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…”)



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



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”



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



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)




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)()(~




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




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



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



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)



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



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



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



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



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



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



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



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



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



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



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)



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 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.

Documents

An Overview of Collective Effects in 3rd Generation Light Sources