## 2 Theory and Calculation Model
### Model Settings  
The commonly used magnetic field configuration of Hall thrusters is the type convex to the anode. For example, a typical magnetic field configuration applied to spt100 is shown in Figure 3a, from which it can be seen that the magnetic field lines are curved in the channel and almost perpendicular to the wall near the exit. In order to simplify the calculation, it is assumed that the magnetic field line is axially symmetric along the center line of the channel, as was performed in Ref. [16]. The angle \(\alpha\) between the lines and the thruster's radial direction can be introduced to describe the characteristics of the magnetic field configuration near the channel exit, which can be simply expressed by:  
\[\alpha=\arctan(\frac{|\mathbf{B}_{z}|}{|\mathbf{B}_{y}|}) \tag{5}\]  
The \(\alpha\) is considered to be zero when the magnetic field line is perpendicular to the wall (i.e., \(\mathbf{E}\bot\mathbf{B}\)), \(\alpha<0\) when the magnetic field line is convex to the anode, and \(\alpha>0\) when the magnetic field lines convex to the exit plane of the channel. The schematic of the simplified magnetic field configuration and the angle of \(\alpha\) is presented in Figure 3b.  
In order to ensure that the ECDI caused by the radial magnetic field and axial electric field (\(\mathbf{E}_{z}\times\mathbf{B}_{y}\)) does not affect the simulation results as the magnetic field changes, the axial electric field \(\mathbf{E}_{z}\) and the radial magnetic field \(\mathbf{B}_{y}\) remain unchanged in all cases. Thus, the  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Symbol** & **Parameter** & **Value** \\ \hline electronic quality & \(m_{e}\) & \(9.109\times 10^{-31}\) kg \\ ion mass & \(m_{i}\) & \(6.63\times 10^{-26}\) kg \\ area size & \(L_{r}\times L_{\theta}\times L_{z}\) & \(8.4\) mm \(\times\)\(3.36\) mm \(\times\)\(8.4\) mm \\ grid length & \(\Delta x\) & \(4.2\times 10^{-5}\) m \\ Time Step & \(\Delta t\) & \(6.4\times 10^{-12}\) s \\ Electric field & \(\mathbf{E}_{z}\) & \(2\times 10^{4}\) V/m \\ Magnetic induction & \(\mathbf{B}_{y}\) & \(0.03\) T \\ Ion density in quasi-neutral region & \(n_{0}\) & \(2.5\times 10^{17}\)/m\({}^{3}\) \\ Number of particles per cell & \(N_{p}\) & \(100\) \\ electron temperature & \(T_{e}\) & \(10\) eV \\ Electronic Debye length & \(\lambda_{De}\) & \(4.2\times 10^{-5}\) m \\ \hline \hline \end{tabular}
\end{table}
Table 1: Numerical parameters used in the 2D3V PIC simulation.  
Figure 3: Schematic diagram of (**a**) typical magnetic field topology of SPT-100; (**b**) the simplified magnetic field configuration and tilt angle of magnetic field lines.  
magnetic field \(B\) is only determined by the magnitude and direction of the axial magnetic field component \(B_{z}\), as shown as follows:  
\[B=B_{x}+B_{y} \tag{6}\]  
By changing the direction and value of the axial magnetic field \(B_{z}\), the value of \(\alpha\) is set as \(\pm 15^{\circ}\), \(\pm 30^{\circ}\), \(\pm 45^{\circ}\), and \(\pm 60^{\circ}\), respectively. Once the value of \(\alpha\) is determined, the magnetic field keeps unchanged during the numerical simulation. Because the parameter variations on the outer wall are the same as those on the inner wall, only the results on the inner wall are presented in this paper [16].