Elsen, E

Xia, G

The University of Manchester - Institutional Repository

Study of Fast Beam-Ion Instability in ILC Damping Ring

CERN Document Server

EUROTeV-Memo-2006-002-1     Study of Fast Beam-Ion Instability in ILC Damping Ring  Guoxing Xia1, Eckhard Elsen1 March  02, 2006    Abstract  By using the update damping ring parameters, the growth time of fast beam-ion instability (FBII) is analytically estimated according to the linear theory. The two different fill patterns are taken into account and the results show that the growth time is larger than the response time of current feedback system, especially when the ion frequency spreads exist along the damping ring lattice.                                                  1 DESY, Hamburg, Germany  - 1 - EUROTeV-Memo-2006-002-1 1 Introduction Fast beam-ion instability was first studied by T. Raubenheimer and F. Zimmermann in 1995. After that, some experiments were carried out to verify the theory of FBII. The ions generated by the head of the bunch train oscillate in the transverse direction and resonantly interact with the betatron oscillation of the subsequent bunches, resulting in the growth of the initial perturbation of the beam. This phenomenon is called fast beam-ion instability. In the damping ring of International Linear Collider (ILC), the beam emittance is smaller and the bunch train is longer compared with some existing storage rings, the number of ions ionized by a single bunch train is large (not trapping from turn to turn), therefore FBII is severe. In this note, we estimate the growth time of FBII by using the update electron damping ring parameters of ILC. 2 Linear Theory of Fast Beam-Ion Instability According to the linear theory, the characteristic rise time of FBII can be described as [1, 2]  ( ) 2/12/32/32/12/122/33341AcLrrnNdyxyseppebbyiongasc σσγσβστ +=                                             (1) where is the density of residual gas, Tkpd bgas /= p is the gas pressure in vacuum chamber,  and T  are Boltzmann Constant and gas temperature respectively. bk ionσ  is the ionization cross section, yβ  is the average vertical beta function,  is the particle number per bunch,  and  are the classical radius of electron and proton respectively,  is the bunch spacing, bNer pr sepLγ  is the relativistic factor, xσ  and yσ  are the horizontal and vertical beam sizes respectively, A  is the ion mass in unit of proton mass and  is the speed of light. cThe ion coherent angular frequency iω  is given by ( )2/1234⎟⎟⎠⎞⎜⎜⎝⎛+= yxyseppbi ALcrNσσσω                                                     (2) Actually, there are some sources which can cause the ion frequency iω spreads along the damping ring lattice structure. One of them is due to the horizontal beam density profile in a flat beam which causes the local ion frequency to depend on the horizontal position. Other sources of spread in ion frequency are due to the nonlinearity of the ion oscillations inside the beam and the presence of multiple ion species, etc. Taking into account of the ion coherent frequency spread, the linear theory gives the coupled bunch motion in the bunch train like ( )ety τ/exp~ ; the growth time is given  ( )rmsitraince lcωττ Δ= 2211                                                          (3) where ( )rmsiωΔ  is the rms spread of ion coherent angular frequency,  is the length of bunch train and  . trainlsepbtrain Lnl =According to Eq.(1)~(3), assuming 10%, 20% and 30% relative coherent frequency spread along the baseline and alternative damping rings, for 0.1nTorr, 1.0nTorr CO gas pressure, the growth times cτ  , eτ  versus bunch index are plotted in Fig.1-8. It can be seen that the growth time becomes less with respect to the bunch index, while considering the ion coherent frequency spread, the growth rates become much slower.    - 2 - EUROTeV-Memo-2006-002-1 3 The Growth Time Estimation of FBII According to the latest configuration of electron damping ring, the fill pattern is changeable in order to alleviate the ion effect. The baseline and alternative damping rings parameters are listed in Table 1 and Table 2 respectively [3]. In order to calculate the growth time of FBII, we use the beam sizes parameters in Frank Zimmermann’s report [4]. Here, only two fill patterns are considered, one is at the left side of Table 1 and Table 2, namely the bunch charge is 2.0×1010 and the number of bunch trains is 60 for baseline damping ring, while for 17km alternative ring, the bunch spacing is 13.85ns and there is 6 bunches per train, the bunch charge is 2.25×1010. The other fill patterns we care about are the data from the right side of these two Tables, namely, the bunch charge is 1.0×1010 and the number of bunch trains is 120 for baseline damping ring, while for 17km alternative ring, the bunch spacing is 4.62ns and there is 18 bunches per train, the bunch charge is 0.75×1010.  We named them mode 1 and mode 2 respectively. The two different vacuum gas pressures 0.1nTorr and 1nTorr are assumed in our calculation. Table 1 Fill patterns of baseline damping ring   Table 2 Fill patterns of alternative damping ring   - 3 - EUROTeV-Memo-2006-002-1 0 10 20 30 40 51E-30,010,11101000 growth time w/o ion frequency spread growth time with 10% ion frequency spread growth time with 20% ion frequency spread growth time with 30% ion frequency spreadGrowth time of FBII / secondBunch index #  Fig.1. Growth time of FBII versus bunch index in 0.1nTorr for baseline  damping ring (Mode 1) 0 10 20 30 40 51E-30,010,11100Growth time of FBII / secondBunch index # growth time w/o ion frequency spread growth time with 10% ion frequency spread growth time with 20% ion frequency spread growth time with 30% ion frequency spread Fig.2. Growth time of FBII versus bunch index in 1nTorr for baseline  damping ring (Mode 1)  0 10 20 30 400,010,1110100100050 growth time w/o ion frequency spread growth time with 10% ion frequency spread growth time with 20% ion frequency spread growth time with 30% ion frequency spreadGrowth time of FBII / secondBunch index #  Fig.3. Growth time of FBII versus bunch index in 0.1nTorr for baseline  damping ring (Mode 2)  - 4 - EUROTeV-Memo-2006-002-1 0 10 20 30 401E-30,010,111050Growth time of FBII / secondBunch index # growth time of FBII w/o ion frequency spread growth time of FBII with 10% ion frequency spread growth time of FBII with 20% ion frequency spread growth time of FBII with 30% ion frequency spread Fig.4. Growth time of FBII versus bunch index in 1nTorr for baseline  damping ring (Mode 2) 0 2 4 60,1110100 growth time w/o ion frequency spread growth time with 10% ion frequency spread growth time with 20% ion frequency spread growth time with 30% ion frequency spreadGrowth time of FBII / secondBunch index #  Fig.5.Growth time of FBII versus bunch index in 0.1nTorr for alternative  damping ring (Mode 1) 0 2 4 60,010,11 growth time w/o ion frequency spread growth time with 10% ion frequency spread growth time with 20% ion frequency spread growth time with 30% ion frequency spreadGrowth time of FBII / secondBunch index #  Fig.6.Growth time of FBII versus bunch index in 1nTorr for alternative  damping ring (Mode 1)  - 5 - EUROTeV-Memo-2006-002-1 0 2 4 6 8 10 12 14 16 18 200,1110100Growth time of FBII / secondBunch index # growth time of FBII w/o ion frequency spread growth time of FBII with 10% ion frequency spread growth time of FBII with 20% ion frequency spread growth time of FBII with 30% ion frequency spread Fig.7.Growth time of FBII versus bunch index in 0.1nTorr for alternative  damping ring (Mode 2) 0 2 4 6 8 10 12 14 16 18 201E-30,010,1110100Growth time of FBII / secondBunch index # growth time of FBII w/o ion frequency spread growth time of FBII with 10% ion frequency spread growth time of FBII with 20% ion frequency spread growth time of FBII with 30% ion frequency spread Fig.8.Growth time of FBII versus bunch index in 1nTorr for alternative  damping ring (Mode 2) 4 Conclusion From the analytical results one can see that the growth time at the bunch train end for baseline and alternative damping rings in both 0.1nTorr and 1nTorr is larger than the revolution period of rings (The revolution period for 6km and 17km rings are 22μs and 57μs respectively. When there is no ion frequency spread, the growth time of FBII is a few hundred microseconds, so it is within the bunch-by-bunch feedback system). Therefore it is possible to cure the fast beam-ion instability by using the fast feedback system in ILC electron damping ring.  Acknowledgement This work is supported by the Commission of the European Communities under the 6th Framework Programme “Structuring the European Research Area”, contract number RIDS-011899.   - 6 - EUROTeV-Memo-2006-002-1 References [1]  T.Raubenheimer and F.Zimmermann, Fast beam-ion instability, I. Linear theory and  simulations, Phys. Rev. E52, no.5, p.5487 (1995). [2]  G.V. Stupakov, T.Raubenheimer and F.Zimmermann, Phys.Rev.E52, no.5,p.5499 (1995). [3]  http://www.desy.de/~awolski/ILCDR/ [4] Frank Zimmermann, Studies of electron cloud, ion effects, and Synchrobetatron resonances for KEKB, Super-KEKB and ILC damping rings, ILC-Asia-2005-24, Nov. 2005.   - 7 - 

Study of Fast Beam-Ion Instability in ILC Damping Ring

Abstract

Similar works

Full text

Available Versions

The University of Manchester - Institutional Repository

CERN Document Server