Kirsten, I

Manglunki, Django

Martini, M

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TRAJECTORY AND OPTICAL PARAMETERSIN A NON-LINEAR STRAY FIELDD. Manglunki, M. Martini, CERN, Geneva, SwitzerlandI. Kirsten, Heidelberg University, Heidelberg, GermanyABSTRACTA new optics for the main CERN Proton Synchrotron(CPS) magnet is modeled to allow a precise descriptionof the ejected beams. For that purpose, field maps of themagnet have been measured for the various operationalcurrent settings. They include the central field, the endstray field and the lateral stray field. In order to get afunctional form which can be inserted in the equations ofmotion for a charged particle in a magnetic field, thediscrete field maps are converted into bi-dimensionalpolynomials of degree up to fourteen. These equationscan be written as a set of four first order differentialequations which are solved simultaneously. Two of themare non-linear and describe the centroid motion, theother two are linear and apply to the betatron motion.The method has been validated by producing extractionconditions which have been verified experimentally withthe 26 GeV/c beam for the future LHC.1. INTRODUCTIONThe proton beams extracted from the CPS experiencethe strongly non-linear stray field from the magnet unitdownstream of the ejection septum. Previous calculationsmodelled each half-magnet multipole component beyondthe sextupole order as a thin element. Multipole coeffi-cients were derived from field measurements at a refer-ence azimuthal magnet location using one-dimensionaltransverse fitting. The approach considered in this paperconsists integrating the equations of motion for a singleproton travelling through a measured discrete field mapconverted into bi-dimensional polynomials. The resultsyield the ejection trajectory and the transfer matricesthrough the field map.This study considers the ejection of a 26 GeV/c protonbeam which will be used to fill the LHC [1]. The sameapproach could similarly be used to handle the otherejection settings (e.g. 24 GeV/c proton slow extraction)and the 1 GeV proton injection in the CPS.2. FITTING OF FIELD MEASUREMENTS2.1 Measured field map on a test magnet unitThe CPS lattice consists of ten super-periods made often combined function magnets, eight 1.0 m and two 2.4m straight sections. Each magnet is composed of twohalf-units with gradients of opposite sign, separated by acentral junction. The half-units are made of five blockswith small gaps in between lined up on the central orbit.Corrections have been installed on the pole profile tocreate independent multipole fields: (i) pole-face wind-ings yielding quadrupolar and sextupolar components toadjust the tunes and chromaticities modified by satura-tion; (ii) a figure-of-eight loop to modify the field distri-bution between the two half-units, allowing additionaltune changes without affecting chromaticities. The vari-ous magnet settings are characterized by their dipolefield level specified by the main coil current and theirpole-face winding and figure-of-eight loop currents. Thelayout of the CPS magnet number 16 is shown in Fig. 1.Fig. 1: CPS magnet unit number 16 including the 26 GeV/cextraction pipe.New magnetic measurements on operational CPSmagnet working points were performed in 1992, includ-ing measurements of the central field, the end and lateralstray fields, and the field in the junction between the twohalf-units [2]. The measured magnet unit was a radiallydefocusing open half-unit (D half-unit) followed by aradially focusing closed half-unit (F half-unit), with theyoke oriented towards the centre of the ring.Measurements were carried out in a Cartesian framewith steps of 20 mm along the z-axis (aligned along thetwo magnet targets and counted positively in the protondirection) and steps of 10 mm along the x-axis (put inthe middle of the central magnet junction and countedpositively towards the outside ring). The field map rangeis: -70 mm ≤ x ≤ 310 mm and -2.55 m ≤ z ≤ 2.73 m (themagnet length is 4.26 m).2.2 Functional form of the discrete field mapThe necessity to extract a 26 GeV/c proton beam in a2.4 m straight section with little angle deflection (≅29mrad) imposes that the downstream half-unit adjacent tothe ejection septum to be open to allow the fitting of theextraction pipe across the magnet aperture. The ejectiontrajectory in this region remains close to the central orbitand thus the aberrations in the magnetic fields are keptat a reasonable value. When traversing the subsequent Fhalf-unit the ejection trajectory moves away from thecentral orbit and field aberrations become strongly non-linear: the beam experiences a field gradient with a re-verse sign, yielding large horizontal betatron functionvalues at the magnet end. Reduction of the non-linearaberrations was done by shimming the F half-unit.Straight parallel shims have been mounted at differentradial positions on the five blocks to shape a constantmagnetic field over the ejected beam width [3].Magnetic measurements have been done on a labora-tory magnet unit in the absence of shims, thus the meas-ured field map has to be corrected to consider the shim-ming effect. Field calculations have been carried out onthe five blocks equipped with shims using the two-dimensional Poisson program [4] with appropriatemeshing of the field region [5]. Polynomial fittings up todegree twenty-five in x (in the range -0.1 m ≤ x ≤ 0.5 m)of Poisson output have been carried out to get a func-tional form of computed field (see Fig. 2). Hence, a cor-recting field map function w(x,z) may be defined asw x zf x z ii( , ) ( )= if  is in block  of the F half-unit otherwise                                    1where fi(x) is the ratio of the computed field function onshimmed block i over the corresponding function derivedwithout shim insertion.Multiplying the measured field map values by w(x,z)yields the field map relevant for the stray field in thepresence of shims, which is of importance for good ex-traction modeling.-0.1 0 0.1 0.2 0.3 0.4x [m]0.250.500.751.001.251.501.752.00B [T]Fig. 2: Calculated fields for the five shimmed and un-shimmedblocks over the range -0.1 m ≤ x ≤ 0.5 m.-202z [m]00.10.20.3x [m]00.511.5B [T]Fig. 3: Fitted field map for the 26 GeV/c working point aftermixing of the shim model into the measured data.Least squares fit of the discrete field map after mixingof the shim model has been performed using two-dimensional polynomials to get a functional form By(x,z)which can be handled by the differential equations forthe betatron motion. Polynomials up to degree fourteenin x and z have been retained to reach a good agreementwith the measured field (accuracy within ±0.01 T). Thefitting was performed using the standard Mathematica fitfunction [6] (see Fig. 3).3. BEAM PATH, TRANSFER MATRIXThe radial centroid motion of the beam in any fieldBy(x,z) expressed in a Cartesian frame is given by thenon-linear differential system [7]dxdz= tanφ ( )ddzepBysinφ= −where x and φ are the trajectory’s position and angle.Small deviations ξ, ψ, ∆p (respectively η, χ) in posi-tion, angle and momentum about the horizontal(respectively vertical) particle trajectory are given by thelinear systemddzξ ψφ= cos2( )ddzK ppxψ φ ξρcos= − +1 ∆ddzη χφ= cos ( )ddzK Kx zχ φ η= − tanwhere ρ is the local curvature radius and Kx, Kz are thenormalized strengths defined asK epdBdxxy= K epdBdzzy=Solutions of the linear system may be expressed as atransfer matrix from the field map entrance to its exit. Inthe horizontal plane the change of variables q=ξcosφ isrequired for the determinant of the transfer matrix to beunity.4. BEAM PARAMETERSThe CPS and the TT2 transport channel which links itto the SPS have been modeled using the MAD program[8]. The 26 GeV/c proton ejection process begins withCPS optics calculation (including a local orbit deforma-tion at the ejection septum and the kick of the ejectionkicker), follows by the  transport through the stray field(i.e. from ejection septum exit), and ends with the TT2transfer line optics calculation [9]. Ejection trajectory,shown on Fig. 4, and transfer matrix computationsthrough the magnet stray field have been performed us-ing the built-in Mathematica numerical differentialequation solver with initial conditions given by MAD.The beam centroid enters the field map with coordi-nates x=91.6 mm, φ=62.6 mrad, and exits the field mapat   x=345.0 mm, φ=36.4 mrad. For comparison the an-gle of the ejection pipe with respect to the z-axis in the Fhalf-unit number 16 is 43 mrad. Calculation of thetransfer matrices using the above linear system (withdifferent values of q, ψ, ∆p/p for the horizontal planeand η, χ for the vertical plane) yields:q qppppψ ψ∆ ∆ =out in1350 5 705 0 0790 083 1090 0 0260 0 1. . .. . .ηχηχ = −out in0 665 4 8780 081 0 909. .. .The determinants of these matrices are not exactlyequal to unity because the Mathematica numerical equa-tion solver is not symplectic.-2 -1 0 1 2-0.100.10.20.30.4z [m]x [m]Fig. 4: Ejection (solid line) and central trajectories (dottedline) through the field map.5. RESULTS AND CONCLUSION5.1 Comparison with the previous modelThe optical parameters have been derived from theabove transfer matrix [7] and compared with previousmodels which consider the MAD stray field descriptiongiven by dipole, quadrupole and sextupole coefficientsdistributed over the magnet length [9]. Both results,which agree reasonably well, are shown in Table 1.βx[m]αx Dx[m]Dx’ βy[m]αyField map 31.25 -2.71 3.10 0.26 7.11 0.74MAD model 33.79 -3.37 3.25 0.32 6.13 0.85Table 1: Optical parameters at the exit of the stray field.5.2 Experimental verificationOptical parameters from the obtained field map at  thelocation of SEM-grid beam profile detectors in the TT2transport channel are shown in table 2.SEM-gridlocationβx[m]αx Dx[m]Dx’ βy[m]αyMSG 257 13.31 1.14 0.90 -0.21 27.19 -1.20MSG 267 10.99 0.20 -1.41 -0.07 32.25 -1.71MSG 277 23.07 0.94 -2.51 0.12 42.05 -2.03Table 2: Optical parameters at the three beam profile detectorsin the CPS-SPS transfer channel (TT2).Transverse emittance matching in the TT2 transferchannel is obtained from beam profiles measured at threeSEM-grid detectors. Using the optical parameters in Ta-ble 2, the mismatch derived from measurements [10] wasfound to be less than 15% for the horizontal plane (seeFig. 5) and less than 10% for the vertical plane. This is afairly good result (the best achieved so far) consideringthat an error on optical parameters transforms into alarge mismatch error. More studies will be carried out onthe present ejected proton beam trajectories to furtherimprove the knowledge of the matching.Fig. 5: Beam profile, matching and emittance measurements inthe PS-SPS transfer channel.6. ACKNOWLEDGMENTSThe authors wish to thank B. Autin, R. Cappi, D. Cor-nuet, E. Métral, T. Risselada, J.P. Riunaud and J.M.Roux for stimulating discussions on the subject.REFERENCES[1] The LHC Study Group, “The Large Hadron Collider, Con-ceptual Design”, CERN/AC/95-05 (LHC).[2] D. Cornuet, Z. Sharifullin, “Magnetic Measurements on thePS Magnet Unit U17 with Hall Probes”, CERN technicalnote AT/MA Note 92-23.[3] P. Bossard, “Shim for Ejection 16”, MPS/SI/Note MAE/68-8, 1968.[4] K. Halbach, “A Program for Inversion of System Analysisand its Application to the Design of Magnets”, Proc 2ndConf. On Magnet Technology, Oxford, England, 1967.[5] E. Métral “Modélisation d’une Ligne de Transport de Fais-ceaux”, to be published.[6] S. Wolfram, “Mathematica, A System for Doing Mathe-matics by Computer”, Addisson-Wesley, 1991.[7] B. Autin, “Introduction to Beam Optics”, CERN/PS/84-18(AA), 1984.[8] H. Grote, F.Iselin, “The MAD program, Version 8.19”,http://hpariel.cern.ch/fci/mad/mad.html.[9] T. Risselada, “Beam Optics Data for the PS Ring and itsTransfer Lines”, PS-PA Note 92-04.[10] M. Arruat, M. Martini, “The New Standard Method toMeasure Emittances in the PS Transfer Lines”, CERN/PS92-59 (PA).

Trajectory and optical parameters in a non-linear stray field

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