## 2 Theory and Calculation Model  
### PIC Simulation Model  
In this paper, a 2D3V IFE-PIC numerical model is established to simulate the radial-azimuthal plane near the exit of the Hall thruster. The model is based on the Cartesian coordinate system, as shown in Figure 2. The simulated domain consists of the dielectric wall and a portion of the discharge channel. Its upper part is set to be quasi-neutral, corresponding to the quasi-neutral mainstream area of the discharge channel, and the lower part is considered BN dielectric wall. For clarity of presentation, the upper surface of the wall is taken as the starting of the radial direction of the simulated region, which is denoted as \(y=0\). The size of the simulation domain is \(R=\left[L_{x}\,\times\,L_{y}\,\times\,L_{z}\right|200\lambda_{\text{De}}\, \times\,80\lambda_{\text{De}}\,\times\,200\lambda_{\text{De}}\,\right]\), where \(\lambda_{De}\) denotes the Debye length. The model ignores the curvature of the wall in the azimuth and has a finite length in both \(Ox\) and \(Ox\) directions. A magnetic field \(B_{y}\) along the radial direction \(Oy\) and a constant electric field \(E_{z}\) along the axial direction \(Ox\) are applied in the numerical model. Thus, the electrons will drift in the azimuthal direction of \(Ox\) (i.e., \(E_{x}\times B_{y}\)). In the numerical calculation, only singly charged electrons and ions are simulated. The secondary electron emission (SEE) and the collisions between particles are not considered. The propellant type used here is argon. A quasi-neutral region of width \(5\lambda_{\text{De}}\) is set near the incident boundary [17]. At each time step, a fixed number of ions (determined by the ion density \(n_{0}\) of the quasi-neutral region) enter the simulation region from the incident boundary, and then the numbers of electrons and ions in the quasi-neutral region are  
Figure 2: Geometry structure of the PIC model in azimuthal-radial plane. counted. If the number of electrons is less than that of the ions, then the corresponding electrons are replenished to make them consistent. If the electron number is greater than that of the ions, no replenishment is required. In addition, the velocity of the incident electrons obeys a Maxwellian distribution, while the ions are considered to be cold, and the incident velocity is chosen equal to the Bohm velocity. The electrons and ions are treated as PIC particles in the simulation area. The position and velocity of the particles are tracked in all three directions, while the electric field is calculated only in the (_Ox-Oy_) plane by solving [PERSON]'s equation, which can be expressed as  
\[-\
abla\cdot(\varepsilon\
abla\Phi)=e(n_{i}-n_{e}) \tag{1}\]  
where \(\varepsilon\) is the vacuum permittivity, \(\Phi\) is the electric potential, \(e\) is the elementary charge, and \(n_{e}\) and \(n_{i}\) are the density of electrons and ions. The charge densities \(n_{i}\) and \(n_{e}\) are calculated by weighting all particles to the 2D radial-azimuthal (_Ox-Oy_) simulation area. The movements of particles are determined from the integration of the equation of motion, which can be described as [18]:  
\[m\frac{d\mathbf{v}}{dt}=q(\mathbf{E}+\mathbf{v}\times\mathbf{B}) \tag{2}\]  
where \(m\) and \(\mathbf{v}\) are the mass and velocity of the particles, \(\mathbf{B}\) is the magnetic field, and \(\mathbf{E}\) is the electric field determined by \(\mathbf{E}=-\
abla\Phi\). Periodic boundary conditions are applied for [PERSON]'s equation and for the particles at the azimuthal boundaries along the _Ox_ direction. At the positions of \(x=0\) and \(x=L_{x}\), the electric potential satisfies. If the particles exceed the boundaries in this direction, they are allowed to re-inject into the simulated region from the opposite side. The potential at the incident boundary (i.e., \(y=L_{y}\)) is supposed to be \(\varphi=0\). The electrons and ions that run out of this boundary are removed from the simulation domain. At the position of \(y=0\), the dielectric wall is adopted.