Bunched beam longitudinal stability

Stability diagrams are calculated for various phase space distributions. It is found that distributions without tails are unstable in the negative mass regime (inductive impedance below transition or capacitive impedance above transition). It is also found that longitudinal instability thresholds of the (usually neglected) higher-order radial modes are lower than expected. For example, the next-to-lowest dipole mode has a lower threshold than the lowest sextupole mode, even though the latter has the larger growth rate in the absence of Landau damping.<>