The problem of long-term losses in hadron colliders such as the Large Hadron Collider (LHC) is considered. A previous formula that provides the reduction of dynamic aperture with the number of turns is generalized to include also the relevant cases of off-momentum and tune ripple. The dynamic aperture turns out to shrink with a power of the inverse logarithm of the number of turns. Long-term tracking data of the LHC are analysed in this framework. The formula proves to hold in all cases, and the possibility of using its extrapolation to predict long-term losses are explored

Scandale, Walter

Todesco, Ezio

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LONG-TERM STABILITY IN HADRON COLLIDERS IN PRESENCE OFSYNCHROTRON OSCILLATIONS AND TUNE RIPPLEW. Scandale, E. TodescoCERN, LHC Division, Geneva, SwitzerlandAbstractThe problem of long-term losses in hadron colliders suchas the Large Hadron Collider (LHC) is considered. A pre-vious formula that provides the reduction of dynamic aper-ture with the number of turns is generalized to include alsothe relevant cases of off-momentum and tune ripple. Thedynamic aperture turns out to shrink with a power of the in-verse logarithm of the number of turns. Long-term trackingdata of the LHC are analysed in this framework. The for-mula proves to hold in all cases, and the possibility of usingits extrapolation to predict long-term losses are explored.1 INTRODUCTIONIn this paper we analyse the long-term stability in largehadron accelerators. In the LHC [1], one has to estimatethe stability for at least 107 turns, corresponding to the in-jection plateau before energy ramping. Long-term stabilityis determined by intricate relations between nonlinearities,resonances, and tune dependence on amplitude and mo-mentum; its evaluation cannot be worked out analytically,and is based on tracking codes. The resulting particle lossesmay also occur after many millions of turns [2, 3, 4, 5, 6, 7].Long-term losses are drastically enhanced if the betatrontune is modulated by some external causes, such as thepower supply ripple, or by synchro-betatron coupling, viathe residual uncompensated chromaticity [6, 7, 8, 9]. Fora realistic model of the LHC lattice, simulations up to 106turns are very onerous.A pragmatic approach is based on plotting survival timesprovided by tracking at 105   106 turns versus the ini-tial amplitude (survival plots [2, 3, 5, 7]). In a previ-ous paper [5] we shown for the purely four-dimensionalcase that if the initial amplitude is averaged over the phasespace [10], survival plots can be interpolated by a two-parameter formula. Here we review the results of analysingrealistic models of the LHC also in presence of tune mod-ulation. It turns out that the dynamic aperture is well inter-polated by the three parameters formula:D(N) = A+BlogN: (1)In the followings, we discuss the validity and the dynamicalmodel underlying the above formula, and we extrapolate itto predict long-term stability.2 LHC MODELThe lattice of the LHC is described in Ref. [11]. The field-shape errors are described by thin-lens multipoles up toorder eleven. For every magnet, each multipolar compo-nent is determined using a random number generator withgaussian distribution, truncated at 3 r.m.s. deviations. Theselected realisation of the random imperfections has a dy-namic aperture at 105 turns close to the average value ina set of 64 random realisations. A set of sextupoles anddecapoles is used to correct the non-linear imperfections.Two additional families of sextupoles are used to correctthe chromaticities. To partially take into account the op-erational difficulty of this correction in a real machine, wedecided, somehow arbitrarily, to set Q0 = 2. We disregardlinear imperfections.The tune modulation is obtained by summing up sevensine-waves, with the same frequencies kand amplitudeskobserved in the SPS spectrum (see Table 1). The am-plitude 1of the main frequency is set to 10 4, and all theamplitudes kare varied by a multiplicative factor  thatranges from 1 to 8. The horizontal and the vertical tunesare affected by a synchrotron modulation of the same orderof magnitude (i.e., p=p = 10 4).k kk1 2=868:12 1:000  10 42 210:218  10 43 310:708  10 44 610:254  10 45 710:100  10 46 1010:078  10 47 1210:218  10 4Table 1. Parameters of the tune modulation frequencies3 DYNAMIC APERTURE EVALUATIONIn a previous work [10] we have proposed a definition ofdynamic aperture D(N) as a function of the number ofturnsN as the first amplitude where particle loss occurs be-fore N turns, averaged over the phase space. Particles arestarted along a 2D polar grid (; ) in the coordinate spacex =  cos , y =  sin : with respect to the approach usedin several long-term simulations (see for instance [2, 4]),where a fixed value of  is considered in order to speedup simulations, this definition provides a smoother depen-dence of D on N , thus allowing to derive interpolating for-mulae and to extrapolate them to predict long-term particleloss. An error estimate due to the finite-size grid used tostart initial conditions can be worked out (see Ref. [7] formore details); this estimate is crucial to determine the reli-ability of the fit.4 DYNAMIC APERTURE PREDICTIONWe propose to interpolate the long-term dynamic apertureD(N) versus the number of turns N according to the fol-lowing empirical formula:D(N) = A+BlogN: (2)This equation can be justified [5] for a four dimensionalmapping in terms of the KAM (Kolmogorov-Arnol’d-Moser) and of the Nekhoroshev theorems, using a phasespace model that features two regions: an inner KAM do-main where almost all the initial conditions give rise to reg-ular orbits, stable for infinite times, and an outer chaoticregion where particles diffuse according to the Nekhoro-shev exponential estimate. Numerical simulations basedon long-term tracking and frequency analysis have con-firmed this scenario for 4D mappings [5]. In the case oftune modulation, this theoretical justification does not holdany more. Nevertheless, we used the above formula to in-terpolate tracking data, with good results.The fitting procedure has been carried out using the stan-dard approach based on least-squares minimization. Somecare has to be taken since the fit is nonlinear. The proce-dures used to work out the confidence limits for the threeparameters A;B and  are described in [7]. The extrapola-tion is carried out by using the best parameters; the error isobtained by extrapolating through all the parameters insidethe confidence limits, and selecting the maximum and theminimum extrapolation value.5 NUMERICAL RESULTSThe dynamic aperture is given in mm normalized atmax= 182m. Onerous simulations were carried out upto 106 turns, with a scan over 17 angles and 100 steps inthe radius. The relative error in the dynamic aperture is ofthe order of 2%.We interpolated the dynamic aperture versus the num-ber of turns according to Eq. (2). The value of 2, andof the parameters  and A, with the error estimated witha confidence level of 90%, are given in Table 2. The dy-namic aperture estimate through tracking with the associ-ated errors (bars), the best fit through Eq. (2) (solid line),and the extrapolation to infinity (dotted line) are shown inFigs. 1-3 for three different cases. The off momentum isp=p = 10 4.The interpolation is very good: the best fit has a 2 thatranges from 0:4 to 2:0. Both  and A decrease as the mod-ulational amplitude  gets larger: the effect of the modula-tion is to shrink the stable core and to slow down the escaperate of the initial conditions in the outer region. For  = 1,A becomes negative and therefore according to the extrap-olation all initial conditions will be lost sooner or later. Theexponent  is determined within 0:5   1:0. The errors onA and B become larger when the modulation is increased:A, that denotes the extrapolation of the dynamic aperturefor infinite number of turns when  is positive, is rathersharply defined for  = 0, but becomes rather loose when is increased. When  change sign in the interval of 90%confidence level (i.e.,  = 1), it becomes impossible to as-sociate an error to A and B since our formula contains asingularity for  = 0.  2 A0 0 0.4 1:9+1:1 1:212:0+0:3 1:70 1 1.0 0:8+1:0 1:19:61 1 1.4 0:3+0:9 1:03:42 1 2.0  0:1+0:9 0:8424 1 1.0  0:1+0:8 0:7478 1 1.3  0:2+0:5 0:533Table 2. Main fitting parameters of Eq. (2) for the LHCFigure 1: Dynamic aperture D versus number of turnsN for the LHC on momentum, without modulation ( =0). Tracking data (error bars), interpolation accordingto Eq. (2) (solid line) and extrapolation at infinity (ver-tical dotted line), prediction through Lyapunov exponent(squares).We use tracking data from 102 up to 105, to evaluate thethree parameters of Eq. (2), and then we extrapolate at 106.The results (see Table 3) are good: all the extrapolationsare in agreement with direct tracking, and rather precise(within 5%); there are some indirect indications that dataup to 105 can be safely extrapolated at 107, even thoughFigure 2: Dynamic aperture D versus number of turns Nfor the LHC off momentum ( = 1), without modulation( = 0). Tracking data (error bars), interpolation accord-ing to Eq. (2) (solid line) and extrapolation at infinity (ver-tical dotted line), prediction through Lyapunov exponent(squares).Figure 3: Dynamic aperture D versus number of turns Nfor the LHC off momentum ( = 1), with modulation( = 2). Tracking data (error bars), interpolation accord-ing to Eq. (2) (solid line) and prediction through Lyapunovexponent (squares).no direct check has been carried out for lack of computingpower (see Ref. [7] for more details).In comparison with the results of Ref. [5], where the casewithout modulation was analysed, we note that a largernumber of turns is required to obtain a reliable prediction.This is due to the fact that the effect of the modulation onthe beam stability requires a longer time to become evident.We also give the comparison with the long-term estimateprovided by a method based on the Lyapunov exponents(see Refs. [7, 5]). The prediction of the Lyapunov exponentare marked as stars in Figs. 1-3; one can observe that with-out modulation the extrapolation of the dynamic apertureat infinity A agrees with the Lyapunov exponent prediction(see Fig. 1), thus supporting previous results on simplifiedmodels [7]. When a tune modulation is considered (seeFigs. 2 and 3), it is hard to say whether the Lyapunov ex-ponent predicts a finite stability domain or not and it seemsvery hard to extract quantitative information on the long-term stability.  Extrap. Track.0 0 12:3+0:1 0:412:3+0:2 0:20 1 11:7+0:2 0:411:5+0:2 0:21 1 11:4+0:4 0:411:1+0:2 0:22 1 11:1+0:5 0:510:7+0:2 0:24 1 10:6+0:5 0:510:4+0:2 0:28 1 10:0+0:5 0:710:1+0:2 0:2Table 3. Comparison between extrapolation of dynamicaperture at 106 and tracking for the LHC6 CONCLUSIONSWe have proposed an empirical formula to analyse survivalplots. Using a definition of dynamic aperture that involvesaverages in phase space, the dynamic aperture turns out toshrink with an inverse power of the logarithm of the num-ber of turns. This numerical evidence confirms a scenariothat features a hard core of phase space stable for infinitetimes and an outer chaotic region where the escape rate canbe evaluated. When modulation is added, one reaches alimit where all the phase space becomes unstable. Besidesgiving a phenomenological framework to interpret trackingdata, this formula allows one to extrapolate the dynamicaperture of at least one order of magnitude in the numberof turns to predict long-term stability.We wish to thank M. Giovannozzi for very relevant con-tributions to this work.7 REFERENCES[1] The LHC Study Group, CERN 95-05 (1995).[2] Y. Yan, SSC 500 (1991).[3] A. Chao, AIP Conf. Proc. 230, (1990) 203.[4] F. Schmidt et al., Part. Accel. 35, (1991) 249–256.[5] M. Giovannozzi et al., Part. Accel. 56, (1997) 195–225.[6] W. Fischer et al., Phys. Rev. E 55, (1997) 3507–20.[7] M. Giovannozzi et al., Phys. Rev. E 57, (1998) 3432-41.[8] D. Brandt et al., in EPAC 90, edited by P. Marin and P. Man-drillon (Edition Frontie`res, Gif sur Yvette, 1991) pp. 1438-40.[9] O. S. Bru¨ning, Part. Accel. 41, (1993) 133–51.[10] E. Todesco, M. Giovannozzi, Phys. Rev. E 53, (1996) 4067.[11] W. Scandale et al., in PAC 95, (IEEE, Piscataway, 1995) pp.2844–46.

Long term stability in hadron colliders in presence of synchrotron oscillations and tune ripple

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