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 Gγ = k, when the number of spin precessions per revolution, Gγ, equals a harmonic of the depolarizing field, k. Intrinsic resonances occur at Gγ = nP±νZ, 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 Gγ = nP±k', and are strongest when the driven betatron oscillation is largest. The effect was most dramatic at the strong Gγ = 27 resonance. Since νZ = 8.8 for the AGS, and there is a major nP = 36 AGS periodicity, a strong beat resonance should exist at Gγ = 36-9 = 27. Applying a 27th harmonic correction directly was unsuccessful, but a 9th harmonic correction removed the depolarization.