Longitudinal Beam Dynamics

2.2 MB·

by  F. Tecker

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