Lattices for Light Sources

Transcription

1 Andreas Streun Swiss Light Source SLS, Paul Scherrer Institute, Villigen, Switzerland Contents: Global requirements: size, brightness, stability Lattice building blocks: magnets and other devices Emittance: lattice cells, minimum emittance, vertical emittance Other lattice parameters: damping, tune, chromaticity etc. Acceptances: lifetime and injection; physical, dynamic and energy acceptance Lattice errors: misalignments, multipolar errors A.Streun, PSI 1

2 Global requirements Number and length of straight sections minimum size Available area maximum size Synchrotron radiation properties radiation spectrum, insertion device types beam energy brightness emittance, coupling, energy spread Stability requirements current stability beam lifetime acceptances orbit stability mechanical tolerances, correction schemes Flexibility and future upgrade potential Budget A.Streun, PSI 2

3 Lattice Design Interfaces Magnet Design: Technological limits, coil space, multipolar errors Vacuum: Impedance, pressure, physical apertures, space Radiofrequency: Energy acceptance, bunchlength, space Diagnostics: Alignment: Beam position monitors, space Orbit distortions and correction Mechanical engineering: Girders, vibrations Design engineering: Assembly, feasibility = Beam current Vacuum, Radiofrequency = Space requirements Magnet, Vacuum, RF, Diagnostics, Engineering A.Streun, PSI 3

4 Space requirements A lattice section... (top)...as seen by the lattice designer (bottom)... as seen by the design engineer (right)... and how it looks in reality A.Streun, PSI 4

5 Lattice building blocks Lattice components Parameters Purpose Bending magnet b 1! = ρ 1, φ, b 2, ζ in,out... form a ring Quadrupole b 2, L focussing Sextupole b 3, L chromaticity correction RF cavity λ rf, V rf acceleration, long. focussing Septum magnet position & width injection Kicker magnet b1 (t)dl injection Correctors b1 dl, a 1 dl orbit correction BPM passiv orbit measurement Skew Quadrupole a2 dl coupling correction Undulator λ u, N u, B u, g synchrotron light A.Streun, PSI 5

6 Lattice building blocks Iron dominated dipole magnet Hds = jda = Coil cross section A A = B 2j c ( Siron µ o µ r + g µ o ) µ r 1 A Bg 2j c µ o S iron coil A L total g L iron L eff A.Streun, PSI 6

7 Lattice building blocks Magnet poletip fields and apertures coil width poletip field aperture L tot L eff 2 [mm] B pt [T] R [mm] Bending magnets: (=g/2) Quadrupoles: Sextupoles: (data from various light sources) Magnet power R n = Apertures: As small as possible As large as necessary acceptance A.Streun, PSI 7

8 Lattice building blocks Lattice Design Code Model: complete set of elements, correct methods for tracking and concatenation, well documented approximations Elementary functions: beta functions and dispersions, periodic solutions, closed orbit finder, energy variations, tracking, matching Toolbox: Fourier transforms of particle data ( resonance analysis), minimizaton routines ( dynamic aperture optimization, coupling suppression), linear algebra package ( orbit correction) User convenience: editor functions, graphical user interface, editable text files Extended functions: RF dimensioning, geometry plots, lifetime calculations, injection design, alignment errors, multipolar errors, ground vibrations Connectivity: database access, control system access ( real machine) A.Streun, PSI 8

9 Emittance Natural horizontal emittance Flat lattice: ɛ xo [nm rad] = 55 hc 32 3 m e c 2 }{{} m Lattice invariant (or dispersion s emittance ): γ 2 I 5 J x I 2 = 1470 (E[GeV]) 2 H/ρ3 J x 1/ρ 2 H(s) = γ x (s)d(s) 2 + 2α x (s)d(s)d (s) + β x (s)d (s) 2 Horizontal damping partition J x 1... lattice average... mag magnets average Simplification for isomagnetic lattice: ɛ xo [nm rad] = 1470 (E[GeV]) 2 H mag ρj x A.Streun, PSI 9

10 E, Lattice Cells Building low emittance lattices... = > N > O A B C D A.Streun, PSI 10

11 Lattice Cells DBA example: ESRF high beta straight sections low beta DBA 6 GeV A.Streun, PSI 11

12 Lattice Cells TBA example: SLS 11.5 m straight 4 m straight 7 m straight GeV A.Streun, PSI 12

13 Lattice Cells Combined function example: SLS booster 2.4 GeV A.Streun, PSI 13

14 Emittance Minimum Emittance d H(α xc, β xc, D c, D c) mag = 0 = Minimum emittance: ɛ xo [nm rad] = 1470 (E[GeV])2 J x Φ [rad] magnet deflection angle (Φ/2 1) Φ 3 F β D β D D c β c β f L F = 1 F = 3 β xc = 1 2 L D 15 c = 1 24ρ L2 s f = 3 L β 3 8 xf = L 320 s f A.Streun, PSI 14

15 Emittance Minimum emittance 2 Deviations from minimum: b = d = β xc β xc,min ɛ xo,min F = ɛxo D c D c,min 135 F=4 F=5 Relative emittance F : F=2 F= (d 1)2 + (b F) 2 = F 2 F=1 Phase advance in cell: ( ) 6 b Ψ = 2 arctan 15 (d 3) F = 1 = Ψ = A.Streun, PSI 15

16 Emittance Minimum emittance cell 10 gradient free sector bend, b=d=1, E = 3 GeV = F = 1: Theoretical minimum emittance (Ex)= 1.5 nm rad Tune advance (Qx)= Ideal phase advance Ψ = A.Streun, PSI 16

17 Emittance Damping times τ i = 6.67 ms C [m]e [GeV] J i U [kev] J x = 1 D J y = 1 J s = 2 + D D = 1 2π mag D(s) [b 1 (s) 2 + 2b 2 (s)] ds Energy loss per turn: U [kev] = 26.5(E[GeV]) 3 B[T] Stability requirement: 2 < D < 1 Separate function bends: D 1 in light sources. Combined function bending magnets: Adjust gradients! Option: Vertical focusing in bending magnet: b 2 < 0 J x 2: half emittance! A.Streun, PSI 17

18 Emittance Energy spread and Beam size r.m.s. natural energy spread: σ e = B[T] E[GeV ] J s J s 2 Beam size and effective emittance: σ x (s) = ɛ x β x (s) + (σ e D(s)) 2 σ y (s) = ɛ y β y (s) ɛ x,eff (s) = ɛ 2 xo + ɛ xo H(s)σ 2 e A.Streun, PSI 18

19 Emittance Vertical emittance Ideal flat Lattice: H y 0 ɛ y = 0 Real Lattice: Errors as sources of vertical emittance ɛ y Vertical dipoles (a 1 ): Skew quadrupoles (a 2 ): Dipole rolls Quadrupole rolls roll = s-rotation Quadrupole heaves Sextupole heaves heave = y displacement Vertical dispersion (D y ) Linear coupling (κ) orbit correction skew quadrupoles for suppression Emittance ratio g = ɛ y ɛ x ɛ x = 1 1+g ɛ xo ɛ y = g 1+g ɛ xo Coupling corrected lattices: g 10 3 BUT: Diffraction limitation Brightness 1/g only for hard X-rays Touschek lifetime (bunch volume) g A.Streun, PSI 19

20 Lattice parameters Circumference and periodicity Circumference C Area minimize Optics relax Spaces reserve RF harmonic number C = hλ rf h = h 1 h 2 h 3... Ritsumeikan PSR LEP C = 98 cm C = 27 km Periodicity N per Advantages of large periodicity: simplicity: design & operation stability: resonances cost efficiency: few types DORIS: N per = 1 APS: N per = 40 A.Streun, PSI 20

21 Lattice parameters Working point Betatron resonances: aq x + bq y = p Tune constraints: NO integer NO half integer NO sum resonance NO sextupole resonances Multiturn injection: and more... order: n = a + b systematic: N per /p = integer regular: b even, skew: b odd (a, b, k, n, N per, p integers) Q x;y = k dipolar errors Q x;y = (2k + 1)/2 gradient errors Q x + Q y = p coupling Q x = p, 3Q x = p, Q x ± 2Q y = p dynamic acceptance frac(q) 0.2 septum A.Streun, PSI 21

22 Lattice parameters Working point: Example Ideal lattice Real Lattice A.Streun, PSI 22

23 Acceptance Acceptance: 6D volume of stable particles decoupling: horizontal, vertical and longitudinal 2D-acceptances Physical acceptance Dynamic acceptance Linear lattice vacuum chamber known Nonlinear lattice separatrix unknown Longitudinal acceptance RF energy acceptance (bucket height) Lattice energy acceptance = δ-dependant horizontal acceptance Dynamic aperture = local projection of dynamic acceptance acceptance [mm mrad] aperture [mm] Design criterion: Dynamic acceptance > physical acceptance A.Streun, PSI 23

24 Acceptance Acceptance Lifetime Single particle processes exponential decay Interaction with residual gas nuclei: elastic scattering & bremsstrahlung Two particle processes (= space charge effects) hyperbolic decay Touschek effect (= intrabeam scattering) (Colliders: beam beam bremsstrahlung) 1/2 1/e hyperbolic exponential T=T=1 T el γ 2 A y P T bs δ 0.2 acc P T t γ3 σ s I sb ɛ xo g [δacc (s)] β(s)... c A y = vertical acceptance, P = pressure, σ s = bunch length, I sb = bunch current δ acc = energy acceptance, ɛ xo = natural emittance, g = emittance ratio A.Streun, PSI 24

25 Acceptance Acceptance Injection TURN 0 TURN 1 Septum stored beam horizontal acceptance TURN 2 injected beam TURN 4 A.Streun, PSI 25

26 Acceptance Physical acceptance Linear lattice (quads and bends only): infinite dynamic acceptance Particle at acceptance limit A x : x(s) = A x β x (s) cos(φ(s)) + D(s) δ Particle loss: x(s) a x (s) somewhere. Acceptance ( (ax (s) D(s) δ ) 2 ) A x = min β x (s) ( ax (s) 2 ) A y = min β x (s) A x invariant of betatron motion. Projection: x max (s) = ± A x β x (s) + D(s) δ y max (s) = ± A y β y (s) A.Streun, PSI 26

27 Acceptance Dynamic acceptance Separatrix for stable motion (no physical limitations) stable test particle not lost in tracking sufficient number of turns: ( 1 damping time, 10 synchrotron oscillations) machine model for tracking: correct & complete & realistic Available aperture = dynamic aperture with physical limitations Dynamic aperture should be wider than beampipe have little distortions = careful balancing of sextupoles = set tolerances for magnets and IDs alignment and multipolar errors x physical dynamic combined x A.Streun, PSI 27

28 Acceptance Energy acceptance Horizontal acceptance A x = 0 for δ > min(a x (s)/ D(s) ) BUT: Scattering processes energy change of core particles: X = ( 0, 0, 0, 0, δ, 0) Betatron oscillation around dispersive orbit with amplitude A x A x = γ xo (D o δ) 2 + 2α xo (D o δ)(d oδ) + β xo (D oδ) 2 = H o δ 2 β xo := β x (s o ) etc., s o = location of scattering event! Maximum value of betatron oscillation: x(s) = ( ) A x β x (s) + D(s)δ = Ho β x (s) + D(s) δ A.Streun, PSI 28

29 Local energy acceptance: δ acc (s o ) = ± min Acceptance ( a x (s) Ho β x (s) + D(s) ) Energy acceptance for different lattice locations (a x (s) = a x ): In dispersionfree section: H o = 0 δ acc = ±a x /D max At location of maximum dispersion: H o = γ o D 2 max δ acc = ±a x /(2D max ) A.Streun, PSI 29

30 Acceptance Local energy acceptance and Touschek lifetime linear δ RF nonlinear (2 Sextupole Families) +δ δ +T T Test Lattice: ESRF standard cell (dispersionfree straight sections) δ acc = min{+δ Lattice acc, δ Lattice acc, δ RF acc} T t A.Streun, PSI 30

31 The real lattice Lattice Imperfections Magnet misalignments Closed Orbit distortion and correction BPMs and correctors Correlated misalignments: magnet girders and dynamic alignment concepts Ground waves and vibrations: orbit feedback Beam rotation and coupling control Multipolar errors (Magnets and Undulators): Dynamic acceptance Gradient errors: Beta-beat quadrupole current control = Stability requirements (photon beam on sample) < 1 µm A.Streun, PSI 31

32 Outlook Trends in lattice design reduce magnet gap Mini-gap undulators define acceptance anyway Example: 5 mm gap 2 m length, β y,max 25 m dipole chamber wall 3 mm dipole gap 25 mm (instead of > 40 mm). relax on flexibility progress in computing, engineering and manufacturing calculated optics will become reality! exploit J x 2 ɛ 1 2 ɛ, σ e 2σ e D = 0 in straights, vertical focussing in bends. try octupoles 1 st order attack of 2 nd order sextupole terms A.Streun, PSI 32