# Analysis of Injected Electron Beam Propagation in a Planar Crossed-Field Gap
## 3 Crossed-Field Simulation Model  
The simulation geometry dimensions were selected carefully, so that the physical geometry and dimensions can be used in planned experiments. The experiments will include in situ measurements of the collector and anode currents and will use diagnostic probes to measure the electron beam in the gap. Future work will directly compare theory, simulation, and experiment. Along these lines, this simulation study and future experiments will consider a simple 10 cm wide and 15 cm long planar electrode system with a variable gap from 0.5 cm to 2 cm, as shown in the side and top views in Figure 1a,b, respectively. This design uses a separate cathode in a beam injection mode to launch electrons into the crossed-field space between a sole electrode and an anode. This simulation primarily studies three parameters: magnetic flux density \(B\), injected current \(I\), and magnetic field tilt angle \(\psi\). Table 1 provides simulation parameters and the corresponding variations examined in the study. We performed the simulations using the particle tracking solver, CST Studio Suite 2020 [27]. Table 2 shows the simulation input parameters. The initial simulation was carried out using the simple crossed-field model described before. A space charge limited emission model based on the 1-D Child-Langmuir law [28] with fixed beam current was used. To model the planned experiment, eight elements of discrete emitters embedded inside a dielectric holder were modeled, as shown in Figure 2a. The particle area sources were defined on the emitter surfaces. These emitters were designed based on a practical gated field emitter array (GFEA) reported elsewhere [29; 30] and planned for our experiment. A predefined magnetic field was applied perpendicular to the anode-sole gap. A hexahedral mesh optimization study was carried out, and over four million mesh cells were used to achieve solution convergence. Figure 2b shows the beam coming out from the emitter surface without a magnetic field. The sole electrode was fixed at 0 V while the electron \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameters** & **Value** & **Type of Study** \\ \hline Anode-Sole distance & 20 mm & Effect on minimum stable current density \\ Magnetic field & 0.01 to 0.05 T & Minimum magnetic field to insulate anode \\ Injected current & 1.5 mA to 1 A & Effects along with the operable range of \\ Magnetic field tilt & \(0^{\circ}\) to \(5^{\circ}\) & Effect of tilted magnetic field on beam stability \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Simulation parameters. source emitter was fixed at +200 V relative to the sole to ensure no cathode collection. The anode and the extractor were fixed at 3000 V in the simulation. Two models, with and without axial end-hats, were used to analyze beam spreading. Figure 3 shows cases with and without end-hats. The top-down view shows the electron trajectories and their spread from space charge. The highly cycloidal beams demonstrate the necessity of including end-hats to confine the beam axially. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Part** & **Applied Voltage** & **Injected Current** \\ \hline \hline \begin{tabular}{c} \(\mathrm{Sole}\) \\ \(\mathrm{Anode}\) \\ \(\mathrm{Emitter}\) \\ \end{tabular} & \
\begin{tabular}{c} \(\mathrm{0V}\) \\ \(\mathrm{3000V}\) \\ \(\mathrm{200V}\) \\ \end{tabular} &
\begin{tabular}{c} \(\mathrm{1.5\,mA}\) to \(\mathrm{1\,A}\) \\ \(\mathrm{200V}\) \\ \end{tabular} \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Applied parameters. Figure 3: Top down view of beam trajectories for CST model for (**a**) without and (**c**) with end-hats. Corresponding 2-D beam cross sections equally spaced in five locations over the sole is shown (**b**) without and (**d**) with end-hats. The applied magnetic field is 0.01 T with \(D=20\) mm and \(I=1.5\) mA. The color intensity of the trajectories indicates the kinetic energy of the electrons. Figure 2: Planar crossed-field model used in CST with \(D=20\) mm and \(I=1.5\) mA.