# Geant4 Simulations of a Scintillator Cosmic-Ray Detector
## 3 Estimation of Sensitivity
### Mapping Detector  
Particles can hit different regions of a detector and this has a significant impact on the number of scintillation photons that reach the SiPM [21]. It is important to keep in mind that scintillators have some thickness, which means that particles may also hit their sides when they do not come directly from above. In those cases the distance they traverse within the scintillator is shorter but some of them can still produce detectable signals. The impact of this effect depends on inclination angles, \(\theta\) and \(\phi\), from which particles arrive and on the shape of the device. Thus, one has to consider in simulations not only particles that hit the top surface of the scintillator but also those that are distant from it. For a flat wavefront of particles coming from the same direction, this causes an effective increase in the area of the detector's surface, \(A_{eff}\), as presented in Figure 4.  
Figure 4: _Cont._  
A common way to characterise cosmic rays is by the local number of particles per flat unit of area. Therefore, it is useful to know the surface of the projection of the shape of the scintillator into a plane parallel to the top of the detector. In the case of a cuboid detector of the dimensions \(x\ \times\ y\ \times\ r\), this increase can be described using the following formula:  
\[A_{eff}(\theta,\phi)\ =\ A_{0}+(|\sin\phi|x+|\cos\phi|y)\cdot r\cdot\tan\theta, \tag{6}\]  
where \(A_{0}\) is the area of the surface of the simulated scintillator at \(\theta\ =\ \phi\ =\ 0^{\circ}\). This formula is valid for \(\theta<90^{\circ}\).  
Another important effect is that particles can scatter and even produce secondary ones through interaction with material above the scintillator, especially with the shield. This means that some of the particles that miss the detector if there is nothing above it can produce a signal because they change their trajectory or the detector is hit by products of their interactions. Due to this and the previously described geometrical effect, it was necessary to simulate particle beams in an area that not only covered the top surface of the detector but also its surroundings. To study this, beams of 100 particles on a grid were simulated, to create maps of the detector's geometry projection on a plane placed on the top of the scintillator. Exemplary maps for the setup with a 1.5 mm steel plate placed 3 cm above the detector are presented in Figures 5 and 6.  
Figure 4: Effect of increasing of detector’s effective area with greater inclination angle of particle wavefronts: (**a**) side view; (**b**) top view.  
Figure 5: Maps of the sensitivity to muons (**left**), electrons (**middle**), and photons (**right**) of \(p\ =\ 0.1\) GeV at \(\theta\ =\ \phi\ =\ 0^{\circ}\) for a detector placed at \(d\ =\ 3\) cm behind a 1.5 mm steel plate. The value of each point was calculated from a beam of 100 particles aiming at the \(x\) and \(y\) coordinates of a plane placed on the top of the scintillator. Dotted lines represent the border of the scintillator while the solid square indicates the position of the SiPM.  
From these maps, the effective sensitivity for the whole detector was calculated, \(\eta_{eff}\). This value was estimated using the following formula:  
\[\eta_{eff}\ =\ \frac{1}{A_{0}}\iint_{S}\eta(x,y)\,ds, \tag{7}\]  
where \(A_{0}\ =\ 25\) cm\({}^{2}\), S is the whole simulated area, and \(\eta(x,y)\) is local sensitivity for particles that are aimed at a small fragment, \(ds\), of the area \(S\). In the performed simulations, \(ds\) was a 0.25 cm \(\times\) 0.25 cm square. In the presented parametrisation, where integrated efficiency was divided by the constant \(A_{0}\) without including an additional effective area from the sides of the non-flat scintillator, the effects described above may have resulted in an effective sensitivity, \(\eta_{eff}\), larger than 100%, especially for large theta.