accelerator physics and ion optics summary damping and cooling

Transcription

1 accelerator physics and ion optics summary damping and cooling Sytze Brandenburg sb/accphys2007_7/1

2 conservation laws Liouville theorem conservative forces Hamiltonian system phase space density conserved adiabatic damping phase space defined in coordinates x and p dx emittance defined in coordinates x and ds acceleration: p increases, p constant dx decreases, emittance decreases ds normalized emittance ε N = εβγ conserved sb/accphys2007_7/2

3 electrons: synchrotron radiation emitted power proportional to E 4 /ρ photon spectrum scales with E 3 /ρ energy loss in circular electron accelerators (e.g. LEP) E = 100 GeV E rad = 2.86 GeV/turn damping of betatron and synchrotron motion smooth approximation: friction, cooling longitudinal cooling time = 0.5 x transverse cooling time discrete emission of photons: heating equilibrium many applications: solid state physics, chemistry, biology special synchrotron radiation sources spectrum manipulation: wigglers, undulators, FEL sb/accphys2007_7/3

4 electron cooling cold electron beam + hot ion beam thermal contact ion beam cools 1 interaction 2 v high power electron beam (up to MW) independent of ion current rel transfer of emittance from ions to electrons cooling effect enhanced by solenoidal guiding field electrons Lorentz contraction electron energy distribution (longitudinal) applications low energy moderate emittance sb/accphys2007_7/4

5 stochastic cooling correct orbit of a sample of particles orbit individual particle corrected via sample mixing average corrections due to other particles in samples 0 applications high energy large emittance squeeze empty space out of emittance cooling time proportional to number of particles to be cooled 1/bandwidth sampling + correction system (typ. GHz) sb/accphys2007_7/5

6 accelerator physics and ion optics cyclotrons Sytze Brandenburg sb/accphys2007_7/6

7 outline classical cyclotron revisited azimuthal field modulation orbit stability small modulation: compact AVF-cyclotron maximum modulation: separated sector cyclotron injection beam centring vertical stability extraction stripping extraction precessional extraction sb/accphys2007_7/7

8 classical cyclotron revisited original idea (Lawrence & Livingston 1931) homogeneous magnetic field isochronous (non-relativistic) 2 mv R = qvb mv Bq R = ν orb = Bq 2πm centrifugal force Lorentz force sb/accphys2007_7/8

9 classical cyclotron revisited original idea (Lawrence & Livingston 1931) homogeneous magnetic field isochronous (non-relativistic) 2 mv R mv Bq = qvb R = ν orb = Bq 2πm accelerate with RF electric field with ν RF = h ν orb (h integer) sb/accphys2007_7/9

10 classical cyclotron revisited original idea (Lawrence & Livingston 1931) homogeneous magnetic field isochronous (non-relativistic) 2 mv R mv Bq = qvb R = ν orb = Bq 2πm accelerate with RF electric field with ν RF = h ν orb (h integer) sb/accphys2007_7/10

11 classical cyclotron revisited original idea (Lawrence & Livingston 1931) homogeneous magnetic field isochronous (non-relativistic) 2 mv R mv Bq = qvb R = ν orb = Bq 2πm accelerate with RF electric field with ν RF = h ν orb (h integer) why it should not work fieldindex n = 0 Q z, ν z = 0; no vertical stability linear growth of vertical beamsize Q r, ν r = 1; resonance no stable orbit due to imperfections no longitudinal stability + relativistic mass increase loss of synchronisation with accelerating voltage sb/accphys2007_7/11

12 why it works after all (to some extent) fringe field effects: fieldindex n = ε > 0 Q z, ν z > 0; marginal vertical stability large beamsize bad transmission Q r, ν r < 1; no resonance weak focussing (see lecture 2 on ion optics) loss of synchronisation with accelerating voltage gradual acceleration possible over limited number of turns maximum energy dependent on acceleration voltage sb/accphys2007_7/12

13 transverse dynamics (Steenbeck 1935; Kerst1941) field in vicinity of reference orbit at radius R 2 γmv restoring force Fr ( r) = qvby ( r) r x orbit deviation x : r R x R = + = 1+ R Taylor expansion in first order 1 1 x = 1 r R R By ( R) R By ( R) x By ( r) = By ( R) + x = By ( R) 1+ x By ( R) x R By ( r) By ( R) 1 n x = R 2 γmv x x Fr ( r) = 1 qvby ( R) 1 n R R R sb/accphys2007_7/13

14 transverse dynamics 2 γmv at reference orbit F r (R) = 0 : = evb z ( R ) R 2 2 γmv x d x Fr ( x ) = ( 1 n) = γm 2 R R dt particle oscillates around reference orbit with ω r = ω0 1 n for n > 1 particle orbit becomes unstable (imaginary ω r ) nomenclature oscillation around reference orbit: betatron oscillations Q r, ν r = ωr ω orb : betatron frequency, number of betatron period per turn sb/accphys2007_7/14

15 transverse dynamics for vertical stability similar reasoning Fz ( z) = qvb x B x B z B = 0 = z x 2 in first order B x ( z) = nb z ( R ) z Fz ( z) = γmn v z R R R particle oscillates around reference orbit with ω z = ω0 n for n< 0 particle orbit becomes unstable (imaginary ω z ) simultaneous radial and axial stability 0 < n < 1: weak focussing sb/accphys2007_7/15

16 longitudinal dynamics homogeneous B(r) = B(0) reality: B(r) slightly smaller than B(0) acceleration phase φ(t) = ω RF t - θ(t) dφ = ω dt RF ω orb energy gain per turn E = 2qV sinφ de ωorb = 2qV sinφ dt 2π dφ de ω ω π ω RF orb RF = = ω orb qv sinφ ωorb π qv sinφ 1 sb/accphys2007_7/16

17 π ω RF π E dcosφ = 1 de = ω ω RF 1 2 de qv orb qv qbc v qbc qbc γ m = qvb γmω = qb ω = = 2 R γmc E RF RF 2 cosφ = cosφ T T qv ωorb0 2qVE0 ωorb0 acceleration when 0 < φ < π (sin φ > 0) T max reached for cos φ = -1 π optimization T max : choose ω RF < ω r0 and cos φ 0 start acceleration close to φ= π/2 choose ω RF < ω r0 to reach maximum phase close to φ= -π/2 then accelerate until φ = π/2 ω π ω sb/accphys2007_7/17

18 maximum energy T = T max : cos φ = -1 T = 0.5 T max : cos φ = 1 ω ω orb0 RF T = 1+ 2mc max 2 Tmax 16qVmc 2 π acceleration voltage V 100 kv protons T max 22 MeV deuterons T max 31 MeV sb/accphys2007_7/18

19 maximum energy strategy: choose ω RF < ω orb(0) and φ 0 close to π (little acceleration) first part acceleration: φ 0 minimum φ > 0 (so acceleration does not stop) second part acceleration φ π acceleration stops at φ = π maximum achievable energy for V = 100 kv protons T max 22 MeV deuterons T max 31 MeV sb/accphys2007_7/19

20 how to make it really work restore isochronism compensate relativisic mass increase with increasing field correction coils decrease magnet gap with radius 2 γ(r)mv (r) = qvb(r) r qb(r) ω = B(r) = γ(r)b(0) γ(r)m r db z(r) 2 n(r) = = 1 γ (r) < 0 B (r) dr z vertical defocussing: exponential growth beamsize restore vertical focussing with azimuthal field modulation (Thomas, 1938) sb/accphys2007_7/20

21 accuracy magnetic field isochronous = orbital period independent momentum no longitudinal stability high accuracy of magnetic field needed phase slip determined by deviations from theoretical field 2πh 2Ema x B cosφ( R) = 2 ( r ) rdr nqv r B R max 0 theo cos φ = 1 acceleration stops: beam lost cos φ = 0.1 φ = 25 ; acceptable plug in some numbers (AGOR) h = 2; V = 80 kv; q = 1; n = 6 ; E max = 190 MeV; r max = 0.9 m < B/B theo > < 2 x 10-5 for cos φ = 0.1 impossible to achieve by design measurement + tuneable field corrections needed sb/accphys2007_7/21

22 azimuthal field modulation modulation magnet gap field modulation valley: large gap, weak field, large curvature hill: small gap, strong field, small curvature orbit not perpendicular to hill-valley edge vertical focussing + radial defocussing on edge sb/accphys2007_7/22

23 azimuthal field modulation modulation magnet gap field modulation valley: large gap, weak field, large curvature hill: small gap, strong field, small curvature orbit not perpendicular to hill-valley edge vertical focussing + radial defocussing on edge sb/accphys2007_7/23

24 azimuthal field modulation modulation magnet gap field modulation valley: large gap, weak field, large curvature hill: small gap, strong field, small curvature orbit not perpendicular to hill-valley edge vertical focussing + radial defocussing on edge sb/accphys2007_7/24

25 edge focussing: vertical particle trajectory not perpendicular to pole edge magnet vertical effect fringe field outside median plane: radial field component in particle coordinate system vertical force towards median plane vertical focussing thin lense with focal distance R/tan(β) sb/accphys2007_7/25

26 total force on particle px = Byvzdt = Bydz = B sinβ By dz = d ξ / cosβ p = tanβ B dξ x ξ ξ integrate over closed path through complete fringe field Bids = 0 Bids = B x + B dξ 0 ξ p x = -B 0 x tan β x = -B 0 x tan β/pq = - x tan β/r sb/accphys2007_7/26

27 edge focussing: radial radial effect magnet with perpendicular incidence + magnetic wedge defocussing angle kick α x = x tan(β) B 0 /pq = x tan(β)/r thin lens with focal distance -R/tan(β) warning: inconsistent definition coordinate systems in analysis vertical and radial case sb/accphys2007_7/27

28 how many sectors needed radial stability for number of sectors N 3 easy to demonstrate for separated sector cyclotron rule and compass drawing homogenous field (n = 0) but...true for any configuration two sectors bending angle 180 per sector vertically stable for α > ~72 radially unstable stable for any α sb/accphys2007_7/28

29 how many sectors needed radial stability for number of sectors N 3 easy to demonstrate for separated sector cyclotron rule and compass drawing homogenous field (n = 0) but...true for any configuration three sectors bending angle 120 per sector vertically stable for α > ~46 radially stable for α > 60 sb/accphys2007_7/29

30 how many sectors needed radial stability for number of sectors N 3 easy to demonstrate for separated sector cyclotron rule and compass drawing homogenous field (n = 0) but...true for any configuration four sectors bending angle 90 per sector vertically stable for α > ~27 radially stable for any α sb/accphys2007_7/30

31 separated sector cyclotron N-fold symmetry: sector angle ϑ s = 2π/N bending angle = sector angle transfer matrix of one sector T = Drift Edge-Focussing Dipole Edge-Focussing Drift radial and vertical motion analyzed separately stability: write T in terms of betatron-functions cosµ + α sinµ β sinµ T = γ sinµ cosµ α sinµ requirement for stability : µ real Tr(T) = 2 cos µ Tr(T) 2 betatron tune Q = N µ/ 2π sb/accphys2007_7/31

32 compact cyclotron weak modulation: f << 1 hill B H = B 0 (1+ f) R H = R 0 (1 - f) ϑ H =π/n (1 + f) valley B V = B 0 (1 - f) R V = R 0 (1 + f) ϑ V =π/n (1 - f) edge focussing: β = πf/2n, fieldstep 2fB 0 focal length -R/2f tan β γ/2 = π/2n(1 - f) sb/accphys2007_7/32

33 betatron frequencies compact cyclotron N-fold symmetry: sector angle ϑ s = 2π/N transfer matrix of one sector T = B[ϑ s (1-f)/4] EF(β) B [ϑ s (1+f)/2] EF(β) B[ϑ s (1-f)/4] analysis as for separated sector cyclotron take f << 1 radial motion: effect on Q r neglegible vertical motion: transfer matrix for half a sector f π πr f π 1 tanβ 1 tan N N β 2N πf T = cosµ = 1 tanβ 2f f π N tanβ 1 tanβ r N β = πf/2n << 1 tan β β, cos µ = 1 - (πf/n) 2 /2 1 - µ 2 /2 µ = πf/n = 2β Q z =2Nµ/2π = f sb/accphys2007_7/33

34 spiralled sectors goal: additional vertical focussing by edge focussing effects valley-hill transition β = β + ξ: stronger vertical focussing hill- valley transition β = β - ξ: weaker vertical focussing net effect: stronger vertical focussing (cf. F + D quadrupole) if ξ >> β (f << 1) for full sector 2πf µ tan ξ N Nµ Qz = = f tanξ 2π sb/accphys2007_7/34

35 field modulation analysis sofar: hard edge with step B = 2 f B 0 reality: smooth transition how to define f? ( ) ( ϑ) B( r) B( r) B r, F( r) = dϑ B r = dϑb r, ϑ F r = for hard edge ( ) ( ) 2 B( r) 2 B r B r ( ) 2 2 ( ) ( ) 1 B = B B = B 1+ f + 1 f = B 1+ f ( ) ( ) ( ) B 1+ f B F = = f B sb/accphys2007_7/35

36 overall picture different contributions to betatron frequencies should be added quadratically (cf. equation of motion x + ω 2 x = 0) detailed analysis (Hagedoorn en Verster, 1962) radial motion 2 2 3N 2 Q 1 n + F 1+ tan ξ + ( N 1)( N 4) r 2 2 ( ) vertical motion 2 2 N 2 Qz n + F( 1+ 2tan ξ ) + 2 N 1 ( ) n( r) = 1 γ 2 ( r) however. accurate values require tracking trajectories in calculated or measured fields higher order effects are important in cyclotrons spread in betatron frequencies (cf.chromaticity in rings) sb/accphys2007_7/36

37 radial injection pre-accelerated beam with q << Z injection energy constraints Bρ injected beam ~ Bρ final beam high enough to strip off many electrons q acc ~ 3-4 q inj charge state distribution stripping: low efficiency Chalk River (Canada) superconducting cyclotron tandem pre-accelerator Catania (Italy) superconducting cyclotron tandem pre-accelerator MSU (USA) superconducting cyclotrons sb/accphys2007_7/37

38 injection separated sector cyclotron radial injection in valley sector bend beam into equilibrium orbit dipole with B > B cyc correction coils + electrostatic deflector to center orbit sb/accphys2007_7/38

39 axial injection low energy (< 30q kev) beam from external ion source pass beam through magnet yoke conventional: weak field in axial hole (iron not saturated) additional focussing needed in yoke superconducting: strong axial field in hole (iron saturated) no additional focussing needed electro-static inflector in center combined effect electric + magnetic field in general injected orbit not centered centering error ~ radius of curvature 10 mm problems at extraction sb/accphys2007_7/39

40 why beam centering is important centering error coherent betatron oscillation oscillations in turn separation sb/accphys2007_7/40

41 why beam centering is important centering error coherent betatron oscillation oscillations in turn separation 6 turn separation (mm) MeV α radius [mm] sb/accphys2007_7/41

42 why beam centering is important centering error coherent betatron oscillation oscillations in turn separation RF phase width radial spread of particles with same # turns at one radius: particles with different # turns mixing of betatron phase centring error increase beamsize/effective emittance bad extraction efficiency large emittance extracted beam sb/accphys2007_7/42

43 how to achieve beam centering optimize geometry acceleration electrodes in center in central region ν r ~ 1 use first harmonic field with tuneable amplitude and phase observables for optimization turn separation (if visible) current density higher when smaller turnseparation measure with scanning wire sb/accphys2007_7/43

44 how to achieve beam centering optimize geometry acceleration electrodes in center in central region ν r ~ 1 use first harmonic field bump amplitude and phase tuneable observables for optimization turn separation (if visible) current density higher when smaller turnseparation measure with scanning wire sb/accphys2007_7/44

45 central region symmetry properties magnetic field n = 0 and F = 0 in center cyclotron ν r = 1 coherent betatron oscillations by field errors no vertical focussing acceptance problems, beam losses solutions field bump in center n > 0 vertical focussing by RF accelerating field sb/accphys2007_7/45

46 vertical focussing with RF-field DC-field focussing in first half of gap defocussing in second half of gap velocity lower in first half gap net focussing effect sb/accphys2007_7/46

47 vertical focussing with RF-field DC-field focussing in first half of gap defocussing in second half of gap velocity lower in first half gap net focussing effect RF-field field changes while particle traversing gap accelerate on falling slope of sinewave field stronger in first half gap focussing depends on RF- phase matching to periodic solution at larger r not possible increase in effective emittance sb/accphys2007_7/47

48 stripping extraction pass beam through foil to remove electrons from ion change of Q/A large magnetic perturbation beam kicked into unstable region of phasespace beam leaves accelerator favourite technique for isotope production cyclotrons accelerate negative H-ions, strip to positive ions curvature changes sign beam moves outward extraction efficiency essentially 100 % extract beam at different azimuths and energies simple energy variation (change radius of foil) small fringe field effects (radial defocussing) sb/accphys2007_7/48

49 IBA 30 MeV H - cyclotron sb/accphys2007_7/49

50 TRIUMF 500 MeV H - cyclotron low field to limit magnetic stripping large machine (R extr ~ 8 m) sb/accphys2007_7/50

51 stripping extraction: positive ions stripping increases Q/A radius of curvature smaller beam bend towards cyclotron center stripper foil at hill edge trochoidal motion along hill edge typically two-three cycles low field complex motion different for each Q/A and E high field low energy (can not accelerate fully stripped ions) charge state distribution after stripping mostly low extraction efficiency used at JINR, Dubna, Russia sb/accphys2007_7/51

52 precession extraction extraction with same charge state as acceleration beam to be captured by extraction system space needed between circulating and extracted beam radial gain by acceleration mostly insufficient example: 190 MeV protons in AGOR: E = 400 kev/turn; R = 0.89 m R a R E/2E = 0.9 mm/turn; beamwidth ~3 mm mechanism needed to increase turn separation sb/accphys2007_7/52

53 mechanism precessional extraction use passage Q r = 1 resonance to excite coherent betatron motion tuneable (amplitude and phase) 1 st harmonic perturbation amplitude limited by fast passage resonance b1 b1 1 ac = π R neff = πr B B dqr dn ν r 0.9 AGOR 3 He 60 MeV/nucleon radius [m] sb/accphys2007_7/53

54 mechanism precessional extraction use passage Q r = 1 resonance to excite coherent betatron motion tuneable (amplitude and phase) 1 st harmonic perturbation amplitude limited by fast passage resonance b1 b1 1 ac = π R neff = πr 1.2 B B dqr 1.1 dn 1.0 ν r 0.9 AGOR 3 He 60 MeV/nucleon radius [m] sb/accphys2007_7/54

55 mechanism precessional extraction use passage Q r = 1 resonance to excite coherent betatron motion tuneable (amplitude and phase) 1 st harmonic perturbation amplitude limited by fast passage resonance b1 b1 1 ac = π R neff = πr B B dqr dn typical values AGOR n eff = 10 (dq r /dn 0.01/turn); R = 0.88 m; b1 = 0.2 mt; B = 3T a c = 1.9 mm coherent betatron motion orbit center rotates around machine center with Q r precession frequency Q r -1 maximum radius difference Rc = 2acsin π( 1 Qr ) sb/accphys2007_7/55

56 mechanism precessional extraction coherent betatron motion orbit center rotates around machine center with frequency Q r after one turn displaced by (Q r -1) 2π maximum radius difference Rc = 2acsin π( 1 Qr ) sb/accphys2007_7/56

57 turn separation due to precessional motion ( ) = ( ) +β ( ) π( νr ) + ϕ 0 r ( n) r0 ( n) ( n) sin 2 ( r 1 ) 2 ( r 1) ( n) cos 2 ( r 1) r n r n n sin 2 1 = + β π ν + ϕ + π ν β π ν + ϕ ( ) 0 ( ) ( ) T n r n r0 ( n ) = acceleration 2T n ( ) ( r ) ( r ) ( ) ( r ) β n sin 2π ν 1 + ϕ increase of coherent oscillation due to first harmonic 2 π ν 1 β n cos 2 π ν 1 + ϕ precession movement sb/accphys2007_7/57

58 precessional extraction dynamics showing the increased turn separation sb/accphys2007_7/58

59 single turn vs. multi turn extraction single turn: all extracted particles same # of turns multi turn: extracted particles different # of turns determining factor: RF phase width of beam φ RF = 5 spread E/turn 0.1% 500 turns E beam = 0.5 E/turn all particles simultaneously at extraction φ RF = 20 spread E/turn 1.5 % 500 turns E beam = 7.5 E/turn different # of turns to arrive at extraction assumption: acceleration around φ RF = 0 (top of cosine) field imperfections phase slip acceleration on slope spread in # of turns increases rapidly φ RF = 30 spread E/turn 20 % good isochronism ( B/B theo < 10-5 ) very important sb/accphys2007_7/59

60 extraction system AGOR ESD 100 kv/cm EMC1 dipole 0.25 T gradient 13 T/m correction coils EMC2 dipole 0.4 T gradient 22 T/m correction coils QPOOL gradient 35 T/m gradient: compensate fringe field correction coils: compensate errors in acceleration region due to extraction channel sb/accphys2007_7/60