# Plume Divergence and Discharge Oscillations of an Accessible Low-Power Hall Effect Thruster
## 1 Introduction
### Theory of Operation
This results in an increased plume divergence that would not otherwise be present in a space environment. Additionally, corrections due to secondary electron emission (SEE) from the probe collector surface, probe geometry, and thruster distance must be made to avoid excessive error in the analyzed results from Faraday probe data. Reference [12] was utilized substantially for analysis of the Faraday probe data collected in this experiment, and we convey the primary processing methods in the following paragraphs. Current density, \(j\), was calculated by  
\[j=\frac{I_{\text{FP}}}{A_{c}+\kappa_{G}}\kappa_{\text{SEE}}. \tag{2}\]  
\(\kappa_{G}\) is a correction factor that accounts for probe geometry, and \(\kappa_{\text{SEE}}\) is a correction factor that accounts for SEE as a result of ion charge species impacting the tungsten probe surface. \(I_{\text{FP}}\) is the current to the Faraday probe, and \(A_{c}\) is the Faraday Probe collector area. An ExB probe (Wein filter) was unavailable for charge species analysis of the WHT44, so the \(I_{\text{FP}}\), was assumed to consist only of \(Xe^{+}\). The total beam current, \(I_{B}\), and the component of the beam current in the axial direction, \(I_{A}\), are calculated using Equations (3) and (4), respectively. \[I_{B}=\pi R^{2}\int_{-\pi/2}^{\pi/2}j[\theta]\frac{\kappa_{D}}{\kappa_{A}}sin( \theta)d\theta \tag{3}\]  
\[I_{A}=\pi R^{2}\int_{-\pi/2}^{\pi/2}j[\theta]\frac{\kappa_{D}}{\kappa_{A}}cos (\alpha_{A})sin(\theta)d\theta \tag{4}\]  
where \(R\) is the radial distance between the Faraday probe and the thruster centerline, and \(\alpha_{A}\) is a piecewise function defined as  
\[\alpha_{A}=\left\{ \begin{array}{cc}\theta-\alpha_{N}=tan^{-1}\left(\frac{ sin(\theta)\mp\frac{D_{\text{T}}-W}{d\kappa}}{cos(\theta)}\right)&sin^{-1}\left(\frac{D_{ \text{T}}-W}{2R}\right)\leq\theta\leq 90^{\circ}\\ 0&0\leq\theta\leq sin^{-1}\left(\frac{D_{\text{T}}-W}{2R}\right)\end{array} \right. \tag{5}\]  
where \(\alpha_{N}\) is the probe angle, \(D_{\text{T}}\) is the thruster diameter, and \(W\) is the discharge channel width. The beam divergence half-angle, \(\lambda\), is defined as  
\[\lambda=cos^{-1}\left(\frac{I_{A}}{I_{B}}\right), \tag{6}\]  
and \(\kappa_{D}\) and \(\kappa_{A}\) are correction factors for measuring ions coming from a two-point source in a hemispherical coordinate system. \(\kappa_{D}\) corrects for the difference in radii formed due to the fact that the thruster annulus is not an ideal point source. \(\kappa_{A}\) corrects for the change in incidence angle of ions impacting the probe face as angle, \(\theta\), and radius are swept. There is a second method for calculating \(\lambda\) based on the angle that includes 90% (or 95%) of the total ion current. To maintain consistency between the methods of analysis for the Faradayprobe data, Equation (6) was used in this work. The difference between the two methods at varying background pressure and thruster diameter downstream (TDD) was examined for the WHT44 in reference [13]. The method outlined using Equation (6) consistently showed smaller half-angles compared with the method that uses 95% of the total ion current by \(\approx 10^{\circ}\). Faraday probe data were used to determine the beam current and the axial component of the beam current using Equations (3) and (4), from which three thruster loss mechanisms are determined. These are the current utilization efficiency, \(\eta_{C}\), mass utilization efficiency, \(\eta_{M}\), and beam utilization efficiency, \(\eta_{B}\) defined with Equations (7)-(9).