Influence of an external static magnetic field on prebreakdown electron emission and heating

TL;DR

This study analyzes vacuum arcing under concurrent strong electric and static magnetic fields, focusing on anodic initiation driven by magnetic focusing of field-emitted electrons. It combines analytic solutions for electron motion in parallel and perpendicular field configurations with a multi-physics PIC-MCC and FEM workflow to quantify anode heating in a Cu plate-to-plate geometry ($d=1.5$ mm, $V_0=80$ kV). Results show that magnetic fields of order $10$–$30$ T can dramatically increase local current density on the anode and raise temperatures to the range of $7{,}000$–$11{,}700$ K, enabling evaporation and potential plasma formation, particularly in the $E \parallel B$ configuration. The findings suggest anodic arcing as a plausible pathway for vacuum breakdown in high-field environments such as muon-collider RF cavities and provide design guidance to mitigate breakdown risk by considering field orientation and heating dynamics.

Abstract

High magnetic fields can increase the occurrence of vacuum arcing, suggesting that both electric and magnetic fields can play a role in the vacuum arcing process. The mechanism of vacuum arcing in high magnetic fields is believed to involve both the cathode and the anode, with the cathode serving as the originator of field-emitting nanoprotrusions or tips, while the anode serves a secondary role. Significant heating of the anode surface can be achieved by magnetic focusing of the emitted electron beam, leading to increased heat flux due to greater current density. We simulated the emitted electron beam in different configurations of the electric and magnetic fields using the particle-in-cell (PIC) and finite element methods (FEM). The heating caused by the impacting electron beam was simulated for magnetic fields ranging from 0 T to 30 T. The directions of the electric and magnetic fields were found to play a major role in the focusing of the electron beam. We found that a sufficient temperature increase on the anode surface for evaporation can be reached at magnetic fields on the order of 10-30 T, suggesting the possibility of plasma initiation on the anode side.

Figures (9)

• Figure 1: The geometry of the system. The parallel electrodes (the cathode is shown in blue and the anode is in red) are separated by the gap length $d$. The blue and red arrows show the directions of electric $\bm{E}$ and magnetic $\bm{B}$ fields. A nanotip is placed on the surface of the cathode. The inset shows the height $h$ and the curvature radius $r_\text{tip}$ of the nanotip, the $X$ and $Y$ indicate the Cartesian coordinates used in the calculations. The initial velocity of the electrons $v_{r0}$ is due to the field enhancement around the nanotip.
• Figure 2: Top and side view schematic representation of the electron beam dynamics ejected from the cathode tip (blue) for both cases: $\bm{E} \parallel \bm{B}$ and $\bm{E} \bot \bm{B}$, with theoretical parameter illustrations. The purple lines represent the trajectories of individual electrons. The electric field component $E_y$ and the charge-to-mass ratio $q_e/m_e$ are negative, while the magnetic field components $B_y$ (for $\bm{E} \parallel \bm{B}$) and $B_z$ (for $\bm{E} \bot \bm{B}$) are positive. The initial velocity vectors $\bm{v}_{r0}$ exhibit radial symmetry around the center of the tip in both cases. $r_\text{max}$ is the radius of the amplitude of the helix trajectory of the electrons or the maximal deviation of electron trajectory from the axis of the tip for $\bm{E} \parallel \bm{B}$, $y_{\text{max}_\text{u}}$ and $y_{\text{max}_\text{l}}$ define the maximum thickness of the spiral trajectory for for $\bm{E} \bot \bm{B}$ and $\alpha_\text{max}$ is the beam's opening angle for $\bm{E} \bot \bm{B}$.
• Figure 3: Electron beam quantities in the parallel case ($\bm{E} \parallel \bm{B}$) on the anode surface, calculated based on theory by assuming an emitted current of $I_e = 1 \ \text{mA}$.
• Figure 4: Modulation of the electron beam shape at different magnetic fields in the case of parallel electric and magnetic fields ($\bm{E} \parallel \bm{B}$). In the legend, "Theory" refers to the results of Eq. \ref{['eq:parallel']} and $r_\text{max}$ is the radius of maximal radial deviation from the tip axis for the $\bm{E} \parallel \bm{B}$ geometry (Fig. \ref{['fig:schem']}).
• Figure 5: Modulation of the electron beam shape at different magnetic fields in the case of perpendicular electric and magnetic fields ($\bm{E} \bot \bm{B}$). The top and bottom rows show the side and top views of the beam shapes for the corresponding values of the magnetic fields. In the legend, "Theory" refers to the results of Eq. \ref{['eq:perpendicular']} and $y_\text{max}$ is the maximal vertical deviation of the electron trajectory for the $\bm{E} \bot \bm{B}$ geometry (Fig. \ref{['fig:schem']}).