High Brightness Electron Beam Diagnostics and their ... - CASA
High Brightness Electron Beam Diagnostics and their ... - CASA High Brightness Electron Beam Diagnostics and their ... - CASA
as: measurementstationofoneelectronwithlongitudinalpositionzwithrespecttothebunchcenter 1497MHzitis20:05cm),thedispersionatthebeamprolemeasurementlocation,andtheaverage thecavitiesusedduringthemeasurement(i.e.operatedatzero-crossing),theRF-wavelength(for beamenergyattheentranceoftherstcavityoperatedatzero-crossing.dE E0arerespectivelythepurebetatroncontributiontotheposition,thetotalacceleratingvoltageof energyspreadandthespacechargeenergyspreadinducedasthebeamdrifts.x,VRF,RF,and whereC0isthecontributionfromRF-inducedenergyspreadandC1isthesumofinitialintrinsic x=x+2eVRF RF+dE dzE0zdef =x+(C0+C1)z dzisthelongitudinal (5.49) tor,withthecavitiesrespectivelyturnedoandturnedonattheir90degzero-crossingpoint.Let0x,x,bethebeamsizesmeasuredafterthespectrometerdipoleontheOTRprolemoni- phaseslope;itcanbeexpressedusingthebeammatrixelementasdE Sincethebeamproleontheprolemonitoristheconvolutionofpurebetatroncontributioni.e. theenergypositioncorrelationi.e.56=hzEi. transverseandlongitudinalphasespace,thesebeamsizescanbeexpressedas: dz=56=(2z)where56is Becausethesignoftheproduct2C0C1isalternatedasthecavitiesareoperatedat90deg,this quantitycanbeeliminatedandbycomputingthepuredispersivecontributionduetotheenergy isthehorizontalbetatroncontributiontothebeamspotsize. (x)2=2+(C1C0)22z (0x)2=2+C212z (5.51) (5.50) spreadinducedbythecavitiesat90degi.e.6(XRMS)2=(x)2(0x)2,itisstraightforwardto deduceananalyticalexpressionforthebunchlength: slope,usingtheformula: Notethatinthecaseofsmallenergyspreadtheseformulaereducetotheonederivedinrefer- FinallywecanalsoestimatethecoecientC1,whichcanprovideinformationonthephasespace C1=jC0j z=h(X+RMS)2+(XRMS)2i1=2 2(X+RMS)2(XRMS)2 (X+RMS)2+(XRMS)2 p2jC0j (5.53) (5.52) ence[52]. FromEqn.(5.53)thelongitudinalphasespaceslopeis: dE theenergydistribution.Thermsvalueofafunctiong=fh(istheconvolutionproduct)ishg2i=hf2i+hh2i Insummary,themeasurementofbunchlength(andpotentiallyphasespaceslope)reducestothree beamprolemeasurementsforthreedierentsettingsofthezerophasingcavities(90deg,and 0deg). 6Thebeamtransversedensityattheprolemeasurementstationisaconvolutionofthebetatrondistributionwith dz=VRF RF(X+RMS)2(XRMS)2 (X+RMS)2+(XRMS)2 (5.54)
Asarstapproximation,wecanestimatetransversespacechargeusingtheK-Venvelopeequations bycalculatingtheratiooftheemittancetermwiththespace-chargeterm(weassumethebeamis cylindrical-symmetric): TransverseSpaceChargeEects beapproximately0.6atthecavity#5exit.Thereforespacechargeandemittancetermsareofthe sameorderindrivingthetransversebeamenvelope. Inequation5.51,onemustinsistthatcontainsthetransversespacechargeeect.Inorder whereIpisthepeakcurrent,I0theAlfvencurrent(17000Aforelectrons),andaretheusual relativisticfactors.FortheexpectedvaluesobtainedvianumericalsimulationweestimatedRto R=14Ip I02 ()3x2 (5.55) tovalidatethederivedequationstocomputethebunchlengthandphasespaceslope,wemust makesurethat,asitisimplicitlyassumedintheprevioussection,remainsthesameasthe areturnedwiththeirphasesetat90deg.Hencethetransversespacechargecontributionis inwhichismeasuredwhenthecavitiesareturnedo.Alsoitremainsthesameasthecavities withtheparmelaspacechargeroutineturnedonando.Theeectonthebeamsizebeforethe spectrometer,inallthecases,remainsthesameandincreasesthebeamrmssizebyapproximately 36m.ThereforethetransversespacechargecontributiontothebeamsizeontheOTRisincluded suchassumptionusingtheparmelacode:thebeamenvelopesalongthebeamlineareplottedin zerophasingcavitiesphasedareturnedonandphasedattheirtwozerocrossing7.Wehaveveried indeeddeconvolvedunambiguouslywhenoneusestheEqn.(5.52)tocomputethebunchlength. gure5.22fordierentcases(dierentnumberofzero-phasingcavitiesused):eachcaseistreated LongitudinalSpaceChargeEect Thelongitudinalspacechargetendstoinducebunchlengtheningwhichinturnrotatesthelongitu- wherehdE theslopeatthedipoleentranceisapproximately: dinalphasespace.Henceonewayofassessingtheassociatedeectistostudyhowthephasespace thespacechargeinducedphasespacerotation. slopeevolvesasthecavitiesarezero-phased.Onecanconceivethatbecauseofthespacecharge AgainweneedtojustifythatjC1jremainsthesameasthezerophasingcavitiessettingsarechanged: namelywemustmakesurethatthespacechargeinducedslopeisthesameinthedierentcases. dE Thiscanbeunderstoodsincethezerophasingcavitiesarenotprovidingenergy.Wehavechecked dziinitisthephasespaceslopeupstreamtherstzerophasingcavity,andhdE dz=dE dzinit+dE dzSC dziSCrepresents (5.56) focusingeect.Sucheectisinvestigatedlaterinthisdissertationandwasanywayfoundtobeverysmallforthe thisusingparmela:theslopeevolutionforthedierentcasesofzerophasingarepresentedin purposeofthepresentdiscussion;thereforeweignoreitforsakeofsimplicity. gure5.24.Foreachcasewecomparetheslopecomputedwiththespacechargeroutineturned 7Thereisanothereectthatcansignicantlyaectthetransversebeamsizeontheprolemonitor:thecavity
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as: measurementstationofoneelectronwithlongitudinalpositionzwithrespecttothebunchcenter<br />
1497MHzitis20:05cm),thedispersionatthebeamprolemeasurementlocation,<strong>and</strong>theaverage thecavitiesusedduringthemeasurement(i.e.operatedatzero-crossing),theRF-wavelength(for beamenergyattheentranceoftherstcavityoperatedatzero-crossing.dE E0arerespectivelythepurebetatroncontributiontotheposition,thetotalacceleratingvoltageof energyspread<strong>and</strong>thespacechargeenergyspreadinducedasthebeamdrifts.x,VRF,RF,<strong>and</strong> whereC0isthecontributionfromRF-inducedenergyspread<strong>and</strong>C1isthesumofinitialintrinsic x=x+2eVRF RF+dE dzE0zdef =x+(C0+C1)z dzisthelongitudinal (5.49)<br />
tor,withthecavitiesrespectivelyturnedo<strong>and</strong>turnedonat<strong>their</strong>90degzero-crossingpoint.Let0x,x,bethebeamsizesmeasuredafterthespectrometerdipoleontheOTRprolemoni- phaseslope;itcanbeexpressedusingthebeammatrixelementasdE Sincethebeamproleontheprolemonitoristheconvolutionofpurebetatroncontributioni.e. theenergypositioncorrelationi.e.56=hzEi. transverse<strong>and</strong>longitudinalphasespace,thesebeamsizescanbeexpressedas: dz=56=(2z)where56is<br />
Becausethesignoftheproduct2C0C1isalternatedasthecavitiesareoperatedat90deg,this quantitycanbeeliminated<strong>and</strong>bycomputingthepuredispersivecontributionduetotheenergy isthehorizontalbetatroncontributiontothebeamspotsize. (x)2=2+(C1C0)22z (0x)2=2+C212z (5.51) (5.50)<br />
spreadinducedbythecavitiesat90degi.e.6(XRMS)2=(x)2(0x)2,itisstraightforwardto deduceananalyticalexpressionforthebunchlength: slope,usingtheformula: Notethatinthecaseofsmallenergyspreadtheseformulaereducetotheonederivedinrefer- FinallywecanalsoestimatethecoecientC1,whichcanprovideinformationonthephasespace C1=jC0j z=h(X+RMS)2+(XRMS)2i1=2 2(X+RMS)2(XRMS)2 (X+RMS)2+(XRMS)2 p2jC0j (5.53) (5.52)<br />
ence[52]. FromEqn.(5.53)thelongitudinalphasespaceslopeis: dE<br />
theenergydistribution.Thermsvalueofafunctiong=fh(istheconvolutionproduct)ishg2i=hf2i+hh2i<br />
Insummary,themeasurementofbunchlength(<strong>and</strong>potentiallyphasespaceslope)reducestothree beamprolemeasurementsforthreedierentsettingsofthezerophasingcavities(90deg,<strong>and</strong> 0deg). 6Thebeamtransversedensityattheprolemeasurementstationisaconvolutionofthebetatrondistributionwith dz=VRF RF(X+RMS)2(XRMS)2 (X+RMS)2+(XRMS)2 (5.54)
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