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Lattice Design C.C. Kuo 郭錦城 July 27 ~ August 6, 2014 安徽省黃山市休寧

Lattice Design C.C. Kuo 郭錦城 July 27 ~ August 6, 2014 安徽省黃山市休寧. outline. Introduction Linear beam dynamics Linear lattice Nonlinear beam dynamics Example Imperfection, correction and lifetime. Accelerator Lattice.

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Lattice Design C.C. Kuo 郭錦城 July 27 ~ August 6, 2014 安徽省黃山市休寧

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  1. Lattice Design C.C. Kuo 郭錦城 July 27 ~ August 6, 2014 安徽省黃山市休寧 OCPA Accelerator School, 2014, CCKuo- 1

  2. outline • Introduction • Linear beam dynamics • Linear lattice • Nonlinear beam dynamics • Example • Imperfection, correction and lifetime OCPA Accelerator School, 2014, CCKuo- 2

  3. Accelerator Lattice • (Magnet) Lattice: The arrangement of the accelerator elements (usually magnets) along beam path for guiding or focusing charged particles is called “Magnet Lattice” or “Lattice”. • Regularity: The arrangement can be irregular array or repetitive regular array of magnets. A transfer line from one accelerator to another is usually irregular. • Periodicity: The repetitive regular array is called periodic lattice. Usually the circular accelerator lattice is in a periodic form. • Symmetry: In a circular accelerator, the periodic lattice can be symmetric. Usually, the lattice is constructed from cells and then super-periods. A number of super-periods then complete a ring. • Design goal: The goal of lattice design is to obtain simple, reliable, flexible, and high performance accelerators that meet users’ request. OCPA Accelerator School, 2014, CCKuo- 3

  4. BEPCII－双环高亮度正负电子对撞机 超导高频腔 IR超导磁铁 Collider: 1.89GeV SR: 2.5GeV e- e+ e- e+ 馬力 第五届OCPA加速器学校 OCPA Accelerator School, 2014, CCKuo- 4

  5. A Lattice with FODO Arc and Doublet straight 散裂中子源 CSNS RCS 王生 OCPA 2010 Lattice consists of FODO arc (with missing gap) and doublet dispersion free straight section • Arcs: 3.5 FODO cells, 315 degrees phase advance • Straights: doublet, 6.5×2+9.3 m long drifts at each straight • Gap of Dipole : 175mm • Max. Aper. of Quadrupoles: 308mm OCPA Accelerator School, 2014, CCKuo- 5

  6. A Lattice with Triplet Cells 散裂中子源 CSNS RCS 王生 OCPA 2010 Lattice consists of 16 triplet cells, with a gap in the middle of arc and dispersion free straight section . • Arcs: Triplet cells as achromatic insertion • Straights: Triplet, 3.85×2+11 m long drifts at each straight • Gap of Dipole : 160mm • Max. Aper. of Quadrupoles: 265mm OCPA Accelerator School, 2014, CCKuo- 6

  7. SSRF • 150 MeVLinac • 3.5 GeV Booster • 3.5 GeV Storage Ring 20 DBA cells 趙振堂等 OCPA Accelerator School, 2014, CCKuo- 7

  8. HLS Hefei Storage Ring 800 MeV 66.13m 劉祖平. OCPA2004_satellite meeting OCPA Accelerator School, 2014, CCKuo- 8

  9. The Hefei Light SourceUpgrade S1 S2 S3 S4 (2012 OCPA School 張闖) E=800MeV Q2 Q1 B Q3 Q4 HLS Upgrading e<40 nm·rad I>300 mA OCPA Accelerator School, 2014, CCKuo- 9

  10. 偏轉磁鐵 Bending Magnet 四極磁鐵Quadrupole Magnet 注射脈衝磁鐵Pulsed Injection Magnet 光束線Beamline 六極磁鐵Sextupole Magnet 高頻系統 RF Cavity 插件磁鐵 Insertion Device 儲存環Storage Ring (1.51 GeV) 線型加速器 LINAC 傳輸線Transport Line 增能環Booster Ring (1.51 GeV) 6 TBA NSRRC TLS FODO cell in TLS booster OCPA Accelerator School, 2014, CCKuo- 10

  11. TPS Energy : 3 GeV Beam current: 500 mA Emittance: 1.6 nm-rad. Straight Section: 7 m (18); 12 m (6) Lattice structure: Double-Bend Circumference: 518.4 m RF: 500 MHz Linac energy : 150 MeV Booster energy: 3 GeV Booster circumference: 496.8 m Booster emittance: 10 nm-rad Lattice structure: MBA Repetition rate: 3 Hz OCPA Accelerator School, 2014, CCKuo- 11

  12. CSRmLayout Accumulator Cooler Synchrotron Fast extraction Slow extraction External target Internal target 12 Tm 12GeV—C6+, 120GeV—U72+ OCPA2012 H1-Hadron Accelerator Complex 夏佳文 OCPA-School-08 OCPA Accelerator School, 2014, CCKuo- 12

  13. Lattice Design Procedure-(1) • For a modern accelerator, lattice design work usually takes some years to finalize the design parameters. It is an iterative process, involving users, funding, accelerator physics, accelerator subsystems, civil engineering, etc. • It starts from major parameters such as energy, size, etc. • Then linear lattice is constructed based on the building blocks. Linear lattice should fulfill accelerator physics criteria and provide global quantities such as circumference, emittance, betatron tunes, magnet strengths, and some other machine parameters. OCPA Accelerator School, 2014, CCKuo- 13

  14. Lattice Design Procedure-(2) • Design codes such as MAD, OPA, BETA, Tracy, Elegant, AT, BeamOptics,…. are used for the matching of lattice functions and parameters calculations. • Usually, a design with periodic cells is needed in a circular machine. The cell can be FODO, Double Bend Achromat (DBA), Triple Bend Achromat (TBA), Quadruple Bend Achromat (QBA), Multi-Bend Achromat (MBA or nBA) or some combination types. • Combined function or separated function magnets are selected. • Maximum magnetic field strengths are constrained. (room- temperature or superconducting magnets, bore radius or chamber profile, etc.) • Using matching subroutines to get desired machine functions and parameters. OCPA Accelerator School, 2014, CCKuo- 14

  15. Lattice Design Procedure-(3) • To get stable solution of the off-momentum particle, we need to put sextupole magnets and RF cavities in the lattice beam line. Such nonlinear elements induce nonlinear beam dynamics and the dynamic acceptances in the transverse and longitudinal planes need to be carefully studied in order to get sufficient acceptances. (for long beam current lifetime and high injection efficiency) • For the modern high performance machines, strong sextupole fields to correct high chromaticity will have large impact on the nonlinear beam dynamics and it is the most challenging and laborious work at this stage. OCPA Accelerator School, 2014, CCKuo- 15

  16. Lattice Design Procedure-(4) • In the real machines, there are always imperfections in the accelerator elements. Therefore, one needs to consider engineering/alignment limitations or errors, vibrations, etc. Correction schemes such as orbit correction, coupling correction, etc., need to be developed. (dipole correctors, skew quadrupoles, beam position monitors, etc) • Make sure long enough beam current lifetime, e.g., Touschek lifetime, in the real machine including insertion devices, etc. • To achieve a successful accelerator, we need to consider not only lattice design but many issues, which might be covered in this school. OCPA Accelerator School, 2014, CCKuo- 16

  17. Lattice Design Process for a Light Source • User requirements • Lattice type (FODO, DBA, TBA, QBA, MBA) • Linear lattice • Nonlinear dynamic aperture tracking • Longitudinal dynamics • Error tolerance analysis • Satisfying user requirements (if NO, go back to step 2) • ID effect, lifetime, COD, coupling, orbit stability, instabilities, injection, etc. OCPA Accelerator School, 2014, CCKuo- 17

  18. Circular machine need dipole magnets Lorentz force = centrifugal force For relativistic particle TLS: 1.5 GeV, B=1.43T, r=3.49m, Circumference=120m; LHC: 7000GeV, B=8.3T, r=2.53km, Circumference=27km • Total required dipole magnet length = 2pr • Synchrotron radiation loss per turn for electron ~ 8.85x10-5β3E4/r [GeV] and for proton ~ 7.78x10-18β3E4/r [GeV] • Usually for the high energy particle with constrained ring circumference, maximum energy is limited by synchrotron radiation power compensation for electron/positron, while proton /heavy ion are limited by dipole magnetic field strength. OCPA Accelerator School, 2014, CCKuo- 18

  19. Basic magnet elements • Need quadrupoles to get restoring force for particle deviating from the ideal (design) orbit. • Quadrupole field can be combined into a dipole magnet and this is called combined-function magnet. • Need sextupoles to provide chromaticity corrections. • Sextupole also can be combined into dipole and/or quadrupolemagnet. • Dipole correctors and skew quadrupoles are used for orbit and coupling control. These elements can be separated or combined into other magnets. • Octupoles can be used to get tune spread as a function of betatron oscillation amplitude and help reduce beam instabilities. • Higher-order magnetic multi-pole fields are usually unwanted and inevitable in real magnets. These fields need to be minimized in design and manufacturing to avoid the shrinkage of the “dynamic aperture”—the maximum allowed particle motion in transverse planes. It is usually around 100 ppm in the good field region. OCPA Accelerator School, 2014, CCKuo- 19

  20. Magnetic field expression For transverse fields in Frenet-Serret (curvilinear ) coordinate system: OCPA Accelerator School, 2014, CCKuo- 20

  21. Frenet-Serret (curvilinear ) coordinate system, negative charge case: Betatron equation of motion (negative charge particle) : OCPA Accelerator School, 2014, CCKuo- 21

  22. Frenet-Serret (curvilinear ) coordinate system, negative charge : Hill’s equation for on-momentum particle : OCPA Accelerator School, 2014, CCKuo- 22

  23. Frenet-Serret (curvilinear ) coordinate system, negative charge : Hill’s equation linear motionfor off-momentum particle (assuming DBz,x=0): Neglecting Chromatic perturbation term OCPA Accelerator School, 2014, CCKuo- 23

  24. Main magnets quadrupole dipole B field limits (example): Dipole < 1.5T Quad poletip < 0.7T Sextpoletip < 0.4T sextupole OCPA Accelerator School, 2014, CCKuo- 24

  25. Matrix formalism in linear beam dynamics: (1) Focusing quadrupole (2) Defocusing quadrupole (3) Drift space K=0 OCPA Accelerator School, 2014, CCKuo- 25

  26. (4) pure sector dipole: non-deflecting plane (ignoring fringe field): OCPA Accelerator School, 2014, CCKuo- 26

  27. (5) pure rectangular dipole due to wedge in both ends: In deflecting plane: In non-deflecting plane (neglecting fringe field): OCPA Accelerator School, 2014, CCKuo- 27

  28. Beam dynamics in transport line • In open transport lines the phase space can be transferred using transfer matrix piecewise. We need initial condition. • In terms of Courant-Snyder parameters, there are relations between initial and final points along the beam path. OCPA Accelerator School, 2014, CCKuo- 28

  29. Courant –Snyder Parameters in periodic cells where transformation is for one period or one turn. Using similarity transformation for any beam line: OCPA Accelerator School, 2014, CCKuo- 29

  30. Floquet transformation Hill’s equation: Transformation matrix in one period: OCPA Accelerator School, 2014, CCKuo- 30

  31. OCPA Accelerator School, 2014, CCKuo- 31

  32. Stability of FODO Cell and Optimum Phase Advance Per Cell FODO cell: Maximum betatron function can be minimized with optimum phase advance per FODO cell: OCPA Accelerator School, 2014, CCKuo- 32

  33. L1:DRIFT,L=1 L2:DRIFT,L=1 QFH :QUADRUPOLE,L=.5/2, K1=0.5 QDH :QUADRUPOLE,L=.5/2, K1=-0.5 BD :SBEND,L=3,ANGLE=TWOPI/96 HSUP :LINE=(QFH,L1,BD,L2,QDH) FSUP :LINE=(HSUP,-HSUP) FODO :LINE=(2*FSUP) RING :LINE=(48*FSUP,) Minimizing beta in one planephase advance of 76.3° each FODO cell Minimizing beta in both planes  phase advance of 90° each FODO cell OCPA Accelerator School, 2014, CCKuo- 33

  34. Normalized coordinate: A simple harmonic oscillation in normalized coordinate. Courant-Snyder Invariant and Emittance: OCPA Accelerator School, 2014, CCKuo- 34

  35. a M. Sands SLAC-121 OCPA Accelerator School, 2014, CCKuo- 35

  36. Off-momentum orbit To the lowest order in OCPA Accelerator School, 2014, CCKuo- 36

  37. Off-momentum orbit pure sector dipole: pure rectangular dipole: thin lens quadrupole: OCPA Accelerator School, 2014, CCKuo- 37

  38. Dispersion for periodic lattice For periodic lattice, the closed-orbit condition: OCPA Accelerator School, 2014, CCKuo- 38

  39. Dispersion in a FODO Cell FODO cell—thin lens approximation: OCPA Accelerator School, 2014, CCKuo- 39

  40. Summary in a FODO cell OCPA Accelerator School, 2014, CCKuo- 40

  41. Integral Representation of Dispersion Function Same as dipole field error representation, we can substitute Jd is constant in the region without dipole. OCPA Accelerator School, 2014, CCKuo- 41

  42. Achromatic Lattice • In principle, dispersion can be suppressed by one focusing quadrupole and one bending magnet, if no strength and space constraints. • With one focusing quad in the middle between two dipoles, one can get achromat condition. Due to mirror symmetry of the lattice w.r.t. to the middle quad, one can find, by simple matrix manipulation, D’ at quad center should be zero to get achromat solution. • Beam line in between two bend is called arc section. Outside arc section, we can match dispersion to zero. This is so called double bend achromat (DBA) structure. • We need quads outside arc section to match the betatron functions, tunes, etc. • Similarly, one can design triple bend achromat (TBA), quadruple bend achromat (QBA), and multi-bend achromat (MBA or nBA) structure. • For FODO cells structure, dispersion suppression section at both ends of the standard cells. OCPA Accelerator School, 2014, CCKuo- 42

  43. DBA Consider a simple DBA cell with a single quadrupole in the middle. In thin-lens approximation, the dispersion matching condition Where f is the focal length of half quad, θ and L are the bending angle and dipole length, L1 is the distance between end of dipole to center of quad. It shows the quad strength becomes smaller with longer distance and the dispersion at quad center becomes higher with longer distance and larger bend angle. By splitting quad into two pieces and being moved away from the center symmetrically, we can reduce the dispersion function and also quad strength. OCPA Accelerator School, 2014, CCKuo- 43

  44. DBA-1 • R. Chasman and K. Green proposed in 1975. • One quad between two bends in the arc. • Also called Chasman-Green lattice. • Doublet or triplet quads in straights for beta function control. • Less flexibility. • Usually emittance is high. • Chromaticity is not so high, sextupole scheme is simple. • Note: both rings with combined-function dipoles. NSLS-Xray ring 2.5GeV, 8-fold NSLS-VUV ring 0.7GeV, 4-fold OCPA Accelerator School, 2014, CCKuo- 44

  45. DBA-2 • Expanded C-G lattice with 4 quads in between 2 bends as shown for ESRF lattice. (6GeV 32 cells, 6.7 nm-rad) • Chromaticity is high (-131, -31), need more families of sextrupoles for nonlinear beam dynamics corrections • Quadrupole triplets in straights for beta function control. • Emittance reduction by breaking achromat. • Upgrade program for lower emittance and increase of straight length for more IDs. • APS also use same scheme and upgrade program is in progress. ESRF APS 7 GeV, 40-fold ESRF upgrade OCPA Accelerator School, 2014, CCKuo- 45

  46. DBA-3 • Expanded C-G lattice in Elettra (2GeV, 260m, 7 nm-rad) • Three quads in between 2 bends • Can optimize emittance to close to theoretical values. • But the drift space in the arc is too long. ID length is limited. • Chromaticity (-41, -13). Use harmonic sextupoles. OCPA Accelerator School, 2014, CCKuo- 46

  47. DBA-4 SSRF 4-fold 2.86 nm-rad at 3.5 GeV • Components compacted C-G latticeis the trend in the modern light sources. • Breaking achromat (dispersion in the ID straights) is usually adopted. Emittance is extremely low. • Need harmonic sextupole families to optimize beam dynamics effects. • ID lengths are varied for different experimental purpose. Long straight can accommodate more IDs in one straight. TPS Half super-period OCPA Accelerator School, 2014, CCKuo- 47

  48. DBA-5 • Increase useful straights in between 2 bends in the arcs. • Straights can be three types. SOLEIL 2.75 GeV 4-fold ALBA 3 GeV 4-fold OCPA Accelerator School, 2014, CCKuo- 48

  49. TBA • Triple-bend achromat (TBA) structure with better flexibility compared with the C-G lattice. • ALS, TLS, PLS-I, SLS, HLS… adopted this type. • In TLS, ALS, PLS-I,… no harmonic sextupoles needed. • Lower the emittance in these lattices need add harmonic sextupoles for nonlinear optimization. • SLS has 8°—14°—8° bend angle in the TBA structure and lower the emittance. • SLS has three types of straights. • SLS uses harmonic sextupoles. TLS ALS SLS 4.5 nmm 2.4 GeV OCPA Accelerator School, 2014, CCKuo- 49

  50. QBA B1 B2 B1 B2 B1=8.823° B2=6.176° TPS QBA 518.4m Tune x:26.27 Tune y: 13.30 Emittance:3.2 nm Nat. chromaticity x:-65.8 Y: -27.3 OCPA Accelerator School, 2014, CCKuo- 50

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