# Study on Effects of Energy Deposition and Thermal Shock Wave by Electron Beam Irradiation
## 3 Energy Deposition of Electrons in Materials
### The Monte Carlo Calculation Method of Energy Deposition  
The Monte Carlo method can be used to simulate various interactions of electrons, the positions of the interactions, and the energy and direction after these interactions. In the calculation, considering that the electron has a small mass and a unit charge and collides with the material atom many times, it is impossible to simulate every collision because of a staggering number of calculations. Therefore, we adopt the method of condensing history proposed by [PERSON] [29], that is, combine many random collisions into one collision for calculation, which is called multi-scattering small collisions, and assume that the electron energy attenuation in each multi-scattering small collision is continuous and uniform (i.e., continuous slowing-down approximation). On this basis, the interaction process of electrons in matters is simulated by the single-energy large collision which generates secondary electrons. Single large collision and multi-scattering small collision are divided according to the scattering size of electron energy, and cutoff energy when tracking electrons. Thus, a single large collision is a collision whose energy loss is greater than the cutoff energy,and the secondary electrons produced by the collision will continue to be tracked. The small collisions include collisions whose energy-generating secondary electrons are less than those for the cutoff energy and collisions that can excite atoms. Under the continuous slowing-down approximation, the resultant energy losses are considered to be continuously and evenly distributed in the material covered by the trajectory of the electrons between two small collisions [30]. Therefore, a flat target of finite thickness and infinite width is taken as the calculated target, without considering the impact of edge effects, as shown in Figure 4, which shows the geometric model of electron incidence in the material. 1. Initial conditions  
(1) Initial coordinates of electrons, \(x_{0}=0\);  
(2) Initial kinetic energy, \(E_{0}\);  
(3) Initial incident angle, \(\theta_{0}\). 2. Using the direct sampling theorem, the path s between two adjacent large collisions is determined by the following equation:  
\[\zeta=1-\exp(-\int_{0}^{s}\mu_{c}\mathrm{d}s) \tag{8}\]  
\[\mu_{c}=\frac{\rho N_{0}ZC}{AE}[\frac{1}{\varepsilon_{c}}-\frac{1}{1- \varepsilon_{c}}+\left(\frac{\tau}{\tau+1}\right)^{2}(0.5-\varepsilon_{c})- \frac{2\tau+1}{(\tau+1)^{2}}\ln\frac{1-\varepsilon_{c}}{\varepsilon_{c}}] \tag{9}\]  
where \(\zeta\) is the random number; \(\rho\) is the material density; \(C=2.5486\times 10^{-25}/\beta^{2}\); \(\varepsilon_{c}\) is the minimum truncation value of the selected \(\varepsilon\). A single large collision ionizes the target material, producing secondary electrons. The energy lost by the electrons is transferred to the secondary electrons and there is no energy deposition in the material. [PERSON] et al. fitted an analytical calculation formula for the energy share obtained by secondary electrons according to the literature [19]. \[\varepsilon=\frac{1-bH(\varepsilon_{c})\xi+\sqrt{[1-bH(\varepsilon_{c})\xi]^{ 2}-4 aH(\varepsilon_{c})\xi[cH(\varepsilon_{c})\xi+0.5]}}{2 aH(\varepsilon_{c}) \xi} \tag{10}\]  
\[ \begin{array}{l}H(\varepsilon_{c})=\frac{1}{\varepsilon_{c}}-\frac{1}{1- \varepsilon_{c}}+(\frac{\tau}{\tau+1})^{2}(0.5-\varepsilon_{c})-\frac{2\tau+ 1}{(\tau+1)^{2}}\ln\frac{1-\varepsilon_{c}}{\varepsilon_{c}}\\ a=0.355982\tau^{0.401947}-0.138050\\ b=-0.3943631e^{-0.04897}-0.1968629\\ c=0.0033217e^{-0.0477\tau}-0.0025865\end{array}  \tag{11}\]  
where \(\tau\) is the kinetic energy of electrons with \(m_{0}c^{2}\) as the unit. 3. The process between the two adjacent large collisions is composed of a series of statistical collisions. The energy logarithmic partition method is adopted to divide the step size of electrons [31], which can be expressed by the formula:  
Figure 4: Schematic diagram of the geometric model of electron incidence.