ELECTRON DYNAMICS WITH SYNCHROTRON RADIATION

ELECTRON DYNAMICS WITH SYNCHROTRON RADIATION Lenny Rivkin Ecole Polythechnique Federale de Lausanne (EPFL) and Paul Scherrer Institute (PSI), Switzerland CERN Accelerator School: Introduction to Accelerator Physics November 5, 2012, Granada, Spain

Radiation effects in electron storage rings Average radiated power restored by RF Electron loses energy each turn RF cavities provide voltage to accelerate electrons back to the nominal energy Radiation damping Average rate of energy loss produces DAMPING of electron oscillations in all three degrees of freedom (if properly arranged!) Quantum fluctuations Statistical fluctuations in energy loss (from quantised emission of radiation) produce RANDOM EXCITATION of these oscillations Equilibrium distributions U 0 10 3 of E 0 V RF > U 0 The balance between the damping and the excitation of the electron oscillations determines the equilibrium distribution of particles in the beam Electron Dynamics, L. Rivkin, EPFL & PSI, Granada, Spain, November 2012

Radiation is emitted into a narrow cone = 1 e v << c vc Electron Dynamics, L. Rivkin, EPFL & PSI, Granada, Spain, November 2012

Synchrotron radiation power Power emitted is proportional to: P E 2 B 2 P 4 cc E 2 2 P 2 c 3 2 4 2 C = 4 3 r e m e c 2 3 = 8.858 10 5 m GeV 3 = 1 137 Energy loss per turn: U 0 = C E 4 hc = 197 Mev fm U 0 = 4 3 hc 4 Electron Dynamics, L. Rivkin, EPFL & PSI, Granada, Spain, November 2012

RADIATION DAMPING TRANSVERSE OSCILLATIONS

Average energy loss and gain per turn Every turn electron radiates small amount of energy E 1 = E 0 U 0 = E 0 1 U 0 E 0 only the amplitude of the momentum changes P 1 = P 0 U 0 c = P 0 1 U 0 E 0 Only the longitudinal component of the momentum is increased in the RF cavity Energy of betatron oscillation E A 2 A 1 2 = A 0 2 1 U 0 E 0 or A 1 A 0 1 U 0 2E 0 Electron Dynamics, L. Rivkin, EPFL & PSI, Granada, Spain, November 2012

Damping of vertical oscillations But this is just the exponential decay law! A A = U 0 2E The oscillations are exponentially damped with the damping time (milliseconds!) 2ET U 0 0 In terms of radiation power 2E P A A e t 0 the time it would take particle to lose all of its energy 4 and since P E E 3 1 Electron Dynamics, L. Rivkin, EPFL & PSI, Granada, Spain, November 2012

Adiabatic damping in linear accelerators In a linear accelerator: p p x = p p p decreases 1 E p In a storage ring beam passes many times through same RF cavity Clean loss of energy every turn (no change in x ) Every turn is re-accelerated by RF (x is reduced) Particle energy on average remains constant

Emittance damping in linacs: e W e 2 W 2 e 4 W 4 e 1 or 2 4 econst.

RADIATION DAMPING LONGITUDINAL OSCILLATIONS

Longitudinal motion: compensating radiation loss U 0 RF cavity provides accelerating field with frequency f RF h f 0 h harmonic number V RF The energy gain: U 0 URF ev RF Synchronous particle: has design energy gains from the RF on the average as much as it loses per turn U 0

Longitudinal motion: phase stability U 0 V RF Particle ahead of synchronous one gets too much energy from the RF goes on a longer orbit (not enough B) >> takes longer to go around comes back to the RF cavity closer to synchronous part. Particle behind the synchronous one gets too little energy from the RF goes on a shorter orbit (too much B) catches-up with the synchronous particle

Longitudinal motion: energy-time oscillations energy deviation from the design energy, or the energy of the synchronous particle e longitudinal coordinate measured from the position of the synchronous electron

Orbit Length Length element depends on x dl = 1 + x ds dl x ds Horizontal displacement has two parts: x = x + x e To first order x does not change L x e has the same sign around the ring Length of the off-energy orbit L e = dl = 1 + x e ds = L 0 +L L = D s s ds p where = p = E E L L =

Something funny happens on the way around the ring... Revolution time changes with energy T T = L L Particle goes faster (not much!) while the orbit length increases (more!) The slip factor T T = 1 2 dp p = dp p Ring is above transition energy isochronous ring: T 0 = L 0 c d = 1 2 dp p since >> 1 2 = 0 or = tr 1 tr 2 (relativity) L L = dp p

Not only accelerators work above transition

RF Voltage V RF V Vˆ sin h 0 s U 0 here the synchronous phase s U arcsin 0 evˆ

Momentum compaction factor 1 L Like the tunes Q x, Q y - depends on the whole optics D s s ds A quick estimate for separated function guide field: = 1 L 0 0 mag D s ds L mag = 2 0 = 1 L 0 0 D L mag But = D R Since dispersion is approximately D R Q 2 1 typically < 1% 2 Q and the orbit change for ~ 1% energy deviation L = 1 4 10 L 2 Q = 0 in dipoles = elsewhere

Energy balance Energy gain from the RF system: U RF = ev RF = U 0 + ev RF synchronous particle () will get exactly the energy loss per turn we consider only linear oscillations Each turn electron gets energy from RF and loses energy to radiation within one revolution time T 0 e = U 0 + ev RF U 0 + Ue V RF = dv RF d = 0 de dt = 1 T 0 ev RF Ue An electron with an energy deviation will arrive after one turn at a different time with respect to the synchronous particle d dt = e E 0

Synchrotron oscillations: damped harmonic oscillator Combining the two equations where the oscillation frequency the damping is slow: the solution is then: e t =e 0 e e t cos Wt + e d 2 e dt 2 + 2 e de dt + W 2 e = 0 W 2 ev RF T 0 E 0 e U 2T 0 typically e <<W similarly, we can get for the time delay: t = 0 e e t cos Wt +

Synchrotron (time - energy) oscillations The ratio of amplitudes at any instant Oscillations are 90 degrees out of phase = WE 0 e e = + 2 The motion can be viewed in the phase space of conjugate variables e, e ˆ e e E 0, W e E 0 ˆ W

V Vo sin s s Stable regime U Separatrix d( ) dt

Longitudinal motion: damping of synchrotron oscillations P E 2 B 2 During one period of synchrotron oscillation: when the particle is in the upper half-plane, it loses more energy per turn, its energy gradually reduces e U > U 0 U < U 0 when the particle is in the lower half-plane, it loses less energy per turn, but receives U 0 on the average, so its energy deviation gradually reduces The synchrotron motion is damped the phase space trajectory is spiraling towards the origin

Robinson theorem: Damping partition numbers Transverse betatron oscillations are damped with Synchrotron oscillations are damped twice as fast x e z 2ET U ET U 0 0 0 0 The total amount of damping (Robinson theorem) depends only on energy and loss per turn 1 x + 1 y + 1 e = 2U 0 ET 0 = U 0 2ET 0 J x + J y + J e the sum of the partition numbers J x + J z + J e = 4 Electron Dynamics, L. Rivkin, EPFL & PSI, Granada, Spain, November 2012

Radiation loss P E 2 B 2 Displaced off the design orbit particle sees fields that are different from design values energy deviation e different energy: P different magnetic field B particle moves on a different orbit, defined by the off-energy or dispersion function D x γ E 2 both contribute to linear term in P γ e betatron oscillations: zero on average Electron Dynamics, L. Rivkin, EPFL & PSI, Granada, Spain, November 2012

Radiation loss To first order in e U rad = U 0 + U e P E 2 B 2 electron energy changes slowly, at any instant it is moving on an orbit defined by D x after some algebra one can write U = U 0 2 + D E 0 D 0 only when k 0 U du rad de E0 Electron Dynamics, L. Rivkin, EPFL & PSI, Granada, Spain, November 2012

Damping partition numbers J x + J z + J e = 4 Typically we build rings with no vertical dispersion J 1 J 3 z Horizontal and energy partition numbers can be modified via D : J x e J x 1D J e 2 D Use of combined function magnets Shift the equilibrium orbit in quads with RF frequency Electron Dynamics, L. Rivkin, EPFL & PSI, Granada, Spain, November 2012

EQUILIBRIUM BEAM SIZES

Radiation effects in electron storage rings Average radiated power restored by RF Electron loses energy each turn RF cavities provide voltage to accelerate electrons back to the nominal energy Radiation damping Average rate of energy loss produces DAMPING of electron oscillations in all three degrees of freedom (if properly arranged!) Quantum fluctuations Statistical fluctuations in energy loss (from quantised emission of radiation) produce RANDOM EXCITATION of these oscillations Equilibrium distributions U 0 10 3 of E 0 V RF > U 0 The balance between the damping and the excitation of the electron oscillations determines the equilibrium distribution of particles in the beam Electron Dynamics, L. Rivkin, EPFL & PSI, Granada, Spain, November 2012

Quantum nature of synchrotron radiation Damping only If damping was the whole story, the beam emittance (size) would shrink to microscopic dimensions!* Lots of problems! (e.g. coherent radiation) How small? On the order of electron wavelength E =mc 2 = h = hc e = 1 mc h = C e C = 2.4 10 12 m Compton wavelength Diffraction limited electron emittance

Quantum nature of synchrotron radiation Quantum fluctuations Because the radiation is emitted in quanta, radiation itself takes care of the problem! It is sufficient to use quasi-classical picture:» Emission time is very short» Emission times are statistically independent (each emission - only a small change in electron energy) Purely stochastic (Poisson) process

Visible quantum effects I have always been somewhat amazed that a purely quantum effect can have gross macroscopic effects in large machines; and, even more, that Planck s constant has just the right magnitude needed to make practical the construction of large electron storage rings. A significantly larger or smaller value of would have posed serious -- perhaps insurmountable -- problems for the realization of large rings. Mathew Sands

Quantum excitation of energy oscillations Photons are emitted with typical energy at the rate (photons/second) Fluctuations in this rate excite oscillations During a small interval t electron emits photons u ph h typ = hc 3 N = P u ph N = N t losing energy of Actually, because of fluctuations, the number is resulting in spread in energy loss N u ph N N N u ph For large time intervals RF compensates the energy loss, providing damping towards the design energy E 0 Steady state: typical deviations from E 0 typical fluctuations in energy during a damping time e

Equilibrium energy spread: rough estimate We then expect the rms energy spread to be and since e E 0 P and P N u ph e N e u ph e E 0 u ph geometric mean of the electron and photon energies! Relative energy spread can be written then as: e e E e = m h 0 e c 4 10 13 m it is roughly constant for all rings typically E 2 e E 0 ~ const ~ 10 3

Equilibrium energy spread More detailed calculations give for the case of an isomagnetic lattice s = 0 in dipoles elsewhere e E 2 = C q E 2 J e 0 with C q = 55 32 3 hc m e c 2 3 = 1.468 10 6 m GeV 2 It is difficult to obtain energy spread < 0.1% limit on undulator brightness!

Equilibrium bunch length Bunch length is related to the energy spread Energy deviation and time of arrival (or position along the bunch) are conjugate variables (synchrotron oscillations) recall that W s V RF = W s e E e = W s e E Two ways to obtain short bunches: RF voltage (power!) 1 1 V VRF RF Momentum compaction factor in the limit of = 0 isochronous ring: particle position along the bunch is frozen

Electron Dynamics, L. Rivkin, EPFL & PSI, Granada, Spain, November 2012 Excitation of betatron oscillations x x e x 0 e x x x x x e x E D x e e E D x e E D x e Courant Snyder invariant 2 2 2 2 2 2 2 E D DD D x x x x e e

Excitation of betatron oscillations Electron emitting a photon at a place with non-zero dispersion starts a betatron oscillation around a new reference orbit x D e E Electron Dynamics, L. Rivkin, EPFL & PSI, Granada, Spain, November 2012

Horizontal oscillations: equilibrium Emission of photons is a random process Again we have random walk, now in x. How far particle will wander away is limited by the radiation damping The balance is achieved on the time scale of the damping time x = 2 e x e N x D 2 E Typical horizontal beam size ~ 1 mm Quantum effect visible to the naked eye! Vertical size - determined by coupling D E e

Beam emittance Betatron oscillations Particles in the beam execute betatron oscillations with different amplitudes. x Transverse beam distribution Gaussian (electrons) Typical particle: 1 - ellipse (in a place where = = 0) Emittance x 2 Units of e x = e x = e / m rad Area = e x x x e = x x = x x

Equilibrium horizontal emittance Detailed calculations for isomagnetic lattice e 2 x x0 = C qe 2 J x H mag where H = D 2 + 2DD + D 2 = 1 D 2 + D + D 2 and H mag is average value in the bending magnets

2-D Gaussian distribution Electron rings emittance definition 1 - ellipse x x n x dx = 1 2 e x 2 / 2 2 dx x x Area = e x Probability to be inside 1- ellipse P 1 = 1 e 1 2 = 0.39 Probability to be inside n- ellipse P n = 1 e n 2 2

FODO cell lattice Electron Dynamics, L. Rivkin, EPFL & PSI, Granada, Spain, November 2012

Emittance FODO lattice emittance 100 H ~ D 2 ~ R Q 3 10 e x0 C qe 2 J x R 1 Q 3 e E2 J x 3 F FODO 1 0 20 40 60 80 100 120 140 160 Phase advance per cell [degrees] 180

Ionization cooling p p absorber E acceleration similar to radiation damping, but there is multiple scattering in the absorber that blows up the emittance = 2 2 0 + MS 0 >> MS to minimize the blow up due to multiple scattering in the absorber we can focus the beam Electron Dynamics, L. Rivkin, EPFL & PSI, Granada, Spain, November 2012

Minimum emittance lattices e C E 2 q 3 x0 θ Flatt J x F min = 1 12 15 Electron Dynamics, L. Rivkin, EPFL & PSI, Granada, Spain, November 2012

Quantum limit on emittance Electron in a storage ring s dipole fields is accelerated, interacts with vacuum fluctuations: «accelerated thermometers show increased temperature» synchrotron radiation opening angle is ~ 1/g-> a lower limit on equilibrium vertical emittance independent of energy G(s) =curvature, C q = 0.384 pm in case of SLS: 0.2 pm isomagnetic lattice e y 0.09pm y Mag

Seeing the electron beam (SLS) Making an image of the electron beam using the vertically polarised synchrotron light

Vertical emittance record Beam size 3.6 0.6 m Emittance 0.9 0.4 pm SLS beam cross section compared to a human hair: 80 m 4 m

Summary of radiation integrals Momentum compaction factor I 1 = D ds = I 1 2R Energy loss per turn U 0 = 1 2 C E 4 I 2 I 2 = I 3 = ds 2 ds 3 I 4 = D 2k + 1 2 ds I 5 = H 3 ds C = 4 3 r e m e c 2 3 = 8.858 10 5 m GeV 3 Electron Dynamics, L. Rivkin, EPFL & PSI, Granada, Spain, November 2012

Summary of radiation integrals (2) Damping parameter Damping times, partition numbers Equilibrium emittance e x0 = 2 x =C E 2 q I 5 J x D = I 4 I 2 J e = 2 + D, J x = 1 D, J y = 1 i = 0 J i Equilibrium energy spread 2 e C q E 2 = I 3 E J e I 2 0 = 2ET 0 I 2 U 0 C q = 55 32 3 I 1 = I 2 = I 3 = D ds ds 2 ds 3 I 4 = D 2k + 1 2 ds I 5 = H 3 hc m e c 2 3 = 1.468 10 6 H = D 2 + 2DD + D 2 ds m GeV 2 Electron Dynamics, L. Rivkin, EPFL & PSI, Granada, Spain, November 2012

Damping wigglers Increase the radiation loss per turn U 0 with WIGGLERS reduce damping time emittance control P E P wig wigglers at high dispersion: blow-up emittance e.g. storage ring colliders for high energy physics wigglers at zero dispersion: decrease emittance e.g. damping rings for linear colliders e.g. synchrotron light sources (PETRAIII, 1 nm.rad) Electron Dynamics, L. Rivkin, EPFL & PSI, Granada, Spain, November 2012

END