The limitations of the dynamic aperture due to field errors of the super-conducting magnets is a notorious problem for the LHC. Given the large amount of independent studies performed by a sizeable research team it becomes necessary to define a common tracking strategy. The emphasis is placed on an elaborate on - and off - line processing of the tracking data making use of all tools presently available. To manage the very time-comsuming investigations our approach is two-fold: firstly we are maximising the computing power running optimised code on state of the art equipment which is continuously upgraded and secondly we speed up the studies by using reliable and automated early indicators for long-term losses. The procedure is exemplified with a series of tracking runs for the LHC version 4 at injection

Böge, M

Grote, H

Qin, Q

Schmidt, F

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EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCHEuropean Laboratory for Particle PhysicsDYNAMIC APERTURE STUDIES FOR THE LHC VERSION 4M. Boge*,  H. Grote*, Q.Qin☯  and F. Schmidt*The limitations of the dynamic aperture due to field errors of the super-conducting magnets is anotorious problem for the LHC.  Given the large amount of independent studies performed by asizeable research team it becomes necessary to define a common tracking strategy.  The emphasis isplaced on an elaborate on - and off - line processing of the tracking data making use of all toolspresently available.  To manage the very time-comsuming investigations our approach is two-fold:firstly we are maximising the computing power running optimised code on state of the art equipmentwhich is continuously upgraded and secondly we speed up the studies by using reliable and automatedearly indicators for long-term losses.  The procedure is exemplified with a series of tracking runs forthe LHC version 4 at injection.* Division SL☯ On leave of absence from IHEP, Beijing, P.R. ChinaPaper presented at the 5th European Particle Accelerator Conference, EPAC96, Sitges, (Barcelona), Spain, 10-14 June 1996Geneva, Large Hadron Collider ProjectCERNCH - 1211 Geneva 23SwitzerlandLHC Project Report 31Abstract16/7/96DYNAMIC APERTURE STUDIES FOR THE LHC VERSION 4M. Bo¨ge, H. Grote, Q. Qin and F. Schmidt, CERN, Geneva, SwitzerlandAbstractThe limitations of the dynamic aperture due to field errorsof the super–conducting magnets is a notorious problem forthe LHC. Given the large amount of independent studiesperformed by a sizeable research team it becomes neces-sary to define a common tracking strategy. The emphasisis placed on an elaborate on– and off–line processing ofthe tracking data making use of all tools presently avail-able. To manage the very time–consuming investigationsour approach is two–fold: firstly we are maximising thecomputing power running optimised code on state of the artequipment which is continuouslyupgraded and secondly wespeed up the studies by using reliable and automated earlyindicators for long–term losses. The procedure is exempli-fied with a series of tracking runs for the LHC version 4 atinjection.1 PREPARATION1.1 ToolsFor the tracking studies of the LHC the programs MAD [1]and SIXTRACK [2] are used. MAD8 is the LHC workhorsewhile SIXTRACK allows an independent check and is usedfor special purposes. Most results of this report were ob-tained with the latter program.It was recognised that for a comparison between the twoprograms a more automated transfer procedure was needed.This MAD2SIX [3] transfer procedure allows to create thenecessary input files needed for SIXTRACK including a fulllist of all systematic and random magnetic imperfectionsstarting from the MAD data base. Survival plots obtainedwith the two programs for LHC version 4.1 are shown inFig. 1. The agreement is good up to the maximum trackedturn number of 100,000. In fact, a better agreement cannotbe expected due to the chaotic behaviour of the tracked par-ticles.To allow the usage of SIXTRACK for non–experts amenu–driven tool was written [4]. This tool, using theMAD2SIX transfer program, allows to produce error–freeinput files for a series of different tracking runs, to start andcontrol those runs and to post–process the produced track-ing data including interactive graphical display.During the latest tracking campaign it became apparentthat for a successful study of a large sample of different ma-chine set–ups a high level of automationwas needed for bothOn leave of absence from IHEP, Beijing, P.R. Chinatracking and post–processing. For the time being, this hasbeen realized with simple Shell scripts. However, for thenew tracking computer facility [5], to be installed at CERNin the near future, more modern tools will be used. It isplanned to have a graphically structured control of the com-plete tracking study using the facilities of the Open Inventorprogram package. In this way the involved tracking proce-dure (see below) can be provided using any of the two track-ing programs. But at the same time, this graphical structurewill allow modifications and extensions with utmost ease.1001000100001000006 8 10Horizontal Amplitude [sigma]Loss Turn NumberMADSIXTRACKFigure 1: Comparison of MAD and SIXTRACK1.2 Tracking ProcedureDynamic aperture studies for hadron machines are natu-rally very time consuming as the particles have to be fol-lowed over many turns. Usually many machine configura-tions have to be tested and the dynamic aperture itself de-pends on many parameters. To arrive at consistent resultsthere is the need for an elaborate tracking procedure, in par-ticular when the work is done in a team. The objectives andtools 1 are: Detection of pathological cases before long trackingruns are launched.1The program MAD is now providing similar tools [6]. Extensive post–processing of tracking data to under-stand the nonlinear behaviour of the motion and tolimit the number of parameters to be studied. Carry the tracking to the largest possible turn numberfor a minimum set of parameters and record the resultin survival plots. Use early indicators for long–term particle stability.To this end the tracking studies were done in several steps:firstly short–term tracking were done for each case to findthe detuningas a functionof momentum deviationand trans-verse amplitude. A rough estimate of the dynamic apertureand the onset of chaos was determined as well. This allowedto detect unreasonably bad cases and served as a base for thelonger term tracking. Secondly medium–term tracking wascarried out over 100,000 turns over the amplitude intervalbetween short–term losses and chaotic border. Various pa-rameters were tested: Instead of varying the initial phases 1 of the ampli-tude variations (smear) and the maximum and min-imum amplitude were calculated from the trackingdata. The dependence of the dynamic aperture on the ratio ofthe horizontal and vertical amplitudes was evaluated. The effect of the longitudinal oscillation on the dy-namic aperture was evaluated by varying the initialmomentum deviation.Finally some cases were prolonged to 1,000,000 turns whichis still only 10% of the expected LHC injection period.1E+031E+041E+051E+061E+070 2 4 6 8 10 12 14Initial Amplitude [sigma]Turn NumberTrack Limit ChaoticBorderFigure 2: Survival Plot for LHC V4.2Fig. 2 shows a typical example of a survival plot obtainedfor LHC version 4.2. It shows that the loss border is re-duced by one third when the tracking is prolonged from1000 to 1,000,000 turns. It also shows that the chaotic bor-der found using only 100,000 turns stays some 20% belowthe 1,000,000 turn loss border. It has to be mentioned that avery fine amplitude scan is needed to obtain meaningful sur-vival plots. Otherwise large errors may be introduced dueto the fact that at any given amplitude the loss turn numbermay vary by a factor of ten or even more.2 RESULTS FROM TRACKINGTwo tracking campaigns were carried out for the recentsubversions of the LHC lattice 4. Optically these subver-sions are not significantly different, but version 4.2 includesnewly defined multipolar errors. The new error tables havelarge systematic parts which vary from octant to octant dueto different production lines of the magnet vendors.2.1 Version 4.105100 5 10Horizontal Amplitude [sigma]Vertical Amplitude [sigma]100,000 Turn Loss BorderChaosTruly RegularFigure 3: Dynamic Aperture for LHC V4.1The main result for the subversion 4.1 [7] can be found inFig. 3: the tracking was done for 9 random distributions(called “seeds” in the following) and 5 amplitude ratios.For each case the 100,000 loss border, the border of strongchaotic motion and the border of truly regular motion wasdetermined. In the figure the data points are averaged overthe 9 seeds and shown with their 1 error bars. The circlesindicate the average over the 5 data points. We concludedthat the dynamic aperture of LHC version 4.1 was just suf-ficient: the averaged loss border of 10 sigma for 100,000turns, where sigma is the transverse r.m.s. beam size, givesthe safety margin needed to obtain the required long–termdynamic aperture of 6 sigma once all other effects, whichare disregarded in this study, are included.This 6d tracking was done with the initial momentum de-viation at 75% of the bucket size. A reduction of the ini-tial momentum deviation to very small values led to an in-crease of the dynamic aperture of 25%. Finally, it can be re-ported that the detuning with amplitude showed a dramaticincrease, in the off–momentum cases, once the b5spoolpiece correction in the magnet ends was turned off. Thisdemonstrates how the automatic recording of the detuningcan become useful to avoid “blind” long–term studies.2.2 Version 4.2For version 4.2 it was decided that the dynamic apertureshould be determined such that a lower bound can be stated.A dynamic aperture below this bound should be excluded bya 95% probability. To this end a minimum of 60 differentseeds had to be studied.AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA051015201 2 3 4 5 6 7 8 9 10 11 12 13 14Amplitude [sigma]Number of Random DistributionsNoCorrection(Chaos)AAAAAAAANoCorrection(Lost)Correctionof a4 and b4(LOST)Figure 4: Influence of Octupoles on Loss DistributionThe short–term studies showed that the detuning due tononlinear chromaticity stayed within 0:005 at 75% of thebucket in all cases. There was however a considerable trans-verse detuning of up to 0.007 for some seeds at an amplitudeas small as 5 sigma. The smear at 6 sigma was 12% on av-erage, but reached a maximum value of 34% for one seed.Nevertheless, it was found that the initial starting amplitudeagreed to about 3% with the amplitude averaged over 1000turns.The results from long–term tracking are summarised inFig. 4: The averaged loss border is 9.2 sigma ranging from7.2 to 13.8 sigma, while the chaotic border is located on av-erage at 7.4 sigma in a range between 4.6 and 12.5 sigma.The assumption of a Gaussian distributionis consistent witha 2 test in both cases, the standard deviations being 1.4 and1.5 sigma respectively.The conclusion is that the safety margin is no longer suf-ficient for version 4.2 with the large “systematic per arc” er-rors assumed in this study. As shown elsewhere [8] the cul-prits are b4and a4. A simple correlation plot of the dynamicaperture versus the integral values around the machines ofthe octupolar components (see Fig. 5) reveals a moderatecorrelation (r = 0:78). The integral value of octupolescan be corrected with additional (b4; a4) spool pieces placednext to the existing b3and b5spool pieces in the magnetends. This correction leads to a dramatic increase of thedynamic aperture to an average of 12.1 sigma with a lowerbound of 10.2 sigma. For one case the correction was alsoattempted using normal and skew octupoles close to the cellquadrupoles. With this technique the improvement was halfthat obtained with the spool pieces.024681012140 0.02 0.04 0.06 0.08SQRT (  (b42  +  a42 ) / 2  )Loss Border (< 1E5 turns) in [sigma]Figure 5: Dynamic Aperture versus Octupole Strength3 CONCLUSIONSThe general conclusions are that brute force tracking hasto be complemented with extensive post–processing of thetracking data. It is also mandatory to define, in detail, atracking procedure which has to be fully automated makinguse of modern tools which combine flexibility with utmostuser–friendliness.For the LHC lattice version 4 it can be concluded that thedynamic aperture is sufficiently large. However with sys-tematic errors as large as assumed in this study octupole cor-rections are needed in the magnet ends.4 REFERENCES[1] H. Grote and F.C. Iselin, “The MAD Program, User’s Refer-ence Manual”, CERN SL/90–13(AP) (Rev. 4) (1995).[2] F. Schmidt: “SIXTRACK version 1.2”, CERN/SL/94–56.[3] H. Grote, to be published.[4] M. Bo¨ge, “6T – A Tool for Preparation, Controlling andPost–processing of Tracking Runs”, to be published.[5] E. McIntosh, et al., LHC95 Workshop, Montreux, 1995.[6] F.C. Iselin and F. Ruggiero, CERN SL/96–11(AP).[7] F. Schmidt, LHC Project Note 30 (1996).[8] Q. Qin and S. Weisz, LHC Project Note 42 (1996).

Dynamic aperture studies for the LHC version 4

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