The transverse coherent oscillations of the two colliding LHC proton beams are studied via multi-particle tracking, using the beam-beam force of a Gaussian distribution with variable barycenters and rms sizes. In addition to head-on collisions, our simulation optionally includes the effect of long-range collisions and an external impedance. Simulation results are the coherent and incoherent oscillation frequencies, the emittance growth of either beam, and evidence for the existence or absence of Landau damping. We find that with equal beam sizes and equal tunes Landau damping acts on the coherent modes for current ratios smaller than 60%. For equal current ratio, Landau damping of the coherent dipole oscillations is lost. However it can be restored by separating the tunes, provided the external impedance is sufficiently small

Zimmermann, Frank

Zorzano-Mier, M P

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EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCHEuropean Laboratory for Particle PhysicsSIMULATIONS OF COHERENT BEAM-BEAM MODES AT THE LHCM.P. Zorzano and F. ZimmermannThe transverse coherent oscillations of the two colliding LHC proton beams are studied viamulti-particle tracking, using the beam-beam force of a Gaussian distribution with variable barycentersand rms sizes. In addition to head-on collisions, our simulation optionally includes the effect oflong-range collisions and an external impedance. Simulation results are the coherent and incoherentoscillation frequencies, the emittance growth of either beam, and evidence for the existence or absenceof Landau damping. We find that with equal beam sizes and equal tunes Landau damping acts on thecoherent modes for current ratios smaller than 60%. For equal current ratio, Landau damping of thecoherent dipole oscillations is lost. However it can be restored by separating the tunes, provided theexternal impedance is sufficiently small.SL DivisionPresented at the Seventh European Particle Accelerator Conference  (EPAC 2000)26-30 June 2000, Vienna, AustriaGeneva, Large Hadron Collider ProjectAdministrative SecretariatLHC DivisionCERNCH - 1211 Geneva 23SwitzerlandLHC Project Report 408Abstract14 August 2000Simulations of Coherent Beam-Beam Modes at the LHCM.P. Zorzano, F. Zimmermann CERN, Geneva SwitzerlandAbstractThe transverse coherent oscillations of the two collidingLHC proton beams are studied via multi-particle tracking,using the beam-beam force of a Gaussian distribution withvariable barycenters and rms sizes. In addition to head-on collisions, our simulation optionally includes the effectof long-range collisions and an external impedance. Sim-ulation results are the coherent and incoherent oscillationfrequencies, the emittance growth of either beam, and ev-idence for the existence or absence of Landau damping.We find that with equal beam sizes and equal tunes Lan-dau damping acts on the coherent modes for current ratiossmaller than 60%. For equal current ratio, Landau dampingof the coherent dipole oscillations is lost. However it canbe restored by separating the tunes, provided the externalimpedance is sufficiently small.1 MODELWe simulate the collision of two strong proton beams.Our system of normalized variables is x = X=oX ,vx = X 0=oX , y = Y=oY , vy = Y 0=oY whereoX = oY =  are the nominal horizontal and verticalrms sizes, and  the beta function at the interaction point.The prime denotes the derivative with respect to longitudi-nal position s. Each of the beams is represented by a set ofN macroparticles, whose trajectories are followed over nturns, assuming linear betatron motion (rotation with tuneQx on the horizontal plane (x; vx) and tune Qy on the ver-tical plane (y; vy)) from one IP to the next. At the IP, eachparticle in the bunch experiences a deflection in the field ofthe counter-rotating beam with barycentres at (x¯(i); y¯(i))and squared transverse sizes M (i)xx =< (x(i) − x¯(i))2 >,M(i)yy =< (y(i) − y¯(i))2 >. This deflection is computedassuming that the opposing beam has a Gaussian shape.Thus, for M (i)xx > M (i)yy we apply a horizontal beam-beamkick ∆vx(n) =rpN(i)pγ2 Fx(x−x¯(i); y− y¯(i); M (i)xx ; M (i)yy )with rp the classical proton radius, N (i)p thebunch population, γ the relativistic Lorentz fac-tor and Fx is given by Fx(x; y; Mxx; Myy) =√2(Mxx−Myy)=[W(x+iyp2(Mxx−Myy))−√2(Mxx−Myy)e(− x22Mxx −y22Myy)W(xpMyy=Mxx+iypMxx=Myyp2(Mxx−Myy))];where W denotes the complex error function (ifM(i)yy > M(i)xx we substitute x by y on both sides ofthe two equations, and vice versa). The vertical beam-beam force is described by the real part of the sameexpression. In these maps the superindex (i) indicatesvariables of the counter-rotating beam.At the LHC, there are about 16 parasitic encoun-ters on each side of an IP, with a minimum transverseseparation of Lx = 7:5 and Ly = 7:5 (in units ofx). The long-range beam-beam kick is then ∆vx(n) =+npar2rpN(i)pγ2{(x−x(i)−Lx)R2[1− exp(− R2M(i)xx +M(i)yy)]}−npar2rpN(i)pγ2{− 1Lx[1− exp(− L2xM(i)xx +M(i)yy)]}whereR2 = (x − x(i) − Lx)2 + (y − y(i))2: An equivalent ex-pression will be used for vertical long range collisions withseparation Ly. In the simulations we assume the followingtypical LHC parameters: fractional betatron tunes ofQx  Qy = 0:32, bunch population N (1)p = 1:05  1011for beam 1, and N (2)p = r  N (1)p for the second beamwith r varying between 0 and 1, proton beam energy of 7TeV, unperturbed horizontal and vertical rms beam sizes atthe primary collision point  = 16  10−6 m, and an IPbeta function x;y = 0:5 m. The beam-beam parametersare defined by (i)x;y =N(i)p rpx;y2γx,y(x+y)with i = 1 for beam1, and i = 2 for beam 2. With the above LHC parameterswe find   0:0034. For equal beam sizes, the ratio of thebeam currents, r = N (2)p =N (1)p = (2)=(1), determinesthe behaviour of the system [1]. In the simulation, theinitial coordinates (x; vx; y; vy) for two groups of Nmacroparticles representing the two beams are selectedfrom a Gaussian random distribution in each variable with< x >= < vx >= < y >=< vy >= 0 and < x2 >=< v2x >= < y2 >= < v2y >= 1.2 COLLISION WITH EQUAL TUNES2.1 pi- and σ-modes for round beamsFirst we consider the strong-strong case, r = 1, and head-on collisions of two bunches.The statistical fluctuation inthe macroparticle distribution is sufficient to excite the co-herent modes. Fourier analysing the motion of the barycen-tre of one bunch reveals two coupling modes. One is lo-cated at Q; the other has a lower frequency. In Fig. 1 thesimulated frequency spectrum of one beam SA(w) is plot-ted on a logarithmic scale, as determined by an FFT.We can identify the two dipole coherent modes  (at theunperturbed tune) and . Theoretical studies based on thelinearized Vlasov theory predict a tune shift between the and -modes equal to Y  . For the case of round beamsthis factor is predicted to be Y = 1:21.The shift obtained inour simulations corresponds to Y  1:1. The difference isprobably due to the simplifying assumptions of our model,where the beam-beam forces are calculated assuming that1e-061e-050.00010.0010.01-1 -0.5 0wS (w)AFigure 1: Frequency spectrum of the bunch centroid mo-tion (over 217 turns, N = 104 macroparticles) for roundbeams with x = y =  = 0:0034, and Q = 0:32. Thehorizontal axis gives the distance w to the unperturbed tuneQ in units of , i.e., w= ( − Q)=. The vertical axis isthe corresponding amplitude on a logarithmic scale. The- and -oscillation modes are clearly visible.the beams are of Gaussian shape.Between the  and the -mode in Fig. 1 we also see thecontinuum. This is related to the incoherent oscillation fre-quencies of individual particles. Due to the non-linearityof the beam-beam interaction particles with different am-plitudes experience different focusing force. The result isan incoherent tune spread, which extends from 0 to −.We have found a size dependence on the tune. Thesize variations as a function of tune can be fitted by thestandard dynamic-beta effect ˆ=√1−(20sin (2Q))2−20 cot (2Q) with some modification for the strong-strong case. Integrating the beam-beam force over theGaussian distribution, the effective beam-beam force is onehalf that experienced by a single particle near the beam cen-tre. We evaluate then the dynamic beta effect replacing inthis equation 0 by Ξ = 0=2, see Fig. 2.0.850.90.9511.051.11.151.21.250 0.1 0.2 0.3 0.4 0.5 QxFigure 2: Comparison of the tracking results with the dy-namic beta effect. Vertical axis: beam size ratio 2=20 ,horizontal axis fractional tune. Points: data obtained fromtracking; curve: 2=20 = ˆ= dynamic beta theoreticalprediction with Ξ = 0=2.2.2 Landau damping and emittance growthIf the frequency of the -mode lies within the incoherenttune spread its energy is absorbed by individual particleswith similar oscillation frequencies. This phenomenon isknown as Landau damping . The fraction of energy whichis absorbed by the continuum leads to an irreversible emit-tance growth. Decreasing the beam-beam parameter ratior there is a point at which the discrete -mode joins thecontinuum [1, 2]. We launch the two beams with an ini-tial horizontal offset d = 0:2 (in units of x) and studythe spectrum of the -mode at consecutive intervals (notshown). For the case r = 1 the -mode is well outside thecontinuum and the amplitude (after some initial loss of en-ergy which is transfered to the continuum) stays constant.For r = 0:6, the -mode lies at w = −0:9 (inside the con-tinuum) the amplitude of the -mode decreases in time. Forthe case r = 0:3 the -mode is well inside the continuumand rapidly damped. These results confirm the predictionthat for current ratios r  0:6 the -mode frequency fallsin the incoherent tune spread of the weaker beam [1] andis therefore Landau damped. As a consequence an initial-mode oscillation will disappear and the beam emittancewill grow until the -mode energy has been completelyabsorbed. Examples are the two upper curves in Fig. 3.As expected, the final emittance is larger than for r = 1,namely ∆I=0  0:01.00.0020.0040.0060.0080.010.0120.0140 20000 40000 60000 80000 100000 120000 140000nr=0.3r=0.6r=1.0∆ε/ ε0IFigure 3: Irreversible emittance growth ∆I=0 (verticalaxis) as a function of time (horizontal axis, time in turns)for three different current ratios, r = 0:3, r = 0:6 andr = 1. The beams are perturbed by an initial offset ofd = 0:2 (in units of ). For r  0:6 the frequency of the -mode lies in the continuum, and, thus, the mode is Landaudamped, and the intrinsic emittance grows until the -modeenergy has been fully absorbed. For r = 1 the -mode isnot Landau damped, and carries part of the kick energy.There is an initial emittance growth which is significantlysmaller.2.3 Long range collisionsThe vertical and horizontal tunes are chosen equal (Qx =Qy = 0:32). Since the tune shifts from long range col-lisions have opposite signs in the two transverse planes,for the LHC an alternating crossing scheme was proposed,where the beams are separated in orthogonal planes at thetwo main IPs. This reduces the overall incoherent tuneshift and tune spread by cancellation of the tune shift be-tween IPs. We consider two closely spaced bunches perbeam, and two interaction regions (beam 1=[a; b], beam2=[c; d]). First the bunches are collided head-on (a-c andb-d). We then apply a phase advance of 90 to reach thelong range collision region. There the bunch pairs (a-d) and(b-c) are collided with a horizontal separation of Lx and abeam-beam parameter which is npar times stronger thanfor the primary collision, representing the accumulated ef-fect of npar = 32 parasitic collisions around each IP. Sub-sequently, we advance the phase of the beams to reach theother interaction region and evaluate the head-on collisions(a-d) and (b-d). This is followed by another phase advanceof 90 to the long range collision point, where again longrange collisions of the pairs (a-d) and (b-c) are applied butthis time with a vertical separation of Ly.The spectrum of the bunch motion is illustrated in Fig. 4.Coherent modes still survive outside the continuum. Col-lision schemes with and without alternating crossing werecompared in Ref. [2], with similar conclusions.1e-061e-050.00010.0010.010.11-3 -2 -1 0 1 2 3wS   (w)AFigure 4: Spectrum in the case of head-on and long rangecollisions with alternating crossing when each bunch col-lides head-on at two interaction points, and undergoes longrange collisions with npar = 32 bunches behind each IP.The horizontal axis the tune distance to the unperturbedbetatron frequency in units of the new incoherent tune shift2 : w= ( −Q)=2.3 SEPARATED TUNESA. Hofmann pointed out that the coherent frequency shiftscan be reduced by separating the tunes of the two beams.We simulate this situation operating beam 1 with Q(1)x;y =0:32 and beam 2 with Q(2)x;y = 0:31 for  = 0:0034. Ex-ternal impedances cause additional coherent tune shifts.We model the effect of the ring impedance applying inthe vertical plane only, every turn, a localized kick thatdepends linearly on the bunch centroid position ∆vy =−4∆QZ;y < y >. This results in a coherent tune shiftof the centroid motion (dipole mode), but has no effect onthe tunes of individual particles (incoherent spectrum). Thecontinuum of beam 2 extends from (0:31−  = 0:3066) to0:31, and that of beam 1 from (0:32− ) = 0:3166 to 0:32.In the horizontal plane both coupling modes are now insidethe incoherent spread of one or the other beam, and Landaudamping is restored. But in the vertical plane, and due tothe impedance tune shift, the coherent modes emerge fromthe incoherent spectrum, see Fig. 5.00.0020.0040.0060.0080.010.0120.305 0.3075 0.31 0.3125 0.315 0.3175 0.32 0.3225 0.325qxSx(qx)00.00050.0010.00150.0020.00250.0030.00350.0040.00450.305 0.3075 0.31 0.3125 0.315 0.3175 0.32 0.3225 0.325qySy(qy)Figure 5: Spectrum in the case of head-on collision ofbunch 1 with tune Q(1)x;y = 0:32 and bunch 2 with tuneQ(2)x;y = 0:31. Horizontal axis: tune; vertical axis: the cor-responding oscillation amplitude for the horizontal (left)and vertical (right) motion of beam 1. In the horizontalplane Landau damping is restored, but in the vertical planean additional coherent tune shift of +20 10−4 pushes thevertical coherent modes away from the continuum.4 CONCLUSIONSWe have confirmed that, for equal beam sizes and currentratio 0 < r  0:6, the -mode lies within the continuumand is Landau damped. Its energy is transferred to the con-tinuum, leading to an irreversible finite emittance growth.For equal beam-beam parameters of the two beams, we finda -mode tune shift of −1:1 in units of , sufficiently largeto place it outside of the continuum and to lose Landaudamping [1]. We have observed a decrease of the beamsize with increasing fractional tune, which is explained bythe dynamic beta effect.In the case of two equally strong beams with head-onand long range collisions, coherent modes exist outside ofthe continuum, even with alternating crossing at two IPs.However, if the betatron tunes of the two beams are suffi-ciently different, and if the impedance coherent tune shiftis lower than ∆Q = 2  10−3, the frequencies of the co-herent modes are shifted towards the continuum of one orthe other beam and Landau damping can be restored.5 REFERENCES[1] Y.I. Alexahin, On the Landau damping and decoherenceof transverse dipole oscillations in colliding beams. ParticleAccelerators, Volume 59, p. 43 (April 1999).[2] M. P. Zorzano, F. Zimmermann. Coherent beam-beam os-cillations at the LHC. LHC project report 314, 1999.

Simulations of Coherent Beam-Beam Modes at the LHC

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