The dynamic aperture (D.A.) of the LHC is estimated by tracking a thin-lens model of the lattice over a real machine time of 9 seconds. This raw result is then processed to predict the D.A. over longer times and realistic conditions. We discuss the reliability and limits of the method, the phenomena limiting the D.A. and scaling laws used for fast estimates. The long-term dynamic aperture at injection is close to the requirements, with prospects of further improvements

Böge, M

Faus-Golfe, A

Gareyte, Jacques

Grote, H

Koutchouk, Jean-Pierre

Miles, J

Qing, Q

Risselada, Thys

Schmidt, F

Weisz, S

CERN Document Server

SL Division AP GroupEUROPEAN ORGANIZATION FOR NUCLEAR RESEARCHEuropean Laboratory for Particle PhysicsOVERVIEW OF THE  LHC DYNAMIC APERTURE STUDIESBoege, M. ;  Faus-Golfe, A. ; Gareyte,  J.;   Grote, H. ; Koutchouk, J.-P. ;  Miles,  J. ; Qing, Q. ;  Risselada,  T. ; Schmidt, F. ; Weisz, S.The dynamic aperture (D.A.) of the LHC is estimated by tracking a thin-lens model of thelattice over a real machine time of 9 seconds. This raw result is then processed to predict theD.A. over longer times and realistic conditions.  We discuss the reliability and limits of themethod, the phenomena limiting the D.A. and scaling laws used for fast estimates. Thelong-term dynamic aperture at injection is close to the requirements, with prospects of furtherimprovements.Presented at 1997 Particle Accelerator Conference, Vancouver, 12-16 May 1997.Geneva, Large Hadron Collider ProjectCERNCH - 1211 Geneva 23SwitzerlandLHC Project Report 106Abstract2/6/97OVERVIEW OF THE LHC DYNAMIC APERTURE STUDIESM. Bo¨ge, A. Faus-Golfe, J. Gareyte, H. Grote, J.-P. Koutchouk, J. Miles,Q. Qing, T. Risselada, F. Schmidt, S. Weisz,CERN, Geneva, SwitzerlandAbstractThe dynamic aperture (D.A.) of the LHC is estimated bytracking a thin-lens model of the lattice over a real ma-chine time of 9 seconds. This raw result is then processedto predict the D.A. over longer times and realistic condi-tions. We discuss the reliability and limits of the method,the phenomena limiting the D.A. and scaling laws used forfast estimates. The long-term dynamic aperture at injec-tion is close to the requirements, with prospects of furtherimprovements.1 INTRODUCTIONIn the LHC, the accumulation of the beam will take 7 min-utes. Prior to this operation, the machine optics will haveto be tuned. Experience from other colliders indicates thatthis step may take from 30 minutes to several hours, de-pending on the previous history of the magnets. The lumi-nosity lifetime at peak energy is expected to be about 10hours. Consequently the lifetime of a single beam must besignificantly larger than one and ten hours respectively toallow efficient operation. Its estimation is however out ofreach of present computers. Instead we resort to comput-ing the dynamic aperture (D.A.) which is the largest stabletransverse amplitude in a connected domain for a sample of‘typical’ particles circulating in a simplified model of theaccelerator. Even this computation is very demanding andis limited in practice to about 10 seconds of real LHC time.Using the results of additional studies, especially measure-ments of the D.A. in the SPS and HERA, the simulationresults are extrapolated to realistic times and acceleratorcharacteristics.2 MODEL OF THE LHC MACHINEThe D.A. of LHC at injection is dominated by systematicmultipoles in the superconducting dipoles. The sextupoleb3and decapole b5are corrected by small coils at the endof each dipole. After correction with an accuracy betterthan 10%, these multipoles do not limit the D.A. The otherallowed systematic multipoles are weak by design and con-tribute little to the D.A. The non-allowed multipoles, in par-ticular a4, b4, a3may have significantly large bias inside aproduction line and limit seriously the D.A. For the timebeing we assume eight independent production lines, eachone being used to equip one arc. The average values of themultipoles are drawn from a Gaussian distribution cut at1.5. This severe cut reflects the capability of measuringand correcting trends during the fabrication.At injection, the multipole errors considered are thosemeasured about 15 minutes after the end of magnet cycling.The consequences of the decay of the persistent currentsand their spread from magnet to magnet, the ramp-inducedmultipoles related to the inter-strand resistance of the su-perconducting cable and the ramping rate have not beentaken into account so far. Our scaling laws indicate thatthey should not degrade the D.A significantly even thoughthey produce an increase of linear coupling.The model of the machine includes a realistic descrip-tion of the misalignment errors and linear imperfections.The closed orbit and linear coupling are corrected with themethods available in control rooms to 1 mm rms and aresidual closest tune approach of less than 0.001. The frac-tional betatron tunes are chosen to be .28/.31. Tune scanshave yet to be performed for the new optics and error table.3 METHODS FOR CALCULATING THEDYNAMIC APERTUREThe evaluation of the D.A. is limited mainly by the comput-ing resources and our ability to analyse the large quantityof data generated by tracking codes. Most of the choicesabout the methods result from these limitations.Each real magnet is represented by one thin lens, provid-ing symplecticity and fast integration. In collision, a finermodel of the low- triplet quadrupoles (localization of theimperfections and edges properly represented) is yet to beimplemented.Sixty different imperfect machines are tracked to esti-mate the lower bound of the stable motion (D.A.) with aconfidence level of 95%. A coarse tracking is done firstto localize the D.A., followed by fine mesh tracking justbelow the coarse D.A. estimate.To ensure a realistic sampling of initial conditions, 90initial xiand yiare chosen within a range of 3, withxi=yi= 1 for equal emittances and -functions in thetwo planes. The momentum deviation is initialized to 1.7e. Studies [1] showed an insignificant loss of D.A. whengoing closer to the bunch edge. The three correspondingphases are initialized to zero as they naturally sample theirphase space from turn to turn. Each initial condition isin fact a pair of two near-by particles (‘Lyapunov pairs’)whose distance is used to detect the onset of a chaotic mo-tion. In the course of tracking, the variation of the ampli-tude ratio xi=yiis usually small. A coarser tracking is occa-sionally done for five different amplitude ratios and used toestimate the loss of D.A. with respect to xi=yi= 1. A com-puter cluster made of 10 powerful workstations (the NAPProject [2]) is dedicated to tracking with either Sixtrack [3]or MAD [4]. The estimate of the ‘conventional’ D.A. asdefined above, takes less than 2 days of computation.Certain machines are tracked up to 106 or 107 turns (10minutes of LHC time ). The survival plots are fitted to aconjecture law to extrapolate to longer times. The onset ofchaos is used as a qualitative indication and may representa pessimistic estimate of the D.A. The effect of the ripple ofthe magnetic fields was studied for 60 machines [5] and theresult is included in the interpretation of the D.A. providedby tracking.For a fast evaluation of the D.A., the concept of D.A.per multipole dnturns out to be surprisingly effective. TheD.A. is tracked for a relevant value of one multipole n oneat a time [6]. For a -times larger value of multipole n,the equation of motion remains invariant if the amplitudes(and the D.A.) are divided by 1=(n 2). The D.A. for a setof multipoles is obtained by combining the dn’s followingthe conjecture in [7]:dn(bn= ) =dn(bn= 1)1=(n 2)1d4=Xn1d4n(1)So far, these scaling laws are verified to better than 10% bytracking in all cases considered.4 TARGET DYNAMIC APERTURETable 1: Relation between required and target D.A.’s.Source or Uncertainty Impact D.A. in Target D.A. at 105 turns 12.0Finite mesh size -5%Initial to average amplitude -5%Amplitude ratio xi=yi-5%Extrapolation to 4 107 turns -7% 9.6Time dependent multipoles -10%Ripple -10% 7.8safety margin -20% 6.2The halo of the injected beam will be cut at 3 in the in-jector. To allow for injection errors and orbit drifts, theprimary collimator will be positioned at 6 [8]. To avoidperturbing the collimation by non-linearities, we requirethe D.A. to be at least 6. This corresponds to observedgood conditions in SPS and HERA. The particles in thebeam secondary halo survive for up to 100 turns and mayreach an amplitude of 9. The amplitude smear must thenstay within limits to maintain a good collimation efficiency.In Table 1, the target D.A.for tracking is computed backfrom the required actual D.A.. The safety margin reflectsthe discrepancy between calculated and measured D.A.’sas observed in the SPS and HERA. The target D.A. fromtracking should be 12. It could reduce to 10.5 if timedependent effects and safety margins are not independentand combined quadratically.0510152025301 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16Amplitude [σ]Number of Random Distributions   Data     Gaussian  FitLoss BoundaryAverage     :  11.08Mininmum :    9.46Figure 1: Distribution of D.A.’s5 RESULTS AND ANALYSIS5.1 Dynamic apertureThe latest results [5] were calculated for LHC version4.3 [8] which has a lattice without tune split and a cell phaseadvance very close to 90 . The imperfection table 9607was used. Figure 1 shows that the minimum D.A. is 9.5.5.2 Survival plot1001'00010'000100'0001'000'00010'000'00010 11 12 13 14 15Amplitude [σ]Turns    Phase Space Averaged Loss Boundary    Conjecture FitFigure 2: Fitting law for the averaged D.A.After proper averaging in phase space [9], an inverse log-arithmic law fits well the LHC D.A. data (Figure 2) andseems to allow a reasonable extrapolation from 105 to4 107 turns.A further loss of 7% can be predicted.5.3 Contributions to the onset of unstable motionThe linear imperfections meaning corrected closed orbits(1 mm rms), a corrected betatron coupling and a residual-beating of 10%, account for a loss of 5% of D.A.Figure 3 shows that the amplitude detuning is only cor-related with the D.A. when it is sufficiently large (Q >0:005 at the D.A.).The main limitation of the D.A. comes from a4and b4.Figure 4 shows that the D.A. is significantly increased bycorrecting the average values of a4and b4with additionalcoils at the end of each dipole. The difference coupling res-onances (2Qx  2Qyand Qx  Qy) driven by these mul-tipoles appear well correlated with the D.A. results [10],78910111213140 0.01 0.02 0.03 0.04 0.05 0.06Amplitude [σ]Detuning Coefficientx Detuningz DetuningFigure 3: Relation between D.A. and amplitude detuning@Q=@J in mm 2suggesting that an improvement will be obtained from thenew optics with a tune split.05101520251 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16Amplitude [σ]Number of Random Distributions   Nominal b4 a4   After Correction of b4 a4Figure 4: Effect of a4, b4on the D.A.5.4 Influence of the second beam at injectionThe beams cross at an angle of 150 rad and are sep-arated by 1mm in the orthogonal plane. This gives analmost uniform separation of about 7. Optionally thescheme can be rotated by 450 to minimize the beam-beamtune footprint. Tracking has revealed a clear reduction ofthe 105-turn D.A. from over 10 to less than 7 (Figure 5)with 4 interaction regions. In the new version 5 of theLHC optics, an additional quadrupole allows a reductionof the maxin the triplet quadrupoles, thereby increasingthe beam separation to 11.5.5 D.A. at collision energyThe aim of the first tracking campaign carried out at colli-sion energy was to evaluate the requirements for the fieldquality in the low- triplet. For that purpose, an extremesituation with four physics insertions ( = 0:5 m) and thenominal crossing angle of 100rad was considered.Tuning D.A. ()= 6 m 37.7= 0:5 m 10.00.0 1.5 3.0 4.5 6.0 7.5 9.0 10.5 12.0 13.5 15.0DYNAPX<------ without beam-beamwith beam-beam --->0.01.53.04.56.07.59.010.512.013.515.0DYNAPYFigure 5: D.A. with/without a second beam at 7The beam-beam effect was disregarded. With the first esti-mates of the multipole field errors in the low- triplet, 6Dtracking [11] shows clearly that the D.A. is only limited bythe triplet errors and more precisely by a large b10compo-nent which will be reduced by design.6 CONCLUSIONSGiven the cost of safety margins, considerable care is be-ing taken to describe in the most realistic way the LHCmagnetic fields and the particle dynamics. The D.A., com-puted by element-by-element tracking, appears to be justsufficient at injection. We aim to gain a safety margin of20% by reducing or correcting the dominant errors a4, b4and a3. The effect of the systematic normal and skew oc-tupoles could be reduced in the new optics with tune split.At collision energy, without beam-beam effect, the D.A. islimited to 10 by the b10of the low- quadrupole whichwill be reduced.7 REFERENCES[1] F. Schmist, LHC Project Note 30, 1996.[2] E. McIntosh et al., Montreux 95.[3] F. Schmidt, CERN SL/94-56(AP),1994.[4] H. Grote and F.C. Iselin, CERN/SL/90-13rev4(AP), 1995.[5] M. Bo¨ge, F. Schmidt, this conference.[6] Q. Qin, S. Weisz, LHC Project Note 42, 1996.[7] V. Ziemann, Particle. Acc. Vol.55,pp.141-150, 1996.[8] LHC Conceptual Design (CERN/AC/95-05(LHC), 1995.[9] M. Giovannozzi et al.,LHC Project report 45, 1997.[10] Y. Cai, J.P. Koutchouk, LHC Project Note 75, 1996.[11] A. Faus-Golfe & A. Verdier, EPAC96, 1996.

Overview of the LHC dynamic aperture studies

Overview of the LHC dynamic aperture studies

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