Landau Damping - CERN · 2019-06-18 · I.Karpov, V.Kornilov, O.Boine-F., PRAB 19, 124201 (2016)...
Transcript of Landau Damping - CERN · 2019-06-18 · I.Karpov, V.Kornilov, O.Boine-F., PRAB 19, 124201 (2016)...
Vladimir Kornilov, The CERN Accelerator School, London, UK, Sept 3-15, 2017 1
Landau Dampingpart 2
Vladimir KornilovGSI Darmstadt, Germany
Vladimir Kornilov, CAS, June 9-21, 2019, Denmark
Vladimir Kornilov, The CERN Accelerator School, London, UK, Sept 3-15, 2017
energy transfer toresonant particles
2
Landau Damping
PERTURBATION
the wave
G
suppressiondriving field
suppression
Main ingredients of Landau damping:ü wave−particle collisionless interaction: Impedance driving fieldü energy transfer: the wave ↔ the (few) resonant particles
Incomplete (!) mechanism of Landau Damping in beamsfor the end of the first part
Vladimir Kornilov, CAS, June 9-21, 2019, Denmark
Vladimir Kornilov, The CERN Accelerator School, London, UK, Sept 3-15, 2017 3
Existence of Landau damping
driving dipole impedance here
Still, the beams are often stable without an active mitigation
In any accelerator, there are many Re(Z) sources
In any beam, there are many unsuppressed eigenmodes
There must be a fundamental damping mechanism in beams
Vladimir Kornilov, CAS, June 9-21, 2019, Denmark
Vladimir Kornilov, The CERN Accelerator School, London, UK, Sept 3-15, 2017 4
Existence of Landau damping
The beams are stable and absorb some energy
Additionally to Re(Z), deliberate excitation is often applied(tune measurements, optics control, …)
Energy is directly transferred to the beam,mostly at the beam resonant frequencies
There must be a fundamental damping mechanism in beams
Tune measurements at PS,kick every 10 ms,M.Gasior, et al, CERN
Vladimir Kornilov, CAS, June 9-21, 2019, Denmark
Vladimir Kornilov, The CERN Accelerator School, London, UK, Sept 3-15, 2017 5
Damping
Basic consideration of a collective mode stability
change the parameters and the source of the
driving mechanism
use and enhance the intrinsic damping
mechanism
InstabilityStabilized modeDriven (unsuppressed) modeMode suppressed by its drive
Vladimir Kornilov, CAS, June 9-21, 2019, Denmark
Vladimir Kornilov, The CERN Accelerator School, London, UK, Sept 3-15, 2017 6
Another Leading Actor:Decoherence
K.Y.Ng, Physics of Intensity Dependent Beam Instabilities, 2006A.Hofmann, Proc. CAS 2003, CERN-2006-002A.Chao, Phys. Coll. Beam Instab. in High Energy Acc. 1993A.W. Chao, et al, SSC-N-360 (1987)
Vladimir Kornilov, CAS, June 9-21, 2019, Denmark
Vladimir Kornilov, The CERN Accelerator School, London, UK, Sept 3-15, 2017 7
V.Kornilov, O.Boine-F., PRSTAB 15, 114201 (2012)
Decoherence
Collective beam oscillations after a short (one turn or shorter) kick
• Usually the beam displacement is comparable to the beam size
• In reality, signals can be very complicated
• A common diagnostics• Decoherence is a process
affected by many effectsand mechanisms
A bunched beam Ar18+ in SIS18.Transverse signal after a kick
Vladimir Kornilov, CAS, June 9-21, 2019, Denmark
Vladimir Kornilov, The CERN Accelerator School, London, UK, Sept 3-15, 2017 8
Pulse Response
-1-0.8-0.6-0.4-0.2
0 0.2 0.4 0.6 0.8
1
0 5 10 15 20
x / x
0
ωt
⟨x⟩ 8 particles with different frequencies
Betatron oscillations:frequency spread
The Pulse Response is the Fourier image of BTF Vladimir Kornilov, CAS, June 9-21, 2019, Denmark
Vladimir Kornilov, The CERN Accelerator School, London, UK, Sept 3-15, 2017 9
Phase-Mixing (Filamentation)
x /
x x
x x
x /
x /
x /
1. 2.
3. 4.
Phase-mixing of non-correlated particles with a tune spread
• Beam offset oscillations decay
• Beam size increases (blow-up)
Vladimir Kornilov, CAS, June 9-21, 2019, Denmark
Vladimir Kornilov, The CERN Accelerator School, London, UK, Sept 3-15, 2017 10
Phase-Mixing, Coasting Beam
-1
-0.5
0
0.5
1
0 2 4 6 8 10
Beam
Offs
et ⟨x
⟩ / x
0
Turn Number
-1
-0.5
0
0.5
1
0 2 4 6 8 10
Beam
Offs
et ⟨x
⟩ / x
0
Turn Number
Gaussian Distribution Lorentz Distribution
Q0=10.3, ξ=1, δp=0.5×10−3 Q0=10.3, ξ=1, δp=0.5×10−3
This is the case without any collective interactions:phase-mixing of non-correlated particles
Vladimir Kornilov, CAS, June 9-21, 2019, Denmark
Vladimir Kornilov, The CERN Accelerator School, London, UK, Sept 3-15, 2017 11
ξ=0.5ξ=1.4Ns=100Qs=0.01
Phase-Mixing, Bunched Beam
-1
-0.5
0
0.5
1
0 50 100 150 200
bunc
h of
fset
/ σ
x0
time (turns)
Due to the particle synchrotron motion,the initial positions return after a synchrotron period
In literature, named as “decoherence” and “recoherence”
In reality, the decoherence is very different (other effects)
dashed lines:amplitude
solid lines:offset
Vladimir Kornilov, CAS, June 9-21, 2019, Denmark
Vladimir Kornilov, The CERN Accelerator School, London, UK, Sept 3-15, 2017 12
DecoherenceLandau Damping
Phase-mixing
Space-charge
Impedances
Other nonlinearities
Chromaticity
Octupoles
Image charges
Decoherence
EffectsProcessMechanisms
K.Y.Ng, Physics of Intensity Dependent Beam Instabilities, 2006A.Hofmann, Proc. CAS 2003, CERN-2006-002A.W. Chao, et al, SSC-N-360 (1987)V.Kornilov, O.Boine-F., PRSTAB 15, 114201 (2012)I.Karpov, V.Kornilov, O.Boine-F., PRAB 19, 124201 (2016)
Vladimir Kornilov, CAS, June 9-21, 2019, Denmark
Vladimir Kornilov, The CERN Accelerator School, London, UK, Sept 3-15, 2017 13
DecoherenceLandau Damping
Phase-mixing
Space-charge
Impedances
Other nonlinearities
Chromaticity
Octupoles
Image charges
Decoherence
EffectsProcessMechanisms
After a kick, we observe the decoherence, which is a mixture of effects.
We do not observe pure Landau Damping.(infinitesimal perturbations)
Vladimir Kornilov, CAS, June 9-21, 2019, Denmark
Vladimir Kornilov, The CERN Accelerator School, London, UK, Sept 3-15, 2017 14
Landau Damping:the wave & the tune spread
Vladimir Kornilov, CAS, June 9-21, 2019, Denmark
Vladimir Kornilov, The CERN Accelerator School, London, UK, Sept 3-15, 2017
0
0.2
0.4
0.6
0.8
1
-3 -2 -1 0 1 2 3ψ
p
∆Q / δQξ
15
Landau Damping
0
0.2
0.4
0.6
0.8
1
-3 -2 -1 0 1 2 3
ψp
10-3 ∆p / p0
resonant particles from both sides contribute to the energy transfer,thus f(ω)
Momentum spread translates into tune spread
Vladimir Kornilov, CAS, June 9-21, 2019, Denmark
Vladimir Kornilov, The CERN Accelerator School, London, UK, Sept 3-15, 2017 16
Landau Damping
Loss of Landau damping due to coherent tune shift
0.0 0.2 0.4 0.6 0.8 1.0 1.2Im(∆Q) / δQ
ξ
-4
-2
0
2
4
Re(∆
Q) /
δQ
ξ
STABLE
UNSTABLE
0
0.2
0.4
0.6
0.8
1
-3 -2 -1 0 1 2 3 4 5 6
Parti
cle
Dis
tribu
tion
∆Q / δQξ
there is still tune spread,but no resonant particles for this wave → no Landau damping
impedance tune shift
Vladimir Kornilov, CAS, June 9-21, 2019, Denmark
Vladimir Kornilov, The CERN Accelerator School, London, UK, Sept 3-15, 2017
-6
-4
-2
0
2
4
0 0.2 0.4 0.6 0.8 1
Re(∆
Q) /
δQξ
Im(∆Q) / δQξ
∆Qsc = 2 δQξ
no space charge
17
Landau Damping
impedance tune shift
Loss of Landau damping due to space-charge
-
0
0.2
0.4
0.6
0.8
1
-6 -5 -4 -3 -2 -1 0 1 2
Parti
cle
Dis
tribu
tion
∆Q / δQξ
no sp
ace-
char
ge
with
spac
e-ch
arge
there is still tune spread,but no resonant particles for this wave → no Landau damping
Vladimir Kornilov, CAS, June 9-21, 2019, Denmark
Vladimir Kornilov, The CERN Accelerator School, London, UK, Sept 3-15, 2017 18
Landau Damping
Loss of Landau damping due to space-charge
Coherent dipole oscillations of four O8+ bunches in the SIS18 of GSI Darmstadt at the injection energy 11.4 MeV/u
O.Boine-F., T.Shukla, PRSTAB 8, 034201 (2005)
there is still tune spread,but no resonant particles for this wave → no Landau damping
Vladimir Kornilov, CAS, June 9-21, 2019, Denmark
Vladimir Kornilov, The CERN Accelerator School, London, UK, Sept 3-15, 2017 19
Landau Damping
no antidamping
Some analytical models or simulations predict Landau antidamping:
unstable oscillations for zeroor negative impedance
No antidamping for Gaussian-like distributions!
D.Pestrikov, NIM A 562, 65 (2006)V.Kornilov, et al, PRSTAB 11, 014201 (2008)A.Burov, V.Lebedev, PRSTAB 12, 034201 (2009)
Remark 1
Vladimir Kornilov, CAS, June 9-21, 2019, Denmark
Vladimir Kornilov, The CERN Accelerator School, London, UK, Sept 3-15, 2017 20
Landau Damping
a tune spread does not automatically means Landau Damping
Remark 2
0
0.2
0.4
0.6
0.8
1
-4 -3 -2 -1 0 1 2 3 4
Parti
cle
Dis
tribu
tion
∆Q / δQξ
k=1
Head-tail modes in bunches:
• there is a tune spread(due to chromaticity ξ)
• the coherent frequency overlaps with the incoherent spectrum
still, there isNO Landau Damping!
Vladimir Kornilov, CAS, June 9-21, 2019, Denmark
Vladimir Kornilov, The CERN Accelerator School, London, UK, Sept 3-15, 2017
energy transfer toresonant particles
decoherence
21
Landau Damping of 1st Type
eigenmodePERTURBATION
the wave
⟨x⟩
suppression
G
beam offset
driving field
suppression
suppression
Main ingredients of Landau damping:ü wave−particle collisionless interaction: Impedance driving fieldü energy transfer: the wave ↔ the (few) resonant particles
Vladimir Kornilov, CAS, June 9-21, 2019, Denmark
Vladimir Kornilov, The CERN Accelerator School, London, UK, Sept 3-15, 2017 22
Landau Damping:”Immune System”
Permanent processof keeping the beams stable
Vladimir Kornilov, CAS, June 9-21, 2019, Denmark
Vladimir Kornilov, The CERN Accelerator School, London, UK, Sept 3-15, 2017 23
Landau Damping in Beamsof 2nd type
Vladimir Kornilov, CAS, June 9-21, 2019, Denmark
Vladimir Kornilov, The CERN Accelerator School, London, UK, Sept 3-15, 2017
For example, an octupole magnet:
First-order frequency shift ofthe anharmonic oscillations
Amplitude-dependent betatron tune shifts
24
Landau Damping of 2nd type
Different situation from ξ+δp:Tune spread due to amplitude-dependent tune shifts
Schematic yoke profileof an octupole magnet
N
N
N
N
S
S
S
S
Vladimir Kornilov, CAS, June 9-21, 2019, Denmark
Vladimir Kornilov, The CERN Accelerator School, London, UK, Sept 3-15, 2017 25
Landau Damping of 2nd type
Thus, octupoles provide an ingredient forLandau damping: tune spread
Vladimir Kornilov, CAS, June 9-21, 2019, Denmark
Tune footprint due to octupoles for a Gaussian bunch
Color: the distribution density (norm. units)
Vladimir Kornilov, The CERN Accelerator School, London, UK, Sept 3-15, 2017 26
Decoherence
0
0.5
1
1.5
2
2.5
3
0 20 40 60 80 100
∆x0 = 2 σx∆x0 = σx
ε / ε
0
turn number
0
0.5
1
1.5
2
0 20 40 60 80 100
∆x0 = 2 σx
∆x0 = σx
A ⟨x⟩
/ σ
x
turn number
A.W. Chao, et al, SSC-N-360 (1987)V.Kornilov, O.Boine-F., PRSTAB 15, 114201 (2012)I.Karpov, V.Kornilov, O.Boine-F., PRAB 19, 124201 (2016)
Phase-Mixing is very differentfrom the chromaticity case
• No recoherence• Kick amplitude dependent• Analytical model for:
Oscillationamplitude
Transverseemittance
Vladimir Kornilov, CAS, June 9-21, 2019, Denmark
Vladimir Kornilov, The CERN Accelerator School, London, UK, Sept 3-15, 2017
Amplitude-dependent betatron tune shifts in a lattice:
Every particle has a different amplitude (Jx, Jy)→ tune spread → Landau damping?
27
Landau Damping of 2nd type
-100
-50
0
50
100
0 50 100 150 200
x iωt
∆ωi / Ω = 0.03∆ωi / Ω = 0.01ωi = Ω
Tune spread due to amplitude-dependent tune shifts
Ωcoh
The resonant particles drift away in tunefrom the resonance as they get excited
-0.04
-0.02
0
0.02
0.04
0.06
0 0.5 1 1.5 2 2.5 3 3.5 4
∆Q
x
ax / a0
Vladimir Kornilov, CAS, June 9-21, 2019, Denmark
Vladimir Kornilov, The CERN Accelerator School, London, UK, Sept 3-15, 2017
-0.04
-0.02
0
0.02
0.04
0.06
0 0.5 1 1.5 2 2.5 3 3.5 4
∆Q
x
ax / a0
28
Landau Damping of 2nd type
0
0.2
0.4
0.6
0.8
1
0 0.5 1 1.5 2 2.5 3 3.5 4
ψx
ax / a0
a(Ωcoh)
Particle excitation for amplitude-dependent tune shifts
a(Ωcoh)
We already guess: the distribution slope (df/da) might be involved
-100
-50
0
50
100
0 50 100 150 200
x i
ωt
∆ωi / Ω = 0.03∆ωi / Ω = 0.01ωi = Ω
Once the particle is driven away from the resonance, the energy is transferred back to the wave
Vladimir Kornilov, CAS, June 9-21, 2019, Denmark
Vladimir Kornilov, The CERN Accelerator School, London, UK, Sept 3-15, 2017 29
Landau Damping of 2nd type
L.Laslett, V.Neil, A.Sessler, 1965D.Möhl, H.Schönauer, 1974J.Berg, F.Ruggiero, CERN SL-96-71 AP 1996
ΔQcoh : coherent no-damping tune shift imposed by an impedanceΔQex(Jx,Jy) : external (lattice) incoherent tune shifts
The resulting damping is a complicated 2D convolution of the distribution df(Jx,Jy)/dJx and tune shifts ΔQext(Jx,Jy)
The dispersion relation for coasting beam
Vladimir Kornilov, CAS, June 9-21, 2019, Denmark
Vladimir Kornilov, The CERN Accelerator School, London, UK, Sept 3-15, 2017 30
Landau Damping of 2nd type
energy transfer toresonant particles
decoherenceeigenmodePERTURBATION
the wave
⟨x⟩
suppression
G
beam offset
driving field
suppression
suppression
Main ingredients of Landau damping:ü wave−particle collisionless interaction: Impedance driving fieldü energy transfer: the wave ↔ the (few) resonant particles
Vladimir Kornilov, CAS, June 9-21, 2019, Denmark
Vladimir Kornilov, The CERN Accelerator School, London, UK, Sept 3-15, 2017 31
Landau Damping of 2nd type
-0.2
-0.1
0
0.1
0.2
-0.2 -0.1 0 0.1 0.2
∆Q
y ( ×
10−
3 )
∆Qx ( × 10−3 )
0 0.005
0.01 0.015
0.02 0.025
0.03 0.035
0.04
-0.6 -0.4 -0.2 0 0.2 0.4 0.6Im
(∆Q
) ( ×
10−
3 )Re(∆Q) ( × 10−3 )
stable
unstable
Octupole Tune Footprint Resulting Stability Diagram
An example of the application: Beam stability studies for FCC
V.Kornilov, FCC Week 2017, Berlin
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Vladimir Kornilov, The CERN Accelerator School, London, UK, Sept 3-15, 2017 32
Landau Damping of 2nd type
DynamicAperture
LandauDamping
A balance is necessary
Octupoles are the essential part of the beam stability in many machines
Drawback: octupoles, like other nonlinearities, can reduce Dynamic Aperture and Lifetime.
Vladimir Kornilov, CAS, June 9-21, 2019, Denmark
Vladimir Kornilov, The CERN Accelerator School, London, UK, Sept 3-15, 2017 33
X.Buffat, et al, PRSTAB 17, 111002 (2014)
Amplitude-Dependent Detuning
Another source of amplitude-dependent tune-shifts:Tune-spread due to Beam-Beam effects produces Landau Damping
Tune Footprints Resulting Stability Diagrams
Vladimir Kornilov, CAS, June 9-21, 2019, Denmark
Vladimir Kornilov, The CERN Accelerator School, London, UK, Sept 3-15, 2017 34
Landau Damping in Beamsof 3rd type
Vladimir Kornilov, CAS, June 9-21, 2019, Denmark
Vladimir Kornilov, The CERN Accelerator School, London, UK, Sept 3-15, 2017 35
Landau Damping of 3rd type
Ωcoh
ωinc Electric fieldof the self-fieldspace charge
Different situation:Wave−Particel interaction is due to space-charge
0
0.5
1
1.5
2
0 1 2 3 4 5 6
∆Q
sc / ∆
QKV
ax / σx
Space-charge tune shift
For the resonant particles Qinc≈Qcoh ,wave↔particles energy transfer should be possible
tune spread
Vladimir Kornilov, CAS, June 9-21, 2019, Denmark
Vladimir Kornilov, The CERN Accelerator School, London, UK, Sept 3-15, 2017 36
Landau Damping of 3rd type
L.Laslett, V.Neil, A.Sessler, 1965D.Möhl, H.Schönauer, 1974
f(Jx,Jy,p)ΔQcoh : no-damping coherent tune shift imposedΔQex(Jx,Jy,p) : external (lattice) incoherent tune shiftΔQsc(Jx,Jy) : space-charge tune shift
The dispersion relation
The resulting damping is a complicated 2D convolution of the distribution df(Jx,Jy)/dJx and tune shifts ΔQsc(Jx,Jy), ΔQext(Jx,Jy)
Vladimir Kornilov, CAS, June 9-21, 2019, Denmark
Vladimir Kornilov, The CERN Accelerator School, London, UK, Sept 3-15, 2017 37
Landau Damping of 3rd type
V.Kornilov, O.Boine-F, I.Hofmann, PRSTAB 11, 014201 (2008)E.Metral, F.Ruggiero, CERN-AB-2004-025 (2004)
Solution examples
Vladimir Kornilov, CAS, June 9-21, 2019, Denmark
Vladimir Kornilov, The CERN Accelerator School, London, UK, Sept 3-15, 2017 38
Landau Damping of 3rd type
L.Laslett, V.Neil, A.Sessler, 1965D.Möhl, H.Schönauer, 1974
V.Kornilov, O.Boine-F, I.Hofmann, PRSTAB 11, 014201 (2008)A.Burov, V.Lebedev, PRSTAB 12, 034201 (2009)
ΔQex=0: no pole, no damping!
Momentum conservation in a closed system
The dispersion relation
Even if Ωcoh is inside the spectrum,and there are resonant particles Qinc≈Qcoh ,
there is no Landau damping in coasting beams only due to space-charge
Vladimir Kornilov, CAS, June 9-21, 2019, Denmark
Vladimir Kornilov, The CERN Accelerator School, London, UK, Sept 3-15, 2017 39
Space Charge
Vladimir Kornilov, CAS, June 9-21, 2019, Denmark
This is space charge
Space charge is a beam-internal interaction
Baron Münchhausen
Vladimir Kornilov, The CERN Accelerator School, London, UK, Sept 3-15, 2017 40
Space Charge
Vladimir Kornilov, CAS, June 9-21, 2019, Denmark
This is an external force (octupoles, chromaticity,…)
Vladimir Kornilov, The CERN Accelerator School, London, UK, Sept 3-15, 2017 41
Landau Damping
a tune spread does not automatically means Landau Damping
Remark
Two examples:1. Head-tail modes and the chromaticity tune-spread2. Nonlinear space-charge in coasting beams
• there is a tune spread• the coherent frequency overlaps with
the incoherent spectrum
still, there is NO Landau Damping! 0
0.2
0.4
0.6
0.8
1
-4 -3 -2 -1 0 1 2 3 4
Parti
cle
Dis
tribu
tion
∆Q / δQξ
Vladimir Kornilov, CAS, June 9-21, 2019, Denmark
Vladimir Kornilov, The CERN Accelerator School, London, UK, Sept 3-15, 2017 42
Landau Damping of 3rd type
energy transfer toresonant particles
decoherenceeigenmodePERTURBATION
the wave
⟨x⟩
suppression
beam offset
over space charge
suppression
Main ingredients of Landau damping:ü wave−particle collisionless interaction: E-field of Space-chargeü energy transfer: the wave ↔ the (few) resonant particles
Vladimir Kornilov, CAS, June 9-21, 2019, Denmark
Vladimir Kornilov, The CERN Accelerator School, London, UK, Sept 3-15, 2017 43
Landau Damping of 3rd type
Landau damping in bunches
• There is damping due to only space charge
• Here: space-charge tune spread due to longitudinal bunch profile
• Additional spread due to transverse profile
A.Burov, PRSTAB 12, 044202 (2009)V.Balbekov, PRSTAB 12, 124402 (2009) V.Kornilov, O.Boine-F, PRSTAB 13, 114201 (2010)
Vladimir Kornilov, CAS, June 9-21, 2019, Denmark
Vladimir Kornilov, The CERN Accelerator School, London, UK, Sept 3-15, 2017 44
Landau Damping of 3rd type
Vladimir Kornilov, CAS, June 9-21, 2019, Denmark
Tune footprint due to space charge for a Gaussian bunch
Color: the distribution density (norm. units)
The role of space charge is still under study
Vladimir Kornilov, The CERN Accelerator School, London, UK, Sept 3-15, 2017 45
Landau Damping
Vladimir Kornilov, CAS, June 9-21, 2019, Denmark
Electron lenses for Landau damping (V. Shiltsev et al, Phys Rev Lett 119, 134802, 2017)
RF Quadrupoles for Landau damping (M. Schenk et al, PRAB 20, 104402, 2017)
A new IOTA facility for more Landau damping (S. Antipov et al 2017 JINST 12 T03002)
Other concepts for Landau damping
W.Fischer et al, BB2013, arXiv:1410.6393
Vladimir Kornilov, The CERN Accelerator School, London, UK, Sept 3-15, 2017 46
Landau Damping in beams
• Waves: collective oscillations in a medium of particles• Landau: Wave ↔ particles collisionless interaction• Dispersion relation and Stability Diagram• Decoherence is a result of Phase-Mixing, Landau damping, etc• Collective Frequency ↔ Tune Spread• Different types of Landau damping (explanatory help here)• “Immune system”• A tune spread does not automatically means Landau damping• A special role of space charge
Vladimir Kornilov, CAS, June 9-21, 2019, Denmark