Part III. The Synchro-Betatron Hamiltonian

Transcription

1 Part III Kevin Li Synchro-Betatron Motion 43/ 71

2 Outline 5 Kevin Li Synchro-Betatron Motion 44/ 71

3 Canonical transformation to kinetic energy (1) Hamiltonian H(u, P u, s 0, H s ; s) = ( 1 + x ) (E 0 β0 H s qv ) /c m c R P0 Pu 1 ( 1 + x ) A s (9) B 0 R R P = (H 0 qv ) m c, P c 0 = E 0β 0 c Kevin Li Synchro-Betatron Motion 45/ 71

4 Canonical transformation to kinetic energy (1) Hamiltonian H(u, P u, s 0, H s ; s) = ( 1 + x ) (E 0 β0 H s qv ) /c m c R P0 Pu 1 ( 1 + x ) A s (9) B 0 R R P = (H 0 qv ) m c, P c 0 = E 0β 0 c Kinetic energy E 0 β 0E = E 0 β 0H s qv (10) Kevin Li Synchro-Betatron Motion 45/ 71

5 Canonical transformation to kinetic energy () Generating function Z F (u, P u, s 0, E; s) = u P u + H s(s 0, E) ds 0 Z V ds 0 = u P u + Es 0 + q E 0β 0 F u = Pu = P u q Z E E 0β0 u ds 0, F F = H s, = se = s0 s 0 E F s = HK = q E 0β0 1 + x R Z E s ds 0 F P u = ũ = u E u = u V, Es = 1 V 1+x/R s Kevin Li Synchro-Betatron Motion 46/ 71

6 Canonical transformation to kinetic energy (3) Hamiltonian H(u, P ( u, s E, E; s) = R Ē s = 1 Es β 0 ds c 0 E 0 β4 0 E /c m c P 0 ) 1 + x R ( P u q E 0 β 0 E u ds 0 ) 1 ( 1 + x ) (A s B 0 R R Ēs) (11) Kevin Li Synchro-Betatron Motion 47/ 71

7 Canonical transformation to kinetic energy (3) Hamiltonian H(u, P ( u, s E, E; s) = R Ē s = 1 Es β 0 ds c 0 E 0 β4 0 E /c m c P 0 ) 1 + x R ( P u q E 0 β 0 E u ds 0 ) 1 ( 1 + x ) (A s B 0 R R Ēs) (11) The transverse fields are usually small and vanish upon averaging P u q E 0 β0 E u ds 0 P u Kevin Li Synchro-Betatron Motion 47/ 71

8 Canonical transformation to kinetic energy (4) We immediately relabel all -variables and can write Hamiltonian P 0 = E 0β 0 c ( H(u, P u, s E, E; s) = 1 + x ) β0 R E 1 γ0 P β u 0 1 ( 1 + x ) (A s B 0 R R Ēs) (1) Kevin Li Synchro-Betatron Motion 48/ 71

9 Canonical transformation to reference orbit system (1) Hamiltonian P 0 = E 0β 0 c ( H(u, P u, s E, E; s) = 1 + x ) β0 R E 1 γ0 P β u 0 1 ( 1 + x ) (A s B 0 R R Ēs) (13) Reference orbit σ = s s E Kevin Li Synchro-Betatron Motion 49/ 71

10 Canonical transformation to reference orbit system () Generating function Z F 3 (ũ, P u, σ, E; s) = ũp u s E (σ, E) de = ũp u + σe σ 1 β0 se+s 1 β0 F 3 ũ = P F 3 u = P u, = u = ũ P u F 3 σ = δ = E 1 «β0 = 1 γ β0 1 γ 0 «F 3 s = δ = E 1 β0 «, F 3 E = s E E = H 0 qv E 0 β 0 Kevin Li Synchro-Betatron Motion 50/ 71

11 Canonical transformation to reference orbit system () Generating function Z F 3 (ũ, P u, σ, E; s) = ũp u s E (σ, E) de = ũp u + σe σ 1 β0 se+s 1 β0 F 3 ũ = P F 3 u = P u, = u = ũ P u F 3 σ = δ = E 1 «β0 = 1 β0 F 3 s = δ = E 1 «β0 γ γ 0 1 «, F 3 E = s E E = H 0 qv E 0 β 0 Kevin Li Synchro-Betatron Motion 50/ 71

12 Canonical transformation to reference orbit system (3) We immediately relabel all -variables Hamiltonian ( H(u, P u, σ, δ; s) = 1 + R) x ( β0 δ + 1 ) β0 1 γ0 P β u 0 1 ( 1 + x ) (A s B 0 R R Ēs) + δ (14) Kevin Li Synchro-Betatron Motion 51/ 71

13 Expansions around reference orbit system (1) We set Γ(δ) = β 0 Hamiltonian ( ) δ + 1 β 1 0 γ0 β 0 and obtain ( H = 1 + x ) Γ(δ) R Pu 1 ( 1 + x ) (A s B 0 R R Ēs) + δ (15) Kevin Li Synchro-Betatron Motion 5/ 71

14 Expansions around reference orbit system () Taylor expansion in P u : Γ P u Γ 1 Γ P u Taylor expansion in δ: ( β0 δ + 1 ) β0 1 γ0 β δ δ γ 0 + δ3 γ 0 Taylor expansion in δ: β 0 1 ( ) δ + 1 β 1 0 γ0 β 0 ( 1 δ ) γ0 δ Kevin Li Synchro-Betatron Motion 53/ 71

15 Canonical transformation to closed orbit system (1) Hamiltonian ( H 1 + x ) ( Γ 1 ) R Γ P u 1 B 0 R = Γ x R Γ + 1 Γ P u + 1 x Γ R P u 1 B 0 R ( 1 + x R ) (A s Ēs) + δ ( 1 + x ) (A s R Ēs) + δ Kevin Li Synchro-Betatron Motion 54/ 71

16 Identification of terms Hamiltonian H = Γ x R Γ + 1 Γ P u + 1 x Γ R P u 1 ( 1 + x ) (A s B 0 R R Ēs) + δ Dispersion: linear synchro-betatron coupling Chromaticity: terms P u δ k Magnetic field terms Eliminate the first order synchro-betatron coupling by moving to the closed orbit system (ˆx, ˆp x, ˆσ, ˆδ) where all first order synchro-betatron terms ˆxδ k and ˆp xδ k cancel; this will naturally introduce dispersion Kevin Li Synchro-Betatron Motion 55/ 71

17 Identification of terms Hamiltonian H = Γ x R Γ + 1 Γ P u 1 x + Γ R P u 1 ( 1 + x ) (A s B 0 R R Ēs) + δ Dispersion: linear synchro-betatron coupling Chromaticity: terms P u δ k Magnetic field terms Eliminate the first order synchro-betatron coupling by moving to the closed orbit system (ˆx, ˆp x, ˆσ, ˆδ) where all first order synchro-betatron terms ˆxδ k and ˆp xδ k cancel; this will naturally introduce dispersion Kevin Li Synchro-Betatron Motion 55/ 71

18 Identification of terms Hamiltonian H = Γ x R Γ + 1 Γ P u 1 x + Γ R P u 1 ( 1 + x ) (A s B 0 R R Ēs) + δ Dispersion: linear synchro-betatron coupling Chromaticity: terms P u δ k Magnetic field terms Eliminate the first order synchro-betatron coupling by moving to the closed orbit system (ˆx, ˆp x, ˆσ, ˆδ) where all first order synchro-betatron terms ˆxδ k and ˆp xδ k cancel; this will naturally introduce dispersion Kevin Li Synchro-Betatron Motion 55/ 71

19 Identification of terms Hamiltonian H = Γ x R Γ + 1 Γ P u 1 x + Γ R P u 1 ( 1 + x ) (A s B 0 R R Ēs) + δ Dispersion: linear synchro-betatron coupling Chromaticity: terms P u δ k Magnetic field terms Eliminate the first order synchro-betatron coupling by moving to the closed orbit system (ˆx, ˆp x, ˆσ, ˆδ) where all first order synchro-betatron terms ˆxδ k and ˆp xδ k cancel; this will naturally introduce dispersion Kevin Li Synchro-Betatron Motion 55/ 71

20 Identification of terms Hamiltonian H = Γ x R Γ + 1 Γ P u + δ + 1 K x + 1 g ( x y ) (K g + f) ( x 3 3 x y ) K = 1 R, g = B 1 B 0 R, f = B B 0 R Dispersion: linear synchro-betatron coupling Chromaticity: terms P u δ k Magnetic field terms Eliminate the first order synchro-betatron coupling by moving to the closed orbit system (ˆx, ˆp x, ˆσ, ˆδ) where all first order synchro-betatron terms ˆxδ k and ˆp xδ k cancel; this will naturally introduce dispersion Kevin Li Synchro-Betatron Motion 55/ 71

21 Identification of terms Hamiltonian H = Γ x R Γ + 1 Γ P u + δ + 1 K x + 1 g ( x y ) (K g + f) ( x 3 3 x y ) Dispersion: linear synchro-betatron coupling Chromaticity: terms P u δ k Magnetic field terms Eliminate the first order synchro-betatron coupling by moving to the closed orbit system (ˆx, ˆp x, ˆσ, ˆδ) where all first order synchro-betatron terms ˆxδ k and ˆp xδ k cancel; this will naturally introduce dispersion Kevin Li Synchro-Betatron Motion 55/ 71

22 Canonical transformation to closed orbit system () Generating function F (x, ˆp x, σ, ˆδ; nx s) = xˆp x + σˆδ + (X k (s)x P k (s)ˆp x + S k (s)) δ k k=1 nx x = ˆx + P k (s)δ k p x = ˆp x + k=1 nx X k (s)δ k k=1 Kevin Li Synchro-Betatron Motion 56/ 71

23 Canonical transformation to closed orbit system () Generating function F (x, ˆp x, σ, ˆδ; s) = xˆp x + σˆδ + σ = ˆσ δ = ˆδ F s = nx nx (X k (s)x P k (s)ˆp x + S k (s)) δ k k=1 nx k (X k (s)ˆx P k (s)ˆp x + S k (s)) δ k 1 k=1 k=1 l=1 n X k=1 mx kx k (s)p l (s)δ k+l 1 `X k (s)x P k(s)ˆp x + S k(s) δ k Kevin Li Synchro-Betatron Motion 56/ 71

24 Canonical transformation to closed orbit system (3) Hamiltonian H = Γ x R Γ + 1 Γ p x + 1 K x + 1 g ( x y ) (K g + f) ( x 3 3 x y ) + δ K = 1 R, g = B 1 B 0 R, f = B B 0 R Insert expansions for Γ, 1/Γ and x, p x, σ, δ Fix a k, collect all terms ˆxδ k and ˆp x δ k and determine the coefficients X k (s), P k (s) and S k (s) such, that these terms vanish Hamiltonian in closed coordinate system Kevin Li Synchro-Betatron Motion 57/ 71

25 First order dispersion function k = 1: setting the coefficienty to vanish yields First order dispersion function X 1(s) + (K + g)p 1(s) K = 0 P 1(s) + X 1(s) = 0 D(s) = P 1(s) D (s) + (K + g)d(s) = K Kevin Li Synchro-Betatron Motion 58/ 71

26 Second order dispersion function k = : setting the coefficienty to vanish yields Second order dispersion function X (s) + K + (K + g)p (s) + K γ0 X1(s) + Kg + f «P 1(s) = 0 P (s) X 1(s) + X (s) + KX 1(s)P 1(s) = 0 D (s) + (K + g)d (s) + D (s) K D (s) K D(s) + D(s) = P 1(s), D (s) = P (s) K 3 + Kg + f «D(s) = K γ0 Kevin Li Synchro-Betatron Motion 59/ 71

27 Closed orbit Hamiltonian (1) Hamiltonian up to third order in closed orbit coordinates H(ˆx, ˆp x,, ŷ, ˆp y, ˆσ, ˆδ; s) = 1 η ˆδ ν ˆδ3 + 1 ˆp x + 1 ˆp y + 1 K ˆx + 1 g (ˆx ŷ ) (K g + f) (ˆx 3 3 ˆx ŷ ) q P 0 (A rf s Ēs) (16) Kevin Li Synchro-Betatron Motion 60/ 71

28 Closed orbit Hamiltonian () Variable annotation ˆx = x Dδ ˆp x = p x D δ ˆσ = σ Dp x + D x + DD δ ˆδ = δ η = KD 1 γ0 ν = KD γ0 1 D (s) KD (s) 3 γ0 Kevin Li Synchro-Betatron Motion 61/ 71

29 Dynamic electric field (1) Assumptions: The RF cavity to be placed in a straight section: x R = 0 The RF cavity is placed in a low dispersion region: D(s) 0 The RF field is given by E rf = V 0 δ h (s) sin(ω rf t) with δ h (s) a periodic delta-function with the period πr. Then, the dynamic field term in the Hamiltonian is q (A rf s P Ēs) = q ( 0 P 0 t Arf s ) s V dt = q E rf dt P 0 = qv 0 (δ h (s) cos(ω rf t) cos(ϕ 0 )) qv 0σ P 0 ω rf πr sin(ϕ 0 ) P 0 ω 0 Kevin Li Synchro-Betatron Motion 6/ 71

30 Dynamic electric field () We can decompose δ h (s) cos(ω rf t) to δ h (s) cos(ω rf t) = 1 4πR n= ( exp ih s ) ( R + iω rft + exp ih s ) R iω rft Kevin Li Synchro-Betatron Motion 63/ 71

31 Averaging the Hamiltonian (1) Averaging over many turns, only the terms for which h = ω rf ω 0 δ h (s) cos(ω rf t) 1 πr cos = 1 πr cos ω rf t ω rf hσ R «ω 0 s R contribute 3, hence «It follows that q A rf s P Ēs = 0 with ϕ = hs/r. = qv 0 cos ««hσ cos(ϕ 0) R qv0σ πr P 0ω 0 sin(ϕ 0) πrp 0ω rf qv 0 (cos(ϕ ϕ0) cos(ϕ0) (ϕ ϕ0) sin(ϕ0)) πrp 0ω 0h 3 ω 0 = β 0c R the revolution frequency Kevin Li Synchro-Betatron Motion 64/ 71

32 Averaging the Hamiltonian () In addition the variables we encountered earlier become η = 1 η ds = 1 D(s) C C R ds 1 γ0 = α 1 ν = 1 C γ 0 = β 3 γ 0 = 1 γ T ν ds = 1 C 1 γ 0 ( D(s) Rγ 0 1 D (s) D ) (s) ds 3 R γ0 η and ν the first and the second order slippage factors. α and β the first and the second order momentum compaction factors. (17) (18) Kevin Li Synchro-Betatron Motion 65/ 71

33 The final decoupled synchro-betatron Hamiltonian Synchro-betatron Hamiltonian H(ˆx, ˆp x, ŷ, ˆp y, σ, δ; s) = 1 ηδ νδ3 qv 0 πrp 0 ω rf ( cos ( ) ) hσ cos(ϕ 0 ) R + 1 ˆp x + 1 ˆp y + 1 K x + 1 g (ˆx ŷ ) qv 0σ πr P 0 ω 0 sin(ϕ 0 ) (K g + f) (ˆx 3 3 ˆx ŷ ) (19) Kevin Li Synchro-Betatron Motion 66/ 71

34 The final decoupled synchro-betatron Hamiltonian Synchro-betatron Hamiltonian H(ˆx, ˆp x, ŷ, ˆp y, σ, δ; s) = 1 ηδ νδ3 qv 0 πrp 0 ω rf ( cos ( ) ) hσ cos(ϕ 0 ) R + 1 ˆp x + 1 ˆp y + 1 K x + 1 g (ˆx ŷ ) qv 0σ πr P 0 ω 0 sin(ϕ 0 ) (K g + f) (ˆx 3 3 ˆx ŷ ) (19) Kevin Li Synchro-Betatron Motion 66/ 71

35 Longitudinal Hamiltonian We relabel q e, V 0 V m, P 0 p 0, σ ζ. Should look familiar... Longitudinal Hamiltonian H(s) = 1 qv 0 ηδ πrp 0ω rf H(t) = 1 βcηδ + h = ω rf ω 0, ω 0 = βc R evm πhp 0 cos cos hσ R hζ R ««cos(ϕ 0) «cos(φ s) «+ qv0σ πr P 0ω 0 sin(ϕ 0) evm sin(φs) ζ p 0C Kevin Li Synchro-Betatron Motion 67/ 71

36 Longitudinal Hamiltonian Kevin Li Synchro-Betatron Motion 67/ 71

37 Longitudinal Hamiltonian We relabel q e, V 0 V m, P 0 p 0, σ ζ. Should look familiar... we have recovered Giovanni s formula! Longitudinal Hamiltonian H(s) = 1 qv 0 ηδ πrp 0ω rf H(t) = 1 βcηδ + h = ω rf ω 0, ω 0 = βc R evm πhp 0 cos cos hσ R hζ R ««cos(ϕ 0) «cos(φ s) «+ qv0σ πr P 0ω 0 sin(ϕ 0) evm sin(φs) ζ p 0C Kevin Li Synchro-Betatron Motion 67/ 71

38 Longitudinal action Synchrotron tune qv 0 ηh Q s = πe 0 β0 cos(ϕ 0 ) Kevin Li Synchro-Betatron Motion 68/ 71

39 Conclusions Summary A hopefully comprehensive overview of why and how to derive a general practical Hamiltonian for circular accelerators starting from the most basic first principles Kevin Li Synchro-Betatron Motion 69/ 71

40 Books Goldstein, Herbert, Poole, Charles, and Safko, John: Classical Mechanics; 3rd ed., Addison-Wesley, 00 Jackson, John David: Classical Electrodynamics, 3rd edition, New York : Wiley, cop., 1999 Huang, Kerson: Statistical Mechanics; nd ed., Wiley, 1987 Peskin, Michael E, and Schroeder, Daniel V: An Introduction to Quantum Field Theory; 1995 ed., Westview, 1995, Includes exercises Kevin Li Synchro-Betatron Motion 70/ 71

41 Articles Suzuki, T.: Hamiltonian Formulation for Synchrobetatron Resonance Driven by Dispersion in RF Cavities, IEEE Trans. Nucl. Sci. 3, volume 3, 7375, 1985 Symon, Keith R.: Derivation of Hamiltonians for Accelerators, ANL/APS/TB-8, 1997 Tzenov, Stephan I: Contemporary Accelerator Physics, World Scientific, New Jersey, NJ, 001 Kevin Li Synchro-Betatron Motion 71/ 71