High Brightness Electron Beam Diagnostics and their ... - CASA
High Brightness Electron Beam Diagnostics and their ... - CASA High Brightness Electron Beam Diagnostics and their ... - CASA
computedusingthelatticeset-upusedduringthemeasurement;thebeamcentroidinducedbythe locationofeachBPMsfx;ygi=1:::N: angularperturbationarecomputedusingtheR12transfermatrixelement. Foratransverseresponselatticemeasurement,thetransversematrixelementsarenumerically xy!i=1:::N= xy!i=1:::N x0 y0!i=1:::N (3.5) ofthetransfermatrixbetweenthedispersiongeneratorexitandtheconsideredlocation: Fordispersionmeasurement,theenergychangeisimpressedusingthelastpairofcavitiesinthecryomodule.Unfortunatelythismethoddoesnotprovidevaluableinformation(seetheexperimental sectionformoreexplanation)andwehadtouseanothertechniquetoperformsuchmeasurement. Intherstplacedispersionalwaysresultsinanon-zerolocalR16transfermatrixelemente.g.due tothepresenceofadipolemagnet.Afterithasbeengenerated,itcanpropagateinregionwithno magneticeld;forinstanceifattheexitofthedispersiongeneratorthedispersionanditsderivative are0and00,thenatadownstreamlocation,thedispersioncanbecomputedfromtheknowledge duetoarelativeenergychangeisequivalenttoarelativemagneticeldvariation: dipoleB(B=ecp),wehaveafterdierentiation(p)=p=(B)=B.Henceatransverseoset Becauseinthedispersiongeneratorthebeammomentumpisrelatedtothemagneticeldofthe Notethatinthecasethedispersiongeneratorisachromatic,wehave0=0and00=0sothat 0. =R110+R1200 (3.6) wherethesubscript0indicatetheenergychangedisimpressedbeforethemagneticsystem,and thehdBiistherelativemagneticeldvariationofthemagneticsystemdownstreamwhichthe dispersionismeasured. x(s)=(s)dp p0(s)dBB (3.7) beampositionmonitor.Thequadrupolesinbetweenthesetwoelementswerenotpowered.So Fromtheaforementionedtechniquetoassesswhethertheopticallatticeisperformingaccordingly tothemodel,weneedtohaveanaccurateknowledgeoftheangularexcitationprovidedbyacorrectormagnet.Inordertoestimatesuchangularkickandsinceallthecorrectormagnetsareofthe sametype,wehaveuseacorrectorlocatedinthebacklegtransportlinewiththenextupstream 3.3.3ResultsonTransverseResponse thetransfermatrixbetweentheBPMandthecorrectoryieldsanangularkickprovidedbythe thetransfermatrixbetweentheelementistheoneofadriftspaceoflength2.80m.Fordierent correctorexcitation,wemeasuredthebeampositionaspresentedingure3.4.Thebeamposition islinearlydependentonthecorrectorstrengthintherangeinpositiontheBPMwasused[-4mm, +4mm].Alinearinterpolationofthedatapresentedinthegure,alongwiththeknowledgeof correctormagnetofapproximately0:64mrad=(100Gauss:cm)thisvalueisveryclose,within3%,
5 4 3 2 Figure3.4:Exampleofcalibrationofacorrector.Theslopeofthelinearinterpolationis 1 0:0183mm=(G:cm)whichcorrespondstoanangulardeectionof6:54rad=(G:cm) 0 −1 −2 −3 −4 alongthebeamline;typicalvalueusedduringtheacquisitionofdierenceorbitmeasurementare ofthecalculatedvaluededucedfromthecorrectormagneticeldmapmeasurement3whichgivea −5 excitationofthecorrector)alltheBPMreadbacksalongthebeamlineareacquiredthreetimes BPMvaluealongthebeamline,insuchawaythatthekickprovideasignicantpositionchange approximately100G:cm.ForagivencorrectorchangeBnom+B,(Bnomisthenominalmagnetic Practically,thecorrectorstrengthissetdevisobylookingattheon-linehistogramplotofthe kickof0:65mrad=(100Gauss:cm) −300 −200 −100 0 100 200 300 Corrector Field Integral (G.cm) oflinearityofthesystem,theBPMreadbackshouldbetheoppositeoftheonemeasuredforthe latteroperationallowtovalidatethemeasurement,sinceforthelattercorrectorsetting,because (toquantifythebeampositionjitter).ThenthecorrectorissettothevalueBnomB.This 2F04Vand2F08V).Thecorrectorsarechosensothattherelativebetatronphaseadvancebetween forthehorizontalplane(2F00H,2F04Hand2F08H)andthreefortheverticalplane(2F00V, probedierentpartofthelattice.Inourpresentstudy,weusesixdierentcorrectors:three zeroforalltheBPMs. Theuseofonlyonecorrectortostudytheresponseofthelatticeisnotsucientsinceitonly\probe" thelatticeatlocationthathavearelativebetatronphaseadvanceofapproximately90deg4.Hence itispreferabletouseatleasttwocorrectorsseparatedbytheproperphaseadvancesothatthey rstmeasurement.Thereforethecomputationofthesumofthetwomeasurementsshouldgive istherelativebetatronphaseadvancebetweenthepointssands0. 3G.H.Biallas,privatecommunication,June99 4InthegeneralTwisstransfermatrixformalismonehas:Rs0!s 12=p(s)(s0)sin()where=(s)(s0) ∆ x 2.80 meters downstream (mm)
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computedusingthelatticeset-upusedduringthemeasurement;thebeamcentroidinducedbythe locationofeachBPMsfx;ygi=1:::N:<br />
angularperturbationarecomputedusingtheR12transfermatrixelement. Foratransverseresponselatticemeasurement,thetransversematrixelementsarenumerically xy!i=1:::N= xy!i=1:::N x0 y0!i=1:::N (3.5)<br />
ofthetransfermatrixbetweenthedispersiongeneratorexit<strong>and</strong>theconsideredlocation: Fordispersionmeasurement,theenergychangeisimpressedusingthelastpairofcavitiesinthecryomodule.Unfortunatelythismethoddoesnotprovidevaluableinformation(seetheexperimental sectionformoreexplanation)<strong>and</strong>wehadtouseanothertechniquetoperformsuchmeasurement. Intherstplacedispersionalwaysresultsinanon-zerolocalR16transfermatrixelemente.g.due tothepresenceofadipolemagnet.Afterithasbeengenerated,itcanpropagateinregionwithno magneticeld;forinstanceifattheexitofthedispersiongeneratorthedispersion<strong>and</strong>itsderivative are0<strong>and</strong>00,thenatadownstreamlocation,thedispersioncanbecomputedfromtheknowledge<br />
duetoarelativeenergychangeisequivalenttoarelativemagneticeldvariation: dipoleB(B=ecp),wehaveafterdierentiation(p)=p=(B)=B.Henceatransverseoset Becauseinthedispersiongeneratorthebeammomentumpisrelatedtothemagneticeldofthe Notethatinthecasethedispersiongeneratorisachromatic,wehave0=0<strong>and</strong>00=0sothat 0. =R110+R1200 (3.6)<br />
wherethesubscript0indicatetheenergychangedisimpressedbeforethemagneticsystem,<strong>and</strong> thehdBiistherelativemagneticeldvariationofthemagneticsystemdownstreamwhichthe dispersionismeasured. x(s)=(s)dp p0(s)dBB (3.7)<br />
beampositionmonitor.Thequadrupolesinbetweenthesetwoelementswerenotpowered.So Fromtheaforementionedtechniquetoassesswhethertheopticallatticeisperformingaccordingly tothemodel,weneedtohaveanaccurateknowledgeoftheangularexcitationprovidedbyacorrectormagnet.Inordertoestimatesuchangularkick<strong>and</strong>sinceallthecorrectormagnetsareofthe sametype,wehaveuseacorrectorlocatedinthebacklegtransportlinewiththenextupstream 3.3.3ResultsonTransverseResponse<br />
thetransfermatrixbetweentheBPM<strong>and</strong>thecorrectoryieldsanangularkickprovidedbythe thetransfermatrixbetweentheelementistheoneofadriftspaceoflength2.80m.Fordierent correctorexcitation,wemeasuredthebeampositionaspresentedingure3.4.Thebeamposition islinearlydependentonthecorrectorstrengthintherangeinpositiontheBPMwasused[-4mm, +4mm].Alinearinterpolationofthedatapresentedinthegure,alongwiththeknowledgeof correctormagnetofapproximately0:64mrad=(100Gauss:cm)thisvalueisveryclose,within3%,
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