Self-Stabilization of Higher-Multipole Modes of Coherent Transverse Oscillations due to Nonlinear Coulomb Tune-Shift

Stability of coherent betatron oscillations of an intense coasting beam in a synchrotron is studied. Local space-charge field is treated by the Green’s function method. Analysis is made of the dipole mode, as well as of any higher-multipole mode of coherent oscillations in the trans-verse phase-plane. Having the same source, coherent and incoherent space-charge effects are studied concurrently, i.e. the problem is approached in a self-consistent way. Generalized threshold map technique shows that given the absence of nonlinearity of external guide field: (1) it is the dipole mode exclusively which may turn unstable, (2) coherent betatron tune shift exactly compensates the in-coherent one, and (3) the higher-multipole modes always occur well below Landau damping threshold, the latter be-ing introduced by the incoherent nonlinear tune shift itself.