An Overview of Collective Effects in 3rd Generation Light Sources
description
Transcript of An Overview of Collective Effects in 3rd Generation Light Sources
An Overview of Collective Effects in 3rd Generation Light Sources
At John Adams Institute, 03 February 2011, Oxford UK
R. Nagaoka, Synchrotron SOLEIL, Gif-sur-Yvette, France
Content:
1. Introduction
2. Induced EM self-field
3. Notion of Wake field
4. Geometric wake field and numerical (GdfidL) calculations
5. Impedance
6. Beam spectra
7. Equations of collective motions
8. Beam spectra overlap with impedance
9. Single bunch instabilities
10. Multibunch instabilities
11. Numerical methods of instability studies
12. Summary
R. Nagaoka An Overview of Collective Effects in 3rd Generation Light Sources …
At JAI, Oxford, 03 February 2011 02/26
R. Nagaoka An Overview of Collective Effects in 3rd Generation Light
Sources …At JAI, Oxford, 03 February 2011 03/26
1. I ntroduction
Higher accelerator perf ormance
Common demand f or a higher beam current
“Luminosity”, “Brilliance”
Single particle motion and the external guide fi eld Collective f orce = Whatever else influencing the single particle motion
= Due to wake fi elds, beam-ion interactions, … Collective f orce Collective motion Beam instability
Beam instability must be avoided to achieve the designed machine perf ormance
What could be the origins of collective forces?
(Resonant) interactions with self-induced EM fields (resistive-wall/geometric/CSR)
Beam-ion interaction
(YH. Chin, “Experimental study of FBII at PLS”, BIW 2000)
(AW Chao, “Physics of collective beam instabilities…”)
R. Nagaoka An Overview of Collective Effects in 3rd Generation Light Sources …
At JAI, Oxford, 03 February 2011 04/26
Why do they become an issue for 3rd generation light sources?
High average/bunch current aimed
Small aperture all around the ring (low gap ID sections/higher magnetic fields)
0
5
10
15
20
25
30
0 2 4 6 8 10
E [GeV]
b0
[m
m]
SOLEIL
SLSBESSY
ALS
ELETTRA
NSLS
ESRF
APS
SPring8
E*b0^3 = const
Vertical half aperture (standard) of some light sources
10 mm gap ID (Insertion Device) chamber at SOLEIL
Low emittance optics and its consequence on instability
Low dispersion low Short bunch length Wider spectra
Stronger interactions with high frequency wakes
R. Nagaoka An Overview of Collective Effects in 3rd Generation Light Sources …
At JAI, Oxford, 03 February 2011 05/26
The present talk mainly f ocuses on collective eff ects due to wake fi elds I mpedance (wake fi eld) describes coupling between the beam and its environment
thus becomes the main ingredient (input) f or instability studies I nstability exists in both longitudinal and transverse
Short-range wakes induce single bunch instabilities Long-range wakes induce multibunch instabilities Forms a “2×2 problem”
R. Nagaoka An Overview of Collective Effects in 3rd Generation Light Sources …
At JAI, Oxford, 03 February 2011 06/26
2. I nduced EM self- field What is space charge f orce?
This was an issue f or low energy proton rings
ra
eEr
022
ra
evH
22
ws Es
abeE
204
)/ln21(
Laslett tune shif t and space charge limit
I ncoherent (mean fi eld) eff ect created by a collective motion
ra
eevBeEF rr
20
2
2 1
2
322
0
24
1
Qa
RNrdskQ
R. Nagaoka An Overview of Collective Effects in 3rd Generation Light Sources …
At JAI, Oxford, 03 February 2011 07/26
I ts important dependence on energy
Almost perf ect cancellation of electric and magnetic f orces f or high energy beams
Self -fi eld of a relativistic particle is Lorentz contracted (angular spread ~-1)
What then breaks this symmetry f or relativistic beams?
Resistive-wall
Beam pipe cross section variations (geometric wakes)
(AW Chao, “Physics of collective beam instabilities…”)
R. Nagaoka An Overview of Collective Effects in 3rd Generation Light Sources …
At JAI, Oxford, 03 February 2011 08/26
3. Notion of Wake field
Mathematical (rigorous) defi nition
dzc
zEW ) (z,
q
1 )( z
)'( )'( ' )( WdeV Superposition to get the f orce (wake potential)
I llustration using the resistive-wall à la A. Chao
Decomposition of beam into azimuthal modes
Analytical solutions f ound
m = 0 longitudinal and m = 1 transverse
0mm
marctsm cos)()(~
(AW Chao, “Physics of collective beam instabilities…”)
R. Nagaoka An Overview of Collective Effects in 3rd Generation Light Sources …
At JAI, Oxford, 03 February 2011 09/26
Large contribution of resistive-wall wakes in light sources
Cubic dependence on the chamber radius
Presence of many low gap sections
Low emittance optics (beta values, symmetry, …)
Asymmetry of the chamber cross section
New (incoherent) detuning eff ect
Some basic characteristics of wake f unctions
cosine like f or L and sine like f or T
Fundamental theorem of beam loading 0)(2
1ctzzqbyseenz EE
Polarity of the wake always hurts a short bunch
(AW Chao, “Physics of collective beam instabilities…”)
R. Nagaoka An Overview of Collective Effects in 3rd Generation Light Sources …
At JAI, Oxford, 03 February 2011 10/26
4. Geometric wake field and numerical (GdfidL) calculations
Numerical solution of Maxwell equations in time/ f requency domains
Stream of developments (TBCI , URMEL, ABCI , MAFI A, Gdfi dL,…)
Numerical diffi culties
I mportance of short-range (high
f requency) interaction:
- I mpedance may extend to tens of GHz
- Bunches are short in reality
Wake fi elds are obtained in an indirect way:
- Wake potentials are calculated
- I mpedance is obtained by dividing the
Fourier transf orm by the bunch spectrum
R. Nagaoka An Overview of Collective Effects in 3rd Generation Light Sources …
At JAI, Oxford, 03 February 2011 11/26
3-dimensional structure (no simplifi cation using symmetry & 3D eff ects)
A huge memory size required due to small mesh sizes
Non-smoothness due to meshing brings about artifi cial wakes (cf . tapers)
At SOLEI L, a parallel processing version Gdfi dL is used on the cluster
Big contributors in light sources
Tapers (due to low gap sections)
RF shielded bellows/ Flanges/ BPMs
R. Nagaoka An Overview of Collective Effects in 3rd Generation Light Sources …
At JAI, Oxford, 03 February 2011 12/26
5. Impedance
I ts defi nition: Fourier transform of the wake f unction
deWZ i
)( )( //// , deW-iZ i
)( )(
Equivalence of description using
Wake f unction (time domain) and
I mpedance (f requency domain)
… Of ten easier physical interpretation in terms
of impedance Properties of the impedance
Resistive versus reactive part
I nductive versus capacitive part
Broadband versus narrow band
(From JL Laclare’s lecture note)
R. Nagaoka An Overview of Collective Effects in 3rd Generation Light Sources …
At JAI, Oxford, 03 February 2011 13/26
Some example f rom Gdfi dL calculations
Booster bellows-3
-2
-1
0
1
2
3
0 4 8 12 16f [GHz]
ZT
[k
/m]
ReZT
ImZT
0
10
20
30
40
50
60
70
0 5 10 15 20 25 30
f [GHz ]
ZL
[oh
m]
SOLEIL Flange
(typical example of broadband) (typical example of narrowband)
Higher cutoff s f or modern chambers and needs of knowledge f or higher f requencies due
to short bunches
R. Nagaoka An Overview of Collective Effects in 3rd Generation Light Sources …
At JAI, Oxford, 03 February 2011 14/26
6. Beam spectra
Single particle motion and its spectrum
Time domain signal
)2
(),(00
//
k
tetsk
and )(),(),( // txtsts
Synchrotron and betatron motions
)cos(ˆ)( 00 tt s and ])(cos[ˆ)( 000 tQxtx ( 00Q )
Single particle spectra (Fourier transform)
)()ˆ(2
),( 00)(
0,
0//
0s
mpjm
mp
m mpepJje
s
..])([]ˆ))[((ˆ4
),( )(00000
,
0 00 ccemQpQpJjexe
s pmjsm
mp
mj
NB The role of chromaticity in shif ting the spectrum
R. Nagaoka An Overview of Collective Effects in 3rd Generation Light Sources …
At JAI, Oxford, 03 February 2011 15/26
Bunch spectrum
Superposition of single particle signals with a certain distribution f unction
00//// ˆˆ),ˆ,(),(),( ddttsNtS
0000 ˆˆˆ̂),ˆ,,ˆ,(),(),( ddxddxtxtsNtS - Distribution f unctions of ten used: Gaussian, parabolic, water-bag, …
Notion of perturbation and coherent instability
tjmjm
cmeexhxfgtx )(0000
00)ˆ,ˆ()ˆ()ˆ(),ˆ,,ˆ,( - Mode-decoupled (weak instability) and mode-coupled (strong instability) regimes
R. Nagaoka An Overview of Collective Effects in 3rd Generation Light Sources …
At JAI, Oxford, 03 February 2011 16/26
7. Equation of collective motion Follow the evolution of beam collective motions Use of Vlasov (Collision-f ree Boltzmann) equation
0 )(
vdivt
Formalism developed by F. Sacherer and others in the ‘70s 0 and linearisation w.r.t.
Equations are usually solved in the f requency domain
Explicit f orms of equations
Longitudinal
ˆ
)ˆ(
/2 )ˆ( ˆ)( 0
g
eE
Imgjmj
sm
msc
'ˆ'ˆ)'ˆ()'ˆ()ˆ( )(
'00
''
0',
// dgpJjpJp
pZmm
mm
mp
R. Nagaoka An Overview of Collective Effects in 3rd Generation Light Sources …
At JAI, Oxford, 03 February 2011 17/26
Transverse
]ˆ))[(( )( /2
)ˆ(ˆ )( 0',
QpJjpZ
eQE
cIxmj m
m
mpmsc
'ˆ'ˆ)'ˆ(ˆ )'ˆ( ]'ˆ))[(( '000
'' dxgQpJj mm
m
- Complex and multidimensional eigenvalue problem - Appearance of ˆ/)ˆ(0 g and ppZ /)(// in the longitudinal equation
- Shif t of beam spectra by in the transverse equation
Diff erent solution procedure according to the nature of instability Weak instability regime (low intensity bunch current, multibunch,…)
- Solution on a single mode (complete decoupling) Strong instability regime (TMCI , …)
- Coupling of neighbouring modes taken into account Very strong instability regime (Microwave, post headtail,…)
- All modes retained or no modal decomposition
R. Nagaoka An Overview of Collective Effects in 3rd Generation Light Sources …
At JAI, Oxford, 03 February 2011 18/26
8. Beam spectra overlap with impedance Basic importance of the notion in interpreting instabilities
Cancellation between damping and anti-damping contributions Role of chromaticity in enhancing the asymmetry in transverse motions
- Positive shif ts have the contrary eff ect to negative ones
- A slightly positive is traditionally said to be optimal Q-dependence in the resistive-wall instability
R. Nagaoka An Overview of Collective Effects in 3rd Generation Light Sources …
At JAI, Oxford, 03 February 2011 19/26
Eff ective impedance and 0// / nZ
d
dZ
Zeff2
2
)(
)( )(
represents the eff ective impedance seen by the beam
0// / nZ indicates the total inductive impedance in the longitudinal plane Evolution of bunch spectra with instability
Associated with bunch lengthening What observed in microwave and post
headtail instability studies at the ESRF The beam tends to have the
maximum overlap with the impedance
Beam spectra (eigen solutions)
0
1
2
3
4
5
6
-60 -40 -20 0 20 40 60
Frequency [GHz]
Ts/ = 0.6
Ts/ = 0.06
Ts/ = 1.26
f = 13.5 GHz
Ts/ = 2.5Ts/ = 15
R. Nagaoka An Overview of Collective Effects in 3rd Generation Light Sources …
At JAI, Oxford, 03 February 2011 20/26
9. Single bunch instabilities I nteraction with inductive impedance at low f requencies
Longitudinal: Bunch lengthening and
tune spread in the PWD regime Transverse: Detuning of the dipolar
(m = 0) mode
Vrf = 8 MV, = (0.13, 0.08)
0.38
0.382
0.384
0.386
0.388
0.39
0 0.2 0.4 0.6 0.8 1
I [mA]
Ver
tica
l Tun
e m = 0
m = -1
I nteraction with resistive impedance (at high f requencies)
Longitudinal: Microwave instability Transverse: Headtail, TMCI and
post-headtail instabilities
R. Nagaoka An Overview of Collective Effects in 3rd Generation Light Sources …
At JAI, Oxford, 03 February 2011 21/26
10. Multibunch instabilities Cavity HOMs are traditionally the principal sources of MBI s LMBI s infl uence the operation in many light sources
Cavity temperature regulation and f eedback applied May associate large energy spread that spoils the brilliance of a light source TMBI s are of ten hidden behind LMBI s
R. Nagaoka An Overview of Collective Effects in 3rd Generation Light Sources …
At JAI, Oxford, 03 February 2011 22/26
TMBI s due to resistive-wall tend to be serious in light sources Large chromaticity applied at the ESRF f or prevention Feedback envisaged to be necessary f or SOLEI L
0
100
200
300
400
500
0 20 40 60 80 100
Coupled-bunch modes
Th
resh
old
cu
rren
t [m
A]
Vertical
Horizontal
Zero chromaticityRW only
No in-vacuum IDs
For high current machines (light sources/ colliders), MBI s may be induced due to Other narrow-band objects (fl anges, BPMs, pumping slots, …) Beam-ion interaction
R. Nagaoka An Overview of Collective Effects in 3rd Generation Light Sources …
At JAI, Oxford, 03 February 2011 23/26
11. Numerical methods of instability studies Solution of Haissinski’s equation f or bunch lengthening
'00
2
22
22
2
0 ] )'"( )"(" ' 2
exp[ )(
WddET
LeA s
Bunch length of the self -consistent solution grows as I 1/ 3
Solution of Vlasov-Sacherer’s equation in f requency domain
-2.0
-1.5
-1.0
-0.5
0.0
0.5
1.0
0.0 0.2 0.4 0.6 0.8 1.0
Single Bunch Current [mA]
(T
une
Shif
t)/Q
s
Examples f or microwave (lef t) and TMCI (right)
R. Nagaoka An Overview of Collective Effects in 3rd Generation Light Sources …
At JAI, Oxford, 03 February 2011 24/26
Tracking codes in time domain Example: Single-turn transf ormations f or the transverse single bunch tracking
Advantages and disadvantages of each method
Frequency domain Easier correspondence with theory and interpretation. More diffi cult to handle docoherence, coupling among L/ H/ V and beam fi llings
Time domain Easier simulation of the reality.
Longer cpu times in general. A lot of post-processing f or interpretations
R. Nagaoka An Overview of Collective Effects in 3rd Generation Light Sources …
At JAI, Oxford, 03 February 2011 25/26
R. Nagaoka An Overview of Collective Effects in 3rd Generation Light
Sources …At JAI, Oxford, 03 February 2011 26/26
12. Summary For 3rd generation light sources, maximising the current of the circulating
beam is one of the keys to raising their perf ormance (i.e. brilliance). There are however several mechanisms that render a high beam current
collectively unstable. These instabilities exist in all situations: (single bunch, multibunch) (transverse, longitudinal). A series of methods developed to analyse and help counteract on them. More complicated and/ or new regimes of instabilities appear as we pursue the
limit of performance, requiring us to make new studies and development.