collective effects in particle accelerators
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Transcript of collective effects in particle accelerators
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collective effectsin particle accelerators
Frank Zimmermann
Bodrum Summer School
September 2007
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what is an accelerator?
• charged particles moving in electromagnetic fields• these fields can be
- static or time dependent
- externally applied or beam generated
- linear (restoring force proportional to displacement) or nonlinear
• motion can be classical or quantum mechanical• combinations of all these occur and can be important
examples
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• they have good memory• they won’t forgive you• easily perturbed and mistakes add up
beam particles are like elephants…
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… and they are not alone!
particles do not move independently; many of the limits of accelerator performance arise from interactions between beam particles = collective effects
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various types of collective effects• beam instabilities (coherent, motion of many particles
is correlated), due to – self-interaction [space charge]– interaction with vacuum chamber [impedance]– interaction with other beam [beam-beam effects]– interaction with “foreign” particles [ions or electrons]
• poor beam lifetime or emittance growth (incoherent), due to – scattering of individual particles off each other [intrabeam
scattering, Touschek effect]– motion of individual particles in the nonlinear electromagnetic
field generated by one of the four interactions above [space charge, beam-beam, incoherent electron cloud,…]
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2 particles at rest or traveling
many charged particles traveling in an unbunched
beam with circular cross-section
K.-H. Schindl
βr
space chargespace chargecartoon - in reality transverseelectric (and magnetic) fields increase for relativstic particles
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Quadrupole(F-type)
FocusingLinear
Uniform
DefocusingLinear
Gaussian
DefocusingNon-Linear
Proton
Beam
K.-H. Schindl
magnet space charge
space-charge force is defocusing in both x & y direction, unlike a quadrupole
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BvEeF srr
22
02
2
0
1
21
2 a
r
c
eI
a
r
c
eIFr
consider as example:uniform round continuous charged beam of radius a & current I
space-charge force:
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schematic of betatron oscillation around storage ringbetatron tune = number of oscillations per turn
interlude: betatron motion
C s
dsCQ
)(2
1
2
)(
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space-charge effect on betatron motion
x
y ring effectivefocusing force
space chargedefocusing force
the global force on each particle shouldbe focusing! in storage rings this is always true.
tunes are dependent on transverse (and longitudinal) position through the global Coulomb-force effect of the beam
find a “jungle” ofcoherent and incoherent effects
G. Franchetti
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2,;
032
,
0, 22 yxNyxyx
NrNrQ
space-charge tune shift - proportional to intensity N- inversely proportional to emittance - proportional to beam brightness N/N
- it decreases like 1/2 or 1/3
(important for low only)- it does not depend on machine radius
dsssKQ yx
C
yxscyx ,
0
,;, 4
1
change of betatron tune due to defocusing force:
from spacecharge force
100% (norm.) beam emittance for our example
cm
FK yxsc
yxsc0
,;,;
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interlude: emittance
x
x’“area in phase space”occupied by the beam= x
rms emittance 222 '' xxxxrms
for Gaussian distributionrms ~ 39%, 4rms ~ 86%, 6rms ~95% of the beam
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incoherent tune shift due to conducting walls
a line charge representing the particle beam between parallel conducting plates of distance 2h; electric field parallel to the conducting plates have to be zero; this is achieved by introducing negative image line charges (right)
K.H. SchindlJ.R. Laslett
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incoherent tune shift due to conducting walls - 2
K.H. SchindlJ.R. Laslett
22
00 41
2
1
2
1
21
hn
y
ynhynhE nniny
y
hy
nhEE
n
inyiy 124
1
4
2
201
2201
0
y
E
x
EE iyixi
xh
Eix 124
2
20
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K.H. Schindl
2
2
223
0
482
1
haec
ICrQ x
x
2
2
223
0
482
1
haec
ICrQ
y
y
direct s.c. imagefeatures: - electric image field is vertically defocusing but horizontally focusing
(typical for most vacuum beam pipes) - field is larger for smaller chamber height h- image effects decrease as 1/, much weaker than 1/3 for direct space charge; they are of some concern for high-energy p machines and e rings
incoherent tune shift due to conducting walls - 3
d.c. beam current is accompanied by dc magnetic field, which is not shielded by beam pipe, but influenced by ferromagnetic boundaries, like magnets, representedby mirror currents → incoherent tune shift due to magnetic images
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incoherent and coherent tune shift K.H. Schindl
incoherent betatron motion of a particleinside a static beam with its center of massat rest
amplitude and phase are distributed at random over all particles
coherent motion of the whole beam after having received a transverse kick
the source of the direct space charge is now moving, individual particles still continue incoherent motion around the common coherent trajectory
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coherent tune shift due to conducting wallexample: round, perfectly conducting beam pipe
coherent oscillation of the beam of the beam inside a circular perfectlyconducting beam pipe and its oscilllating image charge
xb 2
xbxb
xEix 2000
1
2
1
2
1
2
defocusing force
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xe
xFix 20
1
2
22
,0
;, 2
NrQ
yx
cohyx
coherent tune shift due to conducting wall - 2
coherent force
coherent tune shift
for symmetry reasons the force is the same in x and y direction
Features:features: - force is linear in → there is a coherent tune shift- 1/ dependence stems from the fact that the field is proportional to the number of charged beam particles (independent of mass), but their deflection is inversely proportional to their relativistic mass m0- the coherent tune shift is never positive- effect of a thin vacuum chamber with finite coductivity are more subtle
xx
x
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more realistic example: elliptical vacuum chamber,still unbunched beam with uniform density
incoherent and coherent tune shifts given by “Laslett coefficients” and
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most coefficients are larger vertically; coherent coefficients all positive or 0
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most rings storebunched beams;
the s.c. tune shiftfor bunched beams changes withlongitudinal coordinate z
→ tune spread
K.H. Schindl
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space-charge tune spreadspace-charge tune spreadparticlein a storage ring, beam makes many turns
(e.g. PS booster ~106 turns)particles with small deviations from the design orbit oscillate
around this orbit in phase spaceinteger tunes, ½ integer tunes etc. must be avoided since
they lead to resonances and beam loss(particles will “sum up” all machine / magnet imperfection
resonances turn by turn)the space charge reduces the tune, and also leads to a
tune spread Q in the beam (for a real non-uniform and bunched beam particles at large transverse and longitudinal amplitudes will see less tune shift)
once Q becomes too big there will always be some particleson resonance and these will be lost
this is the major problem at low energy hadron accelerators M. Benedikt
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Example for space-chargelimited synchrotron:betatron tune diagram andareas covered by direct tunespread at injection, intermediate energy,and extraction, for the CERN Proton Synchrotron Booster.During acceleration,acceleration gets weakerand the “necktie” areashrinks, enabling the externalmachine tunes to move the“necktie” to a region clearof betatron resonances(up to 4th order)
K.H. Schindl
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nonlinear dynamics and space charge
the problem is complex
detuning resonance condition
particle amplitudegrowth
beam sizegrowth
G. Franchetti
particleloss
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nonlinear dynamics in a bunch
periodic crossingof a resonance
z
x
bare tune
resonance
G. Franchetti
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trapping into resonances during synchrotron motion
periodic crossingof a resonance
z
x
bare tune
resonance
G. Franchetti
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trapping into a resonanceduring synchrotron motion
1 synchrotron oscillation in 6000 turns
G. Franchetti
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single passage through a resonance
3rd orderresonance
+0.15-0.15
Qx = 0.1
Qy
Qx
Bare tune
role of transverse detuning when the stop-band is crossed
Bare tune
Several particles remain on one side of the resonance increasing their amplitude
3 Qx = 13
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particle trapping into a resonanceduring accumulation (at injection energy)
Space charge increased in N turns
3rd orderresonance
Bare tune
Qx = 0.15
Qy
Qx
0.03
“Scattering”
N = 103 turns
Full beam emittance Test particle
N = 5 x103 turns Trapping
G. Franchetti
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adiabatic / non-adiabatic regimescondition for a particle to remain trapped
Tune on theFixed point
Size of the island
Speed of the fixed point
If during 1 revolution around the fixed point the island moves less than its size than the particle can remain trapped
T << 1 characterizes the adiabatic regime
A.W. Chao and Month NIM 121, 129 (1974).A. Schoch, CERN Report, CERN 57-23, (1958)A.I. Neishtadt, Sov. J. Plasma Phys. 12, 568 (1986)
G. Franchetti
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how to overcome the ‘space-charge limit’?
FNAL booster
CERN PS, BNL AGS
1. raise the injection energy!
K.H. Schindl
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how to overcome the ‘space-charge limit’?2. flatten the bunch distribution!
transN
yx FCr
Q
ˆ4 2
0,
maximum line density
transverse form factor
rms emittance
Ftrans=1 for GaussianFtrans=1/2 for transversely uniform distribution
“phase-space painting”, double harmonic rf,…
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how to measure the incoherent tune shift/spread?
K.H. Schindl
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various types of collective effects; space charge
2) wake fields, impedances, beam instabilities, Landau damping
3) beam-beam effects
4) ions and electron cloud effects
collective effects in particle accelerators
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wake fields• the real vacuum chamber (beam pipe) is not a perfectly
conducting pipe of constant aperture• a beam passing an obstacle radiates electromagnetic
fields and excites the normal modes of the object• consequences:
– beam loses energy– energy can be transferred from head to tail of a bunch– the head of the bunch can deflect the tail – energy and deflections can be transferred between bunches if
the high Q (quality factor) normal modes
• the wake fields characterize (“are”?) the beam induced energy losses and deflections
}Instabilities!
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calculation by T. Weiland
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wake-field properties
Particle of charge Q (=1) followed at a distance ct0 by q.Let Q travel on axis (RT=0) and let Ez be the longitudinal electric field at q.Longitudinal wake potential is
dzzEtV z0
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longitudinal wake fieldin the ultrarelativistic limit (→1), V has simple properties:
V
talso, V is independent of rT
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to get V(t0) for a beam, convolute V with the bunch shape
dttttVtV 00
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a
now, let Q travel off axis
cos0'
200
0 tVa
RrtVtV TT
on-axis wake potential modes that have Ez=0 on axis are excited
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V’(t0) must have same qualitative behavior as V(t0)
components azimuthal and radial have ,00' BEr
EV z
→ deflections are possible
dzBvER
tWT
1' 0
transverse wake field
0
'' 0
t
dttVtW
from Maxwell”s equations:
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longitudinal-transverse wake relation
0
'' 0
t
dttVtW Panofsky-Wenzeltheorem
transverse wakeis defocusing
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emittance growth in linacs & linear colliders
• 1st example of impact of wake fields• advantage of 2-particle model for getting insight
single particle injected on axis travels down linac
if injected off-axis, quadrupoles surrounding linac→ oscillation about axis
cartoon- scales are not correct!
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2nd particle follows 1st, wake from 1st deflects 2nd
deflected outward, amplitude grows
amplitude of second particle (bunch tail) grows, effective emittance growth
multi-particle simulationby Karl Bane,
large growth in effectiveemittance
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possible solutions:1.BNS damping (Balakin, Novokhatsky, Smirnov)2. reduce wake fields
BNS damping – analogy classical driven oscillator
response
drive
natural frequency
drive=head, nat=tail
responsedriven with beatsn oscillatio tailinitial 2)
reduced is response 1)
:effects two if tailhead
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multi-particle simulationby Karl Bane,
SLC with BNS damping
(SLC) 2)
CLIC) of version (previous squadrupole rf 1)
:by achieved becan
tailhead
tailhead
EE
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instabilities in circular accelerators
• stick with 2-particle model
• head produces wake that acts on the tail
of charge q/2 each
• head and tail interchange due to synchrotron oscillations
HEADTAIL
WAKE FIELD
HEAD
TAIL
TAIL
HEAD
1/2 Ts later
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example : “fast head-tail” instability
• simplified transverse wake field
W
W
2/
0
122
2
12
1
Wqyyy
yy
for 0 < t < Ts/2: SHM
driven harmonic oscillator
head
tail
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2/
0
212
1
22
2
qWyyy
yy
For Ts/2 < t < Ts: head
tail
look for solutions:
dt
dY
dt
Yd ,
dt
dY ,
2
2
2,12,1 YetYty ti
when 0<t<Ts/2: 1) Y1 is constant2)
8
2/2
12
12
s
titi
TiWqYY
eWqYeYi
when Ts/2<t<Ts: 1) Y2 is constant2)
8
2/2
21
21
s
titi
TiWqYY
eWqYeYi
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• a common technique for assessing stability of this and similar systems is to write as a matrix
02
1
02
1
2
1
1)8/(
01
10
)8/(1
Y
YM
Y
Y
iWqT
iWqT
Y
Y
s
s
Ts
02
1
2
1
Y
YM
Y
Y N
NTs
for N synchrotron periods:
the motion is stable if all elements of MN remain bounded as N→infinity
look at eigenvalues of M. They are exp(+/-i ) where cos = ½ Trace(M) =1/2( 2-(WqTs/(8 ))2)
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• motion is stable if |cos |<1
→ q<16 /(WTs)
note contrast with linac; synchrotron motion in ring gives stability below “threshold”
“fast head-tail instability”(also called “transverse mode coupling instability”)
1. a better calculation2. past and future importance
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1. A better calculation – use many particles and time-dependent wakes
results for PEP
the threshold can be calculated with zero free parameters –wakes are determined by geometryright answer within a factor of two for e+/e- machines
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past & future importance• past: PEP & PETRA & LEP performances were limited
by fast head-tail instability• present: the electron cloud gives rise to a similar
instability limiting KEKB and PEP-II (only the “wake field” is due to electron cloud and not due to the chamber wall)
• future: the single bunch proton current in the LHC is expected to be limited by the fast head-tail instability in the SPS (which serves as LHC injector)
• future: the damping rings of ILC and CLIC may be limited by this instability, in particular the e-cloud driven version
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beam size
center of mass
beam sizew/oSR part
multiparticlesimulationfor PEP by R. Siemann
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multi-bunch instabilities• in the linac example it does not matter whether
the two particles are in the same bunch or not, but for the ring the head-tail exchange is central → particles in same bunch
• if beam induced fields last long enough different bunches can communicate
• an example: Robinson instability
RF
Npart0
00
Tc
turnsprev
tprfrf tV
cm
eN
cm
eV
...
02
0
22
20
arc:
rf:
beam induced voltageacting on beam particle
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turnsprevtp
crf
rfcc
turnsprevtprf
rf
c
tVTcm
eN
Tcm
eV
dt
d
dt
d
tVTcm
eN
Tcm
eV
dt
d
T
dt
d
T
...
0
002
0
22
002
002
2
...
0
02
0
22
02
00
0
0
0
turnsprev
tps tVTcm
Ne
dt
d
...
0
002
0
22
2
2
2/1
002
0
Tcm
eV rfcrfs
simpleharmonicoscillator withfrequency
“force” due to wake field
for e+ or e- beams;[for protons replace c→c-1/2]
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concentrate on force – make some approximation to see essential physics 1.only 1 previous turn2.single mode in cavity with natural frequency N,
quality factor Q,not necessarily N=rf
decay due to QV(tp.t.)
t
rings atN
initial negative valuefrom energy conservation T0+
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0
0
00
0
0
...
.
T
T
dt
dV
dt
dTTV
dt
dVTV
TVturnprevV
neglect constant,it does not affectstability
022
00
2
2
2
0
s
T dt
d
dt
dV
cm
Ne
dt
d
If [ ] > 0: damped motionIf [ ] < 0: unstable motion
A. N=rf, NT0=n 2dV/dt|T=0
B. N>rfdV/dt|T>0, unstable
C. N<rfdV/dt|T<0, damped
stability is extremely sensitive to mode frequency
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multiparticlesimulationby R. Siemann
![Page 60: collective effects in particle accelerators](https://reader036.fdocuments.in/reader036/viewer/2022062301/56814539550346895db2024b/html5/thumbnails/60.jpg)
microwave instability• longitudinal analog of fast head-tail instability
- single bunch - longitudinal- no centroid motion
• self limiting due to nonlinearity of rf wave • important in hadron colliders
+ in SLC damping ring+ B factories+ ILC/CLIC damping rings
![Page 61: collective effects in particle accelerators](https://reader036.fdocuments.in/reader036/viewer/2022062301/56814539550346895db2024b/html5/thumbnails/61.jpg)
observationof microwave instabilityat SPEAR
![Page 62: collective effects in particle accelerators](https://reader036.fdocuments.in/reader036/viewer/2022062301/56814539550346895db2024b/html5/thumbnails/62.jpg)
instability categories
instabilities can be classified as1. single bunch (e.g. fast head tail) or multi-bunch [multiple turn]
instability (e.g. Robinson instability) 2. transverse (fast head tail) or longitudinal (Robinson) instability3. with (Robinson) or without (fast head-tail) centroid motion
all of these 1. occur2. have one or several names – impact on literature3. have different degrees of importance depending on accelerator4. have different cures (basic design strategy is reducing the
size of the wake fields by the design of vacuum vessel and theamount of ions or electrons by surface treatments (e.g. coatingwith low-SEY material) or clearing electrodes
![Page 63: collective effects in particle accelerators](https://reader036.fdocuments.in/reader036/viewer/2022062301/56814539550346895db2024b/html5/thumbnails/63.jpg)
“sawtooth” instability in the SLC damping ring (B. Podobedov, R. Siemann)complex behavior, relaxation oscillations, non-monotonic dependence on N
![Page 64: collective effects in particle accelerators](https://reader036.fdocuments.in/reader036/viewer/2022062301/56814539550346895db2024b/html5/thumbnails/64.jpg)
tWedtiZ
tVedtZ
ti
ti
'1
'||0
remark: Fourier transform of the wake function is the impedance
usually the real part of the impedance is related toinstability growth (or damping) rates; the imaginary part shifts the mode frequencies; energy lossis due to the real part of Z0
||
![Page 65: collective effects in particle accelerators](https://reader036.fdocuments.in/reader036/viewer/2022062301/56814539550346895db2024b/html5/thumbnails/65.jpg)
some references• accelerator-physics lectures by R. Siemann,
1998• CERN summer school lectures by E. Metral
and S. Gilardoni, 2007• several CERN lectures by K.-H. Schindl and
M. Benedikt• summer student talk by I. Santiago 2007• GSI acceleratpr palaver 2005, G. Franchetti• HB2006 talk by G. Franchetti
![Page 66: collective effects in particle accelerators](https://reader036.fdocuments.in/reader036/viewer/2022062301/56814539550346895db2024b/html5/thumbnails/66.jpg)
Why do accelerators work?
• Large number of collective instability mechanisms.
• But the beam seems to be basically stable.
Natural stabilizing mechanism:
![Page 67: collective effects in particle accelerators](https://reader036.fdocuments.in/reader036/viewer/2022062301/56814539550346895db2024b/html5/thumbnails/67.jpg)
Basic idea: swing…
• Stiff frame. Frequency spread.• COHERENT C.M. motion decays quickly compared to INCOHERENT motion of children.• STABLE MOTION
• Flexible frame • INSTABILITY GROWTH RATE higher than FREQUENCY SPREAD.• INSTABILITY
A. Hofmanncoupled oscillators → new eigenmodes with shifted frequencies
![Page 68: collective effects in particle accelerators](https://reader036.fdocuments.in/reader036/viewer/2022062301/56814539550346895db2024b/html5/thumbnails/68.jpg)
Basic idea (still)
![Page 69: collective effects in particle accelerators](https://reader036.fdocuments.in/reader036/viewer/2022062301/56814539550346895db2024b/html5/thumbnails/69.jpg)
Landau dampingeffect occurs in systems consisting of a high number of oscillators having different oscillation frequencies and performing a collective motion
L.D. Landau, 1946, J. Phys. USSR 10 (1946)plasma physicsN.G. van Kampen, 1955, Physica 21 (1955)mathematics C.E. Nielsen, A.N. Sessler, K.R. Symon, HEACC Geneva (1959) accelerator physicsR.D. Kohaupt, “What is Landau Damping? Plausibilities, Fundamental Thoughts, Theory”DESY M-86-02accelerator physics
![Page 70: collective effects in particle accelerators](https://reader036.fdocuments.in/reader036/viewer/2022062301/56814539550346895db2024b/html5/thumbnails/70.jpg)
if a system of many oscillators (protons) with different oscillation frequencies is excited (kicked) their centroid motion decays in time as a result of the frequency spread,which extinguishes the coherent motion
on the other hand, a beam can be driven unstable by self-excited electromagnetic fields (impedance) which act back on the beam; instability rise time g
the beam is Landau damped if the decay time due to thefrequency spread is ‘shorter’ than the instability rise time
g
1
![Page 71: collective effects in particle accelerators](https://reader036.fdocuments.in/reader036/viewer/2022062301/56814539550346895db2024b/html5/thumbnails/71.jpg)
sN
t
edxeqx tiN
k
tik
743
11
01
0
1031060210
10
we can replace discrete sum by integral,for times which are not extremely long
![Page 72: collective effects in particle accelerators](https://reader036.fdocuments.in/reader036/viewer/2022062301/56814539550346895db2024b/html5/thumbnails/72.jpg)
max
min
max
min
,
txdtD
Nd
WDxx 2
frequency distribution
centroid
linear “wake force”
2
WNcoh coherent frequency shift in the absence of
frequency spread
![Page 73: collective effects in particle accelerators](https://reader036.fdocuments.in/reader036/viewer/2022062301/56814539550346895db2024b/html5/thumbnails/73.jpg)
0
11
220
20
NW
eD
NWDDDti
special case: single frequency
ansatz harmonic oscillation
solution exists for arbitrary small W
![Page 74: collective effects in particle accelerators](https://reader036.fdocuments.in/reader036/viewer/2022062301/56814539550346895db2024b/html5/thumbnails/74.jpg)
general case
00
~
~
2
1~
2
1
izttii
C
izttii
etdtDetdtDiD
ezDdeiDdtD
z
C
xdzD ~~
Fourier-Laplace-transform
Bromwich integral(all singularities above the path C)
![Page 75: collective effects in particle accelerators](https://reader036.fdocuments.in/reader036/viewer/2022062301/56814539550346895db2024b/html5/thumbnails/75.jpg)
22
2
1
))0(')0((~
zdW
dz
xizx
zD
)0(')0(~~22 xizxDWxz
C
cut –(max)<Rez<(max)
singularities at zk
analytic otherwise
zk
cut
tizk
ti
tizkcutC
k
k
eceDDd
ec
~~
initial conditions
C
iztezDdtD~
2
1
![Page 76: collective effects in particle accelerators](https://reader036.fdocuments.in/reader036/viewer/2022062301/56814539550346895db2024b/html5/thumbnails/76.jpg)
(1) 1
22
zdW
singularities zk in the upper z-plane describe coherentstability and those in the lower z-plane coherent instability; the zk correspond to solutions of
the integral has a finite value for any reasonablefrequency distribution for all z values
there is no solution to (1) for small W!
interpretation: system cannot organize a collective motion if interaction not strong enough
![Page 77: collective effects in particle accelerators](https://reader036.fdocuments.in/reader036/viewer/2022062301/56814539550346895db2024b/html5/thumbnails/77.jpg)
iz
22
x
N
1
2
1
2
11
2
2
2
11
22
211
11
2
2
11112
1111
2
3
0)Im(,
0Im,22
i
ii
Nii
N
i
iiN
i
i
iiii
iiii
i
iN
iiiii
N
diiii
N
zd
x
x
x
x
xx
xx
res xx
C xx
example:Lorentz spectrum
X
X
X
X
x+i
x-i
i
i Re
Im
![Page 78: collective effects in particle accelerators](https://reader036.fdocuments.in/reader036/viewer/2022062301/56814539550346895db2024b/html5/thumbnails/78.jpg)
i
u x
dispersion relation
>0: instability growth rate, ->0: border of stability;above relation may not apply for <0
11
1
1 22
ui
u
ucoh
consider =0 absolute value of dispersion relation simplifies to |coh|= ; if coherent tune shift less than spread the beam is stable
![Page 79: collective effects in particle accelerators](https://reader036.fdocuments.in/reader036/viewer/2022062301/56814539550346895db2024b/html5/thumbnails/79.jpg)
exampleof Landau-dampedsystem inBodrum?!
![Page 80: collective effects in particle accelerators](https://reader036.fdocuments.in/reader036/viewer/2022062301/56814539550346895db2024b/html5/thumbnails/80.jpg)
various types of collective effects; space charge
wake fields, impedances, beam instabilities, Landau damping
3) beam-beam effects
4) ions and electron cloud effects
collective effects in particle accelerators
![Page 81: collective effects in particle accelerators](https://reader036.fdocuments.in/reader036/viewer/2022062301/56814539550346895db2024b/html5/thumbnails/81.jpg)
beam-beam effectsLHC as example
• incoherent beam-beam effects– lifetime & dynamic aperture
• PACMAN effects– bunch-to-bunch variation
• coherent effects – oscillations and instabilities
(W. Herr, LHC Design Report, Chapter 5)
![Page 82: collective effects in particle accelerators](https://reader036.fdocuments.in/reader036/viewer/2022062301/56814539550346895db2024b/html5/thumbnails/82.jpg)
beam-beam forcecalculation is similar as for space charge, but
- electric and magnetic forces add s.c.: (1-2)=1/2 → beam-beam: (1+2)=2 - beams move in opposite direction; interaction
time is a factor 2 smaller than the time duration of a single bunch passage in lab frame
- interaction is localized at one or a few places around the ring → force does not only cause a a tune shift with amplitude but it also excites many resonances mQx+nQy+p=0
k p
ipek
2
12
![Page 83: collective effects in particle accelerators](https://reader036.fdocuments.in/reader036/viewer/2022062301/56814539550346895db2024b/html5/thumbnails/83.jpg)
2
22
220
2
22
220
2exp12'
2exp12'
yx
yx
yNry
yx
yx
xNrx
beam-beam kick for head-on collision of round Gaussian beams:
force is nonlinear and couples x and y motion;(head-on) beam-beam tune shift from linear expansion
424
2 02
*0
,.
rNNrQ
Nyxyx
the beam-beam tune shift depends only on the beam brightness and is independent of *;NCP collision points → total beam-beam tune shift = Ncp x there is a maximum acceptable value for this tune spread
→ “beam-beam limit”
if beam is notround, there isan analytical expressioninvolving the complex errorfunction(Bassetti-Erskine)
![Page 84: collective effects in particle accelerators](https://reader036.fdocuments.in/reader036/viewer/2022062301/56814539550346895db2024b/html5/thumbnails/84.jpg)
beam-beam deflection curve
W. Herr
head-oncollision
LRcollision
opposite sign of slope for long-range collisions in plane of offset
![Page 85: collective effects in particle accelerators](https://reader036.fdocuments.in/reader036/viewer/2022062301/56814539550346895db2024b/html5/thumbnails/85.jpg)
luminosity at the beam-beam limit
LR reaction rate luminosity
cross section
*0
*2
2 1
44
Nnf
r
NnNfNnfL brev
totb
N
revbrev
total beam-beamtune shiftlimited to ~0.01 (protons)~0.1 (e+/e-)
grows linearlywith energy
beam current
IP beta function
# bunches
maximum luminosity:many bunchessmall *+ max. bunch charge compatible with bb limit
![Page 86: collective effects in particle accelerators](https://reader036.fdocuments.in/reader036/viewer/2022062301/56814539550346895db2024b/html5/thumbnails/86.jpg)
two high luminosityIPs (IP1 ATLAS & IP5 CMS)
two lower-luminosityIPs (IP2 ALICE & IP8 LHCb)
3 (4) head-on collisions &~120 long-range collisions
![Page 87: collective effects in particle accelerators](https://reader036.fdocuments.in/reader036/viewer/2022062301/56814539550346895db2024b/html5/thumbnails/87.jpg)
30 long-range collisions per main IP
partial mitigation by alternating planes of crossing at IP1 & 5 etc.
#LR encounters
SPS 9
Tevatron Run-II 70
LHC 120
![Page 88: collective effects in particle accelerators](https://reader036.fdocuments.in/reader036/viewer/2022062301/56814539550346895db2024b/html5/thumbnails/88.jpg)
2
2
d
nQ LR
LR
N
bpHO
NrQ
4
head-on tune shift(with zero crossing angle)
long-range tune shiftif beams are separatedby d/(thanks to a crossing angle)
head-on & long-range tune shifts & tune spread ~ similar to space charge
HOLR QQ 33.0for nominal LHC:
![Page 89: collective effects in particle accelerators](https://reader036.fdocuments.in/reader036/viewer/2022062301/56814539550346895db2024b/html5/thumbnails/89.jpg)
LHC design criterion J. Gareyte, J.-P. Koutchouk)
avoid resonances < order 13 & |QH-QV|~0.01
→ nominal total tune spread (up to 6in x&y) from all IPs and over all bunches, including long-range effects, should be less than 0.01-0.012
notes: • this limiting value comes from SPS & Tevatron;• 6 is empirical to match results of Ritson & Chou
for “ultimate” LHC, |QH-QV|~0.005, and the total tune spread should be less than 0.015-0.017
![Page 90: collective effects in particle accelerators](https://reader036.fdocuments.in/reader036/viewer/2022062301/56814539550346895db2024b/html5/thumbnails/90.jpg)
nominal 7-TeV collision parameters* offset angle
IP1 0.55 m 16.7 m 0 285 rad (y)
[9.4]
IP2 10 m 70.9 m 355 m
[5]
300 rad (y)
[42.3]
IP5 0.55 m 16.7 m 0 285 rad (x)
[9.4]
IP8 10 m 70.9 m 0 400 rad (x)
[56.4]3 “head-on” collisions with crossing angle1 halo collision with 5- separation at IP260 long-range collisions with on average ~9.5 separation60 negligible long-range collisions
![Page 91: collective effects in particle accelerators](https://reader036.fdocuments.in/reader036/viewer/2022062301/56814539550346895db2024b/html5/thumbnails/91.jpg)
tune footprints due to head-onand long-range collisions in IP1 and IP5 [courtesy H. Grote]
total LHC tune footprint forregular and PACMAN bunch[courtesy H. Grote]
Q from LR collisions is approximately cancelled by alternating crossing[D. Neuffer, S. Peggs, SSC-63 (1986)]
tune footprints & alternating crossing
![Page 92: collective effects in particle accelerators](https://reader036.fdocuments.in/reader036/viewer/2022062301/56814539550346895db2024b/html5/thumbnails/92.jpg)
x
zcR
2
;1
12
“Piwinski angle”
luminosity reduction factor
nominal LHC
crossing angle
c/2
effective beamsize →/R
note: the tune shift is reducedby roughly the same factor
![Page 93: collective effects in particle accelerators](https://reader036.fdocuments.in/reader036/viewer/2022062301/56814539550346895db2024b/html5/thumbnails/93.jpg)
dynamic aperture
simulation by L. Evans = dynamically stable region in phase space
outside: global chaos, rapid diffusion, losses
![Page 94: collective effects in particle accelerators](https://reader036.fdocuments.in/reader036/viewer/2022062301/56814539550346895db2024b/html5/thumbnails/94.jpg)
sepda xx
N
bN
“dynamic aperture”
diffusive aperture due to long-range encounters, new regime of hadron beam-beam
withindependent of * and energy
for nominal LHC: xsep~9.5, xda~6J. Irwin, SSC-223 (1989)Y. Papaphilippou & F.Z., PRST-AB 2, 104001 (1999)Y. Papaphilippou & F.Z., PRST-AB 5, 074001 (2002)
![Page 95: collective effects in particle accelerators](https://reader036.fdocuments.in/reader036/viewer/2022062301/56814539550346895db2024b/html5/thumbnails/95.jpg)
LHC filling pattern
lack of 4-fold symmetry → some bunches encounter abort gap in IP2 or 8 and have missing head-on collisions; in addition IP8 is displaced by 11.22 m and also 3 bunches in each train miss head-on collisions in IP2
![Page 96: collective effects in particle accelerators](https://reader036.fdocuments.in/reader036/viewer/2022062301/56814539550346895db2024b/html5/thumbnails/96.jpg)
All encounters in the straight sections are taken into account. Each bunch in the LHC is represented as a dot. The angular co-ordinate is the initial position of the bunch around the circumference. There is a one-to-one correspondence between beam-beam equivalence class and the radius in the plot. The classes are sorted according to the population of the class. Thus, classes containing a single bunch, of which there are several, lie towards the centre of the plot. Tomake adjacent classes easier to distinguish they are also colored differently (although the colours are used several times over at clearly distinguishable radii).Here there are 171 equivalence classes.
Beam-Beam Equivalence Classes for LHCr [J. Jowett, LHC’99]
only ~half of the bunchesare regular
![Page 97: collective effects in particle accelerators](https://reader036.fdocuments.in/reader036/viewer/2022062301/56814539550346895db2024b/html5/thumbnails/97.jpg)
PACMAN effects
• expect bunch-to-bunch variation of orbit, tune and chromaticity
• partial compensation by alternating crossing in IP1 and 5
![Page 98: collective effects in particle accelerators](https://reader036.fdocuments.in/reader036/viewer/2022062301/56814539550346895db2024b/html5/thumbnails/98.jpg)
bunch-to-bunch orbit variation
beam1
beam2
beam1
beam2
orbit displacements at IP1
HH crossing HV crossing
W. Herr
only half of bunch pattern shown;collisions are head-on in the other plane;in addition ground motion will separate the two beams
by 5 during 8 hours
![Page 99: collective effects in particle accelerators](https://reader036.fdocuments.in/reader036/viewer/2022062301/56814539550346895db2024b/html5/thumbnails/99.jpg)
bunch-to-bunch Q, Q’ variation
HV crossing
HV crossing
HH crossing
HH crossing
W. Herr
first 3 bunchesin each train
abort gap
![Page 100: collective effects in particle accelerators](https://reader036.fdocuments.in/reader036/viewer/2022062301/56814539550346895db2024b/html5/thumbnails/100.jpg)
emittance growth from noise• LHC beams are stored in two separate beam pipes; orbit
perturbations are independent and can steer the beams out of collision
• transverse feedback, rf, wire compensator, crab cavity…• emittance growth due to random beam-beam offset including
decoherence and feedback [Y. Alexahin]:
where g is a feedback gain factor (typically g~0.2), || the total beam-beam tune-shift parameter assumed equal 0.01, x* the horizontal IP beam size, nIP the number of IPs (taken to be two), and s0~0.645
• emittance growth < 1%/hr:→ tolerance: x < 2.6 nm for g=0.2, x < 0.6 nm for g=0.0
22
20
21
1
4
11
g
xsnf
dt
d
xIPrev
consistent withsimulations byK. Ohmi
![Page 101: collective effects in particle accelerators](https://reader036.fdocuments.in/reader036/viewer/2022062301/56814539550346895db2024b/html5/thumbnails/101.jpg)
data from various locations 1989-2001
A. SeryiNanobeam’2002Lausanne
ground motion
HERA
LEP
![Page 102: collective effects in particle accelerators](https://reader036.fdocuments.in/reader036/viewer/2022062301/56814539550346895db2024b/html5/thumbnails/102.jpg)
coherent beam-beam effects
• unlike SPS and Tevatron, LHC will operate in the strong-strong regime
• Y. Alexahin predicted that Landau damping of the mode may be lost
• Landau damping can be restored by symmetry breaking– different intensities– different tunes– broken symmetry for multiple interaction regions
or by overlap with synchrotron sidebands
two-beam system can show dipole-like instabilities (where one beam oscillatesagainst the other) unlike for direct space charge with a single beam
![Page 103: collective effects in particle accelerators](https://reader036.fdocuments.in/reader036/viewer/2022062301/56814539550346895db2024b/html5/thumbnails/103.jpg)
mode mode
continuum
equal intensity intensity ratio 0.55
frequency spectrum of dipole oscillations
mode not Landau damped mode Landau dampedM.P. Zorzano & F.Z., PRST-AB 3, 044401 (2000)W. Herr, M.P. Zorzano, F. Jones, PRST-AB 4, 054402 (2002)
![Page 104: collective effects in particle accelerators](https://reader036.fdocuments.in/reader036/viewer/2022062301/56814539550346895db2024b/html5/thumbnails/104.jpg)
beam-beam tune spread can also do something good – namely provide Landau damping against impedance-driven instabilities
max. octupoles 0.00012
nominalLHC: LRin IP1andIP5
ultimateLHC: HO+LRin IP1andIP5
W. Herrand L. VosLHC ProjectNote 316(2003)
![Page 105: collective effects in particle accelerators](https://reader036.fdocuments.in/reader036/viewer/2022062301/56814539550346895db2024b/html5/thumbnails/105.jpg)
LR RHIC experiments in 2005 and 2006
single off-center collision
one collision with 5-6 offset strongly increases RHIC beam loss rate; worse at smaller offsets
(W. Fischer et al.)
24 GeV
100 GeV
![Page 106: collective effects in particle accelerators](https://reader036.fdocuments.in/reader036/viewer/2022062301/56814539550346895db2024b/html5/thumbnails/106.jpg)
APC meeting, 19.09.03, LRBB J.P. Koutchouk, J. Wenninger, F. Zimmermann, et al.
• To correct all non-linear effects correction must be local.• Layout: 41 m upstream of D2, both sides of IP1/IP5
(Jean-Pierre Koutchouk)
Long-Range Beam-Beam Compensation for the LHC
Phase difference between BBLRC & average LR collision is 2.6o
![Page 107: collective effects in particle accelerators](https://reader036.fdocuments.in/reader036/viewer/2022062301/56814539550346895db2024b/html5/thumbnails/107.jpg)
1st Wire “BBLR” in the SPS, 2001
Tech. Coord. J. Camas &
G. Burtin/BDI
Help from many groups
two 60-cm long wireswith 267 A currentequivalent to 60 LHC LR collisions (e.g., IP1 & 5)
Iwire=Nb e c #LR/lwire
wire lengthwire current
force of current-fed wire mounted parallel to the beamis 1/r, just likethe long-range forcefrom the opposing beam
![Page 108: collective effects in particle accelerators](https://reader036.fdocuments.in/reader036/viewer/2022062301/56814539550346895db2024b/html5/thumbnails/108.jpg)
scaling from LHC to SPS and vice versa
)(
2'
dyec
Ilry wwp
da
wwp
y n
I
ec
lry~)(
2
'
'
for constant normalized emittance the effect in units of sigma is independent of energy and beta function!
relative perturbation:
perturbation by wire:
beam studies are being done in the CERN SPS at much lower beam energy (26-55 GeV) than for the future LHC (7 TeV)
![Page 109: collective effects in particle accelerators](https://reader036.fdocuments.in/reader036/viewer/2022062301/56814539550346895db2024b/html5/thumbnails/109.jpg)
for 2004 two novel 3-wire BBLRs were built; separated from 1-wire BBLR by about 2.6o
(average LR-BBLR phase advance in LHC)
![Page 110: collective effects in particle accelerators](https://reader036.fdocuments.in/reader036/viewer/2022062301/56814539550346895db2024b/html5/thumbnails/110.jpg)
3rd
10th
7th
4th
nearly perfect compensationwhat happens here?
Qx=0.31
1 wire
2 wires
no wire
vertical tune
beam lifetime
lifetime is recovered over a large tune range, except for Qy<0.285
two-wire compensation test at the SPS: tune scan
one wire modelsthe effect of the LHC long-range collisions,the second wireis used for compensation
![Page 111: collective effects in particle accelerators](https://reader036.fdocuments.in/reader036/viewer/2022062301/56814539550346895db2024b/html5/thumbnails/111.jpg)
beam-beam in linac-ring colliders
• The two beams can be optimized differently and independently. While the beam-beam tune shift for the (e+) beam in the ring is limited, the linac beam may encounter much larger beam-beam forces, thereby allowing for much larger luminosity than a ring-ring collider (TAC project)
→ Favors a high bunch charge in the ring
*,0
*2
1
44
rcoll
rtotr
N
lcollrlcoll Nf
r
NNfNNfL
![Page 112: collective effects in particle accelerators](https://reader036.fdocuments.in/reader036/viewer/2022062301/56814539550346895db2024b/html5/thumbnails/112.jpg)
beam-beam in linear colliders
• Linear colliders also have a beam-beam limit but at much higher bunch intensity than a ring collider. In linear colliders, beamstrahlung (synchrotron radiation in the field of the other beam) can lead to an intolerable degradation of the luminosity spectrum.
• Also a kink instability occurs, if particles perform several oscillations in the field of the opposing beam during a collision; this kink instability enhances small offsets and leads to a rapid decrease of luminosity.
• Beam-beam tune shift (ring) → disruption parameter (linear collider); basically the same parameter, but *→4z
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some LHC beam-beam references• J. Poole and F. Zimmermann, eds., Proceedings of Workshop on beam-beam effects in Large Hadron Colliders, CERN/SL 99-039 (AP)
(1999).
• J. Gareyte, Beam-Beam Design Criteria for LHC, Proc. LHC’99
• O. Bruning et al, LHC Design Report, Vol. 1, Chapter 5 (beam-beam section by W. Herr), CERN-2004-003
• Y. Alexahin, On the Landau damping and decoherence of transverse dipole oscillations in colliding beams, Part. Accel. 59, 43 (1998).
• W. Chou and D. Ritson, Dynamic aperture studies during collisions in the LHC, CERN LHC Project Report 123 (1997).
• L. Leunissen, Influence of vertical dispersion and crossing angle on the performance of the LHC, CERN LHC Project Report 298 (1999).
• Y. Papaphilippou, F. Zimmermann, Weak-strong beam-beam simulations for the Large Hadron Collider, PRST-AB 2:104001, 1999
• Y. Papaphilippou & F. Zimmermann, Estimates of diffusion due to long-range beam-beam collisions, PRST-AB 5:074001, 2002.
• M.P. Zorzano, F. Zimmermann, Coherent beam-beam oscillations at the LHC, PRST-AB 3:044401, 2000.
• W. Herr, M.P. Zorzano and F. Jones A Hybrid Fast Multipole Method applied to beam-beam collisions in the strong strong regime, PRST-AB 4, 054402 (2001)
• H. Grote, L. Leunissen, F. Schmidt, LHC Dynamic Aperture at Collision, LHC Project Note 197 (1999).
• J. Jowett, Collision Schedules and Bunch Filling Schemes in the LHC, CERN LHC Project Note 179 (1999).
• M.P.Zorzano, T.Sen, Emittance growth for the LHC beams due to head-on beam-beam interaction and ground motion , LHC Project Note 222 (2000).
• W. Herr, L. Vos, Tune distributions and effective tune spread from beam-beam interactions and the consequences for Landau damping in the LHC, LHC Project Note 316, 2003
• W. Herr, M.-P. Zorzano, Coherent Dipole Modes for Multiple Interaction Regions, LHC Project Report 462 (2001)
• Y. Alexahin, A study of the Coherent Beam-Beam Effect in the framework of the Vlasov Perturbation Theory, NIM A 380, 253 (2002)
• W. Herr, R. Paparella, Landau Damping of Coherent Modes by Overlap with Synchrotron Sidebands, CERN LHC Project Note 304, 2002
• W. Herr, Features and Implications of Different LHC Crosing Schemes, LHC Project Report 628 (2003)
• Y. Alexahin, On the Emittance Growth due to Noise in Hadron Colliders and Methods of its Suppression, NIM A 391, 73 (1996).
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various types of collective effects; space charge
wake fields, impedances, beam instabilities, Landau damping
beam-beam effects
4) ions and electron cloud effects
collective effects in particle accelerators
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INP Novosibirsk, 1965 Argonne ZGS,1965 BNL AGS, 1965
Bevatron, 1971
ISR, ~1972 PSR, 1988
AGS Booster, 1998/99 KEKB, 2000 CERN SPS, 2000
observations of electron cloud at various accelerators
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electron cloud and ionswhere do the e- (or ions) come from? ionization of residual gas
- collisional ionization ~ gas density,typical ionization cross section ~ 1 Mbarn,typically ~10-6 e-/ions per meter per beam particle- tunneling ionization in collective beam fieldif beam is sufficiently small (ILC, CLIC, FFTB)
photoemission from synchrotron radiation- typically 10-4 - 1 e- (ions) per meter per part.
avalanche build up via acceleration in the beam field - electron-cloud build up- pressure bump instability
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R. Cimino, I. Collins, 2003
probability of elastic electron reflection approaches 1 forzero incident energy and is independent of *max
secondarye-yield
max
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electron cloud in the LHC
schematic of e- cloud build up in the arc beam pipe,due to photoemission and secondary emission
[F. Ruggiero]
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e- density at saturation
• equilibrium e- density in multipacting regime:
time average net field at wall ~ 0, or
electron line density ~ beam line density
e ~ Nb/Lsep
• in other cases from balance of e- production and loss rates
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multipacting condition• kick from bunch to e- at the wall:
r’=2 re N/b
• resonance picture: e- cross the vacuum chamber (radius b) in a time equal to the bunch spacing, or r’ Lsep = 2b
• → re Lsep N = b2 (Grobner) ??
• this would suggest Nthresh~1/Lsep
• experimentally however Nthresh~ Lsep !
• threshold is determined by beam line density only
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blue: e-cloud effect observedred: planned accelerators
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effects of electron cloud• heat load (→ quench of s.c. magnets)
• coherent tune shift
• single and multi-bunch instabilities
• large incoherent tune shift due to e- “pinch” and local e- density spikes during a bunch passage (effects similar to space charge, but resonances are crossed only twice per synchrotron period – e- density increases roughly monotonically along the bunch) → poor lifetime, emittance growth
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coherent tune shift from e-cloud
Cr
Q yxe
yxe ,,; 2
coherent tune shift due to e-cloud for a ~round beam
for a flat beam with planar symmetry, the vertical tune shift would be two times larger and the horizontal one about zero
above relation can be used to estimate the average e- densityfrom the measured tune shift
comment: the incoherent tune shift is many orders of magnitudelarger due to the e- pinch (which moves with the bunch)
K. Ohmi et al, APAC’01
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insert this in the formula for fast head-tail threshold from 2-particlemodel (lecture 2):
result in good agreement with experimental data
fast head-tail instability driven by e-cloud
bey NCW /8
Cr
Q
y
sthre
0,
2
estimate of e-cloud wake field acting between bunch head and tail
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incoherent effect of e-cloud
incoherent tune shift increases along the bunchresonances are crossed twice per synchrotron period
G. Franchetti
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“neckties” in tune diagram
G. Franchetti
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multi-particle simulation of e-cloud driven head-tail instability by K. Ohmi; characteristics of “fast head-tail”
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multitude of countermeasures:• multi-bunch & intrabunch feedback
(INP PSR, Bevatron, SPS, KEKB)• clearing electrodes (ISR, BEPC, SNS)• antechamber (PEP-II)• TiN coating (PEP-II, PSR, SNS)• high Q’ (SPS)• octupoles (BEPC)• solenoids (KEKB, PEP-II, SNS)• grooved surfaces (NLC)
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LHC strategy against electron cloud
1) warm sections (20% of circumference) coated by TiZrVgetter developed at CERN; low secondary emission; if cloud occurs, ionization by electrons (high cross section ~400 Mbarn) aids in pumping & pressure will even improve
2) outer wall of beam screen (at 4-20 K, inside 1.9-K cold bore) will have a sawtooth surface (30 m over 500 m) to reduce photon reflectivity to ~2% so that photoelectrons are only emitted from outer wall & confined by dipole field
3) pumping slots in beam screen are shielded to preventelectron impact on cold magnet bore
4) rely on surface conditioning (‘scrubbing’); commissioning strategy; as a last resort doubling or triplingbunch spacing suppresses e-cloud heat load
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e- cloud effect may also be reduced by:• larger bunch spacing• high bunch intensity• superbunches
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predicted heat load in LHC vs. bunch spacing
on a vertical log scalechange in max appears as ~constant vertical shift
WEPLT045, THPLT017
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eV 9.10 E1-9
20 m103.1 cmr
E
eee
saturation of e- build up for high bunch intensities
Nb=4.6x1011
2.3x1011
~average energy of secondaryelectrons
109 m-1
0
e- line density
time10 s
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schematic of e- motion during passage of long protonbunch; most e- do not gain any energy when traversingthe chamber in the quasi-static beam potential
[after V. Danilov]negligible heat load
superbunch
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ion instabilities[conventional: trapped ions]single-pass: fast beam-ion
instability (FBII)
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various types of collective effects; space charge
wake fields, impedances, beam instabilities, Landau damping
beam-beam effects
electron-cloud and ion effects
collective effects in particle accelerators