# Analysis of Injected Electron Beam Propagation in a Planar Crossed-Field Gap
## 4 Simulation Results  
### Effect of Magnetic Field and Current Density  
The instability study entails performing simulations for the parameters listed in Table 1. Specifically, we fixed the anode-sole distance at 20 mm, and characterize the variation in current density by varying the magnetic field strength from 0.01 T to 0.05 T with a step size of 0.01 T. Magnetic field variation was also used to examine the cycloidal beam and Hull cutoff. Figure 4 shows the simulated electron trajectories, demonstrating the cycloidal beam for magnetic field strengths of 0.01 T (\(B<B_{H}\)), 0.015 T and 0.02 T (\(B>B_{H}\)) for \(D\) = 20 mm. Increasing the magnetic field causes the beam to approach the Brillouin state as the cycloid radius decreases; however, the Brillouin instability is not present, and the cycloids are uniform. However, the magnetic field must be sufficiently strong to insulate the anode. We endeavor to operate with a sufficiently low current density to achieve stability for \(B>B_{H}\). For a 3 kV anode voltage, 1.5 mA of injected current, Equation (1) gives \(B_{H}\) = 0.009 T for \(D\) = 20 mm. Using \(D\) = 20 mm provides greater flexibility for experimental and simulation parametric studies. To ensure that we are operating within acceptable current density ranges for \(D\) = 20 mm, we used Equation (2) for 3 kV to show that 27 mA/cm\({}^{2}<J_{c}<94\) mA/cm\({}^{2}\) for our range of \(D\) and \(B\). These values set the range of our simulation runs for 0.01 T \(\leq B\leq\) 0.05 T.  
Comparing theoretical predictions of the instability of the crossed-field configuration to the simulation requires examining the electron trajectories in the simulations. Figure 5 shows three views of the simulation of electron trajectories for a 20 mm anode to sole gap. Figure 4: Beam trajectories for \(D\) = 20 mm, \(I\) = 1.5 mA, and applied magnetic fields of (**a**) 0.01 T, (**b**) 0.015 T, and (**c**) 0.02 T showing the case just below (0.01 T) and above (0.015 T and 0.02 T) \(B_{H}\). The color intensity of the trajectories indicates the kinetic energy of the electrons. Figure 5a provides a top view of the trajectory showing the beam propagating toward the end collector and the cycloidal pattern along both the magnetic field (x-direction) and the direction of propagation (z-direction) where the color intensity (energy in eV) is distributed uniformly. This also indicates stable propagation. Figure 5b,c, the cross-sectional scatter plots at the circuit middle and at the end collector, respectively, show that the trajectories are symmetric in the magnetic field direction with the maximum current density near the bottom of the structure. The converted current densities from the collected current at both of the cross sections are \(\approx\)1.45 mA/cm\({}^{2}\), which also confirms the uniformity. However, this location depends upon where the cut is taken along the cycloidal pattern. Figures 6 and 7 show simulation results using injected current of I = 100 mA and 250 mA, respectively. Figure 6a shows that the cycloidal beam is still relatively uniform over the sole for 100 mA current, and the two cross-sectional views for the circuit center and the end collector, shown in Figure 6b,c, respectively, exhibit only slight changes, indicating transition to instability. However, for the 250 mA injected beam case, the intensity plot (Figure 7a) of the beam shows a significant change in the intensity of trajectories along the device length, which indicates a disturbance in stability when compared to Figure 5. Additionally, the calculated current densities from the 2-D cross section scatter plots at  
Figure 5: Beam trajectories for \(B\) = 0.05 T, \(D\) = 20 mm, \(I\) = 1.5 mA (**a**) Top view, (**b**) cross section at center of the circuit, and (**c**) cross section at the end collector, in the x-y plane. The color intensity of the trajectories indicates the kinetic energy of the electrons. the center (Figure 7b) and the end collector (Figure 7c) differ when they should be similar, indicating instability.