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With a perfect two-fold superperiodicity in a storage ring, the half integer resonance is not excited and can be overlapped by the beam tune spread induced, e.g., by beam-beam collisions in a collider or space charge in a hadron storage ring, without any detrimental effect on the beam.
If the superperiodicity of the quadrupolar lattice is broken, either by design or due to machine imperfections, a resonance stop band is created, which may lead to unstable particle motion.
Guided by historical literature, analytical calculations, and toy simulations, we explore the relation between single-particle motion and optics near the resonance. We discuss possible mitigation approaches including damping and cooling processes that suppress residual instabilities, defining pertinent tolerances for optics errors, and the construction of tuning knobs which can reduce the resonance strength. We also discuss fundamental and practical limits to resonance compensation.
Our discussion here focuses on the linear transverse optics. Chromatic resonance effects can be analysed in a similar way.
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