electron cloud simulations for superkekb y.susaki,kek-accl 9 feb, 2010 kek seminar
TRANSCRIPT
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