## 3 Faraday Cup
### Architecture
The on-axis electron density and temperature in the ISCT200 plume far-field are typically \(\approx 10^{16}\) m\({}^{-3}\) and \(\approx 3\) eV [55]. It gives a Debye length \(\lambda_{D}\) close to 0.1 mm [11]. Due to the relatively large voltage applied to the collector \(eV\gg k_{B}T_{e}\), where \(k_{B}\) is the Boltzmann constant, the sheath thickness can be estimated assuming the current is space charged limited. Therefore, the sheath is governed by the Child-Langmuir law [56], and the sheath thickness \(s\) is given by:  
\[s\approx\lambda_{D}\bigg{(}\frac{|V|}{\tilde{T}_{e}}\bigg{)}^{3/4}\approx 1\, \text{mm}\ll l_{cup}. \tag{1}\]  
A small FC also reduces the amount of sputtered material in the plasma beam and it minimizes flow disturbances. The cylinder plus the collector are maintained at a potential well below the floating potential to capture positively charged particles. The cup design limits the perturbation introduced by the SEE process: the cylindrical geometry allows the  
Figure 5: Cross section view showing the main components of the Faraday cup, collimator geometries and materials used in this study. \begin{table}
\begin{tabular}{c c} \hline
**Diameter** & **Solid Angle** \\
**mm** & \(\bullet\) \\ \hline
10 & 11.4 \\
7 & 8.0 \\
5 & 5.7 \\
3 & 3.4 \\
1 & 1.1 \\ \hline \end{tabular}
\end{table} 
Table 3: Solid angle for a given collimator diameter. capture of most of the secondary electrons emitted by the collector surface. The length minimizes but does not completely compensate SEE effect. The FC is encapsulated into a cylindrical aluminum pod (4) to protect the external PEEK insulator from ion bombardment and from the thermal load and to prevent ambient plasma to disturb cup measurements. The pod is grounded with respect to the local plasma potential (i.e., facilities ground). The pod (or housing) is also used to attach the probe to the rotating arm. The spacers (5) act as electrical insulators. They electrically uncouple the cup from the collimator, as graphite and molybdenum are conductors, same as the housing. The size is chosen to reduce the occurrence of short-circuit between the collimator and the cup. The back of the cup is used to implement and accommodate wires and connections. Figure 5 displays the different designs tested in this work. To prevent confusion a nomenclature (ID) is used to refer to a specific FC design. The ID is split into three parts (\(X.X.X\)). The first component gives information about the material used to equip the front of the FC. It can be either G, Mo, or Al which respectively correspond to graphite, molybdenum and aluminum. The second component refers to the diameter of the inlet aperture. Therefore, one can have numbers 01, 03, 05, 07 and 10. Finally, the last part of the ID indicates information whether the collimator is exposed to or protected from the beam, using the letter E and P respectively. So, designs labelled \(X.X.P\) differ from designs \(X.X.E\) as the housing aperture diameter decreases while the collimator diameter behind becomes larger. Please note that with configuration \(X.X.P\) the collimator can be grounded or biased while in configuration \(X.X.E\) the latter is always grounded (see Section 6.3). As an example, the FC \(G.07.E\) corresponds to a collimator made of graphite with an aperture diameter of \(7\,\mathrm{mm}\) and is exposed to the beam.