Frank Zimmermann, Factories’03 Weak-Strong Model for the Combined Effect of Beam-Beam Interaction and Electron Cloud Frank Zimmermann, Factories’03 Introduction Analytical Model Application

Introduction Why expect an interplay of electron-cloud & beam-beam? Both induce a head-tail ‘wake field’ Both induce a tune shift which varies along the length of the bunch Evidence in simulations Lower threshold for beam blow up was observed in KEKB and PEP-II when both effects were present as compared with the theshold for a single beam or for fewer bunches I review a weak-strong analysis developed in 2001 which was presented at 2-Stream Worksho at KEK (CERN-SL-2001-067) & application to PEP-II; model was inspired by Karel Cornelis. A strong-strong model was later treated by K.Ohmi&A.Chao (PRST-AB 5, 2002)

Simulated beam-size growth and centroid oscillation along the length of a bunch (HEADTAIL, G. Rumolo, 2001) beam size & centroid after 0, 250 & 500 turns, with re=1012 m-3 for SPS-like parameters; e-cloud only only electron cloud

electron cloud & beam-beam, case (1) (HEADTAIL code, G. Rumolo, 2001) beam size & centroid after 0, 250 & 500 turns, with re=1012 m-3 for SPS-like parameters, e-cloud & ‘rotation’ around bunch center due to beam-beam equivalent to x=-0.037; using initial beam size electron cloud & beam-beam, case (1)

electron cloud & beam-beam, case (2) (HEADTAIL code, G. Rumolo, 2001) beam size & centroid after 0, 250 & 500 turns, with re=1012 m-3 for SPS-like parameters, e-cloud & ‘rotation’ around bunch center due to beam-beam equivalent to x=-0.037; using instantaneous electron cloud & beam-beam, case (2)

Analytical Model Few particle model Two primary effects of electron cloud: (1) transverse wake between leading and trailing particles (2) positive tune shift which increases roughly linearly along the bunch Beam-beam effect only represented by additional (quadratic) tune variation along the bunch Not included: ‘wake’-coupling from beam-beam, any nonlinear transverse forces

2-particle model does not yield instability due to a variation of tune with longitudinal position [different from chromaticity, where tune depends on momentum error; see A. Chao’s book, Page 198] minimum number of particles = 3, distributed uniformly in synchrotron phase space:

ordering of particles: (1,3,2), (3,1,2), (3,2,1), (2,3,1), (2,1,3), (1,2,3) 2 1 d 1 2 3 3 z etc.

variation of tune with longitudinal position (a,b>0 for PEP-II) betatron phase of particle j is obtained by integration (assuming unperturbed motion):

Betatron equation of motion for jth particle with e-cloud wake W0: evaluate motion for 1 third synchrotron period; the later 2 thirds are then obtained by permutation of indices ansatz

where if |Dfm-Dfn|<<1, we can expand the exponential if this is not fulfilled, get Bessel functions instead of trigonometric functions

define ec wake ec wake & ec tune shift ec wake & beam-beam tune shift

Rewrite this as

ordering of particles: (1,3,2), (3,1,2) - (3,2,1), (2,3,1) - (2,1,3), (1,2,3) after 1/3 synchrotron period we apply the permutation matrix

stability of the system is determined by eigenvalues of matrix M1/3. Only keeping terms of first and second order and neglecting all higher-order cross products, the matrix simplifies… then the characteristic polynomial becomes e.g., without electron cloud: A=B=C=D=0, (1-l)3=0, and all eigenvalues lie on the unity circle a similar calculation has been done for 4 particles, to explore the dependence on particle number

Example SPS: (1) 3-particle model, e-cloud wake only C=2.4, B=0, A=0, |l|max=6.65,tgrowth=0.88 ms (2) 3-particle model, e-cloud wake&tune shift C=2.4,B=0,A=3.8; |l|max=6.69,tgrowth=0.88 ms (3) 3-particle model, e-cloud wake & tune shift & beam-beam tune shift C=2.4,B=-2.3,A=3.8, |l|max=6.19,tgrowth=0.91 ms (4) 4-particle model, e-cloud wake&tune shift |l|max=5.75,tgrowth=0.71 ms (5) 4-particle model, e-cloud wake & tune shift & |l|max=6.71,tgrowth=0.61 ms changes by 5-10%; however, larger changes for larger beam-beam tune shift eigenvector patterns show larger variation

growth time vs. beam-beam tune shift, SPS like parameters no threshold, unlike 2-particle model (K. Oide)?!

growth rates vs. e-c. wake field, no pinch, no beam-beam apparent ‘threshold’ ~2x105 m-2 2-part. Model threshold ~3.3x105 m-2

PEP-II rough estimate A=0.07, B=0.06, C=0.23 DQec=0.01, DQbb=0.07 (1) 3-particle model, e-cloud wake only |l|max=1.021,tgrowth=4.1 ms (2) 3-particle model, e-cloud wake&tune shift |l|max=1.025,tgrowth=3.3 ms (3) 3-particle model, e-cloud wake & tune shift & beam-beam tune shift, |l|max=1.040,tgrowth=2.1 ms (4) 4-particle model, e-cloud wake & tune shift & beam-beam tune shift |l|max=1.148,tgrowth=0.44 ms

sorry for the horizontal scale! growth time vs. beam-beam tune shift, PEP-II like parameters sorry for the horizontal scale!

growth rates vs. e-c. wake field for PEP-II, no pinch, no beam-beam apparent ‘threshold’ ~6x105 m-2 2-part. Model threshold ~1.1x106 m-2

Summary Beam-beam interaction introduces Gaussian variation of betatron tune along the bunch. Simulations show that this addt’l tune variation enhances e-cloud instability. We developed an analytical model, where a bunch consists of 3 or 4 particles, electron cloud is represented by constant wake and by linear tune shift along the bunch, and beam-beam by parabolic tune shift. Model shows that beam-beam tune shift acts destabilizing Agreement between model and simulation seems to improve with increasing number of particles.