Contribution to 2012 Annual Report, Central Laser Facility (CLF), Rutherford-Appleton Laboratory, UKWe discuss classical and quantum corrections to Thomson scattering between an electron and a laser. For radiation reaction, nonlinear, and quantum effects we identify characteristic dimensionless parameters in terms of which we determine the leading order correction terms

Heinzl, T

PEARL (Univ. of Plymouth)

School of Engineering, Computing and Mathematics Faculty of Science and Engineering 2013-07-01 Corrections to Laser Electron Thomson Scattering T Heinzl School of Engineering, Computing and Mathematics Let us know how access to this document benefits you General rights All content in PEARL is protected by copyright law. Author manuscripts are made available in accordance with publisher policies. Please cite only the published version using the details provided on the item record or document. In the absence of an open licence (e.g. Creative Commons), permissions for further reuse of content should be sought from the publisher or author. Take down policy If you believe that this document breaches copyright please contact the library providing details, and we will remove access to the work immediately and investigate your claim. Follow this and additional works at: https://pearl.plymouth.ac.uk/secam-research Recommended Citation Heinzl, T. (2013) 'Corrections to Laser Electron Thomson Scattering', Retrieved from https://pearl.plymouth.ac.uk/secam-research/1314 This Book is brought to you for free and open access by the Faculty of Science and Engineering at PEARL. It has been accepted for inclusion in School of Engineering, Computing and Mathematics by an authorized administrator of PEARL. For more information, please contact openresearch@plymouth.ac.uk. Corrections to Laser Electron Thomson ScatteringThomas HeinzlSchool of Computing and MathematicsPlymouth UniversityDrake Circus, Plymouth PL4 8AA, UKtheinzl@plymouth.ac.ukAnton IldertonDepartment of Applied PhysicsChalmers University of TechnologySE-412 96 Gothenberg, Swedenanton.ilderton@chalmers.seWe discuss classical and quantum corrections to Thomson scattering between an electron and a laser. Forradiation reaction, nonlinear, and quantum effects we identify characteristic dimensionless parameters in terms ofwhich we determine the leading order correction terms.1 IntroductionThomson scattering is the classical process of light be-ing deflected by a charged obstacle of typical size smallerthan the wavelength of the incident radiation. For defi-niteness we will henceforth assume the charge to be anelectron and the incoming wave stemming from a laserbeam. The scattered radiation may be viewed as re-sulting from a two-step process: (i) the acceleration ofthe charge due to the electromagnetic field it encoun-ters and (ii) the bremsstrahlung emitted in consequence.(In fact, this is what happens when solving the classicalequations in powers of the coupling: at zeroth order,there is only acceleration, while at first order, there isemission.) The instantaneously radiated power is thengiven by Larmor’s formula which, in its nonrelativisticincarnation and using Heaviside-Lorentz units, reads [1]Prad =23e24πc3v̇2 . (1)Expressing the acceleration by means of the nonrel-ativistic Lorentz force (neglecting the v × B term),mv̇ = eE, E being the electric field of the incomingwave, the radiated power becomesPrad =23e44πm2c3E2 . (2)This may be turned into a cross section (units of area)upon dividing by the energy flux, i.e. the modulus of thePoynting vector, which for a plane wave (PW) isS = c |E×B| PW= cE2 . (3)Hence, we obtain the Thomson cross section,σTh :=PradS=8π3(e24πmc2)2=:8π3r2e , (4)with the classical electron radius, re ' 3 fm. At thisdistance, the Coulomb energy between electrons equalsmc2, and (with the benefit of hindsight) one ends upwith a distance scale that is actually typical for stronginteractions (1 fm being roughly the nucleon size). As aresult, the Thomson cross section takes on the numeri-cal value σTh ' 6.6 × 10−24 cm2 ' 0.66 barn which isnot small by particle physics standards.The differential cross section may be derived in asimilar vein. For this one needs the angular distributionof the radiated power (1) in direction l which is [1]dPraddΩ=e2(4π)2c3(l · v̇)2 , (5)or, upon employing the equation of motion, with E =Eε,dPraddΩ=e4(4π)2m2c3E2(1− (l · ε)2). (6)Averaging over polarisations and introducing the scat-tering angle θ one has〈1− (l · ε)2〉pol =1 + cos2 θ2. (7)This angular dependence was originally found by Thom-son when calculating the mean “rate at which energy isstreaming through unit area” [2]. Dividing by the in-coming flux, S = cE2, finally yields the differential crosssection,dσThdΩ= 12r2e (1 + cos2 θ) . (8)Integrating over angles we reobtain (4). Clearly, thedifferential cross section is more useful in general as itconveys more, namely spectral, information.It is instructive to rederive (4) in a fully covariantway. The radiated power may be inferred from theproper time derivative of the wave 4-momentum whichcan be expressed in terms of the electron 4-velocity uµ[3],Ṗµ = −23e24πc5u̇2uµ . (9)Dotting in u, using u2 = c2 and the covariant equationof motion, mu̇µ = (e/c)Fµνuν , one finds the radiatedpower,Prad = u · Ṗ = σThuµFµαF να uν/c =: σThw0 . (10)arXiv:1307.0406v1  [hep-ph]  1 Jul 2013For a plane wave (or, more general, null field), the squareof the field strength tensor coincides with the energy-momentum tensor,FµαF ναPW= c Tµν , (11)so that w0 in (10) may be interpreted as the energyflux density of the electromagnetic wave as ‘seen’ by theelectron in its instantaneous rest frame. This togetherwith (10) yields a nice expression for the Thomson crosssections in terms of Lorentz invariants,σTh =u · ṖuµTµνuν, (12)the numerator and denominator being the field energyrate of change and flux density in the co-moving frameof the electron, respectively.2 Radiation reactionThe two-step procedure mentioned in the introductionbreaks down (or rather, becomes insufficient) when theenergy loss due to radiation becomes substantial suchthat the produced radiation field back-reacts on the par-ticle motion. In this case, one has to solve a modifiedequation of motion taking back-reaction into accountand named after Lorentz, Abraham and Dirac (LAD)[4, 5, 6]. Introducing the time parameterτ0 :=23re/c ' 10−23s , (13)it is most compactly written as [3]mu̇µ = Fµ + τ0müµ =ecFµνuν + τ0müµ . (14)This equation, being third order in time derivatives(ü =...x ), has pathological features such as runaway so-lutions and preacceleration which can be traded for eachother but not entirely removed while insisting on (14)– see the lucent discussion in [7]. The loophole is tomodify the equation via iteration, i.e. by expressing müon the right-hand side in terms of the Einstein-Lorentzforce, mü = Ḟµ+O(τ0) which yields the Landau-Lifshitzequation [8]mu̇µ = Fµ + τ0mḞµ +O(τ20 ) . (15)This equation has been rederived (but only to order τ0)using adiabatic perturbation theory [9] or a sophisti-cated classical regularisation procedure [10]. For planewave backgrounds there is an exact analytic solution [11]Clearly, either equation must lead to a modificationof the Thomson cross section. This has already beenworked out in Dirac’s seminal paper [6] who found thefollowing simple result,σRR =σTh1 + (ω0τ0)2. (16)Here we have introduced the laser frequency in the in-stantaneous rest frame, ω0. For a plane wave dependingonly on the invariant phase k · x with k2 = 0, we canwrite ω0 = k ·u. In this case the solution of (15) impliesω0 = k · u = k · u0 + O(τ0) where u0 is the initial four-velocity [11]. Thus, ω0 is only conserved in the absenceof radiation reaction [12].Assuming that our equations of motion both receivecorrections of order τ20 we should rewrite the cross sec-tion asσRR = σTh(1− ω20τ20). (17)The cross section (16) may alternatively be obtained byconsidering an electron scattering off a bound charge,with the interaction described by an additional har-monic damping force, FRR = −δmv. The cross sectionis (see [1], Ch. 45),σRR =σTh1 + δ2/ω20, (18)which we can match to (16) by identifyingδ = ω20τ0 . (19)For a later comparison with the quantum effects it isuseful to rewrite the small parameter ω0τ0 in terms ofthe fine structure constant, α = e2/4π~c, and the di-mensionless parameterν0 :=~k · u0mc2=~ω0mc2=~eζωmc2, (20)which measures the energy of the laser photons (as seenby the initial-state electron) in units of the electron restmass. In the last expression we have introduced rapid-ity ζ via the electron gamma factor, exp ζ = γ(1 + β),to write ν0 in terms of the lab frequency, ω. Obviously,one hasω0τ0 =23αν0 , (21)where the factors of ~ cancel on the right-hand side asthey must for a purely classical parameter. Let us es-timate the magnitude of this quantity for a relativisticelectron (β ' 1) colliding with an optical laser [12, 13].In this case one has ν := ~ω/mc2 ' 10−6 and eζ ' 2γ.Thus, ω0τ0 ' αγν ' 10−8γ, which implies that classicalradiation reaction will be substantial for electron ener-gies of 102 TeV which is in the far quantum regime. So,unless we find a way of classically boosting the radia-tion reaction its observation will not become feasible. Apossible means towards this end is nonlinearity [14, 15].3 Nonlinear EffectsLaser intensity is measured in terms of the r.m.s. energygained by an electron traversing a laser wave length, inunits of the electron rest energy. This leads to the di-mensionless laser amplitudea0 =eErmsλmc2=eErmsmcω, (22)which may be expressed covariantly using the energydensity of (10) such that [16]a20 =e2〈uµTµνuν〉rmsm2c2(k · u)2. (23)When the laser beam is focussed down to the diffractionlimit this may be turned into a rule-of-thumb estimateof a0 in terms of laser power PL measured in petawatts(PW), namelya20 ' 5× 103 PL/PW . (24)For powers of 10 PW to be expected in the near futurethis implies a0 values of a few hundred. For the sake ofsimplicity, however, we will restrict our interest to thenonlinear corrections of leading order in a0.It is known that an electron in a circularly polarisedplane wave moves along a circle (in the average restframe where there is no longitudinal drift, see [8], Ch. 48,Problem 3). The radiation spectrum for such a chargemotion has been first calculated by Schott [17] [formula(128), Sect. VII.84; see also [1], Ch. 38] and consistsof an infinite sum of higher harmonic contributions la-belled by an integer n. Turning the nth harmonic in-tensity scattered per solid angle into a differential crosssection yields [18]dσnd cos θ=4πnr2ea20(cot2 θ J2n(na0 sin θ) + a20J′2n (na0 sin θ))(25)where n is the harmonic number and θ the scatteringangle. To find the leading correction to linear Thom-son scattering (n = 1, a0 = 0) we expand the Besselfunctions for a0  1 using the first terms in their powerseries,Jn(z) = (z/2)n{1n!− z2/4(n+ 1)!+O(z4)}. (26)Plugging this into (25) for n = 1 results in the differen-tial cross sectiondσ1d cos θ= r2eπ(1 + cos2 θ − 14a20 sin2 θ (3 + cos2 θ)),(27)which nicely exhibits the correction to the Thomsoncross section (8) of order a20. Integrating over the angleθ we find the total cross section (for the fundamentalharmonic, n = 1),σ1 = σTh(1− 25a20). (28)As J2n(na0 sin θ) ∼ a2n0 there is another a20 correctioncoming from the second harmonic. Expanding for n = 2one obtainsdσ2d cos θ= 2 r2eπ a20 sin2 θ(1 + cos2 θ). (29)Interestingly, this has the same overall sin2 θ angular de-pendence as the a20 correction in the fundamental crosssection (27), and hence the two will be difficult to dis-tinguish experimentally. This statement holds for circu-lar polarisation, but for linear polarisation the situationis different and higher harmonics have been unambigu-ously identified from their angular radiation patterns[19].0 Π4Π23 Π4Π0.00.51.01.52.0ΘFig. 1: dσTh/dθ (upper curve), dσ1/dθ (middle curve)and dσ2/dθ (lower curve), in units of r2eπ, for a0 = 0.5,cf. (8), (27) and (29).The angular dependence on θ of both cross sections,(27) and (29) are compared with the Thomson one (8)in Fig. 1. Note that only the Thomson term (a0 = 0)contributes in forward and backward direction (θ = 0and θ = π, respectively) as the a20 corrections are sup-pressed there by the sin2 θ factor.Integrating (29) over θ results in a contribution ofdifferent signσ2 =65 a20 σTh . (30)As a consequence, upon adding (28) and (30), the com-plete order a20 correction in the total cross section, i.e.in the sumσ = σ1 + σ2 = σTh(1 + 45a20), (31)becomes positive.4 Quantum EffectsThe quantum version of Thomson scattering is of courseCompton scattering. Quantum corrections become im-portant when the energy of the radiation becomes com-parable to the electron rest mass. In this case, therewill be substantial transfer of four-momentum and thewhole scattering process is most naturally described us-ing the photon picture, that is, by considering the pro-cess e(p) + γ(k) → e′(p′) + γ′(k′). Energy momentumconservation then reads k + p = k′ + p′ and, in thetechnically simplest case with the electron initially atrest, implies the Compton formula for the scattered fre-quency,ωω′= 1 + ν0(1− cos θ) , (32)with the invariant ν0 defined in (20). In the classicallimit, ν0 → 0, the energy transfer (recoil) vanishes andω = ω′. The differential cross section has first beenobtained by Klein and Nishina [20] and reads [21]dσKNdΩ= 12r2e(ω′ω)2 (ωω′+ω′ω− sin2 θ). (33)The classical Thomson limit (8) is readily obtained bysetting ω = ω′. Integrating (33) over angles and expand-ing for ν0  1 we obtain the leading quantum correctionto the Thomson cross section (4),σKN ' σTh(1− 2ν0) . (34)5 SynthesisIdeally, one would like to have a general theory that in-corporates all corrections. In principle, such a theoryexists (at least to some extent), namely quantum elec-trodynamics (QED) coupled to a strong external planewave field, a particular incarnation of strong-field QED.It was originally developed in a series of papers, themost relevant ones being [22, 23, 24, 25]. The mainchallenge for the present discussion is to go beyond theplane wave model and describe the external laser field,say by a Gaussian beam. In this case one loses the exactsolution of the Dirac equation due to Volkov [26], andbasically no progress has been made along this route. Soone will have to make do with plane wave backgroundsfor the time being.Nevertheless, important achievements have been re-ported. For the case of circular polarisation, the mostcomprehensive collection of results can be found in [27](see also Ch. 101 of [21]), in particular the strong fieldQED generalisation of the classical result (28) whichusually is referred to as ‘nonlinear Compton scattering’(NLC). We will not repeat the lengthy formulae here,but only present our result for the expansion of the fun-damental cross section σ1 which reads,σNLC,1 = σTh(1− 2ν0 − 25a20 +145 ν0a20 + . . .). (35)Comparing with (28) we find the same nonlinear cor-rection of order a20 plus the quantum correction −2ν0from (34). This provides a useful consistency check. Wealso note that higher order corrections mix nonlinearand quantum contributions such as the last term of or-der ν0a20 in (35). In this rather literal sense, the NLCresult (35) may be viewed as a unification of quantumand nonlinear corrections to Thomson scattering. Whatseems to be missing in this unification are the radiationreaction contributions. To incorporate these, one mustunderstand how the classical RR corrections (16) and(17) arise, and then how they can be derived from QED.The classical result (16) comes from inserting the or-bit of a radiating electron into (9) and (10) to calculatethe emitted power. That orbit is calculated, classically,by eliminating the gauge field. The analogous quan-tum calculation would involve summing over all possi-ble numbers of outgoing photons [28, 29, 30]. The NLCcross section (35) contains all quantum and intensity-dependent corrections to Thomson scattering, but onlyin the process of one-photon emission (from the colli-sion of an electron and a laser). To obtain the full result(16) one would need to calculate an appropriate quantityin QED to all orders, which is challenging. Low ordercorrections are more readily calculated; radiation reac-tion appears in the scattered electron momentum follow-ing single photon emission, and in the radiated photonmomentum at higher orders, following e.g. two-photonemission [31, 32]. The lowest order term of Dirac’s re-sult should therefore come from the two-photon emissiondiagram.All this suggests that the full theory should containmixed terms of the form (αν0a0)2n implying the possi-bility of a regime (a0  1) where quantum effects arestill moderate but radiation reaction gets boosted dueto large nonlinearities.References[1] J. Schwinger, L.L. DeRaad, Jr., K.A. Milton andW.-y. 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Corrections to Laser Electron Thomson Scattering

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Corrections to Laser Electron Thomson Scattering

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