electron-cloud effects in fermilab booster
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March 14-15, 2007 ECloud Feedback, IUCF 1
Electron-Cloud Effects in Fermilab Booster
K.Y. NgFermilab
Electron-Cloud Feedback WorkshopIUCF, Indiana
March 14-15, 2007
March 14-15, 2007 ECloud Feedback, IUCF 2
Motivation I
E-Cloud observed at CERN SPS.
Want to know what happens to Fermilab Booster.
CERN SPS Fermi Booster
Nb 8 x 1010 6 x 1010
No to start E-Cloud 20 ??
Bunch Spacing 25 ns 26.4 ns
Einj 26 GeV 1.34 GeV
Vacuum 7.5 x 10-7 Torr ~2 x 10-7 Torr
March 14-15, 2007 ECloud Feedback, IUCF 3
Motivation II
• Fermilab Booster is injected at 400 MeV.
– Space-charge tune shift is ~0.4.
• Sextupole tune spread << 0.4 will be shifted awayfrom coherent frequency.
– Or no Landau damping
• How come sextupole tune spread work in damping coherent instabilities?
• Is it possible that e-cloud cancels part of thespace-charge effect of the beam?
• However, e-cloud effects should not be too large to introduce new instabilities.
March 14-15, 2007 ECloud Feedback, IUCF 4
Simulations with POSINST
• Booster circumference: 474.203 m.
• 80 consecutive bunches + 4 empty buckets.
• Bunch intensity Nb = 6 x 1010.
• Near injection, total energy E = 1.4 GeV.γ = 1.492, β = 0.7422.
• Betatron tunes ≈ 6.8.
• RMS bunch length: σz = 70 cm (3.15 ns).
• Transverse beam sizes: σx = σy = 4.477 mm,(rms normalized emittances ~2 mm mr.)
• Gaussian distribution assumed.
• Vacuum pressure: 2 x 10-7 Torr.
March 14-15, 2007 ECloud Feedback, IUCF 5
Booster Magnets
• F Quad approximatedas 6”x1.64” rectangular opening.
• D Quad approximatedas 6”x2.25” rectangular opening.
• There are also 1.125”long-straight sectionsand 2.125”short-straight sections.
March 14-15, 2007 ECloud Feedback, IUCF 6
• Booster does not have a beam pipe inside the magnets.
• Beam sees magnet laminations, for which we do not know the SEY.
Av. proton linear density
March 14-15, 2007 ECloud Feedback, IUCF 7
Magnets cover only ~60% of Booster Rings.
The rest are cylindrical S.S. beam pipes joining the magnets.
Av. proton linear density
March 14-15, 2007 ECloud Feedback, IUCF 8
Landau Damping in Presence of Sp-Ch
• E. Métral and F. Ruggiero studied Landau dampingwith octupole tune spread in presence of sp-ch.[CERN-AB-2004-025 (ABP), 2004; Möhl earlier]
• They solved a simplified dispersion relation analytically.
• Non-linear incoherent sp-ch tune shift as well asoctupole incoherent tune shift are included.
• They plot ReΔcoh vs. ImΔcoh, showing the stableand unstable regions.
• LHC parameters are used.
March 14-15, 2007 ECloud Feedback, IUCF 9
Stability Contours in Presence of Octopole Tune Spread and Decreasing Space Charge Tune
Spread
Nb/4
Nb=1.15x1011
Nb/2
Nb/10
Outs
ide
unst
able
Insi
de s
table
ΔQoct=0.000056rms
coh
cohcoh
coh
cohcoh
cohcoh
March 14-15, 2007 ECloud Feedback, IUCF 10
Ou
tsid
e u
nst
ab
leIn
sid
e s
tab
le
March 14-15, 2007 ECloud Feedback, IUCF 11
ΔQoct/2
Stability Contours in Presence of Space Charge with Octupole Tune Spread (ΔQoct)
Decreasing
ΔQoct=0.000056
ΔQoct/4 ΔQoct/10
Outs
ide
Unst
able
Insi
de S
table
ΔQoct=0.000056
rms
rms
rms
rms
rms
coh
coh
coh
coh
coh
coh
coh
coh
March 14-15, 2007 ECloud Feedback, IUCF 12
Conclusion
• Without octupole tune spread,
– incoherent sp ch tune spread alone does not provideLandau damping.
• With octupole tune spread,
– damping region is increased in the presence of sp ch to roughly sp ch tune spread,
– there is a big shift of the damping region.
– To be Landau damped, there must be large inductiveimpedance.
• This result has been verified by simulations.(V. Kornilov, O. Boine-Frankenheim and I. Hofmann, HB2006)
March 14-15, 2007 ECloud Feedback, IUCF 13
March 14-15, 2007 ECloud Feedback, IUCF 14
March 14-15, 2007 ECloud Feedback, IUCF 15
Transverse Impedance of Booster
• Left: Computed Z1V of magnet laminations.
• Right: Im Z1V of Booster inferred from tune-
depression measurement (X. Huang).
March 14-15, 2007 ECloud Feedback, IUCF 16
Contribution of Inductive Walls
• From inductive magnet laminations and beam pipe,
= 0.026 at injection
• Inductive tune shift is too small to counteract space charge.
March 14-15, 2007 ECloud Feedback, IUCF 17
Electron Cloud Density (D Quad)
• Electron density is ρσ ~ 2.5 x 1013 m-3, ρc ~ 1 x 1013 m-3.
• Proton density is ρσ ~ 6.4 x 1014 m-3, ρc ~ 1.7 x 1014 m-3.
• Space charge canceled by small amount at bunch center,but more at head and tail.
ρσ
ρcρav
March 14-15, 2007 ECloud Feedback, IUCF 18
Short-Range Wake from E-Cloud• Heifets derived short range wake from e-cloud
depends on cloud/beam trans sizes, (Σy/σy)
p = σy/σx
• Can be approx. by a
resonance:
Σy/σy=2, Q = 6.0, μ =0.9, Wmax = 1.014
March 14-15, 2007 ECloud Feedback, IUCF 19
Impedance from E-Cloud
• Fitted impedance
• Near injection,with ρe = 1013 m-3,ImZ1
V ~ 9.4 MΩ/mat low frequencies.
• ωe/2π~100 MHz is small, because of long σz and large σx, σy.
• ρe = 1012 m-3 is often used for analysis of beam stability??
March 14-15, 2007 ECloud Feedback, IUCF 20
Bunch Length and Electron Bounce Frequency
March 14-15, 2007 ECloud Feedback, IUCF 21
Trans. Microwave (Strong Head-Tail)
ωeL
ωeσ
• ωeσ ≤ ½π but ωeL~ 6 to 11 >> π
• Linear part of e-cloud wake contributes.• Use Métral’s long-bunch formula to compute Upsilon.• Upsilon > 2 implies instability.
March 14-15, 2007 ECloud Feedback, IUCF 22
• Booster cannot operate with ξx = ξy= 0, beam unstable.• With ξx and ξy setting, Upsilon is reduced, but still > 2
when close to transition.• Maybe space charge will help. (Blaskiewicz, PR STAB 044201)
• Maybe peak of ReZ1V is not so sharp (or Q is lower).
• Maybe e-cloud density is much less than 1013 m-3.
March 14-15, 2007 ECloud Feedback, IUCF 23
Summary
• Simulations show that e-cloud accumulation is large.Saturation has been reached.
• ρe~ 1013 m-3 amounts to only 1/10 of proton density.It is unsure whether enough sp-ch will be canceledto ensure Landau damping.
• E-cloud leads to a wake that may cause strong head-tail instability.
– Upsilon >2 close to transition, not good.
– Maybe sp ch will delay 2 azimuthal modes to collide.
– Maybe ReZ1V peak is not so sharp (lower Q).
– Maybe actual e density is smaller, thus lowering Upsilon.
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