Phase space structure for matched intense charged-particle beams in periodic focusing transport systems

Abstract

Test particle motion is analyzed analytically and numerically in the field configuration consisting of the equilibrium self-electric and self-magnetic fields of a well-matched, thin, continuous, intense charged-particle beam and an applied periodic focusing solenoidal magnetic field. The self fields are determined self-consistently, assuming the beam to have a uniform-density, rigid-rotor Vlasov equilibrium distribution. Using the Hamilton–Jacobi method, the betatron oscillations of test particles in the average self fields and applied focusing field are analyzed, and the nonlinear resonances induced by periodic modulations in the self fields and applied field are determined. The Poincaré surface-of-section method is used to analyze numerically the phase-space structure for test particle motion outside the outermost envelope of the beam over a wide range of system parameters. For vacuum phase advance σv=80°, it is found that the phase-space structure is almost entirely regular at low beam intensity (phase advance σ≳70°, say), whereas at moderate beam intensity (30°≲σ≲70°), nonlinear resonances appear, the most pronounced of which is the third-order primary nonlinear resonance. As the beam intensity is further increased (σ≲30°), the widths of the higher-order nonlinear resonances increase, and the chaotic region of phase space increases in size. Furthermore, the many chaotic layers associated with the separatrices of the primary and secondary nonlinear resonances are still divided by the remaining invariant Kolmogorov–Arnold–Moser surfaces, even at very high beam intensities. The implications of the rich nonlinear resonance structure and chaotic particle motion found in the present test-particle studies are discussed in the context of halo formation.