Particle trapping during passage through a high-order nonlinear resonance

Abstract

A theory of particle trapping and transport during passage through a high order one-dimensional nonlinear resonance is developed. It is a dynamic theory of what is generally referred to as the resonance "lock-in" process. The main result is an expression for the trapping efficiency as a function of the resonance excitation width, the nonlinear detuning, and the speed of passage through the resonance. The trapping efficiency shows a characteristic exponential dependence on crossing speed and a dependence on phase*-space area «hich follows from rather general arguments connected with the invariance of local particle density. The question of what phase-space region trapped particles are drawn from is discussed. It is shown how the answer to this question is related to the adiabaticity of the system and the phase-space topology. The phenomenon of particle "explosions", occurring at phase-space amplitudes where nonlinear stabilization becomes ineffective, is investigated. The theory has been verified with a computer simulation for crossing a 5th order resonance.-3-CRISP 74-9 order resonances were sufficiently weak that "fast crossing" was the prevailing mechanism, then the trapping process would substantially decrease in significance, since the trapping quickly diminishes to zero with crossing speed. This, in fact, is probably a good description of the accelerator field before storage rings. With relatively fast tune variations and weak high order resonances in existing accelerators, it is not surprising that the mechanism of trapping, which is enhanced as tune variation diminishes in speed and resonance strength increased, was neglected, if not overlooked. This is not to say that resonances, along with fast crossing, do not cause beam growth. However, this subject has received attention elsewhere and we will not pursue it here.

Adiabatic theory of particle trapping due to Coulomb crossing of one-dimensional betatron resonance

“…For narrow nonlinear resonance the oscillations are characterized by the following parameters: 1) J 0 (value of action corresponding to the separatrix center); 2) Ω s (frequency of the particle linear oscillations around the separatrix center); 3) ΔJ s (the separatrix width). These parameters can be found using the technique given in [5]. Let us present the Hamiltonian H(J, φ) in the following form:…”

“…A character of the process depends on the ®adiabaticity parameter¯K ad = δJ s ΔJ s where δJ s is a shift of J 0 during the period of the particle oscillations around the separatrix center. This process is examined by A. Chao [5] for the particular case of theˇfth-order one-dimensional resonance. An estimate of the ®trapping efˇciency¯is made in assumption that all particles with K ad 1 are trapped and all particles with K ad > 1 are nontrapped.…”

“…Strictly speaking, condition of the adiabaticity is K ad 1; however, here we will use Chao's condition [5] (K ad 1) to determine the capture area. Then we obtain a necessary condition for u 0 with account of the adiabaticity criterion (here b = 3/4πν s u t ):…”

Dynamic aperture and transverse proton diffusion in HERA

“…In this case the tune modulation is called adiabatic. 31 The adiabatic boundary can be derived from the intuitive condition 27, 28…”

Introductory statistical mechanics for electron storage rings

“…These simplifications allow substantial economies in running time which, for programs concerned with many particles and many turns of a large storage 41 ring lattice, can be well worthwhile.…”