J. Budnick

Effects of rf voltage modulation on particle motion

Abstract The effects of rf voltage modulation on synchrotron motion were studied experimentally. The experimental data revealed the resonance islands generated by the rf voltage modulation. With electron cooling, beam particles were observed to damp to the basins of these resonance islands or attractors, which were observed to rotate about the origin of the phase space at a half of the modulation frequency. The measured amplitude of the attractors as a function of the modulation frequency agreed very well with the theoretical prediction. The Poincare maps in the resonance rotating frame were obtained from the experimental data and compared with tori of the Hamiltonian flow. Based on our theoretical formulation, slow beam extraction using rf voltage modulation and a bent crystal are also studied.

Experimental results of the betatron sum resonance

The experimental observations of motion near the betatron sum resonance, /spl nusub x/+2/spl nusub z/=13, are presented. A fast quadrupole (Panofsky-style ferrite picture-frame magnet with 8 pulsed power supplies) producing a betatron tune shift of the order of 0.03 at rise time of 1 /spl mu/s was used. This quadrupole was used to produce betatron tunes which jumped past and then crossed back through a betatron sum resonance line. The beam response as function of initial betatron amplitudes were recorded turn by turn. The correlated growth of the action variables, J/sub x/ and J/sub z/, was observed. The phase space plots in the resonance frame reveal the features of particle motion near the nonlinear sum resonance region.<<ETX>>

A method of detecting coherent synchrotron modes

Abstract A method for measuring coherent longitudinal synchrotron modes is developed and tested at the Indiana University Cyclotron Facility Cooler Ring. This method can be used to detect the onset of coherent instability and can provide important diagnosis for the control of beam brightness. Some possible improvement of this technique is discussed.

Determination of the linear coupling resonance strength using 2D invariant Tori

Experimentally obtained Poincare maps in the resonant precessing frame for particle motion with linear coupling revealed invariant tori of the 2D Hamiltonian. Using these tori, we obtained the linear coupling strength, the tune shift with betatron amplitude coefficient and the proximity parameter to the resonance. The coupling strengh obtained with this method agreed well with that obtained from measuring the betatron tune separation of the normal modes.

Longitudinal phase‐space measurements at IUCF

Synchrotron motion in the IUCF cooler ring was studied using turn‐by‐turn beam tracking on ten‐turn intervals, where the beam phase relative to the rf and the closed‐orbit position in a high‐dispersion region were measured. The synchrotron tune shift with amplitude was measured and is compared with theory. The driven response of the system was also studied using the same techniques, and was found to share many of the same characteristics of other parametric resonant systems. The experimental results did not exhibit any effects of bunch decoherence expected from the tune shift with amplitude.

Betatron coupling correction at the IUCF cooler, leading to improved determination of fourth‐order resonance Hamiltonian

Recently the Hamiltonian of particle motion near a resonance condition was deduced at the Indiana University Cyclotron Facility Cooler Ring. It was found that linear betatron coupling complicated the analysis. A coupling correction scheme is described which reduced the coupling coefficient from 0.03 to 0.0012. When particles were kicked onto resonance islands with the coupling reduced, the island motion was stable for upwards of 1×106 turns. This stable island motion allowed for the determination of higher‐order terms in the Hamiltonian.

Modeling of the IUCF cooler synchrotron

Measurements of lattice parameters for the Indiana University Cyclotron Facility (IUCF) Cooler synchrotron at betatron tunes close to a sum resonance line are discussed. The measured beta functions at 36 quadrupole locations and the dispersion functions at 35 Beam Position Monitor (BPM) locations are compared with calculations. The methods used to make the above measurements and the data analysis are described.

Beam motions near separatrix

Experimental data on particle motion near the separatrix of the one dimensional (1-D) fourth-integer islands are analyzed. When the beam bunch is initially kicked to the separatrix orbit, we observed a strong decoherence in the coherent betatron motion. We find that, through intensive particle tracking simulation analysis, the decoherence has resulted from the beam being split into beamlets in the betatron phase space. However, we also observe an unexpected recoherence of coherence signal, which may result form a modulated closed orbit or the homoclinic structure near the separatrix.

Parametric resonances and stochastic layer induced by a phase modulation

The Hamiltonian system with phase modulation in a higher harmonic RF cavity is experimentally studied on the IUCF cooler ring. The Poincare maps in the resonant rotating frame are obtained from experimental data and compared with numerical tracking. The formation of the stochastic layer due to the overlap of parametric resonances is discussed. The dependence of the stochastic layer on the voltage of the higher harmonic RF cavity, amplitude and frequency of the phase modulation is studied.

Asymmetric Hopf bifurcation for proton beams with electron cooling

We observed maintained longitudinal limiting cycle oscillations, which grew rapidly once a critical threshold in the relative velocity between the proton beam and the cooling electrons was exceeded. The threshold for the bifurcation of a fixed point into a limit cycle, also known as a Hopf bifurcation, was found to be asymmetric with respect to the relative velocity. This asymmetry of Hopf bifurcation was found to be related to the electron beam alignment with respect to the stored proton beam.