Over the two-decade lifetime of the LEP project, techniques for evaluating the quality of optical configurations have evolved considerably to exploit the growth in computer power and improved modelling of single-particle dynamics. These developments have culminated in a highly automated Monte-Carlo evaluation process whose stages include the generation of an ensemble of imperfect machines, simulation of the operational correction procedures, correlation studies of the optical functions and parameters of (both) beams, 4-dimensional dynamic aperture scans and tracking with quantum fluctuations to determine the beam core distribution. We outline the process, with examples, and explain why each step is necessary to realistically capture essential physics affecting performance. The mechanisms determining the vertical emittance, radial beam distribution and dynamic aperture are especially important. As a storage ring in which an unusual variety of optics have been tested, LEP provides a valuable test case for the predictive power of the methodology

Jowett, John M

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REALISTIC PREDICTION OF DYNAMIC APERTUREAND OPTICS PERFORMANCE FOR LEPJohn M. Jowett∗, CERN, Geneva, SwitzerlandAbstractOver the two-decade lifetime of the LEP project, tech-niques for evaluating the quality of optical configurationshave evolved considerably to exploit the growth in com-puter power and improved modelling of single-particle dy-namics. These developments have culminated in a highlyautomated Monte-Carlo evaluation process whose stagesinclude the generation of an ensemble of imperfect ma-chines, simulation of the operational correction procedures,correlation studies of the optical functions and parametersof (both) beams, 4-dimensional dynamic aperture scansand tracking with quantum fluctuations to determine thebeam core distribution. We outline the process, with ex-amples, and explain why each step is necessary to real-istically capture essential physics affecting performance.The mechanisms determining the vertical emittance, radialbeam distribution and dynamic aperture are especially im-portant. As a storage ring in which an unusual variety ofoptics have been tested, LEP provides a valuable test casefor the predictive power of the methodology.1 INTRODUCTIONPersistent disagreement of, say, a factor of two betweencomputed and measured dynamic apertures would invali-date computation as a rational means of assessing the qual-ity of a storage ring optics.However, the potential of an optical configuration onlybecomes clear after some weeks of operational perfor-mance maximisation (and may also require hardwarechanges). Such tests are expensive in large machineslike LEP. Accurate computational appraisal is therefore vi-tal to estimate not only dynamic aperture but also—andconsistently—the gamut of optical quantities culminatingin the beam sizes at the collision points.The conceptual framework and tools brought to bearon the problem of performance prediction for LEP haveevolved considerably over some 20 years. Rather than re-view the history (traceable through the citations), this paperwill describe current best efforts to model the various LEPoptics, emphasising recent high energy operation (LEP2)where the synchrotron radiation effects are strong.2 CONCEPTS AND METHODOLOGYIn this paper, computing the linear machine is shorthandfor the calculation of the 6D closed orbit including radia-tion, the eigenvectors of linear oscillations around it (henceall optical functions and the canonical transformations to John.Jowett@cern.ch, http://wwwslap.cern.ch/ j˜owett/the eigenmodes), the 3 tunes, energy loss, damping rates,emittances, etc. Usually, the eigenmodes correspond ap-proximately to (1,2) horizontal and vertical “betatron” os-cillations and (3) “synchrotron” oscillations. These calcu-lations are implemented in the program MAD [1].In MAD’s tracking, radiation damping and quantum ex-citation arise naturally [2] because all particle momentumchanges are computed locally, according to the distributionof >∼ 300 RF cavities and the canonical co-ordinates of theparticle in each multipole magnet.The dynamic aperture is the basin of attraction of theclosed orbit in deterministic tracking with radiation damp-ing [2] (without quantum fluctuations). This definitionis computationally unambiguous and independent of thenumber of turns tracked beyond a certain minimum (forLEP2, 100 turns are ample).The dynamic aperture of LEP is essentially determinedby three classes of intentional non-linear elements: chro-maticity sextupoles, RF cavities and focusing quadrupoles.Although quadrupoles provide linear focusing, their non-linear radiation terms are responsible for the ultimate dy-namic aperture limit. (Higher order multipole errors areless important.)A dynamic aperture scan is carried out by varying theaction variables (I1, I2, I3) ∈ (R+)3 over a spherical polargrid R+ × [0, pi/2]2. At each point, the phase φ3 ∈ [0, 2pi)of the third mode is also scanned. A dynamic aperture istypically the result of tracking 1000–2000 particles and isreturned as a surface in the 3D action space [3].The correction procedures applied are designed to emu-late the real operational procedures, as responses to mea-sured quantities without knowledge of the imperfections,rather than the best that could be done computationally.In outline, the optics evaluation procedure is:1. Generate ideal machine with detector solenoids, RF,and radiation switched off.2. Install vacuum chamber elements in all quadrupolesto provide aperture limitations in tracking.3. Compute solenoid compensation with tiltedquadrupoles, to be applied in calculations withthe solenoids on.4. Compute linear machine and dynamic aperture scan ofideal machine (with solenoids, RF and radiation on) asa reference.5. Generate an ensemble (typically 30) of machines withtypical random misalignments, tilts and field errors inall dipoles, quadrupoles and sextupoles.For each machine in the ensemble,(a) Correct the non-radiating closed orbit indepen-dently in the two planes (to√〈x2〉 = 0.6mm,0-7803-5573-3/99/$10.00@1999 IEEE. 1680Proceedings of the 1999 Particle Accelerator Conference, New York, 1999-0.004 -0.002 0 0.002 0.004-0.004-0.00200.0020.004-0.002 -0.001 0 0.001 0.002-0.002-0.00100.0010.002-0.05 -0.025 0 0.025 0.05 0.075 0.1-0.04-0.0200.020.04-0.004 -0.002 0 0.002 0.004-0.004-0.00200.0020.004-0.002 -0.001 0 0.001 0.002-0.002-0.00100.0010.002-0.05 -0.025 0 0.025 0.05 0.075 0.1-0.04-0.0200.020.04Figure 1: Shrinking of the dynamic aperture at the ultimate energy: the upper row shows survival plots in the normalisedphase planes (all co-ordinates in √m) of the 3 normal modes at 94 GeV with Jx = 1 and a comfortable VRF = 2961MV.The lower row shows the same imperfect machine at 100 GeV with Jx = 1.5 and VRF = 3265MV just sufficient.Contours of 1 and 5.5σ of the notional gaussian distributions are shown to give the scale. Most of these 6400 particlesrotate anti-clockwise from the initial conditions shown. Initial conditions are coloured with a hue indicating survival time,ranging from violet ( dark central stable zone) to red (loss in one turn).√〈y2〉 = 0.4mm).(b) Switch on solenoids, RF and radiation. For thee+ beam, correct the IP optics (βy ) and tunesusing “knobs” as in operation.(c) Save all the imperfections and their corrections,reverse the ideal machine structure and rebuildthe physically equivalent machine for the e−beam. Thus, some corrections appropriate forthe positrons are applied to the electron beamand may enhance the differences between thebeams. Tune splits between beams can be >∼0.01, depending on the distribution of VRF.(d) Compute the linear machine and carry out a dy-namic aperture scan for each beam.(e) Track with quantum fluctuations [2] (for >∼ 200)damping times) to estimate non-linear effects onthe emittances and beam sizes.This process involves >∼ 1000 multifarious runs ofMAD, all managed by a Mathematica [4] notebook inter-face that prepares the input and absorbs the results into astructured database of function definitions.Other notebooks load the database to display and cor-relate machine parameters or generate comprehensive re-ports [6] on an optics. These include orbit and optics atkey elements, emittances, tunes, survival data and dynamicapertures and differences between beams. The environmentis open-ended and congenial: it is easy to compute derivedquantities such as beam-overlaps or centre-of-mass ener-gies at the experiments; several databases can be loaded tomake comparisons across optics, etc.3 SAMPLE RESULTS3.1 Dynamic apertureLEP has operated with several optical configurations [3, 7]with varying arc cell phase advances (µx, µy). Dynamicaperture measurements have been made, where possible,with two or three different methods.For certain optics, results of tracking with radiation arequantitatively in agreement with earlier calculations with-out radiation. This is the hallmark of certain physicalmechanisms, e.g., detuning with amplitude onto an inte-ger resonance. Including radiation, or even imperfections,makes little quantitative difference although the radiativebeta-synchrotron coupling instability (RBSC) [2, 3] effectalways steps in to accelerate initial amplitude growth. In1681Proceedings of the 1999 Particle Accelerator Conference, New York, 1999ε+2 /nmε−2 /nmJ+2J−2DRMSy /mε−2 /nmFit:ε2nm = 826(√〈D2y〉m)2.3+ 0.11 2 3 4 5 612345670.8 0.9 1.10.80.91.11.20.02 0.04 0.06 0.08 0.10.511.522.53Figure 2: Correlation of vertical emittance and damping partition numbers between e+e−; correlation of vertical emittance(both beams) and RMS vertical dispersion, DRMSy . The fit tests the hypothesis ε2 ' a(DRMSy )2/J2+εc = a(DRMSy )p+εc.other cases, e.g., higher order resonances or pure RBSC, ra-diation and imperfections are important, the dynamic aper-ture has a spread and is less than for an ideal machine. Ra-diation effects are essential in all cases to determine thestability limit of mode 3 (“momentum acceptance”).There is no space to discuss measurements here. How-ever a recent survey [8, 5] showed that calculation and mea-surement agree within about 10 % in all cases where satis-factory measurements have been made.Figure 1 has been computed for the present optics with(µx, µy) = (102, 90) to show the effect of moving froma situation with a reserve of RF voltage to “ultimate” LEPparameters at 100 GeV where VRF is just sufficient ac-cording to the conventional quantum lifetime calculation.All details of this figure can be understood by analysingthe orbits of individual particles. At 94 GeV, the stabilitylimit of mode 1 is determined by the cross-detuning effect:Q2 moves down to the integer resonance with increasingI1. The boundary between stability and loss in one turn issharp. At 100 GeV, the transverse dynamic apertures aresharply reduced because the limit is now determined by theRBSC instability. Since the growth time of this instabil-ity is ' 2/Q3 ' 20 turns, a “ghost” of the former dynamicaperture remains visible. Similar effects are visible in mode2 which was already limited by RBSC.In mode 3, the dynamic aperture becomes simpler inshape although this example shows why a scan of φ3 isessential. It is clear that the approximations underlyingexisting analytic approaches to the calculation of quantumlifetime are inadequate in this regime.3.2 Vertical emittanceThe emittance of mode 2, ε2, is a crucial parameter de-termining luminosity. It is determined almost entirely bymachine imperfections and so can only be estimated statis-tically. Figure 2 shows predictions for the same optics at94 GeV. A mismatch of ε2 between beams can arise, partlybecause the damping partition numbers J2 can be different.Fitting shows that ε2 is determined mainly by the RMS ver-tical dispersion. The dependence is stronger than quadraticbecause J2 tends to decrease with the dispersion (becauseof combined dispersion and closed orbit in quadrupoles).The statistical distribution of vertical emittance in the en-semble is typical of measurements during the life-cycle ofan optics. Operational procedures seem to select correc-tions that reduce ε2 to the smallest values.Synchro-betatron resonances can increase ε2 further.Such effects can be estimated by tracking particles withquantum fluctuations [5].4 CONCLUSIONSBy carefully constructing ensembles of model machines,simulating the operational correction procedures and usinga physically faithful model of single particle dynamics, itis possible to predict the distribution of beam parametersand their differences between the beams in LEP. Dynamicaperture can be predicted to within about 10 % in a varietyof optical configurations.At the highest accessible energies, the dynamic apertureof LEP will be sharply reduced by the RBSC instability.Acknowledgements: I thank A. Verdier for valuable col-laboration, K. Goral for help with software development.5 REFERENCES[1] F.C. Iselin, http://wwwslap.cern.ch/ f˜ci/mad/mad home.html[2] F. Barbarin, et al, Proc. 4th European Particle AcceleratorConference (EPAC’94), World Scientific, p. 193.[3] Y. Alexahin et al, EPAC’96, Sitges, IOP Publishing, 1996.[4] S. Wolfram, Mathematica, Cambridge Univ. Press, 1996.[5] J.M. Jowett, Proc. 8th LEP Performance Workshop, CERNSL/98-006 (1998).[6] J.M. Jowett, CERN-SL Notes 97-84, 98-020, 98-072 (1997–98). See also [5].[7] D. Brandt et al EPAC’98, Stockholm, IOP Publishing, 1998.[8] H. Burkhardt, Proc. 8th Workshop on LEP Performance,CERN-SL-98-006 (1998).1682Proceedings of the 1999 Particle Accelerator Conference, New York, 1999

Realistic prediction of dynamic aperture and optics performance for LEP

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