The LHC dynamic aperture in collision is constrained mainly by the beam-beam encounters, and by the field errors in the low-beta triplet quadrupoles. The nominal field errors were used and have been corrected with a local corrector scheme at each IP. The correction algorithm is explained, and the resulting dynamic aperture is shown. In the calculations, the effect of the crossing angle geometry, the beta-function, the bunch intensity and the pacman bunches on the dynamic aperture are studied. It is, however, also necessary to study how the unavoidable long range beam-beam encounters influence the dynamic aperture

Grote, H

Leunissen, L H A

Schmidt, F

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EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCHEuropean Laboratory for Particle PhysicsLHC DYNAMIC APERTURE INCLUDING THE BEAM-BEAM FORCEL.H.A. Leunissen, H. Grote and F. SchmidtThe LHC dynamic aperture in collision is constrained mainly by the beam-beam encounters, and by thefield errors in the low-beta triplet quadrupoles. The nominal field errors were used and have beencorrected with a local corrector scheme at each IP. The correction algorithm is explained, and theresulting dynamic aperture is shown. In the calculations, the effect of the crossing angle geometry, thebeta-function, the bunch intensity and the pacman bunches on the dynamic aperture are studied. It is,however, also necessary to study how the unavoidable long range beam-beam encounters influence thedynamic aperture.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 405Abstract14 August 2000LHC DYNAMIC APERTURE INCLUDING THE BEAM-BEAM FORCEL.H.A. Leunissen, H. Grote, F. Schmidt, CERN, Geneva, SwitzerlandAbstractThe LHC dynamic aperture in collision is constrainedmainly by the beam-beam encounters, and by the field er-rors in the low-beta triplet quadrupoles. The nominal fielderrors were used and have been corrected with a local cor-rector scheme at each IP. The correction algorithm is ex-plained,and the resulting dynamic aperture is shown. In thecalculations, the effect of the crossing angle geometry, thebeta-function, the bunch intensity and the pacman buncheson the dynamic aperture are studied. It is, however, alsonecessary to study how the unavoidable long range beam-beam encounters influence the dynamic aperture.1 INTRODUCTIONThe LHC model studied in this note is the new nominal col-lision machine (ATLAS, CMS, and LHCB with head–oncollisions and halo collisions at ALICE) and we present anextension of previous tracking results [1]. The calculationsare based on LHC version 6.0, with 15 parasitic crossingpoints on either side of each IP. The crossing angles arefixed throughout: ±150µrad vertical at IP1 and IP2 (alter-nating), ±150µrad horizontal at IP5 and IP8 (alternating).−60.0 −40.0 −20.0 0.0 20.0 40.0 60.0distance from IP1 [m]0.05.010.015.0vertical beam separation [σ]Figure 1: Full bunch separation at IP1. The scale with thetick marks indicates the beam-beam encounter positions.Arrow head lines mark the ranges with triplet quadrupoles.The bunch sizes in the opposite beam appearing in thebeam–beam element definition, and the beam separation inFigure 1 were taken from the orbits and beta-functions ofthe two matched rings including the latest separation bumpscheme. The number of beam–beam encounters is some-what arbitrary. However, 15 parasitic crossings are consid-ered a reasonable choice since number 16 is already insidethe beam separation dipole D1.First, we present the low–beta quadrupole field errorsused in the numerical studies. Then we show the correctionformalism of the multipole components. Numerical calcu-lations presented in the following paragraphs determine theinfluence of the crossing angle, β and tune modulation onthe dynamic aperture.2 TRIPLET ERRORSComponent systematic uncertainty randomb3 0 0.72 0.36b4 -0.175 0.83 0.36b6 0.34 0.91 0.21a3 0 0.69 0.34a4 0 0.33 0.34Component systematic uncertainty randomb3 0 0.63 0.34b4 0 0.22 0.34b6 0.21 0.41 0.18a3 0 0.32 0.34a4 0 0.26 0.34Table 1: Low–beta quadrupole field errors for KEK ver-sion 4.x (upper) and FNAL version 3.1 (lower). Values arerelative to the main field at x = 17 mm in units of 10−4.The largest components of the latest triplet errors aregiven in Table 1. The body and end effects have been com-bined into one single number for the thin–lens approachused here: each triplet quadrupole is split into four thin–lens quadrupoles at each of IP1, IP2, IP5, and IP8.3 CORRECTION SCHEME ANDFORMALISMOn either side of IP1, IP2, IP5, and IP8 two correctorgroups are placed as proposed by J. Strait at a CERN-KEK-US meeting, April 2000. Each corrector group containsseveral correction spools such that on either side of each IPone corrector exists for b3, b4, b6, a3, and a4. The correc-tion formalism follows the one outlined by A. Verdier andA. Faus–Golfe [2]. The principle is rather simple: with onecorrector for each multipole component on either side ofeach IP, we compensate the total kick for purely horizon-tal, and purely vertical motion simultaneously.4 DYNAMIC APERTURECALCULATIONSWe present tracking studies for the LHC with and with-out the beam–beam interaction. The maximum radius forwhich the particles are stable for 105 turns is defined as thedynamic aperture (DA). The DA is calculated for five dif-ferent phase space angles, φ = atan(√y/x) with φ =15, 30, 45, 60 and 75 and x, y the horizontal andvertical transverse emittances, respectively. The trackingwas performed with SixTrack [3] for 60 different sets ofthe random multipole distribution, called “seeds” in the fol-lowing, of the triplet errors at collision energy.0246810121416180 15 30 45 60 75 90atan(    εy/εx)Dynamic Aperture [ σ] Average (no correction) Average (with Correction) Minimum (no Correction) Minimum (with Correction)(a)0246810121416180 15 30 45 60 75 90atan(    εy/εx)Dynamic Aperture [ σ] Average Minimum(b)Figure 2: Dynamic Aperture without (part a) and with (partb) beam–beam interaction (triplet correction included).Tracking performed for 105 turns.Figure 2a shows that the triplet errors reduce the averageand minimum dynamic aperture to about 13 σ and 9 σ, re-spectively. This is mainly due to the b6 component of thequadrupoles. The correction system, as described above,raises the dynamic aperture to more than 15 σ and 12 σ,respectively. When including the beam–beam interaction(Figure 2b), and the triplet errors and their correction, thedynamic aperture goes down to values around 8.5 σ and6.8 σ, respectively.Table 2 holds the data of all previously discussed cases.The DA is mainly limited by the parasitic beam cross-ings. This is shown by calculations in which either thehead–on or the parasitic collisions are switched off. Forexample, after the head–on collisions are turned off at IP1,IP2, IP5 and IP8 the DA (for 5 phase space angles) re-mains the same. However, when only head–on collisionsare inserted in the calculations, the minimum DA goes upto more than 13 σ. To increase the dynamic aperture theeffect of the parasitic beam crossings should be reduced.Case DA 15 30 45 60 75Before Min. 11.5 10.6 9.1 9.2 9.2Corr. Av. 14.8 14.4 13.4 12.7 13.1After Min. 13.5 13.0 12.2 12.4 12.4Corr. Av. 17.3 16.5 16.1 15.6 16.2After Min. 6.8 6.7 7.4 7.7 8.0Corr. Av. 8.4 8.1 8.5 9.0 9.5Table 2: Computed DA for the LHC with triplet errorsat collision energy without (upper) and including beam–beam interactions (lower). Each time the minimum of,and the average over 60 seeds are given. The angle isφ = atan(√y/x).246810120 100 200 300 400 500 600 700Dynamic Aperture [   ] Crossing angle [   rad]µσFigure 3: DA as function of the crossing angle at IP1 andIP5. The symbols are represented by: 15 = +, 30 = ×,45 = ∗, 60 = white box, 75 = black box.Changing the crossing angle results in different bunchseparation in the interaction region and changes thestrength of the parasitic crossings. For one seed, the cross-ing angle is varied and the luminosity is kept constant (1034cm−2s−1) by adjusting the bunch intensity accordingly.The crossing angles at IP1 and IP5 are changed simultane-ously while keeping the crossings at IP2 and IP8 constant,see Figure 3. In this plot the five lines indicate the differentphase space angles (15 = +, 30 =×, 45 = ∗, 60 = whitebox, 75 = black box). This result shows that an increase incrossing angle to 400 µrad would improve the DA by 2 σ.The DA increases from 0 to 400 µrad due to the increaseof bunch separation. A further increase of the crossing an-gle results in a decrease of the DA. This is the effect of thelarge particle orbit in the low–beta quadrupoles.The intensity of the bunch influences the beam–beamkick. Figure 4 shows the change of DA as function ofthe bunch intensity (nominal 1011 particles). In this plot0.7 · 1011 particles correspond to a luminosity of 0.4 · 1034cm−2s−1 and 1.4 · 1011 particles per bunch gives a lumi-nosity of 1.6 · 1034 cm−2s−1. The increase of the bunchintensity reduces the DA by approximately 2 σ over theindicated luminosity range. The beam–beam parameterchanges from 0.0023 to 0.0046.0246810120.6 0.8 1 1.2 1.4Dynamic Aperture [  ]Bunch intensity [10     part.]   11σFigure 4: DA as function of the bunch intensity for a cross-ing angle of 300 µrad. For the legend, see Figure 3.For coherent effects it can be important to have the tworings operating with different tunes [4]. Two workingpoints have been selected [5] for operation of the LHC.They are: Qx=0.232, Qy=0.242 (average DA = 7.6 σ) andQx=0.285, Qy=0.295 (average DA = 7.6 σ), respectively.The average is taken over the five different phase spaceangles and the crossing angle is 300 µrad. Tracking hasshown that both working points yield approximately thesame dynamic aperture during collision as with the nom-inal tunes (Qx=0.310, Qy=0.320, average DA = 8.1 σ).02468100.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8Dynamic Aperture [   ] * [m]σβFigure 5: DA as function of β at IP1 and IP5 at constantseparation. For the legend, see Figure 3.Investigation of another tune Qx=0.307, Qy=0.317 isdone to shift the footprint away from the third-order res-onance. This should result in a larger DA. However, thedistance to the 7th-order resonance is decreased resultingin an average DA of 7.7 σ.Figure 5 shows the dependence of the DA on the β–function at the IP. The crossing angle is varied simulta-neously with the β–function so as to maintain a constantseparation at the parasitic collision points. The figure indi-cates that the DA peaks at a β of 0.4 m and reduces to 4 σat a β of 0.2 m.The dependence of the DA on the relative momentumdeviation was also tested. For 60 different seeds the relativemomentum deviation δ was reduced by 50 % (the nominalvalue of the bucket half height is δ = 0.00036). Table 3shows the result for this calculation.φDyn. Aper. 15 30 45 60 75Minimum 6.2 7.4 6.8 6.3 7.2Average 8.9 8.4 8.2 7.8 8.3Table 3: Computed DA for the LHC with triplet errors atcollision energy with half the bucket height.The DA is not influenced by reducing the amplitude of thesynchrotron oscillation.5 CONCLUSIONSTracking has been performed for the LHC lattice version6.0 at collision energy without and including the beam–beam interaction. Each interaction zone carries the tripleterrors of either FNAL or KEK and a correction system hasbeen implemented to counteract mainly the b6 component.The correction has shown to be very effective for the casewithout the beam–beam interaction. But there is also a con-siderable improvement when this interaction is included,resulting in a minimum dynamic aperture of some 6 σ for105 turns. It is therefore advisable to increase the totalcrossing angle to 400 µrad which results in a net gain ofroughly 2 σ. This gain should not lead to a substantial lossin luminosity since the particle intensity can be adjustedwithout noticeable change in the dynamic aperture.6 REFERENCES[1] W. Chou and D. Ritson, “Dynamic Aperture Studies duringCollisions in the LHC”, LHC Project Report 123.[2] A. Verdier and A. Faus–Golfe, “Multipole Compensation inthe LHC low–β Insertions”, PAC97 Vancouver, ConferenceProceedings, and LHC Project Report 116.[3] F. Schmidt, “SixTrack, Version 1, Single particle trackingcode treating transverse motion with synchrotron oscillationsin a symplectic manner”, CERN/SL/90–11(AP) (1990).[4] A. Hofmann, “Beam-Beam Modes for Two Beams withUnequal Tunes”, Proc. Workshop on Beam-Beam Effects,CERN-SL-99-039 AP (1999).[5] J. Gareyte, “Beam-Beam Design Criteria for LHC”, Proc.Workshop on Beam-Beam Effects, CERN-SL-99-039 AP(1999).

LHC Dynamic Aperture Including the Beam-Beam Force

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