Bunches with Local Elliptic Energy Distributions

This distribution fits well with distributions observed in proton synchrotrons and makes several analytical calculations for bunched beams in longitudinal phase space possible. For any shape of the focusing force the line density becomes proportional to the potential well. Self-forces caused by space-charge and inductive wall impedances are thus proportional to the external force, making calculation of bucket area reduction and bunch lengthening easy. The microwave instability threshold, as given by the Keil-Schnell criterion with local values for current and energy spread, is independent of the azimuthal position along the bunch, and again analytical formulae are possible even for strongly non-linear focusing forces. The relative magnitude of the self-force and the microwave threshold turn out to be closely related, as the self-force is always 40% of the external force when the microwave threshold is reached. The classical longitudinal space-charge limit can therefore only be reached within a factor of 0.4. Other calculations with this "natural" distribution include analytical formulae for the rigid dipole mode threshold, and creation of flat-topped bunches with reduced peak line density resulting in a higher transverse space-charge limit.