Recently, a novel multi-turn extraction scheme was proposed, based on particle trapping inside stable resonances. Numerical simulations and experimental tests conirmed the feasibility of such a scheme for low order resonances. While the 3rd order resonance is generically unstable and those higher than 4th order are generically stable, the 4th order resonance can be either stable or unstable depending on the details of the system under consideration. By means of the normal form approach a general formula to control the stability of the 4th order resonance is derived. Numerical simulations confirm the analytical results and show that by crossing the unstable 4th order resonance the region around the centre of phase space is depleted and particles are trapped only in the four stable islands. This indicates that a four-turn extraction could be envisaged based on this technique

Giovannozzi, M

Quatraro, D

Turchetti, G

CERN Document Server

CERN-AB-2008-00927Aug2008STABILITY CHANGE OF FOURTH-ORDER RESONANCE WITH APPLICATIONTO MULTI-TURN EXTRACTION SCHEMESRecently, a novel multi-turn extraction scheme was proposed, based on particle trapping inside stableresonances. Numerical simulations and experimental tests conirmed the feasibility of such a scheme for loworder resonances. While the 3rd order resonance is generically unstable and those higher than 4th order aregenerically stable, the 4th order resonance can be either stable or unstable depending on the details of thesystem under consideration. By means of the normal form approach a general formula to control thestability of the 4th order resonance is derived. Numerical simulations confirm the analytical results andshow that by crossing the unstable 4th order resonance the region around the centre of phase space isdepleted and particles are trapped only in the four stable islands. This indicates that a four-turn extractioncould be envisaged based on this technique.EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCHCERN   -   AB DepartmentGeneva, SwitzerlandAugust 2008AbstractCERN-AB-2008-0091) CERN, Geneva, Switzerland and INFN sezione di Bologna, Italy.2) Università degli Studi di Bologna, Bologna, and INFN sezione di Bologna, ItalyM. Giovannozzi, D. Quatraro1), G. Turchetti2)CERN, Geneva, SwitzerlandPresented atEPAC'08, 11th European Particle Accelerator Conference, Genoa, Italy - June 23-27, 2008STABILITY CHANGE OF FOURTH-ORDER RESONANCE WITHAPPLICATION TO MULTI-TURN EXTRACTION SCHEMESM. Giovannozzi, CERN, Geneva, SwitzerlandD. Quatraro, CERN, Geneva, Switzerland, and INFN sezione di Bologna, Bologna, ItalyG. Turchetti, Universita degli Studi di Bologna, and INFN sezione di Bologna, Bologna, ItalyAbstractRecently, a novel multi-turn extraction scheme was pro-posed, based on particle trapping inside stable resonances.Numerical simulations and experimental tests confirmedthe feasibility of such a scheme for low order resonances.While the 3rd order resonance is generically unstable andthose higher than 4th order are generically stable, the 4thorder resonance can be either stable or unstable depend-ing on the details of the system under consideration. Bymeans of the normal form approach a general formula tocontrol the stability of the 4th order resonance is derived.Numerical simulations confirm the analytical results andshow that by crossing the unstable 4th order resonance theregion around the centre of phase space is depleted and par-ticles are trapped only in the four stable islands. This indi-cates that a four-turn extraction could be envisaged basedon this technique.INTRODUCTIONIn recent years, a novel type of extraction based on par-ticle trapping inside stable islands of the horizontal phasespace was proposed [1]. The beam manipulation is intrin-sically linked with non-linear beam dynamics. The beam isswept through a stable non-linear resonance and wheneverthe crossing is adiabatic, some particles can be trapped in-side the stable islands and then transported towards higheramplitudes. At this stage the extraction proper is per-formed.Such an extraction mode is primarily aimed at replac-ing the current Continuous Transfer extraction [2] fromthe CERN Proton Synchrotron (PS) to the Super ProtonSynchrotron (SPS) as an improved extraction mode, in thesense of losses (that are highly reduced, if not completelysuppressed) and of injection matching. Parenthetically, along measurement campaign was performed [3] to assessthe performance of the proposed method, and a projectsetup for the implementation of such an extraction in thePS machine [4].Further studies, however, showed that the method has amuch broader scope, being applicable to resonances otherthat the 1/4, which is the one selected for the CERN-specific application [5]. More than this, the approach canbe time-reversed and used for a novel injection type [6].In the case of a stable resonance of order n, n+ 1 beam-lets are generated, n corresponding to the islands and oneto the beam core part remaining after the trapping process.The beamlets at the end of the trapping process form twodisconnected structures in phase space: a ribbon closingup after n turns around the machine circumference and thepart, one-turn long, at the centre of phase space.At the level of the extraction system proper, a set of kick-ers should generate a fast bump constant over the n + 1extraction turns, noting that the kickers’ strength should beincreased for ejecting the last turn. This might be a limitingfactor, imposing higher demands on the strength require-ments. In this respect, an unstable resonance, for whichthere is basically no beam left at the centre of phase space,might be advantageous.In Ref. [5] the case of the third-order resonance was stud-ied. However, one could also consider the fourth-order res-onance and make it unstable so to generate a four-turn ex-traction scheme. This can be done by using the tool of Nor-mal Forms. The application of such a tool to the non-linearbetatronic motion was proposed in [7], while a review canbe found in [8]. This approach can be used to change thestability type of a resonance [9]. The theory and the simplemodel used for this study will be presented together withsome results of numerical simulations.THEORY AND MODELTo simulate the adiabatic capture process the composi-tion of two symplectic maps is used:(xpx)n+1= R(ψ2) ·(xpx + k3,2x3)◦R(ψ1) ·(xpx + x2 + k3,1x3)n,(1)where (x, px) are phase space coordinates, R(ϕ) is a rota-tion matrix of an angle ϕ, k3,1 and k3,1 are the strengthsof two octupolar elements. Without any loss of generalitythe sextupolar strength can be set equal to one. During theiteration of the map the value of ψ1 is kept constant lettingψ2 varying with n so to set ω = 2pi ν = ψ1 + ψ2 to theactual value of the linear frequency of the system.Via Normal Forms theory applied to polynomialmaps [8] it is known that the 1/4 resonance could be ei-ther stable on unstable, depending on the value of the firstamplitude detuning term, as the Hamiltonian can be writtenH(θ, ρ) =  ρ+ Ω2 ρ2 +Aρ2 cos 4θ +O(ρ3), (2)where (θ, ρ) stand for the angle-action variable and Ω2 isthe first amplitude-dependent detuning term.A general recurrence to evaluate Ω2 in the case of thecomposition of M polynomial maps of the form(xpx)n+1= R(ψj) ·(xpx + k2,jx2 + k3,jx3)n, (3)is described in Ref. [9]. This case represents a realisticmodel of a machine, whereM sextupoles and octupoles areinstalled. Of course, for the case ofM maps, ω =∑Mi=0 ψiis the global tune.For the model used in the numerical simulations pre-sented in this paper the values ψ1/2pi = 0.62, k3,1 = 1,and k3,1 = −1/3 were used. The variable k3,1 was used toset Ω2 = 0 during the crossing of the 1/4 resonance, whichcorresponds to changing its stability type.A typical sequence of phase space topology obtainedduring the resonance crossing process is shown in Fig. 1.By changing the tune the separatrices related to the hyper-XP X-1.00 1.00-1.001.00XP X-1.00 1.00-1.001.00XP X-1.00 1.00-1.001.00XP X-1.00 1.00-1.001.00Figure 1: Phase portrait of the dynamical system Eq. (1) asthe linear tune ν is changed and approaching the resonantvalue ν = 1/4 from above.bolic fixed points shrink towards the origin, thus making itunstable.NUMERICAL SIMULATIONSThe numerical simulations have been performed usingthe model of Eq. (1) and considering a Gaussian distribu-tion of particles in the horizontal phase space space (x, px)ρ(x, px) =12piσ2e−x2+p2x2σ2 . (4)The tune ν(n) is changed with time as a polynomial func-tion fk of order k, where the key parameters are νs, νr,and νf , the starting, resonant, and final tune values, respec-tively. n1, and N represent the turn at which the resonanceis crossed and the total number turns involved in the cap-ture process. The order of the polynomial can change be-fore and after n1 with the constraint that ν ′(n1) = 0 soto achieve a smooth transition at exact resonance crossing.Henceforth, we refer to the i−j curve if we have a polyno-mial curve of ith order before the resonance crossing and ajth polynomial curve after the resonance crossing.Typically, 2×106 particles were tracked for about 5×104iterations of the map, with νs = 1.253, νr = 1.25,and νe = 1.248, while the resonance crossing occurs forn1 = 3.5× 104. An example of splitting process is shownin Fig. 2 The corresponding tune variation is reported in       Beam distribution after     0 turns-0.7-0.42-0.140.140.420.7-0.7 -0.42 -0.14 0.14 0.42 0.7 020406080100120140 xp x       Beam distribution after  9000 turns-0.7-0.42-0.140.140.420.7-0.7 -0.42 -0.14 0.14 0.42 0.7 020406080100120140xp x       Beam distribution after 34000 turns-0.7-0.42-0.140.140.420.7-0.7 -0.42 -0.14 0.14 0.42 0.7 020406080100120140xp x       Beam distribution after 42000 turns-0.7-0.42-0.140.140.420.7-0.7 -0.42 -0.14 0.14 0.42 0.7 020406080100120140xp xFigure 2: Evolution of the initial distribution during theresonance crossing process.Fig. 3. While the separatrices collapse towards the origin, 1.249 1.25 1.251 1.252 1.2535.0 ⋅1044.0 ⋅1043.0 ⋅1042.0 ⋅ 1041.0 ⋅ 1040Linear tune νnResonant valueFigure 3: Tune variation used for the splitting presented inFig. 2. The tune curve is of type 2− 3.the particles are pushed towards the islands and trapped in-side.In Figs. 4, 5 the results concerning the fraction of trappedparticles in the resonance island and the relative emittancesas a function of the σ of the initial distribution are shown.For the particles left close to the origin of the phase spacethe so-called halo parameter h [11] defined ash =〈x4〉〈x2〉2− 2 (5)is shown in Fig. 4. The bigger is the σ the greater is thefraction of beam trapped into the resonance islands. This 0 5 10 15 20 25 0.065  0.07  0.075  0.08  0.085  0.09 0 0.5 1 1.5 2Fraction of trapped particlesBeam core hσislandsbeam coreh  0 5 10 15 20 25 0.065  0.07  0.075  0.08  0.085  0.09 10 15 20 25 30 35 40 45 50 55Fraction of trapped particlesBeam core hσislandsbeam coreh Figure 4: Fraction of trapped particles as a function of theinitial sigma. Two types of tune curves are used: 1−1 (left)and 2 − 3 (right). The value of the factor h for the beamcore is also given.trend is valid for every type of tune curve. On the otherhand it is also clearly seen that the higher the order of thepolynomial the smaller is the fraction of beam left in thecentre. For the curve 1 − 1, ≈ 10 % of the beam is leftin the core distribution. The best result is obtained for thecurve 2− 3 with a fraction of ≈ 4 % of the whole particlesdistribution. It is worth mentioning that almost no particleloss is observed (≈ 0.01% of the total number of particles).For the numerical simulations of the adiabatic splitting0.600.851.101.35 0.004  0.005  0.006  0.0070.070.090.100.12Island ε x/ εx,0Beam core  εx/ εx,0εx,0islands beam core1.021.031.041.051.061.07 0.004  0.005  0.006  0.0070.050.100.150.200.25Island ε x/ εx,0Beam core  εx/ εx,0εx,0islands beam coreFigure 5: Relative islands emittance as a function of theinitial emittance. Two types of tune curves are used: 1− 1(left) and 2− 3 (right).the condition Ω2 = 0 was set only for the resonant valueof the tune. Comparisons with simulations performed withΩ2 = 0 throughout the whole process showed a decreaseof the fraction of particle left in the centre to ≈ 0.5 %.A set of numerical simulations was made to assess thedependence of the fraction of trapped particles on the totalturn numberN over which the tune variation ∆ν = νf −νiis performed and the σ of the initial distribution. A tunecurve of type 1 − 1 was used. The dependence on σ isperfectly fitted with an exponential function, while thaton N is a linear function and the combined result givesNCore/Npart = Ae−τσ with A(N) = a + bN . The nu-merical results together with the fitted curves are shown inFig. 6. The excellent agreement between data and fit func-tion is clearly visible. From the numerical simulations itemerges clearly that the parameter τ does not depend onNand its value is τ ≈ 22.3. Still, it could be a function of thenon-linear parameters of the system. 13 15 17 19 21 23 25 0.065  0.07  0.075  0.08  0.085  0.09Fraction of trapped particlesσN = 15000N = 17500N = 20000N = 22500N = 25000  85 90 95 100 105 15000  17000  19000  21000  23000  25000A ( N)Na = 131  b = -0.0018Numerical dataA (N) = a + b ⋅ NFigure 6: Fit results of the fraction of trapped particles as afunction of the total number of turnsN with an exponentialfunction (left). The amplitude of such an exponential isfitted as a linear function of N (right).CONCLUSIONSIn this paper the use of Normal Forms for the change ofthe stability type of a non-linear resonance was presented.Applied to a simple model describing the horizontal be-tatronic motion it can be successfully used to generate ascheme for a four-turn extraction. A parametric study of thebeam parameters after splitting was also discussed, show-ing that the process can be accurately controlled.ACKNOWLEDGEMENTSMany fruitful discussions with A. Franchi andS. Gilardoni are warmly acknowledged.REFERENCES[1] R. Cappi, M. Giovannozzi, Phys. Rev. Lett. 88, 104801 2002.[2] C. Bovet et al., in proceedings 1973 PAC, edited by D. W.Dupen (IEEE, New York, 1973) p. 438.[3] S. Gilardoni, M. Giovannozzi, M. Martini, E. Me´tral, A.-S. Mu¨ller, P. Scaramuzzi, R. Steerenberg, Phys. Rev. ST Ac-cel. Beams 9, 104001, 2006.[4] M. Giovannozzi (Ed.) et al., CERN Report, 2006-011, 2006.[5] R. Cappi, M. Giovannozzi, Phys. Rev. ST Accel. Beams 7,024001, 2003.[6] M. Giovannozzi, J. Morel, Phys. Rev. ST Accel. Beams 10,034001, 2007.[7] A. Bazzani, G. Turchetti, P. Mazzanti, G. Servizi, Il NuovoCimento B102, 51, 1988.[8] A. Bazzani, G. Servizi, E. Todesco, G. Turchetti, CERN Yel-low Report, 94-02, 1994 (unpublished).[9] D. Quatraro, Diploma Thesis University of Bologna, July2007 (unpublished).[10] J. Buon, in proceedings of the CERN Accelerator School,CERN 94-01 Vol. 1, p. 89, 1994 (unpublished).[11] C. K. Allen, T. P. Wangler, Phys. Rev. ST Accel. Beams 5,124202, 2002.

Stability change of Fourth-Order Resonance with application to Multi-Turn Extraction Schemes

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