## 3 Numerical Results and Analysis  
### Magnetic Filter Field Simulation Result  
In this paper, the scalar magnetic potential finite-difference method is used to calculate the magnetic filter field. The layout of the magnetic field is as referred to in [22; 33]; that is, the magnets are located in the magnetic boxes on both sides of the chamber, forming a strong magnetic field on the two sides and weak one in the middle. The magnetic field line enters the expansion region from the chamber wall and exits from the other chamber wall, and it does not form a closed magnetic field line, which is a typical structure and characteristic of the negative hydrogen ion source magnetic filter field. Figure 3a shows the distribution of the magnetic filter field \(B_{z}\) on the Y\(-\)Z plane calculated by SMPM. Figure 3b shows the calculation result for the distribution of the magnetic filter field \(B_{z}\) along the X-axis. It shows that the maximum magnetic induction intensity \(B_{z}\) along the X-axis at the center of the chamber is about 7.4 mT, which is in good agreement with the magnetic field intensity distribution curve in [33]. Section 3.1 is a collision-free model. The filtering effect of the magnetic field on the electron beam with a temperature of 10 eV on the X\(-\)Z plane is shown in Figure 4. It can be seen that most of the electrons are effectively filtered by the magnetic filter field, and only a small number of the electrons are allowed to pass through the expansion region to the extraction surface (as shown in Figure 2). The whole simulation time \(T\) is 15 us. The calculation is regarded as steady-state when the number of electrons in the chamber reaches saturation after 15 us. When \(T=9\sim 15\) us, compared with the absence of a magnetic filter field, the proportion of extracted low-energy electrons (\(\epsilon\leq 2\) eV) increases from 2.6% to 10.3%, and the temperature of the extracted electrons decreases from 10 eV to 6.5 eV, but the number of extracted electrons drops from 1.95 \(\times\) 10\({}^{17}\) to 5.37 \(\times\) 10\({}^{13}\). This suggests that the magnetic filter field can effectively filter both \"hot\" and \"cold\" electrons. For the objects to be filtered, \"hot\" and \"cold\" electrons have equal statuses. Since collisions are not considered in this section, the electron energy does not decay in any form, and the mechanism of the magnetic filter field can be regarded as a matter of how electrons are transported in the magnetic filter field. Figure 3: (**a**) The distribution of magnetic filter field \(B_{z}\) calculated by SMPM on the Y\(-\)Z plane; (**b**) Magnetic filter field \(B_{z}\) distribution along the X-axis compared with experimental data. In order to explore the general law of electron transport paths in the expansion region, the electron emission position is fixed at the centre of each grid on the emission surface (i.e., the emitter in Figure 2) to observe the electron trajectory. The energy of all electrons is fixed at 30 eV. Results are shown in Figure 5. Figure 5a shows the projection of all electrons on the X\(-\)Z plane. Figure 5b is the projection of some electrons on the X\(-\)Z plane in the range of the Y-axis [\(-\)4 mm, 4 mm]. Electrons travel in the +X direction from the emission surface (at X = 0 cm). As the magnetic field intensity increases, the electrons gyrate along the magnetic field lines. Figure 5b clearly shows that some of the electrons are reflected by the strong magnetic field on both sides of the chamber as they pass through the expansion region. Such a phenomenon can be explained by the magnetic mirror effect; that is, having a strong magnetic field at both ends and a weak one in the middle causes charged particles to be reflected by the strong field, just as light is reflected by a flat mirror. The magnetic mirror phenomenon can be explained by Equation (1).