/Brookhaven

/Fermilab

Jansson, A.

Kopp, S.E.

Miyamoto, R.

Syphers, M.J.

U., /Texas

An AC dipole magnet produces a sinusoidally oscillating dipole field with frequency close to betatron frequency and excites large sustained oscillations of beam particles circulating in a synchrotron. Observation of such oscillations with beam position monitors allows direct measurements of a synchrotron's nonlinear parameters. This paper presents experimental studies to measure perturbative effects of sextupole and octupole fields, performed in the Fermilab Tevatron using an ACdipole

Syphers, M. J.

Kopp, S. E.

UNT Digital Library

BROOK .VENNATIONA'L LABORATORYBNL-82366-2009-CPNonlinear dynamics studies in the Fermilab tevatronusing an AC dipoleR. Miyamoto,Brookhaven National Laboratory, Upton, NY 11973, USAA. Jansson, M. J. Syphers,Fermi National Accelerator Laboratory, Batavia, IL 60510, USAS. E. Kopp,University of Texas at Austin, Austin, TX 78712, USAPresented at the Particle Accelerator Conference (PAC09)Vancouver, B.C., CanadaMay 4-8, 2009Collider-Accelerator DepartmentBrookhaven National LaboratoryP.O. Box 5000Upton, NY 11973-5000www.bnl.govNotice: This manuscript has been authored by employees of Brookhaven Science Associates, LLC underContract No. DE-AC02-98CH10886 with the U.S. Department of Energy . The publisher by accepting themanuscript for publication acknowledges that the United States Government retains a non-exclusive, paid-up ,irrevocable, world-wide license to publish or reproduce the published form of this manuscript, or allow othersto do so, for United States Government purposes.This preprint is intended for publication in a journal or proceedings. Since changes may be made beforepublication, it may not be cited or reproduced without the author's permission .DISCLAIMERThis report was prepared as an account of work sponsored by an agency of theUnited States Government. Neither the United States Government nor anyagency thereof, nor any of their employees, nor any of their contractors,subcontractors, or their employees, makes any warranty, express or implied , orassumes any legal liability or responsibility for the accuracy, completeness, or anythird party's use or the results of such use of any information, apparatus, product,or process disclosed, or represents that its use would not infringe privately ownedrights. Reference herein to any specific commercial product, process , or serviceby trade name, trademark, manufacturer, or otherwise, does not necessarilyconstitute or imply its endorsement, recommendation, or favoring by the UnitedStates Government or any agency thereof or its contractors or subcontractors.The views and opinions of authors expressed herein do not necessarily state orreflect those of the United States Government or any agency thereofNONLINEAR DYNAMICS STUDIES IN THE FERMILAB TEVATRONUSING AN AC DIPOLE*R. Miyamoto, Brookhaven National Laboratory, Upton, NY 11973, USAA. Jansson, M. J. Syphers, Fermi National Accelerator Laboratory, Batavia, IL 60510, USAS. E. Kopp, University of Texas at Austin, Austin , TX 78712, USAAbstractAn AC dipole magnet produces a sinusoidally oscillat-ing dipole field with frequency close to betatronfrequencyand excites large sustained oscillations of beam particlescirculating in a synchrotron. Observation of such oscilla-tions with beam position monitors allows direct measure-ments of a synchrotron's nonlinear parameters. This pa-per presents experimental studies to measure perturvativeeffects of sextupoleand octupole fields, performed in theFermilab Tevatron usingan AC dipole.INTRODUCTIONwhere n is number of the beam revolution, S and Sac arelongitudinal positionsof the BPM and AC dipole, Ad andX are constants, and v« is driving tune", The constant Ad,determining the amplitude, is proportional to the amplitudeof the ACdipole's currentandto a factor1/ sin[1T"(vd -II) ].We typicallychoose the driving tune lid near the betatrontune II to make large amplitude oscillations. In the ACdipole's oscillation, the amplitude function pCs) and phaseadvance 'iNs) are modulated to I3d(s) and 'ljJd(S) , depend-ing on the tunes v« and v [6, 7].CENTRAL ORBIT DISTORTION DUE TOSEXTUPOLE FIELDS3The definition of I/d (0 S I/d < 1) is the following 16. 7}. We sup--pose that lrev and I ac are frequencies of the beam revolution and the ACdipole field and I/ac is the fractional part of lac/IMV' When the betatrontune 1/ (0 S II < 1) and I/ac satisfy II/ac - 1/1< 11 - I/ac - II I,I/d = I/ac,and when 11 - I/ac - 1/1< II/ac - II I. I/d = 1 - I/ac.The centralorbit distortiondue to the sextupole fields isproportional to strengthsof sextupole fields and A~, whichis proportional to the square of the amplitude of the ACdipole's current. To observe such properties of the effect,we performed an experimental study in the Tevatron us-ing the 150GeV proton beam. In our study, one sextupolemagnetis controlledto make small changesand to test ourmeasurement methods.Fig. 1 showsturn-by-turnpositionof theACdipole's os-cillation observed in the Tevatron. Here. the amplitude ofthe ACdipole's current is adiabatically rampedup (1,000th- 3,OOOth turns)andrampeddown(6,OOOth - 8,OOOth turns),and such adiabatic process preserves the beam emittanceA modem synchrotron uses magnets producing non-linear fields, such as sextupole and octupole magnets, tocompensate chromaticity and instabilities of a high inten-sity beam. Magnet imperfections also produce nonlinearfields. In addition to compensating the corresponding ef-fects, these nonlinear fields perturb the beam's transversemotion. Forinstance,thesextupole fields distortthebeam'scentral orbit and the octupole fields shift the betatron tunevI depending on a beamparticle's amplitude (nonlinear de-tuning effect). These fields also drive modes with tuneshigher than the betatron tune. It is ideal if we could mea-sure and compensate such perturbative effects to fine-tunea synchrotron, but even the measurements are not alwayseasy and not many measurements have been done in theFermilab Tevatron so far [1, 2].An AC dipole magnet, a new diagnostic tool of a syn-chrotron [3], has been implemented to the Tevatron [4, 5,6]. Its oscillating dipole field with frequency close to thebetatronfrequency can excite largesustainedbeam oscilla-tions with no emittance growth. Observation of such oscil-lations with beam position monitors (BPMs) allows mea-surementsof not only linear but also nonlinear propertiesof the beam. In this paper,we discussexperimental studiesto observethe perturbative effectsdue to sextupole and oc-tupolefields. performedin the Tevatron withthe ACdipole.In a linearand uncoupled synchrotron-,turn-by-turn po-sition of the ACdipole's oscillationis givenby [6, 7]Xd(n;s) =4lJ(jd(S) cos [21rVdn + VJd(S) - '¢d(Sac) + X] , (1)2000 4000 6000number of revolutions8000..Work supported by Fermilab Research Alliance. LLC under ContractNo. DE-AC02-07CHlI359 with the United States Department of Energy.1In this paper, 1/ denotes the fractional part of the betatron tune.2We only consider one dimensional motion of the beam particles, here.Figure 1: Turn-by-tum positionof the AC dipole's excita-tion observed in the Tevatron, indicating a shift of the av-erage position during the excitation. The average positionbefore the excitation is subtractedfrom the data points.NONLINEAR DETUNING EFFECT DUETO OCTUPOLE FIELDSdotted line represents the effects of such residual sextupolefields.Fig. 3 shows the measured central orbit distortion, whenthe current of the controlled sextupole magnet is set to var-ious values while holding other quantities constant (left)and when the amplitude of the AC dipole's current is set tovarious various values, changing the oscillation amplitude,while holding other quantities constant. We may observethat the effect is proportional to the current of the givensextupole magnet and also the square of the amplitude ofthe AC dipole's current, as expected. In this way, the ACdipole allows us to observe changes of the central orbit dis-tortion due to the sextupole fields in the Tevatron.2I 1\:'I~ .~~I N0'VrIseaurole locatio1- I_~ . O A In - 1_ .. - lO A2location in the TevatronFe DO co Doo.Figure 2: Central orbit distortions due to sextupole fieldswhen the current of the controlled sextupole magnet is 0 A(nominal) and 30 A. When the current is increased to 30 A,the effect grows as expected (",80 Jim) and shows a cuspstructure, indicating location of the sextupole magnet.2000 3000longilUlinallD sition [m]4The group referred to as OD . consisting of 18 octupole magnets .The nonlinear detuning effect due to the octupole fieldsis proportional to strengths of octupole fields and also A3.To observe such properties of the effect, we performed an-other experimental study with the 150 GeV proton beam inthe Tevatron. Here, strength of one group of octupole mag-netstis controlled to make small changes and to test ourmeasurement methods.As we discussed previously, the amplitude of the ACdipole's oscillation is proportional to the amplitude of theAC dipole's current and the factor 1/ sin [7f (Vd - v)]. Fig.4 shows relations, observed by one BPM, between thebeam's oscillation amplitudes and the amplitude of the ACdipole's current. Here, the beam's amplitude is multipliedby a factor sin 17f(Vd -vo) I, where Vo is the betatron tune ofa beam particle in the limit of a zero oscillation amplitude.In our study's condition, the detuning effect raises the be-tatron tune along with the growth of the beam's amplitude.In such a condition, if the driving tune is smaller than thebetatron tune, the betatron tune moves away along with the[3, 6, 12]. We see that the average position is shifted up-ward while the amplitude is maximum. This shift is thecentral orbit distortion due to the sextupole fields. Fig. 2shows the central orbit distortion measured in this way atall the BPM locations. The dotted and solid lines representthe cases when the current of the controlled sextupole mag-net is 0 A (the nominal value) and 30 A. As expected, themeasurements shows a larger central orbit distortion whenthe current is increased to 30 A and also the cusp struc-ture indicating the source location. In these measurements,the amplitude function and the oscillation amplitude at thelocation of the controlled sextupole magnet, Ssx , are about(Jd(Ssx) ~ 94 m and AdV{Jd(Ssx) ~ 4 mm, and the effec-tive strength of the sextupole magnet is about B"i/ (B p) ~420 nrad/mm? when its current is 30 A [9]. In such a condi-tion, this sextupole magnet is equivalent to a dipole magnetwith a deflection angle ~A~{Jd(Ssx)B"i/(Bp) ~ 1.7 Jirad[6] and the expected central orbit distortion is about 80 Jimat the BPM locations, where the amplitude function is(J(s) ~ 95 m. We may see that this estimate is not far fromthe measurement in the figure . We note that some sextupolemagnets are also used in the condition of our study and the1_ =10(A] witroul delulling(calcuIJIl ioo)J.cipol.: = 280 ABPMt*5! BP~). BPM&t5...... , :..~-<>-~~~~::o::::-~-A~-:::.:. ~~~.sextupole cum:nt (AI1cs. IQ = 30 AIr-- - - - --, - A_BPMltS2 BPJ,C:ISl BPMtS;S4 ......-<>-~--A- -,..uan: of the AC dipole current [A' I1=0.20ri~ D. 15)(~a 0.10'5.~ACdip;>1eeum:ol [AIFigure 3: Central orbit distortions observed by three BPMs.The effect is proportional to the current of the controlledsextupole magnet (left) and the square of the amplitude ofthe AC dipole's current (and hence A~) (right). The differ-ence of the slopes is due to phase dependence of the effect.Figure 4: Amplitude of the driven beam (multiplied by afactor sin 17f(vd - vo)nvs. amplitude of the AC dipole'scurrent. The detuning effect either suppress or enhance thebeam's amplitude. The deviation of the beam's amplitudefrom the linear growth allows to determine the detuningeffect.0.2 0.4 0.6 0.8 1.0tune (fractionalpart)v. _ I.. dipole" 70 [A]- I... dipole" 280 [AIBPMtl93[edu" lOlA]I~ 0.1~ 0.01~.§ 0.001&0.002,.<l0.0000 20 40 60 80 100 120BPM'sFigure 5: Tune shifts measured with all the BPMs when thedriving tune and the maximum amplitude of the AC dipoleare fixed to v« - Va = -0.015 and 280 A, producing 5 mmoscillation amplitude in the arc.Figure 6: Fourier spectra of turn-by-turn positions of theAC dipole's excitations. The peaks at 211d. 1-2Vd. 3Vd, and1-3vd, corresponding to the modes driven by the sextupoleand octupole fields, appear only for the larger excitation.indicating these modes depend on the oscillation amplitudeof the AC dipole's excitation.beam's amplitude growth and the beam's amplitude is sup-pressed through the factor 1/ sin [1r(Vd - v)], compared tothe linear growth in the case of no detuning effect. On thecontrary, if the driving tune is larger than the betatron tune,the betatron tune approaches to the driving tune along withthe beam's amplitude growth and the beam's amplitude isenhanced. By observing such modulations of the beam'samplitude, we can measure the detuning effect [10].Fig. 5 shows the tune shift measured in this way withall the BPMs, when the current of the controlled octupolemagnets are changed from 1 A to 10 A (the nominal is 7 A)and when the maximum amplitude and driving tune of theAC dipole are kept the same to produce ,,-,5 mm oscilla-tions in the arc of the Tevatron. For our octupole magnets,the effective focal length is 260 km when its current is 1 Aand the beam's amplitude is 5 mm [9]. The amplitude func-tions at the locations of these octupole magnets are aboutf3(s) ~ 90 m. Under these conditions, when the currentof these controlled octupole magnets is changed by 3 A,the estimated tune shift is about 0.0015, which is roughlyconsistent with the gaps between curves. These measure-ments are based on the beam's amplitude but the amplitudefunction of the AC dipole's oscillation f3d( s) also dependson the betatron tune. The growth of the deviations overBPMs is expected due to the change of f3d(S) induced bythe detuning effect. In this way, the AC dipole allows us toobserve changes of the nonlinear detuning effect due to theoctupole fields in the Tevatron.MODES WITH HIGHER TUNESIn addition to the effects discussed in previous sections,when the beam is driven by the AC dipole, the sextupoleand octupole fields drive modes with tunes 2Vd and 3Vd, foreach, and magnitudes of these modes nonlinearly dependon the constant Ad. Fig. 6 shows the Fourier spectra of theturn-by-turn positions of the AC dipole's excitation. Wemay see .that the peaks at 2Vd, 1 - 2Vd, 3Vd, and 1 - 3Vd,corresponding to the modes of the sextupole and octupolefields, appear only for the larger excitation, indicating thedependence of these modes on Ad. By measuring heightsof these peaks with BPMs around a synchrotron, we canmeasure corresponding resonance driving terms [11. 12].CONCLUSIONOur studies in the Tevatron demonstrated that the ACdipole allows us to measure changes of the central orbitdistortion due to the sextupole fields and the nonlinear de-tuning effect due to the octupole fields. We also demon-strated that the AC dipole allows us to observe the highertune modes driven by these fields. The measurements pre-sented in this paper are non-destructive to the beam qualityand can be performed on a routine basis, and hence may beuseful for a quick diagnostic of a synchrotron's nonlinearfields.ACKNOWLEDGMENTAuthors would like to thank to members of the FermilabTevatron Department for their supports.REFERENCES[1] A. Chao et al., Phys. Rev. Lett. 61, p. 2752 (1988).[2] Y.Alexahin et al, EPAC'06, Edinburgh, 2006. p. 2140 .[3] M. Bai et al, Phys. Rev. E 56, p. 6002 (1997).[4] R. Miyamoto et al, BIW'06. Batavia,2006, p. 402.[5] R. Miyamoto et al, PAC'07, Albuquerque, 2007, p. 3868.[6] R. Miyamoto, Ph.D. thesis , Univ. of Texas at Austin , 2008.[7] R. Miyamoto et al, Phys. Rev. ST-AB 11,084002 (2008).[8] R. Tomas, Phys . Rev. ST-AB 8,024401 (2005).[9] H. T. Edwards, Ann. Rev. Nucl. Part. Sci. 35, p. 605 (1985).[10] L. D. Landau and E. M. Lifshitz, "Mechanics Third Edi-tion" , Butterworth-Heinemann, Oxford, UK, 1982, pp. 58-95.[11] F. Schmidt et al, CERN AB-Note-2003-031 MD (2003) .[12] R. Tomas et al, Phys. Rev. ST-AB 8,024001 (2005).

Nonlinear dynamics studies in the Fermilab tevatron using an AC dipole

NONLINEAR DYNAMICS STUDIES IN THE FERMILAB TEVATRONUSING AN AC DIPOLE∗R. Miyamoto, Brookhaven National Laboratory, Upton, NY 11973, USAA. Jansson, M. J. Syphers, Fermi National Accelerator Laboratory, Batavia, IL 60510, USAS. E. Kopp, University of Texas at Austin, Austin, TX 78712, USAAbstractAn AC dipole magnet produces a sinusoidally oscillat-ing dipole field with frequency close to betatron frequencyand excites large sustained oscillations of beam particlescirculating in a synchrotron. Observation of such oscilla-tions with beam position monitors allows direct measure-ments of a synchrotron’s nonlinear parameters. This pa-per presents experimental studies to measure perturvativeeffects of sextupole and octupole fields, performed in theFermilab Tevatron using an AC dipole.INTRODUCTIONA modern synchrotron uses magnets producing non-linear fields, such as sextupole and octupole magnets, tocompensate chromaticity and instabilities of a high inten-sity beam. Magnet imperfections also produce nonlinearfields. In addition to compensating the corresponding ef-fects, these nonlinear fields perturb the beam’s transversemotion. For instance, the sextupole fields distort the beam’scentral orbit and the octupole fields shift betatron tune ν1depending on a beam particle’s amplitude (nonlinear de-tuning effect). These fields also drive modes with tuneshigher than the betatron tune. It is ideal if we could mea-sure and compensate such perturbative effects to fine-tunea synchrotron, but even the measurements are not alwayseasy and not many measurements have been done in theFermilab Tevatron so far [1, 2].An AC dipole magnet, a new diagnostic tool of a syn-chrotron [3], has been implemented to the Tevatron [4, 5,6]. Its oscillating dipole field with frequency close to thebetatron frequency can excite large sustained beam oscilla-tions with no emittance growth. Observation of such oscil-lations with beam position monitors (BPMs) allows mea-surements of not only linear but also nonlinear propertiesof the beam. In this paper, we discuss experimental stud-ies to observe the perturbative effects due to sextupole andoctupole fields, performed in the Tevatron using the ACdipole.In a linear and uncoupled synchrotron2, turn-by-turn po-sition of the AC dipole’s excitation is given by [6, 7]xd(n; s) =Ad√βd(s) cos[2πνdn + ψd(s)− ψd(sac) + χ], (1)∗Work supported by Fermilab Research Alliance, LLC under ContractNo. DE-AC02-07CH11359 with the United States Department of Energy.1In this paper, ν denotes the fractional part of the betatron tune.2We only consider one dimensional motion of the beam particles.where n is number of the beam revolution, s and sac arelongitudinal positions of the BPM and AC dipole, Ad andχ are constants, and νd is driving tune3. The constant Ad,determining the amplitude, is proportional to the amplitudeof the AC dipole’s current and to a factor 1/ sin[π(νd−ν)].We typically choose the driving tune νd near the betatrontune ν to make large amplitude oscillations. In the ACdipole’s oscillation, the amplitude function β(s) and phaseadvance ψ(s) are modulated to βd(s) and ψd(s), depend-ing on the tunes νd and ν [6, 7].CENTRAL ORBIT DISTORTION DUE TOSEXTUPOLE FIELDSThe central orbit distortion due to the sextupole fields isproportional to strengths of sextupole fields and A2d, whichis proportional to the square of the amplitude of the ACdipole’s current. To observe such properties of the effect,we performed an experimental study in the Tevatron us-ing the 150 GeV proton beam. In our study, one sextupolemagnet is controlled to make small changes and to test ourmeasurement methods.Dycoyco, during excitation0 2000 4000 6000 8000-4-2024number of revolutionsy-y co,beforeexcitation@mmDFigure 1: Turn-by-turn position of the AC dipole’s excita-tion observed in the Tevatron, indicating a shift of the av-erage position during the excitation. The average positionbefore the excitation is subtracted from the data points.Figure 1 shows turn-by-turn position of the AC dipole’sexcitation observed in the Tevatron. Here, the amplitude ofthe AC dipole’s current is adiabatically ramped up (1,000th- 3,000th turns) and ramped down (6,000th - 8,000th turns),and such adiabatic processes preserve the beam emittance3The definition of νd (0 ≤ νd < 1) is the following [6, 7]. We sup-pose that frev and fac are frequencies of the beam revolution and the ACdipole field and νac is the fractional part of fac/frev. When the betatrontune ν and νac satisfy |νac − ν| < |1 − νac − ν|, νd = νac, and when|1− νac − ν| < |νac − ν|, νd = 1− νac.FERMILAB-CONF-09-224-AD-APCsextupole location0 1000 2000 3000 4000 5000 6000-0.2-0.10.00.10.2F0 A0 B0 C0 D0 E0 F0longitudinal position @mDDy co@mmDlocation in the TevatronIsx = 0 A Isx = 30 AFigure 2: Central orbit distortions due to sextupole fieldswhen the current of the controlled sextupole magnet is 0 A(nominal) and 30 A. When the current is increased to 30 A,the effect grows as expected (∼80 µm) and shows a cuspstructure, indicating location of the sextupole magnet.[3, 6, 8]. We see that the average position is shifted up-ward while the amplitude is maximum. This shift is thecentral orbit distortion due to the sextupole fields. Figure2 shows the central orbit distortion measured in this waywith all the BPMs. The black and red curves represent thecases when the current of the controlled sextupole mag-net is 0 A (the nominal value) and 30 A. As expected, themeasurement for 30 A shows a larger central orbit distor-tion and also the cusp structure indicating the source loca-tion. In these measurements, the amplitude function andthe oscillation amplitude at the location of the controlledsextupole magnet, ssx, are about βd(ssx) ' 94 m andAd√βd(ssx) ' 4 mm, and the effective strength of the sex-tupole magnet is about B′′`/(Bρ) ' 420 nrad/mm2 whenits current is 30 A [9]. In such a condition, this sextupolemagnet is equivalent to a dipole magnet with a deflectionangle 14A2dβd(ssx)B′′`/(Bρ) ' 1.7 µrad [6] and the ex-pected central orbit distortion is about 80 µm at the BPMlocations, where the amplitude function is β(s) ' 95 m.We can see that this estimate is not far from the measure-ment in the figure. We note that some sextupole mag-nets are also used in the condition of our study and theblack curve represents the effects of such residual sextupolefields.Iac dipole = 280 A-40 -20 0 20 40 60-0.15-0.10-0.050.000.050.100.15sextupole current @ADDy co@mmDBPMð52 BPMð53 BPMð54Isextu = 30 A0 20 000 40 000 60 000 80 000 100 000-0.10-0.050.000.050.10square of the AC dipole current @A2DDy co@mmDBPMð52 BPMð53 BPMð54Figure 3: Central orbit distortions observed by three BPMs.The effect is proportional to the current of the controlledsextupole magnet (left) and the square of the amplitude ofthe AC dipole’s current (and hence A2d) (right). The differ-ence of the slopes is due to phase dependence of the effect.Figure 3 shows the measured central orbit distortion,when the current of the controlled sextupole magnet is setto various values while holding other quantities constant(left) and when the amplitude of the AC dipole’s current isset to various various values, changing the oscillation am-plitude, while holding other quantities constant. We canobserve that the effect is proportional to the current of thegiven sextupole magnet and also the square of the ampli-tude of the AC dipole’s current, as expected. In this way,the AC dipole allows us to observe changes of the centralorbit distortion due to the sextupole fields in the Tevatron.NONLINEAR DETUNING EFFECT DUETO OCTUPOLE FIELDSThe nonlinear detuning effect due to the octupole fieldsis proportional to strengths of octupole fields and also A2d.To observe such properties of the effect, we performed an-other experimental study with the 150 GeV proton beam inthe Tevatron. Here, strength of one group of octupole mag-nets4 is controlled to make small changes and to test ourmeasurement methods.without detuningHcalculationLΝd-Ν0 = -0.015HdataLΝd-Ν0 = 0.020HdataLIoctu = 10 @AD0 50 100 150 200 250 3000.000.050.100.150.200.250.30AC dipole current @ADamplitude´sinÈΠHΝd-Ν0LÈ@mmDFigure 4: Amplitude of the driven beam (multiplied by afactor sin |π(νd − ν0)|) vs. amplitude of the AC dipole’scurrent. The detuning effect either suppress or enhance thebeam’s amplitude. The deviation of the beam’s amplitudefrom the linear growth allows to determine the detuningeffect.As we discussed previously, the amplitude of the ACdipole’s excitation is proportional to the amplitude of theAC dipole’s current and the factor 1/ sin[π(νd − ν)]. Fig-ure 4 shows relations, observed by one BPM, between thebeam’s oscillation amplitudes and the amplitude of the ACdipole’s current. Here, the beam’s amplitude is multipliedby a factor sin |π(νd−ν0)|, where ν0 is the betatron tune ofa beam particle in the limit of a zero oscillation amplitude.In our study’s condition, the detuning effect raises the be-tatron tune along with the growth of the beam’s amplitude.In such a condition, if the driving tune is smaller than thebetatron tune, the betatron tune moves away along with thebeam’s amplitude growth and the beam’s amplitude is sup-pressed through the factor 1/ sin[π(νd − ν)], compared to4The group referred to as OD, consisting of 18 octupole magnets [10].0 20 40 60 80 100 1200.0000.0020.0040.0060.0080.010BPM'sDΝIoctupole 1 @AD 4 @AD 7 @AD 10 @ADFigure 5: Tune shifts measured with all the BPMs for dif-ferent currents of the controlled octupole magnets. Thedriving tune and the maximum amplitude of the AC dipoleare fixed to νd− ν0 = −0.015 and 280 A, producing 5 mmoscillation amplitude in the arc.the linear growth in the case of no detuning effect. On thecontrary, if the driving tune is larger than the betatron tune,the betatron tune approaches to the driving tune along withthe beam’s amplitude growth and the beam’s amplitude isenhanced. By observing such modulations of the beam’samplitude, we can measure the detuning effect [11].Figure 5 shows the tune shift measured in this way withall the BPMs, when the current of the controlled octupolemagnets are changed from 1 A to 10 A (the nominal is7 A) and when the maximum amplitude and driving tuneof the AC dipole are kept the same to produce ∼5 mmoscillations in the arc. For our octupole magnets, the ef-fective focal length is 260 km when its current is 1 A andthe beam’s amplitude is 5 mm [9]. The amplitude func-tions at the locations of these octupole magnets are aboutβ(s) ' 90 m. Under these conditions, when the currentof these controlled octupole magnets is changed by 3 A,the estimated tune shift is about 0.0015, which is roughlyconsistent with the gaps between curves. These measure-ments are based on the beam’s amplitude modulations butthe amplitude function of the AC dipole’s excitation βd(s)also depends on the betatron tune. The growth of the de-viations over BPMs is expected due to the change of βd(s)induced by the detuning effect. In this way, the AC dipoleallows us to observe changes of the nonlinear detuning ef-fect due to the octupole fields in the Tevatron.MODES WITH HIGHER TUNESIn addition to the effects discussed in previous sections,when the beam is driven by the AC dipole, the sextupoleand octupole fields drive modes with tunes 2νd and 3νd, foreach, and magnitudes of these modes nonlinearly dependon the constant Ad. Figure 6 shows the Fourier spectra ofthe turn-by-turn positions of the AC dipole’s excitation. Wecan see that the peaks at 2νd, 1 − 2νd, 3νd, and 1 − 3νd,corresponding to the modes of the sextupole and octupolefields, appear only for the larger excitation, indicating theIac dipole = 70 @ADIac dipole = 280 @ADBPMð93Ioctu = 10 @ADΝd1-Νd2Νd 1-2Νd3Νd1-3ΝdΝ1-Ν0.0 0.2 0.4 0.6 0.8 1.010-40.0010.010.1110tune Hfractional partLFourierspectrum@mmDIac dipole = 70 @ADIac dipole = 280 @ADBPMð95Ioctu = 10 @AD0.0 0.2 0.4 0.6 0.8 1.010-40.0010.010.1110tune Hfractional partLFourierspectrum@mmDFigure 6: Fourier spectra of turn-by-turn positions of theAC dipole’s excitations. The peaks at 2νd, 1−2νd, 3νd, and1−3νd, corresponding to the modes driven by the sextupoleand octupole fields, appear only for the larger excitation,indicating these modes depend on the oscillation amplitudeof the AC dipole’s excitation.dependence of these modes on Ad. By measuring heightsof these peaks with BPMs around a synchrotron, we canmeasure corresponding resonance driving terms [12, 13].CONCLUSIONOur studies in the Tevatron demonstrated that the ACdipole allows us to measure changes of the central orbitdistortion due to the sextupole fields and the nonlinear de-tuning effect due to the octupole fields. We also demon-strated that the AC dipole allows us to observe the highertune modes driven by these fields. The measurements pre-sented in this paper are non-destructive to the beam qualityand can be performed on a routine basis, and hence may beuseful for a quick diagnostic of a synchrotron’s nonlinearfields.ACKNOWLEDGMENTAuthors would like to thank to members of the FermilabTevatron Department for their supports.REFERENCES[1] A. Chao et al., Phys. Rev. Lett. 61, p. 2752 (1988).[2] Y. Alexahin et al, EPAC’06, Edinburgh, 2006, p. 2140.[3] M. Bai et al, Phys. Rev. E 56, p. 6002 (1997).[4] R. Miyamoto et al, BIW’06, Batavia, 2006, p. 402.[5] R. Miyamoto et al, PAC’07, Albuquerque, 2007, p. 3868.[6] R. Miyamoto, Ph.D. thesis, Univ. of Texas at Austin, 2008.[7] R. Miyamoto et al, Phys. Rev. ST-AB 11, 084002 (2008).[8] R. Tomás, Phys. Rev. ST-AB 8, 024401 (2005).[9] H. T. Edwards, Ann. Rev. Nucl. Part. Sci. 35, p. 605 (1985).[10] P. M. Ivanov et al, PAC’05, Knoxville, 2005, p. 2756.[11] L. D. Landau and E. M. Lifshitz, “Mechanics Third Edi-tion”, Butterworth-Heinemann, Oxford, UK, 1982, pp. 58-95.[12] F. Schmidt et al, CERN AB-Note-2003-031 MD (2003).[13] R. Tomás et al, Phys. Rev. ST-AB 8, 024001 (2005).

Nonlinear dynamics studies in the Fermilab Tevatron using an ac dipole

https://digital.library.unt.edu/ark:/67531/metadc930671/m2/1/high_res_d/965878.pdf

Nonlinear dynamics studies in the Fermilab Tevatron using an ac dipole

Abstract

Similar works

Full text

Available Versions

UNT Digital Library

UNT Digital Library