Ion instability at SuperKEKB

H. Fukuma (KEK) and L. F. Wang (SLAC)

ECLOUD07, 12th Apr. 2007, Daegu, Korea

1. Introduction

2. Ion trapping

3. Fast ion instability (FII)

4. Effect of the train length on the FII

5. Summary

Growth, noise and feedback, tune shift

1. Introduction

SuperKEKB is an upgrade plan of KEKB.

Luminosity 5 - 10 1035 cm-2 sec-1

Beam current

Beam energy 3.5 / 8.0 GeV

9.4 / 4.1 A

Number of bunches 5018

Circumference 3016 m

Bunch spacing 0.6 m

Emittance 24 nm

Parameters LER / HER

•In SuperKEKB, the electron beam may be stored in LER after LINAC upgrade in order to mitigate the electron cloud effect ("charge switch").

•Comparing with KEKB,

◊Beam energy : 8 3.5 GeV

◊Beam current :1.1 9.4A

Ion instability would be strong enough to degrade the luminosity.

◊Larger vacuum pressure than KEKB : 5 nTorr (CO)

•Requirements for colliders

◊Tune shift along the train by the ions should be small.

◊Residual centroid oscillation by the instability should be small.

◊Number of bunches should be maximized. Small number of train gaps

•Ion instability

1) Ion trapping

2) Fast ion instability

◊Ions are trapped for a long time in a beam potential.

◊A bunch interacts with an ion again and again during many turns.

◊The instability is transient.

◊A bunch interacts with an ion only once.

◊The instability can occur not only in rings but also in linacs or beam transport lines.

2. Ion trapping

⎟⎟⎠

⎞⎜⎜⎝

⎛ −

⎭⎬⎫

⎩⎨⎧

⎟⎟⎠

⎞⎜⎜⎝

⎛−

⋅⎟⎟⎠

⎞⎜⎜⎝

⎛=

10

)(1

1

01

10

1 ττ ph

KM

p

)(

2

yxyion

eb

M

mcrNK

σσσ +=

•Ion motion

•Stability condition 2<MTr

h-p gaps

p bunches

cion

Nb : number of electrons / bunch, m, Mion : electron and ion mass,σx,y : beam size of electron bunch,

nny

yM

y

y⎟⎟⎠

⎞⎜⎜⎝

⎛⋅=⎟⎟

⎠

⎞⎜⎜⎝

⎛

+&&

1

y : ion coordinate

n : turn of a bunch train

kick by a bunch

•According to the linear theory, ion trapping would be avoided with a train gap of 2 % empty RF buckets in SuperKEKB.

•|Trace |M|/2 in SuperKEKB is the same order of magnitude as that in KEKB.

3. Fast ion instability (FII)

•The instability is transient.

•The ions created by the head of the bunch train affect to the tail.

•The FII is the single pass coupled-bunch instability (possibly seen at a ring but also a linac or a beam transport line).

bunchion

◊If damping, such as radiation damping, exists, the oscillation is damped from the head to the tail in the train then the oscillation of all bunches is finally damped (A. W. Chao and G. V. Stupakov). ◊Actually an equilibrium amplitude is determined by the balance of the excitation of the instability by the noises and the damping.

')'('

)',('),(),(

02

2

2

2

dzzzDz

zsyzzsy

czsy

s

z−−=+ ∫ ∂

∂κω

∂∂ β

function edecoherenc : )()'(cos)'( iii fttdttD ωωω −=− ∫ ions offunction on distributi : )( if ω

3-1. Linear theory (G. V. Stupakov et al., P.R.E. 52, 5499)

•The offset of the centroid of the beam y(s,z) is given by

.

s

z

0

)(3

4

yxyb

eion

s

r

σσσγλκ

+= ion line density / bunch

sb: bunch spacing)(3

4 2

0yxyb

pbi As

crN

σσσω

+=

b: bunch population

: Atomic mass number of ions

•Ion frequency

gasionizationbion nN σλ = ,

ngas: gas density

')'(ˆ)',('4

),(0

0 dzzzDzsAzs

zsA zi −= ∫βω

κω∂

∂ cziii

iiefdzD /)( 0)()(ˆ , ωωωω −∫=

czicsi iezsAzsy // 0),(Re),( ωωβ +−⋅=

zezD α−=)(ˆQci

2 , 0ωα =

•Assuming a solution of

•A simple model (Q : quality factor of ion oscillations)

)exp()(),( 0 c

szAzsA

e⋅= zQ

s

cr

yxyb

ione

e

⋅+

=)(3

21 ,

σσσγωλ

β

•If z •A(s,z) has a slow variation in z, the growth is exponential,

))'(exp())'(

(')()exp(),(2

22

000 zzc

zzsIdz

c

szIzzsA

z−−⋅

−+⋅−= ∫ α

α

α

ll

,

turns

log (amplitude)

1 σy

linear regime (exponential growth)

nonlinear regime (linear growth)

◊Linear regime without decoherence of the ions

)exp()( taty ≈

◊Linear regime with decoherence of the ions

)exp()( btty ≈

◊Nonlinear regime (S. Heifets)

ctty ≈)(

•Behavior of the amplitude growth

•Numerical example in SuperKEKB (one long train)

Bunch spacing : 2ns (=0.6m)

Number of bunch : 5120

Bunch current : 1.9 mA

Pressure : 1 nTorr (CO)

Energy 3.5 GeV

Emittance (H/V) : 24 nm/0.96 nm

Beam size (H/V) : 0.6 mm/0.12 mm

Beta function (H/V) : 15 m/15 m

Tune(V) : 43.545

18 sec10 23.1 −=iω

3-910 53.8 −= mκ

246.0/ =⋅= csBiion ωφ

0205.02

==Qciωα

17 sec10 72.2 −=βω

se μα 3.22 == l

bunch)(per 806 1−= mionλ

Q : 10 less than one turn

(revolution time : 10 μs)

2) Clearing electrode ?

1) Better vacuum pressure

A) Decrease the ion density

1) Transverse bunch-by-bunch feedback system

B) Damp the bunch oscillation

3) Short train length (gaps between bunch trains)

2) Tune spread among bunches

3) Octupoles

5) Beam-beam detuning

4) Beam shaking

4) Lower vertical emittance or beam size

3-2. Mitigating method

This talk

1) The amplitude saturates at one sigma of the beam size σy due to the nonlinear effect of a beam-ion force. In the nonlinear regime the growth is slow and would be cured by the feedback.

4-1. Method of the estimation of the amplitude growth

2) Oscillation of σy is not tolerable for SuperKEKB because the luminosity is lost. We should damp the oscillation in the linear regime where we may use the linear theory. 3) Thus our method to discuss the amplitude growth of the FII in SuperKEKB is,

i) Use the analytic linear theory by G. V. Stupakov et al. to obtain acceptable fill patterns.

ii) Perform a simulation to confirm the result of i) and get more realistic results than the analytic estimate.

iii) Estimate the noise and the feedback effect to get the equilibrium amplitude of the oscillation.

4. Effect of the train length on the FII

4-2. Conditions to be taken into account

1) Train gap should be less than 200 ns to avoid the effect of the transient beam loading on the RF system (K. Akai).

2) The vacuum pressure is 5 nTorr for CO and 10 nTorr for H2 to get a lifetime of 10 hr (Y. Suetsugu).

4) Fluctuation of the vertical offset at IP should be less than about ± 0.01 σ*

y which causes 5 % loss of the luminosity according to the Ohmi's beam-beam simulation.

3) Typical damping time of the bunch-by-bunch feedback system is 0.2 ms from the experience of KEKB.

4-3. Growth of the FII in the linear regime

◊Parameters

•Bunch current 2 mA

•Bunch space 0.6 m

•Beta function(H,V) 15 m

•Emittance(H/V) 2.4 10-8/9.6 10-10 m

•Energy 3.5 GeV

•Ion : CO, 1nTorr

•Q : 10

))'(exp())'(

(')()exp(),(2

22

000 zzc

zzsIdz

c

szIzzsA

z−−⋅

−+⋅−= ∫ α

α

α

llzezD α−=)(ˆ

Qci

2, 0ωα =

•The above equation was numerically integrated.

•According to G. V. Stupakov et al.,

e-fold growth time at 10 turns = 3.6 μsec

train length : 3016 m train length : 60.3 m (50 trains)

e-fold growth time at 10 turns = 0.31 msec

The train length of 60 m (50 trains) would be a good starting value for the simulation.

4-4. Growth time by the Simulation

0 200 400 600 800 10000

10

20

30

40

50

60

S (m)

βxy

(m)

βx

βy

Dx*100

1) 2D space charge2) Tracking through elements3) Realistic vacuum model (various pressure and mul

ti-gas species)4) Any beam fill pattern5) Bunch-by-bunch feedback6) Wake of ion-cloud…

•A code developed by L. F. Wang was used in simulations.

◊Features

0 10 20 30 40 50-3

-2

-1

0

1

2

3x 10

5

S (m)

Wxy

bunchspaing=2nsN=2e10 sigx=0.69282mm sigy= 0.00775mmλx=72.6m

λy=7.67m

N=2e9 λy=24.27m

sigx=sqrt(2)*0.69282mm sigy= sqrt(2)*0.00775mmλy=10.85m

50 trains; train Gap=20 bucketsNumber of bunch per train=82Total number of bunch=50*82= 4100Pressure=1nTorr

A) Growth time from the tracking

Growth time=35turns=0.35 ms

B) Growth rate from the ion densitynumber of train =50number of bunch per train=82gap=40ns (20 missing bunch)

emittancex=2.4E-8, emittancey=4.8E-10

pressure : 0.75 nTorr( )rmsii

ye

e

cr

ωωγ

ρβ

τ /

1

23

1

Δ≈

•Estimated growth rate

mse 38.0=

( ) 3.0/ =Δ rmsii ωω

) 0.3 of / and

75.0@ (

)1@(1

1

ii

e

e

Torrdensityionfromestimation

nTorrtracking

ωω

Δ≈ −

−

•This relation in our calculation is valid even if the gap is changed.

•We estimated the growth rate@1nTorr from the ion density @0.75 Torr .

Train length vs. estimated growth rate

gap 20

gap 15

gap 10

damping rate of the feedback

4-5. Total number of bunches

The number of bunches / train

Tot

al n

um

ber

of

bu

nch

Gap : 20

10 % loss of the luminosity

20 % loss of the luminosity

•The total number of the bunches saturates when the train length (i.e. the number of bunches /train) is larger than 150.

•Total number of bunches( luminosity) is calculated when the train length and the train gap are fixed.

∝

◊If the pressure of CO is 1 nTorr and the growth rate should be less than 5 ms-1, 150 bunches in a train would be possible, which leads to the luminosity loss of about 15%.

◊If the pressure of CO is 5 nTorr and the growth rate should be less than 5 ms-1, which is the damping rate of the feedback, the bunches in a train should be less than 35, which leads to the luminosity loss of about 40%.

•From the growth rate vs. the number of bunches / train and the total number of bunches vs. the number of bunches / train,

we can get the relation between the luminosity loss and the growth rate.

•The results are,

4-6. Train length vs. growth rate

),()',(''2

1),(

1),( )'(

00

zsfezsAzdzs

zsAcs

zsA zzz

d

+⋅=+ −−∫ α

∂∂

i

sκωωβ2

0 =

noise random : ),( zsf

function edecoherenc : )( zezD α−= ⎟⎟⎠

⎞⎜⎜⎝

⎛=

Qci

2

ωα

•The equation by A. W. Chao and G. V. Stupakov (MBI97, p. 110) was modified to include the ion decoherence function D(z) as,

feedback of timedamping : d

•The above equation was numerically integrated.

4-7. Noise and Feedback

•Parameters in calculation

Bunch current 1.9 mA

Bunch space 0.6 m

Beta function(H,V) 15 m

Emittance(H/V) 2.4 10-8/9.6 10-10 m

Energy 3.5 GeV

Ion : CO, 1nTorr

Q : 10

Number of bunches 82

E-fold growth time e by the simulation : 0.35 msec

Amplitude of the last bunch up to 1000 turns decoherence : on

Damping is always on.

d=1.44ms d=2.15ms

d=0.96ms red : instability on

green : instability off

The same sequence of random numbers was used in each calculation.

d ~ e, i.e. 0.35 ms, is enough to damp the instability to the noise level.

turns

Ad=0.48ms

50 trains; train Gap=20Number of bunch per train=82Total number of bunch=50*82= 4100P=1nTorrGrowth time=35turns; tune shift=0.0035

4-8. Tune shift•Beam-ion force changes the tune of the bunches.

•The vertical tune shift of the last bunch in the train was estimated using the ion density from the simulation as,

•As the ion density changes along the train, the tune also changes along the train.

dsr

regiontrapped yxy

yiey ∫ +=Δ

)(6 σσσβ

πγλν ions ofdensity line : , iλ

5 nTorr (gap 20)

1 nTorr (gap 20)

1 nTorr (gap 10)

1 nTorr (gap 15)•In case of

bunches / train : 82

gap : 20 bunches

pressure : 5 nTorr

, tune shift at the last bunch ~ 0.009.

◊Tune change of 0.001 affects to the luminosity.

◊Tune change of 0.009 would not be acceptable in SuperKEKB.

◊Vertical tune in LER changes 0.0018 along the train due to the electron cloud.

Tune change of 0.002 along the train would be a good reference which is acceptable in SuperKEKB.

Tune shift of the last bunch

•KEKB

◊If the pressure of CO is 1 nTorr and the tune shift along the train should be less than 0.002, 135 bunches in a train would be possible, which leads to the luminosity loss of 15 %.

◊If the pressure of CO is 5 nTorr and the tune shift along the train should be less than 0.002, the bunches in the train should be less than 25, which leads to the luminosity loss of 45 %.

•The result

◊Actually the first bunch in the train has a tune shift. The tune shift from the head to the tail is about 70 % of the tune change of the last bunch.

tune shift from the head to the tail

tune shift of the last bunch

•Assuming that the pressure of CO is 5 nTorr,

if the growth rate of the FII should be less than the damping rate of the feedback system of 5 ms-1, length of the train would be limited to 35, which leads to the luminosity loss of about 40%. if the tune shift due to the ions should be less than 0.002, length of the train would be limited to 25, which leads to the luminosity loss of about 45%.

•If the pressure of CO is 1 nTorr, the luminosity loss due to the growth rate and the tune shift will be 15 % and 15 %, respectively.•The CO pressure of 1 nTorr will be necessary for SuperKEKB if electrons are stored in LER.

5. Summary

•The train length to mitigate the FII was discussed when electrons are stored in LER at SuperKEKB.

•The tune shift would be as much serious as the amplitude growth. We need a way to decrease it.