A possible mechanism, which explains the diffusion in the phase space due to the ripples in the quadrupole currents, is studied on a simplified version of the SPS lattice used for experiments. We describe the diffusion driven by a single resonance in the s

2

A. BAZZANI, F. BRINI5

experimentally observed in accelerators, one has to introduce other e ects in the model. Recent experiments on the SPS at CERN and in other laboratories have shown that the ripples in the feeding currents of the quadrupoles due to harmonics of the 50 Hz frequency, coupled with the nonlinear components of the magnetic eld, cause a slow decrease of the beam intensity. The presence of ripples introduces a slow modulation in the coe cient of the one turn map, which describes the betatronic motion, so that we have to study the stability of the orbits of a non-autonomous symplectic map. However we can simplify the problem by using the results of the adiabatic theory for Hamiltonian systems, since the ratio between the ripple frequency and the betatronic

frequency is 10? 10? and can be used as a perturbation parameter. The numerical simulations show a qualitative agreement with the experimental data, but a quantitative description of the modulated di usion is still not available. In this paper we present a description of the modulated di usion due to a nonlinear resonance in the phase space: our approach is directly derived from the theory of the changing of the adiabatic invariant due to the crossing of a separatrix developed by A.Neishtadt for Hamiltonian systems. We have applied the Neishtadt's theory to an area-preserving map, directly derived from a simpli ed model of the SPS lattice used in the experiments. The plan of the paper is the following: in section 2 we brie y summarize the results of the adiabatic theory and we introduce the hypothesis necessary to apply the results to our model; in section 3 we describe the model and we discuss the comparison between the numerical results and the analytical approach; the concluding remarks are reported in section 4.6 3 4 7 8,9

We would like to thank prof. A.Neishtadt for several fruitful discussions and prof. G.Turchetti who has suggested and followed this work. 2 ADIABATIC THEORY FOR HAMILTONIAN SYSTEMS The adiabatic theory was mainly developed to describe the evolution of integrable or almost integrable Hamiltonian systems H (q; p; t) perturbed by a slow modulation: 1 and H (q; p; ) integrable and periodic in . The concept of slow modulation means that the period of modulation T= 2= is much longer than the typical time scales of the unperturbed Hamiltonian H (q; p; ). The main idea of the adiabatic theory is to look for special dynamical variables I (q; p; ), called adiabatic invariant (a.i.), which have the following property: (1) jI (q; p; t)? I j< C 8 t< 10

1.1 ACKNOWLEDGEMENTS

where I is the initial value of the a.i. and C is a constant. Far from a resonance region, the action variables for the unperturbed system are a.i. and one can prove0