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 ˆ cos t δ = sin(Ωt +ψ ) ϕ = ϕ (Ω +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. • The over voltage Q is usually large so that the core of the bunch “lives” in the small oscillation region of the bucket. The equation of motion in the phase space are elliptical: 2 2 ϕ 2 ⎛ hω0ηC ⎞ hω η cη Δp + δ ⎜ ⎟ = 1 ϕˆ = 0 C δˆ ⇒ Δs = C ˆ 2 ⎜ ˆ ⎟ ϕ ⎝ ϕ Ω ⎠ Ω Ω p0

• If σp/p0 is the rms relative momentum spread of the gaussian distribution, then the rms bunch length is given by:

3 cη σ p c p β η σ p σ = C = 0 0 C ΔS 2 ˆ Ω p0 2πq h f0 V cos(φS ) p0 • In the case of heavy ions and of most of protons machines, the whole RF bucket is usually filled with particles. The bunch length l is then proportional to the difference between the two extreme phases of the separatrix:

l = (ϕ2 −ϕ1 )λRF 2π 21 • A charged particle when accelerated radiates. • In high energy storage rings transverse acceleration induces significant radiation (synchrotron radiation) while longitudinal acceleration generates negligible radiation (1/γ2).

U P dt energy lost per turn 0 = ∫ SR 4 finite ρ dU 2c re E = −PSR = − 3 2 dt 3 m c 2 ρ ( 0 ) 1 dU 1 d α D = − = PSR (E0 )dt 2T dE 2T dE [∫ ] re ≡ classical electron radius 0 E0 0

ρ ≡ trajectory curvature α DX , α DY damping in all planes

σ p equilibrium momentum spread and emittances p0

ε X ,ε Y • Synchrotron radiation plays a major role in the dynamics of an electron storage ring 22 4 dU 2c re E U 0 = PSR dt energy lost per turn = −PSR = − ∫ dt 2 3 ρ 2 finite ρ 3(m0c ) • For relativistic electrons: 1 2r E 4 ds ds e 0 U 0 = PSR ds = s = βct ≅ ct ⇒ dt = c ∫ 2 3 ∫ ρ 2 c finite ρ 3(m0c ) finite ρ

• In the case of dipole magnets with constant radius ρ (iso-magnetic case):

4 4π re E0 U 0 = 2 3 ρ 3(m0c )

• The average radiated power is given by:

4 U 0 4π c re E0 PSR = = L ≡ ring circumference T 2 3 ρ L 0 3 m0c ( ) 23 Example: Bunch splitting (for the LHC) motivation: using existing accelerators, produce multiple high-current bunches produce ~40 bunch trains of 72 bunches with 1011 protons and 25 ns bunch spacing (LHC) history: debunching of 6-7 high intensity bunches in the CERN PS + capture in higher-f rf system (microwave instability observed in the process leading to non-uniform beam distributions) concept: application of higher-harmonic rf cavities layout of the LHC including the preinjectors

one bunch from the PS booster gets split into twelve bunches in the CERN PS LHC Example: Split factor 1à 3

example: simulation of bunch

triple-splitting in the CERN time PS (courtesy R. Garoby, 1999)

one of 6 bunches from the booster in the CERN PS time example: measurement of bunch triple-splitting in the CERN PS (courtesy R. Garoby, 2001)

Issues: preservation of longitudinal beam emittance stability of initial conditions complicated (then) by B-field drift requires careful synchronization control of longitudinal coupled-bunch instabilities bunch intensity fluctuations stability of initial conditions

Example: Bunch coalescing motivation: combine many bunches into 1 δ bunch for high peak intensity (and luminosity) concept: φ 1) initial condition with multiple bunches in different high frequency rf buckets

2) lower (vector sum) of δ φ cavity voltages

bunches “shear” due to longitudinal δ mismatch φ 3) turn on a subharmonic rf system bunches rotate with new synchrotron frequency δ

4) restore initial rf (with appro- φ priate phase), turn off the lower frequency rf system example: bunch coalescing in the Fermilab Main Ring (courtesy P. Martin, 1999)

initial condition: 11 bunches captured in 53 MHz rf buckets

“paraphrasing” – adiabatic reduction of the vector sum rf voltage by shift of the relative phases between rf cavities

application of higher voltage 2.5 MHz rf system (in practice, a 5 MHz rf system was used to help linearize the rotation)

capture of bunches in a single

53 MHz rf bucket time

peak intensity monitor with successive traces spaced by 6.8 ms intervals “snap coalescing” – fast change in voltage amplitude applied (instead of adiabatic voltage reduction) observed advantage: avoidance of high-current beam instabilities during paraphrasing observed disadvantage: reduced capture efficiency (~10%) • The main sinusoidal RF gives a linear restoring force. Synchrotron motion is a simple damped harmonic oscillator. • For a Gaussian beam energy distribution, the longitudinal bunch shape is Gaussian. • We manipulate the bunch shape if higher harmonic RF is added.

2 Total voltage main RF voltage 3rd harmonic 1 electron bunch

0 Voltage (MV) Voltage

-1

-2 0 1 2 3 4 5 6

Phase (rad) 32 50 • The potential well can become Main RF, V=1.1 MV substantially non-harmonic, Main RF + HHC 0.08 40 long. distribution w/main RF

long. distribution w/HHC Density Normalized

with strong nonlinear detuning. ) 12 0.06 • Sometimes called “Landau” 30 (z) (x10

cavities because the Φ synchrotron frequency spread 20 0.04

provides “Landau damping” of Potential coherent instabilities. 10 0.02

0 0.00 -80 -60 -40 -20 0 20 40 60 Longitudinal position (mm)

In this mode, the bunch is lengthened, and the peak current can be reduced by a factor of 2-4. This reduces the effect of Touschek scattering by the same factor. This is the primary means to improve beam lifetime in high brightness light sources. 33 • During injection, the injected bunch shape and offset can be mismatched to the bucket, resulting in filamentation of the distribution. For an electron ring, the distribution eventually damps to the equilibrium.

34 • For the mismatched bunch shape, the dependence of synchrotron tune on amplitude causes filamentation. For an electron ring, this eventually damps. For a hadron ring, the longitudinal phase space area is “increased”.

35 • Use a streak camera to record the longitudinal profile vs. time. • a) observe quadrupole oscillations from bunch shape mismatch. • b) filamentation

Linac longitudinal dynamics: Bunch compressors

δ ≡ ΔΕ/Ε δ δ under- compression σ zi ‘chirp’ z z z σz

σδi

V = V0sin(kz) Δz = R56δ

RF Accelerating Path-Length Energy- Voltage Dependent Beamline

To compress a bunch longitudinally, trajectory in dispersive region must be shorter for tail of the bunch than it is for the head.

q How short can the bunch be compressed? q Can low emittance be maintained? q How large are the effects of space charge and coherent synchrotron radiation in bunch compression? • Phase stability is necessary to maintain electrons on a stable orbit in a ring. • Synchrotron oscillations can be modeled as simple damped harmonic oscillators: – Longitudinal focusing come from a time-varying accelerating field provided by an RF system. – Electrons with higher energy take a longer path length around the ring (α>0) – Discrete photon emission excites oscillations – Energy dependence of SR gives radiation damping.