Lattices and Emittance

Transcription

1 Lattices and Emittance Introduction Design phases Interfaces Space Lattice building blocks local vs. global Approximations Fields and Magnets Beam dynamics pocket tools Transfer matrices and betafunctions Design code Emittance and lattices Emittance Lattice cells Minimum emittance Vertical emittance Other lattice parameters Acceptance Circumference & periodicity Tune and chromaticity physical momentum dynamic Optimization A.Streun, PSI 1

2 Introduction Lattice Design Phases 1. Preparation Performance Boundary conditions Building blocks (magnets) 2. Linear lattice Global quantities Cells, matching sections, insertions, etc. 3. Nonlinear lattice Sextupoles Dynamic acceptance 4. Real lattice Closed orbit Alignment errors Multipolar errors A.Streun, PSI 2

3 Introduction Lattice Design Interfaces Magnet Design: Technological limits, coil space, multipolar errors Vacuum: Impedance, pressure, physical apertures, space Radiofrequency: Momentum 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 Introduction 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 composition *? I * *? = H J A I E =? O H E? = J D A H + E = J A 6 H = I B H = J E I 6 6! +? = J A = J E 6 H = I = J E I 4 J = J E I. E E C - A A J I - -, H E B J 3 K H K F A * E C = C A J 6 D A = J J E? A * * * *! ! 6 6 6! 6! " Ref.: E.Forrest & K.Hirata, A contemporary guide to beam dynamics, KEK A.Streun, PSI 5

6 N O I N O I Lattice building blocks local global X = (x, p x, y, p y, δ, s) Block X in X out δ := p s = s s o p o = Concatenate = Lattice One turn map: X n X n+1 Closed Orbit = Fixpoint Transfermatrix = Linearization } of one turn map A.Streun, PSI 6

7 Lattice building blocks Approximations Ideal lattice: no translations: closed orbit = block symmetry axis Decoupling: Betatron frequencies Synchrotron frequency: {x,p x,y,p y, δ, s } {{ } fast (MHz) }{{} slow (khz) Horizontal-Vertical coupling κ 1: { {x,p x } }{{} horizontal {y,p y } }{{} vertical } δ, s constant. {δ, s} }{{} longitudinal Linear Lattice: betafunctions, betatron phases, etc. Nonlinear latice: perturbative treatment of nonlinearities } Tracking A.Streun, PSI 7

8 Lattice building blocks Hierarchy of building blocks Error lattice "Bare" lattice Nonlinear lattice Linear lattice Drift Bend Quadrupole Sextupole RF-cavity Injection BPM Corrector [Skew Quad] special devices Solenoid Undulator Separator A.Streun, PSI 8

9 Lattice building blocks Field and multipole definition B y (x, y) + ib x (x, y) = (Bρ) n (ia n + b n )(x + iy) n 1 2n-pole magnet: n = 1, 2, 3... = dipole, quadrupole, sextupole b n regular, a n skew (rotated by 90 /n) Magnetic rigidity: Bρ = p q = βe/e n e c = E[GeV] for electrons Regular multipole: b n = 1 Bρ 1 (n 1)! (n 1) B y (x,y) x n 1 y=0 Poletip field: B pt = (Bρ)b n R n 1 = Rn 1 (n 1)! (n 1) B y (x,y) x n 1 y=0 R pole inscribed radius Conventions: h = 1/ρ = b 1 (Dip.), k = b 2 (Quad.), m = b 3 (Sext.) A.Streun, PSI 9

10 H H Lattice building blocks Bending magnet + O H E? = *? = C A J I A? J H H A B 4 A B A H A? A 2 = H J E? A * H F A K I J > O H H H A B 1 * * O >. * O * H. H A? J = C K = H > Parameters: L, 1 ρ ref! = b 1, b 2, ζ 1, ζ 2, [g, k 1, k 2 ], [b n, n 3] A.Streun, PSI 10

11 Lattice building blocks Quadrupole Parameters: L, b 2, R A.Streun, PSI 11

12 Lattice building blocks Sextupole Parameters: L, b 3, R A.Streun, PSI 12

13 Lattice building blocks Other devices Parameters Purpose 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 13

14 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 14

15 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 current R n = Apertures: As small as possible As large as necessary acceptance A.Streun, PSI 15

16 Beam Dynamics Pocket Tools Local: Transfer matrix of a gradient bend x x y y δ = out c x 1 K s x 0 0 K s x c x 0 0 b 1 K (1 c x) b 1 K s x 0 0 c y 1 b2 s y b 2 s y c y x x y y δ in with: c x [s x ] = cos[sin]( K L), c y [s y ] = cos [sin]( b 2 L), K = b b 2 cos ix = cosh x, sin ix = i sinh x can be focussing or defocussing Special cases: b 2 = 0 Dipole b 1 = 0 Quadrupol b 1 = 0 and b 2 = 0 Drift A.Streun, PSI 16

17 Beam Dynamics Pocket Tools Transfer matrix of a gradient bend, alternative convention x x y y δ = out c x 1 K s x 0 0 h K (1 c x) K s x c x 0 0 h K s x 0 0 c y 1 k s y k s y c y x x y y δ in with: c x [s x ] = cos[sin]( K L), c y [s y ] = cos[sin]( k L), K = h 2 k cos ix = cosh x, sin ix = i sinh x can be focussing or defocussing Special cases: k = 0 Dipole h = 0 Quadrupol h = 0 and k = 0 Drift A.Streun, PSI 17

18 Beam Dynamics Pocket Tools Global: Betafunction and Emittance Practical view: Theoretical view: σ x = Betafunction β }{{} magnet structure H = p2 x 2 + k(s)x2 2 c.t. Emittance ɛ }{{} particle ensemble H = J β(s), ɛ = J Betatron oscillation: Twiss parameters: φ = 1 β Transformation of twiss parameters: β α γ b = C 2 2SC S 2 CC S C + SC SS C 2 2S C S 2 x(s) = 2J x β x (s)cosφ(s) + D(s) δ β ds α = 2 γ = 1+α2 β β α γ a with M a b = Ref.: R. D.Ruth, Single particle dynamics in circular accelerators, AIP Conf.Proc. 153 (1987) 150 ( C S S C A.Streun, PSI 18 )

19 Beam Dynamics Pocket Tools Transfermatrix a b : M a b = T 1 b = Global: Transfermatrix of a lattice section Transformation to normalized phase space at a Rotation by φ = φ b φ a in normalized phase space Backtransformation to real phase space at b ( cos φ sin φ sin φ cos φ ) βb β a (cos φ + α a sin φ) (α a α b )cos φ (1+α a α b )sin φ βa β b T a with T = 1 0 β α β βa β b sin φ β βa β b (cos φ α b sin φ) Periodic structure (a = b) One turn matrix ( µ = 2πQ, Q betatron tune): ( ) ( cosµ + αa sinµ β a sinµ cosµ βa sinµ M a = γ a sinµ cosµ α a sinµ symmetry point 1 β a sinµ cosµ A.Streun, PSI 19 )

20 Beam Dynamics Pocket Tools 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 20

21 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 21

22 E, Lattice Cells Building low emittance lattices... = > N > O A B C D A.Streun, PSI 22

23 Lattice Cells Dispersion suppression by using a half bending magnet: Increased quadrupole strength (lengthes constant) Incrased length before half bend (quadrupoles constant) A.Streun, PSI 23

24 Lattice Cells DBA example: ESRF high beta straight sections low beta DBA 6 GeV A.Streun, PSI 24

25 Lattice Cells TBA example: SLS 11.5 m straight 4 m straight 7 m straight GeV A.Streun, PSI 25

26 Lattice Cells Combined function example: SLS booster 2.4 GeV A.Streun, PSI 26

27 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 27

28 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 28

29 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 29

30 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 30

31 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 31

32 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 32

33 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 33

34 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 34

35 Lattice parameters Working point: Example Ideal lattice Real Lattice A.Streun, PSI 35

36 Lattice parameters Chromaticity Chromatic aberrations: b 2 (δ) = b 2 /(1 + δ) = Q = Q o + ξδ Chromaticity ξ = dq/dδ < 0 : Tune spread, resonance crossings Head-tail instability Correction by sextupoles in dispersive regions: ξ x = 1 [ 2b3 (s)d(s) b 2 (s) ] β x (s) ds 4π C ξ y = 1 [ 2b3 (s)d(s) + b 2 (s) ] β y (s) ds 4π C Sextupole nonlinearity (B y (x) x 2 ) = dynamic acceptance problems A.Streun, PSI 36

37 Acceptance Acceptance Definitions Acceptance: 6D volume of stable particles decoupling: horizontal, vertical and longitudinal 2D-acceptances Physical acceptance Dynamic acceptance Linear lattice vaccum chamber known Nonlinear lattice separatrix unknown Longitudinal acceptance RF momentum acceptance (bucket height) Lattice momentum 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 37

38 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) A x invariant of betatron motion. Projection: x max (s) = ± A x β x (s) + D(s) δ A.Streun, PSI 38

39 Acceptance Momentum acceptance Horizontal acceptance A x = 0 for δ > min(a x (s)/ D(s) ) BUT: Scattering processes momentum 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 39

40 Local momentum acceptance: δ acc (s o ) = ± min Acceptance ( a x (s) Ho β x (s) + D(s) ) Momentum 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 40

41 Acceptance Dynamic acceptance Quadrupole: x = b 2 Lx y = b 2 Ly Sextupole: x = b 3 L(x 2 y 2 ) y = 2b 3 Lxy Quadrupole: chromatic aberration b 2 (δ) = b 2 /(1 + δ) b 2 (1 δ) compensation by sextupole in dispersive region (x Dδ + x, y y): Quadrupole: ( x ) = [b 2 L] δ x ( y ) = [b 2 L] δ y Sextupole: ( x ) = [2b 3 LD] δ x b 3 LD 2 δ 2 b 3 L(x 2 y 2 ) ( y ) = [2b 3 LD] δ y+2b 3 L xy nonlinear kicks: (b 2 L! = 2b 3 LD) Chromaticity correction nonlinear resonance driving horizontal/vertical coupling amplitude dependant tune shift = CHAOS! Restriction of dynamic acceptance A.Streun, PSI 41

42 Acceptance Sextupole effects 9 first order terms: 2 chromaticities ξ x, ξ y 2 off-momentum resonances 2Q x, 2Q y dβ/dδ ξ (2) = 2 Q/ δ 2 2 terms integer resonances Q x 1 term 3 rd integer resonances 3Q x 2 terms coupling resonances Q x ± 2Q y 13 second order terms: 3 tune shifts with amplitude: Q x / J x, Q x / J y = Q y / J x, Q y / J y 8 terms octupole like resonances: 4Q x, 2Q x ± 2Q y, 4Q y, 2Q x, 2Q y 2 second order chromaticities: 2 Q x / δ 2 and 2 Q y / δ 2 Ref.: J.Bengtsson, The Sextupole Scheme for the Swiss Light Source: An Analytic Approach, SLS-Note 9/97, PSI 1997 A.Streun, PSI 42

43 Acceptance Nonlinear Lattice Design Optimization of sextupole patterns: Chromaticity correction: Decoupling: keep strength low Sextupoles in quadrupoles: Non interleaved sextupoles: b 3 = b 2 /D inflexible! I transformer (KEK-B scheme) Multicell cancellation: N cells: N Q x, 3N Q x, 2N Q x, 2N Q y integer! Cancellation between sections: lattice section vs. mirror image = Iterate: linear nonlinear lattice design A.Streun, PSI 43

44 Acceptance Sextupole pattern optimization A.Streun, PSI 44

45 Acceptance Dynamic aperture optimization Y [mm] Physical Aperture (beampipe) X[mm] Dynamic aperture: only chromaticity correction (2 sextupole families) 1st and 2nd order optimization (9 sextupole families) required for injection A.Streun, PSI 45

46 Closing else... Impact on lattice design: The Injection Process: multi turn accumulation. Lattice Errors: 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 A.Streun, PSI 46