Longitudinal Stability of Flat Bunches with Space-Charge or Inductive Impedance
We study the loss of Landau damping for the longitudinal plane via the “Sacherer formalism”. Stability limits are calculated for several longitudinal beam distributions, in particular for two types of flat bunches, which could be of interest to the LHC upgrade. The resulting stability diagrams are computed and displayed 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. The formalism was 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. Contribution to the EPAC’08 Conference, Genoa (Italy), 23-27 June 2008 Work supported by the European Community-Research Infrastructure Activity under the FP6 “Structuring the European Research Area” programme (CARE, contract number RII3-CT2003-506395) Longitudinal Stability of Flat Bunches with Space-Charge or Inductive Impedance I. Santiago Gonzalez ∗, F. Zimmermann, CERN, Geneva, Switzerland
LONGITUDINAL BUNCHED-BEAM COHERENT MODES: FROM STABILITY TO INSTABILITY AND INVERSELY
Solving the dispersion relation for different distribution functions yields different stability boundary diagrams. A better insight into what physically happens is given in the case of an elliptical spectrum, which leads to a circular range of stability, considering only the most critical (dipole) mode. In this case, the coherent synchrotron frequency is expressed analytically as a function of the low-intensity small-amplitude synchrotron frequency, the incoherent and coherent frequency shifts, and the incoherent frequency spread. The evolution of the coherent synchrotron frequency with respect to the incoherent band can then be obtained in both unstable and stable regions. In the unstable region, Besnier’s picture [1] is recovered for the case of a capacitive impedance below transition or inductive impedance above transition. Finally, Sacherer’s stability criterion is extended to include the potential-well distortion. This result is then applied to the case of the LHC at top energy.
Longitudinal symmetric coupled bunch modes
The theory of longitudinal coupled bunch instabilities is discussed. Although some of the results included are not new, it is useful to summarize them here in a unified fashion.