Particle trapping during passage through a high-order nonlinear resonance

New Approach to Resonance Crossing

“…c tÞ þ b n sinðn! c tÞ ; (12) where N h denotes the maximum number of harmonics. The particular values of the complex coefficients a n and b n depend on the detailed shape of the electron pinch and also on the island to which the fixed point belongs.…”

“…Trapping or detrapping of a moving object into a resonance occurs during the separatrix crossing process [1,2]. This mechanism is found in almost all areas of physics, for example, in tokamaks [3], in planetary science [4][5][6], in the orbital evolution of asteroid fragments [7], in fluid dynamics [8], in spacecraft research [9], in ion traps [10], and also in accelerator beam physics [11][12][13][14][15][16].We here consider resonance-crossing phenomena occurring in an accelerator for a proton beam afflicted by an electron cloud [13,17], which are of relevance, e.g., for the facility for antiproton and ion research in Europe project (FAIR) [18] and for the Large Hadron Collider [19]. During the passage of a proton bunch, the cloud electrons perform nonlinear oscillations in the nonlinear beam potential, which induces a particular ''pinch'' structure of the negatively charged electron cloud.…”

“…Trapping or detrapping of a moving object into a resonance occurs during the separatrix crossing process [1,2]. This mechanism is found in almost all areas of physics, for example, in tokamaks [3], in planetary science [4][5][6], in the orbital evolution of asteroid fragments [7], in fluid dynamics [8], in spacecraft research [9], in ion traps [10], and also in accelerator beam physics [11][12][13][14][15][16].…”

“…Historically, particles have been considered trapped into a resonance island if the particle coordinates Y 0 follow the fixed point of the frozen system A 0 ðNÞ [11,12]. Namely, when the migration speed of A 0 ðNÞ is very slow with respect to the maximum speed of rotation, v r ðNÞ ¼ S 0 ðNÞ , of particles inside the island of the frozen system, an adiabatic regime is reached and particles follow the migration of the island [12].…”

“…Namely, when the migration speed of A 0 ðNÞ is very slow with respect to the maximum speed of rotation, v r ðNÞ ¼ S 0 ðNÞ , of particles inside the island of the frozen system, an adiabatic regime is reached and particles follow the migration of the island [12]. The ratio of these two velocities, namely, T ¼ j _ A 0 ðNÞj=v r ðNÞ, defines an adiabatic parameter.…”