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Lecture - Day 8 Fundamentals of Accelerators - 2012 Lecture - Day 8 William A. Barletta Director, US Particle Accelerator School Dept. of Physics, MIT Economics Faculty, University of Ljubljana 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 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 & “kicks” 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 ~αγ 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 % ( + 6#$0m0c ' dt * negligible! 4 c 2 (%&) ! P" = q 2 ' = curvature radius ! 6#$0 ' 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 2c r E 4 = "P = " e $ U = P dt energy lost per turn dt SR 2 3 2 0 % SR 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" r E 4 e2 % 4 U = e 0 = 0 2 3 3 3(m0c ) # $o # The average radiated power is given by: U 4" c r E 4 P = 0 = e 0 where L $ ring circumference SR! T 2 3 L 0 3(m0c ) # US Particle Accelerator School ! Energy loss to synchrotron radiation (practical units) e2" 4 E(GeV)4 Uo,electron (keV ) = = 88.46 Energy Loss per turn (per particle) 3#0$ $(m) e2" 4 E(TeV )4 Uo,proton (keV ) = = 6.03 3#0$ $(m) ! Power radiated by a beam of average ! e" 4 E(GeV)4 I(A) P (kW ) = I = 88.46 current Ib: to be restored by RF system electron b 3#0$ $(m) I T b ! rev 4 4 Ntot = e" E(TeV ) I(A) e Pproton (kW ) = Ib = 6.03 3#0$ $(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) 6#$0% %(m) US Particle Accelerator School ! Frequency spectrum Radiation is emitted in a cone of angle 1/γ Therefore the radiation that sweeps the observer is emitted by the particle during the retarded time period "t # $ ret %c Assume that γ and ρ do not change appreciably during Δt. At the observer ! dtobs 1 "tobs = "tret = 2 "tret dtret # Therefore the observer sees Δω ~ 1/ Δtobs ! c "# ~ % 3 $ US Particle Accelerator School ! Critical frequency and critical angle 3 2 2 2 2 d I e ( 2#& + 20 ' . 3 1 2 2 K 2 ( ) K 2 ( ) = 3 * 2 - ( + ' . ) 2 2 / 3 / + 2 2 1/ 3 / 5 d"d# 16$ %0c ) 3c' , 1 1+ ' . 4 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 Integrate over all angles ==> 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 0.3ωc where the critical photon energy is 3 hc 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/γ2). U0 = PSR dt energy lost per turn 4 # dU 2c re E finite " = "PSR = " 3 2 dt 3 m c 2 # ( 0 ) 1 dU 1 d P E dt " D = # = [ $ SR ( 0) ] 2T0 dE E 2T0 dE re ! classical electron radius ! 0 trajectory curvature " , " damping in all planes ! " ! DX DY " p p equilibrium momentum spread and emittances 0 ! ! 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 ==> Longitudinal (synchrotron) oscillations Longitudinal dynamics are described by 1) ε, energy deviation, w.r.t the synchronous particle 2) τ, time delay w.r.t. the synchronous particle qV0 $ c "'= [sin(#s + $%) & sin#s] and "'= # % L E s Linearized equations describe elliptical phase space trajectories ε ! e dV " ! " '= ! # '= $ c ! T0 dt Es τ 2 !c eV& "s = angular synchrotron frequency T0 E0 US Particle Accelerator School Radiation damping of energy fluctuations dU The derivative 0 (> 0) dE is responsible for the damping of the longitudinal oscillations Combine the two equations for (ε, τ) in a single 2nd order differential equation 2 d ! 2 d! 2 & 4 # + +" ! = 0 + = Ae't /) 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 times The energy damping time ~ the time for beam to radiate its original energy Typically 4" R$ Ti = 3 C# Ji E o $5 $3 Where Je ≈ 2, Jx ≈ 1, Jy ≈1 and C" = 8.9 #10 meter $ GeV ! Note Σ 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 ΔEquantum comes from the deviation of <N γ> emitted in one damping time, τE <N γ> = nγ τE 1/2 ==> Δ<N γ> = (nγ τE ) The mean energy of each quantum ~ εcrit 1/2 ==> σε = εcrit(nγ τE ) Note that nγ = Pγ/ εcrit and τE = Eo/ Pγ 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 E o J% &curv E o & 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 2 3 '* p $ Cq) 0 # " p & Cq) 0 1 * ds 13 % " = = + where C = 3.84 ,10- m % " % ( 2 q & p0 # J S ( $ p0 ' JS + 1 * ds iso ! magnetic case • For the horizontal emittance at equilibrium: ! 2 H $ 3ds # 0 % 2 2 " = Cq 2 where: H(s) = "T D# + $T D + 2%T DD# JX % 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 " = " and " = " with " # coupling factor Y # +1 X # +1 US Particle Accelerator School ! ! Equilibrium emittance & ΔE Set Growth rate due to fluctuations (linear) = exponential damping rate due to radiation ==> equilibrium value of emittance or ΔE #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.
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