The equilibrium LHC beam distribution at large amplitudes is a crucial input to the collimation and machine protection design, as well as to background studies. Its estimation requires a knowledge of the diffusion rates at which beam particles are transported to large transverse or longitudinal amplitudes. Important known mechanisms of particle diffusion include Touschek scattering, synchrotron radiation, intrabeam scattering (IBS) the nonlinear motion due to the long-range (LR) beam-beam (BB) collisions at top energy, persistent-current field errors during injection and at the start of acceleration, and Coulomb scattering off the residual gas. We summarize the expected contributions from different sources, introduce a diffusion model, and illustrate the evolution of the beam distribution at 7 TeV

Assmann, R W

Schmidt, F

Zimmermann, Frank

Zorzano-Mier, M P

CERN Document Server

EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCHEuropean Laboratory for Particle PhysicsEQUILIBRIUM BEAM DISTRIBUTION AND HALO IN THE LHCR. Assmann1, F. Schmidt1, F. Zimmermann1 and M.-P. Zorzano2The equilibrium LHC beam distribution at large amplitudes is a crucial input to the collimation and machineprotection design, as well as to background studies. Its estimation requires a knowledge of the diffusionrates at which beam particles are transported to large transverse or longitudinal amplitudes. Importantknown mechanisms of particle diffusion include Touschek scattering, synchrotron radiation, intrabeamscattering (IBS), the nonlinear motion due to the long-range (LR) beam-beam (BB) collisions at top energy,persistent-current field errors during injection and at the start of acceleration, and Coulomb scattering offthe residual gas. We summarize the expected contributions from different sources, introduce a diffusionmodel, and illustrate the evolution of the beam distribution at 7 TeV..1 CERN, Geneva, Switzerland2 INTA, SpainPresented at the Eighth European Particle Accelerator Conference (EPAC)3-7 June 2002 - La Villette, Paris, FranceGeneva, Large Hadron Collider ProjectAdministrative SecretariatLHC DivisionCERNCH - 1211 Geneva 23SwitzerlandLHC Project Report 592Abstract8 July 2002Equilibrium Beam Distribution and Halo in the LHCR. Assmann, F. Schmidt, F. Zimmermann, CERN, Geneva, Switzerland; M.-P. Zorzano, INTA, SpainAbstractThe equilibrium LHC beam distribution at large ampli-tudes is a crucial input to the collimation and machine pro-tection design, as well as to background studies. Its esti-mation requires a knowledge of the diffusion rates at whichbeam particles are transported to large transverse or longi-tudinal amplitudes. Important known mechanisms of par-ticle diffusion include Touschek scattering, synchrotron ra-diation, intrabeam scattering (IBS), the nonlinear motiondue to the long-range (LR) beam-beam (BB) collisions attop energy, persistent-current field errors during injectionand at the start of acceleration, and Coulomb scattering offthe residual gas. We summarize the expected contributionsfrom different sources, introduce a diffusion model, and il-lustrate the evolution of the beam distribution at 7 TeV.1 INTRABEAM SCATTERINGThe transverse particle position at location s is relatedto the action variable I via x =√2β(s)I cosφ(s), whereφ(s) denotes the betatron phase, and β(s) the beta function.The average action equals the rms beam emittance< I >=. We estimate the action drift due to IBS as〈∆I∆t〉≈ x,0τIBS,x≈ 1.5× 10−15ms−1 , (1)using τIBS,x ≈ 94 hr (at 7 TeV with 16 MV rf voltage) andγx,0 ≈ 3.75 µm, and the associated diffusion coefficientasDIBS(I) ≈ 2I〈∆I/∆t〉 ≈ 2.9× 10−15Ims−1 . (2)At injection, τIBS,x ≈ 44 hr, and DIBS(I) ≈ 6.3 ×10−15Ims−1.2 GAS SCATTERINGThe diffusion due to Coulomb scattering off the residualgas is described byDgs(I) = Iβ〈∆θ2∆t〉= Iβ(14.1MeV/cp)2ρcX0, (3)where β is the average beta function (about 100 m), X 0 theradiation length of the gas and ρ its density.The LHC ‘design gas density’ is derived for a 100 hrlifetime due to nuclear interactions [1]. Consider as anexample an H2 density of 1015 molecules per m3 [1].With X0 ≈ 610 kg m−2, at 7 TeV one finds Dgs(I) ≈6×10−16Ims−1 , smaller than the diffusion expected fromIBS by a factor of 5. The corresponding emittance lifetimeis 500 hrs. If we consider instead CO (X0 ≈ 362 kg m−2)at a density of 1.3 × 1014 m−3, also compatible with the100 hr beam lifetime, we get Dgs(I) ≈ 2× 10−15 I ms−1or an emittance lifetime of 150 hr. At injection energy thiswould decrease to 40 minutes. Fortunately, this CO pres-sure is considered as unlikely to occur in the LHC [1].3 LONG-RANGE BEAM BEAMThe dominant diffusion mechanism at large amplitudesis the effect of the long-range collisions. At 7 TeV the twobeams are separated by about 9.5σ during npar = 30 long-range encounters around each of the two main interactionpoints (IPs).(σx,y)Figure 1: Change of action variance per turn vs. start am-plitude xstart (= ystart) in units of rms beam size for twoIPs [2].As an example, Fig. 1 illustrates the diffusion rate com-puted by a weak-strong beam-beam simulations as a func-tion of amplitude. Note that a ‘1’ on the vertical axisrefers to ∆I2/∆t ≈ 3 × 10−15 m2/s or to ∆x ≈2.3 mm in ∆t ≈ 1 s at β = 100 m.There is a clear threshold (‘diffusive aperture’) at an am-plitude of about 6σx,y. Beyond this aperture, the motion isstrongly chaotic and the diffusion here can be described bythe analytical expression [3]Dlr(I) =K2frev21A− 1f(A) , where (4)f(A) =[A3 −A2 + 4A2√1−A1+A − 6A+ 6− 6√1−A1+A],A =√2I/β∗/θc, and K = 2rpNbnpar/γ. For the LHCβ∗ = 0.5 m, frev = 11 kHz (revolution frequency), θc =300µrad (crossing angle), Nb = 1.1 × 1011, npar = 30(considering 1 IP), rp ≈ 1.5 × 10−18 m, γ ≈ 7461. Atsmall amplitudes the motion is regular, and in this case theestimated diffusion coefficient (4) can be regarded as anupper bound.4 DIFFUSION MODELWe approximate the evolution of the beam distributionf(I) by a diffusion equation∂f∂t=∂∂I(D(I)∂f∂I). (5)Further we assume that below 6σ (I/ < 18) the diffusioncoefficient is dominated by intrabeam scattering, approxi-mated by Eq. (2), and that, if the second beam is present,above 6σ it is due to long-range collision, and describedby Eq. (4). The change in D(I) = DIBS(I) + Dlr(I) isdiscontinuous. The diffusion coefficient so obtained is il-lustrated in Fig. 2.Figure 2: Model diffusion coefficient vs. normalized ac-tion, representing the combined effect of intrabeam scatter-ing and long-range collisions.The primary collimators may be represented by an ab-sorbing boundary, if we neglect the re-scattering probabil-ity of about 1%.5 TIME EVOLUTIONWe assume that the closest collimator is located at 7 σ.For I = 0 we choose a reflecting boundary. The initialbeam distribution is taken to be exponential in the actionvariable of the form exp(−I/) (here  is the rms emit-tance). We integrate Eq. (5) in time, following a proceduresimilar to that described in [4].Figure 3 shows the total number of protons lost on thecollimators as a function of time. Most of the diffusion isdue to intrabeam scattering. It takes about 60 hours beforeparticles are transferred near the diffusive aperture. After90 hours the long-range collisions increase the total lossesby about 30%.The effect of the long-range collisions becomes muchmore pronounced, if we increase the core diffusion rate. Afactor of about two increase in the diffusion rate may beexpected from gas scattering, if the CO molecular density00.010.020.030.040.050.060.070.080.090 10 20 30 40 50 60 70 80 90 100% beam-losst [h]with beam-beam dif.without beam-beam dif.Figure 3: Fraction of protons lost vs. time in hrs with intra-beam scattering only (lower curve) and if diffusion due tolong-range beam-beam forces is also included (top curve).is 1.3×1014 m−3. As a more pessimistic scenario, we con-sider a factor 30 increase in the core diffusion rate. In thiscase, with or without long-range collisions, after roughly6 hours the emittance is 3 times larger than at the begin-ning. From that moment on a much higher particle densityis incident on the collimator. With long-range collisionspresent, after 16 hr 10% of the beam has been lost on thecollimators, which is about two times more than withoutthe long-range collisions. However, radiation damping at 7TeV, not included in the model, will reduce the losses.Figure 4 displays the beam distribution at various timesduring the store for the nominal configuration and for the30 times larger core diffusion rate, respectively. The leftpictures represent the effect of intrabeam or gas scatteringalone, the right picture includes the long-range collisons.The latter clearly deplete the beam halo at large amplitudes.These distributions provide information on the beam loss atthe collimators in case of orbit motion.1e-161e-141e-121e-101e-081e-060.00010.0111000 5 10 15 20 25t=0 ht=23 ht=46 ht=69 ht=92 hI/εf1e-141e-121e-101e-081e-060.00010.0111000 5 10 15 20 25t=0 ht=23 ht=46 ht=69 ht=92 hfI/ε1E-101E-081E-060.00010.0111000 5 10 15 20 25t = 0 ht = 4,5 ht = 9 ht = 13,5 ht = 18 hI/εf1e-101e-081e-060.00010.0111000 5 10 15 20 25t=0 ht=4,5 ht=9 ht=13,5 ht=18 hfI/εFigure 4: Density distributions at various times without(left) and with (right) diffusion due to long-range beam-beam forces, for the nominal case (top) and for enhancedcore diffusion (bottom).6 CHAOTIC DIFFUSION AT 450 GEVAlso at injection, the long-range collisions induce diffu-sion. Since here the beams are separated further (by 12–14σ) the diffusive aperture is found at a larger amplitude(in units of σ) than at 7 TeV. The nonlinear fields of the su-perconducting magnets introduce additional nonlinearities.We have simulated the diffusion due to the combined effectof field errors and LR-BB using the SixTrack code. For 60error random seeds, we have tracked groups of 60 parti-cles launched at different starting amplitudes, with slightlydifferent initial conditions, and computed the evolution ofaction spread and the maximum amplitude growth in eachgroup. Results for the worst random seed are displayed inFig. 5. We show the rise in amplitude in the left part of thefigure. Besides a very steep amplitude increase at ampli-tudes larger than 10 σ there is also a considerable amplitudeincrease and particle loss at 7 σ. It is interesting to note thatthese losses come from particles in small chaotic nests lo-cated within otherwise stable regions and that this effect ismuch enhanced when the long-range collisions are takeninto account. Correspondingly (right part of the figure), thediffusion rates are large in the regions with amplitude riseand there is no diffusion in the stable regime.y = 0.1804x, Lost after 22kTurn y = 0.0215x, Lost before 1000kTurny = 0.0064x, Stable till 1000kTurn02468101214160 20 40 60 80 100Turn Number [kTurn]Amplitude [ σ]Green: No Amplitude Increase, No Loss1E-111E-101E-091E-081E-071E-061E-051E-041E-031E-021E-011E+001E+010 20 40 60 80 100Turn Number [kTurn]( ∆I)2/N [ εo2 /kTurn]  8.5 σ  8.0 σ  6.5 σ  4.0 σ9.45 σ 7.07 σ7.0 σ7.5 σ9.0 σFigure 5: Simulated amplitude increase and diffusionvs. starting amplitude, at 450 GeV, illustrating the effectof long-range beam-beam forces and field errors.7 TOUSCHEK SCATTERINGTouschek scattering is a longitudinal loss mechanismwhich can create a significant component of uncaptured‘coasting’ beam. The Touschek scattering rate is describedby [5] dNb/dt = −αN2b , giving rise to a coasting beamcomponent of Ncoast = αN0t/(1 + αN0t) N0. For roundbeams, the coefficient α equals [6]αrd =πr20cγ4βxβyσxσyV ηg(δqη)withg() =√∫ ∞e−uu3/2(u− 1− 12lnu)duwhere rp (= 1.5 × 10−18 m), V = 8π3/2σxσyσz , η =(∆E/E)max, δq = γσx/βx.The nominal LHC design [7] foresees two rf systemswith total voltages and harmonic numbers equal to Vˆrf,1(h = 35640) = 750 kV (16000 kV), Vˆrf,2 (h = 17820) =3000 kV (0 kV) where the first numbers refer to injectionand the values in parentheses to top energy.This yields [8] αrd ≈ 5.0× 10−19 s−1 at injection, andαrd ≈ 2.0×10−19 s−1 at 7 TeV. These numbers have beenconfirmed [8] using the more general formula by Piwinski[9], which applies to beams of arbitrary aspect ratio.A coasting beam component is produced at rate per pro-ton of 1.8×10−4 hr−1 (inj.) and 8×10−5 hr−1 (at 7 TeV).Particles outside of the rf bucket lose energy due to syn-chrotron radiation at a rate dδ/dt ≈ −2.8 × 10−9 s−1 atinjection, and dδ/dt ≈ −1.0 × 10−5 s−1 at top energy.Since the energy aperture provided by the collimators is3.9× 10−3 [10], a scattered proton is lost after τloss ≈ 390hrs (injection) or τloss ≈ 6.5 minutes (at 7 TeV), and at7 TeV we expect a steady-state coasting beam fraction ofαrdN0τloss ≈ 10−5.8 CONCLUSIONSAfter storing the beam for 1 hour at injection a fractionof 2 × 10−4 of the beam is outside of the rf bucket dueto Touschek scattering. In collision, the Touschek effectleads to a proton loss rate of 10−4 hr−1, and to a steadycoasting-beam component of about 10−5.Beyond the transverse diffusive aperture of about 6 σ, in-duced by the LR-BB interaction, particles are lost rapidly.A transverse diffusion model predicts the loss rate and thetime evolution of the transverse beam distribution causedby the combined action of IBS, gas scattering and LRbeam-beam. The results of a numerical solution indicatehow the LR collisions affect the shape of the beam halo.Finally, an investigation of the diffusion at injection in-cluded the effects of LR encounters and magnet field er-rors. In this case, there is no clear border between reg-ular and unstable regions of phase space, and particles inchaotic ‘nests’ can be lost rapidly. Modelling this situationwill require an extension of the diffusion model to higherdimensions.9 REFERENCES[1] N. Hilleret, private communication (2002).[2] Y. Papaphilippou et al., PRST-AB 2, 104001 (1999).[3] Y. Papaphilippou, F. Zimmermann, “Estimates of Diffusiondue to Long-Range Beam-Beam Collisions,” in preparation.[4] M.-P. Zorzano and T. Sen, LHC-PROJECT-NOTE-222(2000).[5] C. Bernadini et al., PRL, vol. 10, p. 407 (1963).[6] Y. Miyahara, Jap. J. Appl. Phys., vol. 24, p. L742 (1985).[7] E. Shaposhnikova, private communication (2000).[8] F. Zimmermann, M.-P. Zorzano, LHC Project Note 244(2000).[9] A. Piwinski, DESY 98-179 (1998).[10] D. Kaltchev, private communication (2000).

Equilibrium Beam Distribution and Halo in the LHC

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