Magnetically Insulated Diode: Existence of Solutions and Complex Bifurcation. I

TL;DR

This work develops a reduced, self-consistent model for magnetically insulated diodes by starting from a singularly perturbed 1.5D Vlasov–Maxwell system and deriving a limit system for the electrostatic and magnetic potentials. It proves existence of physically admissible nonnegative solutions for the effective potential on the insulated interval and then plunges into the fully insulated regime, where a cubic equation for the insulated potential governs the turning-point location and spacing. By analyzing the discriminant and providing explicit algebraic and trigonometric solutions, the authors classify the bifurcation structure and construct detailed diagrams mapping solution multiplicity as a function of key boundary and regime parameters. The combination of contraction-mapping arguments, Fredholm integral reformulations, and comprehensive numerical bifurcation analysis yields a unified analytical–numerical framework that clarifies the transitions between insulated and noninsulated states and informs practical design of MID devices, including estimation of insulated spacing.

Abstract

In order to avoid the electron oscillation of the cathode and enhance the work efficiency of a vacuum diode, an approach for analyzing the solutions and complex bifurcation has been proposed and used to determine the optimal trajectory of electron motion of the vacuum diode. This work is focusing on the stationary self-consistent problem of magnetic insulation in a space-charge-limited vacuum diode, modeled by a singularly perturbed 1.5-dimensional Vlasov-Maxwell system. We focus on the insulated regime, characterized by the reflection of electrons back toward the cathode at a point $x^{*}.$ The analysis proceeds in two primary stages. First, the original Vlasov-Maxwell system is reduced to a nonlinear singular system of ordinary differential equations governing the electric and magnetic field potentials. Subsequently, this system is further reduced to a novel nonlinear singular ODE for an effective potential $θ(x).$ The existence of non-negative solutions to this final equation is established on the interval $[0, x^{*})$, where $θ(x)>0$. This is achieved by reformulating the associated initial value problem into a system of coupled nonlinear Fredholm integral equations and proving the existence of fixed points for the corresponding operators. The most significant and previously unexplored case occurs when $θ(x)<0$ on the interval $(x^{*}, 1]$, which corresponds to the fully insulated diode. For this regime, we present a novel numerical analysis of complex solution bifurcations, examining their dependence on system parameters and boundary conditions. Bifurcation diagrams illustrating the solution $θ(x)$ as a function of the free boundary $x^{*}$ is constructed, and the insulated diode spacing is determined.

Figures (15)

• Figure 1: (a) Experimental configuration illustrating the magnetically insulated regime, where the cathode and anode are separated by a magnetic field that restricts electron motion. (b) Corresponding schematic of the effective potential $\Theta(x)$. Electrons emitted from the cathode at $x=0$ move within the region where $\Theta(x)>0$, reaching the turning point $x^{*}$ where $\Theta(x^{*})=0$ before being reflected back. For $x>x^{*}$, the potential becomes negative ($\Theta(x)<0$), representing a forbidden region where no electron trajectories can exist.
• Figure 2: $u$ solutions given $\hat{k}=-\sqrt{3}$.
• Figure 3: $u$ solutions given $\hat{\beta}=-\frac{\sqrt{3}}{9}$.
• Figure 4: $u$ solutions given $\hat{k}=0$.
• Figure 5: $u$ solutions given $\hat{\beta}=0$.

Theorems & Definitions (26)

• Definition 4.1
• Proposition 4.1
• proof
• Lemma 4.1
• proof
• Remark 4.1
• Proposition 4.2
• proof
• Lemma 4.2
• proof