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
1. Introduction
 Higher accelerator performance
 Common demand for a higher beam current
“Luminosity”, “Brilliance”
 Single particle motion and the external guide field
 Collective force = Whatever else influencing the single particle motion
= Due to wake fields, beam-ion interactions, …
 Collective force  Collective motion  Beam instability
 Beam instability must be avoided to achieve the designed machine performance
R. Nagaoka
An Overview of Collective Effects in 3rd Generation Light Sources …
At JAI, Oxford, 03 February 2011 03/26
 What could be the origins of collective forces?
 (Resonant) interactions with self-induced EM fields (resistive-wall/geometric/CSR)
(AW Chao, “Physics of collective beam instabilities…”)
 Beam-ion interaction
(YH. Chin, “Experimental study of FBII at PLS”, BIW 2000)
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)
30
ELETTRA
APS
b0 [mm]
25
NSLS
ALS
20
SPring8
BESSY
SLS
15
10
ESRF
SOLEIL
5
E*b0^3 = const
0
10 mm gap ID (Insertion Device) chamber at SOLEIL
0
2
4
6
8
10
E [GeV]
Vertical half aperture (standard) of some light
sources
 Low emittance optics and its consequence on instability
Low dispersion  low a  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 focuses on collective effects due to wake fields
 Impedance (wake field) describes coupling between the beam and its environment
 thus becomes the main ingredient (input) for instability studies
 Instability 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. Induced EM self-field
 What is space charge force?
 This was an issue for low energy proton rings
Er 
e
2a 2 0
ev
H 
r
2
2a
r
Es  
e(1  2 ln b / a ) 
 Ew
s
4 0  2
 Laslett tune shift and space charge limit
 Incoherent (mean field) effect created by a collective motion
e 2 1
Fr  eE r  evB 
r
2a 2 0  2
Q 
R. Nagaoka
Nr0 R
1
k ds  

4
2a 2 Q  2  3
An Overview of Collective Effects in 3rd Generation Light Sources …
At JAI, Oxford, 03 February 2011 07/26
 Its important dependence on energy
 Almost perfect cancellation of electric and magnetic forces for high energy beams
 Self-field of a relativistic particle is Lorentz contracted (angular spread ~-1)
 What then breaks this symmetry for 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
(AW Chao, “Physics of collective beam instabilities…”)
 Mathematical (rigorous) definition
1 
z
W ( ) 
 E z (z,   )dz
q 
c

V ( )  e  d '  ( ' ) W ( ' )

Superposition to get the force (wake potential)
 Illustration using the resistive-wall à la A. Chao
 Decomposition of beam into azimuthal modes
 Analytical solutions found
 m = 0 longitudinal and m = 1 transverse
R. Nagaoka
 


m 0
m
 m ~  ( s  ct )  (r  a ) cos m
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 effect
 Some basic characteristics of wake functions
 cosine like for L and sine like for T
 Fundamental theorem of beam loading
Ez
seen by q

1
Ez
2
( z ct )0 
 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/frequency domains
 Stream of developments (TBCI, URMEL, ABCI, MAFIA, GdfidL,…)
 Numerical difficulties
 Importance of short-range (high
frequency) interaction:
- Impedance may extend to tens of GHz
- Bunches are short in reality
 Wake fields are obtained in an indirect way:
- Wake potentials are calculated
- Impedance is obtained by dividing the
Fourier transform 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 simplification using symmetry & 3D effects)
 A huge memory size required due to small mesh sizes
 Non-smoothness due to meshing brings about artificial wakes (cf. tapers)
 At SOLEIL, a parallel processing version GdfidL 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
 Its definition: Fourier transform of the wake function
Z // ( ) 

i
 W// ( ) e d ,


Z  ( )  -i  W ( ) e i d

 Equivalence of description using
 Wake function (time domain) and
 Impedance (frequency domain)
… Often easier physical interpretation in terms
of impedance
 Properties of the impedance
 Resistive versus reactive part
 Inductive 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 from GdfidL calculations
70
3
ReZT
2
SOLEIL Flange
60
ImZT
ZL [ohm]
ZT [k/m]
50
1
0
40
30
-1
20
-2
10
Booster bellows
0
-3
0
4
8
12
16
0
5
10
15
20
25
f [GHz]
f [GHz ]
(typical example of broadband)
(typical example of narrowband)
30
 Higher cutoffs for modern chambers and needs of knowledge for higher frequencies 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

s // (t , )  e   (t   
k  
 2k

)
0 0
and
s  (t , )  s // (t , )  x (t )
 Synchrotron and betatron motions
and
 (t )  ˆ  cos( s 0 t  0)
x(t )  xˆ  cos[Q0 0 (t   )       0 ] (    Q0 0

)

 Single particle spectra (Fourier transform)
e 0
s // ( , ) 
2
s  ( , ) 

 j ( p  m 0 )
 (  p 0  m s 0 )
 j m J m ( p 0ˆ)e
p , m  
e 0 j0
xˆe
4

 j m J m [(( p  Q0 ) 0   )ˆ]   [  ( p  Q0 ) 0  m s 0 ]e
p ,m  
j ( m 0  p )
 c.c.
NB The role of chromaticity in shifting 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 function
S // (t , )  N  s // (t , )   ( 0 ,ˆ, t )ˆdˆd 0
S  (t , )  N  s  (t , )   ( 0 ,ˆ,  0 , xˆ , t )ˆxˆdˆdxˆd 0 d 0
- Distribution functions often used: Gaussian, parabolic, water-bag, …
 Notion of perturbation and coherent instability
 ( 0 ,ˆ,  0 , xˆ , t )  g 0 (ˆ)  f 0 ( xˆ )  hm (ˆ, xˆ )e  j (0  m 0 )  e jcmt
- 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-free Boltzmann) equation


 div(v )  0
t
 Formalism developed by F. Sacherer and others in the ‘70s
   0   and linearisation w.r.t. 
 Equations are usually solved in the frequency domain
 Explicit forms of equations
 Longitudinal
j ( c  m s ) j mˆ g m (ˆ)  
maI g 0 (ˆ)
2 s E / e ˆ

Z // ( p )
m'
 
J m ( p 0ˆ) j  J m ' ( p 0ˆ' ) g m ' (ˆ' )ˆ' dˆ'
p
p ,m ' 
0

R. Nagaoka
An Overview of Collective Effects in 3rd Generation Light Sources …
At JAI, Oxford, 03 February 2011 17/26
 Transverse
j ( c     m s ) xˆ m (ˆ)  
cI
2QE / e
 j
m'

 Z  ( p) j m J m [(( p  Q) 0   )ˆ]
p , m ' 

 J m' [(( p  Q) 0    )ˆ' ] g 0 (ˆ' ) xˆ m ' (ˆ' )ˆ' dˆ'
0
- Complex and multidimensional eigenvalue problem
- Appearance of g 0 (ˆ) / ˆ and Z // ( p ) / p in the longitudinal equation
- Shift of beam spectra by   in the transverse equation
 Different 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  shifts have the contrary effect 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
 Effective impedance and Z // / n 0

Z eff 
2
 Z ( )  ( ) d


represents the effective impedance seen by the beam
2
  ( ) d

Z // / n 0 indicates the total inductive impedance in the longitudinal plane
 Evolution of bunch spectra with instability
 Associated with bunch lengthening
Beam spectra (eigen solutions)
 What observed in microwave and post
6
f  = 13.5 GHz
5
headtail instability studies at the ESRF
4
 The beam tends to have the
Ts/  = 0.6
3
maximum overlap with the impedance
Ts/  = 0.06
2
Ts/  = 1.26
1
Ts/  = 2.5
Ts/  = 15
0
-60
-40
-20
0
20
40
60
Frequency [GHz]
R. Nagaoka
An Overview of Collective Effects in 3rd Generation Light Sources …
At JAI, Oxford, 03 February 2011 20/26
9. Single bunch instabilities
 Interaction with inductive impedance at low frequencies
 Transverse: Detuning of the dipolar
(m = 0) mode
Vrf = 8 MV,  = (0.13, 0.08)
0.39
Vertical Tune
 Longitudinal: Bunch lengthening and
tune spread in the PWD regime
0.388
m =0
0.386
0.384
m = -1
0.382
0.38
0
0.2
0.4
0.6
0.8
1
I [mA]
 Interaction with resistive impedance (at high frequencies)
 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 MBIs
 LMBIs influence the operation in many light sources
 Cavity temperature regulation and feedback applied
 May associate large energy spread that spoils the brilliance of a light source
 TMBIs are often hidden behind LMBIs
R. Nagaoka
An Overview of Collective Effects in 3rd Generation Light Sources …
At JAI, Oxford, 03 February 2011 22/26
 TMBIs due to resistive-wall tend to be serious in light sources
 Large chromaticity applied at the ESRF for prevention
 Feedback envisaged to be necessary for SOLEIL
Threshold current [mA]
500
400
300
Zero chromaticity
RW only
200
No in-vacuum
IDs
Vertical
Horizontal
100
0
0
20
40
60
80
100
Coupled-bunch modes
 For high current machines (light sources/colliders), MBIs may be induced due to
 Other narrow-band objects (flanges, 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 for bunch lengthening

 s2
e2L 
2
 ( )  A0 exp[  2 2  
d

'

 d "  ( " ) W ( " ' ) ]
2
2a  
a  ET0 0
'
 Bunch length of the self-consistent solution grows as  I1/3
 Solution of Vlasov-Sacherer’s equation in frequency domain
1.0
(Tune Shift)/Qs
0.5
0.0
-0.5
-1.0
-1.5
-2.0
0.0
0.2
0.4
0.6
0.8
1.0
Single Bunch Current [mA]
 Examples for microwave (left) 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 transformations for the transverse single bunch tracking
 Advantages and disadvantages of each method
 Frequency domain Easier correspondence with theory and interpretation.
More difficult to handle docoherence, coupling among L/H/V and beam fillings
 Time domain Easier simulation of the reality.
Longer cpu times in general. A lot of post-processing for interpretations
R. Nagaoka
An Overview of Collective Effects in 3rd Generation Light Sources …
At JAI, Oxford, 03 February 2011 25/26
12. Summary
 For 3rd generation light sources, maximising the current of the circulating
beam is one of the keys to raising their performance (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.
R. Nagaoka
An Overview of Collective Effects in 3rd Generation Light Sources …
At JAI, Oxford, 03 February 2011 26/26