Yu. Senichev, Coloumb 2005, Italy1 HAMILTONIAN FORMALISM FOR HALO INVESTIGATION IN HIGH INTENSITY BEAM Yu. Senichev, IKP, FZJ

Yu. Senichev, Coloumb 2005, Italy2 The problem definition The aim of this work is the investigation of the behaviour of the smallest part of beam, which we call the halo!!! For investigation of this phenomenon we applied Hamiltonian formalism together with standard theory of perturbation. We investigate the non-linear resonances and their self-stabilising effect at the different initial distributions.

Yu. Senichev, Coloumb 2005, Italy3 The halo definition In case, when Hamiltonian has not the explicit dependence on the time the particle moves along the trajectory When Hamiltonian has the explicit dependence on the time the particles oscillate around the time averaging curve:

Yu. Senichev, Coloumb 2005, Italy4 Model approximations The coasting beam is assumed to have axial symmetry: solenoid and triplet channel The central core (90-95% of the intensity) is unaffected by the halo.

Yu. Senichev, Coloumb 2005, Italy5 The core distribution the distribution is discribed by the binomial polynomial: The space charge electrical field:

Yu. Senichev, Coloumb 2005, Italy6 Equation motion After the longitudinal coordinate normalizing is the periodical coefficient the space charge force:

Yu. Senichev, Coloumb 2005, Italy7 Equation solution Solution is seeked in the form where Envelope equation In case of the space charge

Yu. Senichev, Coloumb 2005, Italy8 Equation solution Envelope oscillation with Particle in core oscillates with

Yu. Senichev, Coloumb 2005, Italy9 Equation solution Using Courant and Snyder formalism together with Floquet method, we have the selfconsistent equations system:

Yu. Senichev, Coloumb 2005, Italy10 Non-linear equation The right side of equation : B-M=>

Yu. Senichev, Coloumb 2005, Italy11 Non-linear equation solution Solution: Linear detuning Non-linear detuning

Yu. Senichev, Coloumb 2005, Italy12 Isolated resonance New variables: and Resonance condition at some

Yu. Senichev, Coloumb 2005, Italy13 New Hamiltonian After canonical transformation with :

Yu. Senichev, Coloumb 2005, Italy14 Resonance width

Yu. Senichev, Coloumb 2005, Italy15 Emittance growth Nr is the resonant harmonic Emittance growth is

Yu. Senichev, Coloumb 2005, Italy16 Phase oscillation The halo phase delay relative the core

Yu. Senichev, Coloumb 2005, Italy17 Numerical simulation Numerical simulations have been done for the periodical FDO channel Maximum size of beam, normalised on in FDO channel at 100 mA (the lower curve) and 150 mA(the upper curve) micro-pulse current.

Yu. Senichev, Coloumb 2005, Italy18 Numerical simulation

Yu. Senichev, Coloumb 2005, Italy19 Conclusion the model of the halo creation was developed the case without an external resonance was considered, and the beating of envelope is the source for the emittance growth the analitical formula for the emittance growth has been derived The analitical and the numerical results have been compared and the good agreement was observed