## 4 Hall Thruster Erosion Modeling  
Predictive Hall thruster erosion modeling is a formidable challenge that promises to enable researchers to accurately and rapidly assess the lifetime of thruster prototypes without the need for expensive and time-consuming life tests. However, predictive modeling requires a precise description of the discharge plasma, wall material, and plasma-wall interaction, as well as the ability to simulate the impact of microsecond-scale events over the course of thousands of hours. Experimental uncertainties in available experimental data and limitations in computing power have thus far prevented the development of truly predictive models. Though they can accurately reproduce the decaying erosion rate trend observed in Hall thruster life tests, most contemporary models are highly sensitive to tunable parameters that must be adjusted to match preexisting thruster erosion data. They are also unable to reproduce experimentally observed surface evolution features, such as anomalous erosion ridges, microcracking, and atomic-level changes in surface composition. This section reviews erosion modeling and highlights important results. Early efforts [79; 80] circumnavigated the need for detailed plasma and sputtering models by simply curve-fitting preliminary SPT-100 erosion data to extrapolate erosion trends and thereby predict thruster life without complete life test data. Though recent work [81] has improved upon this technique by employing machine learning across multiple thruster life test datasets, life test data extrapolation is a fundamentally limited approach. It requires a thruster be constructed and operated before any life prediction is made and therefore does not enable rapid assessment of multiple thruster prototypes in early design stages. More sophisticated models introduced by [PERSON] et al. [82] and [PERSON] et al. [83] in the late 1990s used fluid equations to describe the plasma discharge and determine the ion flux to the wall. The ion wall flux, coupled with a simplified sputtering rate model, gave a first-order calculation of erosion rate. The sputtering rate model used by these authors is given in Equation (1), where Y is the sputtering rate, J\({}_{i}\) is the ion current density to the wall, and S is a sputtering factor that incorporates dependence on ion energy and angle of incidence. \[\mathrm{Y}=\mathrm{J_{i}S} \tag{1}\]  
Later efforts [84; 85; 86; 87; 88; 89; 90; 91] accounted for plasma-wall interactions and implemented increasingly sophisticated discharge plasma and semiempirical sputter yield models tethered to experimental data. Many authors fit variations of the Yamamura and Tawara [92] sputtering rate law for monoatomic solids to the sputtering rate data of [PERSON] et al. [38; 39] or [PERSON] et al. [35]. The general Yamamura and Tawara model is given in Equation (2), where F is a function that describes the dependence of sputtering rate on wall material and ion properties, E is ion energy, E\({}_{th}\) is the threshold sputtering energy of the wall, and b is an adjustable parameter. \[\mathrm{Y}=\mathrm{F}(\ldots)\left(1-\sqrt{\frac{E_{th}}{E}}\right)^{b} \tag{2}\]  
Threshold sputtering energy is typically assumed to lie between 30 and 70 eV. The form of Equation (2) and the values of the associated fit parameters vary as a function of the chosen model, fitted experimental data, and assumed threshold sputtering energy. An important contribution was made by [PERSON] et al. [70], who recovered the exponentially decaying erosion rate observed in life tests by updating the thruster discharge size with wall erosion. Increased discharge volume caused by decreased wall thickness provided ions with greater opportunity to collide and lose energy before striking the wall, thereby decreasing the sputtering rate. The model of [PERSON] et al.