## 3 Results and Discussions
### Wall Erosion When the Value of \(\alpha\) Is Positive
Figure 10 shows the average sputtering erosion rate of the wall corresponding to different positive \(\alpha\), which is obtained by averaging the erosion rate along the \(Ox\) direction after simulation convergence. It can be seen that compared with the case in which the electric field \(E\) is perpendicular to the magnetic field \(B\), the applied axial magnetic field component \(B_{z}\) results in a larger wall erosion rate. The increasing potential drop along the radial direction promotes the divergence of the ion beam in the discharge channel and increases the energy of the ions hitting the wall. Thus, the sputtering erosion of the ions on the wall is aggravated. Moreover, with the increasing value of the positive \(\alpha\), the radial velocity of the ions increases, which makes the wall erosion more serious. Compared with the case with the electric field \(E\) perpendicular to the magnetic field \(B\), when the value of the angle \(\alpha\) is \(15^{\circ}\), \(30^{\circ}\), \(45^{\circ}\), and \(60^{\circ}\), the wall erosion rate is enhanced by 4.9%, 31%, 70%, and 171%, respectively. Figure 10: The average erosion rate of the dielectric wall with different values of the positive \(\alpha\). Figure 9: Ion velocity distribution function for radial component with different values of the positive \(\alpha\). will accumulate near the entrance or escape directly from the incident boundary depends on the magnitude of the radial drift velocity. Figure 12 shows the distribution of electron density corresponding to different negative \(\alpha\). It can be seen that there is a large density gradient in the radial direction. When the value of \(\alpha\) is \(-15^{\circ}\) and \(-30^{\circ}\), the magnitude of the drift velocity away from the wall is small. It makes it difficult for the electrons to enter the simulation region, so the density accumulation can be observed near the entrance in Figure 12a,b. However, when the value of \(\alpha\) is \(-45^{\circ}\) or \(-60^{\circ}\), the drift velocity becomes larger, making it impossible for electrons to remain near the entrance. Most of the electrons escape from the simulation region. As a result, the average electron density near the upper boundary decreases, as shown in Figure 12c,d. In addition, as the value of the negative \(\alpha\) decreases, the velocity of the electrons away from the wall increases, resulting in a decreased number of electrons that can reach the wall. Therefore, as shown in Figure 11, the potential of the dielectric wall increases with the decreasing value of the negative \(\alpha\). When the value of \(\alpha\) is \(60^{\circ}\), there is even a positive wall potential. However, the ions are able to enter the simulation area for the given incident conditions because they are not bound by the magnetic field, thus resulting in a local positive potential in the simulation area. Meanwhile, as the electron density decreases due to the change in \(\alpha\), a higher value of the positive potential is presented due to the growth of the difference in \(n_{e}\) and \(n_{i}\). The electron motion and wall potential distribution have important effects on the behavior of the ions when the value of \(\alpha\) is negative. The decrease in the potential drop between the dielectric wall and the quasi-neutral region makes it difficult for ions to reach the wall. More electrons are trapped in the channel, creating local positive potentials, as shown in Figure 11. The resulting potential gradient along the radial direction further  
Figure 12: The 2D distributions of electron density with different values of the negative \(\alpha\). prevents the ions from approaching the walls, even causing them to show acceleration in the direction away from the wall. In addition, the ion velocity induced by the \"ion trapping\" phenomenon of ECDI is also in the direction away from the wall.