High Brightness Electron Beam Diagnostics and their ... - CASA
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High Brightness Electron Beam Diagnostics and their ... - CASA

High Brightness Electron Beam Diagnostics and their ... - CASA High Brightness Electron Beam Diagnostics and their ... - CASA

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equationthatrequiresomeinitialconditions,thatwewilltaketobeattheemittancesolenoidexit. Letrandzberespectivelythermstransverseandlongitudinalbeamenvelope.Itiswellknow (e.g.seereference[54])thatonecandescribestheevolutionofthebeamenvelopeviatheso-called coupledrmsenvelopeequationthatwrite(extendedfromreference[54]): @2r(s) @s2+k20r(s)3 10p5Nrc 20301 r(s)z(s) 1g22r 202z!~"2r3r=0 (6.1) @2z(s) @s2+k20r(s)3 10p5Nrc 2050g z(s)2~"z(s)2 z(s)3=0 whereg=g(z=r;b=r)afunctionofthebeamrmssizeandthevacuumpipediameterb,de- scribestheeectofthebunchinteractionwithitsimageonthebeamlinevacuumchamber; rc=e2=(40mc3)istheclassicalradiusofanelectronand0isthebunchreducedenergy(from nowonwewillassumeanenergyof350keV,i.e.0=0.8048and0=1.6849.Toconvinceourselves onthenecessityofusingtheaboveequationsystem,wecanstudythedependenceofthe\space chargeoveremittanceratio".Forthelongitudinaldirectionwedenethisratioas: Rrdef =3 10p5Nrc 30gz (~"nr)2 (6.2) Thesamekindoffactorcanbedenedforthetransversedirection: Rzdef =3 10p5Nrc 02r (~"nz)2z 1g22r 202z! (6.3) Theevolutionoftheseratiosalongthebeamlineusingrmsenvelopenumericallycomputedwith parmelaareshowningure6.3.Inthe350keVlineitisseenthatspacechargecontributionin theenvelopeequationcanbeafactor100largerthantheemittancetermcontribution.Evenin the10MeVregion,thereisstillapredominanceofspacechargetermbyafactor10exceptinthe bunchingchicanewheredispersionincreasetransversebeamsizeandthereforelocallyreducespace chargeforce.Ontheotherhand,thelongitudinalratioissignicantlylargerthanunityonlyin the350keVregion.Itisstronglydampedasthebeamisacceleratedinthe10MeVstructureand downstreamthecryounitthelongitudinalenvelopeequationisonlydrivenbytheemittanceterm. Toapplythermsenvelopeequationtothedierentelementswecanusethefollowingsteps: foradriftspace,theexternalfocusingparameters,krandkzaresettozero. thebunchercavityismodeledasa\slopeimpulse":z0buncher !z0+2RFeV mc223z thesolenoidexternalfocusingparameterisestimatedusingtherelationk0=eB0 2mcwhereB0 istheintegratedmagneticeld,whichwehaveestimatedusingapoissongeneratedmagnetic eldprole. 6.1.3Thehighgradientstructure Inthissectionwewouldliketodiscussfewinterestingeectsinducedonthetransversebeam dynamicsbytheCEBAF-typeacceleratingcavities.Thediscussionwillenablethereadertoun- derstandexperimentalresultspresentedinthenextsection.

Space Charge over Emittance Ratio (no unit) 10 2 10 0 200 400 600 800 1000 1200 Distance from the Photocathode (cm) -4 10 -3 10 -2 10 -1 10 0 10 1 Rr tion. EnergyGain Theaccelerationinacceleratingcavitiesisprovidedbythelongitudinalcomponentoftheelectric Figure6.3:\spacechargeoveremittanceratioforthetransverse(Rr)andlongitudinal(Rz)direc- eldofthefundamentalmode.Sucheldcanbewrittenapproximately: Rz betweentheparticleandtheRF-wave.Becauseoftheirenergyattherstcavityentrance,350keV, theelectronsarenotrelativisticandthereforeoneelectronisnotgoingtokeepthesamerelative phasewithrespecttotheRF-wave,sucheectisnamedphaseslippage.Let'sdenethephase E0isthepeakeld,zisthepositionwithrespecttothecavitycenter,andistheosetphase (z)as: Ez=E0cos(kz)cos(!t+)=E0 2(cos(!t+kz)+cos(!t++kz)) (6.4) TheEqns.(6.5)and(6.6)togetherformacoupleddierentialequationsystemthatcanbesolved Moreoverthenormalizedenergygainis: d(z) dz=eE0 (z)def =!tkz=kZz 2mc21(cos((z)+2kz)+cos((z))) 0 p211!dz+ (6.6) (6.5) numericallyusingstandardtechnique.Figures6.4presentstheenergygaininthetwocavitywith

equationthatrequiresomeinitialconditions,thatwewilltaketobeattheemittancesolenoidexit.<br />

Letr<strong>and</strong>zberespectivelythermstransverse<strong>and</strong>longitudinalbeamenvelope.Itiswellknow<br />

(e.g.seereference[54])thatonec<strong>and</strong>escribestheevolutionofthebeamenvelopeviatheso-called<br />

coupledrmsenvelopeequationthatwrite(extendedfromreference[54]):<br />

@2r(s)<br />

@s2+k20r(s)3<br />

10p5Nrc<br />

20301<br />

r(s)z(s) 1g22r<br />

202z!~"2r3r=0 (6.1)<br />

@2z(s)<br />

@s2+k20r(s)3<br />

10p5Nrc<br />

2050g<br />

z(s)2~"z(s)2<br />

z(s)3=0<br />

whereg=g(z=r;b=r)afunctionofthebeamrmssize<strong>and</strong>thevacuumpipediameterb,de-<br />

scribestheeectofthebunchinteractionwithitsimageonthebeamlinevacuumchamber;<br />

rc=e2=(40mc3)istheclassicalradiusofanelectron<strong>and</strong>0isthebunchreducedenergy(from<br />

nowonwewillassumeanenergyof350keV,i.e.0=0.8048<strong>and</strong>0=1.6849.Toconvinceourselves<br />

onthenecessityofusingtheaboveequationsystem,wecanstudythedependenceofthe\space<br />

chargeoveremittanceratio".Forthelongitudinaldirectionwedenethisratioas:<br />

Rrdef<br />

=3<br />

10p5Nrc<br />

30gz<br />

(~"nr)2 (6.2)<br />

Thesamekindoffactorcanbedenedforthetransversedirection:<br />

Rzdef<br />

=3<br />

10p5Nrc<br />

02r<br />

(~"nz)2z 1g22r<br />

202z! (6.3)<br />

Theevolutionoftheseratiosalongthebeamlineusingrmsenvelopenumericallycomputedwith<br />

parmelaareshowningure6.3.Inthe350keVlineitisseenthatspacechargecontributionin<br />

theenvelopeequationcanbeafactor100largerthantheemittancetermcontribution.Evenin<br />

the10MeVregion,thereisstillapredominanceofspacechargetermbyafactor10exceptinthe<br />

bunchingchicanewheredispersionincreasetransversebeamsize<strong>and</strong>thereforelocallyreducespace<br />

chargeforce.Ontheotherh<strong>and</strong>,thelongitudinalratioissignicantlylargerthanunityonlyin<br />

the350keVregion.Itisstronglydampedasthebeamisacceleratedinthe10MeVstructure<strong>and</strong><br />

downstreamthecryounitthelongitudinalenvelopeequationisonlydrivenbytheemittanceterm.<br />

Toapplythermsenvelopeequationtothedierentelementswecanusethefollowingsteps:<br />

foradriftspace,theexternalfocusingparameters,kr<strong>and</strong>kzaresettozero.<br />

thebunchercavityismodeledasa\slopeimpulse":z0buncher<br />

!z0+2RFeV<br />

mc223z<br />

thesolenoidexternalfocusingparameterisestimatedusingtherelationk0=eB0<br />

2mcwhereB0<br />

istheintegratedmagneticeld,whichwehaveestimatedusingapoissongeneratedmagnetic<br />

eldprole.<br />

6.1.3Thehighgradientstructure<br />

Inthissectionwewouldliketodiscussfewinterestingeectsinducedonthetransversebeam<br />

dynamicsbytheCEBAF-typeacceleratingcavities.Thediscussionwillenablethereadertoun-<br />

derst<strong>and</strong>experimentalresultspresentedinthenextsection.

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