## 4 Monte Carlo Simulation
### Geant4 Simulation Results and Analysis  
Geant4 [23] is a Monte Carlo simulation package commonly used in particle simulation technology, providing a variety of particles, models, geometric description classes, and a series of visualization interfaces and other tools, with multi-physics process simulation capabilities, flexible geometric models, open source code, a rich toolbox, and other advantages [24; 25]. In this section, we use Geant4 software to simulate the transport process of \(\alpha\)-particles inside tin spheres by constructing various tin sphere models, obtaining the corresponding surface emission energy spectra, and investigating the internal radionuclide distribution pattern. A schematic of the simulation model for Sn\({}_{63\%}\)Pb\({}_{37\%}\) and SAC305 tin spheres is shown in Figure 10. The outer diameter of the tin sphere is 100 \(\mu\)m, and according to the SRIM simulation results, radionuclides are uniformly set inside the 20 \(\mu\)m thick spherical shell inward from the surface of the tin sphere. The materials of the spherical shells were set as Sn\({}_{63\%}\)Pb\({}_{37\%}\) and Sn\({}_{96.5\%}\)Ag\({}_{33.0\%}\)Cu\({}_{0.5\%}\), and their densities were \(\rho_{\text{Sn63\%}\text{Pb37\%}}=7.34\) g/cm\({}^{3}\) and \(\rho_{\text{SAC305}}=8.84\) g/cm\({}^{3}\), respectively. One million alpha particles with an energy of 5.3 MeV are randomly ejected and their outgoing direction is random. The calculations take into account in detail the secondary electron and nuclear reaction processes in the \(\alpha\)-particle and record the deposition energy distribution and cross-section information in the sensitive region. The physical processes used include: decay, EmStandard Screened, hElasticWEL_CHIPS_HP, particles, G4 GammaLeptoNuclear-Phys, hInelastic FTFP_CEM_HP, stopping, and IonInelasticLAQGSM [26]. Figure 8: Ranges of \(\alpha\)-particles with different energies inside the Sn\({}_{63\%}\)Pb\({}_{37\%}\) tin sphere. Figure 9: Ranges of \(\alpha\)-particles with different energies inside the SAC305 tin sphere. The \(\alpha\)-particle energy spectrum on the surface of the tin spheres obtained by probing is shown in Figure 11:  
(1) The emission energy spectra of the two tin spheres in the region of 0\(\sim\)0.2 MeV almost overlap; the slope of the spectra decreases with increasing \(\alpha\)-particle energy; the peak of the spectra is located at 5.3 MeV;  
(2) Compared with the measurement results (see Figures 2 and 4), the \"tail\" of the simulated spectrum in the low-energy region appears to be higher and longer, and the reasons for this are analyzed as follows: Due to the uniform distribution of internal radionuclides in the simulation model of Figure 10, the energy of \(\alpha\)-particles generated inside the tin sphere is significantly reduced when they are detected on the surface, resulting in a higher number of \(\alpha\)-particles in the low-energy region in Figure 10; therefore, it can be inferred that the internal radionuclides of the tin sphere used for the test are not uniformly distributed, presenting the characteristics of more in the outer layer and less in the inner layer;  
(3) Using SRIM calculations, it was determined that the LET value of \(\alpha\)-particles in the SAC305 tin sphere is 3.12 \(\times\) 10\({}^{-1}\) MeV\(\cdot\)cm\({}^{2}\)/mg, which is higher than its LET value in the Sn\({}_{63\%}\)Pb\({}_{37\%}\) tin sphere (2.67 \(\times\) 10\({}^{-1}\) MeV\(\cdot\)cm\({}^{2}\)/mg), i.e., the \(\alpha\)-particles are more prone to lose their energy in the SAC305 tin sphere, which results in some of the particles failing to be exited to the surface of the tin sphere. As a result, the number of \(\alpha\)-particles detected on the surface of the SAC305 Sn sphere in Figure 11 is higher than that of the Sn\({}_{63\%}\)Pb\({}_{37\%}\) tin sphere. Further simulations were carried out.