Single-pass BBU instability in a solenoidal field within a 4D hypercomplex space

Transverse instability of a multi-bunch beam in the presence of a longitudinal magnetostatic field and hybrid dipole modes is considered analytically within a single-section model. It incorporates resonant interaction with beam harmonics and eigenmodes, degenerated waves of different polarizations, and the Lorentz’ RF force contribution. The analysis is performed in a very compact form using a bi-complex i,j-space including four-component collective frequency ν˜ of the instability. Rotating polarization of the collective field is determined by ImiImjν˜, which is in agreement with experimental data. The other three components represent detuning of the collective frequency ReiRejv˜, the “left-hand”, and “right-hand” increments ImiRejv˜±ImiRejv˜ of the gyro-magnetic BBU effect. The approach can be applied for characterization of complex transverse dynamics and spin transport. The phenomenon mechanism studied here can be used in non-reciprocal, non-ferrite, high-power microwave devices.