Threshold radius of Budker-Chirikov instability in electron and ion beams with strongly different radii

The Budker-Chirikov (1956,1965) instability (BCI) in a two-beam electron-ion system is researched. The BCI is due to the interaction of the slow cyclotron wave of the electron beam with the fast betatron wave of the ion beam. The electron-ion system consists of a high current relativistic electron beam propagating co-axially with a nonrelativistic ion beam in a drift tube. The external radii of the electron and ion beams are considered to be strongly different. The BCI increment decreases with the decrease in the ion beam radius when the electron beam radius is unchanged. The BCI resonance frequency is shifted in this case. Existing the threshold ion beam radius, smaller than that radius BCI does not develop. Now, the cause of threshold radius existing is investigated.