Loss Of Longitudinal Landau Damping in the LHC Injectors

A large number of collective instability mechanisms act on high-intensity beams. It is necessary to determine under what conditions the beam will remain stable. Space charge is the most fundamental mechanism and it represents the main intensity limitation in low-energy machines, while at high energy the inductive chamber impedance is often dominant. Landau damping provides a natural stabilizing mechanism against collective effects, if particles in the beam have a small spread S in their natural frequencies. The purpose of this report is to study the loss of Landau damping for the longitudinal plane via the â€ﾜSacherer formalism”. Stability limits are calculated for several longitudinal beam distributions, including two types of flat bunches, which could be of interest to the LHC upgrade. Landau stability diagrams are computed and presented for different azimuthal modes. A general recipe is given for calculating the threshold intensity in the case of a capacitive impedance below transition or, equivalently, for a purely inductive impedance above transition. Results are finally applied to the case of the PS Booster, as an example of space-charge impedance below transition, and to the SPS, as an example of inductive impedance above transition.