Experimental measurements of a betatron difference resonance

Abstract

The betatron difference resonance, ${\mathit{Q}}_{\mathit{x}}$-2${\mathit{Q}}_{\mathit{z}}$=-6, where the ${\mathit{Q}}_{\mathit{x},}$z are the number of betatron oscillations per revolution, was studied at the Indiana University Cyclotron Facility cooler ring. Measurements of both vertical and horizontal coherent betatron oscillations were made, at a nonlinear resonance, after a pulsed dipole kick. We found that the Poincar\'e surface of section for the nonlinear resonance could be described by a simple Hamiltonian. The resonance strength and phase, as well as the tune shift, as a function of betatron amplitude, were deduced from the experimental data. Attempts to deduce the amplitude and phase of the time dependent fluctuations around the time averaged Poincar\'e surface of section will also be discussed.