Depolarizing ´beat´ Resonances in the Brookhaven AGS

While accelerating polarized protons in the Brookhaven AGS we found a variant of the standard imperfection and intrinsic depolarizing resonances which has some of the properties of both types. Imperfection resonances occur at G¿ = k, when the number of spin precessions per revolution, G¿, equals a harmonic of the depolarizing field, k. Intrinsic resonances occur at G¿ = nP±¿Z, when the AGS gradient periodicities, nP, modulate free vertical betatron oscillations to create the sum and difference frequencies. The variant resonance is a beat between nP and an imperfection driven betatron oscillation of periodicity k´. These occur at G¿ = nP±k´, and are strongest when the driven betatron oscillation is largest. The effect was most dramatic at the strong G¿ = 27 resonance. Since ¿Z = 8.8 for the AGS, and there is a major nP = 36 AGS periodicity, a strong beat resonance should exist at G¿ = 36-9 = 27. Applying a 27th harmonic correction directly was unsuccessful, but a 9th harmonic correction removed the depolarization.