Acceleato Physics G. A. Kafft, A. Bogacz, and H. Sayed Jeffeson Lab Old Dominion Univesity Lectue 9 USPAS Acceleato Physics Jan. 11
Synchoton Radiation Acceleated paticles emit electomagnetic adiation. i Emission i fom vey high enegy paticles has unique popeties fo a adiation souce. As such adiation was fist obseved at one of the ealiest electon synchotons, adiation fom high enegy paticles (mainly electons is known geneically as synchoton adiation by the acceleato and HENP communities. The adiation is highly collimated in the beam diection Fom elativity ct ' γ ct γβ z x ' x y' y z' γβ ct + γ z USPAS Acceleato Physics Jan. 11
Loentz invaiance of wave phase implies k µ (ω/c,k x,k y,k z is a Loentz 4-vecto ω γω γβkc z k x kx k y ky k γβcω + γk z kx + ky k x + k y k z sin θ sin θ cosθ ω / c ω / c ω / c ω / c γω / c + γβ k γ 1 + β cos θ ω / c z z ( ( USPAS Acceleato Physics Jan. 11
θ sinθ sinθ γ β θ ( 1+ β cos Theefoe all adiation with θ' < π /, which is oughly y½ of the emission fo dipole emission fom a tansvese acceleation in the beam fame, is Loentz tansfomed into an angle less than 1/γ. Because of the stong Dopple shift of the photon enegy, highe fo θ, most of the enegy in the photons is within a cone of angula extent 1/γ aound the beam diection. USPAS Acceleato Physics Jan. 11
Lamo s Fomula Fo a paticle executing non-elativistic motion, the total powe emitted in electomagnetic adiation is (Lamo 1 q 1 e P ( t a p & 6πε c 6πε m c 3 3 Lienad s elativistic genealization: Note both de and dt ae the fouth component of elativistic 4-vectos when one is dealing with photon emission. Theefoe, thei atio must be an Loentz invaiant. The invaiant that educes to Lamo s fomula in the non-elativistic limit is P µ e du duµ 6πε c dτ dτ USPAS Acceleato Physics Jan. 11
( P t e 6 c γ & πε β β & β 6 Fo acceleation along a line, second tem is zeo and fist tem fo the adiation eaction is small compaed to the acceleation as long as gadient less than 1 14 MV/m. Technically impossible. β Fo tansvese bend acceleation & β c ˆ ρ ec 4 4 ( β γ P t 6πε ρ USPAS Acceleato Physics Jan. 11
Factional Enegy Loss δe e Θ β γ 6πε ρ 3 4 Fo one tun with isomagnetic bending fields δ E 4π e 3ρ E beam β γ 3 3 e is the classical electon adius:.8 1-13 cm USPAS Acceleato Physics Jan. 11
Radiation Powe Distibution Consulting you favoite Classical E&M text (Jackson, Schwinge, Landau and Lifshitz Classical Theoy of Fields dp d 3 e ω γ 8 ω π ε ρ ωc ω / ω c K 5/3 ( xdx USPAS Acceleato Physics Jan. 11
Citical Fequency Citical (angula fequency is 3 3 c ω c γ ρ Enegy scaling of citical fequency is undestood fom 1/γ emission cone and fact that 1 β ~ 1/( γ t A t B ρ ρ ρ t t B γ c γ c γ c ρ 3 3 γβ c A B ρ ρ ρ + 3 γβc γc γ c 1/γ USPAS Acceleato Physics Jan. 11
Photon Numbe dp 3 e e c c 5/3 ( dω 8π ε ρ 6πε ξ ρ P dω ωγ ξ K x dxdξ γ dn& 1 dp dω hω dω 4 hω dn& hω d ω dω dn& d ω dω 8 15 3 hω c n& 5α c 5α e 1 γ δn γ α 3 ρ 3 Θ 4πε hc 137 USPAS Acceleato Physics Jan. 11
Insetion Devices Often peiodic magnetic field magnets ae placed in beam path of high enegy stoage ings. The adiation geneated by electons passing though such insetion devices has unique popeties. Field of the insetion device magnet B x y z B z y B z B z ( ( ˆ ( ( π λ,, cos / ID Vecto potential fo magnet (1 dimensional appoximation B λ,, ˆ ID A x y z A z x A z sin π z/ λ π ( ( ( ( ID USPAS Acceleato Physics Jan. 11
Electon Obit Unifomity in x-diection means that canonical momentum in the x-diection is conseved ( ea z K vx ( z csin z/ γm γ ( π λ ID v 1 K λ x z dz z v β γ π x ID ( cos( π / λ Field Stength Paamete z z ID K eb λ ID π mc USPAS Acceleato Physics Jan. 11
Aveage Velocity 1 Enegy consevation gives that γ is a constant of the motion ( ( z z x z 1 1 β γ β Aveage longitudinal velocity in the insetion device is 1 1 γ γ β β K z Aveage est fame has 1 γ γ / 1 1 + K β γ USPAS Acceleato Physics Jan. 11
Relativistic Kinematics In aveage est fame the insetion device is Loentz contacted, and so its wavelength is λ λ ID / β γ The sinusoidal wiggling motion emits with angula fequency ω πc / λ Loentz tansfomation fomulas fo the wave vecto k k k k x y z γ k k k x y γ k ( 1 β cosθ k sinθ cosϕ k sinθ sinϕ ( cosθ β USPAS Acceleato Physics Jan. 11
Insetion Device (FEL Resonance Angle tansfoms as Condition ( cosθ β ( 1 β cosθθ k cos θ z k Wave vecto in lab fame has k γ k 1 β cosθ πβ c 1 β cosθ ( λ ( ID In the fowad diection cos θ 1 λ ID λid ( K λ e 1 / γ γ + USPAS Acceleato Physics Jan. 11
Powe Emitted Lab Fame Lamo/Lienad calculation in the lab fame yields P e 4 K π γ β c πε γ λid 1 6 Total enegy adiated afte one passage of the insetion device e δ E π γ β NK 6πε λid USPAS Acceleato Physics Jan. 11
Powe Emitted Beam Fame Lamo/Lienad calculation in the beam fame yields P e π 1 ck 6πε λ Total enegy of each photon is ħπc/λ, theefoe the total numbe of photons adiated afte one passage of the insetion device π N π γ αnk 3 USPAS Acceleato Physics Jan. 11
Spectal Distibution in Beam Fame Begin with powe distibution in beam fame: dipole adiation patten (single hamonic only when K<<1; eplace γ by γ, β by β dp dω e c k a sin Θ 3πε 4 Numbe distibution in tems of wave numbe Solid angle tansfomation dnγ α k + k dω 4 k d y z NK Ω dω ( γ 1 βcosθ USPAS Acceleato Physics Jan. 11
Numbe distibution in beam fame Enegy is simply E ( 4 ( dn γ α sin θsin ϕ+ γ cosθ β NK 4 dω 4 γ 1 βcosθ ( θ πβc h 1 λ ID β θ β θ Eˆ ( θ ( 1 cos ( 1 cos Numbe distibution as a function of nomalized lab-fame enegy dn ˆ γ απ E NK 1 β ˆ 3 + de 4γ β γ USPAS Acceleato Physics Jan. 11
Limits of integation Aveage Enegy ˆ 1 ˆ 1 cos θ 1 E cos θ 1 E 1 β 1+ β Aveage enegy is also analytically ll calculable l E dnγ ˆ E de deˆ γ h πβc/ λid dnγ de ˆ de ˆ E max USPAS Acceleato Physics Jan. 11
Conventions on Fouie Tansfoms Fo the time dimensions i t f% ω f t e dt ( ω ( 1 iω t f ( t f % ( ω e dω π Results on Diac delta functions % ( iω t ( te dt 1 δ ω δ δ ( 1 iω t π t e d ω USPAS Acceleato Physics Jan. 11
Fo the thee spatial dimensions δ f% k f x e d x ( ( ik x 3 ( f x f% k e d k 1 3 ( π ( π ( + ik x 3 x x e d k 3 1 + ik x 3 ( δ ( 3 USPAS Acceleato Physics Jan. 11
Geen Function fo Wave Equation Solution to inhomogeneous wave equation 1 + + x y z c t G( x, t; x, t 4πδ x x δ t t ( ( Will pick out the solution with causal bounday conditions G x, t; x, t t < t (, ;, This choice leads automatically to the so-called Retaded Geen Function USPAS Acceleato Physics Jan. 11
In geneal G ( x, t ; x, t t < t G x, t; x, t ( ( i( k x t ( i( k x t 3 A k e ω ω B k e + + d k t > t because thee ae two possible signs of the fequency fo each value of the wave vecto. To solve the homogeneous wave equation it is necessay that ω k k c ( i.e., thee is no dispesion in fee space. USPAS Acceleato Physics Jan. 11
Continuity of G implies ω A k e B k e ( ( i t iω t t t + ε Integate the inhomogeneous equation between and t t ε 1 G( x, t; x, t 4πδ ( x x c t t + ε ( i( k x ωt ( i( k x+ ωt 3 iω A k e + iωb k e d k 4π c δ x x c A( k e e ω π iω ( ik x i t ( USPAS Acceleato Physics Jan. 11
G ( x, t; x, t c ( π i ik x x c e iω ( t t c e dk π x x π δ ( x x / c t + t + x x 1 i( k ( x x ω( t t i( k ( x x + ω( t t 3 e e d k ω t > t e x ik x x x ( > + iω t t e dk t t Called etaded because the influence at time t is due to the souce evaluated at the etaded time t t x x / c USPAS Acceleato Physics Jan. 11
Retaded Solutions fo Fields 1 ρ + + φ z x y c t ε 1 + + A µ J x y z c t 1 3 ρ ( x, t φ ( x, t d x dt δ x x / c t + t 4πε x x µ 3 J ( x, t A ( x, t d x dt δ ( x x / c t + t 4π x x ( Tip: Leave the delta function in it s integal fom to do deivations. Don t have to emembe complicated delta-function ules USPAS Acceleato Physics Jan. 11
φ (, Retaded Solutions fo Fields 1 3 x t d x dt dω e 8π ε x x (, 3 A x t d x dt d e ρ µ J ω 8 π x x ( x, t iω x x / c ( t t ( x, t iω x x / c ( t t Evaluation can be expedited by noting and using the symmety of the Geen function and using elations such as x f ( t t f ( t t t t f x x f x x x ( ( USPAS Acceleato Physics Jan. 11
φ (, Retaded Solutions fo Fields 1 3 x t d x dt dω e 8π ε x x (, 3 A x t d x dt dω e ρ µ J 8 π x x ( x, t iω x x / c ( t t ( x, t iω x x / c ( t t Evaluation can be expedited by noting and using the symmety of the Geen function and using elations such as x f ( t t f ( t t t t f x x f x x x ( ( USPAS Acceleato Physics Jan. 11
Radiation Fom Relativistic Electons Fom discussion ealy in the couse, in the Loenz gauge the equation fo the potentials is 1 ρ + + φ x y z c t ε 1 + + A µ J x y z c t The solution, using the etaded Geen Function is 1 3 ρ ( x, t φ ( x, t d x dt δ ( x x / c t + t 4πε x x µ 3 J ( x, t A ( x, t d x dt δ ( x x / c t + t 4π x x USPAS Acceleato Physics Jan. 11
φ (, Delta Function Repesentation 1 ( x, t i x x / c ( t t 3 x t d x dt d e ω ω 8π ε x x µ 3 J ( x, t i x x / c ( t t A x t d x dt dω e ω (, 8π x x 3 3 ρ x, t qδ x t J x, t qv t δ x t ( ( ρ ( ( ( ( ( q 1 φ( x, t dt dω e 8π ε x qµ v A( x, t dt dω e 8π x ( t ( t ( t iω x t c t t ( / ( iω x t c t t ( / ( USPAS Acceleato Physics Jan. 11
Lienad-Weichet Potentials φ ( x, t A x t φ (, ( x, t A x t (, q 4πε qµ 4π dt δ ( / ( x ( t ( x t c t t v / dt ( t δ x ( t c ( t t x ( t ( q 1 4πε ( ( ( x t 1 nˆ β t qµ v ( t 4π x ( t ( 1 nˆ β ( t et et USPAS Acceleato Physics Jan. 11
φ (, π ε EM Field Radiated q 1 x t dt dω e 8 x ( t v ( t x ( t iω x t c t t ( / ( qµ iω x ( t / c ( t t A( x, t dt dω e 8 π x t A E φ B A t ˆ nˆ & q n β q E + 4πε ( 3 γ 1 nˆ β R 4πε c ( 1 nˆ 3 β R et B nˆ E / c {( nˆ β β} et USPAS Acceleato Physics Jan. 11
1 nˆ x t nˆ ( x ( t x ( t 1 ˆ β ˆ d / dt + nˆ( nˆ d / dt x ( t x ( t dt x ( t d n c dn dt q nˆ 1 i x t / c x t dt dω e ω (, φ 8π ε ω ( x ( t ( i x ( t / c ( t t qµ v i x ( t / c ( t t A x t dt d e ω (, ω t 8π x ( t iω ( t d ω ( ( e i ω 1 β t n ˆ t e dt i x t / c t t i ω x ( t / c ( t t ( ( ( L USPAS Acceleato Physics Jan. 11
q i x ( t / c ( t t nˆ i n ω ω β E( x, t dt dω e + 8π ε integate by pats to get final esult iω x t c t t ( / ( q e E( x, t dt dω vel 8π ε ˆ ( 1 β n x ( t ( ˆ x ( t cx ( t 1 nˆ ( nˆ β + β + β nˆ β nˆ + β nˆ β nˆ nˆ β + β n ˆ ( β 1 β nˆ + β nˆ ( β nˆ β + ( β nˆ ( ( ( USPAS Acceleato Physics Jan. 11
i x ( t / c ( t t ω q e (, ω acc ( 8π ε c 1 β nˆ x ( t E x t dt d ( & nˆ β nˆ & ˆ ˆ ( ( & β 1 β n β β n ( / ( iω x t c t t q e ˆ & 8 c dt dω n n β β π ε 1 β n ˆ x t ( ( { ( ˆ } USPAS Acceleato Physics Jan. 11
Lamo s Fomula Fo small velocities can neglect etadation q E( x, t ˆ { n nˆ & β } / R acc 4πε c dp dω P q nˆ 16π ε µ c 3 q 16π ε c q 6πε c 3 3 v & v & sin & { } nˆ β θ USPAS Acceleato Physics Jan. 11
Relativistic Peaking In fa field afte shot acceleation nˆ nˆ & β ( ( dp t q dω 16π ε c 1 nˆ β dp t θ ( dω max 1 γ { β } ( 5 q & β sin θ 16π ε c 1 β cosθ Fo cicula motions ( ( & sin cos 1 Ω 16π ε c ( 1 β cosθ γ ( 1 β cosθ dp t q β θ ϕ d 5 3 USPAS Acceleato Physics Jan. 11
Spectum Radiated by Motion de dp 1 dt E H nr ( ˆ dt E E R dt dω dω cµ { & ( } {( &} ˆ ˆ ˆ ˆ 1 q n n β β n n β β ( t ( t ( ( cµ 8π ε c 1 nˆ β 1 nˆ β e iω R 1 n ˆ ( t / R+ ( n ˆ ( t / R / c t+ t iω 1 n ˆ ( t ( ( / R+ nˆ t / R / c t+ t cleaing the unpimed time integal and omega pime e dt dωdt dω dt integal with delta epesntation {( & } ( ˆ ˆ ˆ ˆ & π q n n β β n n β β ( t c ( µ 8π ε c 1 n ˆ β 1 n ( ˆ β iω nˆ ( t / c t+ t iω nˆ ( t / c t+ t e e dt dt dω { } t ( USPAS Acceleato Physics Jan. 11
ˆ d E q n 3 d ω dω 3 π ε c 1 nˆ β {( nˆ β β} ( 1 & e iω nˆ ( t / c t+ t dt d E q ω iω t nˆ ( t / c ˆ ( ˆ n n β e dt 3 dωdω 3π ε c Facto of two diffeence fom Jackson because he combines positive fequency and negative fequency contibutions in one positive fequency integal. I don't like because Paseval's fomula, etc. don't wok! I've witten papes about pefoming this calculation in new egimes of high intensity pulsed lases. USPAS Acceleato Physics Jan. 11