1. 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

2. 1. Introduction SuperKEKB is an upgrade plan of KEKB. Luminosity 5 - 10 10 35 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

3. 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

4. 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.

5. 2. Ion trapping Ion motion Stability condition h-p gaps p bunches cc ion N b : number of electrons / bunch, m, M ion : electron and ion mass,  x,y : beam size of electron bunch, y : ion coordinate n : turn of a bunch train kick by a bunch

6. 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.

7. 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). bunch ion ◊ 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.

8. 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 ion line density / bunch s b : bunch spacing  b : bunch population  : Atomic mass number of ions Ion frequency n gas : gas density

9. Assuming a solution of A simple model (Q : quality factor of ion oscillations) If z A(s,z) has a slow variation in z, the growth is exponential,,

10. turns log (amplitude) 1  y linear regime (exponential growth) nonlinear regime (linear growth) ◊ Linear regime without decoherence of the ions ◊ Linear regime with decoherence of the ions ◊ Nonlinear regime (S. Heifets) Behavior of the amplitude growth

11. 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 Q : 10 less than one turn (revolution time : 10  s)

12. 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

13. 1) The amplitude saturates at one sigma of the beam size  y 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  y 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

14. 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 H 2 to get a lifetime of 10 hr (Y. Suetsugu). 4) Fluctuation of the vertical offset at IP should be less than about ± 0.01  * 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.

15. 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 The above equation was numerically integrated. According to G. V. Stupakov et al.,

16. e-fold growth time at 10 turns = 3.6  sec 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.

17. 4-4. Growth time by the Simulation 1) 2D space charge 2) Tracking through elements 3) Realistic vacuum model (various pressure and multi- gas species) 4) Any beam fill pattern 5) Bunch-by-bunch feedback 6) Wake of ion-cloud … A code developed by L. F. Wang was used in simulations. ◊ Features

18. 50 trains; train Gap=20 buckets Number of bunch per train=82 Total number of bunch=50*82= 4100 Pressure=1nTorr A) Growth time from the tracking Growth time=35turns=0.35 ms

19. B) Growth rate from the ion density number of train =50 number of bunch per train=82 gap=40ns (20 missing bunch) emittancex=2.4E-8, emittancey=4.8E-10 pressure : 0.75 nTorr Estimated growth rate 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.

20. Train length vs. estimated growth rate gap 20 gap 15 gap 10 damping rate of the feedback

21. 4-5. Total number of bunches The number of bunches / train Total number of bunch 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.

22. ◊ 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

23. The equation by A. W. Chao and G. V. Stupakov (MBI97, p. 110) was modified to include the ion decoherence function D(z) as, The above equation was numerically integrated. 4-7. Noise and Feedback

24. 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

25. 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 A  d =0.48ms

26. 50 trains; train Gap=20 Number of bunch per train=82 Total number of bunch=50*82= 4100 P=1nTorr Growth 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.

27. 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

28. ◊ 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

29. 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.