On Tomonaga's Theory of Split-Anode Magnetrons

TL;DR

This paper formulates the equations of motion for electrons in a cavity magnetron using action-angle variables, analyzes resonance conditions, and explores how electron ensembles form spokes and transfer energy to cavity oscillations.

Contribution

It introduces a canonical perturbation theory approach to model electron dynamics and resonance in magnetrons, linking classical electron motion to cavity field excitation.

Findings

Resonance conditions between electron and oscillatory frequencies identified.

Electron ensemble forms spokes that facilitate energy transfer.

Secular changes in electron paths enable electrons to reach the anode.

Abstract

This article is meant to formulate the equations of motion of an electron in a cavity magnetron using action-angle variables. This means following the electron's path on its way from a cylindrical cathode moving toward a co-axial cylindrical anode in presence of a uniform magnetic field parallel to the common axis. After analyzing the situation without coupling to an external oscillatory electric field, we employ methods of canonical perturbation theory to find the resonance condition between the frequencies of the free theory w_r, w_phi and the applied perturbing oscillatory frequency w. A long-time averaging process will then eliminate the periodic terms in the equation for the now time-dependent action-angle variables. The terms that are no longer periodic will cause secular changes so that the canonical action-angle variables (J, delta) change in a way that the path of the electron will deform gradually so that it can reach the anode. How the ensemble of the initially randomly distributed electrons forms spokes and how their energy is conveyed to the cavity-field oscillation is the main focus of this article. Some remarks concerning the importance of results in QED and the invention of radar theory and application conclude the article.