Evolution of Beam Distribution in Crossing a Walkinshaw Resonance

摘要

The third-integer coupling resonance at v_x — 2v_z = ?, known as the Walkinshaw resonance, is important in high-power accelerators. We find that, when the betatron tunes ramp through a Walkinshaw resonance the fractional emittance growth (FEG) is a universal function of the effective resonance strength: G_1.-2.?√∈_xi|Δ(v_x-2v_z)/Δn|~(-1/2), where G_(1-2,?) is the resonance strength; ∈_xi and ∈_zi are the initial horizontal and vertical emittances, respectively; and |Δ(v_x — 2v_z)/Δn| is the resonance crossing rate per revolution. At large effective resonance strengths, the FEG reaches an asymptotic maximum value (FEG)_max ~ 2∈_xi/∈_zi for ∈_xi 》1/2 ∈_zi or ∈_zi/(2∈_xi) for ∈_xi ?1/2 ∈_zl. There is little emittance exchange at ∈_xi = 1/2∈_zi, which can be used to minimize emittance growth in crossing a Walkinshaw resonance.