# Simplified Optimization of the Magnetic Configuration of HEMP-Thrusters
## 3 Results
Here, the radial transport of electrons from the central region populates this region slowly, and therefore, the electron density is reduced. The active volume decreases with the increasing length of the magnet and the passive volume increases, as expected from the magnetic field plots, which can be seen in Figure 3.  
The integration of several field lines, which correspond to the red lines in Figure 3a-c, at equidistant radii \(r_{0}\) delivers a length scale measure \(\ell\) of the magnetic field \(\mathbf{B}\), which is shown in the upper panel of Figure 7. The field line length is nearly proportional to the length \(L\). For small magnet lengths, the lengths \(\ell\) deviate from a linear dependence, since the bigger elongation in the \(z\) direction adds to the path. The average of the 20 field lines seems to be a good indicator for the qualitative behavior of \(\ell\). In future studies, the average alone might be sufficient for an optimization procedure. Further analysis is performed by a weight function reflecting the observed density distribution. A typical HEMP electron density distribution \(n_{e}\) of a PIC simulation for a xenon discharge is plotted in Figure 6. The code for the simulation was developed within the group. Further details about the physics and the PIC code can be found in [21]. PIC calculations can deliver a better understanding of the processes in HEMP. Figure 5: The relative distance of how far the separatrix elongates axially \(\overline{z}=(z_{sep}-L/2)/L\) and the ratio \(\widetilde{V}\) of the active volume \(V_{active}\) to the total volume \(V_{total}\) as a function of \(L\) are plotted. Both affect the performance of a HEMP thruster. The cusps show up as density sinks in the electron density in Figure 6 because the plasma-wall interaction is concentrated at these locations. They are clearly visible at the ends of the channel and at the cusp at z about 20 mm. The cusps act as magnetic mirrors and trap the electrons in between, such that the ionization probability is strongly enhanced, and larger densities in between the cusps are thereby obtained. For an axial position \(z_{0}\) between the cusps, a radial decay in density is observed due to the individual length of the field lines. The higher the radial position \(r_{0}\), the lower the electron density becomes. The weight function used in the following analysis is motivated by the density distribution obtained by PIC, which decays linearly and radially outwards in the logarithmic plot of Figure 6, so that for the contribution of the particle density, an exponential factor \(\exp\) -\(r_{0}\) is assumed. Because of the cylinder symmetry, field line numbers starting at the radial coordinate \(r_{0}\) scale with the circumference and are \(r_{0}\) times more abundant. The mirror effect at the end of the cylindrical magnets determines the electron confinement by limiting the losses and is responsible for enhanced ionization. It is calculated as the ratio of the magnetic fields at the starting point \(B_{0}=B(z_{0},r_{0})\) to the magnetic field at the endpoint \((z_{w},r_{w})\) of the field line close to the wall \(B_{max}=B(z_{w},r_{w})\)  
\[R_{m}=B_{max}/B_{0}\,.\]  
The ratio of the velocity components yields \(\sin(\theta)=1/\sqrt{R_{m}}\) of the velocity perpendicular to the magnetic field \(v_{\perp}\) to the total velocity \(v\). Then, the ratio \(\sin(\theta)\) gives the reflection condition \(\sin(\theta)\geq 1/\sqrt{R_{m}}\)[22; 23]. Magnetic field lines with a high magnetic ratio are likely to reflect electrons moving along these toward the cusps. The time \(\tau\) of the electrons spent between the mirrors is \(\tau=l/v\propto l\cdot\sqrt{R_{m}}\). A confinement time weighted by volume and geometry results in  
\[\tau=\ell\cdot\sqrt{R_{m}}\cdot r_{0}\cdot e^{-r_{0}}\,. \tag{3}\]  
Figure 7 shows individual field line lengths \(\ell\) for three selected field lines and the average of 20 lines. The magnetic field lines' lengths nearly scale linearly with the length of the magnet. Deviations occur for small lengths.