63
PHYSICS AND OPERATION OF THE DAΦNE COLLIDER THIRD SEMINAR " INSTABILITIES" M. Zobov, A. Gallo Frascati, 18/2/2000
PHYSICS AND OPERATIONOF THE DAΦNE COLLIDER
THIRD SEMINAR
" INSTABILITIES"
M. Zobov, A. Gallo
Frascati, 18/2/2000
OUTLINES
1. Wake Fields and Impedances
2. Longitudinal and Transverse Spectrum
- Single particle spectrum- Single particle spectrum with synchrotron
satellites- Many bunches spectrum- Stationary distribution- Perturbation
3. Instabilities due to beam interaction with vacuumchamber
- Head-tail instability- Resistive wall instability- Narrow band resonator- Robinson instability- Beam loading in RF system- Turbulent mode coupling- Bunch lengthening
4. Cures
- Longitudinal feedback system- Landau damping
5. Ion trapping
6. Bibliography.
The Lorenz force experienced by a trailing charge q at a givenposition r,z due to electromagnetic fields E and B produced by aleading charge q1 is:
r r r r r r r r r rF r z r z t q E r z r z t v B r z r z t, , , ; , , , ; , , , ;1 1 1 1 1 1( ) = ( ) + × ( )[ ]
The longitudinal wake function is defined as the energylost by the trailing charge q per unit of both charges q and q1:
w r rU
E r z r z t dz
qqt
z
vV Cz
z
, ,
, , , ;
; /τ τ( ) = = −( )∫
= + [ ]−∞
+∞
∆
1
1 1
1
1
where qEz is the longitudinal component of the Lorentz force.
The coupling impedance or beam impedance is defined asthe Fourier transform of the wake function:
Z w e dzjω τ τωτ( ) = ( )∫ −
Analogously, we introduce the transverse wake functionas a kick experienced by the trailing charge per unit of bothcharges:
wp
q E v B dz
qqt
z
v⊥⊥
⊥−∞
+∞
( ) = =+ ×( )∫
= +τ τ∆
1 1
1;
The transverse coupling (beam) impedance is found bythe Fourier transform of the wake function:
Z j w e dj⊥ ⊥
−∞
+∞−( ) = ( )∫ [ ]ω τ τωτ Ω
Since the transverse dynamics is dominated by the dipoletransverse wakes, we can define the transverse dipole impedanceper unit transverse displacement:
′ = ( ) [ ]⊥⊥Z
Z
rm
ω1
Ω /
LONGITUDINAL SPECTRUM
A single particle of charge e rotating with speed v in thecentral orbit of an accelerator of average radius of curvature of Rcan be described by a time-dependent linear charge density:
λ δte
vt kT
k( ) = −( )∑
=−∞
∞0
where T0 is the revolution time 2πR/v and δ(t) is the impulsivefunction. By expressing (1) as a Fourier series, we can write:
λ ωπ
ωte
vTjn t
e
Rn t
n n( ) = ∑ = ( )∑
=−∞
∞
=−∞
∞
00 02
exp cos
The frequency spectrum is obtained by the Fouriertransform:
Λ ω ωπ
δ ω ω( ) = −( )∑=−∞
∞e
vk
k
002
The line at n = 0 is the DC component of the signal, theremaining lines are successive orbital harmonics spaced by ω0.
Since cos(-ω0t) = cos(ω0t), the negative frequency lines areindistinguishable from those at corresponding positive frequency;the combined amplitude seen by a spectrum analyzer is thus twicethe DC component.
TRANSVERSE SPECTRUM
A BPM signal is actually proportional to the linear dipoledensity d, defined as the product of the linear charge densityλ times the transverse offset z:
d t t z t t z z Q t( ) = ( ) ( ) = ( ) + ( )( )λ λ ωβ0 0cos
where z0 is a stable offset due to a closed orbit distortion or to aBPM misalignment; zb is the oscillatory term due to the betatronoscillation; Q is the transverse betatron tune. Then:
d t ze
Rn T z
e
Rn t Q t
n n( ) = ( )∑ + ( )∑ ( )
=−∞
∞
=−∞
∞0 0 0 02 2π
ωπ
ω ωβcos cos cos
The first terms gives terms similar to the longitudinal caseweighted by the closed orbit. By considering only the secondterm, the linear dipole density can be written as:
d t ze
Rn Q t or
d t ze
Rq t n q t
n
n
( ) = +( )∑
( ) = ( ) + ±( )∑
=−∞
∞
=−∞
∞
β
β
πω
πω ω
2
2
0
0 0
cos ( )
cos cos ( ' )'
where Q=K+q with K the integer part of the tune and q thefractional part and the new index n’=n+K.
Longitudinal Spectrum with Synchrotron Satellites
In the presence of the longitudinal focusing produced by an RFaccelerating field, a particle beam is bunched and the singleparticle undergoes synchrotron oscillations. The time betweensuccessive passages measured at the monitor is:
T TT
tss0 0
01+ = + +( )
τ τ ψcos Ω
where Ω s and τs is the angular frequency and the amplitude ofsynchrotron oscillations, respectively. In this case the lineardensity is:
λ δ τte
vt kT
k( ) = − −( )∑
=−∞
∞0
Using the relation:
exp cos( ) ( )exp( )jx y j J x jmymm
m[ ] = ∑
=−∞
∞
we can express the linear charge density as a Fourier series:
λπ
ω τ ω ψte
Rj J n j n m t mm
n mm s s( ) = −∑ ( ) +( ) +[ ]
=−∞
∞
2 0 0( ) exp,
Ω
Each original line in the spectrum has now degenerated intoinfinite set of satellites right and left at ±Ω s, ±2Ω s,..,mΩ swith the amplitudes modulated by the Bessel function of the firstkind of order m.
Transverse Spectrum with Synchrotron Satellites
Now we must take into account the modulation of the betatrontune due to the energy modulation. In fact the machine latticefocuses differently particles with energy deviating from thenominal value. Then the betatron term can be written as:
z t z t with
Q Qd dQ
Q
( ) = ( )( )
= ≈ + +
ˆ cos
˙
ϕ
ϕ ω ω ωω0 0
0 01
Applying the definitions of the chromaticity and momentumcompaction:
d dp
p
d
dt
dQ
Q
p
dp
ωω
η τ ξ0 0 0
0= − = − =;
we get the expression for the betatron frequency:
˙ ˙ . .
cos cos
ϕ ω τ ξη
ω ω ω τ ψξ
= − −
( ) = − −( ) +( )( )Q i e
z t z Q t Q ts s
0 0
0 0 0 0
1 1
Ω
Finally, after some mathematics, we obtain the dipole density:
λπ
ω ω τ ω ψξtez
Rj J n Q j t mm
n mm s n m( ) = −∑ + −[ ]( ) ( ) +[ ]
=−∞
∞ˆ( ) ( ) exp
,,2 0
with the mode frequency ωn,m= (n+Q)ω0+mΩ s. Here weintroduced the chromatic frequency ωξ = (ξQ0/η)ω0.
Longitudinal Spectrum of Multibunch Beam
A beam of M similar and equally-spaced bunches can oscillatecoherently in M different modes depending on the phase relationbetween the individual oscillations. Consider the over-simplifiedsituation of M equal bunches consisting of one particle on the samephase space orbit τs. Then, the linear density is:
λ δ τte
vt
b
Mk T b
kb
M( ) = − +
−
∑∑
=−∞
∞
=0
1
whereτ τ ψb s s bt= +( )cos Ω
This relation can expressed again in terms of Bessel functions:
λπ
ω τ
ω ψ ψ π
te
Rj J n
j n m t m j mbn
M
m
n mm s
s bb
M
( ) = −∑ ( ) ×
× +( ) +[ ] −
∑
=−∞
∞
=
2
2
0
01
( )
exp exp
,
Ω
If the phase shifts between the two adjacent bunches satisfy:
mp
Mulob bψ ψ π π+ −( ) =1
22, mod
then the Σ is equal to M provided that n = kM + p and is null ifn = kM + p. Here p can be 0,1,2,.., M-1, defining the p- thmode of coherent coupled bunch motion. The spectrum of the p-thmode is at frequencies:
ω ωkmp skM p m= +( ) +0 Ω
STATIONARY DISTRIBUTION
Let us introduce a distribution function which represents theparticle density in the bi-dimensional phase space:
Ψ Ωψ τ τ τ ψ0 0, ˆ, ; ˆ cost ts( ) = +( )The signal of a single bunch is:
S t N t t d d( ) = ( )∫∫ ( )=
=
=
=+∞Ψ ψ τ λ τ τ ψ
ψ
ψ π
τ
τ0
0
2
00
0
0, ˆ, ˆ ˆ
ˆ
ˆ
N is the total number of particles in this bunch. A distributionis called stationary when the density does not change with timearound any point of the phase space. Any function dependingonly on the amplitude τ satisfies the required condition:
Ψ ψ τ τ0 , ˆ, ˆt go( ) = ( )
Then, in the frequency domain we can write the signalspectrum:
S I p J p g dp
pω π δ ω ω ω τ τ τ τ
τ
τ( ) = −( )∑ ∫ ( ) ( )
=−∞
=+∞
=
=+∞2 0 0
00 0
ˆˆ ˆ ˆ ˆ
where I is the intensity in one bunch.
There is no evidence of the internal synchrotronmotion. All of the rich spectrum of the single particlewith harmonics of the synchrotron frequency disappeared.Only multiples of the particle revolution frequency exist.
PERTURBATION
In order to drive an instability, one needs to produce fields atsynchrotron frequency. This will be the role of the perturbation∆Ψ. The actual distribution in the bunch is written:
Ψ ∆Ψψ τ τ ψ τ0 0 0, ˆ, ˆ , ˆ,t g t( ) = ( ) + ( )In order to obtain some perturbation signal at synchrotron
frequency a ψ0 dependence is necessary in ∆Ψ. The form to begiven to the perturbation is suggested by the expression of thesingle particle signal:
∆Ψ ∆ψ τ τ ψ ω0 0, ˆ, ˆ expt g j m tm cm( ) = ( ) − +( ) The perturbation signal in the time domain is defined as:
∆ ∆ΨS t N t t d d( ) = ( )∫∫ ( )=
=
=
=+∞ψ τ λ τ τ ψ
ψ
ψ π
τ
τ0
0
2
00
0
0, ˆ, ˆ ˆ
ˆ
ˆ
So by performing the Fourier transform, we get theperturbation spectrum in the frequency domain:
∆ Ω ∆S I p m
j J p g d
s cmp
p
mm m
ω π δ ω ω ω
ω τ τ τ ττ
τ
( ) = − + +( )( )∑ ×
× ∫ ( ) ( )
=−∞
=+∞
−
=
=+∞
2 0
00
ˆˆ ˆ ˆ ˆ
The perturbation spectrum shows an infinite number offrequency lines at mωs+∆ωcm , distant from the revolutionharmonics. Only a single value of m is present in this caseidentifying the number of azimuthal variations in theperturbation.
PERTURBATION
Since the perturbation is proportional to:
∆Ψ ∆∝ e j tcmω
we have to calculate the coherent frequency shift ∆ωcm:
- The motion is unstable and the perturbation will grow if:
Im ∆ωcm < 0
- The beam is stable and the perturbation will be damped if:
Im ∆ωcm > 0
- The real part of ∆ωcm gives the coherent frequency shift withrespect to mΩs
Usually, ∆ωcm is found by applying the perturbation methodlinearizing Vlasov’s equation:
d
dt t t t
Ψ Ψ Ψ Ψ= + + =∂∂
∂∂τ
∂τ∂
∂∂ψ
∂ψ∂ˆ
ˆ
0
0 0
with respect to the small perturbation ∆Ψ and taking into accountthe equation of motion of a particle:
˙ exp//
//τ τ ηπ
ω ω ω ωω
ω+ = = ( )∫ ( )
=−∞
=+∞Ω ∆s cF
e
RpZ S j t d2
02
where Z// is the machine coupling impedance.
COHERENT FREQUENCY SHIFT
Longitudinal Case:
∆Ω
Ωωφ
ωω
ω
ωω ωcm
s
RF s
p
pm p
p
p
m pp
p p sjm
m
I
B hV
Zh
hpM k m//
//
cos;= −
+ ( )
( ) ( )∑
( )∑= +( ) +=−∞
=∞
=−∞
=∞1 3 3 0
Transverse Case:
∆ Ωω βγ ω
ω ω ω
ω ωω ω
ξ
ξ
cm
k m kk
k
m kk
k k sjm
e
m
I
Q L
Z h
hkM n Q m⊥
⊥=−∞
=∞
=−∞
=∞=+
( ) −( )∑
−( )∑= + +( ) +1
1 20 00;
where the spectral power density for the m-th mode for a Gaussian bunch is given by:
hc cm
mω ωσ ωσ( ) =
−
2 2exp
INSTABILITIES DUE TO INTERACTION WITH VACUUM CHAMBER
SINGLE BUNCH MULTIBUNCH
- Longitudinal Case: - Longitudinal Case:
a) Microwave instability a) Interaction with parasitic HOM;and bunch lengthening; b) Robinson instability
b) Robinson instability
- Transverse Case: - Transverse Case:
a) Head-tail instability; a) Interaction with parasitic HOM; b) Turbulent mode coupling b) Resistive wall instability
instability;c) Resistive wall instability
HEAD-TAIL INSTABILITY
Very often the transverse coupling impedance of anaccelerator can be modeled by the broad-band resonatorimpedance:
ZR
jQ
r
r
r⊥ ( ) =
+ −
ω ωω ω
ωωω
1
with the quality factor Q equal to 1.
-6 -4 -2 0 2 4 6-1.2
-0.8
-0.4
0
0.4
0.8
1.2
Re Z
ωωξ
0-th mode spectrum
ωr
Without sextupoles for the chromaticity correction the modespectrum is shifted towards negative frequencies and the resultingimaginary part of the coherent frequency shift is negative leadingto the instability.
RESISTIVE WALL INSTABILITY
The impedance describing the interaction of a circulating beamwith the surrounding vacuum chamber having finite conductivityσ is given by:
Z jRZ
b⊥ ( ) = +( )ω
µ σω1
203
0
where R is the mean machine radius; b is the vacuum chambercross-section radius; Z0 = 120π Ohm.
-10
-5
0
5
10
-4 -2 0 2 4
stable
unstable ωξ
0-th mode spectrum
Re Z
ω
The first negative spectrum line is closer to the origin than thefirst positive one, thus coupling to the higher negativeimpedance. The situation leads to the instability. In the singlebunch case the instability can be removed by choosing thebetatron tune slightly above integers. The instability can not beeliminated in the multibunch case.
NARROW BAND RESONATOR
ZR
jQ
r
r
r⊥ ( ) =
+ −
ω ωω ω
ωωω
1
with the quality factor Q >> 1.
-1
-0.5
0
0.5
1
-10 -5 0 5 10
ωξ ωr ω
0-th mode spectrum
Re Z
The bandwidth of the resonator is such that a single betatronline lies inside the resonant curve. The 0-th mode spectrum iscentered in the positive frequency region around ωξ. Mainly, theline at the negative frequency couples to the narrow bandresonator impedance, thus driving the instability.
ROBINSON INSTABILITY
The RF cavity is a typical example of a narrow band resonator.It is necessarily tuned to a resonant frequency close to hω0.
The figure below shows a spectrum analyzer display with anRF cavity tuned to a resonant frequency ωr slightly above hω0.
The upper and lower sidebands of hω0 for mode m = 1 aredrawn. The resistance associated with the lower sideband issmaller than that associated with the upper one. This correspondsto an unstable configuration above transition.
In order to reach stability the cavity has to be tuned below hω0,i.e. the resonant frequency has to be shifted towards lowerfrequencies.
ωrhω0
h +ω0 Ω s-h +ω0 Ωs
upper sidebandlower sideband
spectrum analyser
ω
Re Z / ω//
ROBINSON INSTABILITY
The RF cavity is a typical example of a narrow band resonator.It is necessarily tuned to a resonant frequency close to hω0.
The figure below shows a spectrum analyzer display with anRF cavity tuned to a resonant frequency ωr slightly above hω0.
The upper and lower sidebands of hω0 for mode m = 1 aredrawn. The resistance associated with the lower sideband issmaller than that associated with the upper one. This correspondsto an unstable configuration above transition.
In order to reach stability the cavity has to be tuned below hω0,i.e. the resonant frequency has to be shifted towards lowerfrequencies.
ωrhω0
h +ω0 Ω s-h +ω0 Ωs
upper sidebandlower sideband
spectrum analyser
ω
Re Z / ω//
BEAM LOADING IN THE RF SYSTEM
The task of the RF system in a storage ring is to restore theenergy lost each turn by the particles (mainly due to thesynchrotron radia-tion emission), and to provide longitudinalfocusing to the bunches.
This is done by exciting accelerating resonant modes in RFcavities. An "accelerating mode" is a resonant field distributionshowing a strong longitudinal E-field along the beam trajectory.
Anyway, looking at the beam dynamics, the cavity acceleratingmodes are sharp and strong longitudinal impedances kept (onpurpose) well tuned with respect to a certain revolutionharmonics of the beam.
In short machines (like DAΦNE) the cavity accelerating modeimpedance interacts only with the synchrotron sidebands of the"rigid" (or "barycentric") motion of the bunches. In longmachines (such as the B-factories), being the revolutionfrequency much lower, the bandwidth of the accelerating moderesonances may cover several revolution harmonics, affectingalso the motion of coupled bunch modes different from the rigidone.
The beam dynamics implications of the cavity acceleratingmodes can be studied again through the concept of thelongitudinal coupling impedance. The only complication is that,in this case, the presence of an active system (the RF system),working around the cavity accelerating mode, has to be taken intoaccount.
The effects on the RF system of the voltage induced by thebeam in the cavity accelerating mode impedance are known as"beam loading".
The instability of the beam barycentric motion driven by thecavity accelerating mode impedance is known as "Robinsoninstability".
The analysis of the response of the RF system to the beamcoherent synchrotron oscillation has to include also the effects ofthe active RF control electronics which is normally implementedto automati-cally control the accelerating voltage and the cavitytuning.
CAVITY DETUNING UNDER BEAM LOADING ANDSTABILITY IMPLICATIONS
The general accepted model to describe the interacting betweenthe beam and the RF system is the following:
CAVITYRF source Beam
Ig Rs CL
Rs/β Ib = Σ qn δ(t-nTb)Vc
The bunches are current pulses crossing the accelerating voltageVc at the synchronous phase φs = Acos(Vr/Vc). The acceleratingvolta-ge negative slope gives the longitudinal focusing of thebeam.
-0 .8
-0.4
0
0.4
0.8
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
t /TRFF
Vccs
IbIb
Vrr
- RF harmonicsIb
φ
Since the beam current RF harmonics follows the acceleratingvoltage with a delay equal to φs , the beam is equivalent to anextra admittance loading the RF system:
1
Zbeam =
2Ib e-jφs Vc
= 2Ib cos φs
Vc -j
2Ib sin φs Vc
The beam loads the RF system as an extra resistance in parallelto an extra inductance which are both inversely proportional tothe total current:
CAVITYRF source Beam
Ig Rs CL
Rs/β Vc
Vc2Ib
cos φs
RbLb
Rb=
Vc2Ib
sin φs
Lb=
The resistive part of the beam equivalent impedance describesthe energy transfer from the RF source to the beam and it gives acondition to optimize the coupling coefficient β for the bestmatching:
β = 1 + 2 Rs Ib cos φs/Vc
The inductive part of the beam equivalent impedance tends toshift up the resonant frequency of the cavity + beam system, andbeyond some current threshold the system is so largely detunedthat the RF source can not sustain the required accelerating fieldanymore.
To avoid that, a tuning system automatically changes theresonant frequency of the cavity, shifting it toward lower valuesto compensate the beam inductance.
Since the complex admittance of the loaded cavity is given by:
1
ZcavL =
1 RsL
(1 + j QL δ)
the compensation of the beam reactance, if the cavity starts froma perfect tune at zero current, is simply given by:
QL δ = 2 Ib RsL sin φs / Vc = Y sin φs
If the cavity starts from a non-zero detune at zero current, theexpression above has to be corrected as follows:
QL δ = Y sin φs + QL δ0 (1 + Y cosφs)
The cavity detuning increases linearly with Y, which isproportional to Ib and inversely proportional to Vc .
The following plot shows the expected detuning of the DAΦNEcavities for a typical accelerating voltages of 150 kV.
0
0.2
0.4
0.6
0.8
1
- 2 0 - 1 5 - 1 0 - 5 0 5
∆f [in units of cavity bandwidths]
|Zc| @ Ib=0 |Zc| @ Ib=0.5 A|Zc| @ Ib=1 A
|Zc| @ Ib=1.5 A
fRF
Vc = 150 kVtuning angle = 0
The beam loading can detune the cavity by several bandwidths.The synchrotron sidebands interact with both the real and theimaginary part of the cavity impedance strongly perturbing thevalues of both damping factor and coherent frequency of thebarycentric motion.
The global set-up of the RF system should be such that thedamping factor never gets negative (Robinson first limit) and thecoherent synchrotron frequency never gets to 0 (Robinson secondlimit) over the full operating current range.
To do that sometimes it is necessary to implement specialfeedback systems (feedforward, direct RF feedback, phasemodulation "woofer" systems, ...) to reach high current values.In this case the mutual interactions among these feedbacksystems, the synchrotron coherent motion and the ordinary RFservo loops (amplitude, phase and tuning automatic controls)have to be analyzed to prevent sorts of global instabilities.
DAΦNE 0-mode coherent frequency shift
0.4
0.6
0.8
1
10 100 1000
fc/fi@150kVfc/fi@200kVfc/fi@250kV
Ib [mA]
tuning angle ≈ 22 deg
0
0.2
0.4
0.6
0.8
1
1.2
1 10 100 1000
fc/fi @ PhiRob
≈ 0°
fc/fi @ PhiRob
≈-22°
fc/fi @ PhiRob
≈-45°
Ib [mA]
VRF
= 120 kV
TURBULENT MODE COUPLING
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
-10 -5 0 5 10
Im Z
m=0 mode spectrumm=-1 mode spectrum
ωω = 0ξ
0
-1
ω /Ωcm s
m = 0
m = -1~I
POTENTIAL WELL DISTORTION(bunch lengthening)
-1
-0.5
0
0.5
1
0 0.5 1 1.5 2 2.5 3
VRF
φVind
V + VRF ind
The stationary bunch shape is defined by self-consistentHaissinski’s equation:
λ τ τσ
πφ ω σ
τ λ ττ
( ) = − −( )
−( ) ( )∫
−∞K
I
hVdtS tb
RF s
expcos
2
02
02
022
2
with
S dt d Z j tτπ
ω ω ωτ
( ) = ∫ ( )∫ −∞
+∞1
2 0exp
The normalization constant K is defined by:
dt tλ( )∫ =−∞
+∞1
The line density λ(t) is Gaussian for vanishing bunch currentand the Gaussian distribution can be substantially deformed athigh bunch current.
BUNCH LENGTHENING FOR Z(ω) = jωL
BUNCH LENGTHENING FOR Z(ω) = R
Digital sampling OscilloscopeTEKTRONIX 11801A
SD-24 Samplinghead (20 GHz)
ANDREW FSJB-50A, ≈ 8 m long Cable
Broad-bandButton
GP-IBPC (off-line analysis)
Ib(ω) =Zbut(ω) αcable(ω)
Vosc(ω)
Zbut(ω) = Zb 1 + j ω / ω0
j ω / ω0 + High Order Resonances
αcable(ω) = exp[-(ω / ωcable)1/2]
Bunch Length Measurement Schematics
Ib(t)
Vosc(t)∝ Ib(t)
Bunch Lengthening in DAΦNEat RF voltages 150 kV (a) and 200 kV (b)
Comparison between the measurement results (dots) andthe numerical simulation (line) of the bunch lengtheningdue to parasitic electromagnetic interaction of a bunch
with different elements of the DAΦNE vacuum chamber
Σ
Pickup
Bunched Beam
$ $Power
Amplifier
LongitudinalKicker Cavity
AmplitudeModulator
CombGenerator
PhaseDetector
LP Filter
MasterOscillator6 (4) * fRF
Timing/ControlA
DC
/Dow
nSam
pler
DA
C/H
old
Buf
fer
CarrierOscillator
VMEDSP Farm
2.75 (3.25)* fRF
LONGITUDINAL FEEDBACK SYSTEM- BLOCK DIAGRAM
QPSKModulator
LANDAU DAMPING
The beam dynamics study deals with a population of particlesinteracting through a variety of mechanisms, the most importantone being described by the wake and the impedance concepts.
The beam collective behavior can be analyzed by assuming thatall the particles oscillates with the same natural frequency, i.e.the spectrum of the natural frequencies is a δ-function. Thisassumption generally simplifies the analysis, but it may lead totoo pessimistic results.
In a real world the natural frequencies of a particle populationare always spread over a continuos and finite spectrum. Then onemay find that some collective effects that were expected to bringthe beam to instability under the oversimplified hypothesis ofδ-function spectrum, are in reality too weak to drive theinstability once the real spectrum has been took into account.
The stabilization due to the frequency spread of real systems isknown as "Landau Damping". A simple example illustrating howthe Landau Damping works is reported in the following.
Let's consider a population of N damped oscillators all subjectto the same driving force:
xi + 2α xi + Ωi2 xi = A e jωd t
where Ωi is the natural frequency of the i-th oscillator, α is thedamping constant (common to all the oscillators), and A and ωdare the amplitude and the frequency of the driving term. Theregime solution of the differential eq. above gives for the i-thoscillator:
xi ( t ) = A
Ωi2 − ωd
2 + 2α jωd
e jωd t
The coherent signal Σc ( t ), observable macroscopically with adedicated monitor, is the sum of the oscillations over the entirepopulation:
Σc ( t ) = xi ( t ) =
i=1
N
∑ Ae jωd t 1Ωi
2 − ωd2 + 2α jωdi=1
N
∑
Let's consider first the case of all oscillators having the samenatural frequency Ω0 . In this case we obtain:
Σc ( t ) = NAe jωd t
Ω02 − ωd
2 + 2α jωdωd →Ω0 ,α →0
→ ∞
which gives us the non surprising result that driving a set oflossless oscillators at their natural frequency, the coherent signalgrows without limit.
Let's consider now what happens if the oscillator frequenciesare distributed according to a normalized function G(Ω) (like agaussian or a lorentzian, or any other sort of reasonabledistribution function) centered around a certain frequency Ω0 . Inthis case we have:
Σc ( t )NAe jωd t = G(Ω)dΩ
Ω2 − ωd2 + 2α jωd
−∞
+∞
∫ ≈ 1
2Ω0
G(Ω)dΩΩ − ωd + jα
−∞
+∞
∫ = 1
2Ω0
⋅
⋅G(Ω)(Ω − ωd )dΩ(Ω − ωd )2 + α 2
− jα G(Ω)dΩ(Ω − ωd )2 + α 2
−∞
+∞
∫−∞
+∞
∫
α →0
→PV(ωd ) − jπ G(ωd )
2Ω0
where:
PV(ωd ) = P.V.G(Ω)dΩ(Ω − ωd )
−∞
+∞
∫ = limε→0
G(ωd + Ω) − G(ωd − Ω)
Ωε
+∞
∫ dΩ
is the Cauchy's "principle value" that has to be considered tointegrate around the singularity in ωd . It is easy to demonstratethat PV(ωd ) exists and is finite for any ωd value as long as G(Ω) is a regular and smooth function.
In the end we got a less obviuos result that a set of losslessoscillators, whose frequency spectrum is a continuous functioncentered at some Ω0 , all subject to the same driving harmonicforce, produces a finite coherent signal for any location of theexciting frequency in the spectrum.
Moreover, the coherent motion has an imaginary part, whichmeans that the "velocity" of the coherent motion has a componentwhich stays in phase with the driving force draining energy fromthe source like in a frictional system (even in the limit α → 0 oflossless oscillators!!!).
This "frictional" term is proportional to the spectrum density atωd and its nature can be hardly discussed in a short, introductorypresentation. The most convincing physical picture is that thedrained energy is not dissipated but stored in a band ofoscillators, close to ωd , which is continuously narrowing withtime.
These basic relations give a glance of the nature of the "Landaudamping" concept. To apply this concept to the beam dynamicsone has to add the fact that the driving term is, in general, notexternal but self-generated by the interaction among theparticles , and consistent equations of motion have to be derived.
Where does the synchrotron and betatron tune spread comefrom? Due to the radiation quantum emission, the bunch in astorage ring has its own equlibrium distribution in thelongitudinal and transverse phase spaces (whose areas representthe energy spread and the emittance respectively). Since thefocusing forces are not perfectly linear (RF voltage non linearity,lattice multipolar terms), the particles that (temporarily) occupy aposition in the tail of the distributions have oscillation frequencieswhich deviate from the unperturbed values. The energy spread,through the chromaticity factor ξ , gives also a direct additionalcontribution to the betatron tune spread.
Synchrotron and betatron tune spread values of ∆νs νs ≈ 1 ⋅10 −3
and ∆νβ νβ ≈ 5 ⋅10 −3 are typical numbers in DAΦNE.
Effetto dell'interazionefascio-fascio
in operazione di multibunch
Smorzamentodiun'instabilita'trasversaverticale pereffettodell'interazionefascio-fascio
ION TRAPPING
After passage of one period (electron bunch + gap betweensuccessive bunches) the new position y2 and speed y2 of an ionare obtained applying the matrix M:
y
yM
y
y
T y
y
yy
y
y y y
b2
2
1
1
1
1
11
0 1˙ ˙
cos sin
sin cos˙
=
=( ) ( )
− ( ) ( )
−
ω τω
ω τ
ω ω τ ω τ
τ
Here Tb is the time bunch separation; τ is the bunch length andω is the angular ion frequency inside the bunch:
ωτσ σ σy
p
y x y
r Nc
A2 2
=+( )
where N is the number of particles per bunch; rp the classicalproton radius; A the molecular number of the ion; σx,y thetransverse bunch sizes.
The ion motion is stable if |Sp M|/2 < 1. Usually, ωyτ << 1and τ << Tb so that the stability condition can be written as:
ω τ ω τ ω τy b y y bT T−( ) ≅ <2 4
The ions with the atomic number higher than Ac are trapped inthe electron beam potential:
Ar NC
nwhere T
C
n ccp
b x y x yb
b=
+( ) =2 σ σ σ,
Bad Consequences of Ion Trapping:
a) Pressure increase and bad lifetime:
b) Emittance growth;
c) Tune shift and tune spread;
d) Beam losses trough excitation of resonances;
e) Two stream coherent instability;
f) Fast ion instability.
Possible Cures:
a) DC clearing electrodes;
b) Beam shaking at the ion frequency;
c) Clearing gaps in the bunch train.
BIBLIOGRAPHY
1. L. Palumbo, V. G. Vaccaro and M. Zobov, “Wake fieldsand Impedance”, CAS CERN Accelerator School, FifthAdvanced Accelerator Physics Course, Rhodes, Greece, 20September-1 October 1993, CERN 95-06.
2. M. Serio and M. Zobov, “Measurement of Transverseand Longitudinal Spectra”, Invited talk at 1st EuropeanWorkshop on Beam Instrumentation and Diagnostics forParticle Accelerators, Montreux, Switzerland, 3-5 May 1993.LNF-93-042-P, Aug 1993. 22pp.
3. J. L. Laclare, “Bunched Beam Coherent Instabilities”,CAS CERN Accelerator School, Advanced AcceleratorPhysics, The Queen’s College, Oxford, England, 16-27September 1985. CERN 87-03
4. A. Chao, “Physics of Collective Beam Instabilities inHigh-Energy Accelerators”, New York, USA: Wiley(1993) 371 p.
5. M. Serio et. al., “Multibunch Instabilities and Cures”,Invited Talk at 5th European Particle Accelerator Conference(EPAC 96), Sitges, Spain, 10-14 Jun 1996. In *Sitges 1996,EPAC 96* 148-152.
6. S. Bartalucci et. al., “Analysis of Methods forControlling Multibunch Instabilities in DAΦ N E ” ,Published in Part.Accel.48: 213-237, 1995.
7. D. Boussard, “Design of a Ring RF System”, CERN-SL-91-2, Jan 1991. 33pp. Published in Julich Accelerator School1990: 294-322.
