Nonlinear wave phenomena in coasting beams

Abstract

In beams which are suf®ciently close to the linear stability limit, a variety of nonlinear wave phenomena are readily observed which can be used to diagnose aspects of the beam dynamics and the machine impedance. We have found that debunched beams in both the Fermilab Main Ring and the Tevatron are marginally stable to longitudinal oscillations and exhibit nonlinear three-wave coupling as well as nonlinear Landau damping and the formation of soliton-like perturbations. In addition, we have generated classical nonlinear echoes using twofrequency excitation in the Fermilab Accumulator. These phenomena can be used as diagnostic tools to determine diffusion rates in the beam as well as the longitudinal impedance. Moreover, aspects of these effects are likely involved in the approach to equilibrium as a longitudinal instability saturates under the in¯uence of a driving impedance. We present experimental results and analytical models based on perturbation techniques. In addition, we describe particle simulations which model the fully nonlinear evolution of these phenomena.

Summary of instabilities and damping group

“…In this regime, one may see either saturation or linear growth depending as Z; also wave-breaking and sub/super-harmonic generation effects (reminiscent of the work of Colestock [43,44,45,46]) are possible. Further, as is obvious, when [Z] > 0 one sees a slow decrease in the beam mean energy due to a.c. losses.…”

Hamiltonian dynamics with a weak noise and the echo effect for the rotator model

“…The suggested procedure consists in displacing a Gaussian beam, invariant over one turn, with a dipole kick, and to apply a quadrupole kick at time τ ; the echo signal appears at time 2τ in this case. The measurement of echo signals to investigate the longitudinal motion of the beam was also proposed and experimentally achieved [9] [10]. The decay law of the echo amplitude in the presence of noise was established [11].…”