Report

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 outline • • • • • • Introduction Linear beam dynamics Linear lattice Nonlinear beam dynamics Example Imperfection, correction and lifetime OCPA Accelerator School, 2014, CCKuo- 2 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 BEPCII－双环高亮度正负电子对撞机 超导高频 腔 IR超导磁 Collider: 1.89GeV 铁 SR: 2.5GeV e- e+ e+ e馬力 第五届OCPA加速器学校 OCPA Accelerator School, 2014, CCKuo- 4 王生 OCPA 2010 散裂中子源 CSNS RCS A Lattice with FODO Arc and Doublet straight 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 王生 OCPA 2010 散裂中子源 CSNS RCS A Lattice with Triplet Cells 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 SSRF 150 MeV Linac 3.5 GeV Booster 3.5 GeV Storage Ring 20 DBA cells 趙振堂等 OCPA Accelerator School, 2014, CCKuo- 7 HLS Hefei Storage Ring Length(m) 25 0 2 4 x 20 y x,y(m) 15 6 8 10 12 14 16 Theoretical Curve of x Measured Value of x Theoretical Curve of y Measured Value of y 10 5 0 Q1 -5 Q4 SF Q2 B1 SD Q3 SF Q5 B2 800 MeV 66.13m 劉祖平. OCPA2004_satellite meeting OCPA Accelerator School, 2014, CCKuo- 8 Q6 SD Q8 B3 Q7 The Hefei Light Source Upgrade (2012 OCPA School 張闖) E=800MeV S1 S2 Q1 Q2 S3 B Q3 S4 Q4 HLS Upgrading e<40 nm·rad I>300 mA OCPA Accelerator School, 2014, CCKuo- 9 NSRRC TLS 6 TBA 偏轉磁鐵 Bending Magnet 四極磁鐵 Quadrupole Magnet 注射脈衝磁鐵 Pulsed Injection Magnet 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) FODO cell in TLS booster OCPA Accelerator School, 2014, CCKuo- 10 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 Layout Accumulator Cooler Synchrotron Fast extraction Slow extraction CSRm CSRe Circumference (m) 161.0014 128.8011 Average radius (m) 8RSSC=34RSFC=25.62416 4/5RCSRm =20.499328 Geometry Race-track Race-track 2800 (p) 1100 (C6+) 500 (U72+) 2000 (p) 750 (C6+) 500 (U92+) 0.81/12.05 0.50/9.40 0.10/1.60 0.08/1.60 Ramping rate (T/s) 0.05 ~ 0.4 -0.1 ~ -0.2 Repeating circle (s) ~17 (~10s for Accumulation ) Acceptance Fast-extraction mode Max. energy (MeV/u) External target Internal target B (Tm) B(T) A h A v ( mm-mrad) ( mm-mrad) p/p (%) 200 (p/p = 0.3 %) Normal mode 150 (p/p =0.5%) 40 75 1.4 2.6 (eh= 50 mm-mrad) (eh= 10 mm-mrad) 12 Tm 12GeV—C6+, 120GeV—U72+ OCPA2012 H1-Hadron Accelerator Complex 夏佳文 OCPA Accelerator School, 2014, CCKuo- 12 OCPA-School-08 Lattice Design Procedure-(1) 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. 2) It starts from major parameters such as energy, size, etc. 3) 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 Lattice Design Procedure-(2) 4) Design codes such as MAD, OPA, BETA, Tracy, Elegant, AT, BeamOptics,…. are used for the matching of lattice functions and parameters calculations. 5) 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. (roomtemperature 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 Lattice Design Procedure-(3) 6) 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) 7) 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 Lattice Design Procedure-(4) 8) 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) 9) Make sure long enough beam current lifetime, e.g., Touschek lifetime, in the real machine including insertion devices, etc. 10) 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 Lattice Design Process for a Light Source 1) 2) 3) 4) 5) 6) 7) 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) 8) ID effect, lifetime, COD, coupling, orbit stability, instabilities, injection, etc. OCPA Accelerator School, 2014, CCKuo- 17 Circular machine need dipole magnets Lorentz force = centrifugal force m0v 2 FL evB Fcentr m0 v e p B e For relativistic particle B[T ] B(Vs / m2 ) [1 / m] e e , E in GeV 9 p E 10 (eV ) / c 1 Note : for ion 1 [1 / m] 0.3 1 [1 / m] 0.3 B[T ] E[GeV / u ] Z B[T ] A E[GeV / u ] TLS: 1.5 GeV, B=1.43T, =3.49m, Circumference=120m; LHC: 7000GeV, B=8.3T, =2.53km, Circumference=27km • Total required dipole magnet length = 2 • Synchrotron radiation loss per turn for electron ~ 8.85x10-5β3E4/ [GeV] and for proton ~ 7.78x10-18β3E4/ [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 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 quadrupole magnet. • 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 Magnetic field expression For transverse fields in Frenet-Serret (curvilinear ) coordinate system: 1 1 ( Bz jBx ) (bn jan )(x jz ) n , B0 n 0 1 n Bz 1 n Bx bn | , an | x z 0 n x z 0 n B0 n! x B0 n! x : posit ivecharge(upper),negat ive(lower) b0 : dipole Bz B0b0 a0 : vert.dipole b1 : quad Bz B0b1 x, Bx B0b1 z a1 : skew quad b2 : sext Bx B0 a0 Bz B0 a1 z , Bx B0 a1 x Bz B0b2 ( x 2 z 2 ), Bx 2 B0b2 xz a2 : skew sext Bz 2 B0 a2 xz, Bx B0 a2 ( x 2 z 2 ) kn n 1 n Bz Bz ( x) |z 0 B0 x , kn B0 x n n n! OCPA Accelerator School, 2014, CCKuo- 20 define focusing: kn 0 Frenet-Serret (curvilinear ) coordinate system, negative charge case: F dp / dt ev B B Bx xˆ Bz zˆ 2 v 2 s Bz x ( x ) B0 2 v B z s x B0 Betatron equation of motion (negative charge particle) : d 2x s vt, x v x, x 2 ds 2 x Bz p0 x 2 x ( 1 ) 2 B0 p z Bx p0 (1 x ) 2 B0 p OCPA Accelerator School, 2014, CCKuo- 21 Frenet-Serret (curvilinear ) coordinate system, negative charge : x Bz p0 x 2 x ( 1 ) 2 B0 p z Bx p0 (1 x ) 2 B0 p Hill’s equation for on-momentum particle : Bz B0 ( 1 1 Bz B x z ) B0 x B0 Bz B0 B0 (k x x Bz ) B0 B z x K ( s ) x x B0 z K ( s ) z Bx z B0 1 1 Bz 1 Bx K x (s) 2 , K z ( s) , B0 x B0 z OCPA Accelerator School, 2014, CCKuo- 22 Frenet-Serret (curvilinear ) coordinate system, negative charge : x Bz p0 x 2 x ( 1 ) 2 B0 p z Bx p0 (1 x ) 2 B0 p Bz B0 ( 1 1 Bz B x z ) B0 x B0 Bz B0 B0 (k x x Bz ) B0 Hill’s equation linear motion for off-momentum particle (assuming Bz,x=0) : x ( k x ( s) 1 1 Bz p ) x , k ( s ) , x 2 (1 ) 1 (1 ) B0 x p0 then x ( K x (s) K x (s))x 0 where K x ( s) 1 2 D ( K x ( s) K x ( s))D k x ( s), K x [ Neglecting Chromatic perturbation term K x (s) Let x x (s) D(s) 2 2 k x ( s)] o( 2 ) x ( s) K x ( s) x ( s) 0 D( s) K ( s) D( s) 1 x (s) OCPA Accelerator School, 2014, CCKuo- 23 1 o( ) Main magnets Bx B1 z , Bz B1 x B1 dipole B 2 0 NI R2 quadrupole 0 NI h B field limits (example): Dipole < 1.5T Quad poletip < 0.7T Sext poletip < 0.4T Bx B2 xz , Bz 1 B2 ( x 2 z 2 ) 2 sextupole B2 60 NI / R 3 OCPA Accelerator School, 2014, CCKuo- 24 Matrix formalism in linear beam dynamics: y ( s0 ) y(s) M ( s, s0 ) y (s) y ( s0 ) (1) Focusing quadrupole (2) Defocusing quadrupole (3) Drift space K=0 1 sin K cos K M ( s , s0 ) K , K 0, s s0 K sin K cos K 0 1 1 , 0, f (focallengt h) lim 0 | K | 1/ f 1 1 sinh | K | cosh | K | M ( s , s0 ) |K| , K 0, s s0 | K | sinh | K | cosh | K | 0 1 , for 0 1 / f 1 1 M ( s, s0 ) , s s0 0 1 OCPA Accelerator School, 2014, CCKuo- 25 (4) pure sector dipole: sin cos M x ( s , s0 ) 1 , sin cos / 1 M x ( s, s0 ) for small / 0 1 non-deflecting plane (ignoring fringe field): 1 M z ( s, s0 ) 0 1 OCPA Accelerator School, 2014, CCKuo- 26 (5) pure rectangular dipole due to wedge in both ends: In deflecting plane: 1 M x ( s, s0 ) 1 / f x 1 M x ( s, s0 ) 0 0 cos 1 1 sin sin 1 cos 1 / f x 0 1 sin 1 1 1 where tan( ) fx 2 In non-deflecting plane (neglecting fringe field): 1 M z ( s, s0 ) 1/ f z 0 1 1 1 0 1 1 / f z 1 fz M z ( s, s0 ) 2 2 f z fz 0 1 1 , where tan( ) 1 fz 2 cos 1 sin 1 fz sin if cos OCPA Accelerator School, 2014, CCKuo- 27 Beam dynamics in transport line • In open transport lines the phase space can be transferred using transfer matrix piecewise. We need initial condition. x x M (sn , sn1 )...M (s3 , s2 )M (s2 , s1 ) x' sn x' s1 • In terms of Courant-Snyder parameters, there are relations between initial and final points along the beam path. x x M (s2 , s1 ) x' s 2 x' s1 2 (cos 1 sin ) 1 M ( s2 , s1 ) 1 1 2 2 sin 1 cos 1 2 1 2 OCPA Accelerator School, 2014, CCKuo- 28 1 (cos 2 sin ) 2 1 2 sin Courant –Snyder Parameters in periodic cells M 11 M 21 M J M 12 M 22 M sin cos sin I cos J sin cos sin sin , T race(J ) 0, J 2 I 1 2 eigenvalues ei cos i sin M k ( I cos J sin )k I cosk J sin k cos1[(M11 M 22 ) / 2] M12 / sin (M11 M 22 ) / 2 sin where transformation is for one period or one turn. Using similarity transformation for any beam line: M(s2 ) M(s2 , s1 )M(s1 )M(s2 , s1 )1 2 2 M 11 2 M 11M 21 M 2 21 2 2M 11M 22 M 11M 22 M 12 M 21 2M 21M 22 1 M 12 M 22 1 M 222 1 OCPA Accelerator School, 2014, CCKuo- 29 M 122 Floquet transformation y K (s) y 0, K (s L) K (s) Hill’s equation: y(s) aw(s)e i ( s ) w( s L) w( s) 1 w K ( s ) w 3 0 w 1 w2 (s L) (s) Transformation matrix in one period: cos ww sin cos sin w2 sin sin 2 M 1 ( ww) sin cos ww sin sin cos sin 2 w (s) 2 1 ( s) (s) w2 (s), (s) , ( s) 2 ( s) OCPA Accelerator School, 2014, CCKuo- 30 y( s) a e i d ds , ( s s0 ) , ( ) (s) s0 0 s L 2 (cos 1 sin ) 1 M( s2 , s1 ) 1 1 2 2 sin 1 cos 1 2 1 2 2 M( s2 , s1 ) 2 2 0 cos 1 sin 2 1 (cos 2 sin ) 2 sin cos 1 2 sin 1 1 1 1 OCPA Accelerator School, 2014, CCKuo- 31 0 1 Stability of FODO Cell and Optimum Phase Advance Per Cell FODO cell: M FODO L2 1 2 2 f * 1/ f 1 1 QF L QD L QF , f 1 / k, half quad length 2 2 L ) f * 2 2 with f f f d f , 1 / f 2(1 L / f )(L / f ) L 1 2 2 f 2 L(1 x (middle of QF) x L ( 1) x (middle of QD) x L 1 ( 1) 2 with f / L 1 f L 2 1 Maximum betatron function can be minimized with optimum phase advance per FODO cell: d x 0 0 76.345 d OCPA Accelerator School, 2014, CCKuo- 32 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,) 1 1 QF L QD L QF , f 1 / k, half quad lengt h 2 2 0 1 L 1 0 1 L 1 0 1 0 1 2 / f 1 0 1 1 / f 1 FODO half cell : 1 M 1/ f L2 L 1 2 2 2 L(1 ) f f M wit h f f f d f , 1 / f * 2(1 L / f )(L / f 2 ) 2 L * 1 / f 1 2 f2 sin cos sin sin cos sin (1 sin 2 ) L2 L cos 1 2 , sin 2 , 2 L , 0 f f sin f 1 /(0.25* .5), L 5.5 86.86507 OCPA Accelerator School, 2014, CCKuo- 33 Minimizing beta in one planephase advance of 76.3° each FODO cell Minimizing beta in both planes phase advance of 90° each FODO cell BetatronT une x, z 1 2 ds x , z ( s) General solution: y(s) a cos( 0 ) Normalized coordinate: w w( ) a cosv( 0 ) y , d ( ) 1 s d 2w 2 w 0 A simple harmonic oscillation in normalized coordinate. 2 d Courant-Snyder Invariant and Emittance: C ( y, y ' ) 1 ( y 2 (y y ' ) 2 ) C ( y, y ' ) y 2 2yy' y '2 e emittance 2 J a 2 Beam size : (s) e (s) ( D(s) ) 2 Beam divergence: ' (s) e (s) ( D' (s) ) 2 OCPA Accelerator School, 2014, CCKuo- 34 1 dw 1 (x x ' ) d a w x/ M. Sands SLAC-121 OCPA Accelerator School, 2014, CCKuo- 35 Off-momentum orbit x ( k x ( s) 1 1 Bz ) x , k ( s ) x 2 (1 ) 1 (1 ) B x Let x x (s) D(s) then x (K x (s) K x ) x 0 D ( K x ( s) K x ) D 1 o( ) where K x ( s ) To the lowest order in D K x ( s) D 1 2 k x ( s ), K x [ 2 2 k x ( s )] o( 2 ) 1 D(s L) D(s), D' (s L) D' (s), K (s), (s) are periodic D( s) D ( s0 ) D ( s0 ) M ( s, s0 ) d D( s0 ) M D( s0 ) D( s ) 0 1 1 1 1 1 (1 cos K x s) K x if K 0, d x 1 sin K s x K x 1 (1 cosh | K x |s) | Kx | if K 0 d x 1 | K | sinh | K x |s x OCPA Accelerator School, 2014, CCKuo- 36 Off-momentum orbit D( s) D ( s0 ) D ( s ) M D ( s ) 0 1 1 3 3 matrix pure sector dipole: where K x ( s) cos M x (1 / ) sin 0 sin cos 0 1 2 (1 cos ) sin 1 1 / 2 0 1 0 0 1 0 Mx pure rectangular dipole: Mx 1 0 0 sin 1 (1 cos ) 0 1 2 t an 2 1 thin lens quadrupole: M x -1/f 0 1 / 2 0 1 0 0 1 0 Mx 0 0 1 0 0 1 1 M z 1/f 0 OCPA Accelerator School, 2014, CCKuo- 37 0 0 1 0 0 1 Dispersion for periodic lattice For periodic lattice, the closed-orbit condition: D M 11 D M 21 1 0 M 12 M 22 0 M 13 D M 23 D 1 1 D M 13 (1 M 22 ) M 12 M 23 2 M 11 M 22 D M 13 M 21 (1 M 11 ) M 23 2 M 11 M 22 Solving M13 , M 23 sin (1 cos sin ) D sin D cos sin M sin cos sin sin D (1 cos sin ) D 0 0 1 OCPA Accelerator School, 2014, CCKuo- 38 Dispersion in a FODO Cell FODO cell—thin lens approximation: f 1 / k, half quad length,L half cell length M 1/ 2 FODO 1 1 / f 0 0 0 1 L L2 /(2 ) 1 1 0 0 1 L / 1/ f 0 1 0 0 1 0 D- D 0 M 1/ 2 FODO 0 1 1 0 0 1 0 0 1 1 L / f L L / f 2 1 L / f 0 0 D 0 f2 L D ( 1 ) 2f 2 f L D (1 ) 2f OCPA Accelerator School, 2014, CCKuo- 39 L2 /(2 ) ( L / )(1 L / 2 f ) 1 Summary in a FODO cell f QF f QD f ( Note: for full quad length) 2L 2L L (1 sin 2 ), xD (1 sin 2 ), sin 2 sin sin 2f L L DxF 2 (1 12 sin 2 ), DxD 2 (1 12 sin 2 ) sin 2 sin 2 xF f QF f1 , f QD f 2 (not e: for full quad st rengt h) xF 1 L2 1 L2 (2 L ), xD (2 L ) sin x f2 sin y f1 yF 1 L2 1 L2 (2 L ), yD (2 L ) sin x f2 sin y f1 L L L2 cos x 1 f1 f 2 2 f1 f 2 L L L2 cos y 1 f1 f 2 2 f1 f 2 OCPA Accelerator School, 2014, CCKuo- 40 Integral Representation of Dispersion Function Same as dipole field error representation, we can substitute B p 1 p , xco D( s ) B0 p0 p0 x ( s) D( s) 2 sin x Jd s C s x (t ) cos( x | x (t ) x ( s ) |) dt 1 1 1 H ( D, D ' ) [ D 2 ( x D' x D) 2 ] dispersion action 2 2 x Jd is constant in the region without dipole. OCPA Accelerator School, 2014, CCKuo- 41 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 DBA Consider a simple DBA cell with a single quadrupole in the middle. In thin-lens approximation, the dispersion matching condition 1 0 0 1 L1 0 1 L L / 2 0 Dc 0 1 /( 2 f ) 1 0 0 1 0 0 1 0 1 0 0 1 0 0 1 0 0 1 1 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. 1 1 1 f ( L1 L), Dc ( L1 L) 2 2 2 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 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 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. APS 7 GeV, 40-fold OCPA Accelerator School, 2014, CCKuo- 45 ESRF ESRF upgrade 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 DBA-4SSRF 4-fold 2.86 nm-rad at 3.5 GeV • Components compacted C-G lattice is 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 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 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 QBA B2 B1 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 FODO Lattice (missing dipole cases) Focusing and defocusing strengths are different TLS booster 119 nm-rad at 1.3 GeV OCPA Accelerator School, 2014, CCKuo- 51f Dispersion suppression in FODO type structure • For FODO cells structure, dispersion suppression section at both ends of the standard cells. • Usually, the solution can be with one bend (same or smaller angle) and proper phase advance. It can be with two bends with smaller angle. The figure shown below is the case with two bends and same quadrupole strengths and relations are as : 1 ) 4 sin 2 1 1 ( ) 2 4 sin 1 2 2 (1 and : Bend angle and phase advance in normal half FODO cell OCPA Accelerator School, 2014, CCKuo- 52 Dispersion suppression in FODO type structure 1 1 M 2f 0 0 0 1 L 1 0 0 1 0 1 0 0 L2 1 2f 2 L L2 M 2 4f3 2f 0 L 1 2 1 f 1 0 L 2 L1 2f L2 1 2f 2 0 0 D 0 M M 1 2 0 1 1 L 1 2 1 2f 1 0 L 2 L 1 4 f L L2 2 1 2 4 f 8 f 1 0 0 1 L 1 0 0 1 0 1 0 0 0 0 1 0 0 1 1 L (1 sin( / 2)) L 2 D , sin( / 2 ) , L 2 sin ( / 2) 2f 1 ( 1 ) 2 4 sin 2 , 1 2 , 2 1 1 ( ) 4 sin 2 OCPA Accelerator School, 2014, CCKuo- 53 Dispersion suppression in FODO cell θ1 θ1 θ2 θ2 θ θ 1 ( 1 ) 2 L 4 sin 2 , 1 2 , 2 , sin 1 2 f 1 ( ) 2 4 sin OCPA Accelerator School, 2014, CCKuo- 54 Dispersion suppression Same dipole Separated functions dipole OCPA Accelerator School, 2014, CCKuo- 55 MBA:PEP-X (SLAC) PEP-X, 6GeV, 11pm-rad 2.2 km OCPA Accelerator School, 2014, CCKuo- 56 MBA: MAX-IV (Sweden) MAX-IV, 3GeV, 0.3 nm-rad 20-cell, 528m OCPA Accelerator School, 2014, CCKuo- 57 MBA: Sirius (Brazil) Sirius (Brazil) 5BA with superbend 3 GeV, 518 m, 0.28 nm BAPS 20*13BA 徐刚 5GeV, 1272m, 36pm-rad (bare) 10/10 pm-rad with damping wiggler/full coupling OCPA Accelerator School, 2014, CCKuo- 59 ESRF(France) Hybrid Multi-Bend (HMB) lattice ESRF existing (DBA) cell • Ex = 4 nmrad, 7GeV • tunes (36.44,13.39) • nat. chromaticity (-130, -58) • multi-bend for lower emittance • dispersion bump for efficient chromaticity correction => “weak” sextupoles (<0.6kT/m) • no need of “large” dispersion on the three inner dipoles => small Hx and Ex Proposed HMB cell • Ex = 160 pmrad, 7GeV • tunes (75.60, 27.60) • nat. chromaticity (-97, -79) 60 Andrea Franchi ESRF Momentum compaction C 1 1 D 1 C ds , c ds Di i C C C i ( DF DD ) 2 1 FODO cell : c, FODO 2 2 2L sin ( / 2) x D DBA cell : D0 D0 c ( ) Lm 6 L 2 3 1 Lm : half DBA module L : dipole lengt h D0 D0 1 negative L 2 6 c , DBA 2 6R R : averageradius of ring • For small bend angle and large circumference DBA lattice, the 1storder momentum compactor can be around 10-4. • Higher-order momentum compaction will play important role in the longitudinal motion if the 1storder is small. • One can design very low alpha or negative alpha lattice by properly adjusting dispersion in the dipole so that the integration is small. • Inverting dipole bend also can get low alpha. OCPA Accelerator School, 2014, CCKuo- 61 Low alpha lattice c C 1 1 C C D ds 1 2 3 2 1 105 ~ 106 TPS High emittance C.C. Kuo ipac2013 Low emittance Low alpha lattice can have shorter bunch length (a few ps) for time-resolved experiments or THz source Need sextupole/octupole to control 2 and 3 to near zero NewSUBARU invert bend OCPA Accelerator School, 2014, CCKuo- 62 Chromaticity Effects For off-momentum particle, gradient errors: K x k x ( s ) , x ( K x ( s) K x ( s))x 0 K z k z ( s ) K x ( s) 1 -1 x K ds x k x ds x x 4 4 1 -1 z K ds z k z ds z z 4 4 K x [ Chromaticity: Natural chromaticity: 1 2 2 2 k x ( s), k x ( s)] o( 2 ) nat -1 kds 4 FODO cell (N cells): FODO nat -1 N(βmax βmin )/f 4 OCPA Accelerator School, 2014, CCKuo- 63 f (p / p0 0) f (p / p0 0) f (p / p0 0) Sextupoles x D>0 p / p0 0 sextupole s quadrupole sextupole p / p0 0 Focal length Need focusing strength proportional to displacement k ( x) x Sextupole field Bx Bz B2 B ( x 2 z 2 ), 2 xz, B2 2 Bz / x 2 | x z 0 B0 2 B0 B0 B0 OCPA Accelerator School, 2014, CCKuo- 64 Chromaticity correction For a large (negative) natural chromaticity, the tune shift is large with typical energy offset and can cause beam storage lifetime reduction induced by transverse resonances. We need sextupole magnets installed in the storage ring to increase the focusing strength for larger energy beam. Two types of sextupoles, i.e., focusing and defocusing sextupoles, are properly situated. To avoid head-tail instability, a slightly positive chromaticity is preferred. B Bz B B 2 ( x 2 z 2 ), x 2 xz, B2 2 Bz / x 2 |x z 0 B0 2B0 B0 B0 Let x x D , to thefirst order of and Bz [ SD ]x B0 Bx B [ SD ]z where S 2 B0 B0 ΔK x SD , ΔK z SD -1 x 4 x [k x S F DF S D DD ]ds -1 z z [k z S F DF S D DD ]ds 4 OCPA Accelerator School, 2014, CCKuo- 65 B z x K x ( s ) x B 0 z K ( s ) z Bx z B0 1 1 Bz 1 Bx K x (s) 2 , K z ( s) , B0 x B0 z FODO cell 2L 2L 2L (1 sin 2 ), D (1 sin 2 ), sin 2 sin sin f L L 1 1 DF ( 1 sin ), D ( 1 sin D 2 2 2 2) 2 2 sin 2 sin 2 F sin 2 1 1 S F , 2 1 fDF 2 f 1 2 sin 2 sin 2 1 1 SD 2 fDD 2 f 1 12 sin 2 If f1 f 2 S F 1 1 , SD f1 DF f 2 DF OCPA Accelerator School, 2014, CCKuo- 66 Design Rules for sextupole scheme • To minimize chromatic sextupoles strengths, it should be located near quadrupoles at least for FODO cells, where βxDx and βzDx are maxima. • A large ratio of βx/βz for the focusing and βz/βx for the defousing sextupoles are needed. • The families of sextupoles should be arranged to minimize resonance strengths. • For strong focusing (low emittance) lattice, strong chromatic sextupole fields are needed to correct chromaticity and such strong nonlinear fields can induce strong nonlinear chromatic effects as well as geometric aberrations. Phase cancellations can help reduce nonlinear chromatic aberrations. Harmonic sextupoles are installed at proper positions to further reduce resonance strengths of geometric aberrations. OCPA Accelerator School, 2014, CCKuo- 67 Nonlinear Effects of Chromatic Sextupoles x K x ( s) x S ( s)(x 2 z 2 ) z K z ( s) z S ( s) xz, x x D , Normalized coordinate: w w w , w x x D ~ , w D 1 2 5/ 2 2 w / kw 0 Sw w 2 1 2 5/ 2 ~ 2 ~ 2 ~ 2 1/ 2 2 2 ~ D 0 D 0 / 0 kD 0 SD , 2 2 0 2 0 3/ 2 2 0 2 1 w 02 w 02 2 kw 02 2 SDw 02 5 / 2 Sw2 2 OCPA Accelerator School, 2014, CCKuo- 68 x 1 2 5/ 2 2 w kw SDw 0 Sw w 2 2 0 2 0 2 2 0 2 • The first two terms cancel the chromatic aberrations to first order locally. However, in real machine, local cancellation might be not perfect and beta-beat and higher harmonics of chromatic terms in the circular machine still exist . • The third term is betatron amplitude dependent perturbation and called geometric aberrations. -I transformation scheme is to put sextupoles in (2n+1) π phase advance apart in a periodic lattice and this scheme can compensate the effects. OCPA Accelerator School, 2014, CCKuo- 69 Chromatic Aberrations Chromatic gradient errors with sextupoles K (s) (k SD) Beta-beat: (s) 2 sin 0 (t )(k (t ) S (t ) D(t )) cos[2 0 ( | ( s) (t ) |]dt If chromaticity is corrected locally, there is no induced betabeat and no half integer resonances. However, in reality, we might need several families of sextupoles to get small betabeat Colliders need to have local chromatic aberration correction with sextuples at interaction point to ensure small beta-beat at IP and keep beam size as small as possible (high luminosity). OCPA Accelerator School, 2014, CCKuo- 70 -I transformation 1 0 0 0 0 1 0 0 M I 0 0 1 0 0 0 0 1 1 x S( x 2 y 2 ), y Sxy 2 x0 1 0 0 0 1 x 1 x0 0 1 0 0 Sx0 x ( I ) M 2 s y y0 0 0 1 0 0 y y 0 0 0 1 1 0 0 Adding second sextupole: x0 x0 x0 x0 y y (2n 1) phase advance 0 0 y y 0 0 0 0 1 Sy0 2 0 1 0 Sx0 1 x0 0 x0 1 2 2 0 x0 S( x0 y0 ) x0 2 0 y0 y0 1 y0 Sx0 y0 y0 Cancellation of geometric aberration after a complete transformation matrix through one unit. OCPA Accelerator School, 2014, CCKuo- 71 Real machine For periodic lattice, compensation can be across one or several cells. Thick-lens sextupoles together with errors in real machine such as COD, quad gradient errors, and working point selection always lead some limitation in its effectiveness. Diamond phase cancellation scheme. Horizontal tune ~ 29.12 due to long straight perturbation Selection of working tune is different from above. Diamond Light Source nominal lattice working tune (27.23, 12.36) OCPA Accelerator School, 2014, CCKuo- 72 Nonlinear Hamiltonian S ( s) 3 1 2 2 2 2 ( x 3xy 2 ) H ( p x , x, p y , y ) ( p x K x x p y K y y ) 2 2 H H 0 V3 ( x, y, s) S ( s) 3 1 2 2 2 2 ( x 3xy 2 ) H 0 ( p x K x x p y K y y ), V3 2 2 x 2 J x x cos( x ( s) x ) 2 1/ 2 J x J y x1/ 2 y S ( s )[2 cos x cos( x 2 y ) cos( x 2 y )] 4 2 3/ 2 3/ 2 J x x S ( s)[cos3 x 3 cos x ] 12 V3 s ds 0 x x x x ( s) x , x ( s) s ds 0 y y y y ( s) y , y ( s) ( J x ,x ), ( J y , y ) are pairs of conjugate phase-space OCPA Accelerator School, 2014, CCKuo- 73 Nonlinear Hamiltonian Periodic in s, Fourier expansion: H ( J x , x , J y , y ) x J x y J y G3, 0, J x3 / 2 cos(3 x 3,0, ) G1, 2, J 1x / 2 J y cos( x 2 y 1, 2, ) G1, 2, J 1x / 2 J y cos( x 2 y 1, 2, ) ... G3,0, e j 2 j [ 3 x ( s ) ( 3 x ) ] 3/ 2 S ( s ) e ds x 24 OCPA Accelerator School, 2014, CCKuo- 74 Nonlinear Effects of Chromatic Sextupoles Sum resonance: νx 2νy Difference resonance: Parametric resonance: νx 2νy νx , 3νx Other higher-order resonance: 4νx , 2νx 2νy ... Concatenation of sextupoles perturbation to the betatron motion can induce nonlinear betatron detuning. ν x ν x 0 xx J x xy J y ν y ν y 0 xy J x yy J y Detuning coefficient: xx , xy , yy OCPA Accelerator School, 2014, CCKuo- 75 Sextupole Hamiltonian In the single resonance approach, first 9 terms of first-order driving sources: 4 Chromatic terms, 5 Geometry terms 9 families can get first-order optimal solutions (ref: J. Bengtsson, The sextupole scheme for the SLS, SLS-Note 9/97) Nsxt h jklmp (b2 L) n jk 2 xn l m 2 yn e l m 2 yn Dnp e i[( j k ) xn ( l m ) yn ] n [ Nquad (b1L) n jk 2 xn i[( j k ) xn ( l m ) yn ] ] p 0 n For the first 12 terms: 4 Chromatic terms(P9~P12), 5 Geometry terms(P1~P5),3 Amplitude dependent terms (P6~P8) (second-order terms and critical) need to be minimized (by Powell’s method in OPA). f p1 | h21000 ( x ) |2 p2 | h10110 ( x ) |2 p3 | h30000 (3 x ) |2 p4 | h10020 ( x 2 y ) |2 p5 | h10200 ( x 2 y ) |2 p6 | d x / dJ x |2 p7 | d x / dJ y |2 p8 | d y / dJ y |2 p9 | h11001 ( x ) |2 p10 | h00111 ( y ) |2 p11 | h20001 ( 2 x ( )) |2 p12 | h00201 ( 2 y ( )) |2 OCPA Accelerator School, 2014, CCKuo- 76 Objective Function Including Higher-order Driving terms Usually, we can get good solution for the first 12 terms optimization. However, we can further reduce 2nd-order terms. 13 terms in 2nd -order of sextupole strength 3 tune shift with amplitude 8 octupole-like driving terms 4νx , 2 ν x±2 ν y , 4νy , 2 ν x , 2 ν y 2 terms generating second-order chromaticity This nonlinear optimization is very important for the low emittance, high chromaticity lattice. Iterations between linear and nonlinear schemes are proceeded to get acceptable solution. OCPA Accelerator School, 2014, CCKuo- 77 TLS Storage Ring For TLS, do we need more than two sextupole families? Lattice type TBA Operational energy 1.5 GeV Circumference 120 m Natural emittance 25.6 nm-rad (achromat) Natural energy spread 0.075% Momentum compaction factor 0.00678 Damping time Horizontal 6.959 ms Vertical 9.372 ms Longitudinal 5.668 ms Betatron tunes horizontal/vertical 7.18/4.13 Natural chromaticities Horizontal -15.292 Vertical -7.868 Radiation loss per turn (dipole) 128 keV Only two family of sextupoles for chromaticity correction. SD=-5.3(m-2), SF=7.62(m-2) OCPA Accelerator School, 2014, CCKuo- 78 TLS Storage Ring 10 10 8 8 Y [mm] Y [mm] If only put sextupoles in one section (locally) to control chromaticity, SD=-23.7 (1/m2) and SF=26.6 (1/m2), dynamic aperture reduce significantly due to lack of phase cancellation. 6 4 6 4 2 2 0 -30 -20 -10 0 X [mm] 10 20 30 Dynamic aperture for distributed sextupoles 0 -30 -20 -10 0 X [mm] 10 20 Local sextupoles only OCPA Accelerator School, 2014, CCKuo- 79 30 TLS Storage Ring What is the RF energy acceptance? ( eV sin s p 2 ) acc 0 ( q 1 cos1 (1 / q) p0 h c cp0 q eV0 1 U 0 sin s T LS:U 0 128keV , No ID RF Energy acceptance (%) 3 2 Beta and tune change vs energy 1 0 0 0.5 10 Y [mm] 8 6 4 2 0 -30 -20 -10 0 X [mm] 10 20 30 Phase space Dynamic aperture OCPA Accelerator School, 2014, CCKuo- 80 RF1Voltage (MV) 1.5 2 2.5 Betatron Tune and Nonlinear Resonance Tune selection in the lattice very important. Tune should be away from integer, half integer and third-order resonances. Introduction of sextupoles and nonlinear field errors in the magnets can drive higherorder nonlinear resonances. Particle tracking study and fequency (tune)-map analyses can further provide optimization information in tune selection. Systematic resonances Random resonances n x m y 6 p n x m y p n, m, p are integers n, m, p are integers OCPA Accelerator School, 2014, CCKuo- 81 Linear Lattice Matching • Computer design codes are usually used in the matching. (for example MAD) • Matching method such as simplex command is to minimize the penalty function by simplex method. Make sure enough varying parameters for the selected constraints. • Starting from unit cell and impose constraints on optical functions such as D, D’ at both ends of dipoles, local and global betatron functions, phase advance per cell, etc. Weighting factors are also given. • Construct super-period structure and do the same matching process with different constraints. • Maximum strengths of quads are limited. • Ring tunes are matched. • Not always able to find stable solutions and need change initial conditions for matching. • Examine the global parameters and fitted parameters. If satisfactory, go to nonlinear optimization. OCPA Accelerator School, 2014, CCKuo- 82 Linear matching Objective function F T f1 f1 ... f n , f i f i (k1 , k2 ,...k j ,..km ) ki : quad strength F f1 k1 f2 k2 . and K . . . k fn m f1 k1 f 2 A k1 .. f n k 1 f1 k 2 f 2 k 2 .. f n k 2 f1 k m f 2 .. .. k m .. .. .. f n .. .. k m .. .. Fideal F0 A( K K0 ) m=n, with equal weighting factor and only linear functions considered, K K0 A1 ( Fideal F0 ) Iteration needed due to nonlinearity. With different weightings and constraints of objective functions, or m<n case, least square or nonlinear optimization methods can be used. Design codes usually provide useful tools to get solutions but maybe not what one wants. In MAD code, minimization by gradient or simplex method can be employed. Local minimum maybe the unwanted solutions. Initial values of quad strength should be changed. Some tools using genetic algorithm to get desired linear optics. OCPA Accelerator School, 2014, CCKuo- 83 TPS Lattice Nonlinear optimization Quality factor: Tune shift with amplitude, Phase space plots,Tune shift with energy, Dynamic aperture (on and off momentum, 4D, 6D), Frequency Map Analysis Codes: OPA, BETA, Tracy-2, MAD, Patricia, AT, elegant, etc. 8 families of sextupoles are used. Chromaticities are corrected to slightly positive. Weighting factors, sextupole families, positions are varied. Effects on the dynamic aperture in the presence of ID, field errors, chamber limitation, alignment errors, etc. are studied. OCPA Accelerator School, 2014, CCKuo- 85 TPS Sextupole scheme OPA S1 S2 SD SD SF S4 S3 S5 S6 SD SD OCPA Accelerator School, 2014, CCKuoSF86 S6 S5 Only Chromatic Sextupoles OCPA Accelerator School, 2014, CCKuo- 87 Nonlinear Optimization with Sextupoles OPA 8 families of sextupoles for nonlinear optimization. Chromaticity of +5 in both planes are still with acceptable dynamic aperture and energy acceptance. OCPA Accelerator School, 2014, CCKuo- 88 Phase Space Betatron Function vs Energy Tune Shifts vs. Amplitude and Energy Dynamic Aperture x,y vs. X x,y vs. Y OCPA Accelerator School, 2014, CCKuo- 89 x,y vs. dp/p RF Energy Aperture 3.5 MV RF (no chamber limitation) TPS Momentumcompactionfactors 1 2.4 10 4 , 2 2.1103 1 1 0 ds L0 2 1 0 1 2 ds L0 2 1 2 2 p / p Tracy-2 ( p 2 eV sin s ) 0 ( q 1 cos 1 (1 / q) p0 h c cp 0 q acc eV0 1 U 0 sin s TPS : U 0 0.85MeV , No ID OCPA Accelerator School, 2014, CCKuo- 90 analytical Dynamic Aperture and Frequency Map Analysis 4uy=53 ux+3uy=66 3uy=40 2ux+2uy=79 3uy=40 4uy=53 6ux=157 With multipole errors B 104 at R 25mm B Tune diffusion rate: D= log10((x)2+ (y)2)1/2 for tune difference between the first 512 turns and second 512 turns OCPA Accelerator School, 2014, CCKuo- 91 Liouville theorem foundation of accelerator physics “Beam phase space is a constant” Joseph Liouville, 1809 – 1882 • Liouville’s theorem :the density of points representing particles in (6-D) (x, p) phase space is conserved if any forces conservative and differentiable . i.e., the forces must be divergence free in momentum space (p-divergence =0). p p 0 • radiation and dissipation do not satisfy the p-divergence requirement, but magnetic forces and (Newtonian) gravitational forces do. • No or very slow time dependence in the Hamiltonian system. • Note: acceleration keeps (x,p) phase space constant, but reduces (x, x’) phase space , no violation of Liouville theorem i.e. normalized emittance e N e 0 , v / c, 1 / 1 2 e 0 geometricemittance OCPA Accelerator School, 2014, CCKuo- 92 Non-conservation of emittance • Coupling (horizontal-vertical, chromaticity,..) • Scattering (gas, intra-beam, beam-beam, through foil) • Damping (synchrotron radiation, electron cooling, stochastic cooling, laser cooling) • Filamentation (nonlinearities, phase space conserved, but emittance growth) • Instabilities (wake-fields,…) • Space charge effects OCPA Accelerator School, 2014, CCKuo- 93 Transverse beam Emittance • Transverse emittance in a linac is determined by the beam source (gun) emittance. The normalized emittance is a constant, but geometric emittance is reduced during acceleration due to adiabatic damping. • In a booster synchrotron, the geometric beam emittance is damped first due to adiabatic damping during energy ramping and then reach equilibrium natural emittance. • The natural emittance in the storage ring is determined by the equilibrium between radiation damping and quantum excitation of synchrotron radiation. • In proton or heavy-ion rings, e-cooling or other damping schemes are needed. • The natural emittance in a ring is determined by the lattice design. • The natural emittance can be reduced by adding damping wigglers, Robinson wigglers, or change the bend radius, using longitudinal gradient dipole, etc. • The diffraction limited emittance is e E (keV) = 1.24 / λ (nm) 4 OCPA Accelerator School, 2014, CCKuo- 94 Synchrotron radiation and damping • Charged particle under acceleration (through EM field, etc.) will radiate EM wave. • The radiation in synchrotron accelerator is called synchrotron radiation. Properties of synchrotron radiation: High Intensity, Continuous Spectrum, Excellent Collimation, Low Emittance, Pulsed-time Structure, Polarization. Energy loss due to synchrotron radiation in circulating particle is replenished by rf cavities with longitudinal electric field. Higher energy particle loss more energy and in average the energy is damped to the equilibrium energy. (longitudinal damping) In transverse planes, radiation in a cone with some angle compensated by the longitudinal electric field and also particle is damped in transverse phase space. (transverse damping) Radiation is a quantum process and cause diffusion and excitation. energy spread in equilibrium Longitudinal and transverse motions are coupled through dispersion function natural emittance in equilibrium OCPA Accelerator School, 2014, CCKuo- 95 Damping of Synchrotron Oscillations Relativistic electron loss energy due to radiation c E4 P C 2 2 U 0 Pdt U0 c E4 P C T0 2R d (E ) 1 (eV WE ) dt T0 (t ) Ae 2 ds 2 C E0 4 C 8.846105 m /(GeV )3 d 2 d 2 e V W 2 2 0 c E s E , s dt 2 dt 2T0 T0 E E cos(st 0 ) D parameter : 4 dU U ( E ) U 0 WE , W | E E0 dE d (E ) eV ( ) U ( E ) d E , and cT0 dt T0 dt E Plus gain energy from rf: E t C E0 U W 0 ( 2 D) 2T0 2T0 E Isomagnetic D( s) 1 ds D ( 2 2 K ( s ))ds 2 1 1 D ( s )( 2 K ( s ))dipole ds 2 2 1 Isomagnetic separated function ring OCPA Accelerator School, 2014, CCKuo- 96 U 0 P E ET0 E Damping of Vertical Betatron Oscillation z A cos , z A sin , A2 z 2 ( z) 2 , A a After emitting radiation and rf acceleration: p u p E U AA 2 z z ) ( z ) 2 0 E z z p z p p z Average one turn s (z) A / 2 2 2 A2 U 0 AA 2 E U0 1 dA 1 A A dt T0 A 2 ET0 P U0 z 2ET0 2E OCPA Accelerator School, 2014, CCKuo- 97 Damping of Horizontal Betatron Oscillations E E x xe , x x D x xe E E u u x x D , x x D After emitting radiation : e e E E Change in betatron amplitude and average in betatron phase: u AA x x 2 x x ( Dx 2 Dx ) E x P 2 dB AA x D(1 x ) ds B dx cE x x D Average one turn: 1 U A U 0 D 1 1 2 K 2 ds 2 ds D 0 A 2 E 2E Including rf acceleration: U A (1 D ) 0 A 2E x (1 D) P U0 (1 D) 2ET0 2E OCPA Accelerator School, 2014, CCKuo- 98 Robinson theorem Radiation damping coefficients U0 x (1 D) J x 0 2T0 E U0 z J z 0 2T0 E U0 s ( 2 D ) J E 0 2T0 E U0 0 , Jz 1 2 ET0 For separatedfunctionstrongfocusing ring : D 1, J x J z 1, J E 2 For combinedfunctionstrongfocusing ring : D 2, J x 1, J z 1, J E 4 still works for hardronmachine(U 0 /E verysamll) Robinson damping partition theorem Jx Jz JE 4 D( s) 1 ds D ( 2 2 K ( s ))ds 2 1 1 D ( s )( 2 K ( s ))dipole ds 2 2 1 But used for leptonmachine(emittanceconcerned), need changeD by changingorbit or using Robison wiggler, e For combinedfunctionelectronring, stabilitycondition: 1 D 2 OCPA Accelerator School, 2014, CCKuo- 99 Energy spread Synchrotron oscillation without fluctuation and damping: E A cosst With radiation and damping: d A2 A2 2 Nu 2 dt E In equilibrium: Define N : rateof photonemission 1 A2 Nu 2 E 2 2 A 1 E2 Nu 2 E 2 4 3 55 3 P c 3 where N u 2 24 3 1 / 2 2 ( E E ) 2 Cq 2 J E 1 / 2 1 / 3 Cq For isomagnetic ring: E E 2E J E P 55 3.831013 m 32 3 m c 2 ( ) Cq E JE 2 OCPA Accelerator School, 2014, CCKuo- 100 Horizontal Emittance Radiation emission results in change of betatron coordinates: x D u u , x D E E Change of Courant –Snyder invariant after average: 2 x 1 2 H x Dx x Dx Dx x 2 2 a 2 N u 2 H s Including damping term a 2 and average one turn: dt x E2 3 55 3 P 2 N u c 3 Equilibrium: 2 2 2 2 1 / 24 3 N u H 1 x x 2 s u 2 2 a H ( ) E a x 2 E 2 x Natural emittance 3 x2 H / | | 2 13 x ex Cq , C 3 . 83 10 m q 2 x J x 1 / x OCPA Accelerator School, 2014, CCKuo- 101 Vertical emittance For an ideal flat accelerator without errors, due to the finite emission angle of synchrotron, we can get natural vertical emittance as: β y / | ρ |3 13 ε y Cq 10 m rad 2 J y 1/ρ However, due to spurious vertical dispersion by errors, ε y Cq γ 2 H y / | ρ |3 J y 1/ρ 2 1013m rad, H y γ y Dy2 2α y Dy Dy β y Dy2 Betatron coupling from skew quads, etc., also generate vertical emittance coupled from horizontal emittance. OCPA Accelerator School, 2014, CCKuo- 102 Radiation integral and electron beam properties Radiation integral I1 I2 I3 x ds 1 2 (2) Emittancee x Cq 2 I 5 /( I 2 I 4 ) Cq (55 / 32 3 ) / m c 3.831013 m ds 1 || 3 (3) Energyspread ( E / E ) 2 Cq 2 I 3 /(2 I 2 I 4 ) ds x 1 ( 2 K )ds 2 H I5 ds, | |3 I4 whereH (1) Mom.comp. c I1 / 2R 1 x ( x2 ( x x x x' ) 2 ) (4)Energyloss per turnU 0 C E 4 I 2 / 2 C (4 / 3)r0 /(m c2 ) 3 8.85105 m /(GeV )3 (5) J x 1 I 4 / I 2 , J E 2 I 4 / I 2 , D I 4 / I 2 OCPA Accelerator School, 2014, CCKuo- 103 FODO cell F 2L 2L 2L (1 sin ), D (1 sin ), sin sin 2 sin 2 f 2 DF L 1 L 1 ( 1 sin ), D ( 1 sin ) D sin 2 ( / 2) 2 2 sin 2 ( / 2) 2 2 cos( / 2) 1 2 ( 1 sin ) 3 sin ( / 2)(1 sin( / 2) 2 2 cos( / 2) 1 2 2 H D L (1 sin ) 3 sin ( / 2)(1 sin( / 2) 2 2 H F L 2 1 2 1 2 ( 1 sin ) ( 1 sin ) cos( / 2) 2 2 2 2 H 3 2 sin ( / 2) (1 sin ) (1 sin ) 2 2 2 1 ( 3 / 4 ) sin ( / 2) 1 2 3 e x FFODO Cq 3 Cq 2 3 sin ( / 2) cos( / 2) J x OCPA Accelerator School, 2014, CCKuo- 104 FODO Lattice 2 1 ( 3 / 4 ) sin ( / 2) 1 2 3 e x FFODO Cq 3 Cq 2 3 sin ( / 2) cos( / 2) J x Let J x 1 e ME FCq 2 3 F~1.2 for phase advance ~ 138° per cell F~ 2.5 for 90° • FODO lattices are commonly used in colliders, booster synchrotrons in which emittance is not pushed to extremely small values. • In high energy ring for particle experiments, dipole magnets with FODO cells occupy most of the ring path. • Light source design usually try to have as large percentage of straights for IDs, FODO type lattice usually not adopted for dedicated light sources. OCPA Accelerator School, 2014, CCKuo- 105 Jx 1 Minimum Emittance in DBA DBA: small bend angle At bend entrance,D0=D0’=0 Bend length L ( s) 0 2 0 s 0 s 2 (s) 0 0 s (s) 0 S.Y. Lee, “Emittance optimization in three- and multiple-bend achromats”, Phy. Rev. E, 52, 1940, 1996 and “Accelerator Physics”, World Scientific, 2nd edition, 2004 D(s) D0 D0 ' s (1 cos ) (1 cos ) D' (s) D0 ' sin sin 1 1 H H ( s)ds ( ( s) D'2 ( s) 2 ( s) D( s) D' ( s) D'2 ( s))ds 0 0 2 3 0 0 3 0 C 0 0 0 q H ( ) ex ( ) 3 4 20 J 3 4 20 x e x e x 0 Optimum emittance: 0 0 e MEDBA Cq 2 3 0,min (6 / 15), 0,min 15 Cq 2 3 0.0645 Jx 4 15J x Each MEDBA module with horizontal phase advance across each dipole of 156.70, dispersion matching 1220, and about 1.2 unit of horizontal tune per DBA cell. OCPA Accelerator School, 2014, CCKuo- 106 Minimum Emittance in DBA For non-achromat lattice, minimization of I5 lead to 1 ex 12 15 Cq 2 3 Jx T ME(non- achromat) (s / 2) 0, D(s / 2) 0 8 1 2 1 0 , 0 15, D0 , D0 6 2 15 only one type of dipole. beta and dispersion functions are symmetric at dipole center OCPA Accelerator School, 2014, CCKuo- 107 Design Emittance • In practice, real machine will not be able to reach theoretical minimum emittance because of the constraints in betatron function limitation, tune range, nonlinear sextupole scheme, and engineering limitation, etc. • Usually, a few factor larger in real machine is feasible. • For 24-cell 3GeV, theoretical minimum emittance is 1.92 nm-rad for the achromatic DBA. In real design, it can reach 4.9 nm-rad. • For non-achromat configuration, we can reach 1.6 nm-rad (as compared with the TME of 0.64 nm-rad) OCPA Accelerator School, 2014, CCKuo- 108 Minimum Emittance in TBA Matching optical functions (with quad) between MEDBA module (outer dipoles L1) and ME (central dipole L2) results in: (ref. S.Y. Lee, Accelerator Physics, World Scientific) L32 L13 3 2 2 ρ2 ρ1 and phase advance 1 Cq 1 Cq , e TME 4 15 J x 12 15 J x 3 1 2 e METBA isomagnetic ring: 1 2 3 1 1 1/ 3 , 2 ( 2 3 )1 2 1 2 1/ 3 3 1 Cq 1 Cq 12 15 J x 12 15 J x 2 e TME 127.76 3 1 2 ( M 2)3 3 2 : T otalbend angle in a cell, M 2 OCPA Accelerator School, 2014, CCKuo- 109 3 Emittance comparison • One can compare the minimum emittance among DBA, TBA and QBA if same number of dipoles are used in the ring. • From the formula in the previous slide, we can get 3 e METBA 3 e MEDBA 0.66e MEDBA 1/ 3 23 3 e MEQBA 2 e MEDBA 0.55e MEDBA 1/ 3 1 3 OCPA Accelerator School, 2014, CCKuo- 110 Effective emittance With dispersion in the long straights, we can reduce the natural emittance by a factor of 3, but the effective emittance is At the symmetry point of e x x' xx' the long straights, 2 2 2 2 x ,eff e x ,eff e x2 H ID E2 e x e eff e x 1 H ID x Dx2 2 x Dx Dx' x Dx'2 in ID section TPS emittance emittance (nm-rad) 5 natural emittance 4 effective emittance 3 2 1 0 0 0.05 0.1 0.15 0.2 dispersion (m) OCPA Accelerator School, 2014, CCKuo- 111 ( E Dx ) 2 e xx Closed Orbit Distortion In reality, dipole field errors distributed around the ring: yco ( s ) s c s B( s ' ) G ( s, s ' ) ds ' B (s) (s' ) where G ( s, s ' ) cos( | ( s ) ( s ' ) |) 2 sin In dipole angular kick form: (s) N yco ( s ) i ( si ) cos( | ( s ) ( si ) |) 2 sin i 1 s y co,rms N rms 2 2 | sin | due to quadrupole misalignment xco,rms av N q xrms , 2 2 f av | sin | i B(si ) dsi B (Bz / x ) y y B f The coefficient in curly brackets is called sensitivity or amplification factor. OCPA Accelerator School, 2014, CCKuo- 112 COD correction, SVD method: y co ( s j ) A ji (s j ) N 2 sin i 1 (s j ) i ( si ) cos( | ( s j ) ( si ) |) ( si ) cos( | ( s j ) ( si ) |) 2 sin The distorted orbit can be minimized at least at the BPM to the desired value so that n A yco,m 1 OCPA Accelerator School, 2014, CCKuo- 113 ID effect Insertion devices cause some effects on beam dynamics like betatron tune shifts, optical functions perturbation, emittance variation, multipole field effects, etc. The field of a wiggler can be modeled as: Bx 0 By Bw cos(kws) cosh(kw y) Bs Bw sin(k w s) sinh(k w y) The equations of motion: d 2x 1 cosh(k w y) cos(k w s), 2 w ds 2 2 d y sin (k w s) sinh(2k w y) p x sinh(k w y) sin(k w s), 2 2k w w w2 ds sin 2 (k w s ) average focusing strength: vertical tune shift: y y Lw w2 1 2 w2 8 w2 Beta beat in vertical plan can be evaluated w20 Tune-shift with gap for w20 at TLS Vertical beta perturabtion due to Wiggler 20 in TLS (Kuo, et al PAC95) OCPA Accelerator School, 2014, CCKuo- 114 Correction Algorithm 1. 2. 3. 4. By the following linear relations, construct the response matrix A of bare lattice. Use SVD method to restore the perturbed optics back to the optics of the bare lattice as much as possible by minimizing (AΔk +b). Where Δk is the tuning strength of the quadrupoles, and b is the perturbed optics. One can choose arbitrary positions to restore optics with different weighting. Current (A) λ (mm) Nperiod By (T) Bx (T) L (m) Gap (mm) Total power (kW) w1 w2 x1 x1 y1 y1 k1 k 2 x1 A wn y1 k nq Qx Q y SW60 0.4 60 8 3.5 0.45 17 13.39 OCPA Accelerator School, 2014, CCKuo- 115 Beta Beating and phase beating correction (a) Horizontal beta-beating with SW60 (c) Horizontal phase-beating with SW60 (b) Vertical beta-beating with SW60 (d) Vertical phase-beating with SW60 OCPA Accelerator School, 2014, CCKuo- 116 Emittance with IDs 8Bw U w 8 0 U w , planar undulator 3 f U 3 f B U I 5w h w 0 h 0 0 I 50 I 5 w 2 Emittance with ID e x Cq B U I 50 0 U w I 20 I 40 I 2 w I 4 w w w , helical undulator f h w U 0 f h B0 U 0 1 2 I 50 f hH ID 3 ds f hH ID 2 , 0 8Bw U w ( 1 w 3f B U ) H dipole 2 2 fh , H x x 2 x x x x x h 0 0 , planar undulators H ID U (1 w ) 3 4 H L /( 3 ) , planar undulator 1 ID w w w U0 ex I 5 w H ID ds 3 3 w| | H ID Lw / w , helicalundulator Bw U w e x0 4 ( 1 ) U 0 C E / 0 w f h B0 U 0 , helical undulators C E 4 Lw /(4w2 ), planarundulator U Uw (1 w ) 4 2 C E Lw /(2w ), helicalundulator w U0 Emittancewithout ID e x 0 Cq 2 I 50 I 20 I 40 If Bw > (3πfh/8)B0 then emittance will increase for planar undulator Installing IDs in the non-dispersive straights, where fh is very large, emittance always decreases. For a TME lattice, fh=0.25, and for a tyical well-designed distributed dispersion lattice, fh=0.5~0.8 OCPA Accelerator School, 2014, CCKuo- 117 Energy spread with IDs 4 Lw /( 3 w3 ), planar undulator I 30 2 , I 3 w 3 0 Lw / w , helical undulator 8 Bw U w 8 0 U w , planar undulator 3 U 3 B U I 3w w 0 0 0 I 30 0 U w Bw U w , helical undulator B0 U 0 f w U 0 I3 1 I 3 w / I 30 ( E ) 2 C q 2 ( E ) 02 E (I2 I4 ) E 1 I 2 w / I 20 2 8 Bw U w Uw ( 1 ) /( 1 ), planar undulators 3 B U U w w E 2 E 2 0 0 0 ( ) ( )0 E E (1 Bw U w ) /(1 U w ), helical undulators w B0 U 0 w U0 U 0 C E 4 / 0 4 2 C E L /( 4 ), planar undulator w w Uw 4 2 C E L /( 2 ), helical undulator w w If Bw > (3π/8)B0 then energy spread will increase for planar undulator. OCPA Accelerator School, 2014, CCKuo- 118 Touschek scattering • Transverse momentum can be transferred to longitudinal plane by Coulomb scattering in the same bunch and energy loss/gain is boosted by a factor of p=px = p x /x. • Particle can be lost due to energy acceptance (rf or transverse) limitation or aperture(dynamic or physical) limitation. • The effects were first recognized by B.Touschek in 1963 in Frascati e+-e- storage ring ADA. • Touschek lifetime can be expressed as: Bruck formula e (s) 2 C acc' x (s) 1 r e2 cN 1 L . ds 3 ' 2 T 1 8 l L 0 x (s) z (s) x (s) e acc (s) 2 where V=83/2 xyL, =[ eacc/x’ ]2, N is the number of electrons per bunch. C ( ) 1.5e ln u 2 e u e du 0.5(3 ln 2) du u u u OCPA Accelerator School, 2014, CCKuo- 119 Touschek lifetime vs RF gap voltage summary Lattice design of accelerators is an iterative process. Different options might be presented for comparison and evaluation. FODO lattice usually adopted in rings for proton and heavy ions. DBA, TBA, or MBA are adopted for light sources which require extremely small emittance Linear lattice design is to match requirements of betatron, dispersion functions, betatron tunes, and other global parameters. Nonlinear optimization with sextupole scheme is more complicated in modern light sources. Correction schemes for orbit, optics, coupling should be studied. Make sure enough aperture (energy, dynamic and physical ) for injection and lifetime with the existence of errors (magnetic field errors, alignment errors, etc.) Effects in the presence of insertion devices need to be well investigated. Instability issues are also very important and make sure that impedances of vacuum components are well controlled. Feedback systems for fast orbit correction, instabilities, etc., are required. OCPA Accelerator School, 2014, CCKuo- 121 Some of the materials, figures, etc., are from references: 1. S.Y. Lee, Accelerator Physics, World Scientific, 2nd edition, 2004. 2. Helmut Wiedemann, Particle Accelerator Physics, Springer, 3rd edition, 2007. 3. Klause Wille, The Physics of Particle Accelerators, Oxford University Press, 2000 4. A. W. Chao and Maury Tigner, Handbook of Accelerator Physics and Engineering, World Scientific, 3rd edition, 1999. 5. And other resources, public web pages, etc. These lecture notes are only for this School. OCPA Accelerator School, 2014, CCKuo- 122 Thank you for your attention OCPA Accelerator School, 2014, CCKuo- 123