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.

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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.

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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.

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• 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

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Magnets cover only ~60% of Booster Rings.

The rest are cylindrical S.S. beam pipes joining the magnets.

Av. proton linear density

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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

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Ou

tsid

e u

nst

ab

leIn

sid

e s

tab

le

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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)

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March 14-15, 2007 ECloud Feedback, IUCF 14

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Transverse Impedance of Booster

• Left: Computed Z1V of magnet laminations.

• Right: Im Z1V of Booster inferred from tune-

depression measurement (X. Huang).

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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.

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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

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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??

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Bunch Length and Electron Bounce Frequency

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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.

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• 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.

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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.