Empirical condition of betatron resonances with space charge

This paper addresses the generalization of the single-particle betatron resonance condition derived by Courant and Snyder more than a half century ago. A two-dimensional resonance condition including the effect of space-charge interaction was recently conjectured from one-dimensional Vlasov predictions made by Sacherer, Okamoto, and Yokoya [K. Ito et al., Phys. Rev. Aced. Beams 20, 064201 (2017)]. The condition is remarkably simple which only contains a few measurable quantities and indicates the possibility that twice as many resonance stop bands as expected from the conventional incoherent picture may exist at high beam density. Self-consistent multiparticle simulations are performed systematically to locate low-order stop bands in the tune diagram. The proposed betatron resonance formula is shown to explain the basic feature of numerical observations, which suggests that no serious incoherent resonance is activated inside the phase-space core of a dense beam matched to the external linear focusing potential. It is confirmed that the coherent tune-shift factor of any collective mode is less than unity and practically considered as a constant over the whole tune space in a typical high-intensity storage ring. The procedure for finding the optimum operating point of the ring is discussed on the basis of the coherent picture instead of the commonly used picture relying on the concept of incoherent tune spread. Despite years of theoretical efforts by many researchers, the coherent resonance concept is still not being employed for the construction of a stability tune diagram. We here provide a simple prescription to draw the diagram quickly. The present study also indicates the possibility of complete suppression of emittance exchange on particular difference resonances by choosing a proper emittance ratio.