## 3 Numerical Results and Analysis
### Influence of Different Collision Reactions on Electron Energy Distribution  
The temperature of the electron beam is 10 eV, and the electron velocity distribution follows a Gaussian distribution. Table 2 can be divided into four reaction types, namely: Type-A, Elastic collisions (index: 1 in Table 2); Type-B, Elastic and various excitation collisions (index: 1 and 9-18); Type-C, All collisions (index: 1-19); and Type-D, No collisions. They are applied to the electron transport process in the expansion region containing the magnetic filter field to analyze the electron energy distribution on the extraction surface. Figure 7 is a diagram of the proportion of electron energy at the extraction surface under the four reaction types when \(T=9\sim 15\)\(\upmu\)s. More relevant parameters of the extracted electrons are shown in Table 3, where the temperature of the extracted electron is derived from the Maxwellian velocity distribution function. Type-D, i.e., the no collisions model, is considered as the reference model. For Type-A, elastic collisions theoretically cause almost no decay of electron energy. As shown in Table 3, \(N_{ext}\) increases by a factor of about three, but \(\overline{\varepsilon}\) decreases from 10 eV to 6.9 eV, indicating that elastic collisions promote the mobility of low-energy electrons (\(\varepsilon\leq 2\) eV) to a certain extent. For Type-B, \(N_{ext}\) is the largest, the proportion of low-energy electrons in Figure 7 is the highest, and \(\overline{\varepsilon}\) was the lowest, indicating that excitation collision plays the most important role in the electron energy decay mechanism; for Type-C, \(T_{ext}=3.2\) eV, which is consistent with the experimentally measured 3 eV [31] of a negative hydrogen ion source with the same magnetic field structure. Collision reactions and types in Table 2 basically conform to the main electron collision reactions of the negative hydrogen ion source. The numerical results above show that the magnetic filter field system contributes to the reduction in electron temperature and that electron excitation collision is the main reason for electron energy decay. Figure 7: Diagram of the proportion of electron energy at the extraction surface under the four types of reaction types when \(T=9\sim 15\)\(\upmu\)s with the emission electron beam at a temperature of 10 eV. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Type**} & **Electron Mean Energy** & **Extracted Electron Temperature** & **Extracted Electron Number** \\  & \(\overline{\varepsilon}\) (eV) & \(T_{ext}\)(eV) & \(N_{ext}\) \\ \hline A & 6.9 & 4.4 & \(1.72\times 10^{14}\) \\ B & 3.6 & 2.2 & \(1.98\times 10^{14}\) \\ C & 5.3 & 3.2 & \(1.02\times 10^{14}\) \\ D & 10.0 & 6.5 & \(5.37\times 10^{13}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The related parameters of extracted electrons during \(T=9\sim 15\)\(\upmu\)s.  
Figure 8 shows the spatial distribution of \"cold\" electrons on the X\(-\)Z plane and the average density distribution along the +X direction under the four types of reactions at \(T=15\) us; the discrete red points represent \"cold\" electrons. As shown in Figure 8a,d, elastic collision leads to the change in scattering angle, which makes the diffusion effect of the electrons more pronounced. Results from modelling [29; 30] show that a gradient drift occurs in the IPP prototype source. Likewise, the vertical drift of electrons due to gradient drift is observed in Figure 8d. In Figure 8d, there seems to be electron bunching at \(Z=0\) cm, which is actually the gradient drift of electrons caused by the magnetic field [40]. In all models, magnets are magnetized in the \(-\)Z direction, and the direction of \(\
abla\vec{B}\) is in the \(-\)Y direction. However, since the direction of the magnetic field is parallel to the \(Z\)-axis at the central axis \(Z=0\) cm, the vertical drift of electrons in the \(-\)Y direction is more pronounced, creating the phenomenon of \"electron bunching\" on the X\(-\)Z plane as shown in Figure 8d.