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

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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.

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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.

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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.

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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

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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 %

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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

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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

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

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

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

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

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13 Growth times vs. vacuum pressures

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Vertical amplitudes vs. feedback number 0.23 nT Case A

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15 Vertical amplitudes vs. bunch intensity 0.23 nT

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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)

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

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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

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

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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.

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