A Deterministic Limit for RF Multipactor Avalanche: Deriving the Critical Swarm Radius via Surface Recombination

The prediction of catastrophic multipactor discharge in high-power radio frequency (RF) and microwave vacuum components—such as satellite communications payloads and particle accelerator waveguides—relies heavily on empirical Hatch-Williams susceptibility charts and statistical secondary electron yield (SEY) curves. While these frameworks estimate aggregate breakdown zones based on voltage, frequency, and gap distance, they fail to deterministically define the exact spatial boundary where localized secondary emission triggers an irreversible plasma avalanche. This paper introduces a strict continuum framework for vacuum electrodynamics. By modeling the RF cavity as a dynamic kinematic balance between the spatial capacity for charge absorption (surface recombination) and the localized rate of electron multiplication (secondary emission), we derive a universal critical multipactor radius (Rmultipactor). We demonstrate that RF breakdown in a vacuum is not a probabilistic resonance, but an exact deterministic limit where localized electron multiplication strictly overpowers the advective charge-absorption capacity of the surrounding metallic geometry. We propose a blueprint for Active Multipactor Telemetry (AMT) using ultra-fast phase-noise monitoring to provide real-time spatial prediction and signal dithering prior to catastrophic payload failure.