Specifications for Third-Integer Resonant Extraction from the Cornell Synchrotron

Cornell’s proposed dark photon search (EAGER) requires a high-intensity positron source to generate statistically significant results. One available option is direct extraction of these positrons from the Cornell Synchrotron, which provides a potential 10 positrons per cycle, at 60 cycles per second. This corresponds to a time-averaged current of about 1 nA. However, single-turn extraction of the entire beam would overwhelm the detector by generating many concurrent events. Therefore, it is necessary to extract the beam over many turns. This report discusses the potential benefits and drawbacks of third-order resonant extraction, which is one of several potential slow extraction methods. When a particle beam has a tune near a third-integer, a localized sextupole moment can drive a resonance that shrinks the stable region of phase space in a controlled manner. This resonance can be understood intuitively by examining the horizontal transverse (or x-direction) phase space trajectory of a single particle with third-integer tune. Assume that the particle is initially located immediately before the sextupole element with an x betatron phase of zero. It first receives a strong positive kick from the sextupole, which has an x field dependence. However, since the particle advances by a phase of 2π/3 each turn, we can see that the next two kicks cancel each other out. After completing three full turns, the particle receives another strong positive kick from the sextupole. Allowing the emittance and initial betatron phase of the particle to vary, we find that the resonance limits the stable phase space to a triangular region which shrinks as the sextupole strength is increased or the accelerator tune is moved toward the third-integer. (In principle, this resonance has no stopband – there is always some region of stable phase space unless the tune is exactly a third-integer.) Since a particle beam will have a range of phases and emittances, we can use this fact to selectively extract particles from the beam at a rate of our choosing. As the stable phase space shrinks, particles with large emittances are extracted first, followed by particles with smaller emittances. If the process is sufficiently adiabatic, particles with the threshold emittance become trapped at fixed points located at the corners of the stable triangle. A small decrease in the region of stable phase space then causes these particles to stream out along well-defined trajectories, allowing for controlled extraction of the beam.