# Simplified Optimization of the Magnetic Configuration of HEMP-Thrusters
## 3 Results  
After the validation of the package, further numerical diagnostics can be used to characterize the properties of the magnetic configurations with respect to different aspects of physics. In the following, different geometries are calculated, where the length \(L\) varies between \(10\) mm and \(90\) mm in \(2\) mm increments for a fixed radius \(R=20\) mm. For each geometry, the field and \(20\) selected field lines are computed. Visually, this procedure is transcribed in Figure 3. Two measures are introduced in order to characterize the geometry. The subsequent figures demonstrate different properties of the magnetic flux \(\mathbf{B}\). Electrons in the HEMP-Thruster are magnetized and follow the field lines. Therefore, long field lines and the confinement by the magnetic mirror effect at the cusps increase the ionization and thereby the efficiency of the thruster. The different geometries are compared with respect to the field lengths by field line tracing. The geometries shown in Figure 3 illustrate the field configuration change when the length \(L\) is varied. One observes different properties of the magnetic field for the short geometry, Figure 3a, compared to intermediate, Figure 3b, or long magnets, Figure 3c. Relative to the length of the ring magnet, the field lines protrude more into the vacuum outside the magnet for smaller magnets. A measure \(\widetilde{z}\) of how far the concerning field lines elongate along the \(z\) axis is plotted in Figure 5. The \(z\) coordinate at the axial symmetry (\(r=0\)) where the axial magnetic field \(B_{z}\) changes direction defines the position \(z_{sep}\) of the separatrix of the magnetic field. Thus, a relative distance \(\widetilde{z}\), how far the separatrix moves from the channel axially, is defined with respect to the channel length (\(z_{sep}-L/2\))/\(L\). The position of \(z_{sep}\) has been marked for the three cases in Figure 3 with a black star (\(\bullet\)). For the three panels, the position relative to the exit \(\widetilde{z}\) varies greatly with length \(L\). The blue curve \(\widetilde{z}(L)\) in Figure 5 displays the relation for the whole scan. For smaller lengths, the relative elongation increases quickly. Longer designs have a nearly negligible protrusion. The exact placement influences and shapes the exit field of a HEMP-Thruster, which in turn will influence the angular distribution of the expelled ions and consequently the thrust [20]. Figure 4: Ratio of the radial to the total magnetic field \(\sin(\alpha)=B_{r}/B\) averaged over analytical sampling points within the box area (\(r\leq 0.1\cdot R\), \(z\leq 0.1\cdot L\)) for various configurations \(L\) and fixed radius \(R=20\) mm. The olive vertical line denotes the analytical solution, where the field lines are particularly parallel. A second observation from the field line plots in Figure 3 is the appearance of regions that are not accessible from field lines starting at the center of the cusp. These regions will have less filling by parallel transport and will depend strongly on radial transport effects. This is normally detrimental, but having such a small region here can even be beneficial for a HEMP-Thruster, because it can further reduce the wall contact of the plasma just to the cusp region, which is one of the design goals. The region that is not accessed by the field lines is called the passive region, whereas the region covered by field lines is called active. The active volume \(V_{active}\) is calculated as the volume spanned by the closest diagnostic field line to the axis. The ratio \(\widetilde{V}\) of the active volume to the total volume within the magnetic ring is also displayed in Figure 5. It is a measure of the volume that will be filled by electrons due to parallel transport along field lines connected to the near-axis region. The passive volume is filled by short field lines without contact with the central near-axis region.