strong–strong simulation study for the wire compensation of long-range beam–beam effect in lhc

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Nuclear Instruments and Methods in Physics Research A 550 (2005) 6–13

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Strong–strong simulation study for the wire compensation oflong-range beam–beam effect in LHC

Lihui Jin, Jicong Shi�

Department of Physics and Astronomy, The University of Kansas, Lawrence, KS 66045, USA

Received 25 May 2005; received in revised form 6 June 2005; accepted 7 June 2005

Available online 6 July 2005

Abstract

The wire compensation of long-range beam–beam interactions in LHC was studied with numerical simulation of 106

macro-particles up to 105 turns. In the simulation, head-on beam–beam interactions at two high-luminosity interaction

points of LHC were calculated self-consistently with the particle-in-cell (PIC) method and long-range beam–beam

interactions calculated with the soft-Gaussian approximation. Including also the multipole field errors in the LHC

lattice, the effectiveness of the compensation was evaluated in terms of the emittance growth. The robustness of the

compensation to static and random errors in the electric currents on the wires was also investigated.

r 2005 Elsevier B.V. All rights reserved.

PACS: 29.27.Bd; 29.20.Dh; 41.85.�p

Keywords: Beam–beam interaction; Emittance growth; Strong–strong simulation; Particle-in-cell; Large-hadron collider

Many studies have shown that the long-rangebeam–beam interactions in LHC due to parasiticcollisions inside interaction regions have signifi-cant adverse effects on the beam stability [1,2]. Awire compensation scheme has been proposed forLHC [3] to compensate the long-range beam–beam effect by using electric currents. Because oflarge beam separations at the parasitic collisions,the long-range beam–beam interaction is domi-

e front matter r 2005 Elsevier B.V. All rights reserve

ma.2005.06.039

ng author. Tel.: +1785 864 5273;

5262.

ss: [email protected] (J. Shi).

nated by a force that has the form of 1=r in thephase-space regions occupied by beams (the phase-space regions near closed orbits), where r is thedistance from a particle in one beam to the centerof the counter-rotating beam. In the wire compen-sation scheme, an electric wire is thus installed oneach side of an interaction point (IP) to providethe beam with a force similar to 1=r in strengthsbut opposite in signs to the long-range beam–beamforce. A numerical simulation with a strong–weakmodel of beam–beam interactions showed that thewire compensation could improve the beamstability in LHC [4]. In that study, the effect of

d.

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L. Jin, J. Shi / Nuclear Instruments and Methods in Physics Research A 550 (2005) 6–13 7

the wire compensation was examined based on astudy of the particle diffusion in the phase-spaceregion relevant to the dynamical aperture and theparticle diffusion was studied by tracking 1000particles launched at different amplitudes. It wasfound that the wire compensation reduces thediffusion rate in the phase-space region withamplitudes between 6 and 8s, but increases thediffusion rate in the phase-space region beyond 8s,where s is the r.m.s. beam size. Note that theaverage of beam separations at the parasiticcrossings of LHC is about 9s. In the phase-spaceregion with amplitudes that are close to or largerthan the beam separations, the 1=r force does notdominate the long-range beam–beam interactionand the principle of the wire compensation is nolonger valid. The wire compensation may thereforefail to improve and actually worsen the stability inthe phase-space region of a large amplitude.Since the wire compensation is intended toimprove the linearity of the phase-space regionoccupied by the beam that is within about 4sin amplitudes, it is necessary to examine itseffectiveness in that region. With a smallnumber of particles and limited tracking time, itis however difficult, if not impossible, to study thediffusion rate in that phase-space region. More-over, for the long-term beam behavior such as thediffusion rate, the use of the strong–weak approx-imation for head-on beam–beam interactionscould be problematic and needs to be cross-checked with a self-consistent beam–beam simula-tion [5]. In order to examine the direct effect of thewire compensation on the dynamics of beams, inthis paper, the emittance growth in LHC wasstudied with a beam–beam simulation of 106

macro-particles for the two colliding beams. Inthe simulation, the head-on beam–beam interac-tions were calculated self-consistently using theparticle-in-cell method.The use of a large numberof macro-particles is necessary to ensure a correctcalculation of head-on beam–beam interactionsand to reduce computational noises for theemittance growth [5]. As in the time scale that isaccessible for the numerical simulation the emit-tance growth is fairly small, the reduction ofnumerical noises is crucial to the study of theemittance growth.

The lattice used in this study is the LHCcollision lattice [6] in which the fractional partsof horizontal and vertical tunes are ðnx; nyÞ ¼

ð0:31; 0:32Þ. In the simulation, both head-on andlong-range beam–beam interactions at two high-luminosity interaction points (IP1 and IP5) andmultipole field errors in the lattice of LHC wereincluded. The crossing angle of two counter-rotating beams was taken to be 300 mrad withthe vertical crossing at IP1 and the horizontal atIP5. For LHC, the design beam–beam parameteris x ¼ 0:0034. In this study, cases of x ¼

0:0034–0.02 were studied in order to be moreconservative. For the long-range beam–beaminteractions, there are 15 parasitic collisions onthe each side of an IP. In the simulation, the firstfive parasitic collisions were calculated individu-ally and the rest of parasitic collisions, due to analmost zero phase advance among them, werelumped into one single kick at the location of thesixth parasitic collision point. Because of a largebeam separation at the parasitic collision points,the momentum kick in the transverse phase spacedue to the long-range beam–beam interaction doesnot sensitively depend on the details of beamparticle distributions and, therefore, can be calcu-lated with the Gaussian-beam formula,

D~p ¼ G0

~r þ~r0j~r þ~r0j

21� exp �

j~r þ~r0j2

2s2

� �� , (1)

where ~r ¼ ðx; yÞ is the transverse coordinate, ~r0 isthe horizontal and vertical beam separation, and sis the r.m.s. beam size at the parasitic collisionpoint. s was updated in each turn during thetracking. For LHC, j~r0j�9s. The kick strength G0

is related to the beam–beam parameter x byG0 ¼ 8ps�2x=b�, where s� and b� ¼ 0:5m are thenominal transverse beam size and beta function atIP, respectively. Since j~r0jbs, the long-rangebeam–beam interaction is dominated by the j~r þ~r0j

�1 term in the phase-space region occupied bythe beam (j~rjo4s). In this simulation model, anelectric wire is placed on each side of an IP beforethe first insertion quadrupole (MQX) Q1 [6] forthe wire compensation. The horizontal or verticaldistance from the wire to the beam is 9:5s that isthe average beam separation at parasitic collision

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(a)

L. Jin, J. Shi / Nuclear Instruments and Methods in Physics Research A 550 (2005) 6–138

points. The maximal phase advance from thelocation of the wire to the location of the parasiticcollision points is about 3. An almost perfect localcompensation can therefore be achieved in thephase-space region where j~r þ~r0jbs. The currenton the wire provides the beam a kick equivalent tothe total contributions of 15 parasitic collisions.Dipole components of all the long-range beam–beam kicks and wire kicks were subtracted duringthe tracking. The head-on beam–beam interac-tions were calculated at each IP with our self-consistent (strong–strong) beam–beam simulationcode based on the particle-in-cell method. Thecode was fully tested and presented in detail in aprevious paper [5]. In this code, each beam isrepresented by a large number of macro-particleswith a given initial distribution in the transversephase space. In this study, the initial beams have around Gaussian distribution in the normalizedtransverse phase space with standard deviation s0and truncated at 4s0, where s0 ¼ s�=

ffiffiffiffiffib�

p. The

computational parameters of the particle-in-cellmethod were carefully tested for the LHC case toensure the computational convergence. Fig. 1 plotsthe example of the convergence for the number ofmacro-particles used in the simulation. It showsthat in the LHC case when the number of macro-particle used is smaller than 4� 105, the simula-

Fig. 1. Evolution of the average horizontal and vertical beam

size in LHC with head-on and long-range beam–beam

interactions. The number of macro-particles used for each

beam in the simulation is (a) 1� 106, (b) 4� 105, (c) 2� 105,

(d) 1� 105, (e) 5� 104. (a) and (b) overlap each other. s0 is theinitial beam size and x ¼ 0:02.

tion produces a significant artificial beam-sizegrowth. In this study, 5� 105 macro-particleswere therefore used for each beam.Fig. 2 plots the beam tune spread without and

with the wire compensation when x ¼ 0:02. Thetune spreads were calculated with the Fouriertransformation (FFT) over the betatron motionsof 420 particles that have initial amplitudes up to4s. It shows that the wire compensation eliminatesthe tune spread of the long-range beam–beam

(b)

Fig. 2. Beam tune spread in LHC including both the head-on

and long-range beam–beam interactions without (a) or with (b)

the wire compensation. The solid lines are the resonance lines

up to the 10th order. The cross indicates the lattice tunes of (nx,

ny) ¼ (0.31, 0.32). x ¼ 0:02.

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Fig. 4. The same as in Fig. 3 but x ¼ 0:0034.


interactions. This is consistent with the result ofRefs. [3,4]. Note that the compensation of the tunespread was actually first studied in Ref. [3]. InFigs. 3 and 4, the evolutions of the beam sizeswithout or with the wire compensation wereplotted for the cases of x ¼ 0:02 and 0.0034,respectively. For a comparison, the case withoutthe long-range beam–beam interactions was alsoplotted in both figures and it overlaps with the onethat includes both the long-range beam–beaminteractions and the wire compensation. Thebeam-size growth due to the long-range beam–beam interactions is therefore eliminated by thewire compensation. As shown in Figs. 3 and 4, thebeam-size growth due to the beam–beam interac-tions consists of a quick short-term blowup thatoccurs within the first thousand turns and followedby a slow long-term growth. The short-term beam-size blowup is partially due to the linear mismatchbetween the initial beams and the linear ringsincluding the linear beam–beam effects (the changeof the twiss parameters due to the beam–beaminteractions) since in these cases, the initial beamswere matched with the linear rings without thebeam–beam interactions. To examine the signifi-


size in LHC (a) without the long-range beam–beam interac-

tions; (b) with the long-range beam–beam interactions but

without the wire compensation; (c) with both the long-range

beam–beam interactions and the wire compensation; and (d)

the same as (b) but the initial beams are matched with the linear

ring that includes the linear effects of the long-range

beam–beam interactions. (a) and (c) overlap each other. s0 is

the initial beam size and x ¼ 0:02.

cance of this linear beam–beam effect, the case ofthe initially matched beams was also plotted ascurve d in Fig. 3, where the change of the twissparameters due to the long-range beam–beaminteractions including the feed-down effect werecalculated exactly by expanding the long-rangebeam–beam interactions around the beam separa-tion at parasitic collision points [7]. Note that theremains of the short-term beam-size blowup of thematched beams is due to the nonlinear beamfilementation resulted from the phase-space dis-tortion of the nonlinear beam–beam perturba-tions. A comparison of the beam-size growthbetween the matched and mismatched beams(curve d and b in Fig. 3) shows that the linearmismatch changes only the degree of the short-term beam-size blowup but no effect on the long-term beam-size growth. In both the cases, theincrease in the short-term beam-size blowup due tothe long-range beam–beam interactions is com-pletely eliminated by the wire compensation (seecurve c in Figs. 3 and 4).The effectiveness of thewire compensation is therefore not affected by thelinear beam–beam interactions. On the other hand,for the long-term beam stability and the beamlifetime, the rate of the slow beam-size growth ismore important than the short-term beam-sizeblowup. A comparison of the slops of the curves inFigs. 3 and 4 shows a significant increase of thebeam-size growth rate due to the long-rangebeam–beam interactions. This increase of thebeam-size growth rate is the dominant long-range

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Fig. 5. Projection of the phase space in the action–angle

variables for single-particle dynamics in the cases of Fig. 3,

where �0 ¼ s20 is the transverse emittance. (a) Without the long-

range beam–beam interactions, (b) with the long-range

beam–beam interactions but without the wire compensation,

and (c) with both the long-range beam–beam interactions and


beam–beam effect for the long-term beam dy-namics (over millions turns). As shown in Figs. 3and 4, this long-term effect of the long-rangebeam–beam interactions is completely eliminatedwith the wire compensation as well.

Figs. 5 and 6 are examples of projections of thedistorted phase space in the action–angle variablesfor the cases of Figs. 3 and 4, respectively. Theaction and angle variables for the horizontaldirection are defined as Ix ¼ ðX 2 þ P2

xÞ=2 andfx ¼ arctanðPx=X Þ, where X and Px are thenormalized horizontal coordinate and its conju-gated momentum, and similarly for the verticaldirection. Six trajectories with initial amplitudesranging from 1 to 4 s were plotted for the first4000 turns during the beam–beam simulation. Toreduce the effect of linear coupling, Ix þ Iy insteadof Ix or Iy was plotted as a function of the anglevariables. Note that for a linear ring, Ix þ Iy is aconstant of motion and the trajectories in theaction–angle phase space are straight horizontallines. Ix þ Iy was plotted as a function of fy inFig. 5 and of fx in Fig. 6 in order to have examplesof the projections of the phase space in the verticalas well as horizontal direction. In Figs. 5a and 6a,the phase-space distortion is the result of the head-on beam–beam interactions. Since in usual opera-tions of storage-ring colliders, the injected beamsare only matched with the ring without the head-on beam–beam interactions, the linear effect of thehead-on beam–beam interactions (the change ofthe beta functions) was not included in thecalculation of the normalized variables. Themismatch between the beams and this distortedphase space results in the initial beam-size blowupas shown by curve a in Figs. 3 and 4. Figs. 5b and6b plot the cases that includes both the head-onand long-range beam–beam interactions, wherethe changes of the twiss parameters due to thelong-range beam–beam interactions were includedin the calculation of the normalized variable-s.Comparisons between Figs. 5a and b andbetween Figs. 6a and b show that the additionalnonlinear perturbation from the long-rangebeam–beam interactions further distorts the phasespace. Moreover, the long-range beam–beaminteractions result in a large chaotic band in thephase-space region near the beam tails

the wire compensation. x ¼ 0:02.

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Fig. 6. The same as in Fig. 5 but for the cases of Fig. 4 as

x ¼ 0:0034.


(ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2 þ y2

p�4s) in the case of x ¼ 0:02. For

x ¼ 0:0034, such enhanced chaotic regions shouldalso exist but are too small to be visible clearly inthe phase-space region of interest. Note that chaosis generic in nonintegrable Hamiltonian systems.Both the short-term beam-size blowup and thelong-term beam-size growth are therefore en-hanced by the long-range beam–beam interac-tions. As shown in Figs. 5c and 6c, the wirecompensation eliminates the phase-space distor-tion due to the long-range beam–beam interac-tions in both the cases of x ¼ 0:02 and 0.0034. Theincrease in the short-term beam-size blowup due tothe long-range beam–beam interactions is there-fore completely eliminated by the wire compensa-tion. Moreover, as the wire compensationeliminates the chaos due to the long-rangebeam–beam interactions, it should significantlyreduce the slow chaotic diffusion (chaotic trans-port) for beam particles crossing different chaoticbands in the phase space [8] and, consequently,reduces the slow beam-size growth and improvesthe beam lifetime.In the application of the wire compensation,

there are always fluctuations or errors in thecurrent on the wire. The applicability of the wirecompensation depends on the robustness of thecompensation to the errors in the current. Cases ofthe wire compensation with different strength ofthe static and random errors were thus studied.Fig. 7 shows the beam-size growth at the 104thturns as a function of the static error in the currentfor the compensation when x ¼ 0:02, where thebeam-size growth is scaled by that for the case ofthe head-on collisions only. For a comparison, thecase including both head-on and long-rangecollisions but without the wire compensation isalso plotted. It shows that the wire compensationwith a static error in the range of 0 to �10% has aperfect compensation in terms of the beam-sizegrowth. The asymmetry of this window of theperfection for the compensation could be due tothe small phase difference between the location ofthe wire and the parasitic collision points. More-over, in cases of the wire compensation with up to50% static error in the current, there is only up to20% increase in the beam-size growth due to thelong-range beam–beam interactions. Without the

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size for cases of different strength of current fluctuation in the

wire compensation when x ¼ 0:02. (a) is without the long-rangebeam–beam interactions, (b) is with the long-range beam–beam

interactions but without the wire compensation, and (c)–(f) are

the cases with both the long-range beam–beam interactions and

the wire compensation. The current fluctuation measured in a

percentage of the current amplitude is (c) 0%, (d) 0.5%, (e)

1.0%, and (f) 2.5%. (a) and (c) overlap each other.

Fig. 7. The average horizontal and vertical beam-size growth

Ds at the 104th turn vs. the static error of the current DI for the

wire compensation when x ¼ 0:02. Ds1 is the beam-size growth

without the long-range beam–beam interactions at the 104th

turn.


wire compensation, on the other hand, the long-range beam–beam interactions result in a 60%increase in the beam-size blowup when comparedwith the case of the head-on collisions only. Thisindicates that the wire compensation with up to50% error in the current always improves thelinearity of the phase-space region occupied bythe beam. Previous study on the improvement ofthe stability region by the wire compensationsuggested that the static error in the range of 0 to�10% appears acceptable [4]. Our study on thebeam-size growth, however, indicates that thetolerance of the static error in the current couldbe loosened. Fig. 8 plots the evolution of the beamsize for different magnitude of current fluctuation(power ripple) in the wires when x ¼ 0:02. Thecurrent fluctuation used in the simulation is anuniform random noise (white noise). It shows thata very small current fluctuation even less than1:0% can heat up the beam and result in a largergrowth rate of the beam size when compared withthe case without the wire compensation. A muchstrict requirement on the current fluctuation hastherefore to be imposed in order to use the wirecompensation scheme. The requirement of thecurrent fluctuation should at least be o0:5% or

smaller. The study of Ref. [4] also showed a similarstrict tolerance on the current fluctuation.The significance of the wire compensation of

long-range beam–beam interactions is to improvethe linearity of the phase-space region near theclosed orbit. This numerical study demonstratedthat the wire compensation of long-range beam–beam interactions in LHC is very effective inreducing the beam-size growth due to long-rangebeam–beam interactions. The compensation is notsensitive to static current errors but is sensitive tocurrent fluctuations. A power ripple of 0:5% ormore can heat up the beams and make thesituation worse than that without the compensa-tion. The use of the wire compensation thusrequires a strict tolerance on the power ripple.For the wire compensation in LHC, the currentneeded is about 80A for an electric wire of 1mlong on each side of an IP when x ¼ 0:0034. Therequirement of less than a 0.5% (or even less)ripple in the current is not difficult to achieve withcommercially available power supplies. The wirecompensation scheme is a good option for thecompensation of long-range beam–beam interactions

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due to localized parasitic collisions such as in theLHC interaction regions. It is, however, difficult toapply this scheme to the case where there is a largenumber of parasitic collisions distributed aroundring such as in Tevatron. To reduce the nonlineareffects of such non-localized long-range beam–beam interactions, a global compensation of long-range beam–beam interactions with magneticmultipole correctors based on a minimization ofnonlinearities in one-turn maps has been studiedand found to be effective [7,9]. For LHC, thismultipole compensation of the long-range beam–beam interactions is equally effective as comparedwith the wire compensation [7]. It should be notedthat to control the beam–beam effects, theelectron-beam compensation of the beam–beamtune spread has also been explored [10]. Studieshave shown that a reduction of the tune spreadwith electron beams increases the possibility of theonset of the coherent beam–beam instability and,therefore, it is not a good option for thecompensation of the beam–beam effects [11].

This work is supported by the US Departmentof Energy under Grant no. DE-FG02-04ER41288.We would like to thank the Center for Advanced

Scientific Computing at the University of Kansasfor the use of the Supercomputer.

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strong–strong simulation study for the wire compensation of long-range beam–beam effect in lhc

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