Analytical formulae and computer simulation results are presented for the emittance growth caused by small asymmetries of the beam-beam force, caused by small fluctuations of the phase, small offsets between the beams and fluctuations in the size of the opposite beam

Ellison, J A

Sen, T

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Particle Accelerators, 1996, Vol. 55, pp. [293-302] /47-56Reprints available directly from the publisherPhotocopying permitted by license only© 1996 OPA (Overseas Publishers Association)Amsterdam B.V. Published in The Netherlands underlicense by Gordon and Breach Science Publishers SAPrinted in MalaysiaEMITTANCE GROWTH DUE TO TUNEFLUCTUATIONS AND THEBEAM-BEAM INTERACTIONTANAJI SEN! and JAMES A. ELLISON21DESY, 85 Notkestrasse, 22607 Hamburg, Germany2Dept. ofMathematics, University ofNew Mexico, Albuquerque, NM 87131, USA(Received 6 February 1996; infinalform 6 February 1996)Analytical formulae and computer simulation results are presented for the emittance growthcaused by small asymmetries of the beam-beam force, caused by small fluctuations of thephase, small offsets between the beams and fluctuations in the size of the opposite beam.Keywords: Dynamic aperture.1 INTRODUCTIONConsider an ensemble ofparticles such that the phase is uniformly distributedover the whole range 0 ~ 1/1x ~ 2n amongst this ensemble. Then the phaseaverage ofthe beam-beam force (F (x) } 1/1x over this ensemble vanishes. In thispaper we will consider the diffusion and emittance growth that result if wedestroy this symmetry a little. This symmetry of the beam-beam force can bedestroyed by small fluctuations in (i) the phase, (ii) offsets between the beams,and (iii) size of the opposing beam. The main source of tune fluctations arethe quadrupole strength fluctuations, either directly by power supply noise inquadrupoles or indirectly through closed orbit fluctuations through non-linearmagnets and mechanical vibrations of non-linear magnets. There are alsosources of "pure" tune fluctuations which arise when the chromaticity isnon-zero and there is momentum diffusion either due to Intra-beam scattering,RF noise or beam gas scattering. Offset fluctuations between the beams atthe interaction points can occur due to power supply noise either in thedipoles or in misaligned higher order multipole magnets, especially the highfJ quadrupoles in the interaction regions. Beam size fluctuations can occur[293]/47[294]/48 T. SEN and lA. ELLISONfor example due to RF noise and dispersion in the RF cavities. An exampleof this occurred during the November 1993 luminosity run in HERA.1The principal source of random variations is noise in the power supplieswhich can have a fairly complicated spectrum. Analysis shows2 that theoutput power noise density falls as 1//2 over a wide range of frequenciesfor a high quality oscillator which can be modelled as a linear oscillator.At the lowest frequencies the spectrum has a 1//3 dependence due tothe intermodulation of carrier noise and flicker noise while at the highestfrequencies, the output noise has a flat spectrum. For our analysis we willconsider Ornstein-Uhlenbeck noise which has a 1//2 spectrum so as toreasonably model the noise spectrum that particles are subjected to and alsobecause it is the only stationary Gaussian Markov process.3Simulations of the emittance growth with a linear lattice and the beam-"beam force in the strong-weak approximation show a dramatic influence ofnoise, specially near low order resonances. Figure 1 shows the emittancegrowth of 1000 particles over 107 turns with the tune set to VQ = 1/4, bothwithout and with noise. Without noise, there is no evidence of any emittancegrowth. With tune noise of amplitude ~Vr = 10-4 there is a large emittancegrowth. Most of this emittance growth occurs due to a small fraction ofthe total number of particles going to large amplitudes. Notice also the factthat the emittance grows only slowly over the first 106 turns and increasessubstantially after that. In one space dimension, the emittance growth seenwith noise is significant only for low order resonances (less than 1Oth order)while in two space dimensions, the emittance growth is significant even forhigh order resonances such as the 14th.42 TUNE FLUCTUATIONS OFF RESONANCEIn action angle (J, 1/J) variables, the Hamiltonian describing the beam-beaminteraction is(1)where VQ is the nominal tune, U is the beam-beam potential and Dp is theperiodic delta function. The beam-beam potential experienced by a protondue to a Gaussian bunch with Nb particles of charge e isEMITTANCE GROWTH [295]/492e-05-4e-05-6e-05-8e-05 L..-----'-_....L-------L_~ _ _'______'__ ___'__.L..____L____'o 1e+06 2e+06 3e+06 4e+06 5e+06 6e+06 7e+06 8e+06 ge+06 1e+07TurnsO~~_-'-----L_---J.....--....J'--------'--_-'--------'-_-'-----'o 1e+06 2e+06 3e+06 4e+06 5e+06 6e+06 7e+06 8e+06 ge+06 1e+07TurnsFIGURE 1 Relative change in the emittance of 1000 particles due to the one dimensionalbeam-beam interaction at the fourth integer resonance Vo = 0.25. Top: without external noise.Bottom: with tune noise of amplitude Ll Vr = 10-4 .Substituting x = J2f3* J cos l/t where f3* is the beta function at theinteraction point, we obtain the Fourier expansion(3)a = f3* J /(20'2) is a dimensionless measure of the particle amplitude andC = Nbrp/Yp. The Fourier coefficients are Fo(a) = foa[l- e-zIo(z)]/z dz,Fk(a) = (_l)k+l f; e-Z1k(Z)/Z dz. Integrating the equations of motionover a tum, we obtain the one tum beam-beam map, ~ l/t = 21l'vo +au/aJ,~ J = - au/al/t. As mentioned earlier, in the absence of time dependent[296]/50 T. SEN and J.A. ELLISONeffects and far from resonances, the phase averaged change in action dueto the beam-beam force vanishes (~J) = 0, so that J can be considered aconserved quantity in the averaged sense.Now we suppose that the phase advance 1/1 has a fluctuating componentfrom turn to turn. At turn n the phase is 1/I(n) = 2nnv(J) + 1/10 +E7=0 ~1/Ir(i) where v(J) accounts for the action dependence of the tune dueto the beam-beam potential, 1/10 is the initial phase and ~ 1/Ir (i) is the randomcontribution at turn i. The change in action to first order in the random phaseisdD.]r(n) = d1/r D.](n)D.1/rr(n) + O(D.1/r;)a2= - a1/r2 U(1/r(n))D.1/rr(n) + O(D.1/r;) (4)We will assume that (i) ~ J « J and (ii) the noise process is stationary sothat we can define a correlation function K v (n) as(5)where ~vr is the amplitude of the tune noise. The correlation function K vis related to the spectral density S(f) by the cosine transform, Kv(n) =2J~ S(f)cos(2nfn/fo)df, where fa is the revolution frequency. Thediffusion coefficient due to the fluctuating tune is defined as D v (J)limN---.+oo [J(N) - J(0)]2 / N. We obtain00Dv(J) = 128fo(nC~vr)2 Lk4 FlS(2kfov)k=1(6)This result states that tune noise at only the even harmonics of thebetatron frequency cause diffusion in the action when the nominal tune va issufficiently far from resonances.For the Ornstein-Uhlenbeck process the correlation function for randomvariables with unit variance is Kv(n) = e-n/ rc , Tc is the correlation timemeasured in units of the revolution time of the particle around the ring. Thespectral density S(f) r-v 1/(4n2f2 / f5 + l/T;), falling to half its maximumvalue at a frequency fl/2 = fa / (2n Tc ). Substituting in this correlationfunction leads toEMITTANCE GROWTH [297]/51The diffusion coefficient can also be calculated by a simulation. Thesimulation model is a linear lattice with the beam-beam force at a singleinteraction point. 100 amplitudes are chosen from an exponential distributionand at each amplitude 10 particles are chosen with different initial phases.The diffusion coefficient at each amplitude is obtained by averaging theten diffusion coefficients of these particles. Chirikov's method5 is used tocalculate "diffusion coefficients" Dl, D2 over two intervals f1Nl, f1N2 asfollows:where Nlf1Nl = N2f1N2 = N, N being the total number of turns. Jk(m)is the average of J over the mth interval of f1Nk. For motion that is onlyoscillatory and bounded, these coefficients differ by two or three orders ofmagnitude while for true diffusive motion these coefficients are fairly close.With an external noise source present, one expects the motion to bediffusive and indeed the two coefficients Dl, D2 we calculate with f1Nl =105 , f1N2 = 105, N = 107 are within 10% of each other. Figure 2 shows acomparison of D2 with the analytically calculated diffusion coefficient fromEquation (7) at a correlation time of Tc = 1.1 (corresponding to nearly whitenoise). The agreement is very good. The simulations have been repeated forlarger correlation times of 10, 100 and 10000. The agreement between theanalytical and numerical diffusion coefficients is reasonably good in all cases,with the analytical coefficient larger at all amplitudes.3 EMITTANCE GROWTH DUE TO TUNE FLUCTUATIONSSimulations show that with tune noise for example, there is emittance growthof say 1000 particles over a few minutes. The emittance growth of a bunchwith 1010 protons over the storage time ofa day cannot however be estimatedby simulations. If we make certain assumptions about the noise process,1.5 2 2.5Turns[298]/52 T. SEN and lA. ELLISON1e-20 l..---'--'-_---I.-_--'---_-'-_~~_ ____l.__ ____'oFIGURE 2 Comparison of the diffusion coefficient from Equation (7) and the numericallycalculated Chirikov diffusion coefficient due to tune flucutations with amplitude LlV r = 10-4 .Correlation time of the Omstein-Uhlenbeck noise process is!c = 1.1, the tune is Vo = 0.291.then the evolution of the beam density and hence moments of the beam canbe followed by integrating the Fokker-Planck equation. Assuming then thatthe diffusive growth in the action J is Markovian, the evolution of the densityp is given by(9)where for D(J) the diffusion coefficient we use the analytic expression givenby Equation (7). We will use the method of lines to numerically integratethis ID Fokker-Planck equation.6 The boundary condition is that particlesare lost at an action Jb corresponding to the position of the beam pipe,i.e. p (Jb, t) = 0 at all times. The diffusion coefficient Dv (J) and its firstderivative (from Equation (7)) both vanish at the origin. From these propertiesit follows that the density at the origin obeysp(J = 0, t) = Po(J = 0), !-p(] = 0 t) = eD t/ tI2 !-po(] = 0)aJ ' aJ (10)where Po is the initial density. We introduce dimensionless time, actionand density variables (r, X, U) respectively defined as r = t / to, X =J / ]b, U = JbP. The parameter to is interpreted as the number of revolutionsaround the ring per time step ofthe numerical integration. The scaled diffusionEMITTANCE GROWTH15Time (Hrs)Te = 1.1Te = 10Te = 100[299]/53FIGURE 3 Change in the relative action (J) due to tune fluctuations with nominal tuneVo ==== 0.291 over a time of 30 hours obtained by integrating the Fokker-Planck equation.Shown are the changes due to noise at three correlation times Tc == 1.1 (nearly white noise), 10(/1/2=750 Hz) and 100 (/1/2=75 Hz). Tune noise amplitude is /:;,. V r == 10-4 .coefficient is VeX) = toD(J)/J;. The Fokker-Planck equation in thedimensionless variables is numerically integrated for the required numberof turns by appropriately choosing to.Once the density p(J, t) is known, the average action of the distributionat any time is found fromJb(J) = 1 / Jp(J, t)dJ.foJb P(Jb, t)dJo(11)Figure 3 shows the relative change in the average action (J) due to tunenoise at three different correlation times with the higher correlation timescorresponding to noise with less high frequency content. With ~Vr = 10-4and nearly white noise (Te = 1.1) acting on the beam we see that the emittancedoubles over a period of 10 hours before decreasing due to particles beinglost at the beam-pipe. Diffusion is slower for the other two noise realizationsso particles do not reach the beam-pipe in 30 hours and the emittance growsnearly linearly to a maximum ofabout 70% forTe = 10 and 10% forTe = 100respectively. Due to both filtering of power supplies and attenuation in themetallic lining of beam pipes, a correlation time of Te = 100 is likely to bethe most realistic model of the noise spectrum experienced by the beam. AtTe = 100, the spectral density at twice the betatron frequency (the part of thenoise spectrum responsible for the diffusion) is smaller by a factor of7 x 10-6[300]/54 T. SEN and J.A. ELLISONthan the spectral density near zero frequency. The emittance growth increasesnearly linearly with the tune noise amplitude 1::1 Vr since Dv (J) <X (1::1 Vr )2. AtTc = 100 and 1::1 Vr = 10-3, the emittance increases initially by 90% in about7 hours before decreasing and reaching a value about 70% greater than theinitial value at the end of 30 hours.4 TUNE FLUCTUATIONS NEAR RESONANCEThe dynamics with noise near low order resonances are significantly differentthan that far away from such resonances. A detailed analytical treatment ofthe effects of noise near resonances will appear elsewhere.7 The phase spacestructure is specially important when the noise is sufficiently weak that thedetails of the resonance islands have not been completely smeared out. Fornoise strength below a certain value, particles can be trapped within theseislands and be thereby transported in phase space as the islands move, leadingto emittance growth. Trapping occurs when the time for a resonance to diffusea distance equal to its width is greater than the time period of motion aroundthe resonance island. Let 1::1 Vr be the rIDS amplitude of tune fluctuations, thentrapping occurs for 1::1Vr < 1::1v;,where the trapping limit amplitude is givenbyT 2 1 2(1::1 Vr ) = -VisZ(l::1vW) .2 (12)VisZ is the island tune and 1::1Vw is the island width in tune space. For thebeam-beam interaction, these parameters for a resonance island of order 2kare given byVisZ = k~vw (13)where Is is the action at the stable fixed point. Putting these back we obtainthe trapping conditionFor typical HERA parameters, ~v; rv 2.5 x 10-4 for the 4th integerresonance (k = 2).EMITTANCE GROWTH [301]/55In the other extreme limit, the tune fluctuations may be strong enough towash out the resonance structure. When the phase is completely random fromtum to tum, the random contribution is of the same magnitude or more thanthe nominal tune, ~vfR > Ij(2k). In this situation the random tune can movea particle from one resonance island to the neighbouring .resonance islandand beyond in one tum. In the not so extreme limit but still fast diffusion is thecase when the random tune causes the particle to diffuse from one resonanceisland to the next in the time it takes for a particle to go around the first island.This occurs for ~Vr > ~vI where(15)For typical HERA parameters, the maximum value of the island tune is 0.008at the 4th integer resonance. The corresponding lower limit for fast tunediffusion is ~vl rv 0.016.The frequencies of the tune fluctuation are also important in determiningthe dynamical response. When resonance islands are present, motion in thevicinity of these islands slows down and the island tunes are much smallerthan the nominal tune. Exactly at the resonance tune the islands disappear(because the fixed points are at infinity) but the periods keep increasing withamplitude. Figure 4 shows the phase portrait of two particles at the 4th integerresonance. The emittance growth seen in Figure 1 is due to particles diffusingfrom the inner regions onto the curves with long extensions along the axes inFigure 4. Since the motion on these latter curves has a long period, it can takea long time before the emittance growth is observed, as is the case in Figure 1.A detailed analysis7 shows that near resonances, low frequency noise whichis resonant with the slow motion of the linear invariant is responsible fordiffusion. This is in complete contrast to the off-resonance case. A similartopological reason should also explain the emittance growth in the two degreeof freedom case.5 SUMMARYWe have analysed the effects of tune fluctuations on the emittance growthof a hadron beam colliding with another beam. Off low order resonances,tune noise at even harmonics of the betatron frequency is responsible for[302]156 T. SEN and lA. ELLISON0.0015 ,...------,------,------,-----r-----,------,-----,------,0.001-0.0015 '--------'-~----'-------'--------'------'-----'-----'-8FIGURE 4 Phase portrait of two particles at the tune Vo = 0.25 without external fluctuations.particle diffusion and emittance growth. Near low order resonances, lowfrequency noise causes diffusion. Other fluctuations which lead to emittancegrowth include beam offset fluctuations and fluctuations in the size of abeam. A detailed analysis can be found in a forthcoming publication.7 Brieflystated, we find that off resonances, offset fluctuations at odd harmonics of thebetatron tune and beam size fluctuations at even harmonics of the betatrontune lead to diffusion. Both of these fluctuations lead to greater emittancegrowth than tune fluctuations in the off-resonance case for reasonablevalues of the fluctuating amplitudes. These three fluctuations which arealmost always present might explain a large part of the observed emittancegrowth. They also underline the importance ofchoosing high precision powersupplies, especially for quadrupoles and dipoles in the interaction regions.Future work will focus on understanding the effects of these fluctuations inthe two degree of freedom case.References[1] A. Pauluhn, DESY HERA Technical Note 94-02 (1994).[2] W.P. Robins, Phase Noise in Signal Sources, Peter Peregrinus Ltd., London, U.K. (1982).[3] J.L. Doob, Ann. Am. Stat., 15,229 (1944).[4] R. Brinkmann, DESY-HERA report 89-24 (1989).[5] B.V. Chirikov, Phys. Rep., 52, pg 265 (1979).[6] H.-J. Shih, J.A. Ellison, B.S. Newberger, R. Cogburn, Part. Ace. 43, 159 (1994).[7] T. Sen and J. Ellison, manuscript in preparation.

Emittance Growth Due To Tune Fluctuations and the Beam-beam interaction

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