High Brightness Electron Beam Diagnostics and their ... - CASA
High Brightness Electron Beam Diagnostics and their ... - CASA High Brightness Electron Beam Diagnostics and their ... - CASA
60 0.25 40 20 emittances~"x;y,andrmsbeamsizesx;y)versustheoperatingacceleratingphaseofthelinac. Figure6.18:Evolutionofbeamparameters(bunchlengthz,rmsenergyspreadE,transverse 0 0 6 1 x x 4 y 0.5 y 2 thelatterequationclearlyshowsthat,comparedtotheconstantenergyspreadequation,thereis whichcanbesolvedusingthestandardGreenfunctionperturbativetechnique[59]11Thesolution 0 0 anincrementinangleandpositionof: oftheaboveequationis: x(s)=cos(s=x)x(0)+xsin(s=x)x0(0)+x(1cos(s=x))(0)+Zs x0(s)=1=xsin(s=x)x(0)+cos(s=x)x0(0)+cos(s=x)(0)+Zs 0xsin(~s=x)(~s)d~s 0cos(~s=x)(~s)d~s(6.24) −20 −15 −10 −5 0 −20 −15 −10 −5 0 ∆φ (RF−Deg) ∆φ (RF−Deg) achromaticbendingsystem,theachromaticcharacterisbrokenbecauseofEqn.(6.25).Another interestingpointisthatdependingonthebendingsystemdesign,onecanconceiveawayofmaking andhxx0i)andsubstitutethemintheeqn.(6.20).Itisinterestingtonotethatinthecaseofan Tocomputetheemittancegrowthweneedtocomputethesecondordermoments(hx2i,hx02i x(s)=Zs x0(s)=Zs 0xsin(~s=x)(~s)d~s theaboveintegralverysmall(orideallyzero)sothatthenetemittancegrowthisnegligible.Such amethodhasbeendiscussedindetailinreferences[60]and[61]. 0cos(~s=x)(~s) (6.25) principalsolutions(S(t)andC(t))accordinglyto: equationwritesx(t)=Rt 11Ifweconsidertherighthandsideofthepreviousequationasaperturbationtermp(t;s)thesolutionofthis 0p(~t)d~tG(t;~t)whereG(t;~t)isaGreen'sfunctionthatcanbeconstructedfromthetwo G(t;~t)=S(t)C(~t)C(t)S(~t) σ z (mm) ε x,y (mm−mrad) 0.5 ∆E (keV) σ x,y (mm) 100 80
6.4.4BunchSelfInteractionviaCoherentSynchrotronRadiation CSRisalongstandingtopicinseveralsubjects,especiallyinAcceleratorPhysics.Therst comprehensivestudywasperformedbyJ.S.NodvickandD.S.Saxon[4]in1954.Theseauthors accelerator.Itisaconsequenceofthegenerallylongbunchthatarecirculatinginsuchaccelerator: aswewillseeinthischapter,CSRemissionoccursatwavelengthcomparabletothebunchlength. Therefore,forbunchlengthoftheorderofcentimeters(asitiscurrentincircularaccelerator), studiedtheinteractionofchargedparticlemovingonacurvedpathbetweentoperfectlyconducting planeandshowedhowCSRemissioncouldbepartiallysuppressatagivenwavelengthbythemeans ofthetwoconductingplanethatactasashielding.Indeed,tothebestofourknowledge,CSReect ontheBeamDynamics,andCSRemission,haveneverbeenobservedinstorageringorcircular theTohokuUniversitybyT.Nagazato[66].Thisgroupshowedexperimentallyhowitwaspossible toinferthebunchlengthandbunchstructureusingthefrequencyspectrumofCSR,usingthe beampipechamber,whichserveasawaveguidefortheCSRpropagation,arealsooftheorderof centimetersandsoistheircutowavelength.ThereforetheCSRemissionis\shielded"bythe theemissionofCSRshouldoccurinthemicrowaveregion:unfortunately,thesizeofthevacuum region,andinthefar-infra-redwavelengthhasbeenobservedina100MeVlinearacceleratorof sametechniquewepresentedinChapter4forthetransitionradiation.Theyalsodemonstrate beampipe,i.e.itdoesnotpropagate.Infact,onlyveryrecently,CSRemissioninthefareld thepossibleshieldingofCSRemissionusingtwoparallelconductingplanewithvariablegap[67]. HowevertheanticipatedeectsofCSRontheBeamDynamics,i.e.transverseemittancedilution, hasneverbeenobserveduptonow. Asimplemodel:steadystateinfreespace WeoutlineinthepresentsectionasimplepictureoftheCSRphenomenon.Forsuchapurposewe startwiththeLienard-Wietchertretardedelectriceld[8]: !E=e"bn! inthemovingframe,theradiusofcurvature,andtheanglebetweentheminthelaboratory orequivalentlyby=2sin(=2)withbeingtheanglebetweenthetwoelectrons timet0.Becauseofcausalitytheretardedt0andpresentttimesarerelatedbyt=t0+R(t0)=c Thesubscriptretmeansthatthequantitiesinsidethebracketsmustbeevaluatedattheretarded !RisavectorfromS0toS,and1bn!=1cos(=2)and=6(! 2(1bn:!)3R2#ret+ec24bn^(bn!)^!_ (1bn!)3R35ret OS0;! OS)(seegure6.19). (6.26) frame. Theproblemhasbeentreatedinseveralreferences(e.g.Ref.[69]),itrstconsistsofcalculatingthe electriceldemittedattheretardedtimeandlocationS0atthepresenttimeandlocationS.This electriceldinducesanenergychangeonS,V(ss0),thatdependsontherelativepositions,sand alongthebunch.TheenergychangeofareferenceparticleSisgivenbythesuperpositionofthe s0,ofthetwoparticles.InessenceCSRisverysimilartowakeeld:ityieldsanenergyredistribution radiationforceofallthebackparticles: d(ct)=Zs dE1(s0)V(ss0)ds0 (6.27)
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6.4.4BunchSelfInteractionviaCoherentSynchrotronRadiation CSRisalongst<strong>and</strong>ingtopicinseveralsubjects,especiallyinAcceleratorPhysics.Therst comprehensivestudywasperformedbyJ.S.Nodvick<strong>and</strong>D.S.Saxon[4]in1954.Theseauthors accelerator.Itisaconsequenceofthegenerallylongbunchthatarecirculatinginsuchaccelerator: aswewillseeinthischapter,CSRemissionoccursatwavelengthcomparabletothebunchlength. Therefore,forbunchlengthoftheorderofcentimeters(asitiscurrentincircularaccelerator), studiedtheinteractionofchargedparticlemovingonacurvedpathbetweentoperfectlyconducting plane<strong>and</strong>showedhowCSRemissioncouldbepartiallysuppressatagivenwavelengthbythemeans ofthetwoconductingplanethatactasashielding.Indeed,tothebestofourknowledge,CSReect onthe<strong>Beam</strong>Dynamics,<strong>and</strong>CSRemission,haveneverbeenobservedinstorageringorcircular<br />
theTohokuUniversitybyT.Nagazato[66].Thisgroupshowedexperimentallyhowitwaspossible toinferthebunchlength<strong>and</strong>bunchstructureusingthefrequencyspectrumofCSR,usingthe beampipechamber,whichserveasawaveguidefortheCSRpropagation,arealsooftheorderof centimeters<strong>and</strong>sois<strong>their</strong>cutowavelength.ThereforetheCSRemissionis\shielded"bythe theemissionofCSRshouldoccurinthemicrowaveregion:unfortunately,thesizeofthevacuum region,<strong>and</strong>inthefar-infra-redwavelengthhasbeenobservedina100MeVlinearacceleratorof sametechniquewepresentedinChapter4forthetransitionradiation.Theyalsodemonstrate beampipe,i.e.itdoesnotpropagate.Infact,onlyveryrecently,CSRemissioninthefareld thepossibleshieldingofCSRemissionusingtwoparallelconductingplanewithvariablegap[67]. HowevertheanticipatedeectsofCSRonthe<strong>Beam</strong>Dynamics,i.e.transverseemittancedilution, hasneverbeenobserveduptonow. Asimplemodel:steadystateinfreespace WeoutlineinthepresentsectionasimplepictureoftheCSRphenomenon.Forsuchapurposewe startwiththeLienard-Wietchertretardedelectriceld[8]: !E=e"bn!<br />
inthemovingframe,theradiusofcurvature,<strong>and</strong>theanglebetweentheminthelaboratory orequivalentlyby=2sin(=2)withbeingtheanglebetweenthetwoelectrons timet0.Becauseofcausalitytheretardedt0<strong>and</strong>presentttimesarerelatedbyt=t0+R(t0)=c Thesubscriptretmeansthatthequantitiesinsidethebracketsmustbeevaluatedattheretarded !RisavectorfromS0toS,<strong>and</strong>1bn!=1cos(=2)<strong>and</strong>=6(! 2(1bn:!)3R2#ret+ec24bn^(bn!)^!_ (1bn!)3R35ret OS0;! OS)(seegure6.19). (6.26)<br />
frame. Theproblemhasbeentreatedinseveralreferences(e.g.Ref.[69]),itrstconsistsofcalculatingthe electriceldemittedattheretardedtime<strong>and</strong>locationS0atthepresenttime<strong>and</strong>locationS.This electriceldinducesanenergychangeonS,V(ss0),thatdependsontherelativepositions,s<strong>and</strong> alongthebunch.TheenergychangeofareferenceparticleSisgivenbythesuperpositionofthe s0,ofthetwoparticles.InessenceCSRisverysimilartowakeeld:ityieldsanenergyredistribution radiationforceofallthebackparticles: d(ct)=Zs dE1(s0)V(ss0)ds0<br />
(6.27)
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