Assadi, S

Mishra, C S

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Particle Accelerators, 1996, Vol. 55, pp. [407-418] /161-172Reprints available directly from the publisherPhotocopying permitted by license only© 1996 OPA (Overseas Publishers Association)Amsterdam B.Y. Published in The Netherlands underlicense by Gordon and Breach Science Publishers SAPrinted in MalaysiaSIMULATIONS OF BEAM-BEAM EFFECTSIN TEVATRON AND NONLINEARMODE COUPLING ANALYSISC.S. MISHRA and S. ASSADIFermilab, * Batavia, IL 60510, USA(Received 29 February 1996; infinalform 29 February 1996)The Fermilab accelerator complex is in the middle of an upgrade plan, Fermilab III. In thelast phase of this upgrade the number of protons and antiproton bunches will increase from6 x 6 to 36 x 36 and finally to 108 x 108 bunches. The luminosity of the Tevatron willincrease by at least one order of magnitude. The Tevatron beam dynamics has been studiedfrom the "Weak-Strong" representation of the beam-beam effects. A single particle is trackedusing TEAPOT in the presence of the other beam.The nonlinearities introduced by beam-beamforce introduces two unwanted effects: (a) It excites nonlinear resonances; (b) It introduces adispersion of the tunes with amplitude. The strength ofnonlinear three mode coupling satisfyingthe sum rules Q3 = Q 1 + Q2 is measured. These higher orders spectral analysis results will bepresented.Keywords: Beam-beam effects; computer simulations.1 INTRODUCTIONThe Fermilab Tevatron is a 1.8 TeVIc center of momentum proton-antiprotoncollider,.delivering a peak luminosity greater than 2.E31 cm-2sec-1. In thecurrent collider operation, six proton and antiproton bunches collide at BO(CDF) and DO interaction points. The two beams are kept separated ina helical orbit at ten other possible interaction locations by electrostaticseparators, with average 50' separation. During the Run 1b the averageintensities of proton and antiproton bunches were increased by about a factorof two to 25E10 and 8E10 respectively. 1 Every attempt were made to keep thelifetime of the colliding beam high by adjusting the feed-down sextupoles.*Operated by University Research Association under contract with the Department ofEnergy.[407]/161[408]/162 C.S. MISHRA and S. ASSADIThis allowed us to keep the tunes of proton and antiproton away from nearbyresonances.The calculations presented in this paper were started at the time whenwe had only one model2 to study the Beam-Beam effects in the Tevatron.Preliminary results from these two model calculations for the Run II runningconditions has generated considerable interest in the measurements andsimulations of Beam-Beam effects at Tevatron.The Fermilab III accelerator complex upgrade, including the Main Injectorwill increase the peak luminosity to 2E32 cm-2sec-l . Higher luminosity willbe achieved by injecting more proton and antiproton bunches with similarintensities to present bunches. Number of bunches will increase from 6 x 6to 36 x 36 and eventually to 108 x 108, to keep the number of interactionsper crossing at each high energy physics detector at a manageable level. Wehave studied the change in tunes of particles, change in closed orbit and beamsize as a function of bunch intensity and beam separations. We have studiedthis for 6 x 1 and 36 x·1 case. In this paper we present some general featuresof our calculation and introduce a new technique to analyze the non-linearityin the particle motion.2 NUMERICAL SIMULATIONSThe beam dynamics of particles in the Tevatron has been studied from theBeam-Beam effect point of view in a "Weak-Strong" representation. Thisis modeled by tracking a single particle in the presence of a constant fieldof the other beam like those of any other fixed beam-line elements. All thecalculations presented here are performed at 900 GeV for the setting whentwo beams are not colliding head-on at BO and DO. The thin element trackingprogram TEAPOT3 has been used for these simulations. The code TEAPOTsimulates the passage of a single particle in the presence of an oncomingbeam by a BEAM-BEAM element. The code takes as an input the transversebeam size, number of particles in the fixed beam, horizontal and verticaloffset from the ideal orbit.2.1 Tevatron LatticeThe Tevatron Lattice includes standard magnetic elements dipoles, quadru-poles, sextupoles, correction elements, and measured higher order multipolesBEAM-BEAM EFFECTS IN THE TEVATRON [409]/163TABLE I Magnetic multipole errors used in the simulationsNormal Errors Skew ErrorsMagnet Type Multipole Order (bn) O'bn (an) O'anDipole quad 0.10 0.50 0.20 0.60sext 0.48 2.66 0.12 1.198 -0.20 0.81 -0.10 1.4210 -0.64 1.33 -0.10 0.4512 -0.06 0.32 -0.06 0.5514 5.35 0.46 -0.12 0.2918 12.46 0.30 -0.08 0.36Arc Quadrupoles quad 19.2 0.25 0.45sext 1.98 3.69 2.83. 3.41oct 1.25 0.93 -0.45 1.9710 -0.26 0.74 -0.72 0.8012 -1.91 1.70 0.21 0.4414 0.05 0.28 0.18 0.2918 -0.03 0.26 -0.10 0.2220 -1.66 0.32 0.19 0.29Low Beta quad 19.2 0.25 0.45Quadrupoles sext 0.62 1.88 -0.43 3.11oct 0.13 0.76 -0.30 0.9910 0.15 0.48 -0.25 0.6512 -3.38 1.62 -1.91 1.7014 0.05 0.30 0.11 0.37for dipoles and quadrupoles. The higher order multipoles include bothnormal and skew components up to 18 poles for dipoles,4 20 poles for arcquadrupoles5 and up to 14 poles for low beta quadrupoles.6 The higher ordermultipoles are distributed about its mean value with a sigma, cut off at 30' .Table I shows the magnetic multipoles as used in the lattice.In TEAPOT there is no direct provision for electrostatic separator, soits function is achieved by providing a horizontal and vertical kick to theparticles at separator locations. The Tevatron has four RF cavities per beam,each operating at Vrf= 0.2125 MV. The RF frequency is set to 53 MHzcorresponding to a harmonic number of 1113. We introduce the RF cavity inthe lattice to study the coupling between the transverse betatron oscillationsand the longitudinal synchrotron oscillations. These coupling resonancesget excited by single beam effects arising from dispersion in accelerating[410]/164 C.S. MISHRA and S. ASSADIcavities, transverse fields that vary over the bunch length and beam-beameffects.Misalignment of all the magnetic elements and beam position monitor hasbeen included in these calculations. The sigma of the alignment error withrespect to closed orbit is 0.25 mm in both horizontal and vertical directions.The dipoles and quadrupole magnets have a roll angle deviation of 0.5 mrad.The horizontal and vertical correctors are used to correct for closed orbiterror due to these errors.Quadrupole strengths and separator kicks are input from the Tevatroncontrols page. The strengths of the trim quadrupoles were adjusted to setthe Tevatron tune to the desired value (Qx,Qy) =(20.583,20.574), as isdone in the experiment. The two sextupole families are used to set thechromaticity of the Tevatron to (Cx,Cy) = (20,24). All the launched particleshave ~p/p = 0.0001. The Tevatron has four sets of skew quadrupoles.Coupling is introduced in the lattice error fields present in dipole andquadrupole magnets and rolled quadrupoles. Skew quadrupoles are used toglobally decouple the Tevatron. Decoupling at the interaction regions are alsoperformed using the skew quadrupoles present near the interaction regionsand other skew quadrupoles in the lattice.2.2 Calculations and ResultsThe beam size in the Tevatron is measured at E1710cation using flying wire.At 900 GeV the beam dimensions are measured to have a RMS of 0.35 mmin both planes. We have used the beta-function of the Tevatron to calculatethe beam size at all the points where two beams crosses each other. Theseparation of the two beams is maintained on the helical orbit by offsettingthe beam-beam element as if it was an opposing beam. We study the effectof antiproton bunches on a single proton. In 6 x 1 and 36 x 1 calculations thesingle tracked proton crosses the antiproton bunches at 12 and 72 locationsrespectively. The particles are launched with these amplitudes at maximumbeta of 125 meters and with x amplitude equal to y amplitude. The largeamplitude particles are closer to the fixed beam beam elements and particleswith small amplitudes radially move away from it. Simulations of the 36 x 1case takes into account one of the present loading scheme planned for Run IIoperations.Figure 1 shows the change in horizontal and vertical tunes of a smallamplitude (1.0 mm at maximum beta) particle as a function ofbeam intensity.BEAM-BEAM EFFECTS IN THE TEVATRON [411]/165G---------E) 6X1 , X Plane AG- - EJ 6X1, Y Plane //0.030 0--036X1, X Plane A<DC !r - ----6, 36X1, Y Plane I:::Jl- I.~<D 0.020 f{0) /Cm / -0.c A'U v0.010 // -'P __ -0/' __@~-:::- -B'::'" - --0- - ---G- --0.000 ,~.y-0.0 1.0 2.0 3.0 4.0 5.0 6.0Bunch Intensity (E11)FIGURE 1 Change in betatron tune as a function of bunch intensity.4.0e-04I G----E) X Plane3.0e-04 G--fJ Y Plane:.c0Q)C/)0 2.0e-04C3.~Q)0>C 1.0e-04ca..c()5.01.0 2.0 3.0 4.0Bunch Intensity (E11)O.Oe+OO ('4------'-------'---'------'----------'-----'------'--'------'-------'0.0FIGURE 2 Change in closed orbit as a function of bunch intensity.These changes in tunes are measured with respect to the calculated tune of thesame particle for no beam-beam case. The change in tune is first order due tobeam-beam effects. As expected the beam-beam tune shifts is considerablelarger in 36 x 1 case, compared to 6 x 1. This large tune shift will require usto adjust the tune of the Tevatron during the opposing beam injection process.We have also studied the change in tune as a function of beam separation.The change in tunes of the particle increases as we reduce the two beamseparation indicating a possible need of higher separation of the two beams.Figure 2 shows the change in closed orbit in x and y plane due to beam-beamforce for 36 x 1 case. At a bunch intensity of 3Ell the change in y closed[412]/166 C.S. MISHRA and S. ASSADIorbit is of the order ofhalf the beam size. The beam-beam interaction distortsthe phase space and increases the emittance of the beam. We have studiedthe change in transverse beam ,size as a function of intensity and find that itincreases with the bunch intensity of opposing beam.3 NONLINEAR MODE COUPLING ANALYSIS IN THETEVATRONThe analytic theory of synchro-betatron motions was first studied in con-nection with chromaticity. The dispersion in RF cavities was found tocause beam loss by coupling, and this effect has been studied analyticallyand computationally by a number of physicist. However, no experimentalevidence of considerable wave-wave coupling of the broad band turbulencehas been reported so far. Whereas the standard linear spectral analysis (powerspectrum) provides experimental information on the amplitude and phasebehavior of the individual Fourier components and the transport inducedby fluctuations, it does not give any information about the coupling amongdifferent spectral components. The use of the bispectral analysis allows oneto discriminate between oscillations spontaneously excited by the beam andthose generated by nonlinear coupling.We investigate the combined effect of long range interactions and higherorder multipoles to calculate the non-linearly generated frequency spectrum.In this paper we discuss three cases. First we restrict ourselves to synchro-betatron and betatron-betatron coupling of a single beam with betatrontunes close to 7/12 resonance when higher order multipoles are present.Second, beam-beam without error fields are considered. Finally beam-beamwith multipoles are analyzed. Tum by tum data was generated using theTevatron Lattice described above. The bicoherence spectrum has beencomputed to study the strength of the three-wave coupling contributing tothe characteristics of beam-beam coupling in the Tevatron.The bicoherency is defined asWhere R(Vl, V2)2 is the bispectrum defined as R(Vl, V2) = (XVI X V2 X v*),P(v) the autopowerspectrum P(v) = (XVI X V2*), Xv is the Fourier transformof the tum by tum data, X(t) and 0 means ensemble averaging over manyBEAM-BEAM EFFECTS IN THE TEVATRON [413]/167statistically similar realizations. The bicoherency measures the fraction of thefluctuating power at a frequency v that is phase correlated with the spectralcomponents at frequency VI and V2 obeying the summation rule V = VI +V2.The bicoherency thus quantifies the degree of coupling among three waves.The bicoherency is bound between 0 and 1: When b2 (VI, V2) is equal to 1then the oscillations at frequency V are completely coupled with frequencycomponents at frequency VI and V2 and completely decoupled for the zerovalue. The use of the bicoherence as a measure of three-wave couplingis independent of any closure assumptions. The bicoherence is nonzero ifthere is a statistically significant phase relation between the three modes.Such bispectral analysis has been advanced by Ritz, Powers, and Bengstonfor plasma research, and applied in the frequency domain to electrostaticfluctuations.7In order to predict a particle's behavior in the presence of the beam-beaminteraction in the Tevatron, modified TEAPOT simulation code was rununder three conditions. The predicted position and phase of a particle aftera designated 20000 turns is used to calculate the bicoherencies. In all cases,the base tunes for all conditions are (Qx,Qy) = (20.5836,20.5826). The twosextupole families are used to set the chromaticity of the Tevatron to a desiredvalue (minimum chromaticities to keep the beam stable at 900 GeV); thechromaticity setting is adjusted to Qx =20 and Qy =24. The launched particlehas ~p / p = 1.0 E -3. Beam intensity used in these simulations were 3.0E11per bunch. The initial position and phase are the same for all cases. In a firstapproximation, when the intensity of the opposite beam is relatively lowand the tunes are far enough from the low order resonances, the beam-beameffects appear as a nonlinear dependence of the frequencies of transverseoscillations on their amplitude. For larger intensity, or when the tunes areclose to the low-order resonances, isolated nonlinear resonances, creating aperiodic energy exchange between the transverse planes, come into play. Thisis the regime we have performed the simulations. For still larger intensity, theresonances can overlap to form stochastic regions where the particle dynamicis diffusive.Figure 3 shows the normalized phase space plot for the single beam. Theonly source ofnonlinearity is the higher order multipoles. The above standardnormalized phase space plot does not show any structure that could revealthe sources of nonlinearities if any exist. Figure 4 shows the normalizedphase-space plot for the beam-beam with the long range kicks for 6 x 1case. The higher order multipoles are not present. Twelve islands associated[414]/168 C.S. MISHRA and S. ASSADI1.00.5'0 0.0x-0.5-1.0-1.0 -0.5 0.0 0.5 1.0x10-31.00.5'0 0.0><-0.5-La-1.0 -0.5 0.0 0.5 1.0x10·3FIGURE 3 Single beam normalized phase space plots with higher order multipoles.with 12 parasitic crossing is observable. This non-linearity is small for smallamplitude particles.The same turn by tum data is analyzed to calculate bicoherency. Figure 5shows the calculated bicoherence which clearly shows the three wavescoupling. The horizontal axis denote horizontal ·and vertical tunes. Thevertical axis denotes the bicoherence for V3 == VI + V2. Peak "~' is the onlysignificant non linearity observed which is due to the dispersion present in thecavities. This synchro-betatron coupling can be removed from the simulationby removing dispersion. This type of analysis is not possible by simple studyof traditional normalized phase space plots.BEAM-BEAM EFFECTS IN THE TEVATRON [415]/169FIGURE 4 Normalized phase space plot for beam-beam without higher order multipoles.Figure 6 represents the bicoherence spectrum of particle tracked in thepresence of beam-beam elements without the higher order multipoles. Weobserve nonlinearly generated modes due to dispersion and also along thedifference resonance vH - vv. When both the beam-beam interaction andhigher order multipoles are present together some resonances gets excitedwith higher amplitude and other gets suppressed. Figure 7 shows that thephase space of the particle get fully diluted when both the higher ordermultipoles and beam-beam effects are present. No obvious non-linearity isobserved in the normalized phase space plot. When the same data for 6 x 1beam-beam with 3El1 particles per bunch and higher order multipoles is[416]/170 C.S. MISHRA and S. ASSADIFIGURE 5 Bicoherence, b(Vl, V2, V3), of single beam with all higher order multipoles present..v.VFIGURE 6 Bicoherence, b(Vl, V2, V3), of beam-beam without all higher order multipoles.BEAM-BEAM EFFECTS IN THE TEVATRONX-X'-1 0Position x10·3[417]/1711.00.5'0x~ 0.0~-0.5-1.0-1.0 -0.5 _ 0.0 - 0.5 1.0Position x10·3FIGURE 7 Normalized phase space plots for beam-beam with higher order multipoles.analyzed for Bicoherence analysis Figure 8, we observe resonances whichwere not present either in the case of single beam with higher order multipolesFigure 5 or just the beam-beam Figure 6.4 SUMMARYIn this paper we have presented initial results from a study to understand thebeam-beam effects in the Fermilab Tevatron. A closer attention needs to bepaid by changing the Tevatron operating procedures to overcome the effectsof increasing beam-beam force. We have also introduced a new technique toanalyze the particle dynamics.[418]/172 C.S. MISHRA and S. ASSADIFIGURE 8 Bicoherence, b(Vl, V2, V3), of beam-beam with all higher order multipoles.AcknowledgementsWe thank R. Talman for providing the tracking code TEAPOT and answeringquestions relating to these calculations. We would also like to thank NormanGelfand and B.C. Brown for providing magnetic multipoles file for theTevatron magnets.References[1] Bharadwaj, V., Halling, M., Lucas, P., McCrory, E., Mishra, S., Pruss, S. and Werkema, S.Run 1b Accelerator Performance, Fennilab Technical Memo.[2] Michelotti, L. Fermilab Technical Memo TM-1738.[3] Schachinger, L. and Talman, R. (1987). Particle Accel., 22, 35.[4] Hanft, R. Proceedings of the 1984 Summer Study on the Design Utilization of theSuperconducting Super Collider, Donaldson, R. and Jorge Morfin editors, p. 342.[5] Schmidt, E.E. et al. (1983). Fermilab TM-1184.[6] Hanft, R. et al. (1991). Particle Accel Conf: IEEE, 2233-2235.[7] Ritz, Ch.P. et al. (1989). Phys. Fluids, BI, 153.

Simulations of Beam-beam Effects in Tevatron and Nonlinear mode Coupling analysis

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