Hanke, K

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Betatron Matching and Dispersion MatchingK. Hanke, CERN, Geneva, SwitzerlandAbstractThe TT2-TT10 transfer line optics has been modeled andoptimized in order to minimize blow-up at injection intothe SPS. Betatron and dispersion matching has been per-formed for the fixed-target proton and ion beams, as wellas for the future LHC beam. Based on the model, tuningknobs were developed to tune independently Twiss param-eters, dispersion and dispersion derivative. Coupled to themeasurement of the Twiss parameters in the line or in theSPS, they can be used for on-line mismatch correction. Thecorrection mechanisms are discussed and first experimentalresults are presented.1 INTRODUCTIONRecently there has been renewed interest in the optics ofthe TT2-TT10 transfer line between the PS and the SPSmachines at CERN. The main reason is that the line will bean important part of the LHC injection chain [1]. Given thetight emittance budget allowed for the LHC beam, Twissparameters and dispersion must be matched accurately inorder to prevent blow-up of the beam at injection into theSPS. Good matching is also of interest for the fixed-targetproton and ion beams which use the same beamline: min-imizing blow-up at injection results in an increased trans-mission through the SPS and hence in an increased inten-sity delivered to the targets. In addition, minimizing beamlosses means less activation of machine components.Proper matching relies both on a consistent model as wellas on a precise knowledge of the input parameters at thebeginning of the line. Considerable effort was spent onthese two items during the 1998 SPS run. In a first step, thecomplete TT2-TT10 line was modeled using the programpackage MAD [2]. The geometry of the model was cross-checked versus the official CERN survey data [3] and thecorrect magnetic behaviour of the elements was confirmedin a series of measurements [4]. Then, the initial parame-ters of all the beams concerned were measured and used asan input for the model.Based on the measured initial conditions and the verifiedmodel, the line was successfully rematched for the LHCbeam as well as for the fixed-target proton and ion beams.In the following sections we discuss the matching proce-dure as well as the results obtained for the various beams.In the last section we present a tuning tool to correct for theunavoidable residual discrepancy between simulation andreal machine.2 LHC BEAMExtensive studies were carried out during the 1998 SPS runusing the 26 GeV/c proton beam which is used for LHCstudies. The beam was provided on its own cycle and mea-surements could be done independently at a minimum in-terference with the physics program. The relevant beamparameters can be summarized asmomentum: p = 26 GeV/cmomentum spread: dp/p = 10−3normalized horizontal emittance: εx ≈ 3.0 µmnormalized vertical emittance: εy ≈ 3.0 µmSince for the LHC injection a maximum emittance growthof 0.5 µm is allowed from the PS to the SPS1, accurateTwiss parameter and dispersion matching is mandatory. Inparticular for this type of beam, where a small beam sizecoincides with a large momentum spread, blow-up due todispersion mismatch is a major concern.2.1 Betatron MatchingThe 1998 run was started with an optics matched to ini-tial conditions obtained from a MAD simulation of the fastextraction from the PS machine. The geometrical beta-tron mismatch obtained from a multi-grid measurement inTT10 was about 1.5 in the horizontal and 1.15 in the ver-tical plane2. During the 1998 run, Twiss parameters anddispersion were measured and tracked back to the PS ex-traction point. With this set of input parameters and thenewly generated set of MAD files a matching of the com-plete TT2-TT10 line was performed. The mismatch factordecreased in one iteration to 1.3 in the horizontal and 1.0in the vertical plane.2.2 Dispersion MatchingIn order to determine the initial values of D and D0, themomentum in the PS machine has to be varied. Typicallyfive different settings are applied. The displacement of thebeam in both planes is then observed at all monitors in theinjection line and in the SPS. Considering the first turn ofthe SPS as a continuation of the transfer line allows to makeuse of a large number of monitors the transfer matrices be-tween which are well known. A fit taking into account the1Note, that only a fraction of this is assigned to mismatch.2These values correspond to the geometrical mismatch factor. Themismatch factor after filamentation is much smaller, see L. Vos and G. Ar-duini, these proceedings.Chamonix IX110-20-15-10-50510152011906124091300920209210092170622309230093020931009318093240933009402094100941707422094300950209510095170752209530096020961009616066220963009102091100911833Ring positionDispersion [m]Figure 1: Measured horizontal dispersion of the LHC beam in the SPS before matching the injection line. The trianglesrepresent the measured data, the squares the theoretical values.-20-15-10-50510152011906124091300920209210092170622309230093020931009318093240933009402094100941707422094300950209510095170752209530096020961009616066220963009102091100911833Ring positionDispersion [m]Figure 2: Measured horizontal dispersion of the LHC beam in the SPS after matching the injection line. The trianglesrepresent the measured data, the squares the theoretical values.measured dispersion at all monitors and the transfer matri-ces between them yields the initial dispersion at the begin-ning of the line.Figure 1 shows the horizontal dispersion function in theSPS before and Fig. 2 after matching the injection line. Thedispersion amplitude was brought down to the design val-ues in one iteration. In the vertical plane, the measureddispersion was close to zero which corresponds as well tothe theoretical value.3 FIXED-TARGET PROTON BEAMMatching the line for the 14 GeV/c fixed-target protonbeam3 turned out to be more complicated than in the case ofthe LHC beam. The optics for this beam includes an emit-tance exchange insertion in the TT10 part of the line. Thequadrupoles in this section have a fixed setting and cannot3This beam is frequently referred to as ’continuous transfer’ or ’ct’beam.be used for matching. Another technical complication is,that the MAD model presently used for this beam splits upthe line into two parts before and after the emittance ex-change. The computation is then done in two steps whichmakes the matching even more complicated.The beam parameters for 1998 were as follows:momentum: p = 14 GeV/cmomentum spread: dp/p = 0.5 · 10−3normalized horizontal emittance: εx ≈ 10.0 µmnormalized vertical emittance: εy ≈ 7.5 µmThe fixed-target proton beam was provided on the ’maincycle’ of the SPS. In order not to interfere with the physicsprogram, measurements could only be done during dedi-cated MD time.111Chamonix IX3.1 Betatron MatchingRunning the line with the 1997 optics, the mismatch factorwas determined from a multi-grid measurement in TT10 toabout 2.0 in both planes. The same effort as in the caseof the LHC beam was made to determine a set of initialbeam parameters to be used as input for a rematching ofthe line. The Twiss parameters were determined using theSEM monitors in TT2 and TT10 and tracked back to thebeginning of the line. Based on the measured initial Twissparameters, the line was rematched. Several iterations werenecessary to achieve a reasonable result. For the best op-tics, we found a horizontal mismatch of 1.3 and a verticalmismatch of 1.1.3.2 Dispersion MatchingThe dispersion along the line and the SPS first turn wasmeasured and the initial conditions determined as in thecase of the LHC beam. The horizontal dispersion was al-ready before rematching close to the theoretical values. Itdid therefore not improve any more. The vertical disper-sion, which was of the order of 3 m before changing theoptics, decreased to almost zero after dispersion matching.4 FIXED-TARGET LEAD ION BEAMThe optics model used for the fixed-target lead ion beamcorresponds exactly to the one used for the LHC beam, thatis the optics without emittance exchange. The beam param-eters of the lead ion beam can be summarized as follows:momentum (per nucleon): p = 5.11 GeV/cmomentum spread: dp/p = 1.5 · 10−4normalized horizontal emittance: εx ≈ 3.4 µmnormalized vertical emittance: εy ≈ 1.9 µm4.1 Betatron MatchingThe 1998 run was started with the optics used during thelast lead ion run in 1996. The mismatch factor was deter-mined to about 1.8 - 1.9 in the horizontal and 2.1 in thevertical plane. Betatron matching was performed as for theother beams, and for the matched optics a mismatch factorof 1.2 in the horizontal and 1.3 in the vertical plane wasfound.Loading the matched optics resulted immediately in an in-crease of transmission through the SPS of about 20%. Fig-ure 3 shows the average number of ions per cycle duringthe day when the optics was changed. A step can clearly beseen at the time when the optics was put into operation.4.2 Dispersion MatchingSince the momentum spread of the beam was extremelysmall, dispersion mismatch was not a major concern. How-00:00 02:00 04:00 06:00 08:00 10:00 12:00 14:00 16:00 18:00 20:00 22:00Time0*1.0E10*2.0E10*3.0E10*4.0E10*5.0E10*6.0E10*7.0E10*8.0E10*Figure 3: Average number of Pb ions per cycle logged over24 hours. The step function in efficiency at 16:25 is due tothe installation of a new, matched optics in TT10. The dropat 14:00 is due to a power supply failure.ever, it was decided to measure the dispersion and to per-form a full Twiss parameter and dispersion matching. Be-fore matching, the horizontal dispersion had values of upto 15 m to be compared with a maximum theoretical valueof 5 m. It decreased to a maximum amplitude of 10 m afterrematching, which is still about twice the design value. Inthe vertical plane, a significant decrease of the dispersionamplitude from about 6 m to about 1 m was found. This isto be compared with the theoretical value of zero verticaldispersion in the SPS.5 TUNING TOOL FOR ON-LINECORRECTIONIn the previous sections we have shown the results ofglobal betatron and dispersion matching. Although accu-rate matching was performed in the simulation, some resid-ual mismatch is found in all cases. This is both due to anunavoidable discrepancy between model and real machineand due to errors in the measurement of the initial beamparameters.To correct for these effects, tuning knobs were developedto tune independently the beam parameters. They are basedon the inversion of the matrix(∂∆i∂Kj)(i, j = 1, ...8) where∆i = (αh(v), βh(v), Dh(v), D0h(v)) and Kj is the strengthof the jth matching quadrupole. In a first attempt, onlythe eight independent quadrupoles of the TT10 matchingsection have been considered. The 64 coefficients of thematrix were obtained from the MAD simulation. Invertingthe matrix yields immediately the change of quadrupolestrength that has to be applied in order to obtain a givenchange of any of the beam parameters.Frequently, as in our case, the coefficient matrix for a givenoptics will be either singular or numerically close to singu-lar. In these cases, it cannot easily be inverted. SingularChamonix IX112−20 −15 −10 −5 0 5 10 15 20708090100110120130140150160170180change of βx [m]β x [m]theoreticalMAD        measured   Figure 4: Horizontal β-function at the TT10 injection pointversus variation applied using the tuning knob. Startingfrom a matched optics (∆β = 0), the β-function is de-creased by 10 m and 20 m. The expected, simulated andmeasured behaviour of the β-function is shown.−20 −15 −10 −5 0 5 10 15 2011.522.5change of βx [m]horizontal mismatch factorFigure 5: Measured horizontal mismatch factor versusvariation applied using the tuning knob. Starting from amatched optics (∆β = 0), the β-function is decreased by10 m and 20 m.value decomposition (SVD) algorithms provide a tool todiagnose a matrix and to solve systems of equations of thegiven form even for ill-conditioned matrices [6]. An SVDalgorithm was applied to recondition the coefficient matrixand the resulting tuning tool was tested with the model.The simulation showed that the knob worked very well forall beam parameters except the horizontal β-function.−5 0 515202530change of βy [m]β y [m]theoreticalMAD        measured   Figure 6: Vertical β-function at the TT10 injection pointversus variation applied using the tuning knob. Startingfrom a matched optics (∆β = 0), the β-function is increasedand decreased by 2 m and 4 m. The expected, simulatedand measured behaviour of the β-function is shown.−5 0 511.051.11.151.21.251.31.351.4change of βy [m]vertical mismatch factorFigure 7: Measured vertical mismatch factor versus varia-tion applied using the tuning knob. Starting from a matchedoptics (∆β = 0), the β-function is increased and decreasedby 2 m and 4 m.5.1 Test with BeamThe tuning tool was tested with beam, using the SPS mis-match monitor [7]. This system is based on a turn-by-turnmeasurement of the beamsize with an OTR screen and afast CCD camera. The oscillation of the beamsize indicatesbetatron mismatch at injection into the SPS. Preliminarytests by drastically detuning the TT10 optics had provedthe sensitivity of the monitor. While the oscillation of thebeamsize gives only a qualitative measure of the mismatch,it is important to obtain also the values of α, β and the mis-match factor. This can be done by measuring the beam113Chamonix IXbeam β mismatch β mismatch dispersion mismatch dispersion mismatch(geometrical) (filamentation) (geometrical) (filamentation)LHC 1.3 (h) / 1.0 (v) 1.03 (h) / 1.00 (v) 4.63 (h) / 1.18 (v) 1.66 (h) / 1.00 (v)ct 1.3 (h) / 1.1 (v) 1.03 (h) / 1.00 (v) 1.46 (h) / 1.59 (v) 1.02 (h) / 1.03 (v)Pb 1.2 (h) / 1.3 (v) 1.02 (h) / 1.03 (v) 1.27 (h) / 1.04 (v) 1.01 (h) / 1.00 (v)Table 1: Measured betatron and dispersion mismatch factors for the various beams. The relevant quantity is the mismatchfactor after filamentation.profile at three successive turns in the SPS. The Twiss pa-rameters can then be obtained in the same way as from amulti-grid measurement in a transfer line.Since measurement of the Twiss parameters can be per-formed quickly, while measurement of the dispersion istime consuming, it was decided to detune the horizontaland vertical β-funcions starting from a matched setting andto measure Twiss parameters and mismatch factor using theSPS mismatch monitor. The β-funcions were detuned by±10% and ±20%. Figure 4 shows the result for the hor-izontal plane. The plot shows the expected change of theβ-function (dashed line), the result obtained from the simu-lation and the measured values. It can be seen, that alreadyin the simulation the expected variation is not achieved. Asfar as the measurement is concerned, only the points for∆β = 0, -10 m and -20 m could be measured due to a tech-nical problem. For the initial setting (∆β = 0), the mea-sured value lies already below the theoretical one. Thismeans that this setting is not perfectly matched, which isconsistent with a measured mismatch factor of 1.3 - 1.4. Ingeneral, the fluctuation is very large due to a horizontallyunstable beam which makes it impossible to draw a con-clusion.From the measured Twiss parameters, the mismatch factorcan be computed. Figure 5 shows the horizontal mismatchfactor versus change of βx for the same measurement. Thelarge fluctuations in the horizontal plane make a conclusionimpossible.The same measurement was done in the vertical plane. Thebeam was much more stable and all five settings could bemeasured. Figure 6 shows theoretical, simulated and mea-sured values of βy for five different settings of the tuningknob. All data agree within the statistical error.From the same measurement, the vertical mismatch fac-tor was computed. The result is shown in Fig. 7. For thematched optics (∆β = 0), a vertical mismatch factor of 1.1is found which is in perfect agreement with the result ob-tained from a multi-grid measurement in TT10. Detuningthe β-function at the injection point in both directions leadsto an increase of the mismatch factor as expected.6 CONCLUSIONAccurate measurement of the beam parameters in the TT2-TT10 injection line and consequent matching of the opticsto the measured values has significantly improved the SPSperformance during the 1998 run.For the LHC beam, the geometrical mismatch factor de-creased in one iteration from 1.5 (h) and 1.15 (v) to 1.3 (h)and 1.0 (v). The dispersion with the matched optics is nowin both planes close to the theoretical values.For the fixed-target proton beam, the mismatch factor couldbe decreased from 2.0 in both planes to 1.3 (h) and 1.1 (v).The horizontal dispersion was already close to the theo-retical values and did not improve any more. The verticaldispersion, which had a maximum amplitude of 3 m beforematching, is now close to zero (theoretical value).A spectacular success was achieved for the fixed-targetlead ion beam. By accurate Twiss parameter matching, thetransmission through the SPS and hence the intensity deliv-ered to the targets could be increased in one step by about20%. The measured mismatch factors were 1.9 (h) / 2.1 (v)before and 1.2 (h) / 1.3 (v) after matching. The dispersioncould slightly be improved in the horizontal plane and sig-nificantly improved in the vertical plane.On top of the matched optics, a tuning tool was developedand successfully tested for the LHC beam.Table 1 shows the measured mismatch factors for betatronand dispersion mismatch for the various beams. Whilethe measured values correspond to the geometrical mis-match factor, the mismatch after filamentation is muchsmaller [5]. A critical value is the blow-up due to disper-sion mismatch for the LHC beam. This problem is likely tobe overcome, either by another iteration or by applying thetuning tool which has been shown to work excellent for Dand D0 in the simulation.7 ACKNOWLEDGEMENTSThe author would like to thank the SL-OP team for theirsupport during the measurements as well as the colleaguesfrom the CERN PS division for the excellent collaboration.I would furthermore like to thank G. Arduini and M. Gio-vannozzi for their contribution to this work and R. W. Ass-mann, P. Raimondi and J. Uythoven for fruitful discussions.8 REFERENCES[1] P. Collier (Ed.), The SPS as Injector for LHC, ConceptualDesign, CERN-SL-97-07 DI (1997).[2] H. Grote, F. C. Iselin, The MAD Program, User’s ReferenceManual, CERN-SL-90-13 AP (1990).[3] G. Arduini, M. Giovannozzi, K. Hanke, J.-Y. He´mery,M. Martini, MAD and BeamOptics Description of theChamonix IX114TT2/TT10 Transfer Lines - Part I: Optics without EmittanceExchange Insertion, CERN PS-Note-98-014 CA and CERNSL-Note-98-040 OP (1998).[4] G. Arduini, M. Giovannozzi, K. Hanke, J.-Y. He´mery, Studyof the TT2/TT10 Transfer Line Optics via Transfer MatrixMeasurement, CERN PS-Note-98-020 CA and CERN SL-Note-98-056 MD (1998).[5] G. Arduini, these proceedings; L. Vos, these proceedings.[6] Numerical Recipes in Fortran, The Art of Scientific Comput-ing, 2nd edition, Cambridge University Press (1986).[7] C. Bovet, R. Jung, A New Diagnostic for Betatron PhaseSpace Matching at Injection into a Circular Accelerator, LHCproject report 3, rev. (1996);H. Schmickler, R. Jung, these proceedings.115Chamonix IX

Betatron-Matching and Dispersion Matching

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