Nonlinear coupling between breathing and quadrupole-like oscillations in the transport of mismatched beams in continuous magnetic focusing fields

Abstract

A nonlinear analysis of the transport of breathing beams considering nonaxisymmetric perturbations is performed. It is shown that large-amplitude breathing oscillations of an initially round beam may couple nonlinearly to quadrupole-like oscillations, such that the excess energy initially constrained to the axisymmetric breathing oscillations is allowed to flow back and forth between breathing and quadrupole-like oscillations. In this case, the beam develops an elliptical shape with a possible increase in its size along one direction as the beam is transported. This is a highly nonlinear phenomenon that occurs for large mismatch amplitudes on the order of 100% and is found to be particularly relevant for space-charge-dominated beams with K≳k0ϵ, where K is the beam perveance, k0 is the vacuum phase advance per unit axial length, and ϵ is the emittance of the beam. A simple model based on mapping techniques is used to clarify the mechanism that leads to the energy exchange between the modes and is tested against results from direct integration of the envelope equations.

Combined centroid-envelope dynamics of intense, magnetically focused charged beams surrounded by conducting walls

“…Figure 2͑c͒, in fact, tells us that the halo component of the beam becomes a little distorted toward an elliptical shape as times advances. This is expected and has to do with nonlinear anisotropic instabilities ͑as opposed to the more wellknown linear instabilities [11][12][13] ͒ occurring for largely mismatched beams, 14 but does not seem to largely affect the agreement between simulations and estimates.…”

Chaos and relaxation to equilibrium in systems with long-range interactions

“…This is often the case for gravitational systems whose initial particle distribution has a spherical symmetry and satisfies the generalized virial condition. On the other hand, if the initial distribution is spherically symmetric, but far from virial, strong density oscillations during the process of violent relaxation can lead to symmetry breaking [19][20][21]. This means that even if the initial distribution is spherically symmetric, the particle distribution and the static mean-field potential of the QSS will lack this symmetry.…”

Chaotic particle heating due to an obliquely propagating wave in a magnetized plasma

“…This allows for the integration of the resulting dynamical equations over one wave period and the derivation of a map to describe the particle evolution. The map facilitates the analysis not only because it largely speeds up the computation of the system evolution, but also because it may yield a series of exact analytical results to work with [11][12][13][14][15][16][17][18].…”