Nonlinear beam dynamics

The nonlinear beam dynamics are organised around a three-prong approach: devise new beam manipulations based on nonlinear effects; study of dynamic aperture; diffusive models for nonlinear beam dynamics.

Nonlinear beam manipulations

The starting point is the Multi-turn Extraction (MTE) that is used to transfer high-intensity proton beams from the PS to the SPS for the fixed-target physics. The beam is split in five beamlets by crossing adiabatically the fourth-order resonance. Stable islands are created by sextupoles and octupoles. This is the prototype of all other nonlinear beam manipulations. In addition to studying in more detail the splitting by means of stable islands, in view of further improvements for MTE, alternatives are being studied, such as splitting using AC-dipoles.

The use of nonlinear effects leads to the manipulation of the transverse invariants of a beam distribution, i.e. the beam emittances. Indeed, by crossing a 2D resonance, it is possible to share, almost at will, the values of the beam emittances, very much like what is done when crossing the coupling resonance in a linear system.

Cleary, manipulating the invariants means also looking at the possibility of reducing them, hence generating a cooling of the beam emittance. This topic is under active research now with encouraging results obtained so far.

A future topic will be devising new beam manipulations combining the use of stable islands and crystals.

Dynamic aperture

The dynamic aperture is the region in phase space in which bounded motions occur. Its extent is related with the strength of the nonlinear effects acting on the beam dynamics. Intense numerical simulations are needed to compute the dynamic aperture of a given system. For this reason, efforts have been devoted to the development of model describing the evolution of dynamic aperture with time. These models are all based on the time-stability estimate of the Nekhoroshev theorem. In addition, a link has been established between the dynamic aperture an beam losses in a circular accelerator. The current research activities are linked with the application of these scaling laws to reproduce the losses measured in the LHC at injection. Moreover, means to improve the quality of the fit of the dynamic aperture models based on Machine Learning are actively pursued.

Diffusive models

The use of dynamic aperture to describe the losses in circular accelerators has some limitations as it does not allow simulating the evolution of the beam distribution, which is essential in several aspects, e.g. to estimate emittance growth. This shortcoming can be overcome by developing appropriate diffusive models. In fact, in a diffusion framework, the evolution of the beam distribution is determined by imposing a number of boundary conditions. The idea behind the proposed diffusive models is to impose a functional form for the diffusion coefficient that is based, once more, in the Nekhoroshev theorem. This is being carried out successfully, and of course, it opens the study of understanding the relationships between the approach based on dynamic aperture and diffusive models, given that the underlying ideas are all based on Nekhoroshev theorem. The diffusive approach implies the study of stochastic Hamiltonians and entails also the study of indicators of chaos to assess under which conditions the diffusive models are valid.

For all these three fields, experimental activities are envisaged in order to probe the results obtained by theory and simulations in real life accelerators.