The beam breakup instability in quadrupole and solenoidal electron-beam transport systems

Dispersion relations are derived to determine the growth rate, dominant wavelength, and group velocity of disturbances caused by the beam breakup instability. Considerations include weak and strong focusing, x-y coupling in solenoidal transport, the spacing of accelerator cavities, and periodically pulsed beams. Beam breakup growth is minimum when the cavity spacing equals an integral number of half-betatron wavelengths for quadrupole focusing, and an integral number of betatron wavelengths for solenoidal focusing. Minimum growth is also found for periodic pulses separated by an integral number of half-periods of the TMllo cavity mode. Expressions for beam breakup growth at the minima are obtained.