Resonance Excitation by Distortion of the Ã\x9f-Function Coupled with a Single Beam Space Charge Force

It is shown how a single beam space charge force can combine with a lattice design having large ß variations, such as occurs in machines with low ß insertions, to induce resonance behavior. Although the single beam space charge force may be highly nonlinear, it will, by itself, excite no resonances, since its azimuthal Fourier decomposition is essentially composed of 0th harmonic. However, low periodicity, large ß variations are harmonically rich and can provide the necessary azimuthal harmonics for resonance excitation. The resonance characteristics of this type of system are developed. A strength parameter involving the linear tune shift and the ßmax value is introduced. In particular, two cases are discussed: (1) where the force arises from the beam self-field and (2) where the force is induced by images in the surrounding boundaries. A comparison is made with the beam-beam force in terms of both the strength parameters (related to the linear tune shifts), and the nonlinear resonance behavior.