Simple method to calculate the operating frequency of a helical resonator-RF discharge tube configuration

A practical technique is described to estimate the resonant frequency of a helical resonator surrounding a coaxially mounted discharge tube. A simple formula is derived that relates the resonant frequency of the cavity to dimensions and dielectric properties of the discharge tube and its contents. An experimental test is performed on a resonator used to power an RF discharge, designed to produce an atomic hydrogen beam for an atomic collision experiment. The formulation adequately predicts the resonant frequency for a helical resonator that contains a discharge tube and cooling water jacket.