# Plume Divergence and Discharge Oscillations of an Accessible Low-Power Hall Effect Thruster
## 1 Introduction
### Theory of Operation  
1.1.1. Hall Effect Thruster  
HETs are electrostatic propulsion devices consisting of a hollow cathode, a discharge channel with an anode, and a system to generate a magnetic field radial to the discharge channel. Electrons are inhibited from traveling from the cathode to the anode by the radial magnetic field and, instead, form an azimuthal Hall current in the ExB direction. This Hall current ionizes the neutral propellant injected through the anode into the discharge channel. The more massive ions are mostly unaffected by the radial magnetic field and are instead accelerated by the axial electric field. While HETs are mechanically simple and classified as electrostatic devices, they rely upon complex plasma physics and magnetic field topologies to operate. HETs exhibit several oscillatory modes over a wide range of frequencies, from 1 kHz to 60 MHz [7]. The effects of these oscillation modes on thruster lifetime, thrust, and efficiency are not fully understood. The two most studied oscillation modes are the breathing mode and the azimuthal spoke mode. The breathing mode is an ionization driven oscillation from the predator-prey relationship between neutral atoms and ions in the plasma plume. According to reference [7], we should expect to see these oscillations at frequencies between 15-35 kHz. Breathing mode oscillation frequencies, \(f_{i}\), can be modeled in 1-dimension by solving a form of [PERSON]'s predator-prey equations, resulting in Equation (1). \[2\pi f_{i}=\frac{\sqrt{u_{i}u_{n}}}{L_{i}} \tag{1}\]  
where \(L_{i}\) is the ionization zone length, and \(u_{i}\) and \(u_{n}\) are the ion and neutral velocity, respectively [8; 9]. Spoke mode oscillations, when identified with high-speed cameras, are regions of higher plasma light intensity, indicative of an ionization front, most often traveling in the \(E\times B\) direction at velocities around 1 km/s [10; 11]. 1.1.2. Faraday Probe  
Faraday probes are a common diagnostic for determining ion flux in plasma plumes from electric propulsion devices. They are used to quantify thruster beam divergence, identify asymmetric behavior, determine total integrated beam current, and identify various utilization efficiencies (i.e., mass utilization, current utilization, and beam utilization). Faraday probe data can also be collected at a variety of facility background pressures in order to attempt extrapolation of the results to flight-like operation [12]. Fundamentally, Faraday probe data are collected through the use of a planer surface electrode of known area, \(A_{p}\), oriented using a cylindrical or spherical coordinate system. A cylindrical coordinate system is one wherein the probe is mounted parallel to the thruster's axial centerline and swept radially. This can be accomplished with either one or two linear stages mounted orthogonally with respect to the thrust axis. In a hemispherical coordinate system, the probe is mounted on a rotary stage such that the probe is always facing in the direction of the thruster center point, a point created by the intersection of the thruster discharge exit plane and the thruster centerline. A hemispherical arrangement was used exclusively for this study. The Faraday probe is biased negatively such that it is in the linear ion saturation regime of the plasma. Repeatedly repositioning the probe and collecting data gives the operator a current density contour of the thruster plume [12]. Facility interactions that influence ion current density are present in any ground based experiment. There are two primary interaction mechanisms that exist during ground based testing related to background neutrals in the vacuum facility. First, background neutrals may be ingested into the thruster, ionized, and accelerated, creating excess thrust. Second, charge exchange (CEX) collisions occur when fast ions strike slow neutrals to produce slow ions. CEX collisions scatter beam ions and have the effect of increasing current density in the plume periphery.