Collective Effects and Instabilities

G. Wang

Outline• Introduction

– Collective effects, collective instabilities• Wakefields and Impedances (Ultra-relativistic

model)– Wake functions– Panofsky-Wenzel theorem– Cylindrical symmetric structure– Wake potentials, loss factor and kick factor– Impedances

• Single bunch beam breakup (Two particle model)

What are collective effects?• In the single particle dynamics, the E&M fields due to the

charged particle themselves are neglected when considering their motions.

• As the number of the charged increases, the particles’ own fields (and fields induced by them) can start to affect its behavior, which is generally called the collective effects.

IBS, Touschek effects or ion cloud

Collective instabilities• The particle beam interacts with its surroundings to generate an

electromagnetic field, known as wakefield. This field then acts back on the beam, perturbing its motion.

• Under unfavorable conditions, the perturbation on the beam are continously enhanced by the wakefield, leading to the collective instabilities. First turn Second turn Third turn

Example 1: multi-pass BBU in ERL

Example 2: single bunch BBU

• For the rest of the lecture, we will focus on a wakefield model developed for an ultra-relativistic beam, 1γ >>

Ultra-relativistic beam and cylindrical perfect conducting beam pipe

For → ∞, interaction among the particles and their images from the wall vanishes if1. the wall is perfectly conducting, and2. there are no discontinuities (cavities, bpms, bellows…).(It is also assumed that particles go straight, i.e. no radiations from particles)

At the limit of → ∞ (Homework)2 *3

04qRE

Rπε γ=

R*2 = s2 + x2 /γ 2

Fz = qEz = −qes

4πε0γ2 s2 + x2 /γ 2( )3/2

( )0

ˆ2

qrE z ctr

δπε

= −

Fx = q Ex − cβBy( ) = qex4πε0γ

4 s2 + x2 /γ 2( )3/2

qes vt z= −

Wake Functions• Rigid bunch approximation:the motion of particles is not affected while passing through the structure • Impulse approximation:instead of the detailed E&M field in the structure, we care more about the total momentum change to the particles due to the wake field:

Longitudinal wake function*:

Transverse wake function*:

* These definition follow from ‘Impedances and Wakes in High-Energy Particle Accelerators by B. Zotter, which is different from those in ‘ Physics of Collective Beam Instabilities in High Energy Accelerators’ by A. Chao.

wl x, y, s( ) = −cqe

Δpz = −cq

Ez x, y,ct − s,t( )dt−∞

∞

rq

e

[V/C]

[V/C]

Panofsky-Wenzel Theorem

*The derivation follows from USPAS note by K.Y. Ng.

∇s = x̂∂

∂x+ ŷ ∂

∂y− ẑ ∂

∂s ∇ = x̂∂

∂x+ ŷ ∂

∂y+ ẑ ∂

∂z

( ) ( ), , , ,t lw x y s w x y ss ⊥∂ = ∇∂

We assume the B field due to the structure has limited spatial range, i.e. it is localized.

We want to find relation between longitudinal wake function and transverse wake function due to a structure (a piece of beam pipe, bpm, bellow, cavity....)

This is called Panofsky-Wenzel theorem

r

Another Relation at

( )( )

0

0 00

20

ˆ

z

z

F q E q v B

q qv B

q q v vz Et

qq Ec t

q Ec t

ρερ μ ρ εε

ρ βε γ

∇ ⋅ = ∇ ⋅ + ∇ ⋅ ×

= − ⋅ ∇ ×

∂ = − ⋅ + ∂ ∂= −∂

∂≈ −∂

β → 1

0

E ρε

∇ ⋅ =

0 0B j Etμ ε ∂ ∇ × = + ∂

ˆj vzρ=

Cylindrical symmetric structure I

r

For a system with cylindricalsymmetry, it is usually moreconvenient to decomposequantities into azimuthalmodes:

q

e

xy

r

r’

Cylindrical symmetric structure II

∂∂s

Δpr = −∂∂r

Δpz

∂∂s

Δpθ = −1r

∂∂θ

Δpz

∂∂r

rΔpr( ) = − ∂∂θ Δpθ

∂∂r

rΔpθ( ) = ∂∂θ Δpr

( ) ( ) ( )1 1, , ', ',m mm r m mp r s A r s mr B r s r− − −Δ = +

Cylindrical symmetric structure III

( ) ( ) 1, , 'm mm r mqep r s r W s mrv

−Δ =

*Reference: A. Chao ‘Physics of Collective Beam Instabilities in High Energy Accelerators’, eq. (2.35)

( ) ( ) 1, , ', mm r mp r s A r s mr −Δ =

( ) ( ) 1, ', , ', mm mp r r s A r s mrθ −Δ = −

( ), ' ', mm z mp A r s rΔ = −

Analyzing the source term in the Maxwell equations

( ) ( )', 'mm mqeA r s r W sv

=

By analyzing the source term in the Maxwell equations, it can be shown that the driving term has an explicit dependence on r’

,E B

Cylindrical Symmetric Structure IV

wl r ',r,θ, s( ) = −cqe

Δpz r ',r,θ , s( ) = W 'm s( )r 'm rm cos mθ( )m=0

∞

* In many references (by A. Chao, K.Y. Ng ... ), and are called wake functions.Wm s( ) W 'm s( )

m = 0 wl r ',r,θ,s( ) = W '0 s( )m = 1

wt r ',r,θ,s( ) = 0

wl r ',r,θ,s( ) = W '1 s( )r 'r cos θ( )Positive just after the source->deceleration

s

W 'm s( ) Wm s( )

s

Positive just after the source->deflected at the same directionas source

Wake Potential• In practice, usually only monopole mode (m=0) wake is considered

for longitudinal wake field and only dipole mode (m=1) is considered for transverse mode.

w// s( ) = W '0 s( )w⊥ s( ) = W1 s( )

monopole longitudinal wake:

dipole transverse wake:

V/C

V/(C*m)

• Wake potentials are defined to describe the momentum change induced by all particles in a bunch to a test unit charge:

V// z0( ) =−cΔpz z0( )

eQe= λ z1( )w// z1 − z0( )dz1

z0

∞

z

v

z0

* If we observe at and use arriving time, as longitudinal variables, above definition become

z = z* t = 1c

z* − z( )V// t0( ) = λ t1( )w// t0 − t1( )dt1

−∞

t0

λ z( ) is line number density of a bunch

[V/C]

[V/C]

Loss Factor and Kick Factor• Once the longitudinal wake potential is known, the total

energy change of a bunch to the wakefields is given byΔU = − QeV// z( ) Qeλ z( ) dz

−∞

∞

Charge in slice (z,z+dz)

Potential at slice (z,z+dz)

κ // ≡−ΔUQe

2 = V// z( )λ z( )dz−∞

∞

Definition of Loss Factor:

• Similarly, the total transverse momentum change of a bunch to the wakefields is given by

Transverse momentum changeof a particle at slice (z,z+dz). Particle number in slice

(z,z+dz)

[V/C]

[V/C]

Impedances• Although the time domain description of particle-enviroment interaction, the

wake fields, contains all informations, it is often more convinient to describethe interaction in frequency domain (convolution vs multiplication, calculatewakes in frequency domain can be easier some times, solving beaminstability problems...), i.e. the impedances

• The inverse transformations are

Z // ω( ) =1c

w// s( )eiω s/c ds0

∞

Z⊥ ω( ) = −ic

w⊥ s( )eiω s/c0

∞

ds

[s*V/C]=[Ohm]

[s*V/(C*m)]=[Ohm/m]

*The frequency isfrequently allowed tohave an imaginary part,in that case thetransformation is actuallyLaplace transform, whichis only defined for

ω

Im ω( ) ≥ 0

w// s( ) =1

2πZ // ω( )

−∞

∞

e− iωs/cdω

w⊥ s( ) =i

2πZ⊥

−∞

∞

ω( )e−iω s/cdω

* In complex plane, and should not have singularities in the upper half plane, i.e. , in order to satisfy the causality condition:

ω Z // ω( )Z⊥ ω( )

Im ω( ) ≥ 0

Im ω( )

Re ω( )

w// s < 0( ) = 0 w⊥ s < 0( ) = 0

Properties of Impedances

( ) ( )/ / / /*Z Zω ω= −Re Z // ω( ) = Re Z // −ω( ) Im Z // ω( ) = − Im Z // −ω( )

Z⊥ * ω( ) = −Z⊥ −ω( )Re Z⊥ ω( ) = − Re Z⊥ −ω( ) Im Z⊥ ω( ) = Im Z⊥ −ω( )

• Symmetry properties about positive and negative frequency (Homework)

• Relations between real part and imaginary part of impedances

w// s( ) =1

2πRe Z // ω( ) cos

ω sc

− Im Z // ω( ) sinω sc

−∞

∞

dω

wl s < 0( ) =1

2πRe Zl ω( ) cos

ω sc

+ Im Zl ω( ) sinω s

c

dω−∞

∞

= 0 Im Zl ω( ) sinω s

c

dω−∞

∞

= − Re Zl ω( ) cosω s

c

−∞

∞

dω

wl s > 0( ) =2π

Re Zl ω( ) cosω sc

dω0

∞

w// s( ) =1

2πZ // ω( )

−∞

∞

e− iω s/cdω

w⊥ s > 0( ) =2π

Re Z⊥ ω( ) sinω sc

dω0

∞

Kramers-Kronig relations:

Z // ω( ) = −iπ

P.V . Z // ω '( )ω '− ω

dω '−∞

∞

Re Z// ω( ) =1π

P.V .Im Z // ω '( )

ω '− ωdω '

−∞

∞

Im Z// ω( ) = −1π

P.V .Re Z // ω '( )

ω '− ωdω '

−∞

∞

Single pass BBU (Two particle model)

Ne

/ 2Ne / 2Ne

z

Leading particles

Trailing particles

Single pass BBU II

Many pictures and derivations used in the slides are taken from the following references:

[1] ‘Wake and Impedance’ by G.V. Stupakov, SLAC-PUB-8683;[2] ‘Physics of Intensity Dependent Instabilities’ by K.Y. Ng, Lecture Notes in USPAS 2002;[3] ‘Accelerator Physics’ by S.Y. Lee;[4] ‘Physics of Collective Beam Instabilities in High Energy Accelerators’ by A. Chao;[5] ‘Impedances and Wakes in High-Energy Particle Accelerators’ by B. Zotter and S. Kheifets.

Backup Slides

Homework

• Show that the electric field of an ultra-relativistic charged particle with charge q is given by (Hint: you do not need to derive the delta function, just justify the coefficient.)

• Show that the longitudinal and transverse impedances satisfy the following relations:

( )0

ˆ2

qrE z ctr

δπε

= −

( ) ( )// //*Z Zω ω= − Z⊥ * ω( ) = −Z⊥ −ω( )

Electric and magnetic field from a charge moving with constant velocity

( ) ( )( ) ( )( )

( ) ( ) ( )

( ) ( )( ) ( )( ) ( ) ( )

( ) ( )

3 32 20 0

,4 41 1

, ,

r r rr r

r r r r r r r

static rad

n n t t tn t te eE x tt cR t n t t R t n t t

E x t E x t

β ββπε γ πεβ β

× − ×− = + − ⋅ − ⋅

= +

( )( ) ( )( )

( ) ( ) ( )32 2

0

,4 1

r rstatic

r r r

n t teE x tR t n t t

βπε γ β

−=

− ⋅

Rewriting Static Field I:

( )rR t

( )rr t

( )r t

( )x t

β

( )R t

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( )r

r r r r rr

R t R tr t r t c R t R t R t R t R t n t

c R tβ β β− = = − = − = −

( ) ( )( ) ( )1

rr

r

n t R tn t

R tβ

⋅= − ⋅

θ

( ) ( )r rr

R t R tt t t

c cΔ = − = =

( )( ) ( )( )

( ) ( ) ( )( )

( ) ( )3 32 220 0,

4 41

r rstatic

rr r r

n t t R te eE x tn t R tR t n t t

βπε γ πε γβ

−= =

⋅− ⋅

Rewriting Static Field II:

β

( )R t

θ

( )sin rx

R tθ =

( )( )

222 2 2 2 2 2

2sin rr

xd l R t xR t

θ β β= = =

d

( )22 2 2 2 2 2 2 2 2h R t d x s x x sβ γ −= − = + − = +

l s

x

( ) ( ) ( )r rl r t r t R tβ= − =

( ) ( ) ( )rrR t

r t r t c t cc

β β− = Δ =

h( )rR t

( ) ( ) ( ) ( )( ) ( )cosr

rr

R t R tn t R t R t h

R tφ

⋅⋅ = = =

φ

( ) ( ) 2 2 2rn t R t s x γ −⋅ = +

( ) ( )( ) ( )

( )( )

320

3/22 2 2 20

,4

4

static

r

R teE x tn t R t

R tes x

πε γ

πε γ γ −

= ⋅

=+

( ) ( ) ( )

1 1 sin

1 1 1

1 1sin

static static static

static static static

static static

B n E Ec c

d x xE E Ec R t c R t c R t

E Ec c

φ

β β

β ψ β

= × =

= = =

= = ×

Longitudinal Microwave Instability

ψ 0 z,ΔE( ) =ψ 0 ΔE( ) =NC0

f0 ΔE( )

Unperturbed phase space density:

ρ0 z( ) = ρ0 =NC0

V// z0( ) = λ z1( )w// z1 − z0( )dz1z0

∞

= ρ0 W0' z1 − z0( )dz1

z0

∞

= −ρ0W0 0( ) = 0

DC current does not excite wake

Consider perturbation in phase space density:

ψ 1 z,ΔE,0( ) =ψ̂ 1 ΔE( )einz/Rn-th azimuthal mode

*Note that if a perturbation is static,

ψ 1 * z,ΔE,t( ) =ψ̂ 1 * ΔE( )ein z−v0t( )/R =ψ̂ 1 * ΔE( )einz/R−iΩ*t Ω* = nv0 / R = n2πv0 / C = nω 0But the system is not likely to be static and we need to solve Vlasov equation self-consistently to know the answer for and hence Ω ψ 1 s,ΔE,t( )

ψ 1 z,ΔE,t( ) =ψ̂ 1 ΔE( )einz/R−iΩtAnsatz:

∂∂t

ψ 1 z,ΔE,t( ) +dzdt

⋅ ∂∂z

ψ 1 z,ΔE,t( ) +dΔEdt

⋅ ∂∂ΔE

ψ 0 ΔE( ) = 0

C0 = 2π R

R

dzdt

= v ΔE( )

Longitudinal Microwave Instability

w// τ( ) =1

2πZ // ω( )

−∞

∞

e− iωτ dω

dΔE z,t( )dt

= −cΔpz z,t( )

T0= − e

2vT0

ρ1 z,t − τ( )w// τ( )dτ0

∞

cΔpz z,t( ) = −eQeV// z,t( ) = −e2v0 ρ1 z,t1( )w// t − t1( )dt1−∞

t

= −e2v0 ρ1 z,t − τ( )w// τ( )dτ0

∞

T0 =C0v0

is revolution period

ρ1 z,t( ) = ψ 1 z,ΔE,t( )dΔE−∞

∞

= ρ̂1einz/R−iΩt ρ̂1 ≡ ψ̂ 1 ΔE( )dΔE−∞

∞

dΔE z,t( )dt

= −ρ̂1e2v02πT0

einz/R−iΩt dωZ // ω( )−∞

∞

ei Ω−ω( )τ dτ−∞

∞

= −ρ̂1e2v0T0

einz/R−iΩtZ // Ω( )

ρ1v0dt gives particle number in the slice (t,t+dt).

−iΩψ 1 z,ΔE,t( ) + v ΔE( ) ⋅inR

ψ 1 z,ΔE,t( ) − ρ̂1e2v0T0

einz/R−iΩtZ // Ω( ) ⋅∂

∂ΔEψ 0 ΔE( ) = 0

ψ 1 z,ΔE,t( ) =ie2v0Z // Ω( )

T0ρ̂1e

inz/R−iΩt

Ω − ω ΔE( )ndψ 0 ΔE( )

dΔEdΔE

−∞

∞

→

ω ΔE( ) = v ΔE( )R

Longitudinal Microwave Instability1 = ieI0Z // Ω( )

T0

f0 ' ΔE( )Ω − ω ΔE( )n dΔE−∞

∞

ψ 0 ΔE( ) = NC0 f0 ΔE( )

ω ΔE( ) = ω 0 + Δω ΔE( ) = ω 0 −ηω 0Δpzp0,z

= ω 0 −ηω 0β 2

ΔEE0

η = 1γ t

2 −1

γ 2*Phase slip factor:

f0 ΔE( ) = δ ΔE( )Cold Beam:

Dispersion relation:

1 = ieI0Z // Ω( )T0

ηnω 0E0β

2

f0 ΔE( )

Ω − nω 0 +ηnω 0E0β

2 ΔE

2 dΔE−∞

∞

Ω = nω 0 ±ω 0ieI0ηnZ // Ω( )

2πE0β2 ≈ nω 0 ±ω 0

ieI0ηnZ // nω 0( )2πE0β

2

Perturbative appraoch assuming Ω − nω 0nω 0

Longitudinal Microwave InstabilitiesCold beam continued: Taken from ‘Accelerator Physics’ by S.Y. Lee

Warm Beam:

-

inductive

capacitiveη > 0(assuming ) Ω ≈ nω 0 ± ω 0 ieI0ηnZ// nω 0( )2πE0β 2

f0 ΔE( ) =1

2πσ Eexp − ΔE

2

2σ E2

U '− iV ' ≡eI0 Z // nω 0( ) / n E0β 2

ησ E ,FWHM2

U ' Re Z // nω 0( )( ) V ' − Im Z // nω 0( )( )

Taken from S.Y. Lee

Longitudinal Microwave instabilityGaussian with various growth rate,

Contours with for various energy distribution

Keil-Schnell criterion

Z // nω 0( ) / n ≤ 2π η σ E2

E0β2eI0

F

Simplified estimation for stability condition:

F depends on distributiion and forGaussian energy distribution, it is1.

Typical Longitudinal Impedancej = −i

Resonator model (cavities)

− −

− −

−

−

Taken from ‘Coasting beam longitudinal coherent instabilities’ by J.L. Laclare