Suppression of Beam-Ion Instability in Electron Rings with Multi-Bunch Train Beam Fillings Page: 3 of 33
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paper is organized as follows. The transverse distribution and the fields of trapped ion are investigated in
section II. In section III, we describe the interaction force between the electron beam and ions using
conventional wake and give the instability growth rate for uniform and single bunch train filled beam,
respectively. The mechanism of the mitigation of beam instability by using multi-bunch train beam filling
pattern is investigated in great detail in section IV. Simple formulae for the build-up of ion cloud and
instability growth rate are given. The ion-cloud induced tune shift and the suppression of beam instability
using bunch-by-bunch feedback are briefly discussed in section V. Finally, in section VI we come to
II. TRANSVERSE DISTRIBUTION AND FIELD OF AN ION CLOUD
It is important to investigate the distribution and the field of trapped ions. A few of works about the ion
distribution have been done  comparing a larger number of publications in the instabilities [6-9]. For
completeness, we briefly analyze the distribution of trapped ions in steady status in Appendix. One can
refer to Tavares early work for more detail .
The one dimensional transverse distribution of ions trapped by the electron bunches at the equilibrium
p(x)= e ,K 4r 2 (1)
CYe x e4ax
where Ko is the modified Bessel Function of the Second Kind, Ce is the transverse root mean square (rms)
beam size of the electron bunch from which the ions are born. The asymptotic forms of ion distribution
near the beam center is
p(x),~ - e , log( Sc2 )+Y, (2)
) C e x e,x
The same result in Eq. (1) is first given by Tavares . However the asymptotic form in  is incorrect.
Fig.1 compares the ion distributions from analytical and numerical methods. There is an excellent
agreement except at the origin where the analytical result diverges. There is a sharp peak near the center
because the ions are sharply focused to that point and the size of the ion cloud is smaller than that of the
electron bunch. The result given in Eq.(1-2) works in linear regime. Surprisingly, it agrees well with
simulation. This probably is because most ions are focused near the beam center where the linear model
works. Fig. 2 shows the two-dimensional distribution of ion-cloud by simulation. Note that the transverse
position in the figure is in the unit of beam size. The transverse distribution of ion cloud is very flat for
most light sources due to a small coupling.
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Wang, L.; Cai, Y.; Raubenheimer, T.O.; /SLAC; Fukuma, H. & /KEK, Tsukuba. Suppression of Beam-Ion Instability in Electron Rings with Multi-Bunch Train Beam Fillings, article, August 18, 2011; United States. (digital.library.unt.edu/ark:/67531/metadc840877/m1/3/: accessed November 14, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.