Hamiltonian formalism for halo investigation in high-intensity beams

In high-intensity proton accelerators with average current higher than 1 mA the beam losses are restricted on the level of 10−4–10−5. The losses may be due to a number of reasons, for instance, current variation during the macropulse. Such a small fraction is difficult to detect by modern diagnostics and is inaccurately predicted by the most developed numerical simulation. In order to concentrate the problem on the particles expected to be lost in the periodically ideal accelerator, we determine the halo as particles with uncontrolled resonantly growing oscillation due to the explicit time dependence of the Hamiltonian. The Hamiltonian time dependence arises because the space charge force in the core is modulated without changing its distribution. Essentially we investigate the non-linear oscillation of the smallest fraction of the beam in the time-dependent core. In order to investigate this phenomenon we apply the Hamiltonian formalism together with the standard theory of perturbation to the halo-creation problem. We investigate the non-linear resonances and their self-stabilizing effect at different initial distributions. Using our analytical model, we determine the maximum growth of oscillation for the particles in the halo.