Storage Ring Dynamics: Longitudinal Motion John Byrd Lawrence Berkeley National Laboratory 1 • Original concept of phase stability introduced independently by Veksler (1944) and MacMillan (1945) 2 In a particle storage rings, charged particles circulate around the ring in bunches for a large number of turns.! Optics elements" Particle bunches" " dipoles 2 Reference Particle" The longitudinal coordinates are! Particle" ! ΔP/Po! Length and Momentum! ! Lo à Revolution Length of the ΔL! Reference Particle! ! Po à Momentum of the Reference Particle! ! ! ΔL/Lo, ΔP/Po! ! ! 3 Consider two particles with different momentum on parallel trajectories: L0 p = p + Δp 1 0 L1 At a given instant t: ΔL L − L Δβ ⇒ = 1 0 = L1 = (β0 + Δβ )ct L0 = β0 ct L0 L0 β0 3 But: p = β γ m0c ⇒ Δp = m0c Δ(β γ ) = m0cγ Δβ Δp 2 Δβ ΔL 1 Δp ⇒ = γ = 2 p0 β L0 γ p0 • This path length dependence on momentum applies everywhere, also in straight trajectories. • The effect quickly vanishes for relativistic particles. • Higher momentum particles precede the ones with lower momentum. 4 p m c By β γ 0 C ρ = = p > p qBz q Bz A 0 B L0 = Trajectory length between A and B p0 ρ L = Trajectory length between A and C O L − L p − p ΔΔLp 0 ∝ 0 = α where α is constant L0 p0 Lp00 ΔΔΔLpE For γαα>>1 ⇒=≅ LpE000 In the example (sector bending magnet) L > L0 so that α > 0 Higher energy particles will leave the magnet later. 5 Momentum compaction, α, is the change in the closed orbit length as a function of momentum.! ΔΔLp = α Lp ΔE/E = 0" L ΔE/E > 0" 0 D α = ∫ x ds 0 ρ 6 Lose energy in dipoles à Synchrotron Radiation! Gain Energy in the RF Cavity! " dipoles 7 •! Imagine a DC voltage across a gap. –! No way to maintain DC voltage through vacuum chamber! •! Add a DC power supply or insulating gap. - - T /h –! Voltage cancelled for V 0 V round trip around ring. + + •! Add switched DC voltage –! Switch at a time period that is a sub-harmonic of the revolution period T0. (I.e. switching frequency is harmonic of revolution frequency. 9 •! Voltage on “Ds” must reverse every half orbit + - 10 •! The energy gain for a particle that moves from A to B is given by: ds L A ΔE = q E r,t ⋅ds = qV E(r, t) ∫ F ( ) B 0 •! We define as V the voltage gain for the particle. V depends only on the particle trajectory and includes the contribution of every electric field present in the area (RF fields, space charge fields, fields due to the interaction with the vacuum chamber, …) •! The particle can also experience energy variations U(E) that depend also on its energy, as for the case of the radiation emitted by a particle under acceleration (synchrotron radiation when the acceleration is transverse). •! The total energy variation will be given by the sum of the two terms: ΔET = qV +U(E) 8 The energy variation for the reference particle is given by: ΔET (s0 )= qV (s0 )+U(E0 ) For particle with energy E = E0 + ΔE and orbit position s = s0 + Δs: dV dU ΔET (s) = qV (s0 + Δs)+U (E0 + ΔE) ≅ qV (s0 )+ q Δs +U (E0 )+ ΔE ds dE s0 E0 Where the last expression holds for the case where Δs << L0 (reference orbit length) and ΔE << E0. In this approximation we can express the average rate of change of the energy respect to the reference particle energy by: ⎛⎞ ΔE Δ−ΔEsTT( ) Es( 0 ) ΔE1 dV dU ≅ ≅Δ⎜⎟qs+Δ E TT T T⎜⎟ ds dE 00 00⎝⎠sE00 L0 T0 = β0c 9 We obtain the equations of motion for the longitudinal plane: 2 d Δs dΔs 2 s L + 2α + Ω Δs = 0 Δ << 0 dt 2 D dt ΔE << E0 ΔE1 ⎛⎞ dV dU ≅Δ⎜⎟qs+Δ E T T⎜⎟ ds dE 00⎝⎠sE00 Finally, by defining the quantities: 2 1 q dV 1 dU Ω=α α D = − p00 T ds s 2T dE 0 0 E0 We will study the case of storage rings where dV/ds is mainly due to the RF system used for restoring the energy lost per turn by the beam. 10 2 This expression is the well known d Δs dΔs 2 damped harmonic oscillator equation, + 2α D + Ω Δs = 0 dt 2 dt which has the general solution: Δs(t) ≅ e−αDt (Ae iΩt + Be −iΩt ) 2 α D > 0 and Ω > 0 α D > 0 ⇔ damped oscillation e−αD t α D < 0 ⇔ anti − damped oscillation Ω2 > 0 ⇔ stable oscillation ee−ααDDtt((AeAeiΩiΩtt ++BeBe−−iΩiΩtt)) Ω2 < 0 ⇔ unstable motion The stable solution represents an oscillation with frequency 2π Ω and with exponentially decreasing amplitude. 11 • The case of damped oscillations is exactly what we want for storing particles in a storage ring. 1 dU dU α > 0 α D = − < 0 D 2T dE dE 0 E0 E0 • The synchrotron radiation (SR) emitted when particles are on a curved trajectory satisfies the condition. The SR power scales as: 4 2 2 2 2 dU dt = −PSR ∝ −(βγ ) ρ = −(γ −1) ρ ρ ≡ trajectory radius • Typically, synchrotron radiation damping is very efficient in electron storage rings and negligible in proton machines. - • The damping time 1/αD (~ ms for e , ~ 13 hours LHC at 7 TeV) is usually much larger than the period of the longitudinal oscillations 1/2π Ω (~ µs). This implies that the damping term can be neglected when calculating the particle motion for t << 1/αD : 12 Let’s consider a storage ring with reference trajectory of length L0: ˆ VRF (t) = V sin(ωRF t) RF Cavity L0 1 2π T0 = TRF = = β c f RF ωRF TT f RFRF T = hT ⇒ f = RF 0 RF 0 h Synchronicity Condition The integer h is called T0 = 3TRF ⇒ h = 3 the harmonic number 13 For our storage ring: 2 1 q dV ˆ ˆ Ω=α VRF (t) = V sin(ωRF t) = V sin( hω0 t) p T ds 00 s0 ˆ dV 1 dV hω0V s ct = = cos(ωRF t0 ) = β0 ds c dt c s0 β 0 t0 β 0 qhVˆ Ω 22α ν = Ω=ωϕ0 cos( s ) S pc002πβ ω0 synchrotron frequency synchrotron tune ϕs = ωRF t0 ≡ synchronous phase 14 2 d Δs 2 p dsΔ If Additionally: 0 α D << Ω + Ω Δs = 0 ΔEt( ) =− dt 2 α dt p Ω Δs = Δsˆcos(Ωt +ψ ) ΔEs=Δˆ 0 sin( Ω t+ψ ) α A different set of variables: φ Phase : ϕ = φ −φs φ = ωRF t s = β0 ct s = β 0 c ωRF Δp Relative Momentum Deviation : δ = ΔE = β0cΔp p0 ϕˆ Ω Synchrotron Oscillations ϕ = ϕˆ cos Ωt +ψ δ = sin(Ωt +ψ ) ( ) For Δs << L and ΔE << E . hω0ηC 0 0 15 ϕˆ Ω We just found: ϕ = ϕˆ cos(Ωt +ψ ) δ = sin(Ωt +ψ ) hω0ηC 2 2 δ ϕ ⎛ hω η ⎞ ηC > 0 2 ⎜ 0 C ⎟ 1 2 + δ ⎜ ⎟ = ϕˆ Ω ˆ ˆ h ϕ ⎝ ϕ Ω ⎠ ω0ηC This equation represents an ellipse in the ϕ longitudinal phase space {ϕ, δ} ϕˆ δ 0 With damping: ηC > α D > 0 ϕ = ϕˆ e−αDt cos(Ωt +ψ ) ϕˆ Ω ϕ δ = e−α D t sin(Ωt +ψ ) hω0ηC In rings with negligible synchrotron radiation (or with negligible non-Hamiltonian forces, the longitudinal emittance is conserved. This is the case for heavy ion and for most proton machines. 17 Two synchronous phases à one stable one unstable U 0 sinϕ S = qVˆ V But V t Vˆ sin t ΔΔts Δ p RF ( ) = (ωRF ) ==α t TL00 p 0 For positive charge particles: 12 For negative charge Forαϕ>0,⇒SS stable ϕ unstable particles all the phases 12 Forαϕ<0,⇒SS unstable ϕ stable are shifted by π. We define as transition Crossing the transition energy energy the energy at during energy ramping requires which α changes sign. a phase jump of ~ π 18 • A is the synchronous particle and arrives at the right time to receive the right energy gain. • B arrives early and gains too much energy. Next turn it arrives later (for α>0.) • C arrives late and gains too little energy arrives earlier on the next turn. 22 So far we have used the small oscillation approximation where: ˆ dV ˆ ˆ ΔET (ψ ) = qV (ϕ S + ϕ) = qV sin(ϕ S + ϕ) ≅ qV (ϕ S )+ q ϕ = qVϕ S + qVϕ dϕ ϕS In the more general case of larger phase oscillations: ˆ ΔET (ψ ) = qV (ϕS +ϕ) ≅ qV sin(ϕS +ϕ) And by Numerical integration: ϕS ≠ 0 or π δ RF “Buckets” Separatrices ϕS = 0 or π ϕ • For larger amplitudes, trajectories in the phase space are not ellipsis anymore. • Stable and unstable orbits exist. The two regions are separated by a special trajectory called separatrix • Larger amplitude orbits have smaller synchrotron frequencies 19 The RF bucket is the area of the longitudinal phase space where a particle orbit is stable ϕS ≠ 0 or π δ RF Buckets (Δp/p ) . ϕS = 0 or π ϕ 0 ACC The momentum acceptance is defined as the maximum momentum that a particle on a stable orbit can have. 2 2 ⎛ Δp ⎞ 2 qVˆ ⎛ Δp ⎞ F(Q) 2 qVˆ ⎜ ⎟ = ⎜ ⎟ = ⎜ p ⎟ h c p ⎜ p ⎟ 2Q h c p ⎝ 0 ⎠ ACC π ηC β 0 ⎝ 0 ⎠ ACC π ηC β 0 1 qVˆ ⎛ 1 ⎞ Q = = F(Q) = 2⎜ Q 2 −1 − arccos ⎟ sinϕ U ⎜ Q ⎟ s 0 ⎝ ⎠ Over voltage factor 20 • In electron storage rings, the statistical emission of synchrotron radiation photons generates gaussian bunches.