Beam trapping in stable islands of the horizontal phase space generated by non-linear magnetic fields is realized by means of a given tune variation so to cross a resonance of order n. Whenever the resonance is stable, n + 1 beamlets are created whereas if the resonance is unstable, the beam is split in n parts. Experiments at the CERN Proton Synchrotron showed protons trapped in stable islands while crossing the one-third and one-fifth resonance with the creation of 3 and 6 stable beamlets, respectively. The results are presented and discussed

Franchi, A

Gilardoni, S

Giovannozzi, M

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CERN-AB-2008-04004Sep2008EXPERIMENTAL EVIDENCE OF BEAM TRAPPING WITH ONE-THIRD ANDONE-FIFTH RESONANCE CROSSINGBeam trapping in stable islands of the horizontal phase space generated by non-linear magnetic fields isrealized by means of a given tune variation so to cross a resonance of order n. Whenever the resonance isstable, n + 1 beamlets are created whereas if the resonance is unstable, the beam is split in n parts.Experiments at the CERN Proton Synchrotron showed protons trapped in stable islands while crossing theone-third and one-fifth resonance with the creation of 3 and 6 stable beamlets, respectively. The results arepresented and discussed.EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCHCERN   -   AB DepartmentGeneva, SwitzerlandAugust 2008AbstractCERN-AB-2008-040S. Gilardoni, A. Franchi, M. GiovannozziCERN, Geneva, SwitzerlandPresented atEPAC'08, 11th European Particle Accelerator Conference, Genoa, Italy - June 23-27, 2008EXPERIMENTAL EVIDENCE OF BEAM TRAPPING WITH ONE-THIRDAND ONE-FIFTH RESONANCE CROSSINGS. Gilardoni, A. Franchi, M. Giovannozzi, CERN, Geneva, SwitzerlandAbstractBeam trapping in stable islands of the horizontal phasespace generated by non-linear magnetic fields is realizedby means of a given tune variation so to cross a resonanceof order n. Whenever the resonance is stable, n + 1 beam-lets are created whereas if the resonance is unstable, thebeam is split in n parts. Experiments at the CERN Pro-ton Synchrotron showed protons trapped in stable islandswhile crossing the one-third and one-fifth resonance withthe creation of 3 and 6 stable beamlets, respectively. Theresults are presented and discussed.INTRODUCTIONIn 2002 a new extraction scheme was proposed to ejectthe beam over a few turns by means of non-linear magnets,such as sextupoles and octupoles, rather than slicing it ontoan electrostatic septum [1]. During this new multi-turn ex-traction, the beam is split and trapped inside stable islandsof the horizontal phase space, that are generated and sepa-rated by sweeping the horizontal tune through a non-linearresonance while sextupoles and octupoles are powered.The number of generated beamlets depends on the orderof the crossed resonance. In general, a resonance of order ngenerates n+1 beamlets if stable, whereas if it is unstable,the beam is split in n parts, the centre becoming an unsta-ble fixed point in the latter case. In the case of the beamtransfer between the CERN Proton Synchrotron (PS) andthe Super Proton Synchrotron (SPS) at 14 GeV/c, the mostsuitable resonance is the stable one-fourth, i.e. Qx = 6.25.A loss-free beam splitting in five beamlets (four islandsplus the beam core) was already proved [2] and the newfive-turn extraction is due to be commissioned this year [3].The choice of crossing the stable one-forth resonance totransfer the beam from the CERN PS to the SPS is dictatedby the fact that two PS cycles are used to fill the SPS ring,whose circumference is 11 times the PS one. At the endof each PS cycle the beam would be ejected in five turns.The SPS would be then filled in 10/11 of its length (oneempty slot is left to avoid interference between the circu-lating beam and the transient times of the SPS kickers). Thedelicate point of such scheme concerns the equalization ofthe beam parameters between the four beamlets trapped inthe islands and the one remaining in the central area. Thelatter is unavoidably ejected with root-mean-square (RMS)optical parameters slightly different from to ones of the is-lands. As far as the intensity is concerned, the best beamsharing so far achieved is of ∼ 18% in the four islandsand ∼ 28% in the central core. To avoid unwanted tran-sient effects in the SPS, a maximum difference of about5% in the sharing between islands and core shall be pro-vided. Another delicate point is the need of an additionalkick to extract the beam core that might induce a sligthlydifferent trajectory in the trasfer line, compared to the onesof the islands, not to mention the impact on the hardwarerequirements.Dedicated experiments at the CERN PS were carried outin 2007 at 14 GeV/c to exploit the possibility of generat-ing islands by crossing new resonances: the unstable one-third, generating three beamlets only, and the stable one-fifth, thus creating six beamlets, five in the islands plus thebeam core. In the former case, three PS cycles might beused to fill the SPS, leaving two gaps. In the case of theone-fifth resonance, N repeated crossings during the flattop of the same cycle can be envisaged, in order to extractN batches of five identical beamlets each, see Ref. [1]. Inboth cases, the beamlets injected in the SPS would have ex-actly the same intensities and RMS optical parameters. Thebeam remaining in the central area after N crossing of theone-fifth resonance might be eventually ejected towards abeam dump.Octupole magnethorizontal flying wiresMagnetic septumsection 54section 16section 85section 64section 21to the SPSvertical flying wireSextupole magnetsFigure 1: Schematic layout of the PS ring with the elementsused during the experiments.The magnetic elements and the beam instrumentationused in the experimental campaign are shown in Fig. 1.The tune is changed to cross a resonance by means of twofamilies of focusing and defocusing quadrupoles, normallyused to tune the machine at low energy. Two sextupoles andone octupole are used to generate the stable islands. Wirescanners are then used to measure the horizontal beam dis-tribution and to monitor its evolution. The determinationof islands’ position, beamlets’ size, and fraction of trappedparticles is inferred from the recorded beam profile. This isperformed by fitting n or n+1 Gaussian functions, accord-ing to the resonance order n and its stability type. The func-tions used to fit the islands are constrained to have the sameintensity. The reason for such a constrain is twofold: First,the resonance symmetry guarantees that all islands have thesame probability of trapping particles; Second, each mea-sured island profile is actually an average over all islandsprofiles, the wire time of flight being much longer than therevolution period (2.1 μs) and the islands swapping posi-tion turn by turn.CROSSING THE 1/3 RESONANCEThree stable islands were created for the first time on Au-gust 10th 2007 using a single-bunch of∼ 3×1012 protons,whose horizontal RMS normalized emittance at 1σ beforethe resonance crossing was of 5.7 μm. The horizontal tunewas initially set to 6.330, then increased linearly until thevalue of 6.337 in about 100 ms (∼ 4.8 × 104 turns), andeventually brought back to the initial value, as depicted inthe right plots of Fig. 2. The vertical tune was set to 6.205throughout the double resonance crossing.Both the octupole and the sextupoles were powered witha constant current of -421 A (K3 = −155 m−3) and 330 A(K2 = 0.9 m−2), respectively. The main results are plotted-20 -10 0 10 20x [mm]0200400600800Horizontal beam profile [arb. units]      920 960 1000 1040 1080 1120Time [ms]0.330.3330.3360.339Measured Horizontal Tunemeasurement8090100    intensity [%]   resonance crossing-20 -10 0 10 20x [mm]0200400600800Horizontal beam profile [arb. units]     920 960 1000 1040 1080 1120Time [ms]0.330.3330.3360.339Measured Horizontal Tunemeasurement8090100    intensity [%]   resonance crossing-20 -10 0 10 20x [mm]0200400600800Horizontal beam profile [arb. units]     920 960 1000 1040 1080 1120Time [ms]0.330.3330.3360.339Measured Horizontal Tunemeasurement8090100    intensity [%]   resonance crossingFigure 2: Evolution of the horizontal beam profile dur-ing the crossing of the one-third resonance. The profile isshown before the resonance is crossed (upper), right afterthe crossing (centre), and at the end of the crossing (lower).The corresponding phase space portrait obtained from theMADX PS model is superimposed to the measured profile.in Fig. 2, where the horizontal beam profile is shown be-fore the resonance is crossed (upper), right after the cross-ing (centre), at the end of the crossing (lower). Next toeach profile the corresponding position on the tune rampis indicated. In the background of the measured profile, ingray, the phase space portrait computed by the MADX pro-gram [4] for the corresponding working point is shown. Forcompleteness, below the window with the curve of the hor-izontal tune the temporal evolution of the beam intensity isreported. While the resonance is crossed without measur-able losses, at the end of the capture a loss of about 10%occurs, probabily due to a too fast tune variation around1020 ms. The steep descent of the tune after 1050 ms mightbe the reason of the further beam loss, that at the end of theprocess is of about 20%. From the measured profiles itis not possible to conclude whether losses occur becauseparticles trapped in the islands drift away as in the normalslow extraction, or whether they are due to non-trapped (orde-trapped) particles in the chaotic region between the coreand the islands.The beam profiles measured during the capture were fit-ted with a superposition of four Gaussians. Results areshown in Fig. 3. The beam profile measured at the end ofthe capture (1070 ms) is well fitted by three main Gaussiancurves (in green in the picture), each one containing a frac-tion of particle of about 31.8%, see left plot of Fig. 3. Theremaining 4.6% of the beam remains in the central area (inred in the picture). Both the 2D He´non model [1] and pure4D multi-particle simulations predict a complete depletionof the beam core and well separated islands. This is not thecase when chromatic effects (Q′x = 1.2) and tune modula-tion induced by the synchrotron motion (synchrotron tuneQs = 1× 10−2 and momentum spread Δp/p = 1× 10−3)are included: Simulations in this case show that about 6%of the beam remains in the core, while about another 6%is confined in the region between islands and core. Thecomparison between the measured and simulated profilesis shown in the right plot of Fig. 3. Even though the three-peak structure is well reproduced, the beamlets’ positionsslightly differ: the matching is improved is the profile iscomputed 1.4 m upstream the actual position of the wirescanner.-20 -10 0 10 20x [mm]050100150200horiz. beam profile [a.u.]16:18:47 1070 msCore       [~5%]3 Islands [32%]superposition-20 -10 0 10 20x [mm]050100150200hor. beam profile [a.u.]measured profilesimulated profileFigure 3: Measured horizontal beam profile at the endof the capture with superimposed multi-Gaussian fit (left),and comparison with multi-particle MADX simulations in-cluding chromaticity and tune modulation (right).CROSSING THE 1/5 RESONANCEA similar manipulation was performed on July 27th,when five stable islands were created for the first time. Thebeam parameters were the same of the one-third resonancecrossing. The horizontal tune was initially set to 6.190,then increased linearly until the value of 6.213 in about 300ms (∼ 1.3×105 turns), as depicted in the left plot of Fig. 4.The vertical tune was kept constant to about 6.3 throughoutthe the splitting process. Conversely to the one-third reso-nance crossing experiment, the octupole was programmedto impart a large gradient K3 = 99 m−3 (-280 A) during afraction of the crossing (85 ms), to be then reduced at theend of process to K3 = 15 m−3 (-40 A). The sextupolecurrent was fixed to 350 A (K2 = 1.0 m−2) in order notto perturb the machine chromaticity. Data from the currenttransformer clearly show that no losses occur throughoutthe resonance crossing, the beam intensity remaining con-stant at about∼ 2.95× 1012 (see right plot of Fig. 4).900 950 1000 1050 1100 1150Time [ms]0.190.1950.20.2050.210.2150.22Horizontal tunehorizontal tune-300-250-200-150-100-500octupole current [A]octupole currentresonance crossing900 950 1000 1050 1100 1150Time [ms]280290300310320Number of protons per bunch       x10   resonance crossing10Figure 4: Left: measured horizontal fractional tune andtime evolution of the octupole current in the interval wherethe one-fifth resonance is crossed. Right: number of pro-tons per pulse measured by the current transformer.A simple way to scan over the octupole strength withouta repeated and time-consuming programming of the oc-tupole current curve was found by delaying the octupoleturn-on. Several measurements of the horizontal beamprofile were taken at a fixed time (1160 ms) and tune(Qx = 6.212), after retarding the octupole pulsing eachtime of 10 ms. Three cases are shown in Fig. 5. In thefirst case (top row) the resonance is crossed (at 1050 ms)with the octupole weakly powered, while the measurementis taken when its strength is maximum: five islands closeto the beam centre are then expected. In the second case(centre row) the octupole is set to impart the maximumstrength (I = −280 A) when the resonance is crossed. Theprofile in this case is measured when the octupole currentis halved (I = −140 A) yielding more separated islands.In the third and last case (bottom row) the octupole curveis further shifted towards left and the resonance is crossedwhen the octupole current is of about -50 A. The lower cur-rent generates less populated, but more separated, islands.The above considerations are confirmed by displayingthe phase space portraits computed by the MADX program-30 -20 -10 0 10 20 30 40x [mm]0200400600800Horizontal beam profile [arb. units] model phase portaitmeasured profileCore (87%)5 Islands (2.6%)Superposition750 800 850 900 950 1000 1050 1100 1150Time [ms]0.190.1950.20.2050.210.2150.22Horizontal tuneresonance crossinghorizontal tune-300-250-200-150-100-500octupole current [A]octupole current-30 -20 -10 0 10 20 30 40x [mm]0200400600800Horizontal beam profile [arb. units] model phase portaitmeasured profileCore (87%)5 Islands (2.6%)Superposition750 800 850 900 950 1000 1050 1100 1150Time [ms]0.190.1950.20.2050.210.2150.22Horizontal tuneresonance crossinghorizontal tune-300-250-200-150-100-500octupole current [A]octupole current-30 -20 -10 0 10 20 30 40x [mm]0200400600800Horizontal beam profile [arb. units] model phase portaitmeasured profileCore (87%)5 Islands (2.6%)Superposition750 800 850 900 950 1000 1050 1100 1150Time [ms]0.190.1950.20.2050.210.2150.22Horizontal tuneresonance crossinghorizontal tune-300-250-200-150-100-500octupole current [A]octupole currentFigure 5: Horizontal beam profiles (left) measured at theend of the one-fifth resonance crossing (1160 ms) for threedifferent settings of the octupole currents (right).on top of the the measured profiles, and by performing themulti-Gaussian fit, as shown in the left pictures of Fig. 5.The agreement between the fitted Gaussians and the is-lands’ position is excellent and unambiguous for the thirdcase, when the islands are well separated. When instead thedistance is reduced the fit may not be unique (first case),and the phase space portrait can be used to set some con-straints, such as the islands centroids.REFERENCES[1] R. Cappi and M. Giovannozzi, Phys. Rev. Lett. 88, 104801,and Phys. Rev. ST Accel. Beams 7, 024001 .[2] S. Gilardoni et al. , Phys. Rev. ST Accel. Beams 9, 104001,and Nucl. Instr. & Meth. in Phys. Res. A 561, 249.[3] M. Giovannozzi (Ed.) et al. , CERN Report, 2006-011, 2006.[4] F. Schmidt, Proc. of the 2005 PAC Conference in Knoxville,USA, pp.1272.

Experimental Evidence of Beam Trapping with one-third and one-fifth Resonance Crossing

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