Debunching studies at energies close to transition

A very-low-intensity bunch of protons in the Fermilab Main Ring was brought to an energy close to but below transition (frequency-dispersion factor η0 approx-gt - 2 x 10-5) at a front porch. The rf was turned off abruptly. The debunching was recorded in multi-trace oscillograms (''mountain-range plots'') with ∼ 20 ms delay between traces. The front-porch energy was increased in very small steps in subsequent cycles to about η0≅ 8 x 10-5 above transition and the debunching repeated. The transition gamma could be determined very accurately by comparing the debunching rates at different energies. We observed longitudinal spread at every energy, indicating the presence of nonlinear effects in the frequency dispersion. The second term in the momentum-offset expansion of the momentum compaction, namely α1, was determined