The synchrotron sideband spin resonances are shown to arise from the kinematic effect of spin phase modulation with resonance strengths proportional to that of the primary spin resonance. We develop a method to analyze overlapping spin resonances and apply it to fit polarization data of SPEAR and data recently obtained from polarized beam experiments at the IUCF cooler ring. The implication of our analyses is that synchrotron sidebands can only be corrected by correcting its principle resonance. Furthermore, the effect of synchrotron sidebands in proton synchrotrons is to change the resonance phase without affecting the magnitude of the strength. ͓S1063-651X͑96͒04907-0͔
Halo formation for a test particle in a mismatched KV beam is studied. Parametric resonances of the particle Hamiltonian due to envelope modulation are studied with particular emphasis on period 2 resonance which plays dominant role in Halo formation. It is shown that the onset of global chaos exhibits a sharp transition when the amplitude of modulation is larger than a critical value which is a function of a single parameter, , i.e., the ratio of the space charge perveance to the focusing strength.
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