# Analysis of Injected Electron Beam Propagation in a Planar Crossed-Field Gap
## 2 Theory of Crossed-Field Stability  
Figure 1 shows the simple, planar geometry adopted for this study. It consists of an anode, a sole, an extractor, and an injected electron source. The Hull cutoff magnetic field \(B_{H}\), the minimum magnetic field required to insulate the diode such that an electron emitted from the cathode just strikes the anode before turning back toward the cathode, is given by [17]. \[B_{H}=\sqrt{\frac{2 mV}{eD^{2}}} \tag{1}\]where \(m\) is the electron mass (kg), \(V\) is the applied bias voltage (\(V\)), \(D\) is the anode-sole gap distance (m), and \(e\) is the electron charge (C). For \(B>B_{H}\), the maximum current density \(J_{c}\), below which each electron executes a simple, cycloidal orbit in steady form, is given by [5],  
\[J_{c}=J_{\text{CL}}\bigg{(}\frac{9}{8\pi}\bigg{)}\bigg{(}\frac{B}{B_{H}}\bigg{)} ^{3}\bigg{(}1-\sqrt{1-\bigg{(}\frac{B_{H}}{B}\bigg{)}^{2}}\bigg{)} \tag{2}\]  
where \(B\) is the applied magnetic field (\(T\)) and \(J_{\text{CL}}\) is the planar, one-dimensional space-charge-limited current density, given by [25; 26],  
\[J_{CL}=\frac{4}{9}e_{0}\sqrt{\frac{2e}{m}}\frac{V^{3/2}}{D^{2}}, \tag{3}\]  
where \(e_{0}\) is the permittivity of the vacuum. As an example, using the geometry in Figure 1, for a 3 kV anode-sole voltage and a magnetic field of 0.05 T, the maximum stable current density is 94 mA/cm\({}^{2}\) for a gap distance of 20 mm. When \(J>J_{c}\), the cycloidal beam becomes unstable and deviates from a simple cycloidal trajectory. Introducing a magnetic field tilt can eliminate magnetic insulation in 1-D such that emitted electrons reached the anode even when \(B>B_{H}\) and collapse the cycloid orbits for \(J<J_{c}\)[10]. Defining a normalized transit time \(T\), the critical current density for an angle \(\psi\) between the magnetic and electric field is given by [11]  
\[J_{cr}=\frac{1}{f(T)}\frac{me_{0}\Omega^{3}D}{e}\, \tag{4}\]  
where \(f(T)=(\cos^{2}\psi)T^{3}/6+(T-\sin T)\sin^{2}\psi\) and \(\Omega=eB/m\). One may calculate \(T\) by solving  
\[\frac{|B|}{B_{H}}=\frac{f(T)}{\sqrt{g(T)}}\, \tag{5}\]  
where \(g(T)=\big{(}\cos^{2}\psi T^{2}/2+\sin^{2}\psi(1+\cos T)\big{)}^{2}+[\sin\psi (T-\sin T)]^{2}+(\cos\psi\sin\psi\)\((T^{2}/2\)  
Figure 1: The 3-D model for simulation. (**a**) Side view. (**b**) Top view. The sole length is 150 mm and the width is 100 mm. The anode-sole gap was fixed at 20 mm. Eight discrete emitters are embedded inside of the dielectric emitter holder. For the simulation, a magnetic field perpendicular to the anode-sole gap was applied. A similar model will be used for the planned experiment. The complete destabilization of an electron beam by a small tilt in 1-D can be understood by the acceleration of the electron parallel to the electric field, which is given by [11]  
\[a_{\parallel}=\frac{eV}{mD}\cos\psi(\cos\psi,0,\sin\psi) \tag{6}\]  
which shows that an emitted electron is accelerated to the anode when \(\psi<90^{\circ}\). In other words, the magnetic field tilt prevents magnetic field insulation and an emitted electron will eventually reach the anode. Instead of a cycloid, in which the emitted electron always returns to the cathode, as in the PCF case, the emitted electrons form loops as they traverse the gap due to the contribution of both vector components of \(B\). 1-D PIC simulations showed that the resulting loops were highly unstable at the points where the electrons changed direction, corresponding to the points of zero velocity in the electron flow across the gap. Slight perturbations in the applied current could, therefore, lead to instability due to charge buildup at these locations. Interestingly, PIC simulations indicated that increasing the current could lead to other stability bands with a different number of loops. This higher current stability range would also be very sensitive to perturbation in current. Thus, introducing a tilt would tend to cause electron flow to collapse into a near-Brillouin state similar to the PCF case, but without magnetic insulation [11] in 1-D.