## 4 Materials and Methods
### Energy Calibration and Dead Layer Estimation Strategy  
As already discussed, the main scope of the test campaign is to obtain the cluster-level curve of ToT as a function of the deposited energy in the Timepix3 module by performing irradiations with quasi-mono energetic beams directed orthogonally on the sensor (i.e., as shown in Figure 4, with \(\theta=0^{\circ}\)) and stopping within its active layer. For both protons and alphas and for each beam energy, the energy deposited in the active layer of the sensor is effectively given by the difference between the incoming beam energy \(E_{0}\) and the energy lost in the front layers as follows:  
\[E_{dep,eff}(E_{0})=E_{0}-E_{dep,Al}(E_{0},T_{Al})-E_{dep,DL}(E_{0}-E_{dep,Al},T _{DL}) \tag{2}\]  
Both front layers are thin, with \(T_{Al}=500\,\mathrm{nm}\) and \(T_{DL}\) (to be evaluated as part of the test campaign) expected to be at a similar scale, but the fraction of energy that they absorb can be significant, especially for particles with very short penetration depth like alphas (e.g., at \(1\,\mathrm{MeV}\), their penetration depth is just \(4\,\mathrm{\SIUnitSymbolMicro m}\)). The irradiation is performed under vacuum, and thus it is assumed that the beam particle reaches the detector front at its nominal energy \(E_{0}=E_{beam}\). The energy deposited in each layer is calculated via Monte Carlo simulations performed with FLUKA, and presented numerically in the results section in Table 1. The source was considered to be a planar wave with an energy distribution of at most 3% (as communicated by the facility), leading to almost delta-peak energy distributions since the particles for all the beam energies stop within the detector. Given the energy scale of the campaign, the following lowest energy thresholds are used: the PRECISION FLUKA default, characterized by particles transport thresholds at \(100\,\mathrm{keV}\) for hadrons (except for neutrons at \(10^{-5}\,\mathrm{eV}\)), and lowered threshold for the electromagnetic components as follows: electron/positrons at \(1\,\mathrm{keV}\) and gammas at \(100\,\mathrm{eV}\). Given the small Timepix3 detector geometry, in particular the non-sensitive layers at the sub-micron level, the MULSOPT card was additionally used to enable single Coulomb scattering, thereby deactivating the typical condensed history/multiple Coulomb scattering approach in Monte Carlo codes for computational efficiency. The measurement of the thickness of the silicon dead layer \(T_{DL}\) represents a crucial part of the calibration campaign, and it is obtained by irradiating the sensor with a fixed beam type and energy (597 keV protons) at different angles of incidence (from \(\theta=0^{\circ}\) to \(\theta=45^{\circ}\)). Under the well-justified assumption that the average energy deposited by the through-going particles in the front layers \(E_{dep,front}(E_{0},d)\) depends only on the particle energy \(E_{0}\) and on the path length \(d\), from simple geometrical considerations the energy lost in the front layers at different angles can be expressed as \(E_{dep,front}(E_{0},T/\cos\theta)\), where \(T\) is the thickness of the layer under exam. Since the particle stops in the depleted layer, one can express the effective energy deposited \(E_{dep,eff}(E_{0},\theta)\) via the following equation:  
\[E_{dep,eff}(E_{0},\theta)=E_{0}-E_{dep,Al}(E_{0},T_{Al}/\cos\theta)-E_{dep,DL} (E_{0}-E_{dep,Al},T_{DL}/\cos\theta) \tag{3}\]  
where the dependency of \(E_{dep,eff}\) from \(\theta\) is solely due to the different fraction of beam energy lost in the front layers, as the particles are then stopping within the active sensor layer regardless of the incident angle. By assuming a constant stopping power \(dE/dx\) both in the aluminium metalisation and the silicon dead layer (i.e., for a small stopping power gradient in these thin layers at these energies).