March 14-15, 2007ECloud Feedback, IUCF1 Electron-Cloud Effects in Fermilab Booster K.Y. Ng Fermilab Electron-Cloud Feedback Workshop IUCF, Indiana March 14-15, 2007

ECloud Feedback, IUCF2 Motivation I  E-Cloud observed at CERN SPS. Want to know what happens to Fermilab Booster. CERN SPSFermi Booster NbNb 8 x x No to start E-Cloud20?? Bunch Spacing25 ns26.4 ns E inj 26 GeV1.34 GeV Vacuum7.5 x Torr~2 x Torr

March 14-15, 2007ECloud Feedback, IUCF3 Motivation II Fermilab Booster is injected at 400 MeV. –Space-charge tune shift is ~0.4. Sextupole tune spread << 0.4 will be shifted away from 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 the space-charge effect of the beam? However, e-cloud effects should not be too large to introduce new instabilities.

March 14-15, 2007ECloud Feedback, IUCF4 Simulations with POSINST Booster circumference: m. 80 consecutive bunches + 4 empty buckets. Bunch intensity N b = 6 x Near injection, total energy E = 1.4 GeV. γ = 1.492, β = Betatron tunes ≈ 6.8. RMS bunch length: σ z = 70 cm (3.15 ns). Transverse beam sizes: σ x = σ y = mm, (rms normalized emittances ~2 mm mr.) Gaussian distribution assumed. Vacuum pressure: 2 x Torr.

March 14-15, 2007ECloud Feedback, IUCF5 Booster Magnets F Quad approximated as 6”x1.64” rectangular opening. D Quad approximated as 6”x2.25” rectangular opening. There are also 1.125” long-straight sections and 2.125” short-straight sections.

March 14-15, 2007ECloud Feedback, IUCF6 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, 2007ECloud Feedback, IUCF7  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, 2007ECloud Feedback, IUCF8 Landau Damping in Presence of Sp-Ch E. Métral and F. Ruggiero studied Landau damping with octupole tune spread in presence of sp-ch. [CERN-AB (ABP), 2004; Möhl earlier] They solved a simplified dispersion relation analytically. Non-linear incoherent sp-ch tune shift as well as octupole incoherent tune shift are included. They plot ReΔ coh vs. ImΔ coh, showing the stable and unstable regions. LHC parameters are used.

March 14-15, 2007ECloud Feedback, IUCF9 Stability Contours in Presence of Octopole Tune Spread and Decreasing Space Charge Tune Spread N b /4 N b =1.15x10 11 N b /2 N b /10 Outside unstable Inside stable ΔQ oct = rms coh

March 14-15, 2007ECloud Feedback, IUCF10 Outside unstable Inside stable

March 14-15, 2007ECloud Feedback, IUCF11 ΔQ oct /2 Stability Contours in Presence of Space Charge with Octupole Tune Spread (ΔQ oct ) Decreasing ΔQ oct = ΔQ oct /4ΔQ oct /10 Outside Unstable Inside Stable ΔQ oct = rms coh

March 14-15, 2007ECloud Feedback, IUCF12 Conclusion Without octupole tune spread, –incoherent sp ch tune spread alone does not provide Landau 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 inductive impedance. This result has been verified by simulations. (V. Kornilov, O. Boine-Frankenheim and I. Hofmann, HB2006)

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March 14-15, 2007ECloud Feedback, IUCF15 Transverse Impedance of Booster Left: Computed Z 1 V of magnet laminations. Right: Im Z 1 V of Booster inferred from tune- depression measurement (X. Huang).

March 14-15, 2007ECloud Feedback, IUCF16 Contribution of Inductive Walls From inductive magnet laminations and beam pipe, = at injection Inductive tune shift is too small to counteract space charge.

March 14-15, 2007ECloud Feedback, IUCF17 Electron Cloud Density (D Quad) Electron density is ρ σ ~ 2.5 x m -3, ρ c ~ 1 x m -3. Proton density is ρ σ ~ 6.4 x m -3, ρ c ~ 1.7 x m -3. Space charge canceled by small amount at bunch center, but more at head and tail. ρσρσ ρcρc ρ av

March 14-15, 2007ECloud Feedback, IUCF18 Short-Range Wake from E-Cloud Heifets derived short range wake from e-cloud depends on cloud/beam trans sizes, (Σ y /σ y ) p = σ y /σ x Can be approx. by a resonance: Σ y /σ y =2, Q = 6.0, μ =0.9, W max = 1.014

March 14-15, 2007ECloud Feedback, IUCF19 Impedance from E-Cloud Fitted impedance Near injection, with ρ e = m -3, ImZ 1 V ~ 9.4 MΩ/m at low frequencies. ω e /2π~100 MHz is small, because of long σ z and large σ x, σ y. ρ e = m -3 is often used for analysis of beam stability??

March 14-15, 2007ECloud Feedback, IUCF20 Bunch Length and Electron Bounce Frequency

March 14-15, 2007ECloud Feedback, IUCF21 Trans. Microwave (Strong Head-Tail) ωeLωeL ωeσωeσ ω e σ  ≤ ½π but ω e  L ~ 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, 2007ECloud Feedback, IUCF22 Booster cannot operate with ξ x = ξ y = 0, beam unstable. With ξ x and ξ y setting, Upsilon is reduced, but still > 2 when close to transition. Maybe space charge will help. (Blaskiewicz, PR STAB ) Maybe peak of ReZ 1 V is not so sharp (or Q is lower). Maybe e-cloud density is much less than m -3.

March 14-15, 2007ECloud Feedback, IUCF23 Summary Simulations show that e-cloud accumulation is large. Saturation has been reached. ρ e ~ m -3 amounts to only 1/10 of proton density. It is unsure whether enough sp-ch will be canceled to 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.