## 2 Materials and Methods
### Shielding Materials  
Once charged particles enter a shielding material, they interact with the nuclei and electrons along their path, resulting in a gradual loss of energy. Depending on the degree of energy loss, the particles may either pass through the material, lose some energy, or become fully absorbed by it. When a proton interacts with a target substance, ionization and excitation occur due to inelastic collision with electrons. The resulting free electrons then interact with atoms based on their energy and direction. The energy loss rate due to ionization and excitation in a unit plane density can be calculated using the Bethe Formula (6) [27]. The energy lost by a proton in a unit path is referred to as the stopping power of the matter to the charged particle, or \(-\frac{dE}{\rho dx}\). \[\left(-\frac{dE}{\rho dx}\right)_{ion}=\frac{4\pi e^{4}N_{0}Z}{mv^{2}A}\left[ ln\frac{2 mv^{2}}{I(1-\beta^{2})}-\beta^{2}\right], \tag{6}\]  
where \(e\) denotes the Coulomb charge, \(N_{0}\) is the Avogadro constant, \(Z\) represents the atomic number of the substance, \(m\) stands for the proton mass, \(v\) is the proton velocity, \(A\) is the atomic weight of the substance, \(I\) represents the average excitation energy of the substance atom, and \(\beta\) is the ratio of the proton velocity to the speed of light. Unlike proton energy loss mechanisms, due to their large mass and small scattering angle, the path of protons in matter can be approximated as a straight line and bremsstrahlung energy loss can be disregarded. However, aside from ionization and excitation, bremsstrahlung radiation is produced when electrons interact with target materials. This is caused by the electron approaching the nucleus and interacting with its Coulomb field, resulting in energy being converted into radiation. The energy lost by radiation can be expressed as:  
\[\left(-\frac{dE}{\rho dx}\right)_{r}=\frac{4e^{4}N_{0}Z^{2}E_{e}}{137m_{0}c^{4 }A}\left[ln\frac{183}{Z^{1/3}}+\frac{1}{18}\right], \tag{7}\]  
where \(m_{0}\) stands for the electron mass, \(c\) is the electron velocity, and \(E_{e}\) represents the electron energy. The ratio of radiation energy loss rate and ionization energy loss rate of electrons in a substance is approximately:  
\[\frac{\left(-\frac{dE}{\rho dx}\right)_{r}}{\left(-\frac{dE}{\rho dx}\right)_{ ion}}\approx\frac{E_{e}Z}{700}, \tag{8}\]  
The total energy loss rate of the electron in the material is then given by:  
\[\left(-\frac{dE}{\rho dx}\right)_{T}=\frac{4e^{4}N_{0}Z^{2}E_{e}}{137m_{0}c^{4 }A}\left[ln\frac{183}{Z^{1/3}}+\frac{1}{18}\right]\left[1+\frac{700}{E_{e}Z} \right], \tag{9}\]  
Formula (6) shows that the amount of energy protons lose is inversely proportional to the square of their velocity. The rate of energy loss starts increasing before the proton comes to a complete stop. As the proton's energy decreases, the cross-section of interaction increases, resulting in more energy deposition. Therefore, this mechanism dominates before the proton stops completely, and almost all of the energy is deposited near the slice of material at the end of the range. The range in the material increases exponentially with an increase in proton energy. Hence, a shielding material with enough thickness is required to prevent high-energy protons from penetrating. Formula (9) shows that the energy loss rate of electrons is proportional to the energy of the electron, with more energy lost at the beginning of the range. As electrons lose energy, their velocity decreases, leading to an increase in interaction cross-section and more energy deposition. With more energy lost, electrons have less energy deposition. Therefore, two conflicting processes produce maximum energy deposition at the front end of the range. The range in the material increases nearly linearly with an increase in electron energy. A shielding material with a certain thickness can attenuate most of the electron energy.