Simulation of Beam Instabilities in SPring-8

T. Nakamura

JASRI / SPring-8

nakamura@spring8.or.jp

http://acc-web.spring8.or.jp/~nakamura/

Beam Instabilities

Multi-bunch Long range wake >> Tb (bunch spacing)

Resonator(cavity)Resistive-wallIonElectron Cloud,...

Single-bunchShot range wake << Tb

Geometrical (MAFIA) Resistive wall (Analytical)

Electron CloudCSR

Multi-bunch instabilities by ResonatorTransverse

€

V j t − t j( ) = q j x jR

Qωr

ωr

′ ω re

−α t−t j( ) sin ′ ω r t − t j( ) = q j x j Re Weiλ t−t j( ) ⎡

⎣ ⎢ ⎤ ⎦ ⎥

€

Z =ωr

ω

R

1+ iQωωr

−ωr

ω

⎛

⎝ ⎜

⎞

⎠ ⎟

€

˜ V i+1 = ˜ V ieiλTb + qi+1, j xi+1, jWe−iλ t i+1, j

j=1

N i+1

∑

€

W = −iR

Qωr

ωr

′ ω r€

λ =iα + ′ ω r

€ ti,j€ Tb€

Tb+ti+1,j€ ti+1,′ j €

qi,j € qi+1,′ j j-th particle in

i-th bunchj'-th particle in(i+1)-th bunch

€ n+ih ⎛ ⎝ ⎜ ⎞ ⎠ ⎟Tb

€ n+i+1h ⎛ ⎝ ⎜ ⎞ ⎠ ⎟Tb

€

Vi+1 t( ) = Re ˜ V i+1eiλ t

[ ] = Re ˜ V ieiλ Tb +t( ) + qi+1, j xi+1, jWe

iλ t−t j( )

j=1

N i+1

∑ ⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

= Re ˜ V ieiλTb + qi+1, j xi+1, jWe−iλ t j

j=1

N i+1

∑ ⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟eiλ t

⎡

⎣

⎢ ⎢

⎤

⎦

⎥ ⎥

Transverse Kick Voltage produced by a charge

After (i+1)-th bunch,

Multi-bunch instabilities by ResonatorLongitudinal

€

Z =R

1+ iQωωr

−ωr

ω

⎛

⎝ ⎜

⎞

⎠ ⎟

€

V j t − t j( ) = −q jR

Qωre

−α t−t j( ) cos ′ ω r t − t j( ) −α

′ ω rsin ′ ω r t − t j( )

⎛

⎝ ⎜

⎞

⎠ ⎟= q j Re We

iλ t−t j( ) ⎡ ⎣ ⎢

⎤ ⎦ ⎥

€

W = −R

Qωr 1+ i

α′ ω r

⎛

⎝ ⎜

⎞

⎠ ⎟

€

λ =iα + ′ ω r

€

Vi+1 t( ) = Re ˜ V i+1eiλ t

[ ] = Re ˜ V ieiλ Tb +t( ) + qi+1, jWe

iλ t−ti+1, j( )

j=1

N i+1

∑ ⎡

⎣

⎢ ⎢

⎤

⎦

⎥ ⎥

€

˜ V i+1 = ˜ V ieiλTb + qi+1, jWe

iλ −ti+1, j( )

j=1

N i+1

∑

Acceleration Voltage produced by a charge

After (i+1)-th bunch,

Beam Loading

Acceleration Mode = Longitudinal Impedance

Beam

Required Drive Voltage

Cavity Voltage

€

˜ V g = ˜ V c − ˜ V b

€

˜ V b Excitation

€

˜ i b

€

˜ V c Requirement from operation

€

ψ,Δψ

Im

Reψ € Vc

€ Vg€ Vb€ φ€ Δψ€ ψ€ Vb

€ ib€ ig

€ e Vccosφ=U€ θ=φ−Δψ€ θ€ θ+ψ

Application Longitudinal

10

8

6

4

2

0

Energy Oscillation Amp.[x10

-3

]

0.120.080.040.00

Time [s]

1/2 filling

1/5 filling

1/1 filling

(Equal filling)

15.0

14.0

13.0

12.0

Acceleration Voltage [x10

6

V]

50403020100

Bunch No.

1/5 filling

1/3 filling

1/2 filling

1900

1800

1700

1600

Synchrotron Frequency [Hz]

50403020100

Bunch No.

1/5 filling

1/3 filling

1/2 filling

One Turn

One Rad. Damping TimeSimulation

fs

MV

Vacc

Suppression by Acceleration Voltage Modulation by beam loading of Partial Filling (ESRF)

At commissioning of SPring-8 (1997~8) RF Low Level Feedback + beam loading => instability at ~fRF

Growth Rate estimation by Comparison with Simulation

1/5 filling can not suppress => TOO BIG GROWTH-RATE ! => Other Source but HOM

Application Longitudinal

Suppression by Acceleration Voltage Modulation

by Add-on fRF + frev Accelaration 4GeV operation of SPring-8 => Instability

Damping 1/8 of 8GeV (nominal)4MV@fRF + 1MV@(fRF+frev)UVSOR, ETL

Timing of bunchesby Streak Camera

Simulation

0

1

2

3

4

5

6

-100

0

100

200

300

400

500

0 20 40 60 80 100Bunch No.

Synchrotron Frequency

Timing

200ps

1 turn = 4.7us

Optimization of Filling Pattern

unstable

Application TransverseSuppression by Static Chromaticity Reduction Ratio of Growth Rate

Suppression by AC Chromaticity ( fsyn )Introduction of tune spread

200

100

0

-100

1.21.00.80.60.40.20.0

ξ1

Theoretical value forξ1=0

Radiation Damping rate

€

ξ t( ) = ξ0 + ξ1 cosω st

€

σν =1

2ξ1σ δ

€

ξ1 = 0.6, f0 = 209kHz,σ δ =1×10−4

1

τ L

=2

π2πf0

1

2ξ1σ δ

⎛

⎝ ⎜

⎞

⎠ ⎟≅ 2.5 f0ξ1σ δ = 310 / s

(Gaussian)350/s

€

J02 ω0

ωs

ω f

ω0

α + ξ ⎛

⎝ ⎜

⎞

⎠ ⎟ˆ δ

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

1

σ δ2

e−

ˆ δ 2

2σ δ2 ˆ δ d ˆ δ

0

∞∫

= J02 ω0

ωs

0.7 + ξ( ) ˆ δ ⎛

⎝ ⎜

⎞

⎠ ⎟

1

σ δ2

e−

ˆ δ 2

2σ δ2 ˆ δ d ˆ δ

0

∞∫ 0

0.2

0.4

0.6

0.8

1

1.2

0 2 4 6 8 10 12 14 16

Simulation

Growth Rate Ratio

chromaticity

€

ω f = 2π × 6.3GHz

ωs = 2π ×1.7kHz

ω0 = 2π × 209kHz

α =1.46 ×10−4

σ δ =1×10−3

Single-Bunch

Wake by beam pipeMAFIA 2-dim, 3-dimTest Particle Size wake (Green Function)

~ 1mm (meshsize ~ 0.1mm ) ~ 0.2mm (meshsize ~ 0.04mm)

Vacuum Chamber Shapes

< 3mm

20mm< 1mm

20mm< 0.5 mm

< 2mm

< 0.5 mm

20 mm

0.5mm

0.08mm

1mm

2-4mm100mm

0.5mm

20mm

RF shielding fingers

RF shielding fingers

20mm 23mm 20mm 25mm

38mm 96mm 50mm

24mm 20mm

flange 700

weldment 2000

offset 2700

bellows 400

valve 400

Resistive Aluminum Length 1436m pipe radius 20mm

Wake Function

-3 1014

-2 1014

-1 1014

0

1 1014

2 1014

3 1014

0 0.01 0.02 0.03 0.04 0.05

Longitudinal Wake

weldmentflange

m

-3 1014

-2 1014

-1 1014

0

1 1014

2 1014

3 1014

0 0.01 0.02 0.03 0.04 0.05

Longitudinal Wake

cavityoffsetbellowsResistive Wall

m

-3 1016-2 1016-1 1016

01 10162 10163 10164 10165 1016

0 0.01 0.02 0.03 0.04 0.05

Vertical Wake

weldmentflange

m

-3 1016-2 1016-1 1016

01 10162 10163 10164 10165 1016

0 0.01 0.02 0.03 0.04 0.05

Vertical Wake

cavityoffsetbellowsResistive Wall

m

bunch center (1mm rms) bunch center (1mm rms)

x Number x Number x

Single-bunch Equatio of Motion

€

ηi =xi

β

θ =1

ν 0

d ′ s

β

s∫

d2η i

dθ 2+ ν 0 + Δν i( )

2η i = ν 0

2β32 Fi

E0

€

Fi = e q j x jd

dsW⊥ z j − zi , s( )

j=1

N

∑ = eβ1

2 q jη jd

dsW⊥ z j − zi , s( )

j=1

N

∑

€

ηi = Re ai θ( )eiν 0θ[ ] =

1

2ai θ( )eiν 0θ + ai

* θ( )e−iν 0θ( )

Fi = Re fi θ( )eiν 0θ[ ] =

1

2fi θ( )eiν 0θ + fi

* θ( )e−iν 0θ( )

fi θ( ) = eβ12 q ja j

d

dsW⊥ z j − zi , s( )

j=1

N

∑

Single-bunch Time Average

€

d2ai

dθ 2<< 2iν 0

dai

dθ

€

dai

dθ= iΔν iai +

ν 0

2iE0

β3

2 fid ′ θ

2πν 0

⎛

⎝ ⎜

⎞

⎠ ⎟

θ−π ν 0

θ+π ν 0∫ +ν 0

2iE0

β3

2 fi*e−2iν 0 ′ θ d ′ θ

2πν 0

⎛

⎝ ⎜

⎞

⎠ ⎟

θ−π ν 0

θ+π ν 0∫

€

d2η i

dθ 2+ ν 0 + Δν i( )

2η i = ν 0

2β3

2 Fi

E0

€

dai

dθ= iΔν iai +

ν 0

2iE0

β3

2 fid ′ θ

2π

ν 0

⎛

⎝ ⎜

⎞

⎠ ⎟

θ −π ν 0

θ + π ν 0∫

€

ηi = Re ai θ( )eiν 0θ[ ]

Fi = Re fi θ( )eiν 0θ[ ]

fi θ( ) = eβ12 q ja j

d

dsW⊥ z j − zi , s( )

j=1

N

∑

Single-bunch Time Average

€

3

2 fid ′ θ

2πν 0

⎛

⎝ ⎜

⎞

⎠ ⎟

θ−π ν 0

θ+π ν 0∫ = β1

2 fid ′ s

2πs−λ β 2

s+λ β 2∫ =

1

2π

λ β

Cβ

1

2 fid ′ s s−C 2

s+C 2∫

β1

2 fid ′ s s−C 2

s+C 2∫ = β k

1

2

k

∑ fid ′ s k−th element∫

= e β k

k

∑ q ja jWk⊥ z j − zi( )

j

∑ = e q ja j

j

∑ β kWk⊥ z j − zi( )

k

∑

€

dai

dθ= iΔν iai +

e

4iπE0

qjajj =1

N

∑ β kWk⊥ zj − zi( )

k

∑

Single-bunch Difference Equation

€

ri+ = ri

− + Re g i−e−iφi

−

[ ]Δθ

φi+ = φi

− + Δν iΔθ +1

ri−

Im gi−e−iφi

−

[ ]Δθ

gi− =

e

4iπE0

q ja j−

j=1

N

∑ β kWk⊥ z j

− − z i−

( )k

∑

Δν i = ξδ i− + amplitude dependence...( )

€

ai+ = ai

− − ieVa zi

−( )

E0

Im ai−eiν 0u

[ ]ai−eiν 0u

€

ai+ = ai

− + 4ΔT

τ β

ε 0

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

1

2w ie

i2πχ iwi : Gaussian Random Number

i : Uniform Random Number

Damping by Acceleration

Radiation Excitation

€

ai = rieiφi

Numericaly Stable€

dai

dθ= iΔν iai +

e

4iπE0

qjajj =1

N

∑ β kWk⊥ zj − zi( )

k

∑

Particle In Cell (PIC) with Wake Field

€

gi− =

e

4iπE0

q ja j−

j=1

N

∑ β kWk⊥ z j

− − zi−

( )k

∑

= q ja j

j=1

N

∑ δ ′ z − z j( ) β kWk⊥ z j − z( )

k

∑ d ′ z ∫

= q ja j

j=1

N

∑ S ′ z − z j( ) β kWk⊥ z j − z( )

k

∑ d ′ z ∫

= ρ sa ′ z ( )

j=1

N

∑ β kWk⊥ ′ z − zi

−( )

k

∑ d ′ z ∫

€

S z( ) =

−z

Δz

⎛

⎝ ⎜

⎞

⎠ ⎟2

+3

4

z

Δz≤

1

2

⎛

⎝ ⎜

⎞

⎠ ⎟

1

2

z

Δz−

3

2

⎛

⎝ ⎜

⎞

⎠ ⎟

21

2≤

z

Δz≤

3

2

⎛

⎝ ⎜

⎞

⎠ ⎟

03

2≤

z

Δz

⎧

⎨

⎪ ⎪ ⎪ ⎪

⎩

⎪ ⎪ ⎪ ⎪

Comparison with Experiment

Longitudinal Bunch Shape

Simulation

Measured

0

0.2

0.4

0.6

0.8

1

1.2

0 50 100 150 200

Bunch Shape

Charge Density / A.U.

Time / ps

1.09 mA

11.0 mA

4.97 mA8.85 mA

0

0.2

0.4

0.6

0.8

1

1.2

-4 10-10 -3 10-10 -2 10-10

Bunch Shape by Simulation

1mA5mA9mA12mA

Time / s

Energy Spread

0

0.001

0.002

0 5 10 15 20

Energy Spread vs Bunch Current

Vrf = 12MVVrf = 16MVVrf = 12MV Simulation

σΔE/E

Bunch Current / mA

Parasitic loss

Microwave Instability => energy spread

QuickTime˛ Ç∆TIFFÅiLZWÅj êLí£ÉvÉçÉOÉâÉÄ

ǙDZÇÃÉsÉNÉ`ÉÉÇ å©ÇÈÇΩÇflÇ…ÇÕïKóvÇ≈Ç∑ÅB

z

ΔE/E

driven by high frequency resonance of small grooves(weldment, flange)

Comparison with Experiment

Vertical Single-Bunch Instabilities

Prediction by Simulation 1996 (EPAC96 WEP103,WEP104) Impedance model( Inductance, Resistance, Cavitylike) <= Calculated Wake by MAFIA Chromaticity Threshould current (mode-coupling)

0 3mA/bunch 4 10mA/bunch No energy spread increase by model impedance Measurement 1998 Chromaticity Threshould current -4.3 0.5mA/bunch (m=0 head-tail)

0.24 3.5-4mA/bunch (mode-coupling) 4> 16mA/bunch

1.5 times large energy spread at 10mA/bunch

Comparison with ExperimentVertical Single-Bunch Instabilities Simulation based on Calculated Wake Function -4.3 0.5mA/bunch (m=0 head-tail)

0.24 3.5-4mA/bunch (mode-coupling) 4> 16mA/bunch

1.5 times large energy spread at 10mA/bunch

-120

-110

-100

-90

-80

-130

-120

-110

-100

-90

62 10

3

64 10

3

66 10

3

68 10

3

Frequency Response of

Betatron Motion

Relative Height / dB

Frequency / Hz

4.7 mA

6 mA

8.9 mA

6 mA

-130

-120

-110

-100

-90

-80

-140

-130

-120

-110

-100

-90

62 10

3

64 10

3

66 10

3

68 10

3

Frequency Response of

Betatron Motion

Relative Height / dB

Frequency / Hz

0.1 mA

0.5 mA

2 mA

3 mA

10-9

10-8

10-7

10-6

10-5

10-10

10-9

10-8

10-7

10-6

70000 72000 74000 76000 78000 80000

Betatron Frequency Spectrum

3mA

5mA

Frequency / Hz

10-9

10-8

10-7

10-6

10-5

10-10

10-9

10-8

10-7

10-6

70000 72000 74000 76000 78000 80000

Betatron Frequency Spectrum

0.5mA

2mA

Frequency / Hz

10-9

10-8

10-7

10-6

10-5

10-10

10-9

10-8

10-7

10-6

70000 72000 74000 76000 78000 80000

Betatron Frequency Spectrum

6mA

9mA

Frequency / Hz

10-9

10-8

10-7

10-6

10-5

10-10

10-9

10-8

10-7

10-6

70000 72000 74000 76000 78000 80000

Betatron Frequency Spectrum

3mA

5mA

Frequency / HzMeasured

Measured SimulationSimulation

sample of time evolution

10-7

10-6

10-5

0.0001

0.001

0 0.01 0.02 0.03 0.04

Betatron AmplitudeChromaticity = 3.7

Time / s

0.01mA13mA

12mA

1-9mA

10-7

10-6

10-5

0.0001

0.001

0.01

0 0.01 0.02 0.03 0.04

Betatron AmplitudeChromaticity = -4

Time / s

0.01mA

0.2mA

0.4mA

0.6mA

10-7

10-6

10-5

0.0001

0.001

0.01

0 0.01 0.02 0.03 0.04

Betatron AmplitudeChromaticity = 0.24

Time / s

0.01mA

4mA

5mA

2-3mAξ = -4ξ ~ 0

ξ ~ 3.7

head-tail damping

mode-coupling

head-tailinstability

Summary

Next stepHorizontal wake

Electron Cloud, Ion, CSR, ....

Multi-bunch simulation (CISR)Single-bunch simulation(SISR) developed at SPring-8Prediction ~ measured