## 4 Ion Current  
### Total Ion Current and Divergence Angle Determination  
The ion current \(I_{i_{exp}}\) corresponds to a flux of positive charges going through a surface per unit of time. It reads:  
\[I_{i_{exp}}=\int\int j_{i}dS. \tag{2}\]  
The ion current density j (A/m\({}^{2}\)) is assumed to be collinear to the outward pointed unit normal vector to the surface. A spherical coordinate system is often used to determine \(dS\) and compute \(I_{i_{exp}}\). The probe is usually fixed at a distance \(R\) and the thruster is supposed to be a point source at the center of a sphere. To compute \(I_{i_{exp}}\) it is needed to know \(j_{i}(\theta,\phi)\), where \(\theta\) is the latitude and \(\phi\) the longitude. When the current density is solely recorded in  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Material** & **Sputtering Yield (200 V)** & **Sputtering Yield (250 V)** \\  & \(Atoms/ions\) & \(Atoms/ions\) \\ \hline Carbon (Graphite) & 0.0018 & 0.0056 \\ Molybdenum & 0.13 & 0.18 \\ Aluminum & 0.11 & 0.16 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Sputtering yield coefficient computed for different discharge voltages with the software SRIM [64].  
a plane that contains the thruster axis, e.g., following the angle \(\theta\) from \(-\pi/2\) to \(\pi/2\), one can assume a cylindrical symmetry of the ion beam around the thruster axis to determine \(j_{i}(\theta,\phi)\) and compute \(I_{i_{exp}}\). It is in fact easier to use a cylindrical coordinate system to solve Equation (2). In that case the coordinate system is depicted in Figure 7 along with the elementary arc \(ds\). The thruster exit plane points toward the \(x\) axis. Measurements are performed at a fixed distance \(R\) and defined by the angle \(\theta\). Cylindrical symmetry implies a constant \(j_{i}\) inside the element with radius \(y\) and thickness \(ds\). The sum of these elements, each weighted with \(j_{i}(\theta)\), gives the ion current:  
\[I_{i_{exp}}=\int_{0}^{R}j(x,y)2 ryds, \tag{3}\]  
with \(j_{i}(x,y)=j_{i}(\theta)\). With the help of several mathematical conventions and simplifications [66] we obtain the general final form of the ion current:  
\[I_{i_{exp}}=2\pi R^{2}\int_{0}^{\frac{\pi}{2}}j_{i}(\theta)\sin(\theta)d\theta. \tag{4}\]  
This form is also used by [PERSON] et al. [17] in his recommended guidelines for use of Faraday probes.  
The divergence half-angle \(\theta_{div}\) refers to the width of the beam. It quantifies the beam deviation from a straight ion beam. The thrust is directly impacted, and it significantly decreases when the divergence angle gets large. The divergence half-angle \(\theta_{div}\) of the ion beam is defined as the angle for which the ion current corresponds to a given fraction of the total ion current. In general, the ratio is 0.95. Therefore, the half-angle is mathematically related to \(I_{i}\) according to:  
\[I_{i_{\theta_{div}}}=\pi r^{2}\int_{0}^{\theta_{div}}j_{i}(\theta)\cdot\sin( \theta)\cdot d\theta=0.95\cdot I_{i}. \tag{5}\]  
Equation (5) shows that the way \(I_{i}\) is calculated, as well as the treatment of the angular distribution of the ion current density (smoothing, fitting, filtering, interpolation), greatly influence the value of \(\theta_{div}\) for a given dataset.