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Magnetic field study for a new generation high resolution mass separator Marchetto, Marco 2017

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Magnetic Field Study for a NewGeneration High Resolution MassSeparatorbyMarco MarchettoLaurea in Fisica, Universita` Degli Studi di Padova, 2003A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate and Postdoctoral Studies(Physics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)December 2017c© Marco Marchetto 2017AbstractThe work presented in this thesis is part of the design of the high resolutionmass separator for the ARIEL facility under construction at TRIUMF, lo-cated in the UBC campus. This new facility, together with the existing ISACfacility, will produce rare isotope beams for nuclear physics experiments andnuclear medicine.The delivery of such beams requires a stage of separation after productionto select the isotope of interest. The required separation is expressed interms of resolving power defined as the inverse of the relative mass differencebetween two isotopes that need to be separated. The higher the mass thegreater the resolving power required. The challenge is the separation of twoisobars rather than two isotopes that by definition require a much lowerresolving power. A resolving power of twenty thousand is the minimumrequired to achieve isobaric separation up to the uranium mass.The state of the art for existing heavy ion mass separators is a resolvingpower in the order of ten thousand for a transmitted emittance of less thanthree micrometers. The more typical long term operational value is wellbelow ten thousand for larger emittances. The main goal of this project isto develop a mass separator that maintains an operational resolving powerof twenty thousand.Different aspects influence the performance of the mass separator; thetwo main ones are the optics design and the field quality of the magneticdipole(s) that provides the core functionality of the mass separator.In this thesis we worked from the hypothesis that minimizing the mag-netic field integral variation with respect to the design mass resolution isequivalent to minimizing the aberration of the optical system.During this work we investigated how certain geometric parameters in-iiAbstractfluence the field quality, as for example the dependency of the field flatnesson the magnet pole gap. We also developed a new technique to control themesh in the finite element analysis to facilitate particle tracking calculations.Beyond demonstrating our hypothesis, we ultimately produced a finalmagnet design where the field integral variation is less than one part in onehundred thousand.iiiLay SummaryRare isotopes are the new frontier for fundamental studies in nuclear physicsbut also for medical application for the diagnosis and treatment of tumors.Rare isotopes have been produced at TRIUMF since the late nineties. Anew facility, ARIEL, will increase such production three fold. One criticalstep in the delivery of such rare isotopes is the separation and selection.The production includes isobars with mass difference smaller than one partin many thousands. In order to select a desired isotope produced in thenew facility, a new generation mass separator system has been developedcapable of discriminating a mass difference of one part in twenty thousand.The thesis work consists in the design of a high performance magnetic dipolethat is the core component of the separator system. One key achievementis a magnetic field variation of few parts per million.ivPrefaceThe work done in this thesis is a main contribution to the High Resolu-tion Separator (HRS) project funded by Canada Foundation for Innova-tion (CFI). The project is divided into two components: beam dynamicsstudy and magnet design.Dr. James Maloney was in charge to the beam dynamics calculationwhile I was in charge of the magnet design, the main hardware componentsof the HRS system. The work presented in this thesis is related only to themagnet design.A paper was published on the beam dynamics study: “New design stud-ies for TRIUMF’s ARIEL High Resolution Separator”[1]. My contributionto this paper as co-author is related to magnetic field study and the magnetdesign details.A paper dedicated to the design of the magnet will be written based onchapters 2 to 5 of this thesis. I will be the main author with Prof. RichardBaartman and Dr. Maloney as co-author.Chapter 1 contains results from work I did at Canada’s national labora-tory for particle and nuclear physics and accelerator-based science (TRIUMF)in the field of Rare Isotope Beams (RIBs) since 2004. In particular on theissue of separation and delivery of high mass beams as reported in “Progressand plans for high mass beam delivery at TRIUMF”[2] (first author paperand invited oral at the 2012 Heavy Ion Accelerator Technology Conference).On the subject I contributed in developing a technique for mass separationin flight using existing TRIUMF infrastructure as reported in “In flight ionseparation using a Linac chain”[3] (first author paper and invited oral at the2012 Linear Accelerator Conference). I have been working for many years onTRIUMF post-accelerators, that are designed to deliver RIBs; I contributedvPrefacein many aspects including beam dynamics as reported in “Beam DynamicsStudies on the ISAC-II Superconducting Linac”[4] (first author paper) andhardware upgrade.In general I have been working in the field of particle accelerators sincemy university study in Italy, including my physics “laurea” thesis: “Study ofa high-current 176 MHz RFQ as a deuteron injector for the SPES project”[5].All these preliminary works have been instrumental to properly developthe work of this thesis.viTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiList of Acronyms . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxiList of Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxiiiAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . xxivDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxv1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Radioactive ion beam production methods . . . . . . . . . . 21.1.1 In-flight method . . . . . . . . . . . . . . . . . . . . . 31.1.2 ISOL method . . . . . . . . . . . . . . . . . . . . . . 41.2 The ISAC facility at TRIUMF . . . . . . . . . . . . . . . . . 51.2.1 Driver . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.2.2 Target station and mass separator . . . . . . . . . . . 81.2.3 Post accelerators . . . . . . . . . . . . . . . . . . . . . 101.3 Challenges of ISOL production and delivery . . . . . . . . . 15viiTable of Contents1.4 The quest for resolving power . . . . . . . . . . . . . . . . . 191.5 The physics case of 132Sn . . . . . . . . . . . . . . . . . . . . 222 Magnetic Dipole Mass Separator . . . . . . . . . . . . . . . . 252.1 Working hypothesis . . . . . . . . . . . . . . . . . . . . . . . 312.2 Design requirements . . . . . . . . . . . . . . . . . . . . . . . 332.3 Magnetic dipole model . . . . . . . . . . . . . . . . . . . . . 362.4 Reference geometry . . . . . . . . . . . . . . . . . . . . . . . 423 Field Study on the Reference Geometry . . . . . . . . . . . 503.1 Pole gap optimization . . . . . . . . . . . . . . . . . . . . . . 503.2 Flatness versus pole height . . . . . . . . . . . . . . . . . . . 593.3 Flatness versus pole width . . . . . . . . . . . . . . . . . . . 633.4 Magnetic flux balance . . . . . . . . . . . . . . . . . . . . . . 654 Optimization on the Straight Edge Model . . . . . . . . . . 714.1 Nominal geometry . . . . . . . . . . . . . . . . . . . . . . . . 714.2 Sector Rogowski profile . . . . . . . . . . . . . . . . . . . . . 754.3 Purcell-like filter . . . . . . . . . . . . . . . . . . . . . . . . . 824.4 Field clamp . . . . . . . . . . . . . . . . . . . . . . . . . . . . 885 Final Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 945.1 Optimized straight edge model . . . . . . . . . . . . . . . . . 945.2 OPERA-3D R© field for COSY-∞ . . . . . . . . . . . . . . . . 1005.3 Final curved edge model . . . . . . . . . . . . . . . . . . . . 1055.4 Engineering considerations . . . . . . . . . . . . . . . . . . . 1146 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1196.1 HRS system developments . . . . . . . . . . . . . . . . . . . 1216.2 Future upgrades . . . . . . . . . . . . . . . . . . . . . . . . . 1226.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124viiiTable of ContentsI Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130A Hill’s Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 131B Gaussian Beam Distribution . . . . . . . . . . . . . . . . . . . 135ixList of Tables2.1 Simulation statistics for coarse and fine mesh of an HRS ge-ometry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402.2 Chemical composition of the C1006 steel used to manufacturethe HRS dipole magnets. . . . . . . . . . . . . . . . . . . . . . 412.3 Magnetic field integrals for the reference geometry. . . . . . . 474.1 Nominal geometry main parameters. . . . . . . . . . . . . . . 715.1 Final geometry main parameters. . . . . . . . . . . . . . . . . 1075.2 Final geometry calculated integrals. . . . . . . . . . . . . . . 1105.3 Engineering features applied to the final geometry. . . . . . . 115xList of Figures1.1 The TRIUMF site. Highlighted are the ISAC-I (red), ISAC-II(orange) and ARIEL (green) facilities. . . . . . . . . . . . . . 51.2 ISAC facility at TRIUMF. The three experimental areas(low, medium and high) are highlighted. The grey shadedarea is located two stories underground while the remainingis at ground level. . . . . . . . . . . . . . . . . . . . . . . . . . 61.3 TRIUMF H− cyclotron. Multiple beams can be extractedsimultaneously at different energies. Proton beams are ex-tracted at energies up to 500 MeV and up to 100µA for RIBproduction in ISAC (red line) and in ARIEL (blue line - future). 71.4 ISAC target stations and following separation stages. . . . . . 91.5 ISAC linear accelerators: Radio Frequency Quadrupole (RFQ)(top left), Drift Tube Linac (DTL) (top right), superconduct-ing linac SCB (bottom left, crymodules) and SCC (bottomright, cold mass) sections. . . . . . . . . . . . . . . . . . . . . 111.6 Charge state distribution of 16O downstream of the RFQ afterstripping with a 18µm carbon foil. . . . . . . . . . . . . . . . 131.7 ISAC Drift Tube Linac (DTL) . . . . . . . . . . . . . . . . . 141.8 ISAC-II SuperConducting (SC) linac: SCB section cold mass. 141.9 ISAC-II SC linac quarter wave resonators. . . . . . . . . . . . 151.10 Charge state distributions of three isotopes identified as: 69Ga,94Mo, 119Sn. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171.11 Theoretical charge state distribution of 94Rb and 94Mo. Thetwo isobars are separated by a relative mass difference of∆m/m = 1/4405. . . . . . . . . . . . . . . . . . . . . . . . . . 18xiList of Figures1.12 Effect of the carbon foil filtration as measured at the Braggdetector: the left picture correspond to the unfiltered cocktailbeam from the DTL. . . . . . . . . . . . . . . . . . . . . . . . 191.13 Resolving power range required to separate isobaric isotopesfor any given mass. . . . . . . . . . . . . . . . . . . . . . . . . 201.14 Occurrences resolved for a given resolving power as a functionof the isotope mass number. . . . . . . . . . . . . . . . . . . . 211.15 Mass 132 isobars as a function of the neutron number: blackmark correspond to the stable nuclei, red mark correspondto 132Sn. Each division in the vertical scale corresponds to aresolving power of 20000. . . . . . . . . . . . . . . . . . . . . 221.16 Resolving power necessary to separate 132Sn from its iso-bars. Highlighted are: 132Sn (red mark), 132Cs (orange mark),20000 resolving power (dashed blue line). . . . . . . . . . . . 232.1 Artistic representation of particle trajectories (blue and red)inside a magnetic field (orange). . . . . . . . . . . . . . . . . . 262.2 Artistic representation of particle distribution in the trans-verse phase space: w and ϕ are respectively the half widthand the divergence of the beam. . . . . . . . . . . . . . . . . . 282.3 Artistic representation of beam separation (δ = D dmm ) andselection: ideal case of three beams, having the same inten-sity, being selected by a transverse slit (transparent orangesquares) 2w wide. The Gaussian tails, that overlap with theselected beam (green), are truncated at the entrance of theseparator system with a slit. . . . . . . . . . . . . . . . . . . . 302.4 Artistic representation of particle separation for a 180 degreeseparator: the dashed lines represent the particles enteringthe separator with the largest angle ψ defining the width ofthe good field region. . . . . . . . . . . . . . . . . . . . . . . . 32xiiList of Figures2.5 Geometric trajectories. The reference trajectory representedin red has a radius of curvature of 1200 mm. The black solidlines represent the field boundaries of the hard-edge magnet:ϕ is the edge angle. The bend angle θ = 90 degree. . . . . . . 332.6 Schematic layout of the High Resolution Separator (HRS) . . 372.7 Example of parameterized coordinates of the HRS dipole . . 382.8 Rendering of the reference geometry (HRS-120-12Cq2): halfmagnet is represented but only one quarter is simulated. . . . 392.9 BH curves for C1006 and C1010 magnetic steel and relativepermeability (see equation 2.15) for the C1006 steel. Thenarrow side graph is a magnification of the main one aroundthe operational range (orange dotted box). . . . . . . . . . . . 412.10 Reference design Bz vertical magnetic field component: thered line represents the reference geometric trajectory. Thehard edge case would be a constant field at the maximumvalue, dropping to zero at 45 degree. . . . . . . . . . . . . . . 422.11 Reference design Bz vertical magnetic field component mag-nified near the peak. The field is symmetric with respect to0 degree. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432.12 Reference design Bz vertical magnetic field component mag-nified around the fringe field (drop off). . . . . . . . . . . . . 432.13 Reference design field flatness: the red portion of the curverepresents the flatness within the good field region (±160 mmaround the reference geometric trajectory ρ = 1200 mm). . . . 442.14 Reference design magnetic flux density in the steel (half mag-net). The external return yoke surface (yellow) is larger thanthe internal (orange). . . . . . . . . . . . . . . . . . . . . . . . 452.15 Reference design effective field edge location with respect tothe relative hard edge case. The light blue points are outsidethe good field region. If the effective edge aligned with thehard edge case the yellow dotted line (linear interpolation ofthe dark blue data) would be the constant y = 0. . . . . . . . 48xiiiList of Figures2.16 Reference design integral flatness with respect to the relativehard edge case. The light blue sections of the graph are out-side the good field region. If the integral flatness matchedexactly the hard edge case the points would lay on the x-axis(y=0). The green box (2.5 ·10−5 high) represents the flatnessrequirement in the good field region. . . . . . . . . . . . . . . 493.1 Cross section of the reference geometry in the middle of themagnet; the green contour represents the shape of the steel.Magnetic field B (top) and magnetizing field H (bottom) inthe reference geometry. The red loop is the integration circuit∂Σ of equation 3.1. . . . . . . . . . . . . . . . . . . . . . . . 513.2 Field flatness as a function of pole overhang. . . . . . . . . . 533.3 Field flatness dependency on the pole gap; the solid line repre-sents the flatness within the limit of the geometric trajectories. 543.4 Rendering (half magnet) of the “cube” geometry. . . . . . . . 543.5 Cross section of the “cube” geometry in the middle of themagnet (70 mm gap); the green contour represents the shapeof the steel. Field flatness of the “cube” geometry. . . . . . . 553.6 Relative permeability for the low (non-saturated - top) andthe high (saturated - bottom) excitation case. . . . . . . . . . 563.7 Field flatness of the “cube” geometry dependency on the polegap and saturation level: solid and dashed lines represent re-spectively the low excitation (non-saturated) and high exci-tation (saturated) case. . . . . . . . . . . . . . . . . . . . . . 573.8 Reference geometry magnetic flux density in the steel for low(selected operational mode - top) and high excitation (bottom). 593.9 Reference geometry middle section. . . . . . . . . . . . . . . . 593.10 Flatness dependency on the pole height. . . . . . . . . . . . . 603.11 Magnetic flux density in the steel for the two extreme casesof pole height: 90 mm (reference design - top) and 180 mm(bottom). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61xivList of Figures3.12 Flatness dependency on the coil vertical position. The casewith pole height at 180 mm and coil not moved is the sameas in figure3.10. . . . . . . . . . . . . . . . . . . . . . . . . . . 613.13 Flatness dependency on the pole base height. The case withpole height at 180 mm, coil moved −60 mm and base heightunchanged is the same as in figure3.12. . . . . . . . . . . . . . 623.14 Magnetic flux density in the steel for the two extreme casesof pole base height: 169.5 mm (top) and 289.5 mm (bottom). 633.15 Flatness dependency on the pole width. . . . . . . . . . . . . 643.16 Magnetic flux density in the steel for the two extreme casesof pole width: 926 mm (top) and 526 mm (bottom). . . . . . . 653.17 Rendering of the reference surface with the first equaliza-tion method applied; the blue circle indicates the air channelcarved in the outer return yoke steel in order to achieve bal-anced flux. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 663.18 First equalization method configurations comparison. . . . . . 673.19 Equalized geometry (HRS-120-16C3: first method, configura-tion 3) Bz vertical magnetic field component magnified nearthe peak. The field is symmetric with respect to 0 degree.This figure to be compared with figure 2.11. . . . . . . . . . . 673.20 Magnetic flux density in the steel before (left) and after (right)equalization with the first method. . . . . . . . . . . . . . . . 683.21 Second equalization method: the splitting radius (red) is1140 mm while the symmetry one (yellow) is 1200 mm. . . . . 693.22 Magnetic flux density in the steel before (left) and after (right)equalization with the second method. . . . . . . . . . . . . . . 693.23 Field flatness comparison between the split and symmetryradius cases (second method). Note the apparently suddenimprovement with respect to figure 3.18; as we wrote in thetext, this second method was developed later in time on ageometry with a higher level of optimization. . . . . . . . . . 704.1 Nominal geometry rendering. . . . . . . . . . . . . . . . . . . 72xvList of Figures4.2 Nominal geometryBz vertical magnetic field component, mag-nified near the peak, to show flatness. The field is symmetricwith respect to 0 degree. . . . . . . . . . . . . . . . . . . . . . 734.3 Nominal geometry field flatness: the red portion of the curverepresents the flatness within the good field region (±160 mmaround the reference geometric trajectory ρ = 1200 mm). . . . 734.4 Nominal geometry effective field edge location with respect tothe relative hard edge case: the linear interpolation (yellowdashed line) between 1050 mm and 1350 mm gives indicationsabout position and angle. . . . . . . . . . . . . . . . . . . . . 744.5 Nominal geometry integral flatness with respect to the rel-ative hard edge case: the yellow dashed line is a linear in-terpolation between 1050 mm and 1350 mm. The green box(2.5 · 10−5 high) represents the flatness requirement in thegood field region. . . . . . . . . . . . . . . . . . . . . . . . . . 744.6 Rogowski profile approximated with four straight sectors: thedotted light green line represents the pole, the dark green rep-resents the pole base and return yoke, the dotted orange linerepresents the coil of the nominal geometry. The profile hasto be moved inward to accommodate the coil while avoidingan increase of the magnet size. . . . . . . . . . . . . . . . . . 764.7 Geometry with four sectors Rogowski profile (dashed lines). . 774.8 Longitudinal field flatness of the nominal geometry for maxi-mum (solid line) and minimum (dashed line) rigidity. A closeview is represented in the following figure 4.9. . . . . . . . . . 784.9 Close view of the longitudinal field flatness of the nominalgeometry for maximum (solid line) and minimum (dashedline) rigidity around the longitudinal edge of the dipole. . . . 794.10 Longitudinal field flatness of the four sectors Rogowski geom-etry (HRS-120-19C8) for maximum (solid line) and minimum(dashed line) rigidity. A close view is represented in the fol-lowing figure 4.11. . . . . . . . . . . . . . . . . . . . . . . . . 79xviList of Figures4.11 Close view of the longitudinal field flatness of the four sec-tors Rogowski geometry (HRS-120-19C8) for maximum (solidline) and minimum (dashed line) rigidity around the longitu-dinal edge of the dipole. . . . . . . . . . . . . . . . . . . . . . 804.12 Magnetic flux density for the nominal (top) and four sectorsRogowski (bottom) geometries at current excitation relativeto the maximum (left) and minimum (right) beam rigidity. . 814.13 Rendering of the nominal geometry with the full Purcell filter(HRS-120-20C17): the air gap between the pole and pole baseis highlighted by the red loop. . . . . . . . . . . . . . . . . . . 824.14 Field flatness of the geometry with the full Purcell filter. . . . 834.15 Rendering of the nominal geometry with the partial Purcellfilter (HRS-120-20C2): the partial air gap between the poleand pole base is highlighted by the red loop. . . . . . . . . . . 844.16 Field flatness of the geometry with the partial Purcell filter. . 844.17 Rendering of the nominal geometry with the outboard Purcellfilter (HRS-120-20C7): the air slots at the bottom of the poleare highlighted by the red loops. . . . . . . . . . . . . . . . . 854.18 Field flatness of the geometry with the outboard Purcell filter;the dashed curve is ×10 magnified. . . . . . . . . . . . . . . . 854.19 Rendering of the nominal geometry with the windows Pur-cell filter (HRS-120-20C15): the air windows in the pole arehighlighted by the red loop. . . . . . . . . . . . . . . . . . . . 864.20 Field flatness of the geometry with the outboard Purcell filter;the dashed curve is ×10 magnified. . . . . . . . . . . . . . . . 874.21 Rendering of the nominal geometry with the detached partialPurcell filter (HRS-120-20C20): the air gap between the poleand pole base is highlighted by the red loop. . . . . . . . . . . 874.22 Field flatness of the geometry with the detached partial Pur-cell filter; the dashed curve is ×100 magnified. . . . . . . . . . 884.23 Nominal geometry bottom entrance field clamp (dashed line). 89xviiList of Figures4.24 Magnetic field of the geometry with reference clamp versusthe theoretical hard edge case plotted as a function of s (pathlength along a beam trajectory). The outer mechanical edgeof the clamp is at s = 106 mm. . . . . . . . . . . . . . . . . . 904.25 Simple L shape reference clamp. . . . . . . . . . . . . . . . . 914.26 Magnification of the fringe field profiles of the different fieldclamp configurations. . . . . . . . . . . . . . . . . . . . . . . . 924.27 Optimized field clamp for the straight edge geometry. . . . . 935.1 Optimized straight edge geometry. . . . . . . . . . . . . . . . 945.2 Magnetic flux density of the optimized straight edge geometry. 955.3 Sector Rogowski comparison: the most gain in terms of ap-proaching the theoretical curve, is going from four to six sec-tors as seen in figre 5.4. . . . . . . . . . . . . . . . . . . . . . 965.4 Maximum distance limit set to generate the sector Rogowski 965.5 Six sectors Rogowski profile; a scaled version is implementedin the optimized straight geometry . . . . . . . . . . . . . . . 975.6 Pole of the optimized straight edge model with a six sectorRogowski scaled 61% longitudinally and 40% transversely. . . 985.7 Field flatness of the optimized straight edge design. . . . . . . 995.8 Optimized straight edge geometry effective field edge locationwith respect to the relative hard edge case: the linear inter-polation (yellow dashed line) between 1050 mm and 1350 mmgives indications about position and angle. . . . . . . . . . . . 995.9 Optimized straight edge geometry integral flatness with re-spect to the relative hard edge case: the yellow dashed lineis a linear interpolation between 1050 mm and 1350 mm. Thegreen box (2.5 ·10−5 high) represents the flatness requirementin the good field region. . . . . . . . . . . . . . . . . . . . . . 1005.10 Detailed view of the air gap region (yellow dashed line ) stan-dard mesh produced by OPERA R©. Various patches of meshcan be distinguished. . . . . . . . . . . . . . . . . . . . . . . . 101xviiiList of Figures5.11 Bed of nails (A) superimposed on the air gap volume (B)then subtracted from the air gap (C); the remaining structurehas volume-less holes (D). The nails are OPERA R© wire-edgesthat have no volume. . . . . . . . . . . . . . . . . . . . . . . . 1035.12 Air gap volume (yellow dashed line) meshed with the “bedof nails” technique (top). Magnified section of the air gap(central and bottom): evidence (red arrows) that the end ofa given nail (red dashed line) coincides with a node. . . . . . 1045.13 Final geometry. . . . . . . . . . . . . . . . . . . . . . . . . . . 1055.14 Characteristic “C” shape of the second order aberration. . . . 1065.15 Magnetic flux density of the final design in full excitation mode.1075.16 Detail of the pole corner for the optimized straight edge (left)and final curved geometry (right): the curved geometry main-tain the same scaling factor in the vertical plane (different inthe horizontal) producing a simplified geometry of the corner(see orange line for reference). . . . . . . . . . . . . . . . . . . 1085.17 Fringe field of the final geometry. . . . . . . . . . . . . . . . . 1095.18 First derivative of the fringe field of the final geometry. . . . . 1095.19 Final geometry field flatness for the full excitation mode. . . . 1105.20 Final geometry effective field edge location with respect to therelative hard edge case for the full excitation mode: the lin-ear interpolation (yellow dashed line) between 1050 mm and1350 mm gives indications about position and angle. . . . . . 1115.21 Final geometry integral flatness with respect to the relativehard edge case for the full excitation mode. . . . . . . . . . . 1115.22 Magnetic flux density of the final design in low excitationmode; notice the change of scale with respect to the full ex-citation mode. . . . . . . . . . . . . . . . . . . . . . . . . . . . 1125.23 Final geometry field flatness for the low excitation mode. . . 1135.24 Final geometry effective field edge location with respect tothe relative hard edge case for the low excitation mode. . . . 1135.25 Final geometry integral flatness with respect to the relativehard edge case for the low excitation mode. . . . . . . . . . . 114xixList of Figures5.26 Final design with engineering features (HRS-120-23C62eng21).1165.27 Final engineered geometry field flatness with respect to thehard edge case. . . . . . . . . . . . . . . . . . . . . . . . . . . 1165.28 Final engineered geometry effective field edge location withrespect to the relative hard edge case for the full excitationmode. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1175.29 Final engineered geometry integral flatness with respect tothe relative hard edge case for the full excitation mode. . . . 1175.30 First HRS manufactured magnetic dipole (with field clampedges protected by white Styrofoam). Photograph courtesyof Buckley Systems. . . . . . . . . . . . . . . . . . . . . . . . 1186.1 Transverse horizontal phase space at the selection (exit) slitcalculated in COSY-∞ with ideal dipole field (top) and OPERA R©imported field (bottom). . . . . . . . . . . . . . . . . . . . . . 120A.1 Graphic representation of equation A.12 in the Cartesian plane.133B.1 Upright emittances: 1σ (blue) and 4σ (red). . . . . . . . . . . 136xxList of AcronymsANL Argonne National Laboratory (laboratory)ARIEL Advanced Rare IsotopE Laboratory (facility)CANREB CANadian Rare isotope facility with Electron Beam (project)CARIBU CAlifornium Rare Ion Breeder Upgrade (project)CERN Conseil Europeen pour la Recherche Nucleaire (laboratory)CFI Canada Foundation for Innovation (funding)CGS Centimetre-Gram-Second system of units (units)CW Continuous Wave (operational mode)DTL Drift Tube Linac (accelerator)ECR Electron Cyclotron Resonance (ion source)ECRIS Electron Cyclotron Resonance Ion Source (ion source)FEBIAD Forced Electron Beam Induced Arc Discharge (ion source)FRIB Facility for Rare Isotope Beam (facility)HRS High Resolution Separator (separator)ISAC Isotope Separation and ACceleration (facility)ISOL Isotope Separation On-Line (production method)ISOLDE Isotope Separation On Line DEvice (facility)xxiList of AcronymsLIS Laser Ion Source (ion source)MSU Mitchigan State UniversityRIB Radioactive Ion Beam (sometimes also Rare Isotope Beam)RF Radio Frequency (operational mode)RFQ Radio Frequency Quadrupole (accelerator)TRIUMF Canada’s national laboratory for particle and nuclear physicsand accelerator-based science (laboratory1)SC SuperConducting (operational mode)SI International System of units (units)UBC University of British Columbia1historically TRIUMF stands for TRI University Meson Facility.xxiiList of UnitsThis section includes definition of units that are used in the particle accel-erator community and not part of the International System of units (SI)[6].e-A electrical-Ampere: this current definition takes into account the chargestate of the ions that contributes to the electrical current. A submul-tiple of this unit is the e-nA (10−9 e-A).p-A particle-Ampere: this current definition doesn’t consider the chargestate of the ions that contributes to the electrical current. Alterna-tively it can be seen as the current in electrical Amperes divided by thecharge state of the ion; complications may arise though when the elec-trical current is composed of ions with different charge states. Thisunit is preferred to the electrical Ampere definition by the experi-menters since it’s straightforward to infer the number of particles persecond by simply dividing by the elementary charge (1.6 · 10−19 C).pps particle per second: this is the preferred beam intensity unit for anexperiment since they are looking at the number of reactions.xxiiiAcknowledgementsI am always reluctant to make a list of people to thank because inevitablysomeone is left out, nevertheless I will try and risk to upset a few.First and foremost I have to thank Prof. Lia Merminga, my formerTRIUMF accelerator division head and one of my committee members. Itis not an overstatement saying that my PhD would have not happen if itwasn’t for Lia. Lia gave me this once in a life time opportunity to pursuethis degree while maintaining my responsibilities and the possibility of ad-vancing my career at TRIUMF. I own her more than I will ever be able togive back.Another big thank you goes to Prof. Richard Baartman, research co-supervisor and TRIUMF supervisor. I always think of Rick as a colleaguerather than the boss.Thank you to Prof. Andrea Damascelli, research supervisor, and Prof.Vesna Sossi, committee member, for their trust in me.Thanks to Dr. James Maloney for the collaboration in designing the highresolution separator; I hope our effort will not be in vain.Thanks to P.Eng. George S. Clark, former TRIUMF magnets engineer,for his critical approach based on his decades long experience during ourdiscussions about the magnet design.Thanks to Buckley Systems for making my design a realty.Thanks to my colleague Dr. Thomas Planche for his feedback and to myformer supervisor Prof. Robert Laxdal for supporting my PhD.Thanks to the CANREB management to assign me the task of designingthe HRS magnet.xxivDedicationI dedicate this work to Francesca, Orlando, Oliver, Arthur and Tristan. Itwas always a fine balance working and writing the thesis while home wasinvaded by four little Huns. Luckily Francesca was there.xxvChapter 1Introductionfatti non foste a viver come bruti,ma per seguir virtute e canoscenzaDante AlighieriInferno Canto XXVI (1308-1320 d.C.)This thesis is a development in the field of particle accelerator physicsthat studies the behavior of an ion beam as an ensemble of charged particlestransported from a starting point (source) to a final destination (experiment)through electric and magnetic fields contained in a system of beam transportlines.Beams are typically classified in two main categories depending if theyare composed of light or heavy particles. Light particles like electrons orpositrons are practically always relativistic in accelerator systems. Heavierparticles like protons or ions require more accelerating voltage to become rel-ativistic. For the purpose of this thesis we will consider only non-relativisticheavy ions; as a reference 4He (on the light side of the heavy ions) becomesrelativistic (β greater than 0.5) around 600 MeV. This is considered a veryhigh energy for heavy ions.Ion sources produce particles with an initial energy that ranges fromtens to hundreds of keV. For heavy ions with energies above a few hundredsof keV, the unit used is the electronvolts per unified atomic mass unit,in symbol eV/u; in our previous example the 4He energy would be circa150 keV/u. Sometimes the latter are also quoted in eV/A, in this documentwe will use the former unit.The initial energy can be increased, or decreased, using accelerating11.1. Radioactive ion beam production methodsstructures along the beam lines. Two metal plates at different electrostaticpotential separated by a gap constitute a simple accelerating structure. Morecomplex accelerating structures are called particle accelerators that utilizeeither electrostatic or electromagnetic (radio-frequency or RF) fields in orderto provide acceleration. Particles are guided along a defined beam trajectoryby means of electrostatic or magnetostatic fields.A Radioactive Ion Beam (RIB) is an ensemble of radioactive chargedparticles. These beams are used in astrophysics, nuclear and atomic physics,material and medical science experiments. The required final energies forthese experiments range from a few eV for atomic physics to hundreds ofMeV for the nuclear physics.The radioactive elements that compose the beam have half-lives thatcan range typically from ms to hours. Due to their short half-lives, theyare not readily available in nature and therefore they have to be producedin a laboratory. In general the production techniques produce multiple ra-dioactive species. A radioactive ion beam composed of different elements issometimes referred as a cocktail beam. Moreover the ionization stage canintroduce stable ions into the cocktail.Mass selection techniques are necessary to isolate the isotope of interest.Nevertheless it is not always possible to completely separate two differentisotopes. In general only one element in these cocktails is required for theexperiment while the other components are considered contaminants. Thecontaminants can render an experiment unfeasible.1.1 Radioactive ion beam production methodsThere are two commonly used methods for RIB production [7] that we aregoing to introduce in the following sections. Both production methods havein common an accelerator called the driver for the primary beam, a pro-duction target and a separation facility for the selection of the secondaryradioactive beam; post-acceleration is an option to boost the energy of theselected beam.21.1. Radioactive ion beam production methods1.1.1 In-flight methodThe first production method is called the in-flight or fragmentation method.In this case the primary beam is a heavy ion, like 238U, that is acceleratedat high energies in the order of hundreds of MeV/u. The driver is complexand fairly expensive accelerator system like in the case of the MitchiganState University (MSU) new Facility for Rare Isotope Beam (FRIB)[8]. Arelatively thin target intercepts the primary beam to produce the secondaryradioactive particles. The secondary beam retains 90% or more of the pri-mary beam energy[9]. Right downstream of the target the secondary beamgoes through the selection stage consisting of a fairly complex mass separatorsystem. Once selected the beam is sent to the experimental station. Sincethe secondary beam is already produced at high energy, post-acceleration isusually not considered in this case.The production process for the in-flight method occurs via projectilefragmentation, nucleon transfer, fission and Coulomb excitation[9]. Differ-ent target materials can be used such as beryllium, tungsten, nickel or tan-talum as a few examples; the radioactive ion beam production is materialdependent. The thicker the target the higher the probability of multiplescattering, and therefore higher production, but also the higher the energystraggling leading to higher energy spread. This means a reduced beamquality.Beam quality is quantified in terms of transverse and longitudinal emit-tances. The emittance is related to the area the beam occupies in phasespace (see chapter 2): position versus divergence in the transverse case andenergy versus time in the longitudinal case. This area is included insidean ellipse of area pi where  is the emittance of the beam [10] [11]. Thetransverse emittance unit is mm·mrad or µm while the longitudinal emit-tance is expressed in keV/u·ns (or alternatively keV/u·deg). The lower theemittance the higher the beam quality.The typical transverse emittance for the in-flight method is on the orderof 102 µm [12][13][14]; this is considered a large or poor emittance comparedto what an ion source can typically produce. Also the typical energy spread31.1. Radioactive ion beam production methodsis large, in the order of few per cent: that translates into a few MeV/u.The in-flight on the other hand is a fast production method that allowsthe delivery of isotopes with very short half-lives, few µs, where the limita-tion is due to selection rather than production process. The in-flight methodproduces also high intensity beams up to a few 1010 pps or 10−9 pA.1.1.2 ISOL methodThe second production method is the Isotope Separation On-Line (ISOL). Inthis case the driver accelerates light projectiles toward a thick target. Theselight projectiles are usually protons, but studies have been conducted todevelop deuteron or tritium[5] drivers for ISOL production. The projectilesinteract with the target heavy nuclei producing neutral radioactive isotopesvia spallation, fragmentation or induced fission[15]. The target material mayvary from silicon to uranium; the material choice is based on the productionrequirements[16] .The neutral atoms produced in the target migrate into an ion source viadiffusion and effusion processes[15]. Here they are ionized and extracted atsource potential up to a few tens of kV. Different types of sources can beused[17], the simplest being the surface source[18]; the latter works efficientlyfor elements with low ionization potential (less than 6 eV) by transferringenergy through a heated surface. Other sources include: plasma ion sourceslike the Forced Electron Beam Induced Arc Discharge (FEBIAD)[19], Elec-tron Cyclotron Resonance (ECR)[20] and Laser Ion Source (LIS)[21].The transverse emittance produced is an order of magnitude smaller(higher quality) with respect to the in-flight method. An upper limit ex-pected from a FEBIAD ion source is about 20µm, while values of less than10µm are typical of a surface ion source. The energy spread out of thesource is in the order of few eV. This energy spread translates into less thanone part in ten thousand, at least two orders of magnitude lower than theenergy spread produced with the alternative method.The radioactive ions extracted from the source are separated using adipole magnet (mass separator) and selected to be transported to the down-41.2. The ISAC facility at TRIUMFstream experimental stations. Thanks to the production process the beamcan be delivered at energies as low as 10 keV. In order to deliver energieshigher than the extraction voltage, the beam must be post-accelerated. Inthis case energies up to 20 MeV/u have been reached at the isotope separa-tion and acceleration facility (see section 1.2).The ISOL method is limited though in terms of half-lives and beam in-tensity with respect to the in-flight. The relatively slow extraction processlimits the possibility of extracting isotopes with few ms while beam intensi-ties are in the order of few 103 pps or 10−16 pA.1.2 The ISAC facility at TRIUMFRare Isotope Beams (RIBs) are produced at TRIUMF in the Isotope Sep-aration and ACceleration (ISAC) facility using the ISOL method. A planFigure 1.1: The TRIUMF site. Highlighted are the ISAC-I (red), ISAC-II(orange) and ARIEL (green) facilities.51.2. The ISAC facility at TRIUMFview of the TRIUMF facilities is represented in figure 1.1.The ISAC facility counts fifteen experimental stations [22] distributedin three experimental areas characterized by different energy ranges: low,medium and high. The overview of the ISAC facility is represented in fig-ure 1.2 where the three experimental areas are highlighted. Presently onlya single RIB is available and can be sent to one of the fifteen stations at atime.Figure 1.2: ISAC facility at TRIUMF. The three experimental areas (low,medium and high) are highlighted. The grey shaded area is located twostories underground while the remaining is at ground level.The future Advanced Rare IsotopE Laboratory (ARIEL) facility (high-lighted green in figure 1.1) is going to increment the RIB production to threeion beams that can be sent simultaneously to three different experimental61.2. The ISAC facility at TRIUMFstations. The object of this thesis, the High Resolution Separator (HRS)system magnetic dipole, is going to be part of the ARIEL facility.The main components of the ISAC facility are briefly described in thefollowing sections.1.2.1 DriverThe TRIUMF cyclotron is the driver that accelerates H− ions up to an in-tensity of 300µA to a maximum energy of 500 MeV [23][24]. A layout of thecyclotron is represented in figure 1.3. The H− ions move in a counterclock-wise spiral trajectory inside the cyclotron from the center outwards.Figure 1.3: TRIUMF H− cyclotron. Multiple beams can be extracted si-multaneously at different energies. Proton beams are extracted at energiesup to 500 MeV and up to 100µA for RIB production in ISAC (red line) andin ARIEL (blue line - future).In order to extract protons, the H− ions are intercepted with a carbon foilresulting in the stripping of the electrons; the protons then turn clockwiseexiting the cyclotron at specific locations. Since the foil can intercept the71.2. The ISAC facility at TRIUMFH− ions at different radial positions inside the cyclotron, protons can beextracted at different energies. Moreover multiple foils can intercept thebeam leading to multiple simultaneous extractions. Presently three out offour existing extraction ports are in operation, one of which is dedicated toISAC (red arrow figure 1.3) for radioactive beam production.At ISAC the protons are delivered at 500 MeV up to 100µA of current.This corresponds to a beam power of up to 50 kW that allows, for example,the production of the most intense 11Li beam in the world; a productionyield of 2.2 · 104 pps has been measured [25] for this beam.The production capability is going to be expanded by refurbishing thenon-operational port and installing a new extraction beam line (blue arrowin figure 1.3) giving two simultaneous proton beams for RIB production [26].This new beam line, together with a current intensity upgrade [27], is nec-essary to support the future ARIEL facility.1.2.2 Target station and mass separatorThe proton beam can be directed to two independent target stations [28],west and east, as represented in figure 1.4. Only one station can receivebeam at any given time.Each target station is composed of five modules. The entrance modulehouses the diagnostic and protect monitors for the proton beam. The targetmodule contains the target and the ion source that produces singly chargedionized species. Target materials include silicon carbide, tantalum and ura-nium carbide. Two target configurations are available: low and high powerrespectively for proton beam powers up to 20 kW and 50 kW. The targetmodule is routinely removed to change both target and ion source. Thebeam dump module is located downstream of the target module along thedirection of the proton beam. The remaining two modules are the extractionones oriented perpendicular to the proton beam direction; they house theoptics elements to transport the beam downstream to the pre-separator.The two target stations have a pre-separator in common located insidethe target hall (see red object in figure 1.4); this is a dipole magnet capable81.2. The ISAC facility at TRIUMFFigure 1.4: ISAC target stations and following separation stages.of bending the beam from either of two different directions (east and west) toa common (north) beam line leading to the main mass separator. The pre-separator is designed to achieve just isotopic separation in order to containmost of the produced radioactivity inside the shielded target hall. The pre-separator has a low resolving power (m/∆m), in the order of few hundred.An isobaric separation is performed downstream by a second dipole mag-net, the mass separator [28], typically operating at a resolving power of a fewthousands. This device is installed on a 60 kV biased platform to enhancebeam purity in particular from cross contamination [29] generated by resid-ual gas collisions. It is possible in fact that even slow but massive ionizedmolecules arrive at the mass separator entrance with the same momentumof the radioactive ions to be selected; the platform bias is going to changethe momentum such that only the radioactive ions bend around the correcttrajectory.After selection it is possible to boost the single charge state of the ra-91.2. The ISAC facility at TRIUMFdioactive ions by diverting them through an Electron Cyclotron ResonanceIon Source (ECRIS) [30] [31]. This source known as a charge breeder allowsfor the post acceleration of masses greater than 30 [2] by reducing the massto charge ratio to a value compatible with the first stage of post acceleration(see section 1.2.3).1.2.3 Post acceleratorsThe selected radioactive ion beams can be delivered to three experimen-tal areas as represented in figure 1.2: a low energy area where the ionsare accelerated at source potential (up to 60 kV), a medium energy area(β = 1.8%→ 6%) or a high energy area (β = 6%→ 18%) where the ionsare post accelerated with linacs [32] (see figure 1.5) .The first stage of acceleration uses an Radio Frequency Quadrupole(RFQ) as injector [33]; this is a linear accelerator that can accelerate andfocus the beam transversely at the same time. The RFQ accelerator con-cept, developed in 1969 by Kapchinskii and Teplyakov [34], consists of theidea that by modulating longitudinally the electrodes of an electrostaticquadrupole in a sinusoidal like profile, a longitudinal accelerating compo-nent of the electric field is created. A quadrupole is a transverse focusingoptical element with four electrodes arranged 90 degree apart in a clover leaflike configuration; two electrodes facing each other have the same polarity,opposite to the other two. A quadrupole2 has the property that while itfocuses a beam in one transverse direction (e.g. horizontal), it defocusesin the other (e.g. vertical); in order to have an overall focusing transportsystem, it is necessary to arrange at least two quadrupoles longitudinally,separated by an opportune distance, with alternate polarity. In the RFQcase, where the base structure is a single long quadrupole, the alternate fo-cusing is achieved by feeding the electrodes with a Radio Frequency (RF)electric field so the beam sees alternating focusing as it travels along thestructure. This makes the longitudinal component alternate as well, beingaccelerating half of the period and decelerating the other half. This implies2Quadrupoles can be either electrostatic or magnetic; they both generate a transverse(electric or magnetic) field to focus (or defocus) the beam.101.2. The ISAC facility at TRIUMFFigure 1.5: ISAC linear accelerators: Radio Frequency Quadrupole (RFQ)(top left), Drift Tube Linac (DTL) (top right), superconducting linac SCB(bottom left, crymodules) and SCC (bottom right, cold mass) sections.that the ions, in order to be accelerated, have to be bunched in time withinone accelerating period, called the RF accelerating bucket. In an RFQ themodulation of the electrodes is usually done gradually so a continuous (intime) beam can be injected while the output beam is bunched. The outputvelocity, and hence output energy per mass unit of an RFQ is fixed and itdepends on the modulation geometry of the electrodes.The ISAC RFQ (figure 1.5 top-left) boosts the beam energy from 2 keV/uto 150 keV/u. It can accelerate mass to charge ratio (A/Q) in the rangefrom 3 to 30. It is a room temperature machine operating in ContinuousWave (CW) mode at 35.36 MHz with an eight meter long resonant structurecomposed of nineteen split rings supporting the modulated electrodes. The111.2. The ISAC facility at TRIUMFISAC RFQ doesn’t have a typical bunching section; the beam is pre-bunchedat injection by a three harmonic electric buncher, the fundamental harmonicis 11.78 MHz. Because the bunching is performed on a continuous beam, thebunched beam at the entrance of the RFQ presents extended longitudinaldistribution, with about 20% falling outside the accelerating bucket; thisportion of beam exits with incorrect energy and it is stopped into a fixedcollimator downstream of the RFQ [35]. This configuration produces a highquality longitudinal emittance after the RFQ calculated to be 0.22 keV/u·ns.The following acceleration is accomplished by means of a Drift TubeLinac (DTL). A DTL is composed of a series of hollow tubes separated bygaps. The tubes carry an RF voltage; at any given time two adjacent tubeshave opposite voltage that create a longitudinal electric field. As for theRFQ, this electric field is accelerating half of the period and deceleratingthe other half. Because of the tube voltage configuration, two adjacent gapshave opposite field. A particle crossing the first gap with an acceleratingfield is going to drift through the following hollow tube (hence the name ofthe accelerator) while the field in the second gap changes from deceleratingto accelerating. A particle synchronized with the RF is going to see justaccelerating field in all the gaps it crosses.In order to increase the acceleration efficiency the ISAC DTL [36] (top-right in figure 1.5 ) accelerates higher charge state and therefore lower massto charge ratio. The maximum A/Q is 7 so for mass to charge ratio greaterthan this value, the charge state must be increased. The ion charge state isincreased by means of stripping through an 18µm (4µg/cm2) thin carbonfoil downstream of the RFQ. Typically the most populated charge state isselected using a magnetic dipole as long as the mass to charge ratio is withinthe design limits of the DTL. The efficiency of the stripping foil dependson the mass of the stripped ions, ranging between 30% to 50%. Figure 1.6shows the charge state distribution of 16O downstream of the RFQ afterstripping; the percentages in the graph indicate the relative abundance ofparticles in a charge state with respect to the total number of particlesbefore stripping. Also, the width of each peak is not related to the chargestate (that is integer) but to the fact that the beam has a certain transverse121.2. The ISAC facility at TRIUMF0123456789102 3 4 5 6 7Current  (p-nA) Q A/Q= 2.67 - 2.1% A/Q=5.33 - 32% A/Q=4 - 49.8% A/Q=3.2 - 17% Figure 1.6: Charge state distribution of 16O downstream of the RFQ afterstripping with a 18µm carbon foil.distribution (as we are going to explain in chapter 2). When the field of themagnetic dipole is set to the correct value for a desired charge state, thedistribution peaks.The DTL is a variable energy machine covering the entire range of en-ergies between 150 keV/u and 1.8 MeV/u. It is also a separated function(accelerating and bunching) machine composed of five accelerating cavitiesand three energy bunchers located between the first four cavities as repre-sented in figure 1.7.This layout maintains good beam quality at every energy; this trans-lates in a typical energy spread of less than 0.4 % and a time spread at theexperiment in the order of a few nanoseconds. The resonance frequency ofthe cavities and bunchers is 106.08 MHz; they operate at room temperaturein CW mode. Transverse focus through the linac is provided by magneticquadrupole triplets between cavities. The transmission through the linac isgreater than 95%. The DTL delivers beam to the medium energy area butit is also used as injector for the ISAC-II SuperConducting (SC) linac.131.2. The ISAC facility at TRIUMFFigure 1.7: ISAC Drift Tube Linac (DTL)The SC linac [4][37] is the final stage of acceleration before deliveringthe beam to the high energy area. It is composed of eight cryomodules.Figure 1.8: ISAC-II SuperConducting (SC) linac: SCB section cold mass.141.3. Challenges of ISOL production and deliveryEach of the first five cryomodules (identified as SCB, see figure 1.2) housesa cold mass consisting of four superconducting RF cavities and one super-conducting magnetic solenoid in the central position between cavity two andthree as illustrated in figure 1.8. A solenoid, similarly to a quadrupole, is atransverse focusing optical element but with a longitudinal magnetic field.The last three cryomodules (identified as SCC) have cold masses withsix (first two cryomodules) and eight (last cryomodule) cavities. Each ofthem has a superconducting solenoid in the central position as well.The superconducting cavities are bulk niobium quarter wave resonatorsoperating at 4 K. Each cavity has two accelerating gaps separated by adrift tube. The SCB and SCC cavities, represented in figure 1.9, resonaterespectively at a frequency of 106.08 MHz and 141.36 MHz. The ISAC SClinac is capable of accelerating the beam to energy up to 20 MeV/u.SCB low (5.7%)106.08 MHz SCB medium (7.1%)106.08 MHz SCC high (11%)141.44 MHzFigure 1.9: ISAC-II SC linac quarter wave resonators.1.3 Challenges of ISOL production and deliveryRadioactive ion beam production is challenging from several points of view.From a technology point of view, the production chain is composed of manysystems (driver, target, separator, charge breeder and post accelerators) thathave to operate reliably at the same time. The target itself operates at circa151.3. Challenges of ISOL production and delivery2000 ◦C; multiple experiments are expected to run on a single target for aperiod of four to five weeks. A single experiment may run for a two weekperiod when beam is expected to be delivered uninterrupted. Moreover yieldproduction typically peaks at the beginning of the target life cycle and thendegrades, in some cases significantly.From a production point of view, the ISOL method yields low intensity;as an example, the production yield for 11Li is typically in the 104 s−1 range.The number of significant events an experiment can collect are directly pro-portional to the amount of beam on target. Some experiments may alsobe constrained by an intensity threshold below which they cannot run. Itbecomes then critical to minimize the amount of beam loss during transportfrom the source to the experiment. This is true in particular at the selectionstage where a trade-off between resolution and transmission happens.A second limiting factor for an RIB experiment is the presence of con-taminants defined as any other isotope different from the requested. It isexpected that the heavier the isotope (stable or radioactive) to be deliveredthe greater the amount of contamination.A further complication at ISAC arises when the ECRIS is used to in-crease the charge state of masses greater than 30. This type of breederproduces a background of stable species by ionizing residual gasses, vacuumchamber materials and immediate surroundings. The background intensi-ties are orders of magnitude higher than the radioactive species, making theidentification and selection of the desired RIB isotope extremely challenging.This issue has been partially addressed by using the linac chain to filter RIBbeams from the stable background [3].The first stage of selection occurs by exploiting the time of flight sep-aration between the pre-buncher and the RFQ. Since all the radioactiveisotopes are extracted at a fixed voltage, the velocity of any given isotopedepends on its mass. This implies that different masses cover the distancebetween the pre-buncher and the RFQ with different times of flight. Oncethe desired mass is synchronized with the RF, any other mass that falls offthe RFQ acceptance, ∼ 6 ns, is lost during acceleration. Measurements showthat it is possible to achieve a longitudinal selection with a resolving power161.3. Challenges of ISOL production and delivery040080012001600200024003.0 3.2 3.4 3.6 3.8 4.0 4.2 4.4 4.6 4.8 5.0Intensity (a.u.) m/Q 94Mo119Sn69GaFigure 1.10: Charge state distributions of three isotopes identified as: 69Ga,94Mo, 119Sn.of circa 1000.A second stage of selection is achieved by sending the beam through acarbon stripping foil placed downstream of the DTL. This produces a varietyof charge states that further enable particle identification and selection afteran m/Q magnet scan is performed. The goal is to shift the m/Q of thecontaminant outside the acceptance of the downstream SC linac that istuned to the m/Q of the RIB. Various foil thicknesses can be selected,the standard is 195µm (44µg/cm2) thick in order to reach charge stateequilibrium.An example of such scans is represented in figure 1.10. In this case thedesired radioactive beam is 94Rb15+ (m/Q = 6.262) from the charge breeder.A cocktail beam of 94Rb15+ plus three main contaminants survives the firststage of selection: 69Ga11+ (m/Q = 6.266), 94Mo15+ (m/Q = 6.260) and119Sn19+ (m/Q = 6.258).By selecting m/Q = 4.268, mass 94, charge 22, two contaminants, 69Gaand 119Sn, can be removed from the cocktail using a downstream bending171.3. Challenges of ISOL production and deliverysection with a calculated resolving power (this time in terms of mass tocharge ratio, (m/Q)/∆(m/Q)) of circa 320.The two remaining isotopes 94Rb22+ and 94Mo22+ (m/∆m = 4405) cannot be separated, but their ratio can be optimized by a proper selection of thecharge state out of the stripping foil. As shown in figure 1.11, the strippingefficiency ratios between 94Rb and 94Mo at charge state 23+ and 22+ arerespectively 2.5 and 4.5. The charge state choice is in fact a compromisebetween purity and efficiency of the RIB.Clear evidence of filtration is seen at the ∆E−E gas ionization chamber(Bragg detector) installed upstream of the experimental station. Figure 1.12shows the result of applying the second stage of selection. The cocktail hasbeen filtered and optimized to deliver 94Rb22+ radioactive beam, and yetthe final distribution is still dominated by the 94Mo22+ contaminant due toits higher yield.Even though nuclides like 107Ag, 113In or 132Xe are theoretically cut outby the pre-buncher and RFQ filtration, it is still possible that these ions have02468101214161820222416 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33Intensity (a.u.) Q 94Mo94RbFigure 1.11: Theoretical charge state distribution of 94Rb and 94Mo. Thetwo isobars are separated by a relative mass difference of ∆m/m = 1/4405.181.4. The quest for resolving power94Mo/Rb107Ag94Mo/Rb69Ga113In107Ag119Sn132Xe100 200 250 300 350Total E (au)100 200 250 300 350Total E (au)E(au)300250200150100300250200150100E(au)Figure 1.12: Effect of the carbon foil filtration as measured at the Braggdetector: the left picture correspond to the unfiltered cocktail beam fromthe DTL.extended longitudinal distribution tails that are eventually accelerated.In most of the cases the contaminants to RIB ratio can be improvedin favor of radioactive species but the contaminants can not be completelysuppressed. Also the necessary purification of the beam from contaminantshas the side effect of losing part of the produced RIB.1.4 The quest for resolving powerIn general, the heavier the isotope the higher the resolving power neededto produce a pure beam. Figure 1.13 shows the minimum and maximumresolving power needed to achieve isobaric3 separation as a function of theisotope mass number. The graph is produced considering all the knownmasses4 and calculating the resolving power necessary to separate any twoisotopes for a given mass number. As an example for A=12 we have Li, Be,B, C, N, and O; there are fifteen possible combinations of two isobars at a3Isotopic and Isotonic separation only required a maximum resolving power of 3004AME2012 available at http://www.nndc.bnl.gov/masses/191.4. The quest for resolving power1.E+21.E+31.E+41.E+51.E+61.E+71.E+80 20 40 60 80 100ResolvingpowerA - mass numbermin RPmax RP120 140 160 180 200 220 240 260 280 300Figure 1.13: Resolving power range required to separate isobaric isotopesfor any given mass.time. In this case the minimum resolving power is 229 while the maximumis 2820. If the former was the resolving power of the mass separator, thenonly one out of fifteen occurrences could be resolved. If the latter wasthe resolving power of the mass separator then all fifteen occurrences areresolvable.What is then the resolving power required to separate masses up to 238U?It is clear from figure 1.13 that it is unrealistic to think of a mass separatorthat can resolve all cases. For this it would be necessary to have a resolvingpower in the 107 range. The state of the art is the high resolution mass sep-arator of the CAlifornium Rare Ion Breeder Upgrade (CARIBU)[38] projectat Argonne National Laboratory (ANL) where they routinely operate withresolving power of around ten thousand. But a more typical value for thistype of mass separator is less than five thousand. A good compromise, and201.4. The quest for resolving poweryet a challenging goal, is to design a mass separator with a resolving powerof twenty thousands.In order to understand what can be separated by such system, we need tolook at the relative amount of occurrences that can be resolved as a functionof mass for different cases of resolving power, as represented in figure 1.14.The resolving power of three hundred corresponds to the performance ofthe ISAC pre-separator, which is meant to resolve only isotopes; it is clearlyinadequate to resolve isobars with mass number greater than 20. The threethousand resolving power corresponds to the operational performance of theISAC mass separator. In this case the available resolving power can separatemost of the occurrences (> 75%) for light masses (A < 30) but it is a limitingfactor (< 20%) for heavy ions (A > 100).An available resolving power of twenty thousand would be enough to re-0204060801001201400 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300Relative resolution and absolute occrurences A - mass number Occurances (#)RP 300 (%)RP 3000 (%)RP 20000 (%)RP 40000 (%)Figure 1.14: Occurrences resolved for a given resolving power as a functionof the isotope mass number.211.5. The physics case of 132Snsolve more than 50% of the occurrences for masses up to A = 210. Increasingthe resolving power to forty thousand will yield a higher percentage (> 70%for A ≤ 210) but the cost would be at least a factor of four higher sincethe resolving power is, in first approximation, proportional to the bendingradius of the separator dipole magnet, and the surface goes with the squareof the radius.1.5 The physics case of 132SnIn order to validate the choice of twenty thousand resolving power we con-sider the physics case of 132Sn. Figure1.15 shows the isobar of mass 132,including tin (red mark), as a function of the neutron number.The 132Sn isotope falls in the typical range of operation foreseen for theHigh Resolution Separator (HRS), that is around mass 150.131.898131.905131.911131.918131.924131.931131.938131.944131.951131.95765 70 75 80 85 90Mass (u)  N Figure 1.15: Mass 132 isobars as a function of the neutron number: blackmark correspond to the stable nuclei, red mark correspond to 132Sn. Eachdivision in the vertical scale corresponds to a resolving power of 20000.221.5. The physics case of 132SnThe nucleus of 132Sn is particularly important for nuclear structure, aswell as for r-process nucleosynthesis because of its doubly magic (Z = 50,N = 82) nature5. Direct measurements of the binding energy of 132Sn [39]revealed its particularly enhanced stability. Reaction measurements involv-ing 132Sn and the the neighboring 133Sn [40] have brought complementaryinformation on single-particle states confirming the doubly magic nature.It is interesting to point out that the mass measurements required usingmolecular sideband beams at mass 166 (132Sn+34S) because of insufficientresolving power at mass 132. But the use of molecular post-acceleratedbeams is not desirable since it is difficult to control where the moleculesbreak up. Therefore a pure 132Sn beam at mass 132 is more desirable tofurther the study of this key nuclide at future facilities (such as ARIEL).Figure1.16 shows the resolving power necessary to separate 132Sn from0E+01E+42E+43E+44E+45E+46E+47E+48E+49E+41E+565 70 75 80 85 90Resolving power  N Figure 1.16: Resolving power necessary to separate 132Sn from its isobars.Highlighted are: 132Sn (red mark), 132Cs (orange mark), 20000 resolvingpower (dashed blue line).5Private communication with Dr. David Lunney.231.5. The physics case of 132Snits isobars (the resolving power for tin is intentionally set to zero to indicateno need for separation). While some of these isobars may not be produced(depending on the type of source), it is expected that 132Cs will be producedin great quantity.Based on the resolving powers represented in figure1.16, a twenty thou-sand resolving power is adequate to separate the 132Sn beam from 80% ofits isobars.At TRIUMF an experimental proposal6 to study one and two neutrontransfer reactions with neutron-rich Sn beams 132−136 is in the queue since2008 waiting for the HRS capable of delivering a pure beam. The sciencegoals7 for this experiment are to investigate neutron orbitals beyond theN = 82 shell closure in the r-process nucleosynthesis region.Presently there is no facility worldwide capable of delivering such a beam,therefore the HRS of the new ARIEL facility makes possible type of inves-tigation for the first time worldwide.6Letter of Intent LOI-S1187 evaluated high priority by TRIUMF Experiment EvaluationCommittee (EEC).7Private communication with Prof. Ritu Kanungo.24Chapter 2Magnetic Dipole MassSeparatorThe principle of mass separation is based on the fact that particles withdifferent masses have different trajectories when crossing a magnetic field.A particle with charge q traveling through a magnetic field−→B with ve-locity −→v experiences the Lorentz force −→F perpendicular to both the veloc-ity and the magnetic field. In a Cartesian coordinate system (x, y, z), theLorentz force components are:FxFyFz =qvyBz − qvzByqvzBx − qvxBzqvxBy − qvyBx (2.1)If we assume for example that −→v = (vx, 0, 0) and −→B = (0, 0, Bz), thenthe particle is going to perform a circular motion in the (x, y) plane witha bending radius of ρ. The bending radius is related to the mass m of theparticle according to the following:Bρ =mvq(2.2)where B = Bz and v = vx in our example. The product Bρ is referred to asthe magnetic rigidity of the particle.When particles are extracted with the same charge state at a fixed volt-age V from a production target, like for the ISOL targets in ISAC (seesection 1.2.2), the velocity v of the particles depends on the mass m:25Chapter 2. Magnetic Dipole Mass Separatorv =√2 q Vm(2.3)Combining equation 2.2 and 2.3 to eliminate v gives:ρ =1B√2Vqm (2.4)Particles with different masses travel inside the magnetic field with differ-ent radii and, as stated at the beginning of this chapter, this is the principleon which mass separation is based. If these particles enter the magnetic fieldat the same location and direction, as sketched in figure 2.1, they exit thefield with a transverse separation δ equal to:δ = Ddmm(2.5)where D is called the dispersion of the magnetic dipole and mdm is the defi-nition of resolving power.}Figure 2.1: Artistic representation of particle trajectories (blue and red)inside a magnetic field (orange).26Chapter 2. Magnetic Dipole Mass SeparatorA magnetic dipole is a mechanical construction that produces a magneticfield as represented in figure 2.1. The general construction has two steelmagnetic poles facing each other with opposite polarity separated by an airgap known as the pole gap. The steel is used to increase the field in thegap thanks to its high magnetic permeability. The poles are surrounded bycurrent loops (coils) that generate the magnetic field. Particles travel withinthe pole gap and experience a field perpendicular to their velocity.The dispersion is directly proportional to the radius of curvature of thetrajectory:D = ρ (1− cos θ) (2.6)where θ is the bending angle. Notice that for a 90 degree dipole the disper-sion of the magnet is equal to the radius of curvature.Equation 2.5 shows that for a desired resolution and a known dispersion,the maximum transverse size of the beam is δ if we want to separate massm from mass m+ dm.A real beam is an ensemble of particles whose distribution can be rep-resented in the transverse (or longitudinal) phase space as artistically illus-trated in figure 2.2. A reference particle is defined within a beam: this is anideal particle, not necessarily real, that has the momentum which a trans-port system is designed for. Similarly we define the reference trajectory asthe path followed by the reference particle in a transport system.A characteristic quantity of the beam is the emittance; when there areno dissipative forces and no particles are lost or created, then Liouville’stheorem states that the emittance of the beam is conserved8.In the first order (linear) approximation, the emittance is convenientlydescribed in phase space by an ellipse (see Appendix A) as represented infigure 2.2.8see chapter 9 section 1 of [10], or chapter 5 section 3 of [11]27Chapter 2. Magnetic Dipole Mass SeparatorFigure 2.2: Artistic representation of particle distribution in the transversephase space: w and ϕ are respectively the half width and the divergence ofthe beam.In this case the emittance is defined as:ε =Api(2.7)where A is the area of the ellipse.The beam is locally a minimum size if the ellipse is upright, in whichcase ε = wϕ. The separator optics is tuned to create an upright ellipse ata beam-defining slit. Because the emittance is conserved, the smaller thebeam width w the larger the angle ψ and vice versa.The amount of particles inside an emittance ellipse depends on the par-ticle distribution in phase space. A Gaussian distribution can be consideredfor beam dynamics calculations even though a real beam is not Gaussiansince it doesn’t extend to infinity. If the particle distribution is assumedto be Gaussian in both x and x′ then circa 86% of the particles will becontained within 2σ (of the Gaussian distribution) in both axes (see Ap-28Chapter 2. Magnetic Dipole Mass Separatorpendix B). Conventionally the emittance used to quantify a beam is the onecontaining 90% of the particles regardless of the distribution.The separator resolving power is strictly related to the emittance of thebeam. In order to have separation it has to be that:2w ≤ D dmm(2.8)as illustrated in figure 2.3.For an upright ellipse, where ε = wϕ, equation 2.8 can be written as:mdm≤ Dψ2 ε(2.9)The resolving power is inversely proportional to the emittance.More important, for an upright ellipse, the divergence of the beam isknown given the emittance and the maximum beam size, defined by theresolution and the dispersion (see equation 2.8). The divergence determinesthe size of the beam inside the dipole and therefore a lower limit for the sizeof the magnet. As we are going to see in section 2.3, the beam drifts fora certain distance before entering the magnetic dipole; the optics design issuch that the drift is proportional to the radius of curvature ρ. The productof the drift length times the beam divergence gives the width of the beam atthe entrance of the dipole. We have that ρψ = ρ εw = 2ρDεmdm , so the widthof the beam inside the magnetic dipole (in the pole gap) is proportional tothe product of the emittance and the resolving power (ε mdm). This widthdefines the area occupied by the beam, referred in jargon as the good fieldregion (see section 2.2), inside the magnet.The selection process of a particular mass is achieved by interceptingthe beam exiting the separator with a mechanical slit; such beam includesthe mass to be selected as well as all the other masses to be removed. Anexample of such process is artistically represented in figure 2.3 with theideal case of three beams, having the same intensity, being selected by atransverse slit 2w wide centered on the origin. The selection result is that29Chapter 2. Magnetic Dipole Mass SeparatorFigure 2.3: Artistic representation of beam separation (δ = D dmm ) andselection: ideal case of three beams, having the same intensity, being selectedby a transverse slit (transparent orange squares) 2w wide. The Gaussiantails, that overlap with the selected beam (green), are truncated at theentrance of the separator system with a slit.only the central (green) beam is transported beyond the selection location.In order to avoid contamination the Gaussian tails, that overlap with theselected beam, are truncated at the entrance of the separator system witha slit.The size of the beam 2w going through the selection slit is changedaccording to the emittance while maintaining the same beam divergencethrough the separator.302.1. Working hypothesisAccording to equation 2.9, the larger the dispersion D the higher theresolving power achievable. From equation 2.6 we know that the dispersionis proportional to the radius of curvature ρ of the trajectory; in our case theradius of curvature can be at most 1200 mm based on space constraints. Ifour separator was a simple 90 degree (D = ρ) then, based on equation 2.8,the beam width 2w would be 60µm for a 20000 resolving power.Equation 2.9 shows also that the smaller the emittance the higher theresolving power. Notice also that if the beam emittance was zero, and so thewidth of the beam, then the resolving power would be infinite. A reason-ably small emittance to consider is 3µm; in this case with w = 30µm themaximum angle is ψ = 100 mrad. The separator still works for emittanceslarger than 3µm but at a reduced transmission.2.1 Working hypothesisIn the previous section we introduced the concept of how a magnetic dipolegenerates dispersion in order to transversely separate particles of differentmasses at a selection slit.The optics of a separator system is designed to focus the beam at theselection slit so the desired particles of mass m are transported downstreamwhile the contaminant masses m + dm are intercepted by the slit plates.Figure 2.4 is an artistic representation of a 180 degree separator withD = 2ρ;this is just a convenient example that gives the focusing properties needed.The design assumes a constant field (perfectly flat) within degree dipolemagnet; a magnetic dipole with a constant field value B0 between the polesand zero outside is called a hard-edge dipole. Any distortion (aberration)of the intended focus means that particles of mass m are intercepted bythe slit plate while contaminant particles are transported downstream. Thisreduction in performance is a loss of resolving power. The magnetic field isone possible contributor to aberration, in particular because a real magnetdoes not have a perfectly flat field.Based on equation 2.4 we know that, for a 20000 resolving power, avariation of B of 140000 produces the same displacement at the selection slit312.1. Working hypothesisFigure 2.4: Artistic representation of particle separation for a 180 degreeseparator: the dashed lines represent the particles entering the separatorwith the largest angle ψ defining the width of the good field region.as a variation in mass of 120000 . So a variation of this magnitude in the fieldshifts transversely a particle of mass m by the same amount necessary toseparate it from a particle of mass m + dm. Because the curvature is anintegral effect, we are ultimately interested in the field integral variation.From the previous section we also know that the beam occupies a widearea (good field region) within the dipole so we are interested in the integralvariation over the whole area in order to maintain the correct beam focusat the selection slit.In this thesis we work under the hypothesis that minimizing the fieldintegral variation with respect to the system resolution (dmm ) within thegood field region is equivalent to reducing the magnetic field contributionto the system aberration. This means that the system corrections requiredto compensate for magnetic field imperfection are minimized.The objective of the thesis is supporting this hypothesis through themagnetic field study. This study ultimately results in the design of themagnetic dipole for the High Resolution Separator (HRS)9 of the ARIELfacility.9Part of CANadian Rare isotope facility with Electron Beam (CANREB) project fundedby Canada Foundation for Innovation (CFI)322.2. Design requirements2.2 Design requirementsThe final HRS system design has to satisfy specific requirements10; the prin-cipal one is the achievement of a resolving power for the separator systemof 20000 for a 3µm (or mm·mrad) transmitted transverse emittance.The magnetic dipole is the most critical hardware component of theHRS system. Our choice is to design a magnet that minimizes aberrationin order to reduce the corrections necessary to achieve the desire resolvingpower; this is expected to produce a system easy to operate. The goal is todesign a magnet that behave like a hard-edge dipole from an integral pointof view within the good field region.Our unique strategy, adopted to optimize the magnet, needs the formu-lation of the following definitions.We define the reference geometric trajectory, already defined in generalFigure 2.5: Geometric trajectories. The reference trajectory represented inred has a radius of curvature of 1200 mm. The black solid lines representthe field boundaries of the hard-edge magnet: ϕ is the edge angle. The bendangle θ = 90 degree.10ARIEL high resolution spectrometer requirements TRIUMF internal document-74319.332.2. Design requirementsearly in this chapter, as the path followed by the reference particle travelingthrough a hard-edge dipole magnet.The reference geometric trajectory is the red path represented in fig-ure 2.5; this is composed of an arc11 inside the hard-edge boundaries con-nected on the outside to two straight paths tangential to the arc. Figure 2.5shows also that the entrance and exit edges (hard-edge boundaries) have anangle ϕ with respect to a basic magnet where the faces are normal to theincoming and outgoing beam; later in section 2.4 we are going to discuss theentrance and exit edge angle of the reference geometry.We consider also other geometric trajectories composed by an arc con-centric to the reference one that ends at the hard edge boundaries and twostraights parallel to the reference straight paths; these additional geometrictrajectories are not meant to represent particle trajectories, but we choosethem to gauge field quality through the integral calculated along them.We define the field flatness (FF ) as follows:FF =BzB0− 1 (2.10)where B0 is the vertical component (z direction) of the magnetic field at thecenter of the realistic magnet (from OPERA); this center has coordinates(0,1200), see figure 2.7.It is useful to think about the field flatness, and flatness in general, asan error, the smaller the better.We define the integral ratio IRρ for any given geometric trajectory rel-ative to a curvature ρ as follows:IRρ =∫`Bz(s)ds∫arcB0ds(2.11)where Bz(s) is the magnetic field vertical component of the OPERA magnet11The radius of the arc is related to the B0, the mass and the velocity of the referenceparticle assumed to be single charged, as per equation 2.2: ρ = mveB0342.2. Design requirementsalong the considered geometric trajectory of path `. At the denominator theintegral of the hard-edge case is computed only over the arc component ofthe geometric trajectory since the field integrals along the straights are zeroby definition.We lastly define the integral flatness IFρ as follows:IFρ =IRρIR1200− 1 (2.12)where again the index ρ represents a geometric trajectory (ρ = 1200 mmbeing the reference one).The optimized design is achieved by studying the field flatness as a func-tion of selected design parameters and by comparing the field integrals of theOPERA model with respect to the equivalent hard-edge case. Equivalentmeans that the hard-edge case has the same B0 of the OPERA model.The flatness requirements follow from equation 2.4, by calculating thedifferential in B and m we obtain the relationship between the field flatnessand the resolving power:dBB=12dmm(2.13)The first optimization goal is a field flatness in the radial direction at thecenter of the pole of less than 2.5·10−5 inside a region that extends ±160 mmaround the reference geometric trajectory in the middle plane. This regionis defined by optics calculation and it is the area occupied by the beam (fora 3µm emittance); this is referred as the good field region as mentioned inthe opening section.The second optimization goal derive from our working hypothesis and itrequired the integral flatness IFρ to be less than 2.5 · 10−5 within the goodfield region. The integral flatness is a more stringent requirement than thefield flatness because the overall curvature of the beam is an integral effectof the magnetic field.The third optimization goal is related to the position of the effective352.3. Magnetic dipole modelfield edge with respect to the hard-edge case. In a real dipole the magneticfield has a soft transition from B0 to zero (as opposed to the hard-edge);the effective edge is then defined as the integral of the field along somedefined paths divided by B0. In our case the defined paths are the geometrictrajectories.Because we want to design a magnet that reduces the amount of cor-rections necessary to match the beam dynamics designed optics, we requestthat the effective field edge location matches the hard-edge. This guaran-tees that the entrance and exit edge angles are correct. Based on practicalconsiderations, we specify an upper limit for the |IRρ − 1| being less than1 · 10−3. This specific requirement translates into an effective field edge po-sition within 1 mm with respect to the hard edge case for a 2 m path length.It is possible to have the correct integral flatness and edge angle only if theeffective edge match the hard-edge.A final requirement is that the steel of the magnet doesn’t reach sat-uration. This is a soft requirement and mostly driven by experience. Areasonable figure for field saturation is not to exceed 1.5 T. A better re-quirement related to field saturation is in fact to specify that the magnetbehaves sufficiently linearly within the range of operation. The magnet mustbe able to bend beam rigidities (see equation 2.2) between 0.117 T m and0.544 T m, corresponding respectively to 11Li1+ and 238U1+ at 60 keV, andto 0.097 T and 0.453 T magnetic field for the hard edge case and the chosenρ. The linearity requirement means that the magnet has to satisfy the threeprevious requirements at these two extreme cases of operation.2.3 Magnetic dipole modelIn the opening section of this chapter we briefly introduced the concept ofmass separation considering the simplified case of a stand alone magneticdipole. In practice, the simplest possible layout consists of drift (absenceof optical elements) spaces upstream and downstream of a dipole. In thiscase the dispersion of the system, rather than the magnet, is a result of theoverall optical layout. Such a system is usually mechanically defined by the362.3. Magnetic dipole modelelectrostaticquadrupolecorrectorelectrostaticmultipolemagneticdipole 1magneticdipole 2electrostaticquadrupolecorrectorentrancede-magnifyingsectionexitmagnifyingsectionpure separatorobject slitpure separatorimage slitseparatormagnifiedobject slitseparatormagnifiedimage slitFigure 2.6: Schematic layout of the High Resolution Separator (HRS)entrance (of the upstream drift) object slit and the exit (of the downstreamdrift) image, or selection, slit.The final optical layout of the HRS system is represented in figure 2.6;it includes two identical 90 degree magnetic dipoles with a 1200 mm radiusof curvature. Two magnets produced twice the dispersion of a single one.This layout is close to a single 180 degree dipole as seen in section 2.1, but itallows for the cancellation of some high order aberrations; a single 180 degreedipole would also be much more difficult to machine and handle.Different layouts were initially evaluated based on first order beam dy-namics calculations12. The final layout has been developed[1] accounting forhigher order effects using the COSY INFINITY[41] (COSY-∞) code. Theoptical design occurred in parallel with the magnet design. Updates fromthe optics were incorporated in the magnet design as they were issued. The12M. Marchetto, “ARIEL High Resolution Spectrometer First Order Calculations”,TRIUMF internal design note TRI-DN-13-07, document-74265.372.3. Magnetic dipole modelfinal realistic magnetic field is a result of this thesis work. This field wasultimately used in COSY-∞ to validate both the magnet and the opticaldesign.The magnetic dipole design has been developed with the 3D-modeller R©of the OPERA-3D R© software. The field calculation of the developed magnethas been calculated with the TOSCA R© solver part of the same software.The magnet steel, coils and surrounding air volume coordinates areparametrized as functions of the characteristic dimensions of the magnet:reference radius of curvature, angle of entrance and exit face, pole gap, etc.The parametrization reduces the design time and it allows for better trackingof the design parameters to be optimized.Figure 2.7 shows an example of the steel coordinates. The parameterizedcoordinates are coded in a geometry generator file (OPERA COMI file).Figure 2.8 shows a rendering13 of one of the basic dipole configurations; werefer to this as the reference geometry (HRS-120-12Cq2) generated with theCOMI file.a a a b b b c c c cc cc cc m m mm mm n n n nn nn nn o o p p p q q q RR RR RR Split point Split point Split point 0200400600800100012001400160018002000-1400 -1000 -600 -200 200 600 1000 1400y (mm) x (mm) Figure 2.7: Example of parameterized coordinates of the HRS dipole13All the magnet renderings presented in this thesis are an output of OPERA-3DR© soft-ware. The OPERA logo and information frame will be removed from subsequent ren-derings only for presentation reasons.382.3. Magnetic dipole modelFigure 2.8: Rendering of the reference geometry (HRS-120-12Cq2): halfmagnet is represented but only one quarter is simulated.The steel and coils geometry is immersed in an air background whichdimension are at least a factor of five larger than the magnet geometricaldimensions. At the boundaries of this background (far-field boundaries) thefield is null with respect to the field in and around the magnet.By taking advantage of the magnet symmetries, we simulate only onequarter of the full magnet and air background. The xy (z = 0) symmetryplane has a normal magnetic boundary condition while the yz (x = 0) planehas a tangential magnetic boundary condition.We impose total potential everywhere but in the dedicated air volumecontaining the coils where the reduced potential is required.The standard HRS simulations use a tetrahedral element mesh limited insize, within the pole gap (the region of interest), to: 5 mm3 for the volume,5 mm2 for the surface, 5 mm for the edge and 1 mm for the vertices. The steeltetrahedrons are limited to: 10 mm3 for the volume, 10 mm2 for the surface,5 mm for the edge and 1 mm for the vertices. The air background is subdi-vided in various regions where the elements size is progressively increasedfrom the steel to the outer boundaries to: 200 mm3 for the volume, 200 mm2for the surface, 200 mm for the edge and 200 mm for the vertices. OPERAstorage levels are set to prioritize the mesh size to be used at the boundariesbetween two geometrical elements (air gap and steel for example).392.3. Magnetic dipole modelTable 2.1: Simulation statistics for coarse and fine mesh of an HRS geometry.Coarse mesh Fine mesh(simulation HRS23C56) (simulation HRS23C57)Number of:nodes 1.5 M 18.8 Medges 1.9 M 83.7 Mmesh elements 1.5 M 70.4 MMax element size:air gap 6.25 mm3 5 mm3steel 67 mm3 10 mm3background 1333 mm3 200 mm3Iterations 8 12Running time 15 min 10 h 45 minSolution file size 460 MB 12,860 MBA parameter coded in the COMI file allows for scaling the mesh dimen-sions. This feature is used to test the solution convergence; since the typicalsolution (fine mesh) is running at the limit of the available hardware, theconvergence test is done by increasing the mesh size (coarse mesh) on theselected geometry. Some simulation statistics for a late geometry with fineand coarse mesh are listed in table2.1; the solution for the two cases arewithin the same order of magnitude.OPERA solve the Ampere’s circuital law (Maxwell equation) using thenon-linear Newton-Raphson iteration method; the maximum number of it-erations is set to twenty-one.The calculations use a typical steel C1006 BH curve, represented in fig-ure 2.9, for the poles and return yokes. Steel C1006 is the typical softmagnetic steel with low carbon content, less than 0.08%, used in magnetproduction. A complete chemical analysis of the C1006 used to manufac-ture the HRS dipole magnet is reported in table 2.2.In order to probe the final design, the relative geometry is simulatedwith a C1010 property steel (see figure 2.9) as well as with constant relative402.3. Magnetic dipole modelpermeability value of 500 and 2000. The final geometry is also simulatedwith a mix of steels, namely C1010 for the poles and C1006 for the rest ofthe magnet. Simulation results that used different permeability propertiesare consistent with the final geometry solution using C1006 steel.Table 2.2: Chemical composition of the C1006 steel used to manufacturethe HRS dipole magnets.ElementWeightElementWeightElementWeight% % %Fe 99.151 Cu 0.07 V 0.001C 0.06 Si 0.07 Al 0.031Mn 0.39 Ni 0.04 Cb (Nb) 0.001P 0.04 Cr 0.04 N 0.007S 0.02 Mo 0.02 O 0.00400.20.40.60.811.21.41.61.822.22.42.60 100 200B (T) & µr (X1000) H (A/m) 00.20.40.60.811.21.41.61.822.22.42.60 2000 4000 6000 8000 10000B (T) & µr (X1000) H (A/m) C1006 BH curveC1010 BH curveC1006 relative permeabilityFigure 2.9: BH curves for C1006 and C1010 magnetic steel and relativepermeability (see equation 2.15) for the C1006 steel. The narrow side graphis a magnification of the main one around the operational range (orangedotted box).412.4. Reference geometry2.4 Reference geometryThe reference geometry (see figure 2.8) is the starting point of the opti-mization; its dimensions are derived from main parameters like radius ofcurvature (1200 mm), bending angle (90 degree), pole gap (70 mm) and edgeangle (27 degree) used in the beam dynamics calculations and by applyingsome rules of thumb14. The initial coil section of 69× 69 mm2 has a currentdensity of around 2.6 A·mm−2 and it generates a field of about 0.45 T; thefield calculations are done for the maximum rigidity (0.544 T m).The vertical component of the magnetic field for the reference geometryis represented in figure 2.10 as a function of the angle where zero correspondsto symmetry axis of the magnet (x = 0 in figure 2.7). The field fall off issignificantly different from the hard edge case where the field goes to zerowith a discontinuity. Close views of the top and the fringe field profile are-0.0500.050.10.150.20.250.30.350.40.450.5-80 -70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70 80B z (T) Angle (deg) 1000 mm1050 mm1100 mm1150 mm1200 mm1250 mm1300 mm1350 mm1400 mmFigure 2.10: Reference design Bz vertical magnetic field component: the redline represents the reference geometric trajectory. The hard edge case wouldbe a constant field at the maximum value, dropping to zero at 45 degree.14private communication with P.Eng. George S. Clark, TRIUMF magnet engineer.422.4. Reference geometryrepresented respectively in figure 2.11 and figure 2.12.0.43840.43850.43860.43870.43880.43890.4390.43910.43920.43930.4394-50 -40 -30 -20 -10 0 10 20 30 40 50B z (T) Angle (deg) 1000 mm1050 mm1100 mm1150 mm1200 mm1250 mm1300 mm1350 mm1400 mmFigure 2.11: Reference design Bz vertical magnetic field component magni-fied near the peak. The field is symmetric with respect to 0 degree.-5E-3-4E-3-3E-3-2E-3-1E-3      01E-32E-33E-34E-35E-3-90 -85 -80 -75 -70 -65 -60 -55 -50 -45 -40 -35B z (T) Angle (deg) 1000 mm1050 mm1100 mm1150 mm1200 mm1250 mm1300 mm1350 mm1400 mmFigure 2.12: Reference design Bz vertical magnetic field component magni-fied around the fringe field (drop off).432.4. Reference geometry-2.5E-3-2.0E-3-1.5E-3-1.0E-3-5.0E-4          05.0E-41.0E-31.5E-32.0E-32.5E-3800 900 1000 1100 1200 1300 1400 1500 1600B z/B0-1 Radial position (mm) Figure 2.13: Reference design field flatness: the red portion of the curverepresents the flatness within the good field region (±160 mm around thereference geometric trajectory ρ = 1200 mm).The field flatness in the radial direction along the magnet symmetry axisis represented in figure 2.13. The red section of the curve corresponds to theportion within the intended good field region. In this case the field flatnesspresents a strong asymmetry, circa 2 · 10−3; there is a larger magnetic fieldon the outer part of the pole. This field flatness doesn’t meet the 2 · 10−5requirement.The asymmetry can also be seen by looking at the magnetic field distri-bution of the steel, represented in figure 2.14, where the inner return yokepresents a higher level of saturation with respect to the outer one.The return yoke refers to the part of the steel where the magnetic flux isforced to loop around going from the magnetic south to the magnetic northpole, where the orientation depends on the electric current direction flowingthrough the coils. The magnetic field strength follows the equation:B = µH (2.14)442.4. Reference geometry1.51.41.21.00.80.60.40.20Surface contour: B (T)Figure 2.14: Reference design magnetic flux density in the steel (half mag-net). The external return yoke surface (yellow) is larger than the internal(orange).where µ is the permeability of the material (air, steel, etc.) and H is calledthe magnetizing field. In the SI system B is measured in T (teslas), µ inH·m−1 (henrys per inverse meter) and H in A·m−1. The permeability offree space, air for our purpose, is indicated with µ0 and it has a value of4pi · 10−7 H·m−1. More often the permeability is indicated relative to air;equation 2.14 becomes then:B = µrµ0H (2.15)where µr =µµ0is the relative permeability.Incidentally, in the Centimetre-Gram-Second system of units (CGS),where B is measured in G and H in Oe (1 Oe = 1000/4piA·m−1) equa-tion 2.15 is simply:B = µrH (2.16)452.4. Reference geometryAs seen in figure 2.9, the permeability is not a constant and it decreasesfor high magnetization field due to saturation.Consider now the magnetic flux ΦB = B ·S where S indicates the cross-sectional area where the flux is calculated. The reference design (see fig-ure 2.14) has the outer yoke surface larger than the inner one. As the coilcurrent is ramped up (increasing H) the magnetic flux will tend to flowequally through both return yokes, but since the inner one has a smallersurface, the magnetic field is going to be larger with respect to the outerone. Larger magnetic field corresponds to lower permeability.The flux through a return yoke can be written as:ΦB = µrHS (2.17)The outer return yoke has at this point larger surface and higher perme-ability, therefore the magnetic field tends to move toward the outer returnyoke.This behavior has some analogy in Ohm’s law for a conductor, whereH is the voltage, ΦB the current and µ−1r the resistance. If we have tworesistors (µ−1r1 and µ−1r2 ) connected in parallel to a common voltage (H),the current (ΦB) is higher through the lower resistance (higher µr). Themagnetic resistance is called reluctance; the reluctance of a uniform magneticcircuit is described by the following equation:R = lµ0µrS(2.18)where l is the length of the magnetic circuit.The integral ratios for the reference geometry are calculated along thegeometric trajectories represented in figure 2.5. These nine trajectories arespaced 50 mm apart from an inner bending radius of 1000 mm to an outerof 1400 mm, 1200 mm being the reference radius of curvature as per thebeam dynamics. The straight paths extend 1628 mm beyond the hard edgeboundaries; this is to allow for the field to decay low enough so it doesn’t462.4. Reference geometryTable 2.3: Magnetic field integrals for the reference geometry.Geometric trajectory Hard edge Reference geometryIRρρ (mm)∫arcB0ds (T mm)∫`Bz(s)ds (T mm)1000 781.00 800.92 1.02551050 791.83 811.56 1.02491100 803.13 822.81 1.02451150 814.10 834.53 1.02421200 826.77 846.44 1.02381250 838.99 858.92 1.02381300 851.40 871.64 1.02381350 863.98 884.34 1.02361400 876.70 897.70 1.0240contribute significantly to the integral. The results on the magnetic fieldintegrals for the reference geometry are listed in table 2.3. The calculatedfield integral along the reference trajectory is circa 0.846 T m for a field atthe center of the pole, B0, equal to 0.439 T m; for reference,238U1+ requiresa integral field along the reference trajectory of 0.854 T m.The integral ratios give an indication of how the effective edge compareswith the hard edge case. This is easier to visualize if we calculate the amountof effective path length that extends beyond the hard edge case as follows:∆` = (IRρ − 1)arc2(2.19)where the total amount has been divided by two because the effective lengthextend symmetrically on both side of the magnet. The result of such acalculation is shown in figure 2.15; the two extreme cases, 1000 and 1400(light color on the graph) are outside the good field region.The effective field edge is circa 22 mm beyond (outward) the locationof the hard edge case and it presents curvature with respect to the hardedge case (no curvature). The effective edge has also the wrong angle withrespect to the hard edge design. The latter has in fact by design a 27 degreesentrance and exit angle, defined earlier in section 2.2 and shown in figure 2.5;472.4. Reference geometry20.020.521.021.522.022.523.023.524.024.525.0950 1000 1050 1100 1150 1200 1250 1300 1350 1400 1450∆l  (mm) Radial position (mm) Figure 2.15: Reference design effective field edge location with respect tothe relative hard edge case. The light blue points are outside the good fieldregion. If the effective edge aligned with the hard edge case the yellow dottedline (linear interpolation of the dark blue data) would be the constant y = 0.such angle is a beam dynamic feature that provides vertical focusing to thebeam. If the effective angle was correct, the yellow dashed line in figure 2.15,a linear interpolation of the good field region points, would be parallel tothe horizontal axis. In this case the angle error is 0.15 degree (or 2.7 mrad).Lastly the integral flatness of the reference model is calculated for thesame reference trajectories used for the integral ratios. The results areplotted in figure 2.16. The integral flatness confirms the initial outcome ofthe field flatness with the magnet not meeting the requirement representedby the green box (2.5 · 10−5 high) in figure 2.16; the requirement is violatedby a factor of 52.It is interesting to notice that the inner trajectory integrals are higherthan the outer ones even though the opposite occurs when we look at thefield flatness (figure 2.13). However the field flatness is just a local snapshotat the center of the magnet while the integral represents a more global view.482.4. Reference geometryIn order to explain the apparent contradiction we have to look at the fieldof the edge of the magnet in figure 2.11; the magnetic field increases at theedges of the magnet for the inner trajectories rather than dropping off as forthe outer ones. This is due to the unbalance of the field distribution overthe whole pole surface and return yokes as can be seen in figure 2.14; thisissue is going to be addressed later in section 3.4 and section 4.3.-2.0E-3-1.5E-3-1.0E-3-5.0E-4          05.0E-41.0E-31.5E-32.0E-31000 1050 1100 1150 1200 1250 1300 1350 1400Integral flatness (relative) Radial position (mm) Figure 2.16: Reference design integral flatness with respect to the relativehard edge case. The light blue sections of the graph are outside the goodfield region. If the integral flatness matched exactly the hard edge casethe points would lay on the x-axis (y=0). The green box (2.5 · 10−5 high)represents the flatness requirement in the good field region.49Chapter 3Field Study on the ReferenceGeometryBased on the reference geometry initial results presented of the previouschapter, it is important to understand how the steel geometry shapes themagnetic field. This is going to result in producing the optimum geometrythat satisfies our requirements. In this chapter we are going to optimize thereference geometry with straight edges while the final curved edge geometryis going to be discussed in the next chapter.3.1 Pole gap optimizationThe first parameter of the magnet to be optimized is the separation betweenthe two magnetic surfaces, the pole gap, where the beam goes through. Thebending angle and the radius of curvature of the magnet are defined by thebeam dynamics and they are considered given parameters for the magnetdesign. The beam dynamics doesn’t specify the pole gap, but it does seta lower limit based on the vertical envelope of the beam; this envelope iscirca 10 mm for a vertical emittance of 6µm (design value15). A 40 mmvertical clearance for the beam would allow for a 100µm emittance to betransported. The maximum emittance that can be transported is calledacceptance. Another constraint to take in consideration for the pole gap isthe wall thickness of the vacuum chamber.The initial gap dimension of 70 mm takes into account both the verticalbeam envelope and the vacuum chamber dimension (22 mm, top and bottom15Horizontal and vertical emitances can be different.503.1. Pole gap optimizationwall combined). This initial dimension also includes some engineering safetyfactor that allows some spatial contingency for machining error or beammisalignment.There is a simple relationship between the gap, the magnetic field in thegap and the current that generates the magnetic field. This relationship canbe derived using Maxwell’s equation for magnetic field and static electricfield (Ampere’s circuital law - integral form):∮∂Σ−→H · d−→l =∫Σ−→j · d−→S (3.1)By integrating along the circuit (red loop) represented in figure 3.1(top)5.8E+55.0E+53.0E+502.0E+51.0E+54.0E+51.71.61.41.21.00.80.60.400.2B fieldH fieldgMap contour: H (A/m)Map contour: B (T)Figure 3.1: Cross section of the reference geometry in the middle of themagnet; the green contour represents the shape of the steel. Magnetic fieldB (top) and magnetizing field H (bottom) in the reference geometry. Thered loop is the integration circuit ∂Σ of equation 3.1.513.1. Pole gap optimizationwe have that the magnetizing field H can be split into the one inside thegap and the one inside the steel:Hgap g +Hsteel l = I (3.2)where I is the total current flowing trough the magnet coils and g + l isthe total length of the loop. Equation 3.2 can be written in terms of themagnetic field and the permeability of the material:Bgapµ0g +Bgapµl = I (3.3)where we approximate B as the same magnitude in the gap as in the steelwhile µ µ0 for non saturated steel (µr = µµ0  1 as seen in figure 2.9) andtherefore the second term of equation 3.3 can be assumed zero; later in thissection we will address the finite permeability case. We can arrive at thesame conclusion by just looking at the H component in figure 3.1 (bottom).From equation 3.3 we then have that:Bgap g = Iµ0 (3.4)that means reducing the gap allows to run a lower current through the coilsfor the same magnetic field as well as to reduce the steel volume, withoutreaching saturation, leading to an overall less expensive design.If the dipole had an infinite wide pole, and infinite permeability (as wewill see later), the field would be constant everywhere similarly to an infiniteparallel plate capacitor. In this non realistic case the field flatness wouldbe zero (perfect field flatness) like for the hard-edge dipole. In a real dipolewith finite pole width the magnetic field varies as we move from the centerto the pole edges (as in a real capacitor).For an H-frame dipole (like our case) it is expected that reducing thegap [42] increases the normalized pole overhang. The latter is defined as the523.1. Pole gap optimizationextension of the pole width a beyond the good field region normalized tothe pole gap, necessary to reach a certain field flatness within the good fieldregion itself.The approximate dependency between the field flatness in the good fieldregion and the normalized pole overhang ag for an unoptimized pole16 isdescribed by the following empirical equation:∆BB=1100e−2.77( 2ag−0.75)(3.5)where ∆BB is the relative field error (field flatness). Figure 3.2 representsthe dependency described in equation 3.5. As the gap g is reduced thenormalized pole overhang 2ag increases producing a better (smaller) flatness.1E-81E-71E-61E-51E-41E-31E-21E-10 1 2 3 4 5∆B/B 2a/g Figure 3.2: Field flatness as a function of pole overhang.As represented in figure 3.3, our calculations show though that this isnot the case for our magnet . A 50 mm gap required 26% less current butit produces a 40% worse (bigger) flatness. On the other end an 80 mm gaprequired 15% higher current but it results in a 12% better (smaller) flatness.16see section “The “H” Dipole Geometry” chapter “Pole Tip Design” in “Iron DominatedElectromagnets Design, Fabrication, Assembly and Measurements” [42]533.1. Pole gap optimization-4.E-3-3.E-3-2.E-3-1.E-3           01.E-32.E-33.E-34.E-3850 950 1050 1150 1250 1350 1450 1550B z/B0-1 Radial position (mm) 50 mm60 mm70 mm (reference)80 mmFigure 3.3: Field flatness dependency on the pole gap; the solid line repre-sents the flatness within the limit of the geometric trajectories.This result, in contrast with the normalized pole hangover approxima-tion, is made evident by the magnet operating far away from saturation.As seen in section 2.2, this operational mode is necessary in order to havea linear behavior of the magnetic field for a large range of excitation. Thefield flatness dependency on the gap at different excitation levels has beenstudied for a simpler geometry: a “cube” magnet represented in figure 3.4.Figure 3.4: Rendering (half magnet) of the “cube” geometry.543.1. Pole gap optimizationFigure 3.5 shows the magnetic field distribution of the cross section inthe middle of the magnet (70 mm gap) for a low excitation mode, wherethe magnet doesn’t reach saturation in the steel, and high excitation mode,where saturation occurs.2.752.52.001.51.00.51.471.41.21.00.80.60.40.20B field low excitationB field high excitationMap contour: B (T)Map contour: B (T)Figure 3.5: Cross section of the “cube” geometry in the middle of the magnet(70 mm gap); the green contour represents the shape of the steel. Fieldflatness of the “cube” geometry.We apply equation 3.3 to two different loops as shown in figure 3.5: thelarge loop, loop-1 (red solid line), goes through the gap at the center of themagnet while the small one, loop-2 (blue dotted line), goes through the gapat 200 mm from the center of the magnet. For loop-1 we have:B1gapµ0g = I − B1gapµ1(l + ∆l) (3.6)where ∆l is the path length difference with respect to loop-2 and µ1 is the553.1. Pole gap optimizationaverage permeability of the steel along loop-1.Similarly for loop-2 we have:B2gapµ0g = I − B1gapµ2l (3.7)where we made the approximation that the field in the steel is uniform andequal to B1gap.Subtracting equation 3.7 from 3.6 we have:∆BB=B2gap −B1gapB1gap=1g(µ2r(l + ∆l)− µ1rlµ1rµ2r)(3.8)where we introduce the average relative permeability µ1r and µ2r .r at low excitationr at high excitation2.43E32.0E31.5E301.0E35.0E22.43E32.0E31.5E31.0E35.0E20Map contour: B/H/4 ·10^(-7) Map contour: B/H/4 ·10^(-7)Figure 3.6: Relative permeability for the low (non-saturated - top) and thehigh (saturated - bottom) excitation case.563.1. Pole gap optimizationThe relative permeability in the steel for the low (non-saturated) andhigh (saturated) excitation mode is represented in figure 3.6.If we assume the average relative permeability for the two loops to bethe same, a good approximation for the low excitation (non-saturated) case,then the numerator of equation 3.6 is positive with:µ1rµ2r<l + ∆ll(3.9)l+∆ll ∼ 1.2 for the two loops we picked, and equation 3.8 simplifies into:∆BB=∆lgµ1r(3.10)for our case ∆l = 400 mm, g = 70 mm and µ1r = 2200 we have∆BB =2.6 · 10−3; the field at the 200 mm crossing is higher than the field at the-8E-3-6E-3-4E-3-2E-3      02E-34E-30 50 100 150 200 250 300 350B z/B0-1 Radial position (mm) 50 mm low70 mm low80 mm low50 mm high70 mm high80 mm highFigure 3.7: Field flatness of the “cube” geometry dependency on the polegap and saturation level: solid and dashed lines represent respectively thelow excitation (non-saturated) and high excitation (saturated) case.573.1. Pole gap optimizationcenter of the magnet.This is in line with the field flatness results obtained for the cube geom-etry, represented in figure 3.7: here we see that for the 70 mm gap, the fieldat 200 mm is 1 ·10−3 higher with respect to the center of the magnet; in factif we picked a smaller loop that crosses the center and goes through a pathwith higher µr, as the magnetic flux does (lower reluctance), the two fieldflatness values would be closer.If the relative permeability was infinite than we would have a perfectlyflat field, but since it is finite, the flatness depends on how the permeabilityrelated to the gap aspect ratio ∆lg .In the high excitation case, a much less uniform permeability leads toa field distribution in the steel that counteracts the effect described in thelow excitation case. If we have that µ1r is greater that µ2r , as for the highexcitation (saturated) case (see figure 3.6), and that:µ1rµ2r>l + ∆ll(3.11)then the numerator of equation 3.6 is negative. In this case the field at the200 mm crossing is lower with respect to the field at the center as shown infigure 3.7.Magnets used to transport single species at a fixed energy like protons,electrons or heavy ions used as driver beam in the in-flight facility, don’tneed to behave linearly and therefore are often designed to operate closerto saturation with a minimum amount of steel (minimum cost). Beam linesdesigned to transport a variety of heavy ions have to employ magnets thatbehave linearly at the expense of under utilizing the available steel; in ourcase there are pockets of steel with almost no magnetic flux (see figure 3.1).Figure 3.8 shows the reference geometry (70 mm gap) field distribution inthe steel levels for low (selected operational mode) and high excitation.The initial gap dimension of 70 mm is the selected value providing a goodcompromise between current and flatness.583.2. Flatness versus pole height2.52.01.51.00.50Surface contour: B (T)Figure 3.8: Reference geometry magnetic flux density in the steel for low(selected operational mode - top) and high excitation (bottom).3.2 Flatness versus pole heightIn order to study how the flatness changes with the pole height, we considerthe pole composed of two sections as illustrated in figure 3.9: the pole andthe pole base. The boundary of the two regions is the plane that containsthe base of the coil channel.Pole basePoleInner coil channel Outer coil channelFigure 3.9: Reference geometry middle section.593.2. Flatness versus pole heightA first set of simulations is done by fixing the pole base height at229.5 mm, as for the reference geometry, and varying the pole height. Thecoil elevation with respect to the horizontal middle plane is also fixed. Thepole height is changed from 90 mm (reference) to 180 mm in steps of 30 mm.The results of this investigation are displayed in figure 3.10. The thickerthe pole the better the value for the flatness. The flatness difference betweenthe 90 mm and the 180 mm cases is 9 · 10−4; this value drops to 3 · 10−4 inthe good field region.The field distribution in the steel for the two extreme pole height cases isrepresented in figure 3.11. A thicker pole allows for a more uniform magneticfield distribution over the pole surface at interface with the air gap.A second set of simulations is done by fixing both the pole height at180 mm and the unchanged pole base, and varying the vertical position ofthe coil by moving it toward the base of the coil channel. The coil is moveddown in two steps each of 30 mm. The result is shown in figure 3.12.-2.5E-3-2.0E-3-1.5E-3-1.0E-3-5.0E-4           05.0E-41.0E-31.5E-32.0E-32.5E-3850 950 1050 1150 1250 1350 1450 1550B z/B0-1 Radial position (mm) 90 mm (reference)120 mm150 mm180 mmFigure 3.10: Flatness dependency on the pole height.603.2. Flatness versus pole heightSurface contour: B (T)1.51.20.80.60.201.41.00.4Figure 3.11: Magnetic flux density in the steel for the two extreme cases ofpole height: 90 mm (reference design - top) and 180 mm (bottom).-2.0E-3-1.5E-3-1.0E-3-5.0E-4           05.0E-41.0E-31.5E-32.0E-3850 950 1050 1150 1250 1350 1450 1550B z/B0-1 Radial position (mm)  Pole height 180 mm:coil not moved coil moved -30 mmcoil moved -60 mmFigure 3.12: Flatness dependency on the coil vertical position. The casewith pole height at 180 mm and coil not moved is the same as in figure3.10.613.2. Flatness versus pole heightThe flatness difference between the two extreme positions of the coil isalmost one order of magnitude smaller with respect to the first set (3 · 10−5within the good field region).A third set of simulations is done with the pole height and the elevationof the coil fixed and varying the height of the pole base ±60 mm aroundthe starting value of 229.5 mm. Increasing pole base thickness improves theflatness as shown in figure 3.13.The field distribution in the steel for the two extreme pole base thick-nesses is represented in figure 3.14. These can be compared with the fielddistribution in figure 3.11 (bottom picture).-2.0E-3-1.5E-3-1.0E-3-5.0E-4           05.0E-41.0E-31.5E-32.0E-3850 950 1050 1150 1250 1350 1450 1550B z/B0-1 Radial position (mm) Pole height 180 mm coil moved -60 mm:base height decreased 60 mmbase height unchangedbase height increased 60mmFigure 3.13: Flatness dependency on the pole base height. The case withpole height at 180 mm, coil moved −60 mm and base height unchanged isthe same as in figure3.12.The design of a thicker pole improves the field flatness but it results ina heavier and more expensive magnet due to the increase in steel volume.The pole thickness is then going to be a compromise between field qualityand magnet cost.Even though the location of the coil gives marginal improvement of the623.3. Flatness versus pole widthSurface contour: B (T)1.51.20.80.60.201.41.00.4Figure 3.14: Magnetic flux density in the steel for the two extreme cases ofpole base height: 169.5 mm (top) and 289.5 mm (bottom).flatness, it is mechanically practical to sit the coil at the base of its channel.Another consideration is that the coil section imposes a lower limit on thethickness of the pole since the coil channel (see figure 3.9) has to be deepenough to accommodate the coil itself and the latter can not interfere withthe vacuum chamber located in the pole gap.3.3 Flatness versus pole widthAt the edges of the pole the magnetic field drops (edge effect); based on theresults we have obtained so far, it is expected that a field drop at the edgesimproves the field flatness up to the point where the edge effect is so strongthat the flatness starts to worsen (∆B increases).The dependency of the field flatness on the pole width is displayed infigure 3.15. Starting from a reference 826 mm, the pole width is increased upto 926 mm and down to 526 mm in steps of 100 mm. The edge effect starts633.3. Flatness versus pole width-3.5E-3-3.0E-3-2.5E-3-2.0E-3-1.5E-3-1.0E-3-5.0E-4           05.0E-41.0E-31.5E-32.0E-32.5E-33.0E-33.5E-3850 950 1050 1150 1250 1350 1450 1550B z/B0-1 Radial position (mm) 926 mm826 mm (reference)726 mm626 mm526 mmFigure 3.15: Flatness dependency on the pole width.compromising the field flatness at the lowest width value simulated, wherea local maximum can be seen at one extreme of the good field region.The field distribution in the steel for the two extreme pole widths isrepresented in figure 3.16. The 526 mm pole width option has a more uniformsaturation; this is due to the fact that the return yoke inner and outersurfaces combined are almost equal to the pole surface; this issue is discussedin detail in the section 3.4.The best result in terms of flatness is the 526 mm case; this is consis-tent with the findings in section 3.1. Considering equation 3.10, we arereducing ∆l by forcing the magnetic flux into a smaller region (or similarlyconstraining loop-1 and loop-2 to have closer paths).In the 526 mm case though, the good field region is on the edge of thefield fall off. A higher value of the pole width, like 626 mm, is preferable; thisis also in line with the beam dynamics study[1] that looks at the correlationbetween pole width and high order aberration.643.4. Magnetic flux balanceSurface contour: B (T)1.51.20.80.60.201.41.00.4Figure 3.16: Magnetic flux density in the steel for the two extreme cases ofpole width: 926 mm (top) and 526 mm (bottom).3.4 Magnetic flux balanceThe radial symmetry of the reference design (same width of the return yokes)results in a higher magnetic flux through the inner yoke (smaller surface)with respect to the outer one, as represented in figure 2.14 This asymmetry inthe flux distribution causes the field for the outer trajectories (1250 to 1400)to be higher than for the inner ones (1150 to 1000), as seen in figure 2.11.The asymmetry reflects on the field flatness (see figure 2.13). The magneticflux density through each of the return yokes has to be equalized in orderto achieve a balanced field flatness. We propose two different methods ofequalization: in the first method the balance is achieved by controlling themagnetic flux on the inner and outer return yokes alone, while in the secondmethod the balance is achieved by adjusting the flux in the inner and outerreturn yokes in relation to the pole surface.In the first method the equalization is studied by looking at both the653.4. Magnetic flux balanceInner return yokevertical surface edgeInner return yokehorizontal surfaceOuter return yokevertical surface edgeOuter return yokehorizontal surfaceBBFigure 3.17: Rendering of the reference surface with the first equalizationmethod applied; the blue circle indicates the air channel carved in the outerreturn yoke steel in order to achieve balanced flux.horizontal and vertical surface areas of the return yoke as indicated in fig-ure 3.17.Three configurations are considered for the first method. The first con-figuration has the same horizontal surface area for both the inner and outerreturn yoke. The second configuration has the return yoke horizontal surfacesame as the vertical one but different between inner and outer return yoke.The third configuration has all the surfaces (horizontal and vertical) equalfor both return yokes. This is accomplished by carving the steel creating anair channel along the return outer return yoke (see blue circle in figure 3.17).The height of the carved channel is introduced as a parameter so the varioussurface areas can be calculated using the parametrized model of the magnet(see section 2.3).Comparison of the field flatness between the three configurations is shownin figure 3.18. The result seems to indicate that the equality of horizontalsurface areas (inner and outer) is more relevant while the vertical is used tocontrol the steel saturation for the respective return yoke.The last configuration yields the best result; the top magnetic field profilefor this case is shown in figure 3.19 to be compared with figure 2.11.663.4. Magnetic flux balance-2.5E-3-2.0E-3-1.5E-3-1.0E-3-5.0E-4           05.0E-41.0E-31.5E-32.0E-32.5E-3850 950 1050 1150 1250 1350 1450 1550B z/B0-1 Radial position (mm) Reference designCofiguration 1Configuration 2Configuration 3Figure 3.18: First equalization method configurations comparison.0.43770.43780.43790.43800.43810.43820.43830.43840.43850.43860.43870.43880.43890.43900.43910.4392-50 -40 -30 -20 -10 0 10 20 30 40 50B z (T) Angle (deg) 1000 mm1050 mm1100 mm1150 mm1200 mm1250 mm1300 mm1350 mm1400 mmFigure 3.19: Equalized geometry (HRS-120-16C3: first method, configura-tion 3) Bz vertical magnetic field component magnified near the peak. Thefield is symmetric with respect to 0 degree. This figure to be compared withfigure 2.11.673.4. Magnetic flux balanceThe field distribution in the steel before and after equalization with thefirst method (third configuration) is shown in figure 3.20.In the second method, we consider only the horizontal surfaces of thereturn yokes but in relation to the pole face surface (no air channel for verti-cal compensation). On an historical note related to this project, this secondmethod comes later in time and for this reason many of the geometries inchapter 4 still employ the first method (third configuration) of equalization.The pole surface is considered to be of two portions with the dividerbeing the radius of curvature where the magnetic flux is “naturally split”toward the inner and outer return yokes. This splitting radius lays on thetip of the blue cone (no magnetic flux) as represented in figure 3.21 in whichcase the splitting radius is 1140 mm. Our approach is unique to the extentthat the split point is not just taken as the mid point of the pole, that inour case correspond to the 1200 mm radius.The equalization is achieved by making the ratio of the inner returnyoke horizontal surface to the pole inner surface equal to the ratio of theouter counterparts. For this particular geometry (figure 3.21) the ratio isSurface contour: B (T)1.51.20.80.60.201.41.00.4Figure 3.20: Magnetic flux density in the steel before (left) and after (right)equalization with the first method.683.4. Magnetic flux balanceSelected polesplit radius (1140 mm)Pole mid pointradius (1200 mm)Pole externalsurfacePole internalsurfaceFigure 3.21: Second equalization method: the splitting radius (red) is1140 mm while the symmetry one (yellow) is 1200 mm.about 44%. A lower ratio would increase the saturation level of the returnyokes while a higher one would make the magnet unnecessarily larger. Acomparison of the saturation level before and after applying the secondmethod is represented in figure 3.22.Surface contour: B (T)1.51.20.80.60.201.41.00.4Figure 3.22: Magnetic flux density in the steel before (left) and after (right)equalization with the second method.693.4. Magnetic flux balanceOur approach of using the split radius rather than the symmetry oneproduces also a better field flatness as represented in figure 3.23.The second method returns a far better result and it is also easier toimplement from a mechanical point of view.-5.0E-5-4.0E-5-3.0E-5-2.0E-5-1.0E-5          01.0E-52.0E-53.0E-54.0E-55.0E-5850 950 1050 1150 1250 1350 1450 1550B z/B0-1 Radial position (mm) Symmetry radius at 1200 mmSplit radius at 1140 mmFigure 3.23: Field flatness comparison between the split and symmetry ra-dius cases (second method). Note the apparently sudden improvement withrespect to figure 3.18; as we wrote in the text, this second method wasdeveloped later in time on a geometry with a higher level of optimization.70Chapter 4Optimization on the StraightEdge ModelAll the results obtained in the previous chapter are combined into a newgeometry that we identify as nominal (HRS-120-19C1). This geometry isnot yet the final configuration since it still has straight entrance and exitedges; this feature is going to be added as the last step after reaching theoptimization of the nominal.4.1 Nominal geometryThe main dimensions of this new geometry are listed in table 4.1. Theentrance and exit edge angles are relative to the hard edge case. A renderingof the nominal geometry is represented in figure 4.1.Table 4.1: Nominal geometry main parameters.Geometric parameter DimensionBending radius 1200 mmBending angle 90 degreesEntrance and exit hard edge angle 27 degreesPole gap 70 mmPole height 180 mmPole base height 229.5 mmPole width 676 mmEqualizing channel height 44 mmCoil 69× 69 mm2Coil to steel vertical separation 8.5 mm714.1. Nominal geometrySurface contour: B (T)1.51.20.80.60.201.41.00.4Figure 4.1: Nominal geometry rendering.The parameter choice takes also into consideration the production costof the magnet by limiting some specific dimensions; the pole base height isan example of such a dimension. The nominal geometry return yoke flux isbalanced with the third configuration of the first equalization method (seeequalizing channel height in table 4.1). The coil section is also unchangedbut the vertical position above the coil channel is updated to reflect a morerealistic mechanical layout.The vertical field component (top) is represented in figure 4.2. Comparedwith the reference geometry (figure 2.11), the field drops off the edges forboth the inner and outer geometric trajectories thanks to a more uniformfield distribution over the pole surface (see figure 4.1). Still, the field drop ismore pronounced in the outer trajectories; this reflects, as for the referencegeometry, on the integral flatness represented in figure 4.5 despite the fieldflatness.This field flatness profile of the nominal geometry is shown in figure 4.3.This is an improvement with respect to the reference geometry (see fig-ure 2.13) but it is still one order of magnitude too high.The effective fringe field, although still not aligned with the hard edgecase, presents an angle correct to 1 · 10−3 degrees.724.1. Nominal geometry0.43840.43850.43860.43870.43880.43890.439Ϭ0.43910.43920.43930.4394-50 -40 -30 -20 -10 0 10 20 30 40 50B z (T) Angle (deg) 1000 mm1050 mm1100 mm1150 mm1200 mm1250 mm1300 mm1350 mm1400 mmFigure 4.2: Nominal geometry Bz vertical magnetic field component, mag-nified near the peak, to show flatness. The field is symmetric with respectto 0 degree.-2.5E-3-2.0E-3-1.5E-3-1.0E-3-5.0E-4          05.0E-41.0E-31.5E-32.0E-32.5E-3800 900 1000 1100 1200 1300 1400 1500 1600B z/B0-1 Radial position (mm) Figure 4.3: Nominal geometry field flatness: the red portion of the curverepresents the flatness within the good field region (±160 mm around thereference geometric trajectory ρ = 1200 mm).734.1. Nominal geometry22.022.523.023.524.024.525.025.526.0950 1000 1050 1100 1150 1200 1250 1300 1350 1400 1450∆l  (mm) Radial position (mm) Figure 4.4: Nominal geometry effective field edge location with respect tothe relative hard edge case: the linear interpolation (yellow dashed line)between 1050 mm and 1350 mm gives indications about position and angle.-2.0E-3-1.5E-3-1.0E-3-5.0E-40.0E+05.0E-41.0E-31.5E-32.0E-31000 1050 1100 1150 1200 1250 1300 1350 1400Integral flatness (relative) Radial position (mm) Figure 4.5: Nominal geometry integral flatness with respect to the relativehard edge case: the yellow dashed line is a linear interpolation between1050 mm and 1350 mm. The green box (2.5·10−5 high) represents the flatnessrequirement in the good field region.744.2. Sector Rogowski profileThe integral flatness (figure 4.5) is an improvement from the one ofthe reference geometry (figure 2.16) if compared the linear interpolations(yellow dotted line) of the two cases, but it is again far from meeting therequirements (green box).4.2 Sector Rogowski profileIn this section we are going to introduce the Rogowski profile [43]. The orig-inal work by Rogowski [44]17 was related to breakdown of the electrostaticfield in gases for high voltage plates; the same idea can be applied to themagnetostatic field since it also satisfies Laplace equation for the magneticpotential. In the electrostatic case the goal is to avoid breakdown, while inthe magnetic one it is to avoid saturation of the steel.This Rogowski profile is a more complicated design of the profile ofthe edges of the magnet poles compared to what we used so far, a simple45 degrees chamfer at the edge (see figure 3.9 for example). Both theseoptions are used to reduce the saturation of the pole edges (rather thanhaving a 90 degrees corner); the Rogowski profile though is more effectivewith respect to the chamfer option. Avoiding saturation is necessary in ourcase in order to satisfy the requirement of field reproducibility at differentcurrent excitation level (see section 2.2).The Rogowski theoretical profile is a function of the gap g and it isrepresented by the following equation:f(x) = g(12+epixg−1pi)(4.1)where, in our case, x is a transverse direction parallel to the pole surfaceplane and f(x) perpendicular to the same.Even though the theoretical Rogowski is a machinable profile, it is easierto model, simulate and specify a polygonal approximation18. A four straight17The original paper by Rogownski [44] is in German.18A polygonal approximation would be easier to machine only for a 5-axis CNC machine.754.2. Sector Rogowski profile03570105140175210245-160 -120 -80 -40 0 40 80 120 160f(x) (mm) x (mm) Rogowski4 sectorsSECTOR 1 SECTOR 2 SECTOR 4 SECTOR 3 Figure 4.6: Rogowski profile approximated with four straight sectors: thedotted light green line represents the pole, the dark green represents thepole base and return yoke, the dotted orange line represents the coil of thenominal geometry. The profile has to be moved inward to accommodate thecoil while avoiding an increase of the magnet size.sectors approximation is represented in figure 4.6 compared to the theoreticalprofile; in this graph x = 0 represents the magnet middle plane while y = 0corresponds to the side of the initial pole (45 degrees chamfer). Each sectorcoordinates are calculated such that the sector distance from the theoreticalcurve is limited by a defined maximum. In addition the coordinates of allsector are constrained such that the first sectors starts one gap (70 mm)inward from the edge of the original pole while the last sector ends at thebottom of the pole (180 mm high in this case).The four sectors Rogowski that we just defined can not be directly ap-plied as constructed to the nominal geometry since the the coil would notfit as illustrated in figure 4.6. In order to avoid increasing the size of themagnet, the simplest solution is to move the same profile inward. This so-lution does not change the performance of the Rogowski but it does reducethe pole effective width; this issue is going to be discussed further at the764.2. Sector Rogowski profileend of this section.In the longitudinal direction, along the geometrical trajectories, we em-ploy the same Rogowski without elongating the pole for the reason we aregoing to explain at the end of this section; this is done simply by match-ing the end point of the fourth sector to the edge of the base of the initialpole. The geometry with the four sector Rogowski profile is represented infigure 4.7 where the pole width is increased to 800 mm.Figure 4.7: Geometry with four sectors Rogowski profile (dashed lines).In order to evaluate the effectiveness we compare both the nominal caseand the Rogowski geometry at the two extreme excitation levels necessary tobend the maximum and minimum rigidity (see section 2.2). At the lowestexcitation the produced field is around 0.087 T; this value is adequate tobend a particle of mass 10 19.For these two cases we plot the longitudinal field profile along the geo-metric trajectories relative to the reference one; we are going to identify this19This is a slightly stricter requirement with respect to the one specified in section 2.2,but it is due to the approximation of the current scaling input in OPERA.774.2. Sector Rogowski profilequantity as longitudinal field flatness defined by the following:LFF =Bz(s)Bz,1200(s)− 1 (4.2)where we already defined s as the path length along a beam trajectory;notice that for the reference trajectory the longitudinal field flatness is zero.The relative field is plotted as a function of the angle, rather than s20, wherezero degree corresponds to the center of the magnet.Figure 4.8 represents the longitudinal field flatness of the nominal geom-etry for maximum (solid line) and minimum (dashed line) rigidity; a closeview of the same around the longitudinal edges of the dipole is shown infigure 4.9. The longitudinal field flatness for the geometry with the foursectors Rgowski profile for the two extreme rigidities is represented insteadin figure 4.10 and again in close view in figure 4.11.-2.E-4-1.E-4       01.E-42.E-43.E-44.E-45.E-46.E-47.E-48.E-40 5 10 15 20 25 30 35 40 45 50(Bz(s)/B z, 1200(s))-1 Angle (deg) 1000 mm1050 mm1100 mm1150 mm1200 mm1250 mm1300 mm1350 mm1400 mmFigure 4.8: Longitudinal field flatness of the nominal geometry for maximum(solid line) and minimum (dashed line) rigidity. A close view is representedin the following figure 4.9.20As far as the geometric trajectories inside the dipole s = ρ · angle784.2. Sector Rogowski profile-2.E-4-1.E-4        01.E-42.E-43.E-44.E-45.E-46.E-47.E-48.E-426 28 30 32 34 36 38 40 42 44(Bz(s)/B z, 1200(s))-1 Angle (deg) 1000 mm1050 mm1100 mm1150 mm1200 mm1250 mm1300 mm1350 mm1400 mmFigure 4.9: Close view of the longitudinal field flatness of the nominal geom-etry for maximum (solid line) and minimum (dashed line) rigidity aroundthe longitudinal edge of the dipole.-2.E-4-1.E-4       01.E-42.E-43.E-44.E-45.E-46.E-47.E-48.E-40 5 10 15 20 25 30 35 40 45 50(Bz(s)/B z, 1200(s))-1 Angle (deg) 1000 mm1050 mm1100 mm1150 mm1200 mm1250 mm1300 mm1350 mm1400 mmFigure 4.10: Longitudinal field flatness of the four sectors Rogowski geom-etry (HRS-120-19C8) for maximum (solid line) and minimum (dashed line)rigidity. A close view is represented in the following figure 4.11.794.2. Sector Rogowski profile-2.E-4-1.E-4        01.E-42.E-43.E-44.E-45.E-46.E-47.E-48.E-426 27 28 29 30 31 32 33 34 35 36 37 38(Bz(s)/B z, 1200(s))-1 Angle (deg) 1000 mm1050 mm1100 mm1150 mm1200 mm1250 mm1300 mm1350 mm1400 mmFigure 4.11: Close view of the longitudinal field flatness of the four sectorsRogowski geometry (HRS-120-19C8) for maximum (solid line) and minimum(dashed line) rigidity around the longitudinal edge of the dipole.Based on the results, the simple chamfer of the nominal geometry isnot enough to guarantee field reproducibility in the low 10−5 range; forexample at 37.5 degrees the 1250 mm geometric trajectory presents a relativechange of 4 · 10−5 while for the 1300 mm the change is around 1 · 10−4. Bycomparison the Rogowski geometry with four sectors has a reproducibilityof the edge field that is within 2 · 10−5; this means that the optics behaviorof the separator is fairly linear between the two extreme cases of currentexcitation.The different behaviors of the magnet edge for the nominal geometrycan also be seen qualitatively in the saturation of the steel as represented infigure 4.12 (top); in the same picture the Rogowski case (bottom) shows amore uniform field distribution on the edges.The Rogowski profile has a more pronounced edge field effect with respectto the simple 45 degrees chamfer as seen in figure 4.10. In the transversedimension this implies that for the same mechanical pole width we have asmaller effective width in the Rogowski case and therefore a larger pole is in804.2. Sector Rogowski profileSurface contour: B (T)1.51.20.80.60.201.41.00.4Figure 4.12: Magnetic flux density for the nominal (top) and four sectors Ro-gowski (bottom) geometries at current excitation relative to the maximum(left) and minimum (right) beam rigidity.general required. At the same time though this edge effect tends to improvethe field flatness for our mode of operation.The same is true in the longitudinal case where the effective field edgemoves toward the inside because of the more rapid fall off of the field. Sincewe started with an effective field outward with respect to the hard edgecase (see figure 4.4), we applied the four sectors Rogowski profile withoutelongating the pole in order to achieve such a goal. In fact applying theoriginal Rogowski (in the four sectors configuration) leads to the oppositesituation where the effective field edge is inward with respect to the hardedge case. The obvious solution in order to match the hard edge case wouldbe to elongate the pole. We propose instead an alternative that implementsa scaled Rogowski to maintain optical property at different excitations and814.3. Purcell-like filterat the same time to control the location of the effective field edges with-out elongating the pole. Details of the implemented profile are given insection 5.1.4.3 Purcell-like filterA Purcell filter21 is a region within the magnet with relative permeabilityequal to one (air for example) that improves the field uniformity in the polegap. The simplest Purcell filter is a uniform separation between the poleand the pole base, as we will show in our first attempt, but other moreelaborate configurations [45] can be implemented trying to manipulate thefield flatness. Using as base the nominal geometry (no Rogowski profile) wehave developed different original configurations starting from a conventionalPurcell filter.The conventional Purcell filter as applied to the nominal geometry isrepresented in figure 4.13. The pole is completely separated from the polebase with a uniform air gap.The result of the calculation in terms of field flatness, plotted in fig-ure 4.14, shows that this configuration of the filter cancels the low excita-Figure 4.13: Rendering of the nominal geometry with the full Purcell filter(HRS-120-20C17): the air gap between the pole and pole base is highlightedby the red loop.21Patent US2962636 A by inventor Edward M. Purcell - November 29, 1960.824.3. Purcell-like filter-2.5E-3-2.0E-3-1.5E-3-1.0E-3-5.0E-4          05.0E-41.0E-31.5E-32.0E-32.5E-3800 900 1000 1100 1200 1300 1400 1500 1600B z/B0-1 Radial position (mm) Figure 4.14: Field flatness of the geometry with the full Purcell filter.tion effect seen in section 3.1 without saturation of the pole. This solutionthough requires a wider pole to improve the field flatness. The results shownin figure 4.14 are obtained with a 900 mm pole width; despite such a largepole the flatness is still 1.5 · 10−4, one order of magnitude too high.In an attempt to increase the field for the inner and outer trajectories, weadd 38 mm wide legs on both sides of the bottom of the pole. The legs reston the pole base making magnetic contact. The idea is to guide the magneticflux preferentially on the side edges by reducing the local reluctance (no airgap). This first proposal is represented in figure 4.15: partial Purcell filter.The result is such that these legs increase the inner and outer fields toomuch as plotted in figure 4.16, prompting the need to reduce the leg width.On the other hand, the steel of these legs already saturates to a valueover 2.2 T since, as expected, the magnetic flux goes preferentially throughthem. A reduction in width is just going to increase the saturation level.The next configuration takes the opposite approach; instead of resettingthe nominal geometry field flatness starting from the full Purcell filter, weattempt to control it by adding a local small Purcell filter. The idea in this834.3. Purcell-like filterFigure 4.15: Rendering of the nominal geometry with the partial Purcellfilter (HRS-120-20C2): the partial air gap between the pole and pole baseis highlighted by the red loop.-2.5E-3-2.0E-3-1.5E-3-1.0E-3-5.0E-4          05.0E-41.0E-31.5E-32.0E-32.5E-3800 900 1000 1100 1200 1300 1400 1500 1600B z/B0-1 Radial position (mm) Figure 4.16: Field flatness of the geometry with the partial Purcell filter.case is to reduce the field for the inner and outer trajectories by increasingthe reluctance on the side edges of the pole. The second proposed config-uration is shown in figure 4.17: outboard Purcell filter. The pole has two844.3. Purcell-like filterFigure 4.17: Rendering of the nominal geometry with the outboard Purcellfilter (HRS-120-20C7): the air slots at the bottom of the pole are highlightedby the red loops.rectangular slots carved on the bottom sides of the pole.This configuration is successful in bringing the field flatness in the 10−5range as shown in figure 4.18 but the field starts dropping within the goodfield region.-2.5E-3-2.0E-3-1.5E-3-1.0E-3-5.0E-4          05.0E-41.0E-31.5E-32.0E-32.5E-3800 900 1000 1100 1200 1300 1400 1500 1600B z/B0-1 Radial position (mm)  Regularzoom in X10Figure 4.18: Field flatness of the geometry with the outboard Purcell filter;the dashed curve is ×10 magnified.854.3. Purcell-like filterThe narrower base of the pole (due to the outboard filter) translatesinto a reduced effective pole width at the gap level. As a reference the fieldflatness crosses zero at ρ = 1000 mm and ρ = 1390 mm compared with in theprevious configuration where this occurs at ρ = 945 mm and ρ = 1460 mmfor the same pole width. Making the pole wider is not going to improve theflatness since the field at the center would decrease as described in section 3.3(see figure 3.15).Starting from this last configuration we want to try to restore the effec-tive pole width and reduce the field at the edges of the good field region. Thetwo rectangular slots are moved inside the pole generating two air windows(similar to what is outlined in citation [45]). The third proposed configura-tion is shown in figure 4.19: windows Purcell filter.The position and dimensions are optimized to control the strength ofdifferent fields at different geometric trajectories. The result of this con-figuration is represented in figure 4.20. The effective pole width slightlyincreased as expected with the field flatness crossing zero at ρ = 985 mmand ρ = 1405 mm but the field did not improve. Moreover the section ofsteel on the outer side of the windows are at 1.5 T, close to saturation. EvenFigure 4.19: Rendering of the nominal geometry with the windows Purcellfilter (HRS-120-20C15): the air windows in the pole are highlighted by thered loop.864.3. Purcell-like filter-2.5E-3-2.0E-3-1.5E-3-1.0E-3-5.0E-4          05.0E-41.0E-31.5E-32.0E-32.5E-3800 900 1000 1100 1200 1300 1400 1500 1600B z/B0-1 Radial position (mm)  Regularzoom in X10Figure 4.20: Field flatness of the geometry with the outboard Purcell filter;the dashed curve is ×10 magnified.though it is possible to control the fields to some extent, this configurationis not suited to meet the field flatness requirements.The fourth configuration reconsiders the fully detached filter in orderFigure 4.21: Rendering of the nominal geometry with the detached partialPurcell filter (HRS-120-20C20): the air gap between the pole and pole baseis highlighted by the red loop.874.4. Field clampto avoid saturation (as in the last two cases) while increasing the innerand outer fields using the second configuration approach. The result is thegeometry shown in figure 4.21: detached partial Purcell filter.The geometry of the pole is like the partial Purcell but with the polenow detached from the pole base. This last configuration has an 800 mmwide pole, narrower with respect to the full Purcell. The legs create asmaller reluctance by reducing the air gap from the pole base but theydon’t saturate since there is no magnetic contact and therefore no high flux.The leg dimension can be individually set in order to optimize independentlythe inner and outer trajectories. The field flatness for the detached partialPurcell filter is plotted in figure 4.22; this configuration has the capabilityof meeting field flatness requirement (better than 2.5 · 10−5).-2.5E-3-2.0E-3-1.5E-3-1.0E-3-5.0E-4          05.0E-41.0E-31.5E-32.0E-32.5E-3800 900 1000 1100 1200 1300 1400 1500 1600B z/B0-1 Radial position (mm) Regularzoom in X100Figure 4.22: Field flatness of the geometry with the detached partial Purcellfilter; the dashed curve is ×100 magnified.4.4 Field clampIt is important to control the magnetic field fall off at the entry and exitedges of the pole. In section 4.2 we learned how to control the edge field at884.4. Field clampdifferent levels of excitation, but we need also to control the reproducibilityof the field independently from the environment external to the magnet. Inother terms, we need to assure that the magnetic field leaving the pole, fringefield, closes along a predictable path. Longitudinally, along the geometrictrajectories, this is accomplished by the use of steel structures outside thecurrent carrying coil; these structures, located at the entrance and exit ofthe magnet, top and bottom with respect to the middle plane, are called fieldclamps. Transversely such structures are the return yoke that guarantee aclosed path for the magnetic flux.Each pair of field clamps (top and bottom) can be seen as a longitudinalreturn yoke with an aperture that allows the beam to go through; this meansthat the field inside this aperture has the opposite direction of the field insidethe magnet gap, as can be seen in figure 4.26. Figure 4.23 shows the bottomentrance field clamp (not optimized) for the nominal geometry.The the design of the field clamps is optimized based on the followingcriteria:Outer (transverse)return yoke Inner (transverse)return yokeField clamp(longitudinal return yoke)Figure 4.23: Nominal geometry bottom entrance field clamp (dashed line).894.4. Field clamp• The fringe field tail (circa zero field) distance from the magnet shouldbe minimized: this is to avoid that the field profile is going to beaffected by other beam line or surrounding steel elements.• The field profile should minimize the first derivative of the field (sharpchange in the profile): high first derivative contributes to high orderaberration in the beam dynamics.• Saturation of the field clamp should be avoided (magnetic flux to beless than 1.5 T): this is in order to guarantee that the field clampmaintains the same behavior at different excitation levels.Different configurations have been developed starting from the standardone used in the nominal geometry. In order to evaluate a configuration,the vertical field component Bz along the reference geometrical trajectory(ρ = 1200 mm) is plotted as a function of the distance from the entrance(or exit) of the theoretical hard edge case as seen in figure 4.24; the outer-0.050.000.050.100.150.200.250.300.350.400.450.50-300 -200 -100 0 100 200 300B z (T) s (mm)  Geometry withreference clampHard edgeFigure 4.24: Magnetic field of the geometry with reference clamp versus thetheoretical hard edge case plotted as a function of s (path length along abeam trajectory). The outer mechanical edge of the clamp is at s = 106 mm.904.4. Field clamp(with respect to the magnet center) mechanical edge of the field clamp sitsat 120 mm.The field fall off resulting from the different configurations is plotted infigure 4.26, while the configurations are summarized here:1. No clamp: this case is included in the calculations for reference. Inthis case the field falls off with the minimum variation of the firstderivative but it extends the most beyond the edge of the magnet. At400 mm from the edge where other beam line components will reside,the field is still 18 G.2. 70/86 mm: these two numbers refer to the clamp aperture (70 mm,same as the pole gap) and distance from the pole edge (86 mm). Inthis case the clamp has a simple L shape, as in figure 4.25; we aregoing to identify this as reference clamp. The fall off relative to thisconfiguration has a significant overshoot on the negative side and asharp change in the field profile. We already mentioned that the fieldinside the clamp aperture is opposite to the field in the gap, so it isexpected that the former counteracts the latter, but in this case thefield produced by the clamp is too strong.Figure 4.25: Simple L shape reference clamp.914.4. Field clamp-3.E-3-2.E-3-1.E-3       01.E-32.E-33.E-3100 200 300 400 500 600 700 800 900 1000 1100 1200B z (T) Distance from the edge (mm) No clamp70/86 mm (reference clamp)70/129  mm moved outward190/86 mm open70/86 mm chamfer 9/10070/86 mm connected70/86 mm partially detached70/86 mm fully detachedFigure 4.26: Magnification of the fringe field profiles of the different fieldclamp configurations.3. 70/129 mm moved outward: more to the fact that field clamp is actingas a return yoke with opposite field, we consider a configuration wherethe clamp distance is increased (129 mm) while maintaining the sameaperture; since we didn’t change the geometry, the clamp has the samestrength but applied in a weaker (farther away) region. The result isa stronger overshoot with respect to the previous configuration.4. 190/86 mm open: in order to weaken the clamp we increase the aper-ture to 190 mm while preserving the distance (86 mm). The result thistime is as expected with a smoother fall off, but the fringe field fringefield is shifted outward.5. 70/86 mm chamfer 9/100: a better approach to reduce the strengthof the field clamp is applying a chamfer on the edge. This effectivelyreduces the efficiency of the field clamp while maintaining the field falloff close to the clamp.6. 70/86 mm connected: an additional reduction of the field strength can924.4. Field clampbe obtained by connecting the clamp pair (top and bottom) in orderto create a low reluctance path for the magnetic flux. The result is aweaker field through the field clamp aperture.7. 70/86 mm partially detached: a further reduction can be achieved byweakening, from a magnetic point of view, the entire clamp. We insertan opening (filter like) at the base of the clamp where it is connectedto the pole base. This configuration generates a smooth fringe fieldwhile maintaining it close to the magnet.8. 70/86 mm fully detached: as a last check we investigated the option ofa clamp non magnetically connected to the dipole. This results in thefringe field moving away from the magnet.The final design for the straight geometry is represented in figure 4.27and it corresponds to the seventh configuration. The clamp has an apertureequal to the pole gap and a minimum distance from the edge that allowsfor the coil accommodation. The edge has a chamfer and the lower andupper clamp are connected. Finally the clamp has filter-like windows whereit attaches to the pole base of the magnet.Figure 4.27: Optimized field clamp for the straight edge geometry.93Chapter 5Final DesignThe findings presented in the previous three chapters converge to an opti-mized solution of the straight edge magnetic dipole model. This serves asthe basis to design the final geometry for the case with curved entrance andexit edges. A curved edge, as we are going to see in section 5.3, is the wayto implement a second order correction in the beam dynamics [1].5.1 Optimized straight edge modelIn this section we present the final design of the straight edge model. Fig-ure 5.1 is a rendering of the optimized straight edge geometry (HRS-120-21C57).Figure 5.1: Optimized straight edge geometry.945.1. Optimized straight edge modelSurface contour: B (T)1.51.20.80.60.201.41.00.4Figure 5.2: Magnetic flux density of the optimized straight edge geometry.The coil section is changed from a square22 to a rectangular cross sectionto reduce the overall width of the magnet. The steel magnetization is withinthe 1.5 T saturation limit as shown in figure 5.2.A Rogowski-like profile for both the longitudinal and transverse edges ofthe pole is implemented.Figure 5.3 represents a comparison between approximation of the Ro-gowski theoretical curve using 4, 6, 8 and 10 sectors, while figure 5.4 repre-sents the maximum distance limit set to generate the sector Rogowski (seesection 4.2) as a function of the number of sectors; the most gain in termsof approaching the theoretical curve, is going from four to six sectors. Thesix sectors choice is the best compromise between approximation to the the-oretical curve and minimization of the number of sectors for machining andcost purposes.2269 × 69 mm2, see table 4.1.955.1. Optimized straight edge model-180-160-140-120-100-80-60-40-200-60 -40 -20 0 20 40 60 80 100 120 140 160 180 200y (mm) x (mm) 10 sectors8 sectors6 sectors4 sectorsFigure 5.3: Sector Rogowski comparison: the most gain in terms of ap-proaching the theoretical curve, is going from four to six sectors as seen infigre 5.4.00.511.522.532 3 4 5 6 7 8 9 10 11 12Maximum distance limit (mm) Number of sectors Figure 5.4: Maximum distance limit set to generate the sector Rogowski965.1. Optimized straight edge model-180-160-140-120-100-80-60-40-2002040-60 -40 -20 0 20 40 60 80 100 120 140 160 180 200 220 240y (mm) x (mm) Rogowsky6 sectorsSECTOR 6 SECTOR 5 SECTOR 1 SECTOR 3 SECTOR 2 SECTOR 4 Figure 5.5: Six sectors Rogowski profile; a scaled version is implemented inthe optimized straight geometryThe implemented profile is a scaled version of the theoretical Rogowskiprofile (as anticipated at the end of section 4.2) for a 70 mm gap beingapproximated by a six sector segmented line as represented in figure 5.5 (thecurve is rotated for convenience); the sector distance from the theoreticalcurve (see section 4.2) is limited to circa 1 mm.The scaled version of the Rogowski means the coordinate values of thesector end points (see figure 5.5) are scaled: the longitudinal (along the ge-ometric trajectories) and transverse (radially) scaling factors are respective61% and 40%. The scaling of the Rogowski allows to control respectively theeffective field edge and the field flatness. A rendering of the pole is shownin figure 5.6.The model has an optimized detached partial Purcell filter that allows toachieve the field flatness, represented in figure 5.7, well within the require-ment of 2.5 · 10−5 (notice ×100 dashed curve in figure 5.7).The effective field edge profile and the integral flatness of the optimizedstraight edge case are represented respectively in figure 5.8 and figure 5.9.975.1. Optimized straight edge modelFigure 5.6: Pole of the optimized straight edge model with a six sectorRogowski scaled 61% longitudinally and 40% transversely.These two graphs are in fact two interpretations of the same set of data: thefield integrals along the geometric trajectories. In figure 5.8 the integralsare converted in effective length to be compared with the length of thetheoretical hard edge case; this gives a direct information about the positionof the effective field edge. In figure 5.9 the integrals are ultimately (seeequation 2.12) compared against the reference trajectory one; this gives adirect information about the flatness of the integrals.The effective field edge is in the correct location thanks to the optimiza-tion of the sector Rogowski scaling factor (61%) for the longitudinal entranceand exit edges. The effective edge presents though a slight curvature withradius of circa 80 m; this is due to the fall off of the field and it can becorrected by curving the edge outward. The effective edge presents also asmall angle of 0.2 mrad. Both the curvature and the angle are not correctedsince this is not the final geometry.The transverse field fall off is also controlled by optimizing the transversescaling factor (40%). The optimization is achieved by preventing the fieldflatness from dropping within the good field region (see figure 5.7).The field integral result is a significant improvement with respect tothe nominal geometry (see figure 4.5) but it still is outside the requirement(green box on figure 5.9). Since the trend of the integral flatness is consistentwith the effective edge curvature, we could speculate that adjusting the latterwill also fix the former.985.1. Optimized straight edge model-2.5E-3-2.0E-3-1.5E-3-1.0E-3-5.0E-4          05.0E-41.0E-31.5E-32.0E-32.5E-3800 900 1000 1100 1200 1300 1400 1500 1600B z/B0-1 Radial position (mm) Regularzoom in X100Figure 5.7: Field flatness of the optimized straight edge design.-0.5-0.4-0.3-0.2-0.10.00.10.20.30.40.5950 1000 1050 1100 1150 1200 1250 1300 1350 1400 1450∆l (mm) Radial position (mm) Figure 5.8: Optimized straight edge geometry effective field edge locationwith respect to the relative hard edge case: the linear interpolation (yellowdashed line) between 1050 mm and 1350 mm gives indications about positionand angle.995.2. OPERA-3D R© field for COSY-∞-5.0E-4-4.0E-4-3.0E-4-2.0E-4-1.0E-4          01.0E-42.0E-43.0E-44.0E-45.0E-41000 1050 1100 1150 1200 1250 1300 1350 1400Integral flatness (relative) Radial position (mm) Figure 5.9: Optimized straight edge geometry integral flatness with respectto the relative hard edge case: the yellow dashed line is a linear interpolationbetween 1050 mm and 1350 mm. The green box (2.5 · 10−5 high) representsthe flatness requirement in the good field region.5.2 OPERA-3D R© field for COSY-∞In order to have the most accurate beam dynamics model of the HRS sys-tem, it is important to transfer the two dimensional rectangular grid of themagnetic field map of the middle plane from OPERA-3D R© to COSY-∞ (seesection 2.3). It is possible to read such a grid into COSY-∞ from which thecode computes the magnetic field everywhere else using the differential al-gebraic technique [41]; the issue is that COSY-∞ requires that the initialconditions, OPERA R© field on each single point of the grid, are exact solu-tions (within the machine precision) of Maxwell’s equations otherwise theoutcome is not reliable. This is true only if the points happen to coincidewith a node of the tetrahedral mesh produced by OPERA R© while in generalthe field between nodes is interpolated.Figure 5.10 shows a detailed view of the standard OPERA R© mesh in the1005.2. OPERA-3D R© field for COSY-∞Figure 5.10: Detailed view of the air gap region (yellow dashed line ) stan-dard mesh produced by OPERA R©. Various patches of mesh can be distin-guished.air gap region; various patches of mesh indicate that there is not a particularorder in the distribution of the nodes for this complex geometry (a perfectlyrectangular box would be meshed in an orderly fashion).We verified that the use of maps generated by a standard mesh fails toproduce physical results in COSY-∞. So it becomes mandatory to forcethe nodes on the desired COSY-∞ grid, but there is not such a function inOPERA R©.If we manage to force the nodes on a desired rectangular grid (as wewill shortly see), then in principle we only need a coarse mesh in betweennodes as far as COSY-∞ is concerned; on the other hand a fine mesh isstill necessary to calculate the integrals accurately (in the 10−6 range) alongthe geometric trajectories, that don’t follow exactly a rectangular mesh andtherefore include field values from OPERA R© in between nodes. We havein fact run a curved grid that follows the geometric trajectories in order to1015.2. OPERA-3D R© field for COSY-∞compare the integrals against the rectangular grid (for COSY-∞) and nogrid cases while using the same fine mesh. The integrals from the two gridcases are within 1 · 10−6, while integrals between any grid case and no gridare within 8 · 10−6. The use of a grid provide a much ordered mesh thatyields better result without increasing the computation time.It was proposed to create a honeycomb-like structure superimposed onthe air gap volume with the wall (of the honeycomb) intersections being thegrid points, but this actually causes OPERA R© to fail to create a mesh inthe first place because of too many constraints posed by the structure walls.We developed an original technique in order to achieve a COSY-∞ com-patible rectangular mesh. We proposed a “bed of nails” approach whereeach nail is positioned on the coordinates of the grid; the nails are simplyOPERA R© wire-edges that have no volume. The nails layout imposed no par-ticular constraints on the mesher, because of the absence of walls betweennails, other than forcing it to put a node at each end of every nail.The created “bed of nails” is superimposed on the air gap volume, asshown in figure 5.11 (A-B), which base is the middle plane of the magnetwhere we want to extract the COSY-∞ grid. The nails are then subtractedfrom the air volume leaving it with volume-less holes, see figure 5.11 (C-D).The mesh obtained applying the “bed of nails” technique is shown infigure 5.12. Compared with the standard mesh (figure 5.10) the nodes areordered accordingly to the desired grid and the mesh is overall more regu-lar. The end of each and every nail coincides with a node (red arrows infigure 5.12), but of course not every node coincides with the end of a nail.The use of the “bed of nails” allows for the successful transfer of thetwo dimensional map from OPERA R© to COSY-∞. As stated earlier, thetechnique has also the benefit of achieving one order of magnitude more ac-curate solution without further refining the mesh size, that would increasethe demand on the computing system. This has been established by com-paring a solution, for the same geometry, with no nails to two other solutionsusing the nails in two different grids. Using the nails technique is equivalentto increasing the number of mesh points, but with the advantage of notincreasing the computational time.1025.2. OPERA-3D R© field for COSY-∞ABCDFigure 5.11: Bed of nails (A) superimposed on the air gap volume (B) thensubtracted from the air gap (C); the remaining structure has volume-lessholes (D). The nails are OPERA R© wire-edges that have no volume.1035.2. OPERA-3D R© field for COSY-∞Figure 5.12: Air gap volume (yellow dashed line) meshed with the “bedof nails” technique (top). Magnified section of the air gap (central andbottom): evidence (red arrows) that the end of a given nail (red dashedline) coincides with a node.1045.3. Final curved edge model5.3 Final curved edge modelThe final geometry (HRS-120-23C62) is obtained from the optimized straightedge model by applying a curvature to both the pole entrance and exit edgesand the field clamp. The coil follows all the curvature of the pole. The finalresult is shown in figure 5.13.The curvature is necessary to correct the second order aberration; thisaberration has a characteristic “C” shape in the transverse phase space atthe exit of the dipole, as represented in figure 5.14; this means that particleswith extreme angles with respect to the reference trajectory are bent more.In order to limit the bending of such particles, the pole length is graduallyreduced as it moves towards the sides of the pole in such a way that the thefield integral for particles traveling along extreme trajectories (higher angleswith respect to the reference trajectory) is reduced. This is accomplishedgeometrically by curving the edges of the pole.The steel magnetic field distribution of the final design is shown in fig-ure 5.15 when the magnet is run in full excitation mode producing a verticalmagnetic field component B0 = 0.458 T. In this full excitation mode, thefield in most of the steel is less than 1.2 T and it peaks in the field clamp atFigure 5.13: Final geometry.1055.3. Final curved edge model-5 -4 -3 -2 -1 0-60-40-2002040601 2 3 4 5x (mm)x(mrad)´Second order aberration in phase spaceFigure 5.14: Characteristic “C” shape of the second order aberration.1.4 T, below the saturation level of 1.5 T (see figure 2.9) as per requirement.Figure 5.15 shows also the balance between the inner and outer return yokesobtained with the second method (see section 3.4).The main parameters of the final design are listed in table 5.1. Theedge angle and curvature are relative to the hard edge magnet case. It isinteresting to compare the 2.3 m required radius of curvature of the edges,with the natural 80 m curvature in the straight edge model; it is clear nowthat we can define the latter as a slight curvature that has no significanteffect on the beam.The final geometry employs a scaled six sector Rogowski with two differ-ent scaling factors in the horizontal plane (parallel to the pole face) for thelongitudinal and transverse directions, similar to the optimized straight edgegeometry. In the vertical plane (perpendicular to the pole face) though, wemaintain23 the same longitudinal scaling factor also for the transverse direc-tion in order to facilitate the machining process by simplifying the geometry23Suggested by Dr. Thomas Planche.1065.3. Final curved edge modelSurface contour: B (T)1.51.20.80.60.201.41.00.4Figure 5.15: Magnetic flux density of the final design in full excitation mode.of the corners (avoiding unnecessary small steps) where the longitudinal andtransverse directions meet, as represented in figure 5.16. This means thatthe transverse (less critical) profile is no longer a Rogowski; simulationsTable 5.1: Final geometry main parameters.Geometric parameter DimensionBending radius 1200 mmBending angle 90 degreesEntrance and exit hard edge angle 26.5 degreesPole gap 70 mmPole height 185 mmPole base height 205 mmPole width 760 mmHard edge curvature 2238 mmCoil 158× 80 mm2Coil to steel vertical separation 8.5 mm1075.3. Final curved edge modelFigure 5.16: Detail of the pole corner for the optimized straight edge (left)and final curved geometry (right): the curved geometry maintain the samescaling factor in the vertical plane (different in the horizontal) producing asimplified geometry of the corner (see orange line for reference).show that this hybrid profile is still effective in avoiding saturation, as seenin figure 5.15.The field clamps are optimized together with the pole edges in order toachieve the effective field edge corresponding to the hard edge magnet withcurved pole. The final fringe field is shown in figure 5.17 plotted amongstprofiles from other clamp geometries for comparison. The final fringe fieldprofile minimizes the first derivative of the magnetic field (see section 4.4)as shown in figure 5.18.The field flatness of the final geometry is shown in figure 5.19. The finalvalue of 8 · 10−6 within the good field region is three times lower than therequirement of 2.5 · 10−5; it is also a factor of two better with respect toother magnetic dipoles designed for similar high resolution separators [46].The integrals calculated from the final OPERA R© model are listed intable 5.2 together with the hard edge ones.1085.3. Final curved edge model-3.E-3-2.E-3-1.E-3       01.E-32.E-33.E-3100 200 300 400 500 600 700 800 900 1000 1100 1200B z (T) Distance from the edge (mm) No clamp70/86 mm (reference clamp)70/86 mm connected70/86 mm partially detachedFinal (HRS-120-23C62)Figure 5.17: Fringe field of the final geometry.-45-40-35-30-25-20-15-10-505-200 -100 0 100 200 300 400 500dBz/ds (G/mm) Distance from the edge (mm) No clamp70/86 mm (reference clamp)70/86 mm connected70/86 mm partially detachedFinal (HRS-120-23C62)Figure 5.18: First derivative of the fringe field of the final geometry.1095.3. Final curved edge model-2.5E-3-2.0E-3-1.5E-3-1.0E-3-5.0E-405.0E-41.0E-31.5E-32.0E-32.5E-3800 900 1000 1100 1200 1300 1400 1500 1600B z/B0-1 Radial position (mm) Regularzoom in X100Figure 5.19: Final geometry field flatness for the full excitation mode.Table 5.2: Final geometry calculated integrals.Geometric trajectory Hard-edge OPERA R©IRρ−1 IFρρ (mm) (T·mm) (T·mm)1000 802.195 802.147 -6.0·10−5 -6.2·10−51050 819.383 819.382 -1.5·10−6 -3.6·10−61100 835.414 835.417 4.4·10−6 2.2·10−61150 850.275 850.282 8.7·10−6 6.5·10−61200 (reference) 863.955 863.957 2.2·10−6 01250 876.433 876.448 5.2·10−6 3.1·10−61300 887.727 887.731 4.4·10−6 2.2·10−61350 897.796 897.798 1.8·10−6 -4.0·10−71400 906.637 906.636 -1.4·10−6 -3.6·10−6The effective edge profile is represented in figure 5.20; as for the other ge-ometries, the graph represents the distance of the effective edge from the rel-ative curved hard edge case. Figure 5.20 linear interpolation (yellow dashedline) shows that the effective edge matches the hard-edge case in terms ofcurvature and position. The integral flatness is shown in figure 5.21.1105.3. Final curved edge model-0.20-0.16-0.12-0.08-0.04     00.040.080.120.160.20950 1000 1050 1100 1150 1200 1250 1300 1350 1400 1450∆l (mm) Radial position (mm) Figure 5.20: Final geometry effective field edge location with respect to therelative hard edge case for the full excitation mode: the linear interpolation(yellow dashed line) between 1050 mm and 1350 mm gives indications aboutposition and angle.-1E-4-8E-5-6E-5-4E-5-2E-5      02E-54E-56E-58E-51E-41000 1050 1100 1150 1200 1250 1300 1350 1400Integral flatness (relative) Radial position (mm) Figure 5.21: Final geometry integral flatness with respect to the relativehard edge case for the full excitation mode.1115.3. Final curved edge modelAll the values in the good field region (±160 mm around the referencetrajectory) are within the requirement of 2.5·10−5 (green box in figure 5.21).The final model is also run at low excitation to produce a vertical mag-netic field component B0 = 0.098 T; this is close to the nominal value tobend mass 11 at 60 kV (see section 2.2). It should be noted that for suchlow masses a resolving power of 20000 is not necessary (see figure 1.13).Figure 5.22 is a rendering of the magnetic field distribution for the lowexcitation case (HRS-120-23C65).The field flatness for the low excitation mode is represented in figure 5.23,while the effective field edge profile and integral flatness for the same modeare represented respectively in figure 5.24 and figure 5.25.The effective edge shifted about 50µm and present an angle with respectto the hard edge case of circa 0.1 mrad. The shift is acceptable since theeffective edge is still within the requirement. The small angle mismatch,circa 2 · 10−4 of the design angle, translates into a slightly different verticalfocusing of the dipole; this can be corrected if necessary with a dedicatedSurface contour: B (T)0.750.60.40.30.100.70.50.2Figure 5.22: Magnetic flux density of the final design in low excitation mode;notice the change of scale with respect to the full excitation mode.1125.3. Final curved edge modelquadrupole in front of the magnet foreseen in the beam dynamics layout [1].-2.5E-3-2.0E-3-1.5E-3-1.0E-3-5.0E-4          05.0E-41.0E-31.5E-32.0E-32.5E-3800 900 1000 1100 1200 1300 1400 1500 1600B z/B0-1 Radial position (mm) Regularzoom in X100Figure 5.23: Final geometry field flatness for the low excitation mode.-0.20-0.16-0.12-0.08-0.04     00.040.080.120.160.20950 1000 1050 1100 1150 1200 1250 1300 1350 1400 1450∆l (mm) Radial position (mm) Figure 5.24: Final geometry effective field edge location with respect to therelative hard edge case for the low excitation mode.1135.4. Engineering considerations-1E-4-8E-5-6E-5-4E-5-2E-5      02E-54E-56E-58E-51E-41000 1050 1100 1150 1200 1250 1300 1350 1400Integral flatness (mm) Radial position (mm) Figure 5.25: Final geometry integral flatness with respect to the relativehard edge case for the low excitation mode.The integral flatness is still mostly within the requirement but for themost outer trajectory. The overall flatness is compatible with a 14000 re-solving power that, as mentioned, is not required in this low mass range.Most likely the integral flatness in the low excitation case will still be enoughto have the HRS system to perform at a higher resolving power thanks tothe high order multipole corrector [1].5.4 Engineering considerationsIn this section we are going to briefly discuss all the engineering featuresrequired to assemble, transport and operate the magnet. These are detailedengineering aspects of the real magnet that may impact its performance andtherefore we need to confirm their viability in OPERA R©. As a matter of factmany of the implemented features require a re-optimization of some of themagnet parameters. We list them in table 5.3; we don’t provide a detaileddescription of each feature since they are not part of the design optimization1145.4. Engineering considerationsTable 5.3: Engineering features applied to the final geometry.Propose Engineering feature OPERA simulation ResultPole mount HRS-120-23C62eng acceptableYoke mount and hoist rings HRS-120-23C62eng2 acceptableChamfer of the inner yoke HRS-120-23C62eng3 acceptableBuckley coil proposal HRS-120-23C62eng4acceptable butnot adoptedReduced field clamp HRS-120-23C62eng5 acceptableField clamp mount HRS-120-23C62eng6 acceptableField clamp squared HRS-120-23C62eng7 not acceptablePropose single-rib field clamp HRS-120-23C62eng9 acceptablePole holes  = 12 mm HRS-120-23C62eng12 acceptableField clamp connections HRS-120-23C62eng13 acceptable5 mm blended fieldHRS-120-23C62eng14 acceptableclamp knife edgeThermowells HRS-120-23C62eng15 acceptableFinal coil 159× 76 HRS-120-23C62eng16 acceptableMove coil transition fromHRS-120-23C62eng21 acceptableouter to inner coil channel(but rather input from the magnet engineer and draftsman) and because thepurpose is just to show that every aspect of the magnet has been taken intoaccount.The features are applied starting from the final design; each feature issimulated cumulatively, following the same order given in table 5.3, trigger-ing the required parameter re-optimization at each step.The results of each simulation are compared to the final design withrespect to field flatness, effective field position and integral field flatness.The result reported in table 5.3 is in terms of acceptability. All the featuresare simulated as air (voids in the steel); in one case air has been substitutedwith stainless steel (µr = 2, worst case scenario) in order to simulate a bolt,showing no difference in the final result.A rendering of the final geometry with the engineering features (HRS-120-23C62eng21) is represented in figure 5.26. The field flatness, effective1155.4. Engineering considerationsfield edge location and integral flatness of the final engineered geometry areshown respectively in figure 5.27, figure 5.28 and figure 5.29.Figure 5.26: Final design with engineering features (HRS-120-23C62eng21).-2.5E-3-2.0E-3-1.5E-3-1.0E-3-5.0E-4          05.0E-41.0E-31.5E-32.0E-32.5E-3800 900 1000 1100 1200 1300 1400 1500 1600B z/B0-1 Radial position (mm) Regularzoom in X100Figure 5.27: Final engineered geometry field flatness with respect to thehard edge case.1165.4. Engineering considerations-0.20-0.16-0.12-0.08-0.04     00.040.080.120.160.20950 1000 1050 1100 1150 1200 1250 1300 1350 1400 1450∆l (mm) Radial position (mm) Figure 5.28: Final engineered geometry effective field edge location withrespect to the relative hard edge case for the full excitation mode.-1E-4-8E-5-6E-5-4E-5-2E-5      02E-54E-56E-58E-51E-41000 1050 1100 1150 1200 1250 1300 1350 1400Integral flatness (mm) Radial position(mm) Figure 5.29: Final engineered geometry integral flatness with respect to therelative hard edge case for the full excitation mode.1175.4. Engineering considerationsFinally a photograph of the first HRS manufactured24 magnetic dipoleis shown in figure 5.30.Figure 5.30: First HRS manufactured magnetic dipole (with field clampedges protected by white Styrofoam). Photograph courtesy of Buckley Sys-tems.24by Buckley Systems - http://www.buckleysystems.com/118Chapter 6ConclusionThe working hypothesis, and ultimately the magnet design, needs to bevalidated by testing the optical performance of the dipole field within theHRS system.This magnetic field25 is used to run a beam dynamics calculation forbeams transported from the entrance to the exit (selection) slit of the HRS.This calculation is compared against the one that uses an ideal uniform fieldfor a dipole.Figure 6.1 shows the distribution of two beams, green and red, in thehorizontal phase space at the selection slit for the two cases: ideal (top)versus realistic (bottom). The two transported beams have the same 3µmhorizontal emittance and they have a mass difference of one in twenty thou-sand. The green beam is centered around the selection slit (x = 0) with a0.1 mm aperture; this beam is selected and transported to the experimentalstation. The red will be stopped by the 0.1 mm selection slit.The comparison shows that the realistic field performs as well as theideal field. This result confirms that minimizing the integral field variationof a separator dipole, with respect to the system resolution, is equivalentto minimizing aberrations. The result also validates the final design of theHRS magnet.The design of the HRS system and the magnet in particular has beenpresented to an international review committee as standard practice for allTRIUMF projects. Based on the feedback from the committee experts, theHRS project is now proceeding to the manufacturing and installation stage.The two dipoles have been manufactured to the required precision. Atwo dimensional field mapping of the dipoles has been performed and it25Calculated and exported from OPERAR©.119Chapter 6. Conclusionshows the expected field quality.Phase space (x, x´) at selection slit with COSY- map∞-0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25-80-60-40-20020406080Horizontal Position at Exit Slit (mm)AngularDeviationatExitSlit(mrad)-0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25-80-60-40-20020406080Horizontal Position at Exit Slit (mm)AngularDeviationatExitSlit(mrad)Phase space (x, x´) at selection slit with OPERA mapFigure 6.1: Transverse horizontal phase space at the selection (exit) slitcalculated in COSY-∞ with ideal dipole field (top) and OPERA R© importedfield (bottom).1206.1. HRS system developments6.1 HRS system developmentsA further investigation of the field quality of the manufactured HRS dipolesis planned at TRIUMF. Such investigation foresees an integral field mappingrather than a local (points on a Cartesian grid) mapping as performed by themanufacturer. The integral field mapping will measure the field variationwithin a loop around the geometric trajectories; the measurement is a directapplication of Faraday’s law of induction for which a magnetic flux variationthrough an electric circuit (the loop) generates an electromotive force in thecircuit itself. The field variation for different geometric trajectories (loops) isgoing to be compared against the ones extracted from the OPERA R© model.It is expected that these integral measurements are sensitive to the fieldvariation the magnet is designed for, giving indication of the quality of themagnetic field.One characteristic of the HRS system that makes it unique is a completeset of diagnostics devices to characterize the beam. Lack of diagnostic hasbeen identified as an issue for the CARIBU high resolution mass separator(mentioned in section 1.4), in particular diagnostic devices capable of re-constructing the transverse phase space, emittance rigs, at the selection slit.Our system has an emittance rig both at the image and object (selection)slit locations.Even though both the beam dynamics and the magnets are designed tominimize aberrations, it is expected that some level of high order correctionsis necessary to compensate for manufacturing and installation tolerances.In order to compensate for such high order aberrations, a device called amultipole has been designed and its mode of operation is under development.The multipole consists of a set of electrodes that can be adjusted individuallyto a certain voltage to create an electrostatic corrective field in the transversedirection. The multipole, located in between the two dipoles (see figure 2.6),is designed based on the operational experience of the CARIBU system. Ourdesign differs though from the latter in the geometric layout of the electrodesthat follows a rectangular rather than a circular pattern. The rectangularshape better suits the beam envelope and it requires a lower voltage to be1216.2. Future upgradesapplied to the electrodes. This design allows for a better control of thecorrective field.6.2 Future upgradesAs seen in chapter 2, the performance of the HRS depends greatly on thequality of the beam. Our system is designed for a transverse horizontalemittance of 3µm; for bigger value, the emittance can be either cut (meaningbeam losses) or reduced using a device called RFQ cooler. Such device isa gas filled RFQ in which the beam loses transverse moment by stochasticinteractions with the gas atoms. The transverse momentum loss results inan emittance reduction. Even though an RFQ cooler is not in the presentproject budget, the beam lines surrounding the HRS system are designed toaccommodate this future upgrade.6.3 ConclusionIn this work we managed to successfully demonstrate that minimizing thefield integral variation with respect to the system resolution within the goodfield region is equivalent to reducing the magnetic field contribution to thesystem aberration (working hypothesis); this means that the system correc-tions required to compensate for magnetic field imperfection are minimized.We also ultimately produced the design of the magnetic dipole for the HighResolution Separator (HRS) of the new ARIEL facility at TRIUMF.An important issue that we encountered and developed during the workis the relation between field flatness and pole gap (see section 3.1). Theperception that a smaller gap (for a fixed width) produces a better flatnessis not correct. In the large width to gap ratio limit, a smaller gap results inworse flatness; this is a result of the fact that the permeability is finite.We also invented a new technique to force the OPERA mesh nodes on apredetermined grid (see section 5.2). This is not only fundamental to trans-fer the correct values of the magnetic field to the COSY-∞ beam dynamics1226.3. Conclusioncode, but it also improves the mesh quality producing a more accurate so-lution for the same computational time.Once the HRS system is commissioned, ARIEL will be an ISOL facilitycapable of producing and selecting heavy masses RIB beams with high res-olution. The HRS system will finally enable the pursue of experiments like132Sn.Combined with the ISAC capability of delivering high energy, ARIELwill make TRIUMF highly competitive with other ISOL facilities like Iso-tope Separation On Line DEvice (ISOLDE) at Conseil Europeen pour laRecherche Nucleaire (CERN) or CARIBU at ANL.Furthermore the unique capability of delivering three simultaneous ra-dioactive beams will position TRIUMF in a leading role for decades to come.123Bibliography[1] J. Maloney, R. Baartman, and M. Marchetto, “New design studies forTRIUMF’s ARIEL high resolution separator,” Nuclear Instruments andMethods in Physics Research Section B: Beam Interactions with Ma-terials and Atoms, vol. 376, pp. 135 – 139, 2016. Proceedings of the{XVIIth} International Conference on Electromagnetic Isotope Separa-tors and Related Topics (EMIS2015), Grand Rapids, MI, U.S.A., 11-15May 2015.[2] M. Marchetto, “Progress and plans for high mass beam delivery at TRI-UMF,” Heavy Ion Accelerator Technology Conference, Chicago, Illinois,USA, 2012.[3] M. Marchetto, F. Ames, P. Bender, B. Davids, N. Galinski, A. Gar-nsworthy, G. Hackman, O. Kirsebom, R. Laxdal, D. Miller, et al., “Inflight ion separation using a linac chain,” Linear Accelerator Confer-ence, Tel-Aviv, Israel, pp. 1059–1063, 2012.[4] M. Marchetto, A. Bylinskii, R. Laxdal, et al., “Beam dynamics studieson the ISAC-II superconducting linac,” Linear Accelerator Conference,Knoxville, Tennessee, USA, p. 312, 2006.[5] M. Marchetto, M. Comunian, A. Palmieri, A. Pisent, and E. Fagotti,“Study of a high-current 176 MHz RFQ as a deuteron injector forthe SPES project,” European Particle Accelerator Conference, Lucerne,Switzerland, pp. 251–253, 2004.[6] B. N. Taylor and A. Thompson, “The international system of units(SI),” 2008.124Bibliography[7] M. Lieuvin, “Design issues of radioactive ion beam facilities,” EPAC96,Sitges, SPAIN, 1996.[8] D. J. Morrissey, “Status of the FRIB project with a new fragment sepa-rator,” Journal of Physics: Conference Series, vol. 267, no. 1, p. 012001,2011.[9] D. J. Morrissey and B. M. Sherrill, “Radioactive nuclear beam facilitiesbased on projectile fragmentation,” Philosophical Transactions of theRoyal Society of London A: Mathematical, Physical and EngineeringSciences, vol. 356, no. 1744, pp. 1985–2006, 1998.[10] T. P. Wangler, RF Linear accelerators. John Wiley & Sons, 2008.[11] M. Conte and W. W. MacKay, An introduction to the physics of particleaccelerators. world scientific, 2008.[12] H. Geissel, P. Armbruster, K. Behr, A. Bru¨nle, K. Burkard, M. Chen,H. Folger, B. Franczak, H. Keller, O. Klepper, et al., “The GSI projec-tile fragment separator (FRS): a versatile magnetic system for relativis-tic heavy ions,” Nuclear Instruments and Methods in Physics ResearchSection B: Beam Interactions with Materials and Atoms, vol. 70, no. 1-4, pp. 286–297, 1992.[13] D. J. Morrissey, B. M. Sherrill, M. Steiner, A. Stolz, and I. Wiedenho-ever, “Commissioning the A1900 projectile fragment separator,” Nu-clear Instruments and Methods in Physics Research Section B: BeamInteractions with Materials and Atoms, vol. 204, pp. 90–96, 2003.[14] A. Artukh, Y. M. Sereda, S. Klygin, G. Kononenko, Y. G. Teterev,A. Vorontzov, G. Kaminski, B. Erdemchimeg, V. Ostashko, Y. N.Pavlenko, et al., “The COMBAS fragment separator,” Instruments andExperimental Techniques, vol. 54, no. 5, pp. 668–681, 2011.[15] P. G. Bricault, F. Ames, M. Dombsky, P. Kunz, and J. Lassen, “Rareisotope beams at isactarget & ion source systems,” Hyperfine Interac-tions, vol. 225, pp. 25–49, 2014.125Bibliography[16] M. Dombsky and P. Kunz, “Isac targets,” Hyperfine Interactions,vol. 225, pp. 17–23, 2014.[17] R. Kirchner, “Review of ISOL targetion-source systems,” Nuclear In-struments and Methods in Physics Research Section B: Beam Interac-tions with Materials and Atoms, vol. 204, pp. 179 – 190, 2003. 14thInternational Conference on Electromagnetic Isotope Separators andTechniques Related to their Applications.[18] R. Kirchner, “Progress in ion source development for on-line separa-tors,” Nuclear Instruments and Methods in Physics Research, vol. 186,no. 1, pp. 275 – 293, 1981.[19] R. Kirchner and E. Roeckl, “Investigation of gaseous discharge ionsources for isotope separation on-line,” Nuclear Instruments and Meth-ods, vol. 133, no. 2, pp. 187–204, 1976.[20] R. Geller, “Highly charged ECR ion sources: Summary and comments(invited),” Review of Scientific Instruments, vol. 61, no. 1, 1990.[21] J. Lassen, P. Bricault, M. Dombsky, J. Lavoie, C. Geppert, andK. Wendt, “Resonant ionization laser ion source project at TRIUMF,”Hyperfine Interactions, vol. 162, no. 1-4, pp. 69–75, 2005.[22] J. Dilling and R. Kru¨cken, “The experimental facilities at ISAC,” Hy-perfine Interactions, vol. 225, no. 1-3, pp. 111–114, 2014.[23] R. Baartman, F. Bach, Y. Bylinsky, J. Cessford, G. Dutto, D. Gray,A. Hurst, K. Jayamanna, M. Mouat, Y. Rao, et al., “Reliable pro-duction of multiple high intensity beams with the 500 mev TRIUMFcyclotron,” Cyclotrons and Their Application, Lanzhou, China, p. 280.[24] I. Bylinskii and M. Craddock, “The TRIUMF 500 mev cyclotron: thedriver accelerator,” Hyperfine Interactions, vol. 225, no. 1-3, pp. 9–16,2014.126Bibliography[25] P. Kunz, C. Andreoiu, P. Bricault, M. Dombsky, J. Lassen, A. Teigelh-fer, H. Heggen, and F. Wong, “Nuclear and in-source laser spectroscopywith the isac yield station,” Review of Scientific Instruments, vol. 85,no. 5, 2014.[26] G. Dutto, R. A. Baartman, P. G. Bricault, I. Bylinskii, A. Hurst, R. E.Laxdal, Y.-N. Rao, L. Root, P. Schmor, G. Stinson, et al., “Simul-taneous extraction of two stable beams for ISAC,” European ParticleAccelerator Conference, Genova, ITALY, p. 3505, 2008.[27] I. Bylinskii, R. Baartman, G. Dutto, K. Fong, A. Hurst, M. Laverty,C. Mark, F. Mammarella, M. McDonald, A. Mitra, et al., “TRIUMF500 mev cyclotron refurbishment,” Cyclotrons and Their Application,Giardini Naxos, ITALY, p. 143, 2007.[28] P. Bricault, R. Baartman, M. Dombsky, A. Hurst, C. Mark, G. Stan-ford, and P. Schmor, “TRIUMF-ISAC target station and mass separa-tor commissioning,” Nuclear Physics A, vol. 701, no. 14, pp. 49 – 53,2002. 5th International Conference on Radioactive Nuclear Beams.[29] P. G. Bricault and H. Weick, “Use of the chalk river mass separator forISAC,” TRIUMF Internal Design Note, TRI-DN-97-16, 1997.[30] F. Ames, R. Baartman, P. Bricault, K. Jayamanna, M. McDonald,M. Olivo, P. Schmor, D. H. L. Yuan, and T. Lamy, “Charge statebreeding of radioactive ions with an electron cyclotron resonance ionsource at triumf,” Review of Scientific Instruments, vol. 77, no. 3, 2006.[31] F. Ames, R. Baartman, P. Bricault, and K. Jayamanna, “Chargestate breeding of radioactive isotopes for isac,” Hyperfine interactions,vol. 225, no. 1-3, pp. 63–67, 2014.[32] R. Laxdal and M. Marchetto, “The isac post-accelerator,” HyperfineInteractions, vol. 225, no. 1-3, pp. 79–97, 2014.[33] R. Poirier, R. Baartman, P. Bricault, K. Fong, S. Koscielniak, R. Lax-dal, A. Mitra, L. Root, G. Stanford, and D. Pearce, “CW performance127Bibliographyof the TRIUMF 8 meter long RFQ for exotic ions,” Linear AcceleratorConference, Monterey, California, USA, p. 1023, 2000.[34] I. Kapchinskii and V. Teplyakov, “A linear ion accelerator with spa-tially uniform hard focusing,” Prib. Tekh. Eksp., vol. 1970, no. SLAC-TRANS-0099, pp. 19–22, 1969.[35] S. R. Koscielniak, R. Laxdal, R. Lee, and L. Root, “Beam dynamicsstudies on the ISAC RFQ at TRIUMF,” Particle Accelerator Confer-ence, Vancouver, British Columbia, CANADA, vol. 1, pp. 1102–1104,1997.[36] R. Laxdal, G. Dutto, K. Fong, G. Mackenzie, M. Pasini, R. Poirier, andR. Ruegg, “Beam commissioning and first operation of the ISAC DTLat TRIUMF,” Particle Accelerator Conference, Chicago, Illinois, USA,vol. 5, pp. 3942–3944, 2001.[37] R. Laxdal, “Commissioning and early experiments with ISAC-II,” Par-ticle Accelerator Conference, Albuquerque, New Mexico, USA, pp. 2593–2597, 2007.[38] C. N. Davids and D. Peterson, “A compact high-resolution isobar sep-arator for the CARIBU project,” Nuclear Instruments and Methodsin Physics Research Section B: Beam Interactions with Materials andAtoms, vol. 266, no. 1920, pp. 4449 – 4453, 2008. Proceedings of the{XVth} International Conference on Electromagnetic Isotope Separa-tors and Techniques Related to their Applications.[39] M. Dworschak, G. Audi, K. Blaum, P. Delahaye, S. George, U. Hager,F. Herfurth, A. Herlert, A. Kellerbauer, H.-J. Kluge, et al., “Restora-tion of the n= 82 shell gap from direct mass measurements of sn 132,134,” Physical review letters, vol. 100, no. 7, p. 072501, 2008.[40] K. Jones, A. Adekola, D. Bardayan, J. Blackmon, K. Chae, K. Chipps,J. Cizewski, L. Erikson, C. Harlin, R. Hatarik, et al., “The magic natureof 132sn explored through the single-particle states of 133sn,” Nature,vol. 465, pp. 454–457, 2010.128Bibliography[41] K. Makino and M. Berz, “COSY INFINITY version 9,” Nuclear Instru-ments and Methods in Physics Research Section A: Accelerators, Spec-trometers, Detectors and Associated Equipment, vol. 558, no. 1, pp. 346– 350, 2006. Proceedings of the 8th International Computational Accel-erator Physics ConferenceICAP 20048th International ComputationalAccelerator Physics Conference.[42] J. T. Tanabe, “Iron dominated electromagnets: Design, fabrication,assembly and measurements,” IRON DOMINATED ELECTROMAG-NETS: DESIGN, FABRICATION, ASSEMBLY AND MEASURE-MENTS. Edited by TANABE JACK T. Published by World ScientificPress, 2005. ISBN# 9789812567642, 2005.[43] J. F. Ostiguy, “Longitudinal profile and effective length of a conven-tional dipole magnet,” Particle Accelerator Conference, Washington,D.C., USA, pp. 2901–2903 vol.4, 1993.[44] W. Rogowski, “Die elektrische festigkeit am rande des plattenkonden-sators,” Archiv fu¨r Elektrotechnik, vol. 12, no. 1, pp. 1–15, 1923.[45] P. Sarma, A. D. Gupta, C. Nandi, S. Chattopadhyay, and G. Pal,“Unconventional purcell filter in superferric magnets in the facilityfor antiproton and ion research,” Nuclear Instruments and Methodsin Physics Research Section A: Accelerators, Spectrometers, Detectorsand Associated Equipment, vol. 729, pp. 718 – 724, 2013.[46] T. Kurtukian-Nieto, R. Baartman, B. Blank, T. Chiron, C. Davids,F. Delalee, M. Duval, S. E. Abbeir, A. Fournier, D. Lunney, F. Mot,L. Serani, M.-H. Stodel, F. Varenne, and H. Weick, “SPIRAL2/DESIRhigh resolution mass separator,” Nuclear Instruments and Methods inPhysics Research Section B: Beam Interactions with Materials andAtoms, vol. 317, Part B, pp. 284 – 289, 2013. {XVIth} InternationalConference on ElectroMagnetic Isotope Separators and Techniques Re-lated to their Applications, December 27, 2012 at Matsue, Japan.129Part IAppendices130Appendix AHill’s EquationThe transverse motion of the particles in a periodic beam transport channel(beam line) is governed by Hill’s equation26:d2x(s)ds2+ k(s)x(s) = 0 (A.1)where for a periodic transport system k(s) is a periodic function of thevariable s that is the path length along the reference trajectory. The k(s)function describes the focusing strength along the transport channel.The general solution of equation A.1 is:x(s) =√ε0 β˜(s) cos(φ(s) + φ0) (A.2)From equation A.2 we calculate the first derivative dds (indicated with′):x′(s) =12ε0√ε0 β˜(s)β˜′(s) cos(φ(s)+φ0)−√ε0 β˜(s) sin(φ(s)+φ0)φ′(s) (A.3)Equation A.3 can be written as:x′(s) = −α˜(s)√ε0β˜(s)cos(φ(s) + φ0)−√ε0β˜(s)sin(φ(s) + φ0) (A.4)26see chapter 7 of [10], or chapter 5 of [11]131Appendix A. Hill’s Equationwith:α˜(s) = −12β˜′(s) (A.5)andφ′(s) =1β˜(s)(A.6)From equation A.2 and A.4 we derive the following second order terms:x2(s) = ε0 β˜(s) cos2µ(s) (A.7)where µ(s) = φ(s) + φ0.x′2(s) =α˜2(s) ε0β˜(s)cos2µ(s) +ε0β˜(s)sin2µ(s) +2 α˜ ε0β˜(s)cosµ(s) sinµ(s) (A.8)andx(s)x′(s) = −α˜(s) ε0 cos2µ(s)− ε0 cosµ(s) sinµ(s) (A.9)Equation A.8 can be rearranged as follow:x′2(s) =α˜2(s) ε0β˜(s)cos2µ(s) +ε0β˜(s)− ε0β˜(s)cos2µ(s) +2 α˜ ε0β˜(s)cosµ(s) sinµ(s) ==2 α˜2(s) ε0β˜(s)cos2µ(s)− α˜2(s) ε0β˜(s)cos2µ(s) +ε0β˜(s)− ε0β˜(s)cos2µ(s) ++2 α˜ ε0β˜(s)cosµ(s) sinµ(s) ==2 α˜(s)β˜(s)(α˜(s) ε0 cos2µ(s) + ε0 cosµ(s) sinµ(s)) +− ε0β˜(s)cos2µ(s) (α˜2(s) + 1) +ε0β˜(s)=132Appendix A. Hill’s Equation= −2 α˜(s)β˜(s)x(s)x′(s)− ε0 β˜(s)β˜(s)cos2µ(s)α˜2(s) + 1β˜(s)+ε0β˜(s)== −2 α˜(s)β˜(s)x(s)x′(s)− x2(s)β˜(s)γ˜(s) +ε0β˜(s)(A.10)where:γ˜(s) =α˜2(s) + 1β˜(s)(A.11)From equation A.10 we then have:γ˜(s)x2(s) + 2 α˜(s)x(s)x′(s) + β˜(s)x′2(s) = ε0 (A.12)Since γ˜(s) β˜(s) − α˜2(s) = 1, equation A.12 represents an ellipse in theCartesian plane centered in the origin with area piε0 (see figure A.1).Figure A.1: Graphic representation of equation A.12 in the Cartesian plane.133Appendix A. Hill’s EquationThe function α˜(s), β˜(s) and γ˜(s) are called the Courant-Snyder param-eters, while ε0 and φ0 are constants determined by the initial conditions.When α˜(0), β˜(0) and γ˜(0) coincide with the input values of a transportchannel (match condition for the particle), then the Courant-Snyder param-eters have the same periodicity as the function k(s) .For a given ε0, the position of a particle in phase space at a location s lieson an ellipse described by equation A.12. When α˜(s), β˜(s) and γ˜(s) havethe same periodicity as k(s), a particle lies on an identical ellipse at everyperiod but in general on a different position of coordinates (x(s), x′(s)).The angular difference between two different positions on identical ellipsesseparated by one period is φ(s) and it is called the phase advance.134Appendix BGaussian Beam DistributionLet’s consider a particle beam that have a bivariate27 Gaussian distributionin (x,x′) with standard deviation respectively σ1 and σ2, the probabilitydensity function of such distribution is:p(x, x′) =12pi σ1 σ2√1− ρ2x,x′e− (x−µ1)22 (1−ρ2x,x′ )σ21+2 ρx,x′ (x−µ1) (x′−µ2)2 (1−ρ2x,x′ )σ1 σ2− (x′−µ2)22 (1−ρ2x,x′ )σ22(B.1)whereρx,x′ = corr(x, x′) (B.2)If we assumed that the distribution is represented by an upright ellipsecentered in the origin, then the distribution is uncorrelated, ρx,x′ = 0, andµ1 = µ2 = 0. Equation B.2 can then be simplified in:p(x, x′) =12pi σ1 σ2e− x22σ21− x′22σ22 (B.3)If we take:σ1 =√ε β˜(s) (B.4)andσ2 =√ε γ˜(s) (B.5)27http://mathworld.wolfram.com/BivariateNormalDistribution.html135Appendix B. Gaussian Beam Distributionthan we have the distribution of a 1σ emittance as represented by the blueellipse in figure B.1 with no correlation between x and x′.Figure B.1: Upright emittances: 1σ (blue) and 4σ (red).From equation B.3 we can calculate the relative number of particle within1σ emittance as follow:P (x, x′) =12pi σ1 σ2+σ1∫−σ1+σ2∫−σ2e−σ22 x2+σ21 x′22σ21σ22 dx dx′ (B.6)Changing variables:x = σ1 a (B.7)andx′ = σ2 b (B.8)136Appendix B. Gaussian Beam DistributionEquation B.6 become with dx = σ1 da and dx′ = σ2 db:P (a, b) =12pi σ1 σ2+1∫−1+1∫−1e−σ22 σ21 a2+σ21 σ21 b22σ21σ22 σ1 σ1 da db ==12pi+1∫−1+1∫−1e−a2+b22 da db (B.9)Moving to cylindrical coordinates:a = r cos(θ) (B.10)andb = r sin(θ) (B.11)Equation B.6 become with da db = r dr dθ:P (r, θ) =12pi2pi∫0+1∫0e−r2 cos2(θ)+r2 sin2(θ)2 r dr dθ ==12pi2pi∫0+1∫0r e−r22 dr dθ ==[12pi+1∫0r e−r22 dr]2pi0==[−e−r22]10=[1− e− 12]= 0.39 (B.12)So if a beam has a Gaussian distribution in both x and x′, then 39% ofthe beam is contained within the 1σ emittance. If we consider 2σ in both137Appendix B. Gaussian Beam Distributionaxes, then equation B.12 yields that 86% of the beam is contained withinthe 4σ emittance (see red ellipse in figure B.1).138

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