Reply to “Comment on ‘Empirical condition of betatron resonances with space charge”’

Abstract

In the Comment on our paper [Phys. Rev. Accel. Beams 22, 074201 (2019)], Hofmann claims that our conclusions are based on a questionable interpretation of Vlasov's equation and an over-interpretation of our multiparticle simulations. This assertion, however, comes largely from his misinterpretations of the essence of our work and proposed resonance condition. His criticism based on experimental data from some operating machines has also missed the point; we see no essential conflict between our arguments and experimental observations. While most of the questions raised by Hofmann have already been answered in our original paper above and previous publications, we take this opportunity to provide more information and explanation for clarity.

Space charge, nonlinear resonances, and beam loss

“…We finally remark that all this discussion relies on treating the dynamics from a frozen point of view. The role of direct space charge collective modes and collective resonances [22][23][24][25] is not included in this discussion, and their effect on beam loss is open to future studies.…”

Revisiting coherent and incoherent resonances with space charge

“…With the latter they suggest a doubling of the number of lines compared with low intensity (see also critical discussions on this topic in refs. [13,14].…”

Novel tabletop experiment demonstrating the nonlinear resonance excitation observed at the CERN Proton Synchrotron

“…( 2) applies only to the tail particles oscillating around the dense beam core. [13][14][15] Unlike the collective effect in the core, the incoherent effect is insensitive to the details of the beam configuration in phase space. Considering the real-space profile of the beam focused by the quadratic external potential, the incoherent resonance should be relatively severe for even k and ℓ (unless driven by some external source or the beam core is strongly distorted in an asymmetric way).…”