Burov, A

Foster, G W

Shiltsev, V D

CERN Document Server

Space-Charge Compensation in Proton BoostersA.V. Burov, G.W. Foster, V.D. ShiltsevFNAL, P.O. Box 500, Batavia, IL 60510AbstractRecently, it was proposed to use negatively charged elec-tron beams for compensation of beam-beam effects due toprotons in the Tevatron collider [1]. We show that a sim-ilar compensation is possible in space-charge dominatedlow energy proton beams. The idea has a potential ofseveral-fold increase of the FNAL Booster beam bright-ness. Best results will be obtained using three electronlenses around the machine circumference, using co-movingelectron beam with time structure and profile approximatelymatched to the proton beam. This technique, if feasible,will be more cost effective than the straightforward alter-native of increasing the energy of the injection linac.1 INTRODUCTIONDefocusing repulsive forces due to self space charge leadto degradation of high-intensity particle beams. In circularaccelerators, the figure of merit of such an effect is shift ofincoherent betatron oscillations [2]:∆νSC = −NtotrcBf4πεnβpγ2p,where Ntot is total number of particles in the ring, rc =1.53 · 10−18m for protons, εn is rms normalized emittance,βp = vp/c and γp are usual relativistic parameters, andBf ≥ 1 is a peak to average current ratio. Normally, forproton low-energy synchrotrons the tune shift lays in rangeof -0.1...-0.5 (see, e.g.,[4]). Above the threshold, the beamemittance dilute and particles are lost. Due to the accelera-tion, the short time at low energy is enough for developingonly the lowest order resonances.Compensation of space charge forces on average, cansignificantly improve dynamics in low-energy accelerators,and allow higher beam intensities.2 COMPENSATION OF SPACE CHARGEFORCES BY ELECTRON BEAMSNegatively charged electron beams can be used for com-pensation of space-charge effects in low energy protonbeams the same way as in the ongoing Tevatron Elec-tron Lens (TEL) project to compensate beam-beam effectsdue to protons in the Tevatron collider [1]. Protons goingthrough the electron beam experience focusing force whichhas opposite sign to self space charge force and can pre-cisely fully or partially compensate the latter.Practically, the electron beam set-up could occupy onlya small fraction of the ring circumference C . Therefore,the kick experienced by every proton could be compensatedonly in average. In the same time, the proton bunch struc-ture can be matched by fast modulation of the electron cur-rent.Assumingκ being a degree of the compensation, the tuneshift on the electron beam can be presented as∆νe ≡ βr4π(1− βeβp)JeLrceβecσ2eγpβp= −κ∆νSC ,where Je is electron current, σe is the rms electron beamradius, L is interaction length, βr is beta function at the lo-cation of the electron lens.The necessity of fast longitudinal electron current mod-ulation for matching the proton bunch profile requires forelectrons to move in the same direction as protons.Assuming the rms size of the electron beam matching theproton beam size in the location of the electron lens σ2e =εnβpγpβr , then we obtain the necessary electron current inthe lens:Je = κJpBfCLβeγ2pβ2p(1− βeβp),As it is shown in the next Section, optimum compensationrequires κ  0.3.3 COHERENT MODES AND OPTIMALCOMPENSATIONThe beam own space charge forces shift down bothsingle-particle (incoherent) modes and the beam coher-ent modes. An exception is the coherent dipole mode(beam centroid motion) which does not depend on the spacecharge, when the image charges are neglected. If one ofthese modes is in resonance with the focusing lattice (struc-ture resonance), the mode becomes unstable, that leads tothe beam loss and/or growth of the beam emittance. An im-portant point is that the space charge and electron tune shiftsare mode-specific, so if the electron lens compensates thespace charge tune shift for one of the modes, it does notcompensate it for others. Therefore, the electron lens canincrease the proton brightness threshold but can not elimi-nate it. The incoherent tune shift introduced by the electronlens ∆νe can be considered as proportional to the Lasletttune shift ∆νSC, i. e. ∆νe = −κ∆νSC , with compensa-tion coefficient κ to optimize.In the smooth approximation, assuming approximatelyequal vertical and horizontal lattice tunes νx ≈ νy ≈ ν0,and small tune shifts ∆ν << ν0, the single-particle fre-quency goes asν = ν0 + (κ− 1)∆νSC . (1)0-7803-7191-7/01/$10.00 ©2001 IEEE. 2896Proceedings of the 2001 Particle Accelerator Conference, ChicagoIn this case, the coherent tunes of the dipole and two en-velope modes (subscripts ”d”, ”+” and ”-”) are given asνd = ν0 + κ∆νSCν+ = 2ν0 + (2κ− 1)∆νSCν− = 2ν0 + (2κ− 3/2)∆νSC .(2)A consequence of these simplified expressions is that anexact compensation of the space charge tune shift for allthe three tunes, or even for any two of them, is impossi-ble; if one of the modes is compensated, the two others arenot. Instead, there is a best compensation degree κ, with∆ν0 = κ∆ν , where the stopping resonance is reached atthe maximal space charge.Table 3 summarizes data from various space-charge lim-ited low-energy proton synchrotrons [4] - the first rowshows bare lattice tunes νx and νy, the second row containsmaximum empirically achieved Laslett tune shift for eachmachine ∆νexp. We compare experimental values withcalculated distance ∆νinc to the nearest half-integer reso-nances: incoherent (third row) and coherent ∆νcoh (bot-tom row), calculated as in Ref. [3].The table reveals some noticeable features. First, the em-pirically reached Laslett tune shifts significantly exceed thesingle-particle limits ∆νinc. Second, for all machines butAGS and AGS-Booster ∆νexp is close to the threshold ofthe envelope instability ∆νcoh. This leads to a conclusionthat the incoherent resonances may not be important; the ac-tual threshold is rather determined by the coherent instabil-ities. This conclusion agrees with the analysis of Ref.[3].Finally, the resonances at ∆νcoh ≤ 0.1 do not reveal them-selves, perhaps, due to a weak space charge at the resonancecrossing.Behavior of the modes changes when the compensationis applied. Dynamics of the dipole mode is rather differ-ent from the envelope modes, that makes it difficult to evendouble the threshold by varying the compensation degree κ,because one or another mode should cross linear resonance.Threshold Laslett tune shift of about 0.6 can be achievedwith κ = 0.33 (a factor of ≈ 1.6 in comparison with theno-compensation case).An additional option is associated with a possibility tohave the degree of compensation κ variable and dependenton the Laslett tune shift. In that case, modulation of theelectron beam current is almost as easy as for the constantκ strategy. The variable rate allows to jump over the reso-nances, which could additionally increase the space chargelimits.Symmetry of an accelerator focusing lattice is impor-tant for the space-charge compensation. If the lattice con-sists of P identical periods, then strong and wide structurelinear resonances occur at coherent/incoherent frequenciesνcoh, inc = Pm/2, where m is integer.Thus, more than one compensation set-up is needed tokeep nearest linear resonances as non-structure. For exam-ple, the Fermilab Booster has periodicityP = 24, and tunesνx,y = (6.7, 6.8). With two electron lenses symmetricallyplaced in opposite parts of the ring, the effective periodic-ity is reduced to P = 2. Thus, the nearest incoherent res-onance ν = 6.5 is non-structure (as it is without compen-sation devices at P = 24), but the envelope resonances atν−,+ = 13 = Pm/2 = 2 · 13/2 become structure. Asthe result, three identical compensators symmetrically po-sitioned at the orbit are required in order to preserve non-structure weakness of the nearest incoherent and coherentresonances.4 NUMERICAL EXAMPLE: FERMILABBOOSTERLet us consider the space-charge compensation in theFermilab Booster. Parameters of FNAL Booster are pre-sented elsewhere [5, 4]). The space charge tune shift at in-jection is about∆νSC(t) = −0.435·Bf(t)·([εnβpγ2p ](t)/[εnβpγ2p ](t = 0))if parameters are Ntot = 5 · 1012, εn = 1 πmm · mrad,γp = 1.4, βp = 0.7. The period of strong space-chargeaction is comparatively short - injection itself takes about12 turns, then some 20-40 turns the beam circulates with-out RF to wipe off the Linac RF structure - it correspondsto Bf = 1. After that the Booster RF is being adiabati-cally turned on over some 100 turns, and then the beam en-ergy rapidly grows, e.g., to 2.5 GeV after about 3000 turns.Note, that the ratio of peak to average current rapidly in-creases from 1 to Bf (t) ≈ 2 − 3 over a hundred turns,while the factor 1/(βpγ2p) slowly decreases 4 times from0.71 to 0.174 over thousands of turns. If there would be noparticle loss and emittance blow up then the Booster beamspace charge tune shift parameter would reach the maxi-mum value of right after bunching ∆νSC ≈ 0.9− 1.3.Currently, due to space charge effects, the beam losessome 35-40% of particles during that initial period of 5-6 ms, and its transverse emittance blows up about 3 times[6]. From that we can conclude that the beam brightnessdeteriorates quickly if the space charge tune shift exceeds∆νSC ≥ −(0.25−0.4), that is consistent with conclusionsof previous studies in the Booster [7].According to our analysis in the previous Section, the op-timal compensation coefficient is less than 1, κ = 0.33 −0.55, therefore, we can present the required peak electroncurrent asJe = (Bf · κ) · ecNtotLNl· βeγ2p(1− βpβe).As concluded in Section 3 above, it is beneficial to installNl = 3 equally strong lenses in the Booster, so if each isL =4 m long (to fit 6 m long drift sections), then forBf = 3and Ntot = 5 · 1012 we yield Je(κ = 0.33) = 8.3Aof the peak current in each of three 70 kV electron beams,βe = 0.52. The choice of the electron beam energy and βeis made to satisfy condition σs ≈ L(βp − βe)/βp, wherethe rms proton bunch length of σs = 1 m is taken. If the2897Proceedings of the 2001 Particle Accelerator Conference, ChicagoParameter KEK-B FNAL-B ISIS AGS AGS-B CERN PS CERN PS-2νx/νy 2.17 / 2.3 6.7 / 6.8 3.7 / 4.2 8.75 / 8.75 4.8 /4.9 6.22 / 6.22 6.22 / 6.28∆νexp 0.23 0.4 0.4 0.58 0.5 0.27 0.36∆νinc 0.17 0.2 0.2 0.25 0.3 0.22 0.22∆νcoh 0.27 / 0.08 0.36 / 0.08 0.32 0.33 0.07 / 0.2 0.27 0.33Table 1: Laslett tune shift reached empirically ∆νexp at various low-energy synchrotrons as compared with calculatedLaslett tune shifts related to the nearest incoherent ∆νinc and coherent ∆νcoh resonances.ultimate goal is to double the Booster intensity - that isallowed by the FNAL Linac and will require more BoosterRF power - then Je(κ = 0.33)=16.6 A of the electron cur-rent are needed.In general, the Booster Electron Lens (BEL) will bemuch alike the Tevatron Electron Lens (TEL) for compen-sation of beam-beam effects [1].The BEL will consist of solenoid magnets for electronbeam transport, electron gun and collector, high voltagemodulator, vacuum system, beam diagnostics and control.The 4 m long interaction region has to be inside a pair of2-m long 2-4 kG solenoid magnets with opposite polarityof the magnetic field. Such a configuration will providezero field integral∫Bdz = 0 and, thus, will minimize theBooster focusing lattice perturbation. Magnetic field linesin these magnets have to be straight within 1 mm (some20% of the proton beam size); a number of additional dipolecorrector coils are needed for the electron beam steering.Shorter 2-4 kG side solenoids are necessary for electronbeam injection and extraction. It may be necessary to addtwo toroids to make the 90 degrees turns smoother. Thiswill be the subject of more detailed studies. A lot of sub-systems, including the gun and collector, beam diagnos-tics, HV pulser can be just copied and/or upgraded from theTevatron Electron Lens set-up, see Ref.[1] and referencestherein.5 SUMMARYThe use of low-energy high-current electron lenses tocompensate space charge effects in high-brightness pro-ton accelerators looks very promising. Our considerationshows that in optimal configuration few (three) electronlenses can compensate space charge tune shifts as large as∆νSC ≈ 0.6 − 1.2. With further increase of the protonbeam brightness some coherent modes or incoherent mo-tion can become unstable while crossing particular reso-nances.We have considered electron lenses for the compensa-tion in the Fermilab Booster. We found that electron beamsystems are quite feasible for realization of either protonbeam emittance upgrade or two-fold intensity upgrade inthe Booster. These upgrades can significantly improve theperformance of fixed target experiments, e.g., NuMI; and/orwill allow to increase the antiproton production, protonbeam brightness in collision and ultimately increase the lu-minosity of the Tevatron.In principle, there are other tools to compensate the tuneshift but they all have serious disadvantages. For exam-ple, pulsed γt-jump quadrupoles in the Booster can shiftthe tune for about 2, but it does not seem possible to mod-ulate their gradients over the bunch length of few ns. RFquadrupoles can provide necessary modulation along thebunch but will require very powerful RF sources.The proposed electron lens has a number of advantages:easier beam shape control and current modulation, mod-erate power consumption. The parameters of the lens arequite feasible that make the proposal attractive for furtherconsideration and practical implementation.The proposed compensation can also allow a substan-tial brightness increase in low energy proton and ion stor-age rings with electron cooling, as many of them are space-charge dominated.Space charge compensation in low energy (∼ 1 MeV)coasting proton beams by ionization electrons was reportedin [8]. The compensation allowed many-fold increase of theproton intensity compared to the proton space-charge limit.We think that advantages of the well-controlled electronbeams will allow to achieve comparable success in high-energy bunched proton beams as well.We acknowledge helpful discussions with R.Webber,J.Lackey, R. Baartman, J.Marriner, S.Y.Lee, V.Dudnikovand S.Holmes.6 REFERENCES[1] V. Shiltsev, et.al, this conference; see also Phys.Rev.ST-AB,2, 071001 (1999).[2] M. Reiser, Theory and Design of Charged Particles Beams,John Wiley&Sons (1994).[3] R. Baartman, KEK Proceedings 95-7 (1995), 273.[4] W. Chou, in ICFA Beam Dynamics Newsletter, No.20 (1999).[5] E.L. Habbard, FNAL TM-405 (1973).[6] R. Webber, FNAL-Conf-00/193 (2000).[7] C. Ankenbrandt, S. Holmes, in Proc. 1987 IEEE PAC, Wash-ington, D.C., v.2, (1987), 1066S. Holmes, in AIP Conf. Proc. 377 (1995), 42.[8] Yu. Belchenko, et.al, in Proc. IX Int. Part. Accel. Conf., v.2,Protvino (1977), 287.2898Proceedings of the 2001 Particle Accelerator Conference, Chicago

Space-Charge Compensation in Proton Boosters

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