Electron cloud simulations for SuperKEKB

Y.Susaki,KEK-ACCL9 Feb, 2010

KEK seminar

1. Positron beam emits synchrotron radiation2. Electrons are produced at the chamber wall by

photoemission3. The electrons are attracted and interact with the positron

beam4. The electrons are absorbed at the chamber wall after

several 10 nso Secondary electrons are emitted according the

circumferences5. The electrons are supplied continuously for

multi-bunch operation with a narrow spacing

Electron cloud is built up

Electron cloud built-up K.Ohmi, Phys.Rev.Lett,75,1526 (1995)

e-

γSecondary e-e+ beam

y

x

Wake field is left behind in the electron cloud by advanced bunches

The wake field induced by the electron cloud affect backward bunches

Coherent instability occurs when there is resonance between the wake field and the backward bunches • Coupled bunch instability• Single bunch instability

Coherent instability due to electron cloud

e-

Coupled bunch instability

The wake field causes correlations among bunches

Threshold is determined by balance with some damping effects

Independent of emittance, momentum compaction Depends on electron cloud density, distribution and motion

e-

Single bunch instability

The wake filed causes correlations among positrons within a single bunch

Threshold is determined by the balance with Landau damping due to the momentum compaction factor

Depends on emittanceDepends on only local electron cloud density 

e+

e+ e-

The list of parameters

Unit SuperKEKB SuperB

E+/E- GeV 4/7 4/7

I+/I- Amp 3.6/2.6 2.7/2.7

Np ×1010 6.25 4.53/4.53

Nbun 2500 1740

Ibunch mA 1.4/1.0 1.6/1.6

β x,y ave m 12 12

νs Hz 0.012

εx nm 3.2/1.7 2.8/1.6

εy pm 12.8/8.2 7/4

σx mm 0.20/0.14 0.18/0.13

σy μm 12.3/9.9 9.1/6.9

σz mm 6/5 5/5

L m 3016 1400

Number of the produced electrons(1)

Number of the photons emitted by one positron per unit

meter

• SuperKEKB-LER γ=8000, L=3016 　→　 Yγ=0.17 m-1

Bunch population

• SuperKEKB-LER design (3.6A) Np=1011

The quantum efficiency for photoelectrons (np.e./nγ) 　

Energy distribution 10±5 eV

€

Yγ =5π

3

αγ

L α= 1/137 (fine structure const.)

€

η =0.1

Number of the produced electrons(2)

Number of electrons produced by one positron per unit

meter

• SuperKEKB-LER Yp.e= Yϒ η = 0.017 m-1

 Number of electrons produced by one bunch per unit

meter

Maximum secondary emission yield

€

δ2,max =1.0 ~ 1.2€

Yp,eN p =1.7 ×109 m−1

Analysises for coupled bunch instabilities

Increase of electron density re with multi-bunch (simulation results)

• d2,max=1.2 Yp,eNp=1.7×109 (h=0.1) Yp,eNp=1.7×108 (h=0.01) Yp,eNp=1.7×107 (h=0.001)

•  Yp,eNp=1.7×107 (h=0.001) d2,max=1.2 d2,max=1.1 d2,max=1.0

• Theηshould be reduced to 0.001!

Electron density re as functions of quantum efficiencies (h and d2,max)

d2,max=1.2  Yp,eNp=1.7×107 (h=0.001)

η=0.003 w/ antechamber (a simulation result) ρeth=1.1✕1011m-3

The analytical value of the threshold in the case of SBI

Together with solenoid it is expected to reduce the actual η to 0.001 (Suetsugu)

Effect of antechamber

©Suetsugu

Electron distribution and electric potential with d2,max=1.2

• Antechamber

• Cylindrical chamber

Electron distribution and electric potential with d2,max=0

• Antechamber

• Cylindrical chamber

Reduction factor forthe averaged electron density

• The ratio of the densities at the beam pipe of ante-chamber and cylindrical-chamber

• The ratio ≈ 0.03 for δ2,max=0

The antechamber reduces η in 3% effectively!

Wake field induced by electron cloud and beam stability(1)

Coasting beam model• We assume a homogenous stream of the beam• We can apply this model even for bunches if

ωeσz/c>>1

• The position of the center of mass in the transverse direction :

€

yb (s, t)

y

s

Wake field induced by electron cloud and beam stability(2)

EOM for the beam and the cloud

• nb,c : line density of each particle

• rb,c : classical radius of each particle

• F becomes linear near the beam €

∂∂t

+ c∂

∂s

⎛

⎝ ⎜

⎞

⎠ ⎟2

yb (s, t) + ωβ2 yb (s, t) = −

2ncrbc2

γF yb (s, t) − yc (s, t)( )

d2yc (s, t)

dt 2= −2nbrcc

2F yc (s, t) − yb (s, t)( )

€

F(y) =y

σ y σ x + σ y( )

y

s

the betatron oscillation

Wake field induced by electron cloud and beam stability(3)

The eq. for the cloud can be solved as

The eq. for the beam becomes

€

∂∂t

+ c∂

∂s

⎛

⎝ ⎜

⎞

⎠ ⎟2

yb (s, t) + ˜ ω β2 yb (s, t) = ωb

2ωc yb s, t'( )sinωc t − t '( )dt't0

t

∫

˜ ω β2 = ωβ

2 + ωb2 wake force

wake field

€

yc = ωc yb s, t'( )sinωc t − t '( )dt't0

t

∫

€

ωc2 =

2nbrcc2

σ y σ x + σ y( )

€

ωb2 =

2ncrbc2

σ y σ x + σ y( )tune shift Δωβ

Wake field induced by electron cloud and beam stability(4)

• Fourier trans. of the eq. for the beam leads

• Growth rate of instability = Im ω

€

−ω −mω0( )2

+ ωβ2 =

nbrbc2

γT0

Z⊥ ω( )

m : modes

Z⊥ ω( ) = i W t( )−∞

∞

∫ exp −iωt( )dt

Wake field for bunch correlation

Y1eNp=1.7×109 m-1(h=0.1) 1.7×108 m-1(h=0.01) 1.7×107 m-1(h=0.001)

Unstable modes and growth rateY1eNp=1.7×109 m-1(h=0.1) 1.7×108 m-1(h=0.01) 1.7×107 m-1(h=0.001)

Growth rate as a function of ηWe evaluate the growth rate associated with the unstable

modes as a function of η– The growth rate is 0.02 for η=0.001

(Note that η=0.001 corresponds to the threshold of the SBI when η is evaluated as the function of the electron density)

– not so severe that the growth could be suppressed by the feedback from the empirical point of viewThe CBI could be avoided below the threshold of the SBI

Analysis for single bunch instabilities

Stability condition for the single bunch instability

Landau dampingCoherence of the transverse oscillation is weakened by the longitudinal oscillation associated with momentum compaction

The stability condition for ωeσz/c>1•Balance of growth and Landau damping

€

Imωe < 0⇒ U =3λ pr0β

ν sγωeσ z /c

Z⊥ ωe( )

Z0

<1

ωe =λ prec

2

σ z σ x + σ y( )

Threshold of the single bunch instability

Threshold of the electron cloud density

• Qnl depends on the nonlinear interaction

• K characterizes cloud size effect and pinching

• ωeσz/c>10 for low emittance rings

• We use K=ωeσz/c and Qnl =7 for analytical estimation

€

ρe,th =2γν sωeσ z /c

3KQr0βL

€

Q = min Qnl ,ωeσ z /c( )

Threshold for SuperKEKB and SuperB

 

Unit SuperKEKB SuperB

L m 3016 1400

γ 8000 8000

I+/I- Amp 3.6/2.6 2.7/2.7

Np ×1010 6.25 4.53

Ibunch mA 1.4/1.0 1.6/1.6

β x,y ave m 12 12

νs Hz 0.012

σx mm 0.20/0.14 0.18/0.13

σy μm 12.3/9.9 9.1/6.9

σz mm 6/5 5/5

Q 7 7

ωeσz/c 10.9

radiation damping time

ms(turn) 30(3000) 20(4600)

ρe threshold ×1011m-3 1.13

Simulation with Particle In Cell Methodfor the single bunch instability• Electron clouds are put at several positions in a ring• Beam-cloud interaction is calculated by solving  two-

dimensional Poisson equation on the transverse plane• A bunch is sliced into 20-30 pieces along the longitudinal

direction

e+e-

large enough for describing the oscillations

Simulations for instability threshold for SuperKEKB

Profiles of the beam size

ηy=0.2ηy=0

ρe,th≈2.4×1011m-3 ρe,th≈2.2×1011m-3

Bunch and e-cloud profiles at 4000 turn

Coherent motions for SuperKEKB

ηy=0.2ηy=0

FFT spectra below and above the threshold

Unstable modes of the instability for SuperKEKB

stablestable

unstableunstable

ηy=0.2ηy=0

Simulations for instability threshold for SuperBProfiles of the beam size

ηy=0 ηy=0.2

ρe,th≈4.4×1011m-3 ρe,th≈2.6×1011m-3

Bunch and e-cloud profiles at 4000 turn

Coherent motions for SuperB

ηy=0.2ηy=0

FFT spectra below and above the threshold

Unstable modes of the instability for SuperB

stablestable

unstableunstable

ηy=0.2ηy=0

Summary

Multi-bunch numerical simulation • The effective quantum efficiency η should be reduced to 0.001• The antechember alone seems not to be sufficient for achieving

η=0.001, but together with solenoid it is expected to cure the situation (Suetsugu)

• The CBI seems not to be severe with η=0.001

Single bunch numerical simulation• The threshold of the electron cloud density for the stability has

been estimated for SuperKEKB, SuperB