Ng, K Y

Zotter, Bruno W

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SPACE CHARGE AND BEAM STABILITY OF THE PROPOSEDFERMILAB PROTON DRIVER IN PHASE IK.Y. Ng, FNAL, Batavia, IL 60510, USA and B. Zotter, CERN, CH-1211, Geneva 23, SwitzerlandAbstractIssues concerning beam stability of the proposed Fermi-lab proton driver are studied in its Phase I. Although thebetatron tune shifts are dominated by space charge, theseshifts are less than 0.25 and will therefore not drive thesymmetric and antisymmetric modes of the beam envelopeinto instability. The longitudinal space-charge force is largeand inductive inserts may be needed to compensate for thedistortion of the rf potential. Although the longitudinalimpedance is space-charge dominated, it will not drive anymicrowave instability, unless the real part of the impedancecoming from the inductive inserts and wall resistivity ofthe beam tube are large enough. The design of the beamtube is therefore very important in order to limit the flow ofeddy current and keep wall resistivity low. The transverseimpedance is also space-charge dominated. With the protondriver operated at an imaginary transitiongamma, however,Landau damping will never be canceled and beam stabilitycan be maintained with negative chromaticities.1 INTRODUCTIONThe proposed Fermilab proton driver will be a 15 Hzrapid-cycling high-intensity synchrotron that serves a num-ber of purposes in the Fermilab high energy physics pro-gram. In Phase I Stage 1, the proton driver, of circumfer-ence C = 2πR = 711.3 m, accelerates N = 3×1013 pro-tons in nb = 126 bunches with a 53 MHz rf system fromthe injection kinetic energy (K.E.) of 400 keV to the K.E.of 12 GeV to be injected into the Main Injector. In Stage 2,the 3×1013 protons are divided into nb = 18 bunches andaccelerated with a 75 MHz rf system to the K.E. of 16 GeVfor the production of muons destined for a neutrino factory.Because of its low-energy injection, space-charge effectsare significant. Because of its high intensity, collective sin-gle and multi-bunch instabilities are important issues. InStage 2, there is a bunch compression by rotation before ex-traction, which will be covered in detail in Ref. [1].2 SPACE CHARGE AND TUNE SHIFTSThe stainless steel vacuum chamber is of elliptical crosssection with horizontal/vertical radii w = 119 mm andh= 63 mm. The magnet pole faces have vertical half gapg = 64 mm covering about κ= 50% of the ring. The barehorizontal/vertical tunes are ν0x,y=12.428/11.380 so thatthe average betatron functions are 〈βx,y〉=9.109/9.948m.The average dispersion is 〈D〉 = 1.247 m. The beam hasnormalized 90% emittance x,y = 60× 10−6πm. Thus,at injection the beam has elliptical cross section with hor-izontal/vertical radii ax,y = 23.2/24.2 mm, where the fullenergy spread is 7 × 10−4 throughout the cycle. The rfFigure 1: (color) Bunching factor, energy and tune shifts duringone acceleration cycle.frequency is 53 MHz. An acceleration cycle has been de-signed [2] with K.E. and bunching factor Bf as shown inFig. 1. With z=x, y, the incoherent and coherent tune shiftscan be computed according to [3]∆νzinc=− NrpRπβ2γν0z[γ−2−χeBfaz(ax+ay)++(β2 +γ−2−χeBf)1zh2+ κβ22zg2],∆νzcoh=− NrpRπβ2γν0z[γ−2−χeBfξ1zh2+β21zh2+κβ22zg2],where γ and β are the relativistic factors of the protons, rpthe proton classical radius, and χe the neutralization factor.In above, 1z,2z are the Laslett incoherent electric/magneticimage coefficients, while ξ1z are the coherent electric imagecoefficients. For the elliptical beam pipe, we have 1y =−1x=0.1669, ξ1x=0.0948, ξ1y =0.5956. For the mag-netic coefficients, parallel-plate approximation gives 2y =−2x = π2/24. The tune shifts are plotted in Fig. 1. Stage 2will be similar in the early part of the cycle. We see thatthe tune shifts reach their maxima about 1–2 ms after in-jection and capture in the rf field, when the bunching factordecreases sharply fromBf =0.9 but the beam energy is stilllow. We also see that the incoherent tune shifts are less than0.25, although space charge is dominating. However, if thetransverse distribution is not uniform but peaked, the space-charge tune shift can be higher. Simulation shows that theincoherent vertical tune shift is slightly above 0.25 and thehorizontal slightly below [4]. But this is of no concern, be-cause it is the coherent tune shifts that play a role of reso-nant instability. Dipole errors cannot deflect the beam co-herently [5]. Quadrupole errors will shift the 2 quadrupolemodes by 34and 12of the incoherent space-charge tune shiftwhich is less than 0.2. Thus, the choice of the current baretunes will not lead to a half integer resonance.0-7803-7191-7/01/$10.00 ©2001 IEEE. 2887Proceedings of the 2001 Particle Accelerator Conference, Chicago3 SINGLE BUNCH INSTABILITIES3.1 Broad-band impedance estimatesThe largest impedances in the proton driver at low ener-gies are due to space charge [6]:Zsc‖n= −j Z02βγ2gf = −384j Ω , Zsc⊥ x,y =−j 2RZ0β2γ2[1az(ax + ay)− ξ1z−1zb2]=−70j/63jMΩ/m ,where Z0≈377 Ω. With the vacuum chamber made of In-conel having resistivity ρc = 129 µΩcm and relativity per-meability µr = 1.002, the wall resistivity is, at injection,Zrw‖n= (1+j)βµrZ0δh2h= 1.48(1+j)n−1/2 Ω .The transverse impedance is found by multiplication with2R/(βh2) or 0.033/0.118(1+j)|n−νx,y|−1/2 MΩ/m. Inabove, δh=√2ρc/(ωµ0µr) is the skin depth.Other sources of broad-band impedance are the rf cavi-ties loaded with ferrite (or Finemet), kicker tanks, bellows,and variations of vacuum cross section. However, they willcontribute mainly inductive impedance at low frequencies.3.2 Eddy Current and shielding of em fieldsTo prevent excessive eddy current losses, the metallicvacuum chamber can be divided into narrow strips, similarto the wire-cage used in ISIS [7]. The eddy current powerloss per unit length in a metal strip of width w, thicknesst and conductivity σc, at right angle to a changing mag-netic field with time derivative B˙, is approximately givenby σctw3B˙2/12. Since it is proportional to the third powerof the width, it can be reduced from, for example, over8 kW/m for a 1.3 mm thick, 22 cm wide, elliptic Inconelchamber to a few W/m by replacing it with 2 × 50 copperstrips of 4 mm width and 50–100 µm thickness.When the vacuum chamber is thin enough, we have toworry about the field of the beam penetrating into the mag-netic laminations, where the large relative permeability µrtranslates into large Re Zrw‖ . For this purpose, a screen orliner has been proposed, similar to the radiation shield forthe LHC [8]. To minimize eddy current losses, the screenshould be made as thin as possible. Assuming rotationalsymmetry, the shielding condition in the longitudinal direc-tion of a screen of thickness t at radius b, with skin depthδb  t, and an outer wall at radius d, with skin depth δd,can be written as [9]tδb β2γ22 ln (d/b)δbb.In the transverse direction, the shielding condition underthe same assumption is the same but without the factor of2 ln(d/b). Hence, once longitudinal shielding is achieved,transverse shielding is guaranteed.For higher energies, the shielding criterion becomest/δb (d/b)(δb/δd), which is difficult to fulfill. However,at higher energies the beam is more stable and the spacecharge part of the impedance is strongly reduced.3.3 Longitudinal stability criteriaThe Boussard-Keil-Schnell criterion for longitudinalbunched beam stability is [10]∣∣Z‖∣∣n< F|η|BfEeβ2Ib[∆EE]2FWHM,where Ib is the average bunch current, η = γ−2t −γ−2 theslip factor, and the form factor F ≈ 1 for a parabolic dis-tribution distribution, but is actually very much larger fora space-charge dominated beam. With a transition gammaof γt = 27.71j and F = 1, the stability limit starts from24.95 kΩ at injection and drops monitonically to 0.16 kΩat extraction, and is at all times larger than the space-chargeimpedance |Zsc‖ /n|. For Phase I Stage 2, the same current isdivided into only 18 bunches so that the threshold is 7 timessmaller, but is still much larger than |Zsc‖ /n| at all times.3.4 Transverse stability criteriaThe Boussard-Keil-Schnell-like criterion for transversebunched beam stability is (for z=x, y) [11],|Z⊥z| < F 4BfEeβ〈βz〉Ib[∆EE]FWHM|Sz| ,where the effective chromaticity is Sz=(n−νz)η+ξz, withn an arbitrary integer and ξz the chromaticity. Instabilityoccurs only for slow waves with n>νz. The form factor Fdepends on the transverse particle distribution, but is largecompared to unity for a space-charge dominated impedanceas in the longitudinal case. Here, η is always negative. Ifalso the chromaticity ξz is negative, the two terms add andcannot cancel. The lowest value of the spread is obtainedfor n just above the tune νz. Since also |η| ≈ 12 , the firstterm in Sz can be neglected if |ξz| is large enough. ForF = |Sz | = 1, the transverse impedance threshold dipsdown to 11 MΩ/m in Stage 1 about 5 ms into the accel-eration cycle. This is very much larger than |Zrw⊥ z| evenwhen |n−ν| = 1, and requires ξz ≈ −4 to cope with thespace-charge impedance. In Stage 2, however, the thresh-old is 7 times smaller and is still much larger than |Zrw⊥ z|,but requires F |Sz| ≈ 28 to overcome the space-chargeimpedance.To safeguard stability, one may blow up the lon-gitudinal emittance so as to attain a larger energy spread,thus enhacing Landau damping.3.5 Inductive InsertsTo compensate for the large space-charge distortion ofthe rf wave form, inductive inserts can be installed for com-pensation. However, the ferrite cores in the inserts can havelarge Re Z‖ and may drive the beam into microwave insta-bility. Thus, we should choose ferrite with small losses orwe may lower the ferrite losses by heating the inserts [12].4 COUPLED-BUNCH INSTABILITIES4.1 Longitudinal stability criterionLongitudinalcoupled-bunch instabilityare mostly drivenby the parasitic resonances in the rf system. The 53 MHz rfcavities used in the present Fermilab booster and the past2888Proceedings of the 2001 Particle Accelerator Conference, ChicagoFermilab Main Ring have a large number of higher-orderresonances and have driven coupled-bunch instabilities inthese two machines. However, careful passive detuning ofthese resonances has gotten rid of most instabilities. Sincethe total number of protons in the proton driver will beabout 3 times that in the present booster, the coupled-bunchgrowth rates will be 3 times as fast. As a result, past expe-rience in passive damping of parasitic resonances may notbe sufficient to cope with the coupled-bunch instabilities.Further lowering of growth rates can be achieved by blow-ing up the longitudinal emittance of the bunches which al-ters the form factor by decreases the coherency inside thebunches, or by installing a fast bunch-by-bunch damper.In Stage 2, 75 MHz Finemet rf cavities with low qualityfactor Q≈10will be used. As a result, there will not be anyparasitic resonances inside the cavities to drive coupled-bunch instabilities, either longitudinally or transversely.4.2 Transverse stability criterionPossible transverse coupled-bunch instabilities driven byparasitic resonances in the 53 MHz rf cavities can be curedby detuning passively as in the longitudinal situation, orby blowing up the longitudinal emittance of the bunches.However, transverse coupled-bunch instabilities can alsobe driven by the resistive wall impedance at the frequency([νz]−1)f0, where f0 is the revolution frequency and [νz]the noninteger part of the betatron tune. For nb buncheseach with average current Ib, the growth rate is [11]1τ= − 1m+1enbIbc4πνzE∑kRe Z⊥ z(ωk)F ′m(ωk−χz) ,where χz=−ξzω0τL/η is the chromatic phase shift acrossthe full length τL of the bunch. The frequency spectrumfor the m-th mode of transverse oscillations with coupledbunch mode number n (0 ≤ n < nb) is given by ωk=knb+n + νz + mνs. The form factor F ′(ω) expresses thecancellation occurring in the summation over both positiveand negative frequencies of the impedance, weighted by thespectrum of the m-th mode of oscillation. In this situation,however, the contribution is dominated by only one spectralline at the above low frequency and F ′ ≈ 0.8. Fortunatelythe ring is small; with the resistive wall impedance com-puted in Sec. 3.1, the growth times are just 137/37 ms hor-izontally/vertically with ξz=0 at injection in both Stages 1and 2. The growth rate for mode m=0 can be lowered byoperating at a negative chromaticity. However, this may in-troduce instabilities of mode m≥ 1 which can be dampedby a tune spread of order 1/(ω0τL) that usually occurs nat-urally; thus a low-frequency damper may not be necessary.5 ELECTRON-PROTON INSTABILITYWhen a proton beam is partially neutralized, with frac-tional neutralization χe, the electrons in it will start to os-cillate transversely with the bounce frequency given by, forsmall amplitudes, ω2e = 4c2renp, where re is the classicalelectron radius, and np = Nb/(πaxayLb) is the volumedensity of the protons in a bunch with Nb protons, crosssectional area πaxay and full length Lb. The oscillatingelectrons will excite the protons to oscillate with frequencyω2p =4c2rpne=4c2rpχenp. The coupled oscillations leadto a dispersion relation [13] for the e-p oscillation frequencyω as function of the azimuthal mode number n:(ω2e − ω2)[(νzω0)2 + ω2p − (nω0 − ω)2] = ω2eω2p .While the amplitudes of the heavier protons are small,those of the lighter electrons are larger and will growrapidly, reach the vacuum chamber wall, and generate moreelectrons by secondary emission, which can lead to anavalanche effect called multipactoring. This may then leadalso to emittance growth and/or beam loss of the protons.e-p oscillations had been seen in the CERN ISR and hadto be eliminated by better clearing and pumping. They arealso seen in a number of proton storage rings, like the PSRat Los Alamos, where the beam current remained below thedesign goal for many years after many attempts to eliminateor at least reduce the number of electrons failed.The most common method to reduce neutralization of abeam is to leave a gap in the train of bunches, which has alsobeen foreseen for the Fermilab proton driver. A more activemethod is to install clearing electrodes, and eliminate theelectrons by applying transverse electric fields. If the elec-trons are generated by vacuum, better pumping may help—but not if the electrons are coming from the H− strippingfoil. Multipactor at the wall can be reduced by coating witha material with low secondary emission coefficient, such asTi-N. If nothing else helps, a feedback system can be thesolution. For the case of the LANL PSR, all these methodswere tried and failed, and only a combination of higher rfvoltage, sextupoles, partial wall coating and finally an in-ductive insert permitted to reach the design current.For the present proton driver, the injection from the linacis in chopped beams. As the rf voltage is raised, the bunch-ing factor decreases rapidly from Bf =0.9 to 0.44 in 2 msand 0.18 one half into the acceleration cycle. Thus, therewill be large bunch gaps so that trapped electrons should becleared and hence no e-p instability is expected.6 REFERENCES[1] K.Y. Ng, these proceedings.[2] Z. Qian, private communication.[3] J. Laslett, BNL Report 7534, (1963) p. 234; J. Laslett andL. Resegotti, Proc. 6th Int. Conf. High Energy Acc., Cam-bridge 1967, P.150.[4] C. Prior, private communication.[5] R. Baartman, AIP Conf. Proc. 448 (1998) p. 56.[6] D. Moehl, A. Sessler, Proc. 8th Int. Conf. High Energy Acc.,CERN, 1971, p.334.[7] I. Gardner et al., Part. Accel. 31, 227 (1990).[8] F. Caspers et al., Proc. PAC New York, 1999, p. 1408.[9] R. Gluckstern, CERN Report CERN 2000-011, 2000.[10] E. Keil and W. Schnell, CERN/ISR-TH/69-48, 1969;D. Boussard. CERN Lab II/RF/75-2, 1975.[11] F. Sacherer, CERN Report 77-13, 1977.[12] K.Y. Ng, et al., these proceedings.[13] B. Zotter, CERN-ISR-TH/71-58, 1971.2889Proceedings of the 2001 Particle Accelerator Conference, Chicago

Space Charge and Beam Stability of the Proposed Fermilab Proton Driver in Phase I

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