An analytical formula for the band width of a strong nonstructure octupole resonance induced by a space charge

In a high intensity proton synchrotron nonstructure octupole resonances are excited by a space charge force when its designed periodicity is broken by quadrupole errors. This paper gives an analytical formula to calculate a band width of a nonstructure octupole resonance induced by a space charge force. Two kinds of modulations, beam envelope modulation and betatron-phase modulation are taken into account as perturbations. An approximated treatment shows that the phase modulation takes on a more important role. The formula is applied to the KEK-PS, at which the band widths of nonstructure octupole resonances induced by the space charge force are much larger than that produced by magnetic octupole errors.