# Analysis of Injected Electron Beam Propagation in a Planar Crossed-Field Gap
## 4 Simulation Results
### Effect of Magnetic Field Tilt  
We next examined the effect of a tilt in the magnetic field to compare to the analytic 1-D theory [11], which showed that tilting \(B\) in the \(x\)-direction causes an instability at \(I=J_{cr}\) given by Equation (4), which is lower than the PCF critical current density \(J_{c}\) defined in Equation (2). From Equation (4), tilting the magnetic field by \(1^{\circ}\) and \(3^{\circ}\) induces instability at \(0.24J_{c}\) and \(0.3J_{c}\), respectively. Simulations were carried out using a current density just below the maximum stable current density \(J_{c}\) (<100 mA/cm\({}^{2}\)) described in Figure 8, but the magnetic field was tilted with respect to the anode-sole gap. Figure 10 shows beam trajectories for (a) \(0^{\circ}\), (b) \(1^{\circ}\), and (c) \(5^{\circ}\) magnetic field tilts with respect to anode-sole gap. Figure 10(b) shows that a \(1^{\circ}\) tilt of the magnetic field completely destabilizes the beam, and a \(5^{\circ}\) tilt (Figure 10c) almost rotates the beam after the instability described by Equation (4). To demonstrate the transition of propagation from stable to unstable, the case of a \(3^{\circ}\) tilt was studied where Equation (4) gives \(J_{cr}\) = 0.3\(J_{c}\). Two simulations were carried out using input currents of 20 mA and 50 mA for a \(3^{\circ}\) tilt. Figure 11a shows that a 50 mA input current gives \(J\approx 39\) mA/cm\({}^{2}\) in the center of the gap which is greater than \(J_{cr}\approx 28\) mA/cm\({}^{2}\) (adjusted critical current density). The electrons form nonuniform orbits as they traverse to the anode, which ultimately forms an unstable beam trajectory. However, a 20 mA input current yields \(J\approx 18.72\) mA/cm\({}^{2}\), which is lower than the adjusted critical current density \(J_{cr}\). Figure 11b shows that the orbits are nominally stable. Figure 9: Phase space plot at \(y\)-\(z\) plane for (**a**) \(B\) = 0.05 T, \(D\) = 20 mm, \(I\) = 1.5 mA demonstrating a stable orbit and (**b**) \(B\) = 0.05 T, \(D\) = 20 mm, \(I\) = 250 mA demonstrating an unstable orbit indicating unstable beam propagation. The phase space data are obtained at the \(x\) = 0 plane. Additionally, 1-D PIC simulations [11] previously demonstrated the presence of bands of instability and stability for various magnetic field misalignments and current densities. For similar parameters (\(B\) = 0.05 T, \(D\) = 20 mm) and a 5\({}^{\circ}\) tilt, 1-D PIC simulations [11] showed stable flow for \(0<J<0.133J_{c}\) and \(0.33J_{c}<J<0.4J_{c}\), but unstable flow between those bands, where \(J_{c}\approx 100\) mA/cm\({}^{2}\). This means that current densities between 33 mA/cm\({}^{2}\) and 40 mA/cm\({}^{2}\) would be stable for a 5\({}^{\circ}\) tilt. Figure 12 shows the simulation results for an injected beam of (a) 50 mA, (b) 70 mA, and (c) 90 mA, which, at the center of  
Figure 11: CST simulation for \(B\) = 0.05 T, \(D\) = 20 mm, and a 3\({}^{\circ}\) tilt for an input current of (**a**) 50 mA (left: center cross section, right: top view) and (**b**) 20 mA (left: center cross section, right: top view). The color intensity of the trajectories indicates the kinetic energy of the electrons. Figure 10: CST simulation showing a stable beam with \(B\) = 0.05 T, \(D\) = 20 mm, and \(I\) = 150 mA with (**a**) no tilt, (left center cross section. right: top view); (**b**) 1\({}^{\circ}\) tilt (left: center cross section. right: top view) and (**c**) 5\({}^{\circ}\) tilt (left center cross section. right: top view). The color intensity of the trajectories indicates the kinetic energy of the electrons. the gap, produces an effective current density of \(J\approx 26\) mA/cm\({}^{2}\), \(J\approx 36\) mA/cm\({}^{2}\), and \(J\approx 43\) mA/cm\({}^{2}\), respectively, for a 5\({}^{\circ}\) tilt. Among these three current densities, only \(J\approx 36\) mA/cm\({}^{2}\), which falls between \(0.33J_{c}\) (\(\approx\)33 mA/cm\({}^{2}\)) and \(0.4J_{c}\) (\(\approx\)40 mA/cm\({}^{2}\)), produces stable beam propagation, which matches prior 1-D PIC simulation results [11].