Table Of ContentBeam Instabilities
G.Rumolo
CERN,Geneva,Switzerland
Abstract
When a beam propagates in an accelerator, it interacts with both the exter-
nalfieldsandtheself-generatedelectromagneticfields. Ifthelatterarestrong
enough, the interplay between them and a perturbation in the beam distribu-
tion function can lead to an enhancement of the initial perturbation, resulting
6
1 inwhatwecallabeaminstability. Thisunstablemotioncanbecontrolledwith
0 a feedback system, if available, or it grows, causing beam degradation and
2
loss. Beaminstabilitiesinparticleacceleratorshavebeenstudiedandanalysed
n
in detail since the late 1950s. The subject owes its relevance to the fact that
a
J the onset of instabilities usually determines the performance of an accelera-
0 tor. Understandingandsuppressingtheunderlyingsourcesandmechanismsis
2
thereforethekeytoovercomingintensitylimitations,therebypushingforward
] theperformancereachofamachine.
h
p
- 1 Introduction
c
c
The motion of charged particles forming a beam in an accelerator can be studied either individually or
a
. taking into account the electromagnetic interaction between them. In the former case, the beam is re-
s
c garded as a collection of non-interacting particles and the forces acting on them, i.e. the driving terms
i
s in each particle’s equations of motion, are fully prescribed by the accelerator design. The study of the
y
single-particle dynamics is then complicated by all non-linear components of the applied electromag-
h
p netic fields. In practice, this description is sufficient as long as additional electromagnetic fields caused
[ by the presence of the whole beam of particles are not strong enough to perturb significantly the mo-
1 tionimpartedbytheexternalfields. Inmanyapplications,however,beamscarryingahighcharge(high
v
intensity)anddenselypackedinatinyphasespace(highbrightness)arerequired,forwhichtheelectro-
1
magneticfieldscreatedbytheinteractionofthebeamwiththeexternalenvironmentneedtobeincluded
0
2 when solving the particles’ motion. Under unfavourable conditions, these electromagnetic fields act
5
back on the beam distribution itself in a closed loop, such as to enhance a however small initial pertur-
0
. bation. Thissituationeventuallyleadstoaninstability. Themostgeneralexampleofaninstabilityloop
1
isschematicallyillustratedinFig.1.
0
6 The block labelled ‘Interaction between beam and external environment’ has been willingly left
1
vague,asanyfurtherspecificationdependsonthetypeofproblembeingmodelled. Inthemostfrequent
:
v
case,whichwillalsobethesubjectofthisarticle,theinteractionbetweenbeamandexternalenvironment
i
X will be purely electromagnetic, so that it can be expressed in terms of Maxwell’s equations with the
r beamassourcetermandboundaryconditionsgivenbytheacceleratordevicesthroughwhichthebeam
a
propagates. Another case that is frequently the object of study is when the beam generates an electron
orioncloudthatactsbackonthebeamitselfandpotentiallydestabilizesit. Inthiscase, theinteraction
ofthebeamwiththeenvironmentneedstobedescribedwithallofthephysicalprocessesleadingtothe
cloud formation. The additional electric field from the cloud can then be evaluated through Poisson’s
equationandusedasadrivingtermintheequationsofmotionofthebeamparticles.
In practice, a beam becomes unstable when, as a result of the loop described above, at least a
momentofitssix-dimensional(6D)phasespacedistribution,ψ(x,y,z,x(cid:48),y(cid:48),δ),exhibitsanexponential
growth (e.g. typically the mean positions x , y , z or the standard deviations σ , σ , σ ), resulting
x y z
(cid:104) (cid:105) (cid:104) (cid:105) (cid:104) (cid:105)
Mul$-‐bunch
beam
s
Noise
Equa$ons
of
Interac$on
between
mo$on
of
the
beam
and
external
beam
par$cles
environment
E�, B�
� ⇥
Addi$onal
electromagne$c
field
ac$ng
on
the
beam,
besides
RF
and
external
magne$c
fields
Fig.1: Schematicoftheclosedloopthroughwhichabeamcanbecomeunstableundertheeffectofself-generated
electromagneticfields.
in beam loss or emittance growth. Assuming an arbitrary observation point s along the trajectory of a
0
beam inside an accelerator, described through the coordinate s, the full 6D phase space can usually be
decomposedintotransverseandlongitudinalphasespaces. The4Dtransversespaceisdescribedbythe
twopairsofconjugatevariables(x,x(cid:48),y,y(cid:48)),i.e.theoffsetsfromthenominalorbitinthehorizontaland
vertical directions (horizontal is the direction in which the beam is bent), and the relative divergences
from the nominal orbit, x(cid:48) = dx/ds and y(cid:48) = dy/ds. The longitudinal plane is described by the
conjugate pair (z,δ), i.e. a space coordinate proportional to the delay in the arrival time at the selected
locationwithrespecttothesynchronousparticle,z = cτ (theminussignischosensuchthatparticles
−
arriving before the synchronous particle have a positive z), and the relative longitudinal momentum
deviation from the nominal momentum, δ = δp/p . As an example of instability detection, the onset
0
of a transverse instability can be easily revealed by the signal captured from a beam position monitor
(BPM).Aphaseofexponentialgrowthcanbeobserved,usuallyfollowedbysaturationanddecayeither
due to non-linearities or because of beam loss. Figure 2 shows an example of horizontal BPM signals
from two different bunches during the store of a train of 72 bunches with 25 ns spacing in the CERN-
Proton Synchrotron (PS). The signal in Fig. 2(a) is basically BPM noise and represents a stable bunch,
while that in Fig. 2 is from an unstable one. This also highlights how the unstable beam oscillation is
eventuallyassociatedwithacertainamountofbeamlossandisdampedafterthelossoccurs.
The interest in studying coherent beam instabilities arises from the fact that the onset of a beam
instability usually determines the maximum beam intensity that a machine can store/accelerate (i.e. its
performance limitation). Understanding the type of instability limiting the performance, and its under-
lying mechanism, is essential because it allows the source and possible measures to mitigate/suppress
the effect to be identified, or providing the specifications of an active feedback system to prevent the
instability. Beam instabilities occur in both linear and circular machines and can equally affect the lon-
gitudinalplaneorthetransverseplane. Coherentinstabilitiescanaffectthebeamondifferentscales. For
example,atypicalmultibunchinstabilityexhibitsanexcitationpatternextendingoverdifferentbunches
inatrainanddependsonalong-rangecouplingagent. Nevertheless,insomecasestheunstablemotion
of subsequent bunches does not appear as coupled, because the instability can be just the consequence
of a certain mechanism that builds up along the bunch train, but visibly affects only the last bunches of
atrain(e.g.anelectroncloud). Inapuresingle-bunchinstability,usuallythecouplinghappensbetween
2
(a)
Stable
beam
(b)
Unstable
beam
Bunch 10 Bunch 46
0.4 0.4
BCT
signal
I
=
constant
BCT
signal
ΔI
0.3 0.3
0.2 0.2
Hor. position. [a.u.]--000...1210 Hor. position. [a.u.]--000...1210
-0.3 -0.3
-0.4 -0.4
0 5000 10000 15000 0 5000 10000 15000
Turn # Turn #
(a) (b)
Fig.2: Examplesofstable(a)andunstable(b)signalsfromaBPM.Thesignalfromabeamcurrenttransformer
(BCT)isalsosketched,showinghowthestablebeamdoesnotsufferfromanyintensityloss,whileasharpintensity
decreaseisassociatedwiththeriseoftheinstability.
headandtailofthesamebunch. Inthiscase,themechanismthatdrivestheinstabilityonlyneedstoact
ontheshortrange.
Inthefollowingsections, wewillfirstsetthemathematicalframeworktoaddresstheproblemof
beaminstabilitiesdrivenbyself-generatedelectromagneticfields(wakefunctionsandimpedances)and
wewillthenapplytheseconceptstoreducedmodels(one-ortwo-particle)toexplainthephysicsofsome
ofthemostfrequentinstabilitymechanismsinparticleaccelerators. Thereferencethatwillbefollowed
throughoutthisarticleis[1].
2 Thelongitudinalplane
Let us consider two ultra-relativistic charged particles (q and q , travelling at v c, or equivalently
1 2
≈
γ 1)goingthroughanacceleratorstructure,separatedbyadistance z (z = cτ,withτ expressing
(cid:29) | | −
thedelaybetweenthearrivaltimesofthetwoparticlesatanarbitrarylocation). Theleadingparticlewill
beoursourceandthetrailingparticlewillbethewitness. Sincebothparticlesaretravellingbasicallyat
the speed of light, causality imposes that the leading particle cannot be affected by the trailing particle.
As long as source and witness move in a perfectly conducting chamber, the witness does not feel any
force from the source. However, when the source encounters a discontinuity, the electromagnetic field
produced to satisfy the boundary conditions (wake field) can effectively reach the witness particle and
affectitsmotion. Inthisprocess,thesourcelosesenergy,whilethewitnessfeelsanetforceallalongan
effective length, L, of the discontinuity/structure/device that caused the wake. Figure 3 shows a simple
sketchofhowthesituationcouldlooklikeafterasourcehasgonethroughacavity-likeobjectandmodes
aretrappedafteritspassage.
In fact, geometric discontinuities are not the only possible origin of wake fields. For example,
in a chamber with finite conductivity the induced current from a source particle is delayed and can
significantly act back on witness particles within a certain distance range. Generally, electromagnetic
boundaryconditionsotherthanaperfectelectricalconductor(PEC)cangeneratewakefields.
Thelongitudinalwakefunctionassociatedwithacertainacceleratorobject(abletocreateawake
field)isdefinedastheintegratedlongitudinalforce(q E (s,z))actingonthewitnessparticlealongthe
2 s
effective length L of the object (i.e. its energy change, ∆E ), normalized by the source and witness
2
charges:
L
W (z) = 0 Es(s,z)ds = ∆E2. (1)
||
− q −q q
(cid:82) 1 1 2
3
Source,
q
1
Witness,
q
2
z
s
2b
L
Fig.3: Wakefieldfromasourceparticlepotentiallyaffectingawitnesstravellingatdistancezbehindthesource
Theminussignisalsointroducedinthedefinition,sothatW(0) = ∆E /q2 isdefinedpositive
1 1
−
(the source particle can only lose energy, ∆E < 0). The beam loading theorem also proves that the
1
wakefunctionisdiscontinuousinz = 0,withW (0−) = 2 W (0). Intuitively,thistheoremstatesthat
|| ||
·
a particle travelling at the speed of light can only see half of its own wake. Besides, causality imposes
thatW (0+) = 0,andactuallyW (z) = 0forz > 0. Inaglobalenergybalance,theenergylostbythe
|| ||
source,∆E ,splitsinto
1
– Electromagnetic energy of the modes that may remain trapped in the object. This is then partly
dissipated on the lossy walls or into purposely designed inserts or higher order mode (HOM)
absorbers. Partly,itcanbepotentiallytransferredtothetrailingparticles(orthesameparticleover
successiveturns),possiblyfeedingintoaninstability.
– Electromagnetic energy of modes that propagate down the beam chamber (above cut-off), which
willbeeventuallylostonsurroundinglossymaterials.
The energy loss of a beam is very important, because the fraction lost on the beam environment causes
equipment heating (with consequent outgassing and possible damage), while the part associated with
long-livedwakefieldscanfeedintobothlongitudinalandtransverseinstabilities. Thecalculationofthe
energylosswillbethesubjectofthenextsubsection.
The wake function of an accelerator object is basically its Green function in the time domain
(i.e. the electromagnetic response of the object to a pulse excitation). Therefore, it is very useful for
macroparticlemodelsandsimulations,becauseitcanbeusedtodescribethedrivingtermsinthesingle-
particleequationsofmotion,aswewillseeinoneofthenextsubsections. However,wecanalsodescribe
this response as a transfer function in the frequency domain. This gives the definition of longitudinal
beamcouplingimpedanceoftheobjectunderstudy:
∞ iωz dz
Z (ω) = W (z)exp . (2)
|| ||
− c c
(cid:90)−∞ (cid:18) (cid:19)
Typicallongitudinalwake/impedancepairsaredescribedasresonatorsandaredisplayedinFig.4.
The wake function is a damped oscillation with a discontinuity in z = 0, while the beam coupling
impedance spectrum exhibits a peak at the specific oscillation frequency. The width of the peak relates
to the lifetime of the oscillation in the time domain before becoming fully damped, distinguishing be-
tween a narrowband and a broadband resonator, as shown in top and bottom of Fig. 4, respectively. In
more complex cases, several modes can be excited in the object and the beam coupling impedance is a
combinationofseveralpeakslikethoseshowninthesingle-resonanceexamplesdepictedinFig.4. For
4
example, a pill-box cavity with walls having finite conductivity and attached to a vacuum chamber left
and right (Fig. 5(a)) can resonate on all its characteristic modes determined by its geometry. The width
of the excited peaks will be narrower for the modes below the cut-off frequency of the chamber (as the
decayispurelydeterminedbytheresistivelosses),whiletheywillbebroaderforthepeaksabovecut-off,
asadditionallossescomefromthepropagationofthesemodesintothechamber. ThisisvisibleinFig.5
(simulationsdonewithCST® ParticleStudioSuite).
0.6 1.0
W
||
0.8 Re[Z ]
0.4 ||
0.6
0.2 0.4 Im[Z ]
||
0.2
�6z
�5 �4 �3 �2 �1 0.0 ω
�0.2 �0.2
�0.4
�0.4 0 1 2 3 4 5
5
1.0
W
||4 0.8
3 0.6
0.4
2
0.2
1 0.0
ω
�0.2
�6z
�5 �4 �3 �2 �1 �0.4
�1
0 2 4 6 8 10
Fig.4: Wakefunctions(left)andbeamcouplingimpedances(right)fornarrowband(top)andbroadband(bottom)
resonatorobjects.
(a) (b)
Fig. 5: Pill-box cavity: a 3D longitudinal cut of the simulated cavity (a) and the obtained longitudinal beam
couplingimpedance(b).Thecut-offfrequencyofthebeamchamberisshownwithaverticaldashedline.Courtesy
ofC.Zannini.
In beam physics, broadband impedances, such that the associated wake functions decay over the
length of one particle bunch, are only responsible for intrabunch (head–tail) coupling, potentially lead-
ing to single-bunch instabilities. Conversely, narrowband impedances, associated with long-lived wake
functions decaying over the length of a bunch train or several turns, cause bunch-to-bunch or multiturn
5
coupling,leadingtomultibunchormultiturninstabilities.
2.1 Energyloss
By using the concepts so far introduced, we can easily derive an analytical expression for the energy
lost by a bunch with line density λ(z) (see Fig. 6) when it goes through a structure characterized by a
wake function W (z) or beam coupling impedance Z (ω). The energy change ∆E(z) of the witness
|| ||
sliceeλ(z)dzcanbeexpressedastheintegralofthecontributionsfromthewakefunctionsgeneratedby
all the preceding source slices, eλ(z(cid:48))dz(cid:48). Integrating ∆E(z) over the whole bunch provides the total
energylossofthebunch:
zˆ zˆ zˆ
∆E = ∆E(z)dz = eλ(z) eλ(z(cid:48))W (z z(cid:48))dz(cid:48)dz. (3)
||
− −
−zˆ −zˆ z
(cid:90) (cid:90) (cid:90)
By using the Parseval identity and the convolution theorem, the energy loss can be easily written
intermsofbunchspectrumΛ(ω)andbeamcouplingimpedance:
e2 ∞ e2 ∞
∆E = Λ∗(ω) Λ(ω)Z (ω) dω = Λ(ω) 2Re Z (ω) dω. (4)
|| ||
−2π −2π | |
−∞ −∞
(cid:90) (cid:90)
(cid:2) (cid:3) (cid:2) (cid:3)
�(z)
zˆ zˆ
�
�(z)dz �(z )dz
�
Bunch
tail
Bunch
head
z z
�
| � |
Fig.6: Sketchofthebunchandlinedensity. Sourceandwitnessslicesarealsohighlighted
In the last expression, we also took into account that, since W (z) is a real function, Re[Z (ω)]
|| ||
and Im[Z (ω)] are even and odd functions of ω, respectively. Since Eq. (4) represents the total energy
||
lostbythebunchoverasinglepassthroughtheobjectwithbeamcouplingimpedanceZ (ω),itcanalso
||
beinterpretedasthebunchenergylossperturninacircularaccelerator(againduetoasingleobjectwith
beam coupling impedance Z (ω), or the total energy loss per turn if Z (ω) represents instead the total
|| ||
longitudinalbeamcouplingimpedancemodellingthewholering). However,thisisrigorouslytrueonly
aslongasthewakefunctionisshortenoughlivedtobefullydampedafteroneturn, sothatsubsequent
passagesofthebuncharenotcoupledthroughthewake.
In fact, defining C as the circumference of the ring, Eqs. (3) and (4) can be generalized to the
case of a bunch going through a structure that keeps memory of previous passages, assuming that its
longitudinaldistributiondoesnotchangeintime:
zˆ zˆ zˆ ∞
∆E = ∆E(z)dz = eλ(z) eλ(z(cid:48)) W (kC +z z(cid:48))dz(cid:48)dz. (5)
||
− −
−zˆ −zˆ z
(cid:90) (cid:90) (cid:90) k=−∞
(cid:88)
6
Applyingtheidentity
∞ ω ∞ ipω (z z(cid:48))
W (kC +z z(cid:48)) = 0 Z (pω )exp 0 − , (6)
|| || 0
− 2π − c
k=−∞ p=−∞ (cid:20) (cid:21)
(cid:88) (cid:88)
inwhichω = 2πc/C istherevolutionfrequency,wecaneasilyrecastEq.(5)inthefollowingform:
0
e2ω ∞
∆E = 0 Λ(pω ) 2Re Z (pω ) . (7)
0 || 0
− 2π | |
p=−∞
(cid:88) (cid:2) (cid:3)
Equation (7) is very powerful, because it can be applied to the full beam circulating in an accel-
erator ring and can be used for calculating the total beam energy loss per turn. In this case, we would
simply need to replace Λ(ω), the Fourier transform of the single-bunch distribution, with the Fourier
transform of the full beam signal, Λ (ω). For example, we could assume the beam to be a train of M
B
bunchescoveringonlyafractionofthefullcircumference(M < h,hbeingtheharmonicnumberofthe
accelerator)withspacingbetweenbunchesτ = 2π/(hω ):
b 0
M−1 M−1
λB(z) = λ(z ncτb) F ΛB(ω) = Λ(ω) exp( iωτb). (8)
− ⇐⇒ −
n=0 n=0
(cid:88) (cid:88)
Summingupthetermsintheexpressionofthebeamspectrum,weobtain
Mωτ
b
sin
iωτ (M 1) 2
b
Λ (ω) = Λ(ω)exp − (cid:18) (cid:19), (9)
B ωτ
2 · b
(cid:20) (cid:21) sin
2
(cid:16) (cid:17)
whichcanbefinallyinsertedintoEq.(7),yielding
2πMp
∆E = e2ω0 ∞ Λ(pω0) 2Re Z||(pω0) 1−cos(cid:18) h (cid:19). (10)
2π | | · 2πp
p=−∞ 1 cos
(cid:88) (cid:2) (cid:3) − h
(cid:18) (cid:19)
The terms in the summation above are maximum for p = k h, as the ratio in brackets becomes
·
equaltoM2. Thismeansthatnarrowbandimpedancespeakedaroundmultiplesoftheharmonicnumber
oftheacceleratorarethemostefficienttodrainenergyfromthebeam, andconsequentlytheassociated
objects suffer from beam-induced heating. However, this type of impedances, usually associated with
the RF systems and their HOMs, need to be avoided in accelerator design by either detuning them
or including HOM absorbers. In fact, they not only cause equipment heating, but potentially lead to
importantinstabilities(e.g.theRobinsoninstability,seethenextsubsection,ortransversecoupledbunch
instabilities).
Thetotal energylossper turnassociatedwiththe globalacceleratorimpedance needstobe com-
pensatedforbytheRFsystem,sothattheaveragestablephaseshiftsbyanamount ∆Φ givenby
s
(cid:104) (cid:105)
∆E
sin ∆Φ = , (11)
s
(cid:104) (cid:105) MN eV
b m
whereN isthenumberofparticlesperbunchandV istheappliedRFvoltage.
b m
7
2.2 TheRobinsoninstability
Tostudyinstabilities,theeffectofwakefields(orimpedances)mustbeformallyintroducedintheequa-
tionofmotionofthebeamparticles. Resortingtotheconceptsintroducedatthebeginningofthissection,
wecanwritetheequationofmotionofanysingleparticleinthewitnesssliceλ(z)dz undertheeffectof
theforcefromtheRFsystemandthatassociatedwiththewake,whichcanextendtoseveralturns:
d2z ηeV (z) ηe2 ∞ ∞
+ RF = λ(z(cid:48)+kC,t)W (z z(cid:48) kC)dz(cid:48). (12)
dt2 m γC m γC || − −
0 0 z
(cid:90) k=0
(cid:88)
Equation(12)isverygeneralandcanbeusedinmacroparticletrackingprograms,whichsolveit
for each macroparticle, determining self consistently the full beam evolution λ(z,t). It is to be noted
that both the integral and the summation in the above equation can be formally extended from , as
−∞
thewakefunctionvanishesforpositivevaluesofz duetocausality.
Inthefollowing,toillustratethemostbasicmechanismoflongitudinalinstability,i.e. theRobin-
soninstability,wewillmakeuseofthesesimplifications:
– The bunch is assumed to be point-like (carrying the full bunch charge N e) and feels an external
b
linear focusing force (i.e. in absence of the wake forces, it would execute linear synchrotron
oscillationswithsynchrotronfrequencyω ).
s
– Thebunchadditionallyfeelstheeffectofthemultiturnwakefromanimpedancesourcedistributed
over the ring circumference C (the analysis would not change if the impedance source had been
lumped at one ring location, and in reality both the external voltage and the impedance source
shouldbelocalized,makingEq.(12)defactotimediscrete).
Inthiscase,theequationofmotion(12)reducesto
d2z N ηe2 ∞
+ω2z = b W [z(t) z(t kT ) kC]. (13)
dt2 s m γC || − − 0 −
0
k=0
(cid:88)
First of all, we assume that the wake function can be linearized on the scale of the synchrotron
oscillation(i.e. thewakefunctiondoesnotexhibitabruptchangesoverahalf-bucketlength):
W [z(t) z(t kT ) kC] W (kC)+W(cid:48)(kC) [z(t) z(t kT ) kC]. (14)
|| − − 0 − ≈ || || · − − 0 −
WecanusetheaboveexpansioninEq.(13). Theterm W (kC)onlycontributestoaconstant
k ||
terminthesolutionoftheequationofmotion,shiftingthecentreofthesynchrotronoscillationfromthe
(cid:80)
bucketcentretoacertainz . Thistermrepresentsthestablephaseshiftthatcompensatesfortheenergy
0
loss introduced by the wake and will be neglected in the following. The dynamic term proportional to
z(t) z(t kT ) kT dz/dt,instead,isafriction-liketermintheequationoftheharmonicoscillator
0 0
− − ≈
and,undercertainconditions,canleadtoaninstability. Goingtothefrequencydomainthenyields
iN ηe2 ∞
ω2 ω2 = b pω Z (pω ) (pω +ω)Z (pω +ω) . (15)
− s −m γC2 0 || 0 − 0 || 0
0
p=−∞
(cid:88) (cid:2) (cid:3)
At this point, assuming that the wake only introduces a small deviation from the nominal syn-
chrotron frequency, we can easily write the complex frequency shift, which results in a real part (syn-
chrotronfrequencyshift)andanimaginarypart(growth/dampingrate):
e2 N η ∞
b
∆ω = Re(ω ω ) = pω ImZ (pω ) (pω +ω )ImZ (pω +ω ) ,
s − s m c2 2γT2ω 0 || 0 − 0 s || 0 s
(cid:18) 0 (cid:19) 0 s p=−∞
(cid:88) (cid:2) (cid:3)
τ−1 = Im(ω ω ) = e2 Nbη ∞ (pω +ω )ReZ (pω +ω ).
− s m c2 2γT2ω 0 s || 0 s
(cid:18) 0 (cid:19) 0 s p=(cid:88)−∞
(16)
8
The possibility of having an instability is related to a positive value of τ in the second of the
equations in (16). This is determined by the sign of η and that of the weighted summation on ReZ ,
||
whicharetheonlytwotermsthatcanadmitbothsigns.
A relevant situation that can be studied in further detail is when the impedance has a spectrum
peaked at a frequency ω close to the RF frequency hω , or to a multiple of it (i.e. associated with the
r 0
cavityfundamentalmodeorwithaHOM).Inthiscase,outoftheinfinitesummationonlytwotermswill
dominatetheright-handsideoftheequationforthegrowth/dampingrate:
e2 N ηhω
τ−1 = Im(ω ω ) b 0 ReZ (hω +ω ) ReZ (hω ω ) . (17)
− s ≈ m c2 2γT2ω || 0 s − || 0− s
(cid:18) 0 (cid:19) 0 s
(cid:2) (cid:3)
Stability requires that η and the variation of ReZ (ω) around nω have different signs. Figure 7
|| 0
showsthat,assumingω tobesmallwithrespecttothewidthoftheresonancepeak,thiscanbeachieved
s
differentlyaccordingtowhetherω isbeloworabovenω . Inparticular,whenhω < ω (Fig.7(a)),the
r 0 0 r
term ReZ (hω +ω ) ReZ (hω ω ) ispositiveandthereforeηneedstobenegativeforstability
|| 0 s || 0 s
− −
(i.e. themachineshouldbeoperatingbelowtransition). Otherwise,forhω > ω (Fig.7(b)),stabilityis
0 r
(cid:2) (cid:3)
guaranteedonlyabovetransition.
100 100
Re
Z||(ω)
hω0
<
ωr
Re
Z||(ω)
hω0
>
ωr
80 80
60 60
40 40
20 20
0 0
3 4 5 6 7 8 9 10 3 4 5 6 7 8 9 10
hω
ω
ω
hω
0 r r 0
hω
-‐
ω
hω
+
ω
hω
-‐
ω
hω
+
ω
0 s 0 s 0 s 0 s
(a) (b)
Fig.7: SketchofthetwopossiblesituationsfortheRobinsoninstability
OthertypesofimpedancescanalsocauseinstabilitiesthroughtheRobinsonmechanism,following
the general equations (16). However, a smooth broadband impedance with no narrow structures on the
ω scalecannotgiverisetoaninstability,because
0
∞ 1 ∞
(pω +ω )ReZ (pω +ω ) ωReZ (ω)dω 0. (18)
0 s || 0 s ||
→ ω →
p=−∞ 0 (cid:90)−∞
(cid:88)
Physically, this could be expected, because the absence of structure on ω scale in the spectrum
0
impliesthatthewakehasfullydecayedoveroneturnand,therefore,thedrivingtermintheequationof
motion(13)alsovanishes.
Tosummarize,theRobinsoninstabilityaffectsasinglebunchundertheactionofamultiturnwake
field. Itischaracterizedbyatermofcoherentsynchrotrontuneshift(thefirstoftheequations(16))and
9
an unstable rigid bunch dipole oscillation (growth rate given by the second of the equations (16) under
the conditions explained above). It does not involve higher order moments of the bunch longitudinal
phasespacedistribution. Otherimportantcollectiveeffectscanaffectabunchinabeam,forinstance:
– Potential well distortion, resulting in synchronous phase shift, bunch lengthening or shortening,
synchrotrontuneshift/spread.
– Coupledbunchinstabilities.
– Higherordermodeandmode-couplingsingle-bunchinstabilities(e.g. microwaveinstability).
– Coastingbeaminstabilities(e.g. negative-massinstability).
To be able to study these effects, more refined modes of the beam are needed (e.g. the kinetic
model described by the Vlasov equation or macroparticle simulations), but this is beyond the scope of
thisintroductoryarticle.
3 Thetransverseplane
We can start from the same system we have used in the previous section to introduce the concept of
longitudinalwakefunction. Weconsidertwoultra-relativisticchargedparticles,q andq ,goingthrough
1 2
anacceleratorstructure,withthetrailing(witness)particleatadistancez fromtheleading(source)one.
In an axisymmetric structure (or simply with a top–bottom and left–right symmetry) a source particle
travelling on axis cannot induce net transverse forces on a witness particle also following on axis. A
symmetry breaking has to be introduced to drive transverse effects, and at the first order there are two
options,i.e. offsetthesourceorthewitness(seeFig.8). Thetransverse(horizontalorvertical)dipolar
wake function associated with a certain accelerator object (able to create a wake field) is defined as the
integrated transverse force from an offset source (q [E(cid:126)(s,z)+(cid:126)v B(cid:126)(s,z)] ) acting on the witness
2 x,y
· ×
particlealongtheeffectivelengthoftheobject,normalizedbythesourceandwitnesschargesandbythe
offsetofthesourcecharge,∆x or∆y (seeFig.8,top):
1 1
W (z) = 0L E(cid:126)(s,z)+(cid:126)v×B(cid:126)(s,z) x ds = E0 ∆x(cid:48)2,
Dx −(cid:82) (cid:104) q1∆x1 (cid:105) − q1q2 ∆x1
(cid:18) (cid:19)
(19)
L E(cid:126)(s,z)+(cid:126)v B(cid:126)(s,z) ds
W (z) = 0 × y = E0 ∆y2(cid:48) .
Dy (cid:104) (cid:105)
−(cid:82) q ∆y − q q ∆y
1 1 1 2 1
(cid:18) (cid:19)
Thetransverse(horizontalorvertical)quadrupolarwakefunctionassociatedwithacertainaccel-
erator object (able to create a wake field) is defined as the integrated transverse force from an on-axis
source(q [E(cid:126)(s,z)+(cid:126)v B(cid:126)(s,z)] )actingonanoffsetwitnessparticlealongtheeffectivelengthof
2 x,y
· ×
theobject,normalizedbythesourceandwitnesschargesandbytheoffsetofthewitnesscharge,∆x or
2
∆y (seeFig.8,bottom):
2
W (z) = 0L E(cid:126)(s,z)+(cid:126)v×B(cid:126)(s,z) x ds = E0 ∆x(cid:48)2,
Qx −(cid:82) (cid:104) q1∆x2 (cid:105) − q1q2 ∆x2
(cid:18) (cid:19)
(20)
L E(cid:126)(s,z)+(cid:126)v B(cid:126)(s,z) ds
W (z) = 0 × y = E0 ∆y2(cid:48) .
Qy (cid:104) (cid:105)
−(cid:82) q ∆y − q q ∆y
1 2 1 2 2
(cid:18) (cid:19)
For most objects of interest, it can be seen that the wake functions so defined do not depend on
thesourceorwitnessoffsets,providedtheoffsetsaremuchsmallerthanthetransversesizeoftheobject.
For larger offsets, coupling and/or higher order non-linear terms can become important and may need
to be taken into account to describe correctly the particle dynamics. Both the dipolar and quadrupolar
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