LONGITUDINAL BEAM DYNAMICS

Transcription

1 LONGITUDINAL BEAM DYNAMICS Frank Tecker CERN, BE-OP Basics of Accelerator Science and Technology at CERN Chavannes de Bogis, 4-8 November November 013 1

2 Summary of the 3 lectures: Acceleration methods Accelerating structures Phase Stability + Energy-Phase oscillations (Linac) Circular accelerators: Cyclotron / Synchrotron Dispersion Effects in Synchrotron Longitudinal Phase Space Motion Stationary Bucket Injection Matching Two more related lectures: Linacs Maurizio Vretanar RF Systems - Erk Jensen CAS@CERN, 4-8 November 013

3 Main Characteristics of an Accelerator Newton-Lorentz Force on a charged particle: ACCELERATION is the main job of an accelerator. It provides kinetic energy to charged particles, hence increasing their momentum. In order to do so, it is necessary to have an electric field E preferably along the direction of the initial momentum (z). dp dt = ee z BENDING is generated by a magnetic field perpendicular to the plane of the particle trajectory. The bending radius ρ obeys to the relation : p = Bρ e F = d p dt = e E + v B ( ) in practical units: nd term always perpendicular to motion => no acceleration B ρ [Tm] p [GeV/c] 0.3 FOCUSING is a second way of using a magnetic field, in which the bending effect is used to bring the particles trajectory closer to the axis, hence to increase the beam density. CAS@CERN, 4-8 November 013 3

4 Basics of Acceleration Today s accelerators and future projects work/aim at the TeV energy range. LHC: 7 TeV -> 14 TeV CLIC: 3 TeV HE/VHE-LHC: 33/100 TeV In fact, this energy unit comes from acceleration: 1 ev (electron Volt) is the energy that 1 elementary charge e (like one electron or proton) gains when it is accelerated in a potential (voltage) difference of 1 Volt. Basic Unit: ev (electron Volt) kev = 1000 ev = 10 3 ev MeV = 10 6 ev GeV = 10 9 ev TeV = 10 1 ev LHC = ~450 Million km of batteries!!! 3x distance Earth-Sun CAS@CERN, 4-8 November 013 4

5 Electrostatic Acceleration source E ΔV vacuum envelope Electrostatic Field: Force: F = d p dt = e E Energy gain: W = e ΔV used for first stage of acceleration: particle sources, electron guns, x-ray tubes Limitation: isolation problems maximum high voltage (~ 10 MV) 750 kv Cockroft-Walton generator at Fermilab (Proton source) CAS@CERN, 4-8 November 013 5

6 Methods of Acceleration: Time varying fields The electrostatic field is limited by insulation, the magnetic field does not accelerate. From Maxwell s Equations: E = φ A t B = µ H = A The electric field is derived from a scalar potential φ and a vector potential A The time variation of the magnetic field H generates an electric field E or E = B t The solution: => time varying electric fields - Induction - RF frequency fields CAS@CERN, 4-8 November 013 6

7 Acceleration by Induction: The Betatron It is based on the principle of a transformer: - primary side: large electromagnet - secondary side: electron beam. The ramping magnetic field is used to guide particles on a circular trajectory as well as for acceleration. Limited by saturation in iron (~300 MeV e-) Used in industry and medicine, as they are compact accelerators for electrons vacuum pipe side view beam B f iron yoke coil time top view R E beam Donald Kerst with the first betatron, invented at the University of Illinois in 1940 CAS@CERN, 4-8 November B f B

8 Radio-Frequency (RF) Acceleration Electrostatic acceleration limited by isolation possibilities => use RF fields Wideröe-type structure Cylindrical electrodes (drift tubes) separated by gaps and fed by a RF generator, as shown above, lead to an alternating electric field polarity Synchronism condition L = v T/ v = particle velocity T = RF period Similar for standing wave cavity as shown (with v c) D.Schulte CAS@CERN, 4-8 November 013 8

9 RF acceleration: Alvarez Structure g Used for protons, ions (50 00 MeV, f ~ 00 MHz) L 1 L L 3 L 4 L 5 Synchronism condition L ( g << L) = v T = β λ s RF s RF RF generator LINAC 1 (CERN) ω = π RF v s L CAS@CERN, 4-8 November 013 9

10 Resonant RF Cavities - Considering RF acceleration, it is obvious that when particles get high velocities the drift spaces get longer and one looses on the efficiency. => The solution consists of using a higher operating frequency. - The power lost by radiation, due to circulating currents on the electrodes, is proportional to the RF frequency. => The solution consists of enclosing the system in a cavity which resonant frequency matches the RF generator frequency. - The electromagnetic power is now constrained in the resonant volume - Each such cavity can be independently powered from the RF generator - Note however that joule losses will occur in the cavity walls (unless made of superconducting materials) CAS@CERN, 4-8 November

11 E z H θ The Pill Box Cavity From Maxwell s equations one can derive the wave equations: A A ε 0 µ 0 = 0 (A = E or H ) t Solutions for E and H are oscillating modes, at discrete frequencies, of types TM xyz (transverse magnetic) or TE xyz (transverse electric). Indices linked to the number of field knots in polar co-ordinates φ, r and z. For l<a the most simple mode, TM 010, has the lowest frequency, and has only two field components: E z = J 0 (kr) e iωt H θ = i Z 0 J 1 (kr) e iωt k = π λ = ω c λ =.6a Z 0 = 377Ω CAS@CERN, 4-8 November

12 The Pill Box Cavity () The design of a pill-box cavity can be sophisticated in order to improve its performances: - A nose cone can be introduced in order to concentrate the electric field around the axis - Round shaping of the corners allows a better distribution of the magnetic field on the surface and a reduction of the Joule losses. It also prevents from multipactoring effects. A good cavity is a cavity which efficiently transforms the RF power into accelerating voltage. CAS@CERN, 4-8 November 013 1

13 Some RF Cavity Examples L = vt/ (π mode) L = vt (π mode) Single Gap Multi-Gap CAS@CERN, 4-8 November

14 Transit time factor The accelerating field varies during the passage of the particle => particle does not always see maximum field => effective acceleration smaller Transit time factor defined as: T a = energy gain of particle with v = βc maximum energy gain (particle with v ) In the general case, the transit time factor is: for E(s,r,t) = E 1 (s,r) E (t) T a = + " E 1 (s,r) cos ω # $ RF + E 1 (s,r) ds s % v& ' ds Simple model uniform field: VRF E1 ( s, r) = = g const. follows: T a = sin ω RFg v ω RF g v 0 < T a < 1 T a 1 for g 0, smaller ω RF Important for low velocities (ions) CAS@CERN, 4-8 November

15 Disc loaded traveling wave structures -When particles gets ultra-relativistic (v~c) the drift tubes become very long unless the operating frequency is increased. Late 40 s the development of radar led to high power transmitters (klystrons) at very high frequencies (3 GHz). -Next came the idea of suppressing the drift tubes using traveling waves. However to get a continuous acceleration the phase velocity of the wave needs to be adjusted to the particle velocity. CLIC Accelerating Structures (30 & 11 GHz) solution: slow wave guide with irises ==> iris loaded structure CAS@CERN, 4-8 November

16 The Traveling Wave Case E z = E 0 cos( ω RF t kz) k = ω RF v ϕ z = v(t t 0 ) wave number The particle travels along with the wave, and k represents the wave propagation factor. " E z = E 0 cos $ ω RF t ω RF # v φ = phase velocity v = particle velocity v v ϕ t φ 0 % ' & If synchronism satisfied: v = v φ and E z = E 0 cosφ 0 where Φ 0 is the RF phase seen by the particle. CAS@CERN, 4-8 November

17 Energy Gain In relativistic dynamics, total energy E and momentum p are linked by E = E 0 + p c (E = E 0 +W ) W kinetic energy Hence: de =vdp The rate of energy gain per unit length of acceleration (along z) is then: and the kinetic energy gained from the field along the z path is: where V is just a potential. de dz =v dp dz = dp dt =ee z dw =de =ee z dz W =e E z dz = ev CAS@CERN, 4-8 November

18 Velocity, Energy and Momentum 1 normalized velocity v β = c = 1 1 γ Beta 0.5 electrons normalized velocity => electrons almost reach the speed of light very quickly (few MeV range) protons E_kinetic (MeV) total energy rest energy γ = E E Momentum = m m = E = γm 0 c 1 = v 1 c 1 β p = mv = E c βc = β E c = βγm 0 c Gamma total energy rest energy electrons protons E_kinetic (MeV) CAS@CERN, 4-8 November

19 Particle types and acceleration Accelerating system will depend upon the evolution of the particle velocity along the system electrons reach a constant velocity at relatively low energy heavy particles reach a constant velocity only at very high energy» may need different types of resonators, optimized for different velocities Particle rest mass: electron MeV proton 938 MeV 39 U ~0000MeV CAS@CERN, 4-8 November

20 Summary: Relativity + Energy Gain Newton-Lorentz Force F = d p dt = e ( E + v B ) nd term always perpendicular to motion => no acceleration Relativistics Dynamics v β = = c 1 1 γ γ = E E 0 = m m 0 = p = mv = E c βc = β E c = βγ m 0 c 1 1 β RF Acceleration E z =Êz sinω RFt=Êz sinφ ( t ) Ê z dz = ˆ V E de dz = E0 p c de =vdp + =v dp dz = dp dt =ee z de =dw =ee z dz W =e E z dz W =evˆsinφ (neglecting transit time factor) The field will change during the passage of the particle through the cavity => effective energy gain is lower CAS@CERN, 4-8 November 013 0

21 Common Phase Conventions 1. For circular accelerators, the origin of time is taken at the zero crossing of the RF voltage with positive slope. For linear accelerators, the origin of time is taken at the positive crest of the RF voltage Time t= 0 chosen such that: 1 E 1 E φ =ω RF t φ =ω RF t φ 1 φ E 1 (t) = E 0 sin( ω RF t) E (t) = E 0 cos( ω RF t) 3. I will stick to convention 1 in the following to avoid confusion CAS@CERN, 4-8 November 013 1

22 Principle of Phase Stability (Linac) Let s consider a succession of accelerating gaps, operating in the π mode, for which the synchronism condition is fulfilled for a phase Φs. ev s = e ˆ V sin Φ s is the energy gain in one gap for the particle to reach the next gap with the same RF phase: P 1,P, are fixed points. For a π mode, the electric field is the same in all gaps at any given time. If an energy increase is transferred into a velocity increase => M 1 & N 1 will move towards P 1 => stable M & N will go away from P => unstable (Highly relativistic particles have no significant velocity change) CAS@CERN, 4-8 November 013

23 A Consequence of Phase Stability Transverse focusing fields at the entrance and defocusing at the exit of the cavity. Electrostatic case: Energy gain inside the cavity leads to focusing RF case: Field increases during passage => transverse defocusing! Longitudinal phase stability means : V t > E z 0 < z 0 defocusing RF force The divergence of the field is x z x zero according to Maxwell :. E = 0 + = 0 > 0 E x E z E x External focusing (solenoid, quadrupole) is then necessary CAS@CERN, 4-8 November 013 3

24 - Rate of energy gain for the synchronous particle: dw dz = ee w = W Ws = E E s ϕ = φ s [ ( φ + ϕ) sinφ ] ee cosφ ϕ ( small ϕ ) 0 sin s s 0 s. - Rate of change of the phase with respect to the synchronous one: dϕ dz dt dt 1 1 ω RF = ωrf RF dz dz = ω v v s s vs v v Energy-phase Oscillations (1) = c de s dz = dp s dt β β c β φ ( v v ) Since: ( ) ( ) s s s 3 = ee 0 sinϕ s - Rate of energy gain for a non-synchronous particle, expressed in reduced variables, and : s β β w m vγ 0 s s s CAS@CERN, 4-8 November 013 4

25 Energy-phase Oscillations () one gets: dϕ = dz ω m v RF sγ s w Combining the two 1 st order equations into a nd order equation gives the equation of a harmonic oscillator: d ϕ dz + Ωsϕ = 0 with ee cos 0ω φs Ωs = m v γ RF 3 0 s Stable harmonic oscillations imply: Ω s > 0 and real hence: cos φ s > 0 And since acceleration also means: sin φ s > 0 You finally get the result for the stable phase range: 0 φ < acceleration CAS@CERN, 4-8 November s cos (φ s ) π V RF π π φ < π s 3

26 Longitudinal phase space The energy phase oscillations can be drawn in phase space: ΔΕ, Δp/p acceleration ΔΕ, Δp/p move forward move backward reference deceleration φ φ The particle trajectory in the phase space (Δp/p, φ) describes its longitudinal motion. Emittance: phase space area including all the particles NB: if the emittance contour correspond to a possible orbit in phase space, its shape does not change with time (matched beam) CAS@CERN, 4-8 November 013 6

27 Summary up to here Acceleration by electric fields, static fields limited => time-varying fields Synchronous condition needs to be fulfilled for acceleration Particles perform oscillation around synchronous phase visualize oscillations in phase space Electrons are quickly relativistic, speed does not change use traveling wave structures for acceleration Protons and ions need changing structure geometry 4-8 November 013 7

28 Circular accelerators Cyclotron Synchrotron 4-8 November 013 8

29 Circular accelerators: Cyclotron Used for protons, ions B = constant ω RF = constant B g RF generator, ω RF Synchronism condition π ω = ω s ρ = v s RF T RF Ion source Cyclotron frequency ω = q m 0 B γ Ions trajectory Extraction electrode 1. γ increases with the energy no exact synchronism. if v << c γ 1 CAS@CERN, 4-8 November 013 9

30 Cyclotron / Synchrocyclotron TRIUMF 50 MeV cyclotron Vancouver - Canada CERN 600 MeV synchrocyclotron Synchrocyclotron: Same as cyclotron, except a modulation of ω RF B = constant γ ω RF = constant ω RF decreases with time The condition: ω ( t) = ω ( t) = s RF q B m 0 γ ( t) Allows to go beyond the non-relativistic energies CAS@CERN, 4-8 November

31 Circular accelerators: The Synchrotron B R 1. Constant orbit during acceleration. To keep particles on the closed orbit, B should increase with time E 3. ω and ω RF increase with energy RF cavity RF generator RF frequency can be multiple of revolution frequency ω RF = hω r Synchronism condition s T s π R v = = h T h T RF RF h integer, harmonic number: number of RF cycles per revolution CAS@CERN, 4-8 November

32 Circular accelerators: The Synchrotron LEAR (CERN) Low Energy Antiproton Ring EPA (CERN) Electron Positron Accumulator CERN Geneva CERN Geneva Examples of different proton and electron synchrotrons at CERN + LHC (of course!) PS (CERN) Proton Synchrotron CAS@CERN, 4-8 November 013 CERN Geneva 3

33 The Synchrotron The synchrotron is a synchronous accelerator since there is a synchronous RF phase for which the energy gain fits the increase of the magnetic field at each turn. That implies the following operating conditions: B E ^ ev sin Φ Energy gain per turn Bending magnet Φ = Φ s = cte Synchronous particle R=C/π injection extraction ω RF = hω r RF synchronism (h - harmonic number) ρ bending radius ρ = cte R = cte Constant orbit Bρ = P e B Variable magnetic field If v c, ω r hence ω RF remain constant (ultra-relativistic e- ) CAS@CERN, 4-8 November

34 The Synchrotron Energy ramping Energy ramping is simply obtained by varying the B field (frequency follows v): p = ebρ dp = eρ B (Δp) turn = eρ BT r = π eρ R B dt v Since: E = E 0 + p c ΔE = vδp ( ΔE) = ( ΔW ) =π eρrb=e V ˆ sinφ s turn s Stable phase φ s changes during energy ramping sinφ = s π ρ R B V ˆ RF B φ = s arcsin π ρ R V ˆ RF The number of stable synchronous particles is equal to the harmonic number h. They are equally spaced along the circumference. Each synchronous particle satisfies the relation p=ebρ. They have the nominal energy and follow the nominal trajectory. CAS@CERN, 4-8 November

35 The Synchrotron Frequency change During the energy ramping, the RF frequency increases to follow the increase of the revolution frequency : ω r = ω RF h = ω(b, R s) f RF (t) = v(t) = 1 ec h π R s π E s (t) Hence: ( using ) E = (m 0 c ) + p c Since the RF frequency must follow the variation of the B field with the law f RF (t) h This asymptotically tends towards compared to m 0 c / (ecρ) which corresponds to = ρ B(t) p(t) = eb(t)ρ, E = mc R s c! B(t) " π R s #(m 0 c / ecρ) + B(t) v c f r c π R s 1 $ % & when B becomes large CAS@CERN, 4-8 November

36 cavity Dispersion Effects in a Synchrotron Circumference E+δE πr α = E If a particle is slightly shifted in momentum it will have a different orbit and the orbit length is different. The momentum compaction factor is defined as: dl L dp p α = p L dl dp p=particle momentum R=synchrotron physical radius f r =revolution frequency If the particle is shifted in momentum it will have also a different velocity. As a result of both effects the revolution frequency changes: η = d f r fr d p p η = p f r dfr dp CAS@CERN, 4-8 November

37 Momentum Compaction Factor α = p L dl dp ds 0 = ρdθ ds = ( ρ + x)dθ The elementary path difference from the two orbits is: dl ds 0 = ds ds 0 ds 0 definition of dispersion D x = x ρ = D x ρ dp p s s 0 p + dp p ρ θ x dθ x leading to the total change in the circumference: dl = α = 1 L dl = x ds 0 = ρ C C D x (s) ρ(s) ds 0 D x ρ dp p ds 0 With ρ= in straight sections we get: α = Dx m R < > m means that the average is considered over the bending magnet only CAS@CERN, 4-8 November

38 Dispersion Effects Revolution Frequency There are two effects changing the revolution frequency: the orbit length and the velocity of the particle f r = βc π R df r f r = dβ β dr R = dβ β α dp p definition of momentum compaction factor p = mv = βγ E 0 c dp p = dβ β + d ( 1 β ) 1 df r ( 1 β ) 1 ( ) 1 = 1 β γ dβ β df f r r f r = η dp p dp = 1 α η = α γ p γ η=0 at the transition energy γ CAS@CERN, 4-8 November tr 1 = 1 α

39 Phase Stability in a Synchrotron From the definition of η it is clear that an increase in momentum gives - below transition (η > 0) a higher revolution frequency (increase in velocity dominates) while - above transition (η < 0) a lower revolution frequency (v c and longer path) where the momentum compaction (generally > 0) dominates. Stable synchr. Particle for η < 0 η > 0 above transition η 1 γ = α CAS@CERN, 4-8 November

40 Crossing Transition At transition, the velocity change and the path length change with momentum compensate each other. So the revolution frequency there is independent from the momentum deviation. Crossing transition during acceleration makes the previous stable synchronous phase unstable. The RF system needs to make a rapid change of the RF phase, a phase jump. CAS@CERN, 4-8 November

41 Dynamics: Synchrotron oscillations Simple case (no accel.): B = const., below transition φ 1 - The particle B is accelerated - Below transition, an increase in energy means an increase in revolution frequency - The particle arrives earlier tends toward φ 0 γ < γ tr The phase of the synchronous particle must therefore be φ 0 = 0. V RF B φ φ 0 φ 1 φ =ω RF t φ - The particle is decelerated - decrease in energy - decrease in revolution frequency - The particle arrives later tends toward φ 0 CAS@CERN, 4-8 November

42 Synchrotron oscillations A V B time 1 st revolution period CAS@CERN, 4-8 November 013 4

43 Synchrotron oscillations A V B time 100 th revolution period CAS@CERN, 4-8 November

44 Synchrotron oscillations A V B time 00 th revolution period CAS@CERN, 4-8 November

45 Synchrotron oscillations A V B time 400 th revolution period CAS@CERN, 4-8 November

46 Synchrotron oscillations A V B time 500 th revolution period CAS@CERN, 4-8 November

47 Synchrotron oscillations A V B time 600 th revolution period CAS@CERN, 4-8 November

48 Synchrotron oscillations A V B time 700 th revolution period CAS@CERN, 4-8 November

49 Synchrotron oscillations A V B time 800 th revolution period CAS@CERN, 4-8 November

50 Synchrotron oscillations A V B time 900 th revolution period CAS@CERN, 4-8 November

51 Synchrotron oscillations A V B time 900 th revolution period Particle B has made one full oscillation around particle A. The amplitude depends on the initial phase and energy. Exactly like the pendulum This oscillation is called: Synchrotron Oscillation CAS@CERN, 4-8 November

52 The Potential Well A Cavity voltage Potential B Potential well CAS@CERN, 4-8 November 013 5

53 The Potential Well A Cavity voltage Potential B Potential well CAS@CERN, 4-8 November

54 The Potential Well A Cavity voltage Potential B Potential well CAS@CERN, 4-8 November

55 The Potential Well A Cavity voltage Potential B Potential well CAS@CERN, 4-8 November

56 The Potential Well A Cavity voltage Potential B Potential well CAS@CERN, 4-8 November

57 The Potential Well A Cavity voltage Potential B Potential well CAS@CERN, 4-8 November

58 The Potential Well A Cavity voltage Potential B Potential well CAS@CERN, 4-8 November

59 The Potential Well A Cavity voltage Potential B Potential well CAS@CERN, 4-8 November

60 The Potential Well A Cavity voltage Potential B Potential well CAS@CERN, 4-8 November

61 The Potential Well A Cavity voltage Potential B Potential well CAS@CERN, 4-8 November

62 The Potential Well A Cavity voltage Potential B Potential well CAS@CERN, 4-8 November 013 6

63 The Potential Well A Cavity voltage Potential B Potential well CAS@CERN, 4-8 November

64 The Potential Well A Cavity voltage Potential B Potential well CAS@CERN, 4-8 November

65 The Potential Well A Cavity voltage Potential B Potential well CAS@CERN, 4-8 November

66 The Potential Well A Cavity voltage Potential B Potential well CAS@CERN, 4-8 November

67 Longitudinal Phase Space Motion (1) Particle B oscillates around particle A This is a synchrotron oscillation Plotting this motion in longitudinal phase space gives: ΔE higher energy early arrival late arrival Δt (or φ) lower energy CAS@CERN, 4-8 November

68 Longitudinal Phase Space Motion () Particle B oscillates around particle A This is a synchrotron oscillation Plotting this motion in longitudinal phase space gives: ΔE higher energy early arrival late arrival Δt (or φ) lower energy CAS@CERN, 4-8 November

69 Longitudinal Phase Space Motion (3) Particle B oscillates around particle A This is a synchrotron oscillation Plotting this motion in longitudinal phase space gives: ΔE higher energy early arrival late arrival Δt (or φ) lower energy CAS@CERN, 4-8 November

70 Longitudinal Phase Space Motion (4) Particle B oscillates around particle A This is a synchrotron oscillation Plotting this motion in longitudinal phase space gives: ΔE higher energy early arrival late arrival Δt (or φ) lower energy CAS@CERN, 4-8 November

71 Synchrotron oscillations V RF φ φ 0 φ t φ 1 Phase space picture Δp p stable region φ unstable region separatrix CAS@CERN, 4-8 November

72 Synchrotron oscillations (with acceleration) Case with acceleration B increasing γ < γ tr V RF 1 φ φ =ω RF t φ s Phase space picture Δp p φ < φ < π s φ s stable region unstable region separatrix φ The symmetry of the case B = const. is lost CAS@CERN, 4-8 November 013 7

73 Synchrotron motion in phase space ΔE-φ phase space of a stationary bucket Dynamics of a particle Non-linear, conservative oscillator e.g. pendulum Particle inside the separatrix: Particle at the unstable fix-point Bucket area: area enclosed by the separatrix The area covered by particles is the longitudinal emittance Particle outside the separatrix: CAS@CERN, 4-8 November

74 Synchrotron motion in phase space The restoring force is non-linear. speed of motion depends on position in phase-space (here shown for a stationary bucket) CAS@CERN, 4-8 November

75 (Stationary) Bunch & Bucket ΔE Bunch ΔE Bucket Δt (or φ) Δt Bucket area = longitudinal Acceptance [evs] Bunch area = longitudinal beam emittance = π.δe.δt/4 [evs] CAS@CERN, 4-8 November

76 RF Acceptance versus Synchronous Phase The areas of stable motion (closed trajectories) are called BUCKET. As the synchronous phase gets closer to 90º the buckets gets smaller. The number of circulating buckets is equal to h. The phase extension of the bucket is maximum for φ s =180º (or 0 ) which correspond to no acceleration. The RF acceptance increases with the RF voltage. CAS@CERN, 4-8 November

77 Longitudinal Dynamics in Synchrotrons It is also often called synchrotron motion. The RF acceleration process clearly emphasizes two coupled variables, the energy gained by the particle and the RF phase experienced by the same particle. Since there is a well defined synchronous particle which has always the same phase φ s, and the nominal energy E s, it is sufficient to follow other particles with respect to that particle. So let s introduce the following reduced variables: revolution frequency : Δf r = f r f rs particle RF phase : Δφ = φ - φ s particle momentum : Δp = p - p s particle energy : ΔE = E E s azimuth angle : Δθ = θ - θ s CAS@CERN, 4-8 November

78 First Energy-Phase Equation v Δθ θ s R f RF = h f r Δφ = h Δθ with θ = ω r dt For a given particle with respect to the reference one: Δω = r d dt particle ahead arrives earlier => smaller RF phase 1 h d dt ( Δθ ) = ( Δφ ) = 1 h dφ dt Since: η = p s ω rs # $ % dω r dp & ' ( s and p E = E 0 + c ΔE = v s Δp = ω rs R s Δp p ( ) srs d φ one gets: ΔE Δ ω rs = hηωrs dt = psrs hηω rs φ CAS@CERN, 4-8 November

79 Second Energy-Phase Equation The rate of energy gained by a particle is: de = evˆ sinφ ωr dt π The rate of relative energy gain with respect to the reference particle is then: $ π Δ& % E ω r ' ) ( = e ˆV(sinφ sinφ s ) Expanding the left-hand side to first order: Δ( ET ) r EΔT r + T rs Δ E = ΔE T r + T rs Δ E = d dt leads to the second energy-phase equation: ( T rs ΔE) π d dt $ % & ΔE ω rs ' ( ) = e ˆV sinφ sinφ s ( ) CAS@CERN, 4-8 November

80 Equations of Longitudinal Motion ΔE= ωrs psrs hηωrs d ( Δφ) dt = psrs hηω rs φ π d E evˆ dt Δ ω = rs ( sinφ sinφ s) deriving and combining d dt Rs ps hηωrs dφ ˆ + ev dt π ( sinφ sinφ ) s = 0 This second order equation is non linear. Moreover the parameters within the bracket are in general slowly varying with time. We will study some cases in the following CAS@CERN, 4-8 November

81 Small Amplitude Oscillations Let s assume constant parameters R s, p s, ω s and η: φ + s Ωs cosφ s ( sinφ sinφ ) = 0 with Ω s = hηωrsevˆcosφ s πrs ps Consider now small phase deviations from the reference particle: sinφ sinφ s = sin ( φ ) s+δφ sinφ s cosφ sδφ (for small Δφ) and the corresponding linearized motion reduces to a harmonic oscillation: φ + Ω s Δφ = 0 where Ω s is the synchrotron angular frequency CAS@CERN, 4-8 November

82 Stability condition for ϕ s Stability is obtained when Ω s is real and so Ω s positive: Ω s = e ˆV RF η hω s cosφ s Ω s > 0 η cosφ s > 0 π R s p s cos (φ s ) V RF π π 3 π φ Stable in the region if γ < γ tr γ > γ tr γ > γ tr γ < γ tr η > 0 η < 0 η < 0 η > 0 acceleration deceleration CAS@CERN, 4-8 November 013 8

83 Large Amplitude Oscillations For larger phase (or energy) deviations from the reference the second order differential equation is non-linear: φ Ω s cosφ ( sinφ sinφ ) = 0 + s s Multiplying by φ Ω cos φ (Ω s as previously defined) and integrating gives an invariant of the motion: ( cosφ + φ sin φ ) I s s = φs which for small amplitudes reduces to: ( ) φ + Ω Δφ s = I$ (the variable is Δφ, and φ s is constant) Similar equations exist for the second variable : ΔE dφ/dt CAS@CERN, 4-8 November

84 Large Amplitude Oscillations () When φ reaches π-φ s the force goes to zero and beyond it becomes non restoring. Hence π-φ s is an extreme amplitude for a stable motion which in the φ phase space(, Δφ ) is shown as Ω s closed trajectories. Equation of the separatrix: φ Ω s cosφ s m s ( cosφ + φ sinφ ) = ( cos( π φ ) + ( π φ ) sinφ ) cosφ + φ sinφ = cos m s s Ω cosφ Second value φ m where the separatrix crosses the horizontal axis: s ( π φs ) + ( π φs ) sin s Area within this separatrix is called RF bucket. CAS@CERN, 4-8 November s φ s s

85 Energy Acceptance φ φ = φ From the equation of motion it is seen that φ = 0 when, hence corresponding to. reaches an extreme Introducing this value into the equation of the separatrix gives: s φ max = Ω s { + ( φ s π )tanφ } s That translates into an acceptance in energy: " # $ ΔE % E s & ' = β e ˆV π hηe G s max ( φ ) s G( φ s )= $% cosφ s + ( φ s π )sinφ s & ' This RF acceptance depends strongly on φ s and plays an important role for the capture at injection, and the stored beam lifetime. It s largest for φ s =0 and φ s =π (no acceleration, depending on η). CAS@CERN, 4-8 November

86 RF Acceptance versus Synchronous Phase The areas of stable motion (closed trajectories) are called BUCKET. As the synchronous phase gets closer to 90º the buckets gets smaller. The number of circulating buckets is equal to h. The phase extension of the bucket is maximum for φ s =180º (or 0 ) which correspond to no acceleration. The RF acceptance increases with the RF voltage. CAS@CERN, 4-8 November

87 Stationnary Bucket - Separatrix This is the case sinφ s =0 (no acceleration) which means φ s =0 or π. The equation of the separatrix for φ s = π (above transition) becomes: φ φ φ + Ωs cosφ =Ω = Ωssin s Replacing the phase derivative by the (canonical) variable W: W W bk 0 π π φ ps s W = π ΔE = π R ωrs hηω rs φ and introducing the expression for Ω s leads to the following equation for the separatrix: with C=πR s W = ± C c e ˆV E s π hη sinφ = ±W sinφ bk CAS@CERN, 4-8 November

88 Stationnary Bucket () Setting φ=π in the previous equation gives the height of the bucket: W = C c bk evˆ π h E s η This results in the maximum energy acceptance: ΔE max = ω rs π W bk = β s e ˆV RF E s πηh The area of the bucket is: π φ Since: sin dφ = one gets: 0 4 Abk A bk = 8W bk = 16 C c π 0 = W dφ e ˆV E s π hη W bk = A 8 bk CAS@CERN, 4-8 November

89 Effect of a Mismatch Injected bunch: short length and large energy spread after 1/4 synchrotron period: longer bunch with a smaller energy spread. W W φ φ For larger amplitudes, the angular phase space motion is slower (1/8 period shown below) => can lead to filamentation and emittance growth V RF W W ϕ φ φ restoring force is non-linear stationary bucket accelerating bucket W.Pirkl CAS@CERN, 4-8 November

90 Effect of a Mismatch () Evolution of an injected beam for the first 100 turns. For a matched transfer, the emittance does not grow (left). matched beam mismatched beam bunch length CAS@CERN, 4-8 November

91 Effect of a Mismatch (3) Evolution of an injected beam for the first 100 turns. For a mismatched transfer, the emittance increases (right). matched beam mismatched beam phase error CAS@CERN, 4-8 November

92 Bunch Rotation Phase space motion can be used to make short bunches. Start with a long bunch and extract or recapture when it s short. initial beam CAS@CERN, 4-8 November 013 9

93 The LHC5 (ns) cycle in the PS Inject 4+ bunches h = 7 γ tr h = 1 h = 84 Eject 7 bunches Triple splitting after nd injection Split in four at flat-top energy nd injection 1.4 GeV 6 GeV/c (sketched) Each bunch from the Booster divided by = 7 CAS@CERN, 4-8 November

94 Bunch Manipulation in the PS Two times double splitting and bunch rotation: 4-8 November

95 Capture of a Debunched Beam with Fast Turn-On CAS@CERN, 4-8 November

96 Capture of a Debunched Beam with Adiabatic Turn-On CAS@CERN, 4-8 November

97 Bunch Matching into a Stationnary Bucket A particle trajectory inside the separatrix is described by the equation: φ Ω cosφ s φs = s ( cosφ + φsin ) I W W bk W b 0 π π π ˆφ φ φ s = π φ +Ω φ + Ωs cosφ = I The points where the trajectory crosses the axis are symmetric with respect to φ s = π s cosφ =Ω s cosφ m ( cosφ cosφ ) φ = ±Ωs m φ m π-φ m ϕ W = ±W bk cos m cos ϕ cos(φ) = cos φ 1 CAS@CERN, 4-8 November

98 Bunch Matching into a Stationnary Bucket () Setting φ = π in the previous formula allows to calculate the bunch height: W b = W bk φ cos m = W bk sin ˆφ or: W b = A 8 bk cos φ m " # $ ΔE % & ' = E s b " # $ ΔE % φ E s & ' cos m = " ΔE % # $ RF E s & ' RF sin ˆφ This formula shows that for a given bunch energy spread the proper matching of a shorter bunch (φ m close to π, ˆφ small) will require a bigger RF acceptance, hence a higher voltage For small oscillation amplitudes the equation of the ellipse reduces to: W = A bk 16 ˆφ ( Δφ) " 16W % # $ ˆφ & ' + " Δφ % # $ ˆφ & ' =1 Abk Ellipse area is called longitudinal emittance A b = π 16 A ˆφ bk CAS@CERN, 4-8 November

99 Summary Cyclotrons/Synchrocylotrons for low energy Synchrotrons for high energies constant orbit, rising field and frequency Particles with higher energy have a longer orbit (normally) but a higher velocity at low energies (below transition) velocity increase dominates at high energies (above transition) velocity almost constant Particles perform oscillations around synchronous phase synchronous phase depending on acceleration below or above transition bucket is the region in phase space for stable oscillations matching the shape of the bunch to the bucket is important CAS@CERN, 4-8 November

100 Bibliography M. Conte, W.W. Mac Kay An Introduction to the Physics of particle Accelerators (World Scientific, 1991) P. J. Bryant and K. Johnsen The Principles of Circular Accelerators and Storage Rings (Cambridge University Press, 1993) D. A. Edwards, M. J. Syphers An Introduction to the Physics of High Energy Accelerators (J. Wiley & sons, Inc, 1993) H. Wiedemann Particle Accelerator Physics (Springer-Verlag, Berlin, 1993) M. Reiser Theory and Design of Charged Particles Beams (J. Wiley & sons, 1994) A. Chao, M. Tigner Handbook of Accelerator Physics and Engineering (World Scientific 1998) K. Wille The Physics of Particle Accelerators: An Introduction (Oxford University Press, 000) E.J.N. Wilson An introduction to Particle Accelerators (Oxford University Press, 001) And CERN Accelerator Schools (CAS) Proceedings 4-8 November

101 Acknowledgements I would like to thank everyone for the material that I have used. In particular (hope I don t forget anyone): - Joël Le Duff - Rende Steerenberg - Gerald Dugan - Heiko Damerau - Werner Pirkl - Genevieve Tulloue - Mike Syphers - Daniel Schulte - Roberto Corsini CAS@CERN, 4-8 November