Betatron Motion with Coupling of Horizontal and Vertical Degrees of Freedom Part II

Transcription

1 Betatron Moton wth Couplng of Horzontal and Vertcal Degree of Freedom Part II Alex Bogacz, Geoff Krafft and Tmofey Zolkn Lecture 9 Coupled Betatron Moton II USPAS, Fort Colln, CO, June -, 3

2 Outlne Practcal Example: Soft-edge Solenod model Vertex-to-plane adapter for electron coolng Fermlab) Spn otator for Fgure-8 Collder rng Ionzaton coolng channel for Neutrno Factory and Muon Collder Generalzed vertex-to-plane tranformer nert V. Lebedev, A. Bogacz, Betatron Moton wth Couplng of Horzontal and Vertcal Degree of Freedom,, Lecture 9 Coupled Betatron Moton II USPAS, Fort Colln, CO, June -, 3

3 Hard-edge Solenod B z ) B θ h L ) θ L ) h Lecture 9 Coupled Betatron Moton II USPAS, Fort Colln, CO, June -, 3 3

4 Solenod Hard-edge Model Lnear Tranfer Matrx for nfntely long olenod : a<<l M ol cokl) nkl) nkl) -cokl) k k k nkl) cokl) -cokl) nkl) k 4 4 nkl) -cokl) cokl) nkl) k k -cokl) nkl) k nkl) cokl) k 4 4 e k B pc Φ ol e pc B z ) d e pc - B L f ol Lecture 9 Coupled Betatron Moton II USPAS, Fort Colln, CO, June -, 3 4

5 Soft-edge Solenod B z ) B tanh L a Lecture 9 Coupled Betatron Moton II USPAS, Fort Colln, CO, June -, 3 5

6 Solenod Soft-edge Model Non-zero aperture - correcton due to the fnte length of the edge: a L It ntroduce axally ymmetrc edge focung at each olenod end: Moft ol M edge M ol Medge M edge Φ edge Φ edge e k B pc e k a Φedge pc B z) d B L - 8 cokl) nkl) nkl) -cokl) k k k nkl) cokl) -cokl) nkl) k 4 4 Mol nkl) -cokl) cokl) nkl) k k -cokl) nkl) k nkl) cokl) k 4 4 Lecture 9 Coupled Betatron Moton II USPAS, Fort Colln, CO, June -, 3 6

7 Soft-edge Solenod Off-ax Feld Nonlnear focung term ΔF ~ Or ) follow from the calar potental: φr,z) φ B z z d dz B z z r 4 d B z dz ) z z 3 r d 3 B z 3 dz 8 z 4 4 z r 3 r 4 9 Solenod B-feld B z r,z) d B z dz B z r 4 B r r,z) r d dz B z r 3 3 d 6 3 dz B z Nonlnear focung ncluded n partcle trackng Lecture 9 Coupled Betatron Moton II USPAS, Fort Colln, CO, June -, 3 7

8 Axymmetrc otatonal Dtrbuton Electrotatc accelerator Gauan dtrbuton Fermlab electron coolng Solenod otatonal dtrbuton The electron beam dtrbuton axally ymmetrc, and uncoupled at the cathode: Ξ B T γ γ where T rc mktc / P the thermal emttance of the beam Lecture 9 Coupled Betatron Moton II 8 USPAS, Fort Colln, CO, June -, 3

9 Lecture 9 Coupled Betatron Moton II 9 USPAS, Fort Colln, CO, June -, 3 Axymmetrc otatonal Dtrbuton At the ext of the olenod the electron beam dtrbuton tll axally ymmetrc Φ Φ Φ Φ Φ Φ γ γ T B T n Φ Ξ Φ Ξ where Φ Φ Φ c P eb / Φ the rotatonal focung trength of the olenod edge B the olenod magnetc feld.

10 Lecture 9 Coupled Betatron Moton II USPAS, Fort Colln, CO, June -, 3 The egen-vector of the rotatonal dtrbuton: v, v It correpond to u /, / π ν ν Then, the matrx V V Axymmetrc otatonal Dtrbuton

11 Lecture 9 Coupled Betatron Moton II USPAS, Fort Colln, CO, June -, 3 4D-emmtance conervaton: T otatonal emttance etmate ) )Φ Φ Φ θ T rot r r r r Comparng left and rght hand de of the equaton T T n U V UV Ξ / / / / One obtan.,,, Φ Φ Φ Φ Φ Φ Φ Φ >> Φ >> Φ T T T T Axymmetrc otatonal Dtrbuton

12 Spn otator for Fgure-8 Collder ng x [cm] Fgure-8 Collder ng - Footprnt z [cm] total rng crcumference ~ m 6 deg. crong Lecture 9 Coupled Betatron Moton II USPAS, Fort Colln, CO, June -, 3

13 Spn otator Ingredent BETA_X&Y[m] DISP_X&Y[m] BETA_X BETA_Y DISP_X DISP_Y Arc end BL.9 Tela m 8.8 BL 8.7 Tela m 4.4 Spn rotator ~ 46 m Lecture 9 Coupled Betatron Moton II USPAS, Fort Colln, CO, June -, 3 3

14 Locally Decoupled Solenod Par 5 BL 8.7 Tela m BETA_X&Y[m] BETA_X BETA_Y BETA_Y BETA_X olenod 4.6 m decouplng quad nert olenod 4.6 m M C C Hham Sayed, PhD The ODU, Lecture 9 Coupled Betatron Moton II USPAS, Fort Colln, CO, June -, 3 4

15 Locally Decoupled Solenod Par 5 BL 8.7 Tela m BETA_X&Y[m] DISP_X&Y[m] BETA_X BETA_Y DISP_X DISP_Y olenod 4.6 m decouplng quad nert olenod 4.6 m M C C Hham Sayed, PhD The ODU, Lecture 9 Coupled Betatron Moton II USPAS, Fort Colln, CO, June -, 3 5

16 B E T A _ X & 3Y [ M o n S e p 6 7 : 4 4 : 4 5 O p t M - M A I N : - C : \ W o r k 8 B8 E T BA E_ TX DA I_ SY DP I_ SX P _ Y D I S P _ X & Y [ Unveral Spn otator - Optc 3 5 GeV BETA_X&Y[m] DISP_X&Y[m] 88 BETA_X BETA_Y DISP_X DISP_Y BL.9 Tela m 8.8 BL 8.7 Tela m 4.4 Spn rotator ~ 46 m Lecture 9 Coupled Betatron Moton II USPAS, Fort Colln, CO, June -, 3 6

17 Lecture 9 Coupled Betatron Moton II 7 USPAS, Fort Colln, CO, June -, 3 Ionzaton Coolng n an Axally Symmetrc Channel A ngle-partcle phae-pace trajectory along the beam orbt can be expreed a: ), ) ) e ) )) )) e e μ ψ μ ψ v v x One can rewrte the above equaton n the followng compact form ) ) ) a V x where ) ), ), ), ) v v v v V )) n )) co )) n )) co ) μ ψ μ ψ μ ψ μ ψ a

18 Lecture 9 Coupled Betatron Moton II 8 USPAS, Fort Colln, CO, June -, 3 Ionzaton Coolng n an Axally Symmetrc Channel In the cae of axally ymmetrc focung the egen-vector reduce to v, v V here we ued that u /, ν ν π/

19 Prncple of Ionzaton Coolng Aborber Plate Δp ab p n p p Δp z cool out F Δp F p n Each partcle loe momentum by onzng a low-z aborber Only the longtudnal momentum retored by F cavte The angular dvergence reduced untl lmted by multple catterng Lecture 9 Coupled Betatron Moton II 9 USPAS, Fort Colln, CO, June -, 3

20 Ionzaton Coolng n an Axally Symmetrc Channel Coolng Decrpton Ionzaton coolng due to energy lo n a thn aborber can be decrbed a: Δθ Δp θ p θ δ, here the longtudnal energy retoraton by mmedate reacceleraton aumed Ung canoncal varable the above coolng equaton can be wrtten a: x δ δ out x n Lecture 9 Coupled Betatron Moton II USPAS, Fort Colln, CO, June -, 3

21 Lecture 9 Coupled Betatron Moton II USPAS, Fort Colln, CO, June -, 3 Canoncal varable., x y p y x p y x Pc eb / - longtudnal magnetc feld elaton between geometrcal and canoncal varable x x, where,, y x p y p x y x y x x x θ θ, A cap denote tranfer matrce and vector related to the canoncal varable. Canoncal v Geometrc Varable

22 Lecture 9 Coupled Betatron Moton II USPAS, Fort Colln, CO, June -, 3 Ionzaton Coolng n an Axally Symmetrc Channel Employng ampltude vector repreentaton: ) ) ) a V x, one can rewrte the coolng equaton a: out a n V a V δ δ and fnally out a n V V a δ δ )) n )) co )) n )) co ) μ ψ μ ψ μ ψ μ ψ a

23 Lecture 9 Coupled Betatron Moton II 3 USPAS, Fort Colln, CO, June -, 3 Ionzaton Coolng n an Axally Symmetrc Channel Carryng out the above calculaton explctly one obtan: n out a a δ D emttance after coolng are gven by the followng formula: ) [ ] ) [ ] ) [ ] ) [ ] ) n co ) n co 4 3 δ δ φ φ δ δ δ φ φ δ O a a O a a out out out out where ψ ψ μ μ φ

24 Ionzaton Coolng n an Axally Symmetrc Channel If coolng effect of one aborber uffcently mall one can perform averagng over betatron phae. That yeld Δ Δ ) δ )δ d d p d d p dp d dp d Lecture 9 Coupled Betatron Moton II 4 USPAS, Fort Colln, CO, June -, 3

25 Lecture 9 Coupled Betatron Moton II 5 USPAS, Fort Colln, CO, June -, 3 Ionzaton Coolng n an Axally Symmetrc Channel Canoncal momentum of a ngle partcle ) a V Va x x M T T xp y yp x

26 Lecture 9 Coupled Betatron Moton II 6 USPAS, Fort Colln, CO, June -, 3 Ionzaton Coolng n an Axally Symmetrc Channel Second order moment of the Gauan dtrbuton Note that for a ngle partcle - / rm and we ue rm. emttance below) ) ) ),, 4 4,, y x x y y x y x p p xy M yp xp p p yp xp y x

27 Axally Symmetrc FOFO Cell 5 5 BETA_X&Y[m] DISP_X&Y[m] BETA_X BETA_Y 7. ab ab PHASE_X&Y.5 Q_X Q_Y 7. c L[cm]3 B[kG]38. Aperture[cm] c L[cm]8 B[kG]-34.3 Aperture[cm] Lecture 9 Coupled Betatron Moton II 7 USPAS, Fort Colln, CO, June -, 3

28 Perodc Cell - Optc Sat Mar 4 3:6:9 6 OptM - MAIN: - D:\Coolng ng\solng\nake_new.opt Fr Mar 9:55:38 6 OptM - MAIN: - D:\Coolng ng\solng\nake_new.opt PHASE_X&Y.5 7 nward half-cell outward half-cell BETA_X&Y[m] DISP_X&Y[m] BETA_X BETA_Y DISP_X DISP_Y - betatron phae adv/cell h/v) π/π Q_X Q_Y Lecture 9 Coupled Betatron Moton II USPAS, Fort Colln, CO, June -, 3 8

29 Perodc Cell Magnet nward half-cell outward half-cell Sat Mar 4 3:6:9 6 OptM - MAIN: - D:\Coolng ng\solng\nake_new.opt 7 betatron phae adv/cell h/v) π/π - BETA_X&Y[m] DISP_X&Y[m] BETA_X BETA_Y DISP_X DISP_Y olenod: L[cm] B[kG] 5 5 quadrupole: L[cm] G[kG/cm] dpole: $L; > cm $B 5; > 5 kgau $Ang$L*$B/$Hr; >.4996 rad $ang$ang*8/$pi; > 8.68 deg Lecture 9 Coupled Betatron Moton II USPAS, Fort Colln, CO, June -, 3 9

30 Snake Coolng Channel Fr Mar :5:57 6 OptM - MAIN: - D:\Coolng ng\solng\end_nake_new.opt Fr Mar :6:8 6 OptM - MAIN: - D:\Coolng ng\solng\end_nake_new.opt BETA_X&Y[m] DISP_X&Y[m] BETA_X&Y[m] DISP_X&Y[m] BETA_X BETA_Y DISP_X DISP_Y 6 BETA_X BETA_Y DISP_X DISP_Y 6 entrance cell perodc cell ext cell Lecture 9 Coupled Betatron Moton II USPAS, Fort Colln, CO, June -, 3 3

31 Muon Coolng Channel Optc Wed Jun :47:3 6 OptM - MAIN: - D:\Coolng ng\snake Channel\nake.opt 5 BETA_X&Y[m] DISP_X&Y[m] 3 BETA_X BETA_Y DISP_X DISP_Y -3 beam extenon dp. ant-uppr. n perodc PIC/EMEX cell n) dp. uppr. beam extenon F cavty kew quad nake - footprnt x [cm] z [cm] Lecture 9 Coupled Betatron Moton II USPAS, Fort Colln, CO, June -, 3 3

32 Vertex-to-plane Tranformer Inert Solenod Skew-quadrupole ytem Uncoupled axal-ymmetrc dtrbuton otatonal dtrbuton P x x Flat dtrbuton Lecture 9 Coupled Betatron Moton II 3 USPAS, Fort Colln, CO, June -, 3

33 Lecture 9 Coupled Betatron Moton II 33 USPAS, Fort Colln, CO, June -, 3 Vertex-to-plane Tranformer Inert Egen-vector of the decoupled moton n the coordnate ytem rotated by 45 45deg. F F rotaton otatonal egen-vector F F F F

34 Vertex-to-plane Tranformer Inert Focung ytem wth 45 dfference between the horzontal and vertcal betatron phae advance wll tranform the ntal vertex dtrbuton nto the flat one The reultng D emttance are a follow T, Φ Φ Φ Φ T. Lattce mplementaton Tw functon, beam ze etc. Lecture 9 Coupled Betatron Moton II 34 USPAS, Fort Colln, CO, June -, 3

35 Vertex-to-plane Tranformer Inert BETA_X&Y[m] 5 BETA_X BETA_Y BETA_Y BETA_X.6647 Betatron ze X&Y[cm] AlphaXY[-, ] PHASE/*PI).5.5 BETA_X&Y[m] 5 BETA_X BETA_Y BETA_Y BETA_X Ax Ay AlphaXY.6647 Q_ Q_ Teta Teta.6647 E kn MeV, T c. ev, c.5 cm, B ol kg, cm, cm Lecture 9 Coupled Betatron Moton II 35 USPAS, Fort Colln, CO, June -, 3

36 Summary elatonhp between the egen-vector, beam emttance and the beam ellpod n 4D phae pace From the beam ellpod to the egen-vector equvalence of both pcture) New parametrzaton of egen-vector n term of generalzed Tw functon Complete Weyl-lke repreentaton ndependent parameter to fully decrbe the moton tranport lne ambgute reolved Developed oftware baed on th repreentaton allow effectve analy of coupled betatron moton for both crcular accelerator and tranfer lne OptM). Lecture 9 Coupled Betatron Moton II 36 USPAS, Fort Colln, CO, June -, 3