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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

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MODULATED DIFFUSION FOR A SIMPLE LATTICE MODELA. BAZZANIDept. of Mathematics and INFN, Sezione di Bologna, P.zza di Porta San Donato n.5, I-40126 Bologna, Italy.

F. BRINIDept. of Physics and INFN, Sezione di Bologna, v. Irnerio n.46, I-40126 Bologna, Italy.A possible mechanism, which explains the di usion in the phase space due to the ripples in the quadrupole currents, is studied on a simpli ed version of the SPS lattice used for experiments. We describe the di usion driven by a single resonance in the space of the adiabatic invariant (action variable), by using the results of the Neishtadt's theory. Under suitable hypothesis, it is possible to introduce a random walk for the adiabatic invariant, which gives a quantitative description of the di usion. The comparison with the numerical results turns out to be very e ective.

1 INTRODUCTION The study of the long term dynamics aperture in a particle accelerator is related to the stability of the orbits in a neighborhood of an elliptic xed point of a symplectic map . The multipolar components of the magnetic eld correspond to nonlinear terms in the Taylor expansion of the map. The presence of Arnold di usion and overlapping of resonances allows the possibility of nding unstable orbits, near the elliptic xed point. However the numerical simulations show the following scenario: there is a threshold in the phase space after which a fast escape to in nity is observed (short term dynamics aperture); there is a neighborhood of the xed point, where no di usion can be detected; there is a region where some particular orbits escape to in nity after a big number of iterations, but usually the measure of the initial conditions of the unstable orbits is very small. This situation can be explained by taking into account both the extreme slowness and sensitivity to the initial condition of Arnold di usion and the limitation of the region of overlapping of resonances, which is a generic feature for polynomial symplectic maps. As a consequence to explain the slow di usion of particles1,2 3 4

Work partially supported by EC Human Capital and Mobility contract Nr. ERBCHRXCT940480 1

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

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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

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

MODULATED DIFFUSION FOR A SIMPLE LATTICE MODEL

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that the perpetual adiabatic invariance (no limits on t) holds for a large set of initial conditions. As a consequence the instabilities due to a slow modulation are driven by the presence of resonances and in particular of separatrices in the phase space. The adiabatic description of the dynamics near to a resonance is based on the slow pulsation of the separatrix in the phase space due to the slow time dependence of the Hamiltonian (show g. 1 left); starting from an initial condition out of the separatrix the actual trajectory will follow the unperturbed orbits of H (q; p; ) in order to preserve the value of the a.i. (i.e. the area of the unperturbed orbit for a 2D system); but since the separatrix is moving in the phase space, at a certain time it could happen that the orbit crosses the separatrix and starts to be trapped in the resonance region. In such a case the inequality (1) does not hold any more and it is necessary to compute the

changing of the a.i. due to a crossing of a separatrix . The main result of Neishtadt's theory is that if one considers an orbit in the region spanned by the separatrix, after one period of the slow modulation, the value of the a.i. will be changed by a quantity8,9

I1

? I= f (I0

0

;;

0; log )

(2)

where and 0 are two variables 2 0; 1] related to the value of the phase variable at the crossing times (we have two crossings of the separatrix in one period); the leading term in eq. (2) is of order log . In order to understand a possible mechanism for the di usion in the phase space, we consider the following scenario: the unperturbed system H (q; p; ) has a single resonance in the phase space for each value of and the region spanned by the separatrix when= t varies, is free from other resonances (this condition will be not true for quasi integrable systems, but in this case we suppose that the amplitude of other resonances is of order ); the amplitude of the region spanned by the separatrix is much larger than . Let us consider a cluster of initial conditions in the region A spanned by the separatrix; the size of the cluster has to be larger than . The distribution function of the increments for the a.i. can be computed by using eq. (2) and considering; 0 as random variables uniformly distributed in 0; 1] . In spite of the complicated expression of the r.h.s. of eq. (2), we make the operative hypothesis that the increment I? I could be approximated by10 1 0

I1

? I= f^(I; log )0 0

(3)

where is an universal random variable whose distribution can be determined numerically, whereas the function f^ takes into account the dependence on the action of the di usion coe cient and it is zero at the boundary of the region A. The main idea of our approach is to consider successive crossings of the separatrix as a random walk for the a.i. with independent increments; the probability space is de ned by the set of the initial conditions. This approach is correct if we show that the values of phase variable after one period T of the modulation are uncorrelated. According to the results in, the relation between two successive phase values11,12

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

4Z T0

A. BAZZANI, F. BRINI

after a period T, can be expressed in term of the integral@@I

(In?

1

; t In

)(? In? )dt1 1

(4)

where (I; t) is the amplitude dependent rotation number (tune) and In? and In the initial and nal values of the a.i.. As a consequence if the integral (4) is 1, then the increments will be independent. Taking into account that In? In?/ log and that T/ 1=, one can estimate that the integral (4) is 1 if is su ciently small and the derivative@=@I is not too small, which is in general true near a separatrix. We have numerically checked the independence of the increments for the a.i. and our result seems to be generic for symplectic maps; but for 2D symmetric Hamiltonian systems there are examples for which the condition (4) 1 fails and the phases are correlated for

more than one period T . Therefore we can describe the di usion of the a.i. due to the presence of an isolated single resonance by using the random walk^ In? In?= f (In?; log ) (5)1 13 14 1 1

where In is the value of the a.i. after n periods. The equation (5) will be a crude approximation of the di usion in the case that is not small or that the orbits are very close to boundary of the region A. Moreover the di usion described by the random walk (5) is a bounded di usion, since the orbits cannot overcome the region A spanned by the separatrix, so that the nal distribution function for the a.i. will be an uniform distribution in the region A. In order to describe the escape to in nity of an orbit in the phase space we have to consider the overlapping of regions spanned by the separatrices of di erent resonances and to introduce the transition probabilities for the a.i. to go from one region to another. This mechanism is indeed observed in the numerical simulations but up to now there is no analytical description.15

3 DESCRIPTION OF THE MODEL AND NUMERICAL RESULTS In the 1991 experiments at CERN to study the di usion of the beam due to the ripples in the quadrupole magnets, the SPS had a special con guration with eight sextupoles strongly exited to produce intense nonlinear elds . The betatronic motion for a at beam is well described by the composition of eight quadratic area-preserving maps, which take into account the e ect of the strong sextupoles. This is the model we have used for the numerical simulations; the same model was considered to study the di usion of the nonlinear invariant, when a stochastic noise is introduced in the linear tune . We cannot claim that our model is realistic, but we expect that it is su ciently generic to perform all the features of a realistic model in the at beam approximation.5 16

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

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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

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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

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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

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a D 36, 287-316 (1989) 13. A. Bazzani and F. Brini, in NATO ASI on Hamiltonian systems with three or more degrees of freedom, edited by C. Simo ed. (NATO ASI to be published,, 1995) 14. D.L. Bruhwiler and J.R. Cary, Physica D 40, 265-282 (1989) 15. A. Bazzani, S. Siboni and G. Turchetti, in AIP conference proc. 334, edited by S. Chattopadhyay, M.Cornacchia and C. Pelllegrini eds. (AIP press, Woodbury, New York, 1995) 16. A. Bazzani, M. Giovannozzi and G. Turchetti, in AIP conference proc. 334, edited by S. Chattopadhyay, M.Cornacchia and C. Pelllegrini eds. (AIP press, Woodbury, New York, 1995) 17. G. Turchetti, these proceedings.

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