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.