1. 1 Simulations of fast-ion instability in ILC damping ring 12 April 2007 @ ECLOUD 07 workshop Eun-San Kim (KNU) Kazuhito Ohmi (KEK)

2. 2 Introduction  We have performed simulations on the fast-ion beam instabilities in ILC damping ring.  We investigated the effects of various different bunch filling patterns, vacuum pressures and feedback system on the fast-ion instabilities.  Damping ring lattice is included in the simulations.

3. 3 Simulation method (1)  Weak-Strong model - Ions (weak) and beams (strong) are expressed by macroparticles and point charges, respectively. - Barycenter motion in beams is only investigated.  Interactions between a bunch and ions are considered by Bassetti-Erskine formula.  We assume that CO ions exist in the ring and use 1/6 part of the entire ring lattice for the simulations.  Ions are generated at locations that all magnetic components and drift spaces exist. (Ionization in long drift space is examined by every 2 m.)  All electron beams are initially set to zero displacement.

4. 4 Simulation method (2)  New macroparticles are generated at the transverse position (x,x´,y,y´) of beam where ionization occurs.  Incoherent behaviors of ions are obtained by our simulation, but that of the beams, such as emittance growth, can not be computed.  We compute the time evolution of the growth of the dipole amplitude of the beam, where the amplitude is half of the Courant-Snyder invariant J y = (  y y 2 + 2  y y y´ +  y y´ 2 )/2.

5. 5 Simulation method (3) ILC damping ring has a circumference of 6.6 km and trains of 61 to 123, depending on the filling patterns, exist in the ring. One bunch train and 1/6 section of the whole lattice are included for the simulations. for the fast simulations

6. 6 Main parameters of the damping ring Circumference 6.69 km Energy 5 GeV Arc cell type TME Horizontal tune 52.397 Vertical tune 49.305 Natural chromaticity -63, -62 Momentum compaction factor 4.2 x 10 -4 Energy loss/turn 8.69 MeV Transverse damping time 25.7 ms Longitudinal damping time 12.9 ms Norm. emittance 5.04  m Natural energy spread 1.28 x 10 -3 RF frequency 650 MHz Synchrotron tune 0.0958 RF acceptance 2.7 %

7. Filling patterns of the damping ring Case A B C D E Number of train Bunch spacing / bucket Gap between trains / bucket Bunch per train / bucket K b : Time between injection/extraction kicker pulses Bunch per train / bucket Gap between trains / bucket

8. 8 Filling patterns of the damping ring (One example) f 2 bunches in f 2 xn b buckets f 1 bunches in f 1 xn b buckets g 1 buckets g 2 buckets Distance between kicker pulses (pattern of k b buckets repeated p times) 24 buckets n b =2 f 2 =4 f 1 =3 k b =24 g 1 =5g 2 =5 p=1

9. 9 Lattice used in the simulations ~1/6 of the entire ring

10. 10 Vertical amplitudes in different filling patterns Case C shows the fastest exponential growth time. 0.23 nT

11. 11 Vertical amplitudes in different filling patterns 0.23 nT feedback per 50 turns

12. 12 Vertical amplitudes vs. vacuum pressures f 1 bunches in f 1 xn b buckets n b =2 f 1 =49 g 1 =25 ~~

13. 13 Growth times vs. vacuum pressures

14. Vertical amplitudes vs. feedback number 0.23 nT Case A

15. 15 Vertical amplitudes vs. bunch intensity 0.23 nT

16. 16 Different bunch spacing in a bunch train ~ ~ 0.97x10 10 /bunch25 empty buckets ~ ~ 1.94x10 10 /bunch 25 empty buckets ~ ~ 3.88x10 10 /bunch 25 empty buckets bunch spacing (n b ) =2 bunch spacing (n b ) =4 bunch spacing (n b ) =8 (Same total bunch charge)

17. Different bunch spacing in a bunch train No feedback in Case A 0.23 nT (Same total bunch charge)

18. 18 ~ 49 bunches in a train 25 empty buckets 25 bunches in a train has electrons of 0.97x10 10 per bunch. 24 bunches in a train 12 empty buckets empty bucket. One and two trains with same number of bunches Case A

19. 19 damping by gap between trains One and two train with same number of bunches 0.23 nT No feedback

20. 20 Summary  We performed weak-strong simulations to show aspects on the bunch filling patterns of the fast-ion instability in the ILCDR.  The simulation results show that bunch by bunch feedback of ~ 50 turns is enough to suppress the fast-ion instability.  We still need more simulation works to understand fully characteristics, in particular of the filling patterns, of the fast-ion instabilities in the ILC DR.