Unit 11 - Lecture 18 Synchrotron Radiation - I
William A. Barletta Director, United States Particle Accelerator School Dept. of Physics, MIT
US Particle Accelerator School What do we mean by radiation?
Energy is transmitted by the electromagnetic field to infinity Applies in all inertial frames Carried by an electromagnetic wave
Source of the energy Motion of charges
US Particle Accelerator School Schematic of electric field
From: T. Shintake, New Real-time Simulation Technique for Synchrotron and Undulator Radiations, Proc. LINAC 2002, Gyeongju, Korea US Particle Accelerator School Static charge
US Particle Accelerator School Particle moving in a straight line with constant velocity
E field
Q
B field
US Particle Accelerator School Consider the fields from an electron with abrupt accelerations At r = ct, a transition region from one field to the other. At large r, the field in this layer becomes the radiation field.
US Particle Accelerator School Particle moving in a circle at constant speed
dQ = q dl
Field energy flows to infinity
US Particle Accelerator School Remember that fields add, we can compute radiation from a charge twice as long
dQ = 2q dl
The wavelength of the radiation doubles
US Particle Accelerator School All these radiate
Not quantitatively correct because E is a vector; But we can see that the peak field hits the observer twice as often
US Particle Accelerator School Current loop: No radiation
Field is static
B field
US Particle Accelerator School Question to ponder: What is the field from this situation?
US Particle Accelerator School QED approach: Why do particles radiate when accelerated? Charged particles in free space are “surrounded” by virtual photons Appear & disappear & travel with the particles. e
Acceleration separates the charge from the photons and “kicks” the photons onto the “mass shell” Lighter particles have less inertia & radiate photons more efficiently In the field of the dipoles in a synchrotron, charged particles move on a curved trajectory. Transverse acceleration generates the synchrotron radiation
Electrons radiate ~ photons per radian of turning US Particle Accelerator School Longitudinal vs. Transverse Acceleration
2 2 q dp|| P|| = 2 3 2 2 6 m c dt q 2 dp 0 0 P = 2 3 60m0c dt negligible!
4 c 2 () P = q 2 = curvature radius 60
Radiated power for transverse acceleration increases dramatically with energy Limits the maximum energy obtainable with a storage ring
US Particle Accelerator School Energy lost per turn by electrons
dU 2cr E 4 P e U P dt energy lost per turn = SR = 2 3 2 0 = SR dt 3()m0c finite For relativistic electrons: 1 2r E 4 ds ds U P ds e 0 0 = SR = 2 3 2 s = ct ct dt = c c finite 3()m0c finite
For dipole magnets with constant radius r (iso-magnetic case): 4 2 4 4 re E 0 e U0 = 2 3 = 3()m0c 3o The average radiated power is given by:
4 U0 4 cre E 0 PSR = = 2 3 where L ring circumference T0 3()m0c L US Particle Accelerator School Energy loss via synchrotron radiation emission (practical units) e 2 4 E(GeV) 4 Uo,electron (keV ) = = 88.46 Energy Loss per turn (per particle) 30 (m)
e 2 4 E(TeV ) 4 Uo,proton (keV ) = = 6.03 30 (m)
Power radiated by a beam of average e 4 E(GeV) 4 I(A) P (kW ) = I = 88.46 current Ib: to be restored by RF system electron b 30 (m) I T b rev 4 4 Ntot = e E(TeV ) I(A) e Pproton (kW ) = Ib = 6.03 30 (m)
Power radiated by a beam of 4 4 average current I in a dipole of e L(m)I(A)E(GeV) b Pe (kW ) = 2 LIb =14.08 2 length L (energy loss per second) 60 (m)
US Particle Accelerator School Frequency spectrum
Radiation is emitted in a cone of angle 1/ Therefore the radiation that sweeps the observer is emitted by the particle during the retarded time period
t ret c Assume that and do not change appreciably during t. At the observer dtobs 1 tobs = tret = 2 tret dtret
Therefore the observer sees ~ 1/ tobs c ~ 3
US Particle Accelerator School Critical frequency and critical angle
3 2 2 2 2 d I e 2 2 2 2 2 2 = 3 2 ()1+ K 2/3() + 2 2 K1/3() dd 16 0c 3c 1+ Properties of the modified Bessel function ==> radiation intensity is negligible for x >> 1
2 2 3/2 = ()1+ >>1 Higher frequencies 3c 3 have smaller critical 3 c angle Critical frequency = 3 c 2 3 rev
1/ 3 1 c Critical angle c = For frequencies much larger than the critical frequency and angles much larger than the critical angle the synchrotron radiation emission is negligible
US Particle Accelerator School Integrating over all angles yields the spectral density distribution
dI d 3I 3e2 = d = K (x)dx d d d 4 c 5/3 4 0 C /C
1/ 3 1/ 2 2 2 dI e dI 3 e / c << c e >> c d 4 0c c d 2 4 0c c
US Particle Accelerator School Frequency distribution of radiation
The integrated spectral density up to the critical frequency contains half of the
total energy radiated, the peak occurs approximately at 0.3c where the critical photon energy is 3 c = = h 3 c h c 2
50% 50% For electrons, the critical energy in practical units is E[GeV ]3 [keV ] = 2.218 = 0.665 E[GeV ]2 B[T] c [m]
US Particle Accelerator School Number of photons emitted
Since the energy lost per turn is e2 4 U ~ 0 And average energy per photon is the
1 1 c = h c = h 3 3 c 3 2
The average number of photons emitted per revolution is
n 2 fine
US Particle Accelerator School Comparison of S.R. Characteristics
US Particle Accelerator School From: O. Grobner CERN-LHC/VAC VLHC Workshop Sept. 2008 Synchrotron radiation plays a major role in electron storage ring dynamics
• Charged particles radiate when accelerated • Transverse acceleration induces significant radiation (synchrotron radiation) while longitudinal acceleration generates negligible radiation (1/2).
U0 = PSR dt energy lost per turn 4 dU 2cre E finite = PSR = dt 2 3 2 3()m0c 1 dU 1 d D = = [] PSR ()E0 dt 2T0 dE E 2T0 dE re classical electron radius 0
trajectory curvature DX , DY damping in all planes
p equilibrium momentum spread and emittances p0 X , Y
US Particle Accelerator School RF system restores energy loss
Particles change energy according to the phase of the field in the RF cavity
E = eV (t) = eVo sin(RF t) For the synchronous particle
E = U 0 = eV0 sin(s )
US Particle Accelerator School Energy loss + dispersion lead to longitudinal oscillations Longitudinal dynamics are described by 1) , energy deviation, w.r.t the synchronous particle 2) , time delay w.r.t. the synchronous particle
qV0 c '= []sin(s + ) sins and '= L E s Linearized equations describe elliptical phase space trajectories e dV '= '= c T0 dt Es
2 c eV& s = angular synchrotron frequency T0 E0
US Particle Accelerator School Radiation damping of energy fluctuations
Say that the energy loss per turn due to synchrotron radiation loss is U0
The synchronous phase is such that U 0 = eV0 sin(s )
But U0 depends on energy E ==> Rate of change of the energy will be given E eV (t) U (E) = 0 T T 0 0
For E << E and << T0 we can expand
dV dU 0 U 0 (0) + e U 0 (0) + d dt dE e dV 1 dU = = 0 dt T0 To dt T0 dE d = c dt Es
US Particle Accelerator School Energy damping
dU The derivative 0 (> 0) dE is responsible for the damping of the longitudinal oscillations
nd Combine the two equations for (, ) in a single 2 order differential equation 2 d 2 d 2 4 + + = 0 = Aet / s sin 2 t + 2 s s 2 dt s dt s
2 eV& angular synchrotron frequency t / s s = e T0 E0 1 1 dU = 0 longitudinal damping time s 2T0 dE
US Particle Accelerator School Damping Coefficients
4 dU 2cre E 1 dU 1 d = PSR = 2 3 2 D = = []PSR()E0 dt dt 3 m c 2T dE 2T dE ()0 0 E0 0
U 0 By performing the calculation one obtains: D= ()2 + D 2T 0E 0
Where D depends on the lattice parameters. L D=C (<< 1) For the iso-magnetic separate function case: 2 Damping time ~ time required to replace all the original energy
Analogously, for the transverse plane:
U 0 U0 X= ()1 D and Y = 2T 0E 0 2T0 E0
US Particle Accelerator School Damping times
The energy damping time ~ the time for beam to radiate its original energy Typically 4 R Ti = 3 C Ji Eo
5 3 Where Je 2, Jx 1, Jy 1 and C = 8.9 10 meter GeV
Note Ji = 4 (partition theorem)
US Particle Accelerator School Quantum Nature of Synchrotron Radiation
Synchrotron radiation induces damping in all planes. Collapse of beam to a single point is prevented by the quantum nature of synchrotron radiation Photons are randomly emitted in quanta of discrete energy Every time a photon is emitted the parent electron “jumps” in energy and angle Radiation perturbs excites oscillations in all the planes. Oscillations grow until reaching equilibrium balanced by radiation damping.
US Particle Accelerator School Energy fluctuations
Expected Equantum comes from the deviation of
The mean energy of each quantum ~ crit
1/2 ==> = crit(n E )
Note that n = P/ crit and E = Eo/ P
US Particle Accelerator School Therefore, …
The quantum nature of synchrotron radiation emission generates energy fluctuations
1/2 2 E E E Cq o crit o ~ E Eo J curv Eo where Cq is the Compton wavelength of the electron
-13 Cq = 3.8 x 10 m
Bunch length is set by the momentum compaction & Vrf E RE 2 = 2 c o z E eV˙ Using a harmonic rf-cavity can produce shorter bunches
US Particle Accelerator School Schematic of radiation cooling
Transverse cooling:
P less P remains less P less || P|| restored
Passage Acceleration through dipoles in RF cavity Limited by quantum excitation
US Particle Accelerator School Emittance and Momentum Spread • At equilibrium the momentum spread is given by: 2 2 2 3 2 1 ds p C q 0 p Cq0 13 where C 3.84 10 m = = 2 q = p0 J p J S 0 S 1 ds iso magnetic case • For the horizontal emittance at equilibrium:
2 H 3ds 0 2 2 = Cq where: Hs()= D + D + 2 DD J 2 T T T X 1 ds • In the vertical plane, when no vertical bend is present, the synchrotron radiation contribution to the equilibrium emittance is very small • Vertical emittance is defined by machine imperfections & nonlinearities that couple the horizontal & vertical planes: 1 Y = and X = with coupling factor +1 +1 US Particle Accelerator School Equilibrium emittance & E
Set Growth rate due to fluctuations (linear) = exponential damping rate due to radiation
==> equilibrium value of emittance or E
2t / d 2t / d natural = 1e + eq ()1 e
US Particle Accelerator School Quantum lifetime
At a fixed observation point, transverse particle motion looks sinusoidal x a sin t T x,y T = n () n + = Tunes are chosen in order to avoid resonances. At a fixed azimuth, turn-after-turn a particle sweeps all possible positions within the envelope Photon emission randomly changes the “invariant” a Consequently changes the trajectory envelope as well. Cumulative photon emission can bring the envelope beyond acceptance at some azimuth The particle is lost. This mechanism is called the transverse quantum lifetime
US Particle Accelerator School Quantum lifetime was first estimated by Bruck and Sands 2 T exp A2 2 2 T x, y QT DT 2 ()T T = AT Transverse quantum lifetime
2 2 E where T = TT + DT T = x,y E0 transverse damping time DT
exp E 2 2 2 QL DL () A E Longitudinal quantum lifetime Q varies very strongly with the ratio For an iso-magnetic ring: between acceptance & rms size. E 2 J E eVˆ A L 0 2 RF Values for this ratio > 6 2 2 hE U E C 1 0 are usually required. 8 E1 1.08 10 eV US Particle Accelerator School Several time scales govern particle dynamics in storage rings
Damping: several ms for electrons, ~ infinity for heavier particles
Synchrotron oscillations: ~ tens of ms
Revolution period: ~ hundreds of ns to ms
Betatron oscillations: ~ tens of ns
US Particle Accelerator School