## 2 Physics of Hall Thruster Erosion
### Sputtering  
Hall thruster erosion is the gradual sputtering of the discharge chamber walls. Sputtering is the ejection of atoms from a material surface by incident energetic particles. The process is important for many applications and is used as a tool for thin film sputter deposition, ion beam etching, and secondary ion mass spectrometry [32; 33]. One such application, planar magnetron sputter deposition, involves sputtering of a material target with plasma that, as in a Hall thruster, is formed and sustained by crossed electric and magnetic fields [34]. The advancement of sputtering applications has benefited Hall  
Figure 4: Scanning electron microscopy (SEM) image of grade M26 BN-SiO\({}_{2}\). The dark regions are amorphous silica (SiO\({}_{2}\)) and the light regions are orthotropic boron nitride (BN). Figure 3: Simplified illustration of the potential drop in the wall sheath [24]. The potential gradient in the sheath repels electrons (yellow) away from the wall and accelerates ions (blue) toward the wall. truster erosion research because much of the development of sputtering theory has been motivated by these fields. Sputtering is prevalent in plasma-material interactions because many plasmas consist of particles with sufficient energy to release atoms from material surface bonds. These particles can be neutrals, ions, or electrons, but ions with energies below 500 eV are widely expected to be the primary agent of sputtering in the case of Hall thruster wall erosion [35; 36; 37]. Researchers quantifying sputtering for Hall thruster wall materials typically report sputtering rate (or yield) as the measured ejected surface material volume per time-integrated incident ion current, resulting in dimensions of volume per electric charge [35; 38; 39; 40]. However, in other fields, the sputter rate is often given as the average number of surface atoms ejected per incident ion [32; 33]. The transport model provides a satisfactory conceptual picture of the mechanism by which sputtering takes place. Transport theory holds that sputtering happens in three main stages: (1) incident energetic particles transfer energy to material atoms through collisions, (2) impacted material atoms transfer energy to other material atoms in a collision cascade, and (3) some atoms near the material surface gain sufficient velocity in a direction pointing away from the surface to leave the material altogether [32; 33]. Figure 5 illustrates the transport theory sputtering mechanism. Application of the transport model results in a simple relation for the sputter yield that can be corrected for many different materials exposed to a variety of plasma species. The model is fundamentally limited, however, because it assumes the material is a homogenous continuum and relies on adjustable parameters for fitting to experimental data. Sputtering is a far more complex process than what is implied by the transport theory. The sputtered material may have heterogeneous structure, sufficiently high temperatures can cause material sublimation, incident particles may not immediately collide with surface atoms due to ion and energy channeling, and adatom-vacancy pairs and surface island nucleation may form due to energetic atoms not leaving the material surface. These shortcomings may be overcome by individually computing particle trajectories with molecular dynamics simulations [32; 33; 41]. Propulsion engineers must accurately model Hall thruster wall sputtering to predict Hall thruster wall erosion rates and determine the resultant thruster lifetime. This task is challenging because the sputtering rate is highly sensitive to surface properties, such as temperature, roughness, contamination, and charge, and near-wall plasma properties, such as ion density, energy, and incidence angle [32; 33]. To make matters even more complicated, sputtering of the Hall thruster wall changes the surface composition and secondary electron yield. These changes, in turn, alter both the material and plasma properties [22; 23; 39; 42; 43].