Some understandings on radial motion at transition in the Fermilab Booster

The transition crossing is space charge dominated in the Fermilab Booster. Since the longitudinal space charge forces are defocusing below transition and focusing above transition, they generate the mismatch at transition, which causes the longitudinal emittance growth above transition. It's proved by numerical simulation that such mismatch can be partially compensated by a particular radial motion at transition, which is operationally favored by the high intensity beam. The transition crossing in Booster is space charge dominated. Usually, the nonlinear chromatic effect can cause the emittance growth during transition because particles with different energies cross transition at different times. The transition time is set by the synchronous particle; below transition, particles with positive energies relative to the synchronous particle become unstable since they are in the wrong phase, and above transition, particles with negative energies are unstable. The dependence of the transition energy upon the momentum deviation can be adjusted via different sextupole corrector settings such that the emittance growth due to the chromatic nonlinear effect can be greatly reduced. Fortunately, at the corrector setting of Isextl = -97 A and Isexts = 97 A, the dependence can be removed, see Figure 1. Space charge forces are mainly responsible for the longitudinal emittance growth at transition