SL-Note-2000-046 MD

EUROPEANORGANIZATIONFORNUCLEARRESEARCHCERNSLDIVISIONGeneva,Switzerland1stAugust2000SL-Note-2000-046MDSingle-bunchinstabilitybelowtransitionenergyT.Bohl,T.Linnecar,E.ShaposhnikovaSL-Note-2000-046MDSPSSingle-bunchinstabilitybelowtran-sitionenergyT.Bohl,T.Linnecar,E.ShaposhnikovaAugust01,2000Keywords:longitudinal,instability,microwaveNorunnumbersspecified.SummaryIntheCERNSPSmicrowaveinstabilityisanimportantlimitationofsingleprotonbunchintensity.Studiesofthisinstabilitybelowtransitionenergywillbepresentedandcomparedwithearlierresultsobtainedabovetransition.1IntroductionThemicrowaveinstabilityhasbeenunderinvestigationformorethan40years[1],[2].ItbecameanissueintheSPSalreadyafewyearsaftercommissioning(seeforexample[3]).Todayitisofinterest(a)abovetransitionenergy(γt=23.2,becausetheLHCbeam[4]willbeinjectedat26GeV/c(γ=27.7)and(b)belowtransition,becausethehighintensitybeamtobeusedfortheCNGSproject[5]willbeinjectedbelowtransition(γ=14.9).Inthepastmostobservationsofmicrowaveinstabilityhavebeendonebymeasuringintensityde-pendentbunchlengthincrease,emittanceblow-uporbyacquisitionofband-passfiltered(1.3GHzto1.7GHz)longitudinalpick-upsignals.Sinceafewyearsagowehavealsoobservedthelongitudinalbeamspectrumuptoabout4GHz[6]producedbysinglebunchesinjectedaboveinstabilitythresholdwithRFoff.Featuresofthisspectrumcouldbecorrelatedtoresonantfrequenciesofcavity-likestruc-turesaroundtheSPScircumference.AtKEKhighfrequencysignalswereusedtoidentifypossiblesourcesformicrowaveinstabilityjustaftertransitionenergycrossing[7]whensynchrotronmotionisfrozeneventhoughRFison.Thefollowingdiscussionismainlyfocusedononeexperimentatγ=21.3(1999-11-08),belowtransition.Thisenergywaschosentohaveapproximatelythesamedebunchingtime(or|η|∆p/p)asforpreviouslyacquireddataabovetransition(γ=27.7,1999-10-08).Comparisonswillbemadebetweenthesedatasetsandatoneinstancewithdataacquiredwithγ=14.9(1996-11-11).Table1(AppendixA)showstherelevantmachineandbeamparameters.2ExperimentalmethodThemethodusedanditsjustificationisexplainedin[6].Asingleprotonbunchwithanintensitybetween1×1010to6×1010particles(correspondinglocalcurrent100mAto600mA)isinjectedintotheSPS.Thebunchlengthisabout25ns,thelongitudinalemittance0.25eVs.AllRFsystemsintheSPSareoff.TheRFcavitiesareundamped(200MHztravellingwavecavities),passivelydamped(200MHzstandingwavecavities,100MHzstandingwavecavities)oractivelydampedbyfeedback(352MHzsuperconductingstandingwavecavities).Thesignalfromalongitudinalwidebandpick-up1(bandwidth4MHztoabout4GHz)iseitheracquiredwithadigitaloscilloscopeoraspectrumanalyser.Adigitaloscilloscopewithasamplingperiodof250psisemployedtoobtainmountainrangedataintimedomain,whichwillbeusedforFourieranalysisupto2GHz.Forinformationaboutfrequenciesupto4GHz,aspectrumanalyserisusedasatunedreceiver(3MHzbandwidth)toacquiremodeamplitudesversustime.Usingdataacquiredabovetransitionenergy,Figures3-7illustratetheexperimentalmethodused.Fig.3showsamountainrangeplotintimedomain.Fig.4showsonthelefthandsidetheFouriertransformofeachtraceofthepreviousmountainrangeplotintimedomain.OntherightsidetheprojectionofallFouriertransformsisshown(solidline)andalsotheFouriertransformoftheinitialbunchshape(dottedline).Fig.5showstheaverageofseveralprojections.Forfrequenciesabove2GHz,thewallcurrentpick-upsignalisdirectlyacquiredwithaspectrumanalyser,Fig.6.Themaximummodeamplitudewithinatypicalvalueof100msisrecorded.Severalinjectionsleadtoseveralmaximummodeamplitudesatacertainfrequency.ThesevaluesareaveragedandcorrespondtoonepointinFig.7.Thesemodeamplitudespectra(Fig.5,Fig.7)showpeaksatcertainfrequencies.AmajorityofthesepeakscanberelatedtoknownimpedancesintheSPS.3ObservationsintimedomainInthissection,observationsconcerningmicrowaveinstabilityofsinglebunchesbelowtransitionenergy,asobservedintimedomain,willbediscussed.Forthesamebunchintensity(Np=3.5×1010)andinitialbunchlengthFig.8showsamountainrangeplotforacasebelowtransition(leftside,η=−3.5×10−4)andabovetransition(rightside,η=+5.5×10−4).Comparingbothsidesthefirstimpressionisthatbelowtransitionthebunchisbothmoreunstable(asindicatedbythebunchshapemodulationathighfrequencies)anddebunchesfasterthanabovetransition.ThefirstpointisinagreementwiththeKeil-Schnell-Boussardcriterion([2],[8]),sincethethresholdofthemicrowaveinstabilityisproportionaltoγ(∆p/p)2η.Alsofornarrowbandimpedancesthethreshold[9]isproportionaltoγ(∆p/p)2η.Thesecondpointisincontradictionwithwhatonewouldexpectifnointensityeffectsaretakenintoaccount(debunchingtimeisinverselyproportionalto|η|∆p/p,seeTable1).Theplotsofbunchlengthversustime(seeFig.9)forthecasesbelowandabovetransitionarebasedonN=65andN=25acquisitions,respectively.Thebunchlengthisdefinedasthetotalwidthofthebunchat15%ofitspeakamplitude.AmodeltodescribethebunchlengthasafunctionoftimeisshowninFig.10.Uptot=t0theunstablebeamseemstodebunchwithadebunchingtime,td,1duetoitsinitialmomentumspread(∆p/p)1.Ataroundt=t0itblowsup,theeffectofthedistributionchangebecomesvisible.Inthefollowingtimett0itdebuncheswithanewdebunchingtime,td,2asexpectedforanincreasedmomentumspreadof(∆p/p)2.Itseemst