# Analysis of Injected Electron Beam Propagation in a Planar Crossed-Field Gap
## 4 Simulation Results
### Effect of Magnetic Field and Current Density
Determining the critical current density in the model requires converting the beam current to the current density at the center and the end of device. Figure 8 shows the current density at these two locations as functions of input (Figure 8a) and beam (Figure 8b) current. The input current is the current injected at the cathode source; however, some of this current is lost in the scrap off on the anode and electrodes. The beam current was calculated from the simulation at the measurement location. In Figure 8b, the current density is the same at both locations until the total beam current is approximately 150 mA, when the current densities begin to diverge. The local current density at this deviation is just below 100 mA/cm\({}^{2}\), which is close to \(J_{c}=94\) mA/cm\({}^{2}\) from Equation (2), indicating a change in device stability. The sharp transition occurs at a beam current of 275 mA and a current density of 100-125 mA/cm\({}^{2}\), which is also above \(J_{c}\) from Equation (2). Figure 6: Beam trajectories for \(B=0.05\) T, \(D=20\) mm, \(I=100\) mA: (**a**) top view, (**b**) cross section at circuit center and (**c**) cross section at end collector. The center and end cross section particle distributions look different; however, the current density is similar. The color intensity of the trajectories indicates the kinetic energy of the electrons. To quantify the instability from the 3-D particle trajectory, we analyzed a 2-D phase space cross-section (\(y\)-\(z\)), captured at the circuit middle (\(x\) = 0). The complete 2-D beam cycloidal trajectory in \(y\) and \(z\) is shown in Figure 9a,b for injected currents of 1.5 mA and 250 mA, respectively, with \(D\) = 20 mm and \(B\) = 0.05 T for both cases. Figure 9a shows the low current case, which is below the critical current density. Here, the electron orbits vary about a roughly straight line parallel to the z-axis, as indicated by the red line. For the higher input current case, shown in Figure 9b, the electron orbits are moving away from the sole as indicated by the red line. Although the orbits approach an unstable state where the magnetic field starts failing to isolate the beam and the beam starts to lose the uniform cycloidal pattern, this is not a near-Brillouin state [5; 11; 12]. First, this model is for an injected beam and not an emitting sole in which electrons are injected from the sole electrode. Classical theory states that the near-Brillouin state [31] occurs when sufficient space charge density builds up in the gap to make the electric field near the sole zero, causing cycloidal orbits of the newly entering electrons and, therefore, the whole flow, to become nearly laminar. In this injected beam model, electrons cycloid at a location above the sole and do not reduce the electric field at the sole surface. In addition, a new  
Figure 7: Beam trajectories for \(B\) = 0.05 T, \(D\) = 20 mm, \(I\) = 250 mA (**a**) Top view showing that the trajectories become unstable at the end of the structure. (**b**) Beam cross-section at the center of the device. (**c**) Beam cross section at the end collector. The color intensity of the trajectories indicates the kinetic energy of the electrons. study [32] argues that there is practically no threshold current density for this state to occur. Accordingly, the phenomenon observed here is a transition to unstable beam propagation rather than to a near-Brillouin state. The behavior for these cases was characterized by drawing best fit lines through the 2-D electron orbits, as indicated by the red lines. The slope of the line is 0.0047 with an intercept of 0.00531 for the 1.5 mA injected current case; whereas the slope is 0.01396 and the intercept is 0.00458 for the 250 mA injected current case. For a stable beam, the slope of the trajectory line (z-axis) goes through the center of the cycloids. As the cycloid becomes unstable, the slope increases, as shown in Figure 9b.