Longitudinal Beam Dynamics F.Tecker CERN,Geneva,Switzerland Abstract Thecoursegivesasummaryoflongitudinalbeamdynamicsforbothlinearand circularaccelerators. Afterdiscussingdifferenttypesofaccelerationmethods andsynchronismconditions,itfocusesontheparticlemotioninsynchrotrons. 1 Introduction 6 TheforceF(cid:126) onachargedparticlewithachargeeisgivenbytheNewton–Lorentzforce 1 0 dp(cid:126) (cid:16) (cid:17) 2 F(cid:126) = = e E(cid:126) +(cid:126)v×B(cid:126) . (1) dt n a The second term is always perpendicular to the direction of motion, so it does not give any lon- J gitudinal acceleration and it does not increase the energy of the particle. Hence, the acceleration has to 9 1 bedonebyanelectricfieldE(cid:126). Inordertoacceleratetheparticle,thefieldneedstohaveacomponentin the direction of the motion of the particle. If we assume the field and the acceleration to be along the z ] h direction,Eq.(1)becomes p dp = eE . (2) - z c dt c ThetotalenergyE ofaparticleisthesumoftherestenergyE andthekineticenergyW: a 0 . s E = E +W. (3) c 0 i s Inrelativisticdynamics,thetotalenergyE andthemomentumparelinkedby y h p E2 = E02+p2c2, (4) [ fromwhichitfollowsthat 1 dE = vdp. (5) v 1 0 Therateofenergygainperunitlengthofacceleration(alongthez direction)isthengivenby 9 dE dp dp 4 = v = = eE (6) 0 dz dz dt z . 1 andthe(kinetic)energygainedfromthefieldalongthez pathfollowsfrom dW = dE = eE dz: z 0 6 (cid:90) 1 W = e Ezdz = eV, (7) : v i whereV isjustanelectricpotential. X r a 2 Methodsofacceleration 2.1 Electrostaticacceleration The most basic way of acceleration is using an electrostatic field between two electrodes, as shown in Fig.1. TheenergygainW inanelectrostaticfieldisgivenbyW = e∆V,where∆V isthevoltage(or potential)differencebetweentheelectrodes. Thismethodallowstheaccelerationofacontinuousbeam. On today’s energy scale, the maximum energy gain is quite limited by insulation problems, and the maximum voltage is around 10 MV. Nevertheless, this is used for the first stage of acceleration, the particlesources,electronguns,X-raytubes,andlow-energy-ionacceleration. vacuum envelope E source ΔV Fig.1: Electrostaticacceleration 2.2 Induction–thebetatron Insulation issues limit the acceleration by static electric fields. In the general case, the electric field is derivedfromascalarpotentialV andthetimederivativeofavectorpotentialA(cid:126): ∂A(cid:126) E(cid:126) = −∇(cid:126)V − , (8) ∂t ∂B(cid:126) B(cid:126) = µH(cid:126) = ∇(cid:126) ×A(cid:126) ⇒ ∇(cid:126) ×E(cid:126) = − . (9) ∂t From Maxwell’s equations above, it follows that the time variation of the magnetic field generates an electricfield,whichcanaccelerateparticlesovercomingthestaticinsulationproblems. Onemethodbasedonthisprincipleisthebetatron, asshowninFig.2. Thecircularlysymmetric magnet is fed by an alternating current at a frequency typically between 50 and 200 Hz. The time- varyingmagneticfieldB(cid:126) createsanelectricfieldE(cid:126) andatthesametimeguidestheparticlesonacircular trajectory. According to the field symmetry, the electric field generated is tangent to the circular orbit. side view vacuum pipe beam B f iron yoke coil E beam top view R B B f Fig.2: Schematicofabetatron. Thetopshowsthesideview,themiddlethetopview,andthegraphatthebottom themagneticfielddistribution. 2 Betatrons were used to accelerate electrons up to about 300 MeV with the energy reach limited by the saturationinthemagnetyoke. 2.3 Radio-frequencyacceleration One also can overcome the limitations of the electrostatic fields by radio-frequency (RF) acceleration. AnRFoscillatorfeedsalternatelyaseriesofdrifttubeswithgapsinbetweenthem, asshowninFig.3. Insidethetubes,theparticleisshieldedfromtheoutsidefield. Ifthepolarityofthefieldisreversedwhile theparticletravelsinsidethetube,itgetsacceleratedateachgap. Fig.3: Widerøe-typeacceleratingstructure ThisleadstothesynchronismconditionthatthedistanceLbetweenthegapshastofulfil: L = vT/2, (10) wherev = βcistheparticlevelocityandT theperiodoftheRFoscillator. Itisclearthatthisarrange- mentcannotaccelerateacontinuousbeam. Onlyacertainphaserangewillbeacceleratedandthebeam hastobebunched. Astheparticlevelocityincreases,thedriftspaceshavetogetlongerandonelosesefficiency. One canincreasetheradiofrequencytocounteractthiseffectbutalargeamountofpowerwillberadiatedas one goes to higher frequencies. It is then convenient to enclose the accelerating gap in a cavity which holdstheelectromagneticenergyintheformofamagneticfieldandtomaketheresonantfrequencyof thecavityequaltothatoftheacceleratingfield. Severalofthesecavitiescanbeclosetogetherwithacertainphaserelationshipbetweenthem(see Fig.4). The synchronism condition depends on the mode. While it is L = vT/2 for the π-mode, it becomes L = vT for the 2π-mode. In the 2π-mode, the resulting wall current between the cavities is zero,andthecommoncavitywallsareunnecessary. Fig.4: Adjacentcavitieswithdifferentmodes. Left:π-mode–thefieldisoppositeinthegapsofthetwocavities, right: 2π-mode–thefieldisthesameinbothcavities. 3 Fig.5: Alvarez-typeacceleratingstructure Avariantofthatschemeconsistsofplacingthedrifttubesinasingleresonanttanksuchthatthe field has the same phase in all gaps (see Fig. 5). Such a resonant accelerating structure was invented byAlvarez,andthistypeisstillusedfortheaccelerationofprotonswiththeenergyrangingfrom50to 200MeV. 2.4 Transit-timefactor Whentheparticletraversesacavity,thefieldvariesduringthepassageoftheparticlethroughtheaccel- eratinggap. So,theparticlewillnotalwaysseethemaximumfieldandtheeffectiveaccelerationwillbe smallerbyacertainfactor. Thistransit-timefactorT isdefinedas a energygainof particlewithv = βc T = . (11) a maximumenergygain(particlewithv → ∞) It quantifies the reduction of energy gain due to the fact that the particle travels with a finite velocity in anelectricfieldwithasinusoidaltimevariation. Thetransit-timefactorvariesbetween0and1. Inthegeneralcaseforaparticletravellinginthez direction,assumingthattheparticlevelocityis constant,itisgivenby (cid:12) (cid:12) (cid:12) (cid:90)+∞ (cid:12) (cid:12) (cid:12) (cid:12) E (z)eiωRFtdz (cid:12) z (cid:12) (cid:12) (cid:12) (cid:12) T = (cid:12)−∞ (cid:12). (12) a (cid:12) +∞ (cid:12) (cid:12) (cid:90) (cid:12) (cid:12) (cid:12) (cid:12) Ez(z)dz (cid:12) (cid:12) (cid:12) (cid:12) −∞ (cid:12) Forasimplemodel,auniformstandingwavefieldE(z,r,t) = E (z,r)·cos(ω t)withaconstant 1 RF fieldonlypresentinthegap(seeFig.6), E (z,r) = E = const. (13) 1 0 Fig.6: Simpleuniformfieldmodel 4 Foraparticlepassingthecentreofthefieldwhenthefieldismaximum,itfollowsthat (cid:12) (cid:16)ω g(cid:17)(cid:46)(cid:16)ω g(cid:17)(cid:12) T = (cid:12)sin RF RF (cid:12). (14) a (cid:12) 2v 2v (cid:12) Thissimpleexampleshowsthatthetransit-timefactortendstowards1forsmallergapwidthg, smaller radio frequency, and higher velocity v of the particle, which is also true in the general case. The re- ductioninaccelerationbecomesimportantforparticleswithlowvelocities,likelow-energyprotonsand particularlyions. 2.5 Disc-loadedtravellingwavestructures Electrons reach a relativistic β close to unity for a kinetic energy of 10 MeV, while protons reach this onlyatanenergyoftheorderof10GeV.Abovetheseenergies,theparticleshavebasicallythespeedof light,andthedrift-tubelengthremainsconstant. Nevertheless,thedrift-tubelengthwouldbecomevery longunlessthefrequencyisintheGHzrange. So, the idea came up that the ultra-relativistic particles could be accelerated by a travelling wave inawaveguide. Inordertotogetcontinuousacceleration,thephasevelocityv ofthewavehastomatch ϕ the velocity v of the particle. However, rectangular or cylindrical waveguides have electric field modes withanelectricfieldinthedirectionofpropagationwithphasevelocitiesbiggerthanc,sothatthewave does not remain synchronous with the particle. The phase velocity can be adjusted by inserting irises intothewaveguide,andthedimensionsoftheirisesandcellscanbetailoredtomatchthephasevelocity tothevelocityoftheparticle. Figure7showsasketchofsuchadisc-loadedtravellingwavestructure. Fig.7: Cutofadisc-loadedtravellingwavecavity The electric field of an electromagnetic wave travelling in the z direction (see Fig. 8) is given by equation(15). E = E cos(ω t−kz), (15) z 0 RF ω RF k = wavenumber, v ϕ z = v(t−t ), 0 v = phasevelocity, ϕ v = particlevelocity. Fig. 8: Electromagnetic wave travelling in the z direc- tion 5 Thefieldseenbytheparticleis (cid:18) (cid:19) v E = E cos ω t−ω t−φ . (16) z 0 RF RF 0 v ϕ Whensynchronismissatisfiedwithv = v ,theparticleseesaconstantfield ϕ E = E cosφ , (17) z 0 0 whereφ istheRFphaseseenbytheparticle. So,thistypeofstructurewillcontinuouslyacceleratethe 0 particleduringthepassagethroughthestructure. 3 Phasestabilityandenergy-phaseoscillationinalinac Severalphaseconventionsexistintheliterature(seeFig.9): – mainlyforcircularaccelerators,theoriginoftimeistakenatthezerocrossingwithpositiveslope; – mainlyforlinearaccelerators,theoriginoftimeistakenatthepositivecrestoftheRFvoltage. In the following, I will stick to the former case of the positive zero crossing for both the linear and circularcases. 1 E 2 E 1 2 φ=ω t φ=ω t RF RF φ φ 1 2 E (t)= E sin(ω t) E (t)= E cos(ω t) 1 0 RF 2 0 RF Fig.9: Commonphaseconventions: (1)theoriginoftimeistakenatthezerocrossingwithpositiveslope,(2)the originoftimeistakenatthepositivecrestoftheRFvoltage. LetusconsideranAlvarezstructure,wherebydesigntheenergygainforaparticlepassingthrough the structure at a certain RF phase φ is such that the particle reaches the next gap with the same phase s φ . Thentheenergygaininthefollowinggapwillbeagainthesame,andtheparticlewillpassallgaps s at this phase φ , which is called the ‘synchronous phase’. So, the energy gain is eV = eVˆ sinφ . This s s s isillustratedinFig.10bythepointsP andP . 1 3 A particle N which arrives in a gap earlier compared to P will gain less energy and its velocity 1 1 willbesmaller,sothatitwilltakemoretimetotravelthroughthedrifttube. Inthenextgapitwillappear closertoparticleP . TheeffectisoppositeforparticleM , whichwillgainmoreenergyandreduceits 1 1 delay compared to P . So, the points P , P , etc are stable points for the acceleration since particles 1 1 3 slightly away from them will experience forces that will reduce their deviation. On the contrary, it can beseenthatthepointP isanunstablepointinthesensethatparticlesslightlyawayfromthispointwill 2 deviateevenmoreinthenextgaps. So, for stability of the longitudinal oscillation, the particle needs to be on the rising slope of the RFfieldtohavearestoringforcetowardsthestablephase. 6 Fig.10: Energygainasafunctionofparticlephase When we look at the electric field in the accelerating gap between two drift tubes, there is a transverse focusing field at the entrance and a transverse defocusing field at the exit, as illustrated in Fig.11. Inanelectrostaticaccelerator,theeffectofthedefocusingattheexitissmallerthanthefocusing at the entrance, as the particle gains longitudinal momentum inside the gap. So, there is a net focusing effect. IntheRFcasewithstablelongitudinalmotion,thefieldincreasesduringthepassageoftheparti- cle. Asaconsequence,thedefocusingfieldwhentheparticleexitsthegapisstrongerthanthefocusing fieldwhentheparticleenters,resultinginanetdefocusingeffect. Inordertokeepthetransversemotion stable,externalfocusingbysolenoidsorquadrupolemagnetsisnecessary. Inordertostudythelongitudinalmotionitisconvenienttousevariableswhichgivethephaseand energyrelativetothesynchronousparticle(denotedbythesubscripts): ϕ = φ−φ , (18) s w = E −E = W −W . (19) s s Theacceleratingfieldcanbesimplydescribedby E = E sin(ωt). (20) z 0 Therateofenergygainforthesynchronousparticleisgivenby dE dp s s = = eE sinφ (21) 0 s dz dt Fig. 11: Field lines in the gap of a drift-tube accelerator (left), stable phase on the rising slope of the RF field (right). 7 andforanon-synchronousparticle(forsmallϕ) dw = eE [sin(φ +ϕ)−sinφ ] ≈ eE cosφ ϕ. (22) 0 s s 0 s dz Therateofchangeofthephasewithrespecttothesynchronousparticleis,forsmalldeviations, (cid:20) (cid:18) (cid:19)(cid:21) (cid:18) (cid:19) dϕ dt dt 1 1 ω RF = ω − = ω − ≈ − (v−v ). (23) dz RF dz dz RF v v v2 s s s s Usingdγ = γ3βdβ,w becomesinthevicinityofthesynchronousparticle w = E −E = m c2(γ −γ ) = m c2dγ = m c2γ3β dβ = m γ3v (v−v ), (24) s 0 s 0 0 s s 0 s s s whichleadsto dϕ ω RF = − w. (25) dz m v3γ3 0 s s Combining the two first-order equations (22) and (25) into a second-order equation gives the equationofanharmonicoscillatorwiththeangularfrequencyΩ : s d2ϕ eE ω cosφ +Ω2ϕ = 0 with Ω2 = 0 RF s. (26) dz2 s s m v3γ3 0 s s StableharmonicoscillationsimplythatΩ2 > 0andreal,whichmeansthatcosφ > 0. Sinceacceleration s s meansthatsinφ > 0,itfollowsthatthestablephaseregionforaccelerationinthelinacis s π 0 < φ < , (27) s 2 whichconfirmswhatwehaveseenbeforeinourdiscussionabouttherestoringforcetowardsthestable phase. From Eq. (26) it is also visible that the oscillation frequency decreases strongly with growing velocity (and relativistic gamma) of the particle. For highly relativistic particles, the velocity change is negligible, so there is practically no change of the particle phase, and the bunch distribution is not changinganymore. 4 Circularaccelerators–cyclotron The cyclotron is a circular accelerator that has two hollow ‘D’-shape electrodes in a constant magnetic field B (see Fig. 12). When a particle is generated at the source in the centre, it is accelerated by the electricfieldbetweentheelectrodes. Itentersanelectrodeand,whileitisshieldedfromtheelectricfield, thepolarityofthefieldinthegapisreversed. IfthephaseoftheRFiscorrect,theparticleisaccelerated again in the gap and enters the other electrode. The magnetic field creates a spiralling trajectory of the particle. Astheparticlebecomesfaster,theorbitradiusgetsbiggerbuttherevolutionfrequencydoesnot dependontheradius,aslongastheparticleisnotrelativistic. So,thesynchronismconditionisthattheRFperiodhastocorrespondtotherevolutionperiod: T = 2πρ/v (28) RF s withthecyclotronfrequencygivenby qB ω = . (29) m γ 0 8 source RF generator, ω RF B g Extraction electrode trajectory Fig.12: Schematicviewofacyclotronandtheparticletrajectoryinatopview(ontheright) Aslongasv (cid:28) candγ ≈ 1thesynchronismconditionstaysfulfilled. Forhigherenergies,theparticle willgetoutofphasewithrespecttotheRF,eventhoughthereisstillarangefortheinitialparticlephase whereastableaccelerationispossible. In order to keep synchronism at higher energies, one has to decrease the radio frequency during theaccelerationcycleaccordingtotherelativisticγ(t)oftheparticleas qB ω (t) = ω (t) = , (30) RF s m γ(t) 0 which leads to the concept of a synchrocyclotron, which can accelerate protons up to around 500 MeV. Hereanewlimitationoccursduetothesizeofthemagnet. 5 Synchrotron A synchrotron (see Fig. 13) is a circular accelerator where the nominal particle trajectory is kept at a constant physical radius by variation of both the magnetic field and the radio frequency, in order to follow the energy variation. In this way, the aperture of the vacuum chamber and the magnets can be keptsmall. Fig.13: Schematiclayoutofasynchrotron 9 Theradiofrequencyneedstobesynchronoustotherevolutionfrequency. Toachievesynchronism, the synchronous particle needs to arrive at the cavity again after one turn with the same phase. This implies that the angular radio frequency ω = 2πf has to be an integer multiple of the angular RF RF revolutionfrequency: ω = hω , (31) RF r where h is an integer and is called the harmonic number. As a consequence, the number of stable synchronousparticlelocationsequalstheharmonicnumberh. Theyareequidistantlyspacedaroundthe circumferenceoftheaccelerator. Allsynchronousparticleswillhavethesamenominalenergyandwill followthenominaltrajectory. Energy ramping is obtained by varying the magnetic field, while following the change of the revolutionfrequencywithachangeoftheradiofrequency. Thetimederivativeofthemomentum, p = eBρ, (32) yields(whenkeepingtheradiusρconstant) dp = eρB˙. (33) dt Foroneturninthesynchrotron,thisresultsin 2πeρRB˙ (∆p) = eρB˙T = , (34) turn r v whereR = L/2π isthephysicalradiusofthemachine. SinceE2 = E2+p2c2,itfollowsthat∆E = v∆p,sothat 0 (∆E) = (∆W) = 2πeρRB˙ = eVˆ sinφ . (35) turn s s Fromthisrelationitcanbeseenthatthestablephaseforthesynchronousparticlechangesduring theacceleration,whenthemagneticfieldB changes,as (cid:32) (cid:33) B˙ B˙ sinφ = 2πρR or φ = arcsin 2πρR . (36) s Vˆ s Vˆ RF RF As mentioned previously, the radio frequency has to follow the change of revolution frequency andwillincreaseduringaccelerationas f v(t) 1 ec2 ρ RF = f = = B(t). (37) r h 2πR 2πE (t)R s s s SinceE2 = E2+p2c2,theradiofrequencymustfollowthevariationoftheB fieldwiththelaw 0 f c (cid:26) B(t)2 (cid:27)1/2 RF = . (38) h 2πR (m c2/ecρ)2+B(t)2 s 0 This asymptotically tends towards f → c/(2πR ) when v → c and B becomes large compared to r s m c2/(ecρ). 0 10