Rapid Crossing of a Resonance During Tune Modulation by Synchrotron Oscillations
It had become a standard practice to constrain the particle’s tune footprint while designing the ring lattice so that the particle tunes fit between harmful resonances that limit ring dynamic aperture (DA). This approach, known as “tune confinement”, puts tight limits on the magnitude of the tune shifts with amplitude and with momentum. The latter requires labor-intensive optimization of the off-energy dynamic aperture and control over the corresponding tune footprint for the large energy deviations and is a prerequisite to achieving reasonable lifetime if it is dominated by the Touschek scattering. As nonlinearities of the modern ring lattices are much enhanced as compared with the previous generation of synchrotron light sources, it is becoming more and more difficult to keep the off-momentum tune footprint confined. While one could reach reasonable on-energy DA suitable for injection into the machine, the off-energy DA optimization is much more complex and needs to satisfy more constraints so to reach large energy acceptance and secure decent lifetime at the high ring current. In certain cases when the lattice solution with the confined off-energy tune footprint cannot be found, one may ask whether crossing of a resonance stopband from off-momentum particle will lead to loss of this particle. Intuitively, if the particle crosses the stopband quickly, one may expect that the betatron oscillation amplitude will not increase substantially thereby keeping the particle within the machine acceptance. At the same time the tight tolerances of the modern lattice elements as designed and produced afford much narrower resonance stopbands being compared with the machines built decades ago. Under these circumstances the question above may be asked as what are the conditions for operating a ring with a large chromatic tune footprint, which crosses resonances, while not observing particle losses. In this paper we try to answer the questions above by studying the model of a particle with large initial momentum deviation (as, for example, driven by the Touschek scattering event), which crosses stopband of a half-integer resonance while executing synchrotron oscillations in the highly chromatic ring lattice.