We have used the beam-beam simulation code CBI to study the beam-beam interaction for the LHC. We find that for nominal LHC parameters, and assuming only one bunch per beam, there are no collective (coherent) beam-beam instabilities. We have investigated the effect of sweeping one of the beams around the other (a procedure that could be used as a diagnostic for head-on beam-beam collisions), We find that this does not cause any problems at the nominal current, though at higher currents there can be beam blow-up and collective beam motion. (4 refs)

Furman, M A

Krishnagopal, S

Turner, W C

CERN Document Server

STUDIES OF THE BEAM-BEAM INTERACTION FOR THE LHC#S. Krishnagopal * , Centre for Advanced Technology, Indore, INDIAM. A. Furman and W. C. Turner, LBNL, Berkeley, CA, USA# Work supported by the US Department of Energy under contractNo. DE-AC-03-76SF00098.* Email: skrishna@cat.ernet.inAbstractWe have used the beam-beam simulation code CBI tostudy the beam-beam interaction for the LHC. We findthat for nominal LHC parameters, and assuming only onebunch per beam, there are no collective (coherent) beam-beam instabilities. We have investigated the effect ofsweeping one of the beams around the other (a procedurethat could be used as a diagnostic for head-on beam-beamcollisions). We find that this does not cause any problemsat the nominal current, though at higher currents there canbe beam blow-up and collective beam motion.1  THE CODE CBIThe code CBI (for Collective Beam-beam Interactions) is aself-consistent code that models the transverse beam-beamdynamics of beams of arbitrary distribution and ellipticity.It is a Particle-in-Cell (PIC) code that calculates the beam-beam force on a two-dimensional (transverse) Cartesiangrid. The code is evolving and presently has the followingfeatures:(a) there is only one bunch per beam and there is only onecollision point;(b) the beams are ultra-relativistic;(c) longitudinal dynamics is not modeled;(d) arc transport is linear;(e) radiation damping and fluctuations are put in once aturn and at one point in the ring;(f) there is no crossing angle;(g) transverse dimensions and distributions of the beamscan be completely arbitrary.Details of the code can be found in Refs. 1 and 2.The code, as described above, is a strong-strong beam-beam code that is best suited for studying collective beam-beam effects in storage-ring e+e– colliders, particularlyquadrupole effects (that affect the beam sizes). To ourknowledge, quadrupole collective effects have never beenstudied for hadron colliders, and it seemed interesting andrelevant to undertake this study for the Large HadronCollider (LHC). In particular, in light of a proposal forsweeping one beam around the other as a diagnostic forhead-on collisions [3], it seems relevant to look atpossible beam-size blow-up and distortion as aconsequence of quadrupole collective effects.2   RESULTS2.1  Nominal LHC ParametersLHC parameters used in the simulations are given belowin Table 1.Table 1: Parameters for the LHC simulations.Parameter ValueEnergy (TeV) 7.0Revolution period (m s) 88.9Emittance (nm-rad) 0.5Beta function (m) 0.5Tunes (H, V) (0.28, 0.31)Nominal bunch current (mA) 0.2Number of bunches 1As can be seen from Table 1, the simulations use thenominal LHC parameters [4], except for the fact that weassume one bunch per beam, and therefore do not modelparasitic beam-beam collisions. This is a feature we hopeto incorporate into the code in the future.We first ran our simulation for nominal LHCparameters, with the nominal bunch current of 0.2 mA,and with an idealized feedback system that takes out allcollective dipole effects (centroid motion). For reasons ofcomputer time, the simulations were run for only 90,000turns. Figure 1a shows the rms beam sizes for the last20,000 turns of the simulation: it is clear that for nominalLHC parameters there are no collective quadrupole effectsthat could affect the performance. Beam sizes are prettymuch equal to their nominal value, and there is no sizablebeam blow-up. When the current is increased to 1 mA(Fig. 1b) there is some beam blow-up, but all beam sizesare the same, and there is no indication of collectivebehavior.0-7803-5573-3/99/$10.00@1999 IEEE. 1674Proceedings of the 1999 Particle Accelerator Conference, New York, 1999201816σ [µm]90x1038070 turn no.Fig. 1a run #90004I=0.2 mA201816σ [µm]90x1038070 turn no.Fig. 1b run #10029I=1 mAFigure 1:  For nominal LHC parameters, plot of beamsizes as a function of turn number for the last 20,000turns; (a) for the nominal current of 0.2 mA, and (b) for acurrent of 1 mA.2.2  Sweeping One Beam Around The OtherWe next looked at the effect of sweeping one beam aroundthe other. We chose to sweep beam 1 around beam 2. Inthis case, after each turn, as before, the centroids of thetwo beams are zeroed, implementing the idealized feedbacksystem. Then the centroid of beam 1 is displaced a fixedradial distance from the zero position. Two inputparameters govern this displacement: the displacement andthe rotation period (in number of turns). The latter is thenumber of turns taken to sweep beam 1 once around beam2.In the simulations described here, we fixed the rotationperiod at 10 turns, and looked at the effect on the beamsizes of different displacements and different currents. Welooked at two different displacements, s /5 and s /10, wheres  is the nominal size of the beams (15.8 m m). We lookedat five different currents, starting from 0.1 mA, up to 2mA. All simulations were run for 90,000 turns, and withthe idealized feedback system turned on. Note that onlyone bunch is simulated in each beam: there are noparasitic collisions.201816σ [µm]90x1038070 turn no.Fig. 2a run #10032I=0.2 mAsweep=σ/5201816σ [µm]90x1038070 turn no.Fig. 2b run #10023I=1 mAsweep=σ/5x2x1y2y1Figure 2:  Beam size as a function of turn number forbeam 1 being swept around beam 2 with a radialdisplacement of s /5; (a) at I = 0.2 mA,and (b) at I = 1 mA.Figure 2 shows the beam-sizes as a function of turnnumber, for the last 20,000 turns, when the displacementis s /5. One can see that for the nominal current of 0.2mA the beam-sizes are all equal, and equal to the nominalsize of 15.8 m m, indicating that there are no deleteriouscollective beam-beam effects. At a current of 1 mA,however, it is clear that the beams are being blown-up,and by unequal amounts: in other words, the beams arebecoming elliptical, and there is a flip-flop developing.This is a signature of quadrupole collective beam-beameffects. The same picture is seen in Figure 3, for adisplacement of s /10. Again, at a current of 0.2 mA thepicture is benign, but at 1 mA a flip-flop has clearly setin.Table 2 below gives a clearer picture of how thedynamics evolve with current. At a current of 0.1 mA,there is no discernible blow-up of the beam, and nocollective motion; all beam sizes are equal, and equal tothe nominal size of 15.8 m m. When the bunch current isincreased to the nominal LHC value of 0.2 mA, there isslight beam blow-up, but all beam sizes are still equal,which indicates there is no collective motion. At a currentof 0.5 mA there are the first signs of collective motion.All beam sizes are no longer equal: a flip-flop instability1675Proceedings of the 1999 Particle Accelerator Conference, New York, 1999has developed. As the current is increased further, thebeam blow-up as well as the flip-flop become larger.201816σ [µm]90x1038070 turn no.Fig. 3a run #90005I=0.2 mAsweep=σ/10201816σ [µm]90x1038070 turn no.Fig. 3b run #10026I=1 mAsweep=σ/10x2x1 y2y1Figure 3:  Beam size as a function of turn number forbeam 1 being swept around beam 2 with a radialdisplacement of s /10; (a) at I = 0.2 mA,and (b) at I = 1 mA.Table 2: All beam sizes at the end of the simulation (after90,000 turns), at different currents, for a displacement ofs /5. Nominal beam size is 15.8 m m. Recall that beam 1is being swept around beam 2.Current(mA)s x1( m m )s y1 ( m m )s x2 ( m m )s y2( m m )0 . 1  15.8 15.8 15.8 15.80 . 2  16.2 16.1 16.1 16.10 . 5  17.4 17.4 17.1 16.91 . 0  20.6 20.0 20.4 19.62 . 0  28.1 25.4 27.6 25.13  DISCUSSION AND CONCLUSIONSIt is clear from Figs. (1a), (2a) and (3a), that for nominalLHC parameters, particularly the nominal bunch currentof 0.2 mA, there is little beam blow-up, and no collectivemotion, whether or not one beam is swept around theother. When one beam is swept around the other,collective effects are seen from a bunch current of around0.5 mA (over twice the nominal value), though at thiscurrent they are still small.It should be emphasized that our simulations do notmodel multiple bunches and therefore parasitic collisions.These could have a significant impact on collective beam-beam dynamics. We plan to extend the code to handlethese effects.In the simulations reported here dipole motion isremoved by modeling an ideal feedback system in whichthe centroids of the two beams are zeroed every turn. Theconsequences of turning off this feedback, particularly oncollective dipole motion,  need to be explored. We plan todo this in the near future.In conclusion, we have studied the beam-beaminteraction at the LHC using the code CBI. We find thatfor nominal LHC parameters, collective quadrupole effectsshould not be an issue. If one beam is swept around theother, for diagnostic purposes, then collective issues stillare unimportant at the nominal bunch current of 0.2 mA,though they could become important at currents aroundand above twice the nominal value.One of us (SK) would like to thank the other two for aninvitation to visit the Lawrence Berkeley NationalLaboratory during the summer of 1998, when most of thiswork was done. We are grateful to NERSC forsupercomputer support.4  REFERENCES[1] S. Krishnagopal, Phys. Rev. Lett. 76, 235 (1996).[2] S. Krishnagopal, Centre for Advanced Technology Internal Report,CAT-95/5 (1995).[3] Luminosity Instrumentation for the Absorbers in the Low BetaInsertions of the LHC, W. C. Turner, LBNL-42180(Aug. 1998),submitted for publication to PRST-AB.[4] The Large Hadron Collider: Conceptual Design, CERN/AC/95-05(LHC), 20 October 1995.1676Proceedings of the 1999 Particle Accelerator Conference, New York, 1999

Studies of the beam-beam interaction for the LHC

The authors have used the beam-beam simulation code CBI to study the beam-beam interaction for the LHC. We find that for nominal LHC parameters, and assuming only one bunch per beam, there are no collective (coherent) beam-beam instabilities. We have investigated the effect of sweeping one of the beams around the other (a procedure that could be used as a diagnostic for head-on beam-beam collisions). We find that this does not cause any problems at the nominal current, though at higher currents there can be beam blow-up and collective beam motion. consequence of quadrupole collective effects

Krishnagopal, S.

Furman, M. A.

Turner, W. C.

UNT Digital Library

LBNL-43061, CBP Note 308STUDIES OF THE BEAM-BEAM INTERACTION FOR THE LHC #   S. Krishnagopal  ∗ , Centre for Advanced Technology, Indore, INDIAM. A. Furman and W. C. Turner, LBNL, Berkeley, CA, USA                                                # Work supported by the US Department of Energy under contractNo. DE-AC-03-76SF00098. Presented at the PAC99, New York City,March 29-April 2nd, 1999.∗  Email: skrishna@cat.ernet.inAbstractWe have used the beam-beam simulation code CBI tostudy the beam-beam interaction for the LHC. We findthat for nominal LHC parameters, and assuming only onebunch per beam, there are no collective (coherent) beam-beam instabilities. We have investigated the effect ofsweeping one of the beams around the other (a procedurethat could be used as a diagnostic for head-on beam-beamcollisions). We find that this does not cause any problemsat the nominal current, though at higher currents there canbe beam blow-up and collective beam motion.1  THE CODE CBIThe code CBI (for    Collective   Beam-beam  Interactions) is aself-consistent code that models the ransverse beam-beamdynamics of beams of arbitrary distribution and ellipticity.It is a Particle-in-Cell (PIC) code that calculates the beam-beam force on a two-dimensional (transverse) Cartesiangrid. The code is evolving and presently has the followingfeatures:(a) there is only one bunch per beam and there is only onecollision point;(b) the beams are ultra-relativistic;(c) longitudinal dynamics is not modeled;(d) arc transport is linear;(e) radiation damping and fluctuations are put in once aturn and at one point in the ring;(f) there is no crossing angle;(g) transverse dimensions and distributions of the beamscan be completely arbitrary.Details of the code can be found in Refs. 1 and 2.The code, as described above, is a strong-strong beam-beam code that is best suited for studying collective beam-beam effects in storage-ring e+ – colliders, particularlyquadrupole effects (that affect the beam sizes). To ourknowledge, quadrupole collective effects have never beenstudied for hadron colliders, and it seemed interesting andrelevant to undertake this study for the Large HadronCollider (LHC). In particular, in light of a proposal forsweeping one beam around the other as a diagnostic forhead-on collisions [3], it seems relevant to look atpossible beam-size blow-up and distortion as aconsequence of quadrupole collective effects.2  RESULTS2.1  Nominal LHC ParametersLHC parameters used in the simulations are given belowin Table 1.Table 1:  Parameters for the LHC simulations.Parameter ValueEnergy (TeV) 7.0Revolution period (µs) 88.9Emittance (nm-rad) 0.5Beta function (m) 0.5Tunes (H, V) (0.28, 0.31)Nominal bunch current (mA) 0.2Number of bunches 1As can be seen from Table 1, the simulations use thenominal LHC parameters [4], except for the fact that weassume one bunch per beam, and therefore do not modelparasitic beam-beam collisions. This is a fe ture we hopeto incorporate into the code in the future.We first ran our simulation for nominal LHCparameters, with the nominal bunch current of 0.2 mA,and with an idealized feedback system that takes out allcollective dipole effects (centroid motion). For reasons ofcomputer time, the simulations were run for only 90,000turns. Figure 1a shows the rms beam sizes for the last20,000 turns of the simulation: it is clear that for nominalLHC parameters there are no collective quadrupole effectsthat could affect the performance. Beam sizes are prettymuch equal to their nominal value, and there is no sizablebeam blow-up. When the current is increased to 1 mA(Fig. 1b) there is some beam blow-up, but all beam sizesare the same, and there is no indication of collectivebehavior.201816σ [µm]90x1038070 turn no.Fig. 1a run #90004I=0.2 mA201816σ [µm]90x1038070 turn no.Fig. 1b run #10029I=1 mAFigure 1:  For nominal LHC parameters, plot of beamsizes as a function of turn number for the last 20,000turns; (a) for the nominal current of 0.2 mA, and (b) for acurrent of 1 mA.2.2  Sweeping One Beam Around The OtherWe next looked at the effect of sweeping one beam aroundthe other. We chose to sweep beam 1 around beam 2. Inthis case, after each turn, as before, the centroids of thetwo beams are zeroed, implementing the idealized fe backsystem. Then the centroid of beam 1 is displaced a fixedradial distance from the zero position. Two inputparameters govern this displacement: the displacement andthe rotation period (in number of turns). The latter is thenumber of turns taken to sweep beam 1 once around beam2.In the simulations described here, we fixed the rotationperiod at 10 turns, and looked at the effect on the beamsizes of different displacements and different currents. Welooked at two different displacements, σ/5 and σ/10, whereσ is the nominal size of the beams (15.8 µm). We lookedat five different currents, starting from 0.1 mA, up to 2mA. All simulations were run for 90,000 turns, and withthe idealized feedback system turned on. Note that onlyone bunch is simulated in each beam: there are noparasitic collisions.201816σ [µm]90x1038070 turn no.Fig. 2a run #10032I=0.2 mAsweep=σ/5201816σ [µm]90x1038070 turn no.Fig. 2b run #10023I=1 mAsweep=σ/5x2x1y2y1Figure 2:  Beam size as a function of turn number forbeam 1 being swept around beam 2 with a radialdisplacement of σ/5; (a) at I = 0.2 mA, and (b) at I = 1mA.Figure 2 shows the beam-sizes as a function of turnnumber, for the last 20,000 turns, when the displacementis σ/5. One can see that for the nominal current of 0.2mA the beam-sizes are all equal, and equal to the nominalsize of 15.8 µm, indicating that there are no deleteriouscollective beam-beam effects. At a current of 1 mA,however, it is clear that the beams are being blown-up,and by unequal amounts: in other words, the beams arebecoming elliptical, and there is a flip-flop developing.This is a signature of quadrupole collective beam-beameffects. The same picture is seen in Figure 3, for adisplacement of σ/10. Again, at a current of 0.2 mA thepicture is benign, but at 1 mA a flip-flop has clearly setin.Table 2 below gives a clearer picture of how thedynamics evolve with current. At a current of 0.1 mA,there is no discernible blow-up of the beam, and nocollective motion; all beam sizes are equal, and equal tothe nominal size of 15.8 µm. When the bunch current isincreased to the nominal LHC value of 0.2 mA, there isslight beam blow-up, but all beam sizes are still equal,which indicates there is no collective motion. At a currentof 0.5 mA there are the first signs of collective motion.All beam sizes are no longer equal: a flip-flop instabilityhas developed. As the current is increased further, thebeam blow-up as well as the flip-flop become larger.201816σ [µm]90x1038070 turn no.Fig. 3a run #90005I=0.2 mAsweep=σ/10201816σ [µm]90x1038070 turn no.Fig. 3b run #10026I=1 mAsweep=σ/10x2x1 y2y1Figure 3:  Beam size as a function of turn number forbeam 1 being swept around beam 2 with a radialdisplacement of σ/10; (a) at I = 0.2 mA, and (b) at I = 1mA.Table 2: All beam sizes at the end of the simulation (after90,000 turns), at different currents, for a displacement ofσ/5. Nominal beam size is 15.8 µm. Recall that beam 1is being swept around beam 2.Current(mA)σx1(µm)σy1 (µm )σx2 (µm )σy2(µm)0 . 1 15.8 15.8 15.8 15.80 . 2 16.2 16.1 16.1 16.10 . 5 17.4 17.4 17.1 16.91 . 0 20.6 20.0 20.4 19.62 . 0 28.1 25.4 27.6 25.13  DISCUSSION AND CONCLUSIONSIt is clear from Figs. (1a), (2a) and (3a), that for nominalLHC parameters, particularly the nominal bunch currentof 0.2 mA, there is little beam blow-up, and no c llectivemotion, whether or not one beam is swept around theother. When one beam is swept around the other,collective effects are seen from a bunch current of around0.5 mA (over twice the nominal value), though at thiscurrent they are still small.It should be emphasized that our simulations do notmodel multiple bunches and therefore parasitic collisions.These could have a significant impact on collective beam-beam dynamics. We plan to extend the code to handlethese effects.In the simulations reported here dipole motion isremoved by modeling an ideal feedback system in whichthe centroids of the two beams are zeroed every turn. Theconsequences of turning off this feedback, particularly oncollective dipole motion,  need to be explored. We plan todo this in the near future.In conclusion, we have studied the beam-beaminteraction at the LHC using the code CBI. We find thatfor nominal LHC parameters, collective quadrupole effectsshould not be an issue. If one beam is swept around theother, for diagnostic purposes, then collective issues stillare unimportant at the nominal bunch urrent of 0.2 mA,though they could become important at currents aroundand above twice the nominal value.One of us (SK) would like to thank the other two for aninvitation to visit the Lawrence Berkeley NationalLaboratory during the summer of 1998, when most of thiswork was done. We are grateful to NERSC forsupercomputer support.4  REFERENCES[1] S. Krishnagopal, Phys. Rev. Lett. 76, 235 (1996).[2] S. Krishnagopal, Centre for Advanced Technology Internal Report,CAT-95/5 (1995).[3] Luminosity Instrumentation for the Absorbers in the Low BetaInsertions of the LHC, W. C. Turner, LBNL-42180(Aug. 1998),submitted for publication to PRST-AB.[4] The Large Hadron Collider: Conceptual Design, CERN/AC/95-05(LHC), 20 October 1995.

Furman, M.A.

Turner, W.C.

eScholarship - University of California

Lawrence Berkeley National LaboratoryLawrence Berkeley National LaboratoryTitleStudies of the beam-beam interaction for the LHCPermalinkhttps://escholarship.org/uc/item/8rw5p1kpAuthorsKrishnagopal, S.Furman, M.A.Turner, W.C.Publication Date1999-03-01eScholarship.org Powered by the California Digital LibraryUniversity of CaliforniaLBNL-43061, CBP Note 308STUDIES OF THE BEAM-BEAM INTERACTION FOR THE LHC #   S. Krishnagopal  ∗ , Centre for Advanced Technology, Indore, INDIAM. A. Furman and W. C. Turner, LBNL, Berkeley, CA, USA                                                # Work supported by the US Department of Energy under contractNo. DE-AC-03-76SF00098. Presented at the PAC99, New York City,March 29-April 2nd, 1999.∗  Email: skrishna@cat.ernet.inAbstractWe have used the beam-beam simulation code CBI tostudy the beam-beam interaction for the LHC. We findthat for nominal LHC parameters, and assuming only onebunch per beam, there are no collective (coherent) beam-beam instabilities. We have investigated the effect ofsweeping one of the beams around the other (a procedurethat could be used as a diagnostic for head-on beam-beamcollisions). We find that this does not cause any problemsat the nominal current, though at higher currents there canbe beam blow-up and collective beam motion.1  THE CODE CBIThe code CBI (for    Collective   Beam-beam  Interactions) is aself-consistent code that models the ransverse beam-beamdynamics of beams of arbitrary distribution and ellipticity.It is a Particle-in-Cell (PIC) code that calculates the beam-beam force on a two-dimensional (transverse) Cartesiangrid. The code is evolving and presently has the followingfeatures:(a) there is only one bunch per beam and there is only onecollision point;(b) the beams are ultra-relativistic;(c) longitudinal dynamics is not modeled;(d) arc transport is linear;(e) radiation damping and fluctuations are put in once aturn and at one point in the ring;(f) there is no crossing angle;(g) transverse dimensions and distributions of the beamscan be completely arbitrary.Details of the code can be found in Refs. 1 and 2.The code, as described above, is a strong-strong beam-beam code that is best suited for studying collective beam-beam effects in storage-ring e+ – colliders, particularlyquadrupole effects (that affect the beam sizes). To ourknowledge, quadrupole collective effects have never beenstudied for hadron colliders, and it seemed interesting andrelevant to undertake this study for the Large HadronCollider (LHC). In particular, in light of a proposal forsweeping one beam around the other as a diagnostic forhead-on collisions [3], it seems relevant to look atpossible beam-size blow-up and distortion as aconsequence of quadrupole collective effects.2  RESULTS2.1  Nominal LHC ParametersLHC parameters used in the simulations are given belowin Table 1.Table 1:  Parameters for the LHC simulations.Parameter ValueEnergy (TeV) 7.0Revolution period (µs) 88.9Emittance (nm-rad) 0.5Beta function (m) 0.5Tunes (H, V) (0.28, 0.31)Nominal bunch current (mA) 0.2Number of bunches 1As can be seen from Table 1, the simulations use thenominal LHC parameters [4], except for the fact that weassume one bunch per beam, and therefore do not modelparasitic beam-beam collisions. This is a fe ture we hopeto incorporate into the code in the future.We first ran our simulation for nominal LHCparameters, with the nominal bunch current of 0.2 mA,and with an idealized feedback system that takes out allcollective dipole effects (centroid motion). For reasons ofcomputer time, the simulations were run for only 90,000turns. Figure 1a shows the rms beam sizes for the last20,000 turns of the simulation: it is clear that for nominalLHC parameters there are no collective quadrupole effectsthat could affect the performance. Beam sizes are prettymuch equal to their nominal value, and there is no sizablebeam blow-up. When the current is increased to 1 mA(Fig. 1b) there is some beam blow-up, but all beam sizesare the same, and there is no indication of collectivebehavior.201816σ [µm]90x1038070 turn no.Fig. 1a run #90004I=0.2 mA201816σ [µm]90x1038070 turn no.Fig. 1b run #10029I=1 mAFigure 1:  For nominal LHC parameters, plot of beamsizes as a function of turn number for the last 20,000turns; (a) for the nominal current of 0.2 mA, and (b) for acurrent of 1 mA.2.2  Sweeping One Beam Around The OtherWe next looked at the effect of sweeping one beam aroundthe other. We chose to sweep beam 1 around beam 2. Inthis case, after each turn, as before, the centroids of thetwo beams are zeroed, implementing the idealized fe backsystem. Then the centroid of beam 1 is displaced a fixedradial distance from the zero position. Two inputparameters govern this displacement: the displacement andthe rotation period (in number of turns). The latter is thenumber of turns taken to sweep beam 1 once around beam2.In the simulations described here, we fixed the rotationperiod at 10 turns, and looked at the effect on the beamsizes of different displacements and different currents. Welooked at two different displacements, σ/5 and σ/10, whereσ is the nominal size of the beams (15.8 µm). We lookedat five different currents, starting from 0.1 mA, up to 2mA. All simulations were run for 90,000 turns, and withthe idealized feedback system turned on. Note that onlyone bunch is simulated in each beam: there are noparasitic collisions.201816σ [µm]90x1038070 turn no.Fig. 2a run #10032I=0.2 mAsweep=σ/5201816σ [µm]90x1038070 turn no.Fig. 2b run #10023I=1 mAsweep=σ/5x2x1y2y1Figure 2:  Beam size as a function of turn number forbeam 1 being swept around beam 2 with a radialdisplacement of σ/5; (a) at I = 0.2 mA, and (b) at I = 1mA.Figure 2 shows the beam-sizes as a function of turnnumber, for the last 20,000 turns, when the displacementis σ/5. One can see that for the nominal current of 0.2mA the beam-sizes are all equal, and equal to the nominalsize of 15.8 µm, indicating that there are no deleteriouscollective beam-beam effects. At a current of 1 mA,however, it is clear that the beams are being blown-up,and by unequal amounts: in other words, the beams arebecoming elliptical, and there is a flip-flop developing.This is a signature of quadrupole collective beam-beameffects. The same picture is seen in Figure 3, for adisplacement of σ/10. Again, at a current of 0.2 mA thepicture is benign, but at 1 mA a flip-flop has clearly setin.Table 2 below gives a clearer picture of how thedynamics evolve with current. At a current of 0.1 mA,there is no discernible blow-up of the beam, and nocollective motion; all beam sizes are equal, and equal tothe nominal size of 15.8 µm. When the bunch current isincreased to the nominal LHC value of 0.2 mA, there isslight beam blow-up, but all beam sizes are still equal,which indicates there is no collective motion. At a currentof 0.5 mA there are the first signs of collective motion.All beam sizes are no longer equal: a flip-flop instabilityhas developed. As the current is increased further, thebeam blow-up as well as the flip-flop become larger.201816σ [µm]90x1038070 turn no.Fig. 3a run #90005I=0.2 mAsweep=σ/10201816σ [µm]90x1038070 turn no.Fig. 3b run #10026I=1 mAsweep=σ/10x2x1 y2y1Figure 3:  Beam size as a function of turn number forbeam 1 being swept around beam 2 with a radialdisplacement of σ/10; (a) at I = 0.2 mA, and (b) at I = 1mA.Table 2: All beam sizes at the end of the simulation (after90,000 turns), at different currents, for a displacement ofσ/5. Nominal beam size is 15.8 µm. Recall that beam 1is being swept around beam 2.Current(mA)σx1(µm)σy1 (µm )σx2 (µm )σy2(µm)0 . 1 15.8 15.8 15.8 15.80 . 2 16.2 16.1 16.1 16.10 . 5 17.4 17.4 17.1 16.91 . 0 20.6 20.0 20.4 19.62 . 0 28.1 25.4 27.6 25.13  DISCUSSION AND CONCLUSIONSIt is clear from Figs. (1a), (2a) and (3a), that for nominalLHC parameters, particularly the nominal bunch currentof 0.2 mA, there is little beam blow-up, and no c llectivemotion, whether or not one beam is swept around theother. When one beam is swept around the other,collective effects are seen from a bunch current of around0.5 mA (over twice the nominal value), though at thiscurrent they are still small.It should be emphasized that our simulations do notmodel multiple bunches and therefore parasitic collisions.These could have a significant impact on collective beam-beam dynamics. We plan to extend the code to handlethese effects.In the simulations reported here dipole motion isremoved by modeling an ideal feedback system in whichthe centroids of the two beams are zeroed every turn. Theconsequences of turning off this feedback, particularly oncollective dipole motion,  need to be explored. We plan todo this in the near future.In conclusion, we have studied the beam-beaminteraction at the LHC using the code CBI. We find thatfor nominal LHC parameters, collective quadrupole effectsshould not be an issue. If one beam is swept around theother, for diagnostic purposes, then collective issues stillare unimportant at the nominal bunch urrent of 0.2 mA,though they could become important at currents aroundand above twice the nominal value.One of us (SK) would like to thank the other two for aninvitation to visit the Lawrence Berkeley NationalLaboratory during the summer of 1998, when most of thiswork was done. We are grateful to NERSC forsupercomputer support.4  REFERENCES[1] S. Krishnagopal, Phys. Rev. Lett. 76, 235 (1996).[2] S. Krishnagopal, Centre for Advanced Technology Internal Report,CAT-95/5 (1995).[3] Luminosity Instrumentation for the Absorbers in the Low BetaInsertions of the LHC, W. C. Turner, LBNL-42180(Aug. 1998),submitted for publication to PRST-AB.[4] The Large Hadron Collider: Conceptual Design, CERN/AC/95-05(LHC), 20 October 1995.

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https://escholarship.org/uc/item/8rw5p1kp

Studies of the beam-beam interaction for the LHC

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