Impedance and instabilities WANG Na 2016722 Ion instabilities

Impedance and instabilities WANG Na 2016/7/22

Ion instabilities Numerical number based on H-High lumi • Ions of the residual gas accumulate in a potential well of the electron beam => Excite multi-bunch instabilities due to interaction between the beam particles and ions • 1) rough estimation of the limiting current of the electron beam – Linear ion density linear electron density • Tgas and Pgas are the temperature and pressure of the residual gas • i is the cross section of ionization • k. B is the Boltzmann constant • Nb is the number of bunches • Ne=Ib. T 0/e is the number of electrons in a bunch • T 0=1/f 0 is the revolution period • P is the circumference of the storage ring λi=1. 2 E 5[m-1] – The characteristic time of development of instability λe=3. 5 E 8[m-1] τ=0. 5[s] – Restricted by a multiturn electron-ion instability

Ion instabilities • 2) Ions are considered as the source of the wakefields – The interaction is described using the effective impedance Zi( ) ωi=101[MHz] Δνx=0. 0025 Δνy=0. 045 τx=2. 2[ms] τy=0. 1[ms] 2 c/Lgap=37[k. Hz] • Qi<10 is the quality factor of the ion oscillator • qi/qp is the ion charge in the units of a proton charge, Ai is the ion mass in amu • rp is a classical proton radius, Lsep is the space between the bunches – Intensity of the electron-ion interaction can be characterized by the coherent betatron tune shift – Taking into account the spread of the ion frequency induced by varying transverse sizes of the beam along the ring, the instability growth time – A nonuniform distribution of the bunches along the ring is found to be an efficient method for suppression of the multiturn accumulation of ions, since ions are accumulated in the beam at i<2 c/Lgap, where Lgap=P Nb. Lsep.

τcx=10[us] Ion instabilities τcy=0. 5[us] τex=9[ms] • 3) Fast beam-ion instability τey=0. 5[ms] – In facilities with very high beam current and low emittances, the accumulation of ions within a single pass of the beam can excite fast beam-ion instability. – The ion density of the residual gas increases along the train of electron bunches, which leads both coherent oscillations of individual bunches and growth of the beam emittance. – It’s a single turn effect and can't be suppressed by the gap in the bunch train. – When the amplitude of oscillations < transverse size of the beam, growth of the amplitude of oscillations is proportional to exp( ) or • Suppose the ions are produced by the impact ionization of atoms of the residual gas by the beam particles • Suppose the captured particles have small initial velocities – A modified linear theory which consider the frequency spread i due to betafunction beats, yields an exponential growth of instability exp(t/ e)

Ion instabilities • 4) Emittance growth due to the fast beam-ion instability – Due to the coupling of the transverse and longitudinal motion of the beam (head-tail), the ion cloud produced by the head particles of the beam turns out to be shifted with respect to the tail particles, and the electric field of ions deflects the tail particles. – First order perturbation of emittance along the length of the wavetrain Ltrain: • i 6 Ne. Nb. Pgas [torr] is the linear ion density at the end of the train • is the amplitude of the primary perturbation – The initial amplitude can be induced by • Shottky noise • Vertical dispersion Δεy=1. 3 E-28[m]

Summary table for ion instability Parameter Symbol, unit H-High lumi. H-low power Z Beam energy E, Ge. V 120 45. 5 Circumference C, km 54 54 54 Beam current I 0, m. A 16. 9 10. 5 45. 4 Bunch number nb 67 44 1100 Bunch Population Ne 2. 85 1011 2. 67 1011 0. 46 1011 Natural bunch length l 0, mm 4. 1 4. 0 Emittance (horz. /vert. ) x/ y, nm 2. 45/0. 0074 2. 06/0. 0062 0. 62/0. 002 RF frequency frf, GHz 0. 65 h 117081 Natural energy spread e 0 1. 3 E 3 5. 0 E 4 Momentum compaction factor p 2. 5 E 5 2. 2 E 5 3. 5 E 5 x/ y 319. 21/318. 42? s 0. 08 0. 04 ωion, MHz 101 87 323 MHz 0. 037 Fast beam ion without i τcx/τcy, us 10. 0/0. 5 15. 7/0. 9 0. 1/0. 006 Fast beam ion with i τex/τey, ms 9. 0/0. 5 12. 4/0. 7 0. 3/0. 02 m 1. 3 E-28 9. 9 E-29 9. 8 E-27 Harmonic number Betatron tune Synchrotron tune Ion oscillation frequency 2 c/Lgap Emittance growth due to Ion Parameter wangdou 20160325