Theory of Longitudinal Instability for Bunched Electron and Proton Beams

We discuss an original approach for the treatment of the longitudinal stability of high-intensity proton and electron bunches. The general analysis is divided in three steps. First, we search for a stationary bunch distribution which is matched to the external RF forces as well as to the current dependent induced fields. We question the existence of such distribution. Second, we check the stability of the stationary solution by applying a small perturbation and observing whether this is initially damped or not. At this point a stability condition is derived in terms of current, surrounding impedance and bunch size. In the last step one should question what happens to the beam in case the stability condition is not satisfied. The problem here is the determination of the final bunch configuration. We will not deal much with this step. We observe that the "overshoot formula" which is derived from numerical calculation1 is usually applied to proton bunches2, whereas commonly the assumption is made an electron bunch matches always its size to the stability condition3. The originality of our approach stays in the combination of the three steps. All previous theories either consider only the first step4 or combine the second and third ones but disregard the first2,3,5. Sometimes, in the latter case, the modification of the potential well is introduced ad hoc3. Our theory applies to the case of a real frequency independent impedance.