1. 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

2. 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 )
& 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)

3. 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

4. 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)

5. 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)

6. 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

7. 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:

8. 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.

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

10. 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

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

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

14. Rewrite this as

15. 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

16. 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

17. 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

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

19. 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

20. 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

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

22. 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

23. 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.