Accelerator Physics Particle Acceleration. G. A. Krafft Old Dominion University Jefferson Lab Lecture 4

Transcription

1 Aelerator Physis Partile Aeleration G. A. Krafft Old Dominion University Jefferson Lab Leture 4 Graduate Aelerator Physis Fall 15

2 Clarifiations from Last Time On Crest, RI 1 RI a 1 1 Pg RL Pg L V Pg RL RLI Pg RL 1 K Off Crest with Detuning P P g g Z I Z k i = 8 V i i I i 1 tan os sin g b b RL Zk V IR L IR L g 1 os tan sin b b RL V V Graduate Aelerator Physis Fall 15

3 Off Crest/Off Resonane Power Typially eletron storage rings operate off rest in order to ensure stability against phase osillations. As a onsequene, the rf avities must be detuned off resonane in order to minimie the refleted power and the required generator power. Longitudinal gymnastis may also impose off rest operation in reirulating linas. We write the beam urrent and the avity voltage as I V b I e V e i b and set i The generator power an then be expressed as: P V (1 ) IR L IR L g 1 osb tan sinb RL 4 V V Wiedemann Graduate Aelerator Physis Fall 15

4 Optimal Detuning and Coupling Condition for optimum tuning with beam: tan IR V Condition for optimum oupling with beam: Minimum generator power: L IR sin a opt 1 os V b b P g,min V R a opt Wiedemann Graduate Aelerator Physis Fall 15

5 C75 Power Estimates G. A. Krafft Graduate Aelerator Physis Fall 15

6 P (kw) 1 GeV Projet Spes 7-ell, 15 MH, 93 ohms kw MV/m, 46 ua, 5 H, deg 1. MV/m, 46 ua, H, deg 1. MV/m, 46 ua, 15 H, deg 1. MV/m, 46 ua, 1 H, deg 1. MV/m, 46 ua, H, deg Qext (1 6 ) Delayen and Krafft TN mua*15 MV=6.8 kw Graduate Aelerator Physis Fall 15

7 P (kw) 7-ell, 15 MH, 93 ohms MV/m, 46 ua, 5 H, deg 4. MV/m, 46 ua, 5 H, deg 3. MV/m, 46 ua, 5 H, deg. MV/m, 46 ua, 5 H, deg 1. MV/m, 46 ua, 5 H, deg Qext (1 6 ) Graduate Aelerator Physis Fall 15

8 Assumptions Low Loss R/Q = 93*5/7 = 645 Ω Max Current to be aelerated 46 µa Compute and 5 H detuning power urves 75 MV/ryomodule (18.75 MV/m) Therefore mathed power is 4.3 kw (Sale inrease 7.4 kw tube spe) Q ext adjustable to (if not need more RF power!) Graduate Aelerator Physis Fall 15

9 14 Power vs. Qext 1 5 H detuning P g (kw) detuning Q ext ( 1 6 ) Graduate Aelerator Physis Fall 15

10 Frequeny (H) Probability Density RF Cavity with Beam and Mirophonis f f f m The detuning is now: tan Q tan Q L L f f where f and f m is the stati detuning (ontrollable) is the random dynami detuning (unontrollable) Time (se) Probability Density Medium CM Prototype, Cavity #, 6MV/m 4 samples Peak Frequeny Deviation (V) Graduate Aelerator Physis Fall 15

11 Q ext Optimiation with Mirophonis P V (1 ) ItotR L ItotR L 1 os tan sin RL 4 V V g tot tot f tan Q L f where f is the total amount of avity detuning in H, inluding stati detuning and mirophonis. Optimiing the generator power with respet to oupling gives: f opt ItotRa where b os tot V ( b 1) Q b tan f where I tot is the magnitude of the resultant beam urrent vetor in the avity and tot is the phase of the resultant beam vetor with respet to the avity voltage. tot Graduate Aelerator Physis Fall 15

12 Graduate Aelerator Physis Fall 15 Corret Stati Detuning To minimie generator power with respet to tuning: The resulting power is tan tot f f b Q 1 (1 ) 4 m g a V f P b Q R f m g opt opt a opt opt a V f P b b Q R f V b R

13 Optimal Q ext and Power Condition for optimum oupling: and m ( b1) Q f In the absene of beam (b=): opt f opt V f m Pg b 1 ( b 1) Q R a f and P opt f m 1Q f V f opt m g 1 1 Q R a f Graduate Aelerator Physis Fall 15

14 Problem for the Reader Assuming no mirophonis, plot opt and P g opt as funtion of b (beam loading), b=-5 to 5, and explain the results. How do the results hange if mirophonis is present? Graduate Aelerator Physis Fall 15

15 Example ERL Injetor and Lina: f m =5 H, Q =1x1 1, f =13 MH, I =1 ma, V = MV/m, L=1.4 m, R a /Q =136 ohms per avity ERL lina: Resultant beam urrent, I tot = ma (energy reovery) and opt =385 Q L =.6x1 7 P g = 4 kw per avity. ERL Injetor: I =1 ma and opt = 5x1 4! Q L = x1 5 P g =.8 MW per avity! Note: I V a =.8 MW optimiation is entirely dominated by beam loading. Graduate Aelerator Physis Fall 15

16 RF System Modeling To inlude amplitude and phase feedbak, nonlinear effets from the klystron and be able to analye transient response of the system, response to large parameter variations or beam urrent flutuations we developed a model of the avity and low level ontrols using SIMULINK, a MATLAB-based program for simulating dynami systems. Model desribes the beam-avity interation, inludes a realisti representation of low level ontrols, klystron harateristis, mirophoni noise, Lorent fore detuning and oupling and exitation of mehanial resonanes Graduate Aelerator Physis Fall 15

17 RF System Model Graduate Aelerator Physis Fall 15

18 RF Modeling: Simulations vs. Experimental Data Measured and simulated avity voltage and amplified gradient error signal (GASK) in one of CEBAF s avities, when a 65 A, 1 se beam pulse enters the avity. Graduate Aelerator Physis Fall 15

19 Conlusions We derived a differential equation that desribes to a very good approximation the rf avity and its interation with beam. We derived useful relations among avity s parameters and used phasor diagrams to analye steady-state situations. We presented formula for the optimiation of Q ext under beam loading and mirophonis. We showed an example of a Simulink model of the rf ontrol system whih an be useful when nonlinearities an not be ignored. Graduate Aelerator Physis Fall 15

20 RF Foussing In any RF avity that aelerates longitudinally, beause of Maxwell Equations there must be additional transverse eletromagneti fields. These fields will at to fous the beam and must be aounted properly in the beam optis, espeially in the low energy regions of the aelerator. We will disuss this problem in greater depth in injetor letures. Let A(x,y,) be the vetor potential desribing the longitudinal mode (Loren gauge) 1 A t A A Graduate Aelerator Physis Fall 15

21 Graduate Aelerator Physis Fall 15...,..., 1 1 r r r A A r A For ylindrially symmetrial aelerating mode, funtional form an only depend on r and Maxwell s Equations give reurrene formulas for sueeding approximations 1 1 1, 1, n n n n n n d d n A d A d A n

22 Gauge ondition satisfied when da d n i n in the partiular ase n = da d i Eletri field is 1 E A t Graduate Aelerator Physis Fall 15

23 And the potential and vetor potential must satisfy E i E d d i, A d, A 4A 1 d A So the magneti field off axis may be expressed diretly in terms of the eletri field on axis B i r ra E, 1 Graduate Aelerator Physis Fall 15

24 And likewise for the radial eletri field (see also ) E r r 1 r de, d Expliitly, for the time dependene os(ωt + δ) E r E r, r,, t E, os t, t r de, d r B sin os t E r,, t E, t Graduate Aelerator Physis Fall 15

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28 Graduate Aelerator Physis Fall 15 Motion of a partile in this EM field B V E e dt m d V ' ' ' sin ' ' ' ' os ' ' ' - d t G x t d dg x x x

29 The normalied gradient is G ee, m and the other quantities are alulated with the integral equations G' os t' t lim G ' ' os t ' d' d' ' d' Graduate Aelerator Physis Fall 15

30 These equations may be integrated numerially using the ylindrially symmetri CEBAF field model to form the Douglas model of the avity foussing. In the high energy limit the expressions simplify. x a x x x a a ' ' ' ' x' G' a ' ' a x d' ' os t d' Graduate Aelerator Physis Fall 15

31 Graduate Aelerator Physis Fall 15 Transfer Matrix E E I L E E T G G For position-momentum transfer matrix d G d G I / sin sin / os os

32 Kik Generated by mis-alignment E G E Graduate Aelerator Physis Fall 15

33 Damping and Antidamping By symmetry, if eletron traverses the avity exatly on axis, there is no transverse defletion of the partile, but there is an energy inrease. By onservation of transverse momentum, there must be a derease of the phase spae area. For linas NEVER use the word adiabati d mv transverse dt x x Graduate Aelerator Physis Fall 15

34 Conservation law applied to angles, 1 x y / / x x x y y y x y x y Graduate Aelerator Physis Fall 15

35 Phase spae area transformation dx dx dx d dy dy dy d Therefore, if the beam is aelerating, the phase spae area after the avity is less than that before the avity and if the beam is deelerating the phase spae area is greater than the area before the avity. The determinate of the transformation arrying the phase spae through the avity has determinate equal to Det M avity x y Graduate Aelerator Physis Fall 15

36 By onatenation of the transfer matries of all the aelerating or deelerating avities in the reirulated lina, and by the fat that the determinate of the produt of two matries is the produt of the determinates, the phase spae area at eah loation in the lina is dx dy d x d y dx dy d x d y Same type of argument shows that things like orbit flutuations are damped/amplified by aeleration/deeleration. Graduate Aelerator Physis Fall 15

37 Transfer Matrix Non-Unimodular M P P P tot M M 1 M M det M unimodular! M det M M det M tot 1 M PM PM tot M det M M tot 1 an separately trak the"unimodular part"(as before!) and normalie by aumulated determinate 1 Graduate Aelerator Physis Fall 15

38 Solenoid Foussing Can also have ontinuous fousing in both transverse diretions by applying solenoid magnets: B Graduate Aelerator Physis Fall 15

39 Bush s Theorem For ylindrial symmetry magneti field desribed by a vetor potential: ˆ 1 A A, r B ra, r is nearly onstant r r Br, r B r, r A, r Br Conservation of Canonial Momentum gives Bush s Theorem: P mr qra onst for partile with where B, P qr B mr Larmor Beam rotates at the Larmor frequeny whih implies oupling Graduate Aelerator Physis Fall 15

40 d dt k Radial Equation mr mr qr B mr L thin lens foal length 1 f e 4 B d m L L weak ompared to quadrupole for high If go to full ¼ osillation inside the magneti field in the thik lens ase, all partiles end up at r =! Non-ero emittane spreads out perfet fousing! y x Graduate Aelerator Physis Fall 15

41 Larmor s Theorem This result is a speial ase of a more general result. If go to frame that rotates with the loal value of Larmor s frequeny, then the transverse dynamis inluding the magneti field are simply those of a harmoni osillator with frequeny equal to the Larmor frequeny. Any fore from the magneti field linear in the field strength is transformed away in the Larmor frame. And the motion in the two transverse degrees of freedom are now deoupled. Pf: The equations of motion are d dt mr mr qr B mr qa ons P d dt d mr mr dt mr mr qr B qr B mr P mr mr mr L L L L mr P -D Harmoni Osillator Graduate Aelerator Physis Fall 15