## 1 Introduction  
Hall-effect thrusters (HETs) have become the most widely used type of thruster device due to their combined advantages of high thrust density, high specific impulse, system simplicity, and high reliability [1; 2]. Typical HETs consist of an annular ceramic channel. The cross-section and working schematic of the HETs are shown in Figure 1. The anode located upstream in the channel works as the neutral propellant distributor and the external cathode placed in the near-field plume region works as an electron emitter. A large potential difference was applied between the anode and the hollow cathode, creating an electric field along the axial direction that can accelerate the ions produced by ionization and generate thrust. A magnetic field dominated by the radial component along the channel can trap electrons and increase their residence time within the discharge to promote the efficient ionization of neutral propellant gas [3]. With the complexity of spacecraft missions escalating, the service life of the thruster devices becomes more demanding. Therefore, as one of the most important factors limiting the life of HETs, the sputtering wall erosion caused by the flux of energetic ions can lead to the mass loss of the discharge channel and removes HETs from the working order. Much experimental and theoretical research has been carried out to investigate the mechanism of sputtering wall erosion [4; 5; 6; 7]. The magnetic field topology has been found to be a central design issue for HETs because the main working principle of HETs is to use the orthogonal electric and magnetic fields to sustain the ionization process and accelerate ions. Currently, the commonly used magnetic field configuration is the type convex to the anode proposed by [PERSON] et al. [8] for its proven ability to reduce the divergence angle of the ion beam and improve the discharge performance. The characteristics of the configuration are that the main component is along the radial direction and the magnetic field lines near the wall are denser than that in the center of the channel, forming the so-called \"plasma lens\" [9; 10]. A number of studies have been carried out to look for insight into this lens-type magnetic field configuration. [PERSON] et al. [11; 12] investigated the effect of the magnetic field on the plume divergence angle and plasma oscillations by controlling the magnetic field configuration with additional trim coils. It was found that the curvature of the magnetic field line and axial gradient of the main radial magnetic field had a great influence on the performance of the thruster. [PERSON] et al. [13] theoretically analyzed the relationship between magnetic focusing and the plume divergence angle and confirmed the focusing effect of the curved magnetic field line on the plasma. [PERSON] [14] and [PERSON] [15] also confirmed the effect of the magnetic field topology on the process of sputtering erosion by studying the subject of ion divergence and wall erosion conditions under the influence of different magnetic field configurations. [PERSON] et al. [16] studied the ion bombardment and wall erosion in a Hall thruster with magnetic lens design using the PIC method and found that the wall erosion rate as well as the length of the erosion zone decrease monotonically with the increase in the curvature of the lens-type magnetic field. These previous results suggest that the curved magnetic field configuration is helpful in converging the ion beam and reducing the sputtering erosion caused by energetic ions. However, most of the previous studies were conducted using several fixed magnetic field  
Figure 1: Geometry structure of the PIC model in radial-azimuthal plane. configurations, which failed to reach a universal conclusion on the influence of magnetic field line tilt angle on sputtering erosion. Moreover, the effects of the magnetic field line tilt angle on the sheath formation, the particle motion near the wall, and the wall sputtering process have not been well analyzed, and the influencing mechanism is also unclear.