Localized multibunch modes in accelerators

This paper will explore the possibility of the formation of localized modes in the coupled transverse oscillations of bunches in an accelerator. Such modes are characterized by a large amplitude excitation of one of the bunches, with little or no excitation of the other bunches. They are very similar to intrinsic localized modes predicted to appear in lattice excitations of anharmonic condensed matter systems. The existence of the phenomenon in accelerators requires long-range coupling between the bunches, and quartic terms in the effective potential experienced by the bunches. The discussion in this paper considers the specific cases of a long-range coupling due to the resistive wall impedance, and nonlinearities in the transverse force caused by octupoles. The general equations for the localized mode frequencies and amplitudes are derived. The equations are applied to the specific cases of 9 and 21 equally spaced, equally populated bunches in the Cornell electron storage ring. For both cases, with the currently available octupole strength, and with currents of 8 mA per bunch, localized modes with amplitudes of about 2 mm, and with frequency shifts of 50–100s^{-1} relative to the linear coupled-bunch mode frequencies, may be possible.