Evolution of beam distribution in crossing a Walkinshaw resonance

The third-integer coupling resonance at ν(x)-2ν(z)=ℓ, 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,ℓ)√[ε(xi)]|Δ(ν(x)-2ν(z))/Δn|(-1/2), where G(1,-2,ℓ) is the resonance strength; ε(xi) and ε(zi) are the initial horizontal and vertical emittances, respectively; and |Δ(ν(x)-2ν(z))/Δn| is the resonance crossing rate per revolution. At large effective resonance strengths, the FEG reaches an asymptotic maximum value (FEG)(max)~2ε(xi)/ε(zi) for ε(xi)>>1/2ε(zi) or ε(zi)/(2ε(xi)) for ε(xi)<<1/2ε(zi). There is little emittance exchange at ε(xi)=1/2ε(z), which can be used to minimize emittance growth in crossing a Walkinshaw resonance.