** & 1.032 & 2.7 & 1.28 & 2.329 & 0.82 \\
**Thickness [mm]** & 10 & 0.14 & 0.36 & 1 & 0.6 \\
**Composition** & H, C & Al & H, C, Cl & Si & H, C, N \\
**Ratio of elements** & 11:10 & 1 & 3:2:1 & 1 & 15:15:1 \\
**Refractive index** & 1.58 & Figure 2 & 1.54 & 1.59 & 1.467 \\
**Absorption coefficient [cm\({}^{-1}\)]** & 0.001 & 966,850 & 0.069339 & - & 0.0173 \\
**Attenuation length [cm]** & 210 & - & - & - & - \\
**Scintillation yield [MeV\({}^{-1}\)]** & 11,136 & - & - & - & - \\
**Scintillation yield 1** & 0.27 & - & - & - & - \\
**Scintillation constant 1 [ns]** & 2.1 & - & - & - & - \\
**Scintillation constant 2 [ns]** & 14.2 & - & - & - & - \\
**Resolution scale** & 1.0 & - & - & - & - \\ \hline \end{tabular}
\end{table} 
Table 1: Values of parameters used in simulation of standard setup of scintillator detector [17; 18]. Parameters of scintillation are named as in Geant4 version 11.1.2. \begin{table}
\begin{tabular}{l l l l l} \hline
**Parameter** & **Stainless Steel** & **Aluminium** & **Iron** & **Glass** \\ \hline
**Density [g cm\({}^{-3}\)]** & 1.032 & 2.7 & 1.28 & 2.329 \\
**Thickness [mm]** & 0.5, 1.5 & 1.5 & 1.5 & 5 \\
**Composition** & Fe, Cr, Ni, C & Al & Fe & O, Na, Si, Ca \\
**Percent of elements [\%]** & 70.67, 20, 9.25, 0.08 & 100 & 100 & 45.98, 9.64411, 33.6553, 10.7205 \\
**Refractive index** & Figure 2 & Figure 2 & 1.51 \\
**Absorption coefficient [cm\({}^{-1}\)]** & 1.643,500 & 966,850 & 1,643,500 & 0.0021033 \\
**Distance from scintillator, d [cm]** & 0.5, 3, 6 & 3 & 3 & 3 \\ \hline \end{tabular}
\end{table} 
Table 2: Values of parameters used in the simulations for considered types of shields [18]. Geant4 simulations provide the number of scintillation photons of certain wavelengths that reach the SiPM, \(N_{ph}\). The effective number of photons registered by the photomultiplier, \(N_{reg}\), is a convolution of PDE and the spectrum \(N_{ph}\):  
\[N_{reg}\ =\ \int_{300}^{950}PDE(\lambda)N_{ph}(\lambda)\,d\lambda, \tag{1}\]  
where \(\lambda\) is photons' wavelength in nanometers. Integration was performed between 300 and 950 nm as the BC408 plastic scintillator produced photons only in this range. The SiPM is a matrix of many small microcells which are basically Geiger-mode photodiodes in series with an integrated quench resistor. In a cell hit by a photon appears an output voltage, \(\Delta V\), that can be calculated as  
\[\Delta V\ =\ \frac{G(V_{SiPM},V_{br},T)\cdot q}{C}, \tag{2}\]  
where \(G(V_{SiPM},V_{br},T)\) is called gain which is also a function of overvoltage and temperature, \(q\) is the elemental charge of the electron, and \(C\) is the capacitance of such a microcell [19]. Knowing the effective number of photons that are registered by the SiPM, one can calculate how many microcells are activated and thus estimate the total amplitude of the produced signal. There are two more quantities to be taken into account, the number of cells, \(M\), and the percentage of the surface that they cover, \(c_{p}\).