Classical and Quantum Phenomenology in Radiation by Relativistic Electrons in Matter or in External Fields
aa r X i v : . [ phy s i c s . acc - ph ] D ec Classical and Quantum Phenomenologyin Radiation by Relativistic Electronsin Matter or in External Fields
December 8, 2014
Xavier ArtruUniversit´e de Lyon, Institut de Physique Nucl´eaire de Lyon,Universit´e Lyon 1 and CNRS-IN2P3, FranceIn memory of Vladimir Nikolaevich Baierand Vladimir Moisevich Strakhovenko
Abstract
Phenomenological aspects of radiation by relativistic electrons inexternal field, in matter or the vicinity of matter are reviewed, amongwhich: infrared divergence, coherence length effects, shadowing, en-hancement in aligned crystals, quantum recoil and spin effects, elec-tron side-slipping, photon impact parameter and the presence of tun-nelling in the radiation process.
In Classical Electrodynamics (CED), a moving electron can emit radiationby two mechanisms: (A) velocity change : the photon is directly emitted from the electronworld line. This is the case of
Synchrotron Radiation (SR),
Undulator Radia-tion (UR),
Compton Back-Scattering (CBS),
Bremsstrahlung (BR),
CoherentBremsstrahlung (CBR) and
Channeling Radiation (CR).1
B) transient medium polarization :
If the motion is balistic (i.e., rec-tilinear and uniform), but inside or near a medium, the photon is emittedby the polarization currents induced by the travelling Coulomb field of theelectron. This is the case of
Cherenkov-Vavilov Radiation ( ˇCVR),
Transi-tion Radiation (TR) in optical (OTR) or X-ray (XTR) domains,
DiffractionRadiation (DR),
Smith-Purcell Radiation (SPR),
Parametric X-Rays (PXR)and
Polarization Bremsstrahlung (PBR) on individual atoms.(A) and (B) can coexist, e.g., in Diffracted Channeling Radiation.A vast theoretical and experimental work is currently done on the above-listed radiations, specially after the prediction of intense Channeling Radi-ation by relativistic electrons by Muradin Kumakhov [1]. In this paper wereview some of their “universal” properties in a phenomenological approach.We gather these properties under three headings:-
Classical radiation (Section 2), for photon energy ω much smaller thanthe electron energy ǫ ;- Hard photon emission (Section 3), for ω ∼ ǫ ;- Impact parameter description (Section 4).We will assume that the electron is ultrarelativistic ( ǫ/m e ≡ γ ≫
1) andclassical, at least between photon emissions. We will not speak about itsdynamics in the ”radiator” (UR device, amorphous matter, crystal, TR orPXR targets, etc.) which, in crystals, involves many phenomena : chan-neling, dechanneling, volume capture, volume reflection, etc. We supposethat the trajectory of the electron has been calculated beforehand. For thegeneral theory of radiation the reader may consult [2, 3, 4] and for BR, CRand CBR, [5, 6, 7, 8]. Many considerations presented below can be found in[9, 10].We will work with natural unit systems where ~ = c = 1; α = e / (4 π ) ≃ / γ ” can also designate a photon, of momentum k = ω n ; n = | n | is the refraction index. The components X k and X ⊥ of a vector X are parallel and perpendicular to k , whereas X L and X T are relative to theelectron velocity v ( t ), or to h v i for CR and CBR. We consider an electron of classical trajectory r = r ( t ) and suppose thatthe emission of one photon does not modify the trajectory noticeably anddoes not involve the electron spin. This excludes channeling at low energy2 ǫ < ∼
100 MeV), where the number of transverse energy states is low (quantumchanneling regime) and at very high energy, where hard photon emission( ω ∼ ǫ ) takes place. For a given polarization ˆ e , the photon spectrum writes dN (ˆ e ) = α π d k ω |A| . (1)For mechanism (A) in vacuum, A is given in covariant way by A = − ˆ e ∗ · A ( k ) , A = Z T dX exp( iφ ) , φ = k · X . (2) X = ( t, r ), k = ( ω, k ), A and ˆ e are 4-vectors. We take the metric where k · X = ωt − k · r , ˆ e ∗ · ˆ e = −
1. The integral in (2) is along the whole electrontrajectory T . In a non-covariant formulation we take the gauge ˆ e = ˆe ⊥ k , ˆe ∗ · ˆe , and replace − ˆ e ∗ · A by ˆe ∗ · A ⊥ with A = Z T dt d exp( iφ ) d r dt d = iω Z T dt d exp( iφ ) d r dt , φ = ωt d . (3) t d = t − n · r is the detection time up to a constant; d r ⊥ /dt d and d r ⊥ /dt arethe apparent perpendicular velocity and acceleration. There is no bound on | d r ⊥ /dt d | . The second expression of A emphasizes the role of the acceleration.From the QED point of view, the emitted photons form a coherent state .The number of photons in any given k domain has a Poisson distribution.For mechanism (B), or a coexistence of (A) and (B), A can be calculatedusing the reciprocity theorem, related to time reversal. It gives [9] A = Z revers( T ) d r · E (in) − k , ˆe ∗ ( t, r ) , (4)where revers( T ) is the time-reversed trajectory of the electron. E (in) − k , ˆe ∗ isthe complex electric field of the ingoing solution of the homogenous Maxwellequations in matter, for a wave coming from the detector with momentum − k and polarization ˆe ∗ . It is normalized to ˆe ∗ exp( − iωt − i k · r ) in vacuumin the detector direction. Equation (4) is well suited to TR, SPR and PXR.With a complex refraction index n , it takes the absorption in the radiatorinto account. 3n the following we will make the ultrarelativistic and small-angle ap-proximations γ ≫ | v T ( t ) | and | n T | ≪
1. In the X-ray domain, 1 − | n | ≃ ω / (2 ω ) ≪ ω P being the plasma frequency, and dt d /dt ≃ ( γ − + θ + ω /ω ) / ≪ , (5)with ~θ ≡ − v ⊥ ( t ) ≃ n − v ( t ). Equation (3), supplemented by (5), also appliesto XTR, neglecting the refractions of the X-ray. Sudden velocity change.
As a prototype of mechanism (A), we consideran electron of trajectory r = v I t for t < r = v F t for t > A ⊥ = A ⊥ ( v I , k ) − A ⊥ ( v F , k ) , (6)with A ⊥ ( v , k ) = ( iω ) − v ⊥ / (1 − n · v ) ≃ (2 i/ω ) ~θ / ( γ − + θ ) . (7)For v F = 0 we have a suddenly stopped electron, for v I = 0 a suddenly starting electron. They emit the same photon spectrum, dN ( ~ε ) = απ dωω d Ω (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ~θ · ~εγ − + θ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (8)but with opposite amplitudes. It can be understood with the superpositionprinciple: to a suddenly stopping e + , we can add - in thought - an e − comov-ing with the e + at t <
0, but not stopping. This addition does not changethe radiation, and the new system is equivalent to a suddenly starting e − .The spectrum (8) has the following ”universal” properties : • an annular-lobe angular distribution peaked at θ = 1 /γ , • a radial polarization ( i.e., in the ( v , n ) plane), • an infrared divergence dN/ ( dωd Ω) ∼ ω − , related to the semi-infinitebalistic motion, • an ultraviolet divergence, related to the infinite acceleration at t = 0.It does not happen really, because ω < ǫ .4he case | v I | = | v F | corresponds to Bremsstrahlung on one atom, in the Bornapproximation. At large scattering angle ( | v I − v F | ≫ γ − ) we have two well-separated annular lobes. For the so-called dipolar regime | v I − v F | ≪ γ − ,the interference between the lobes is essential.Whether the velocity change is sudden or not, Eqs.(6,7) apply in theinfrared part of the spectrum. It suffices that the motion is balistic for t →±∞ . The annular lobe (8) appears in many other experimental situations:in backward OTR from a metallic foil, in forward OTR from any screen, inthe edge radiation from a bending magnet, etc., although the finite size ofthe machine provides an infrared cutoff.In simulations, one truncates the integral in (3). If we do it in the secondexpression, we introduce a spurious infrared divergence. It is like complet-ing the trajectories by semi-infinite balistic motions. A phenomenologicalinfrared cutoff has to be applied. In [11] such a cutoff was chosen with thehelp of a filtered sum rule [12]. Truncating in the first expression of (3) pro-duces a spurious ultraviolet divergence, as if the electron started and stoppedsuddenly. Longitudinal coherence (LC) [13].
We define the coherence lenght asthe length of a segment of trajectory over which φ increases by ∆ φ = 1 radian.In the X-ray domain and for a straight trajectory in uniform medium, L coh ( v , k ) = dt/dφ ≃ (2 /ω ) ( γ − + θ + ω /ω ) − . (9)It can reach macroscopic values, even though the photon wavelength is mi-croscopic. For a curved trajectory, the ∆ φ = 1 arcs have various lengths.We choose the longest one (where θ is minimum). It gives L coh ≃ min { (2 /ω ) ( γ − + ψ ) − , (24 R /ω ) / } , (10) ψ being angle between the photon and the curvature plane.If the electron undergoes two successive scattering v I → v at X = ( t, r )and v → v F at X ′ = ( t ′ , r ′ ), the XX ′ path contributes to the first form of(3) by A [ XX ′ ] = exp[ iφ ( t )] ( e i ∆ φ − A ( v , k ) , (11)with ∆ φ = L/L coh ( v , k ) and L = | r ′ − r | . When ∆ φ < A [ XX ′ ] is suppressedby the factor ( e i ∆ φ − A ( v , k ). Thisis one part of the explanation of the Landau-Pomeranchuk-Migdal (LPM)5ffect. Similarly, when the electron crosses two radiators ( e.g., a bendingmagnet and a OTR mirror) distant by L < ∼ L coh , the interference between theresulting radiations must be taken into account. LC is destructive in an XTRradiator when the foil spacing is less than L (vac . )coh , or when the foil thicknessis less than L (med . )coh , as expected from a collective response of the atoms.In a radiator of spatial period Λ, LC gathers the radiation spectrumin narrow “resonance” peaks where L coh = Λ / (2 πν ), ν being the harmonicnumber. This interference of order ν is at work in UR, in resonant XTRand in CBR on atomic planes or coplanar atomic strings (“String-Of-String”radiation). From the quantum point of view, L coh = − /q k , where q is themomentum brought by the radiator. For instance, CBR can be consideredas the virtual Compton process e − ( ǫ, p ) + γ V (0 , q ) → e − ( ǫ ′ , p ′ ) + γ ( ω, k ) , (12)where γ V (0 , q ) is a virtual photon of the crystal field and q belongs to thereciprocal lattice. In Undulator Radiation, harmonics appear in the non-dipole regime K ≡ γ h v i > ∼
1, i.e., h p i > ∼ m . In this case we must replace θ by h v ⊥ i = θ + h v i in (9) and ν is the number of virtual photons, eachof momentum 2 π/ Λ, involved in the reaction. The external field then worksbeyond the Born approximation.Independently of its indented spectrum, CBR yields more energy thanBR. This is because, when crossing one plane or string of atoms, the electroncrosses several atoms at nearly equal impact parameters b i , therefore the as-sociated Bremsstrahlung amplitudes add constructively, provided this occurswithin a path L < ∼ L coh . One can then speak of a ”zero-order” interference. Typical frequency.
Photons which mostly conribute to the radiated en-ergy are such that L coh ∼ ∆ L , where ∆ L is the characteristic length of thevariations of the apparent transverse velocity d r ⊥ /dt d . For an almost straighttrajectory, it gives ω ∼ γ / (∆ L ). For a circular trajectory, ∆ L ∼ R/γ and ω ∼ ω c = γ /R . Equivalent Photons, or Weizs¨acker-Williams (WW) Method.
Thevector A ⊥ ( v , k ) of Eq.(7) is, apart from a factor (2 /e ) exp( iωt ), nearly equalto the Fourier transform˜ E T ( v , k ) = ie k T / ( k + γ − k ) exp( − iωt ) (13)6f the transverse component of the electron Coulomb field at t = 0. Accordingto the WW approach [14], the latter evolves into the free radiation field (7)if the electron is suddenly stopped or scattered at large angle. The maindifference between (7) and (13) concerns the dispersion relations, ω = | k | forthe real photons of (7) and ω = v · k for the virtual photons of (13). Thismethod explains in a simple way the spectrum and polarization of backwardOTR, considered as a reflection of the equivalent photons.The WW method can also be applied to the virtual photons of the target ,if their virtuality − q = q is not too high. CBR can thus be treated asCompton scattering (see Eq.12). The crossed process, Coherent Pair Cre-ation on crystal planes, can be viewed as γ + γ V → e + e − .In impact parameter space, the Fourier transform of (13) gives [3] E T ( k L , b ) ≃ − e p /π ( ω/γ ) K ( ωb/γ ) . (14)It has a 1 /b singularity at b = 0 due to the slow decrease at | k T | → ∞ .This singularity is not observed when focusing an optical system on an OTRtarget [15], because of the limit | k T | < ω sin θ max , where θ max is the opticalaperture. Instead one observes a dip at b = 0. The semi-bare electron.
A hard-scattered electron looses its initial pho-ton cloud and is left quasi-bare [16, 17, 18]. The Coulomb field is gradu-ally restored, starting with the short-distance, or high-momentum, compo-nents. For a given k , this process takes place in a formation zone of length L ∼ L coh ( v F , k ), while real photons are created, forming the annular lobearound v F . The same process happens when the electron leaves an opaquematerial, or in beta decay. If one puts a backward OTR or DR foil insidethe formation zone, it will give a very low signal. This is another way tounderstand the destructive LC effect. Shadowing.
A screen whose contour is close to the electron trajectoryintercepts part of the equivalent photon cloud, making a shadow in the for-mation zone [18, 19]. This has been verified in an experiment at Tomsk [20]schematized in Fig.1, left. Shadowing limits in an essential way the Smith-Purcell radiation, since each ridge of the grating makes a shadow on the nextone. A bound dW/dL ≤ α / (2 πb ) for the SPR energy per electron and perunit length has been proposed [21]. 7 ull DRweak DRshadowscreene ! b ! r e " e' " b + b " e + e " Figure 1:
Left : scheme of a shadowing experiment. Right : side-slipping of the electron(upper diagram) and impact parameter of the photon (lower diagram).
Crystal-assisted radiation.
It is weel known that an electron penetratinga crystal at small angle with major atomic planes or strings radiates moreenergy than at random orientation. In short, “CBR and CR > BR”. It isinteresting to have a qualitative explanation of that, without calculating thefull CR or CBR spectrum.A heuristic explanation is that an electron crossing N approximatelyaligned atoms in a relatively small path length L emits Bremsstrahlung as ona ”super-atom” of atomic number N Z , therefore N times more than on onereal atom. This is the ”zero-order” interference mentioned above. Howeversuch atoms are N times less numerous than the real ones, therefore the netgain is only N . This number can be estimated by geometrical arguments,assuming quasi-straight trajectories. By crossing an atomic string at smallangle ψ , N ∼ a/ ( dψ ), a being the Thomas-Fermi radius and d the latticeconstant. By crossing an atomic plane, N ∼ a / ( d ψ ). In the channelingregime ( ψ < ∼ ψ c ) this calculation is not valid, but N is still large. Thus weexpect CBR to be N times more intense than ordinary BR, at least for thefrequencies such that L = a/ψ < ∼ L coh . If not, N has to be reduced by thefactor L coh /L .On the other hand, a classical sum rule predicts, at first sight, that theradiated energy is independent of ψ . From (3) one can derive various sumrules [12, 9] for the photon spectra at fixed n , one of which is Z dω ωdNdω d Ω = α π Z dt d (cid:18) d r ⊥ dt (cid:19) ; (15)8ntegrating over n gives the Li´enard formula for the radiated energy W = 2 α m Z dt ( F + γ F ) , (16) F = F [ r ( t )] being the force acting on the electron. Assuming that it comesfrom the microscopic electric field E m ( r ) in matter, one finds a radiatedenergy dW/dL = (16 πα / m ) γ h E i , which does not depend on the tra-jectory angle, apart from channeling effects. The drawback of this classi-cal prediction comes from applying (15) to a spectrum which overlaps thenon-classical region ( ω ∼ ǫ ), where quantum recoil corrections inhibits theradiation, not to speak of the forbidden region ( ω > ǫ ). Schematically, wecan write F [ r ( t )] = F slow ( t ) + F fast ( t ) . (17)where F slow and F fast generate the soft ( ω ≪ ǫ ) and hard ( ω ∼ ǫ ) radiationsrespectively. Equation (16) applies only to F slow , giving the soft photoncontribution W soft . When applied to F fast , it overestimates W hard . For quasi-straight trajectories, h F i + h F i = h F i is the same at random or alignedcrystal orientation, but not h F i and h F i separately. In the aligned case F slow comes essentially from the Lindhard potential, giving large h F i and W soft , while F fast comes from the residual atomic potentials. At randomorientation, h F i and W soft are small; h F i is larger than in the alignedcase, but W hard cannot increase much because of the recoil corrections.Nevertheless, Eq.(16) has something to tell in CBR : when ψ increasesthe CBR spectral peaks move together toward larger ω , while the energiesunder the peaks remain constant [22]. Thus the total CBR energy does notdepend on ψ . In fact it can be calculated by putting the Lindhard force inEq.(16). When the typical photon frequency discussed above is comparable to ǫ , theclassical theory over-estimates the radiated energy. This happens in a fastvarying field, for instance in ordinary Bremsstrahlung and hard Comptonscattering. It can also happen in uniform but very strong fields. Channeling radiation in QED-strong field.
In an aligned crystal, arelativistic electron ”sees” in its proper frame the field E ∗ T = γ E T , where9 T is the Lindhard field. At high γ , E ∗ T can reach the critical QED valueE crit = m /e = 1 .
32 10 volt/metre. The critical parameter is χ = E ∗ T / E crit .For instance, χ ∼ γ ∼ along the h i axis in germa-nium. In a critical electric field the Schwinger process of spontaneous paircreation should takes place, but in the channeling case it is prevented by theequally strong magnetic field B ∗ = γ v × E ∗ T . Nevertheless, the followingnon-perturbative QED processes take place at χ > ∼ γ → e + e − , first proved in [24] (here χ ≡ ( ω/m ) E T / E crit );- ”photon splitting” γ → γ γ [25].Besides, CR becomes ”hard” ( ω ∼ ǫ ). It leads to a large decrease or “cooling”of the transverse energy ǫ T at each emission, and, as a consequence, to a self-acceleration of the CR process [26, 27, 28, 11, 29]. The χ ∼ e + e − or e − e − linear colliders,producing severe Beamsstrahlung energy loss.
The “magic” Baier-Katkov (BK) formula.
Equations (2-3) do nottake into account the quantum recoil of the electron upon photon emissionand the resulting spectrum may extend beyond the kinematic limit ω = ǫ .They are also blind to the electron spin. These defects are cured by rathersimple modifications introduced by Baier and Katkov [30]:- 1) for the recoil, replace φ = − k · X by Φ = − ( ǫ/ǫ ′ ) k · X = − ( ǫ/ǫ ′ ) ωt d .- 2) A becomes a spin-dependent amplitude h s ′ | A ˆe | s i . In the helicity basis, h + | A − | + i = ˆe ∗− · Z T d r exp( i Φ) , h−| A + |−i = ˆe ∗ + · Z T d r exp( i Φ) , (18a) h−| A − |−i = ( γ/γ ′ ) h + | A − | + i , h + | A + | + i = ( γ/γ ′ ) h−| A + |−i , (18b) h−| A + | + i = −h + | A − |−i = 2 − / (cid:18) γ ′ − γ (cid:19) Z T dt exp( i Φ) , (18c) h−| A − | + i = h + | A + |−i = 0 . (18d)Note that (18a) coincide with the classical formula except for φ → Φ.The substitution φ → Φ can be understood using the sum-over-historiesof Feynman: the phase difference between the emission from two space-timepoints X and X + dX of the trajectory is d Φ = ( k + p ′ − p ) · dX ≡ q · dX , with dX = p dt/ǫ , (19)10nstead of dφ = k · dX . The latter misses the propagation phase of theintial and final electrons. q is the 4-momentum provided by the externalfield. Assuming a slow varying field, we can neglect q and q ⊥ and derive k · p ≃ q · p ′ and q · p/q · p ′ ≃ ǫ/ǫ ′ , from where d Φ /dφ = ǫ/ǫ ′ .What seems ”magic” is that the BK formula does not depend on the fateof the final electron, which may follow a very different trajectory. It only de-pends on the trajectory the electron would have without photon emission. Infact, the BK formula can be obtained in the JWKB approach for an externalpotential which is invariant in the transverse coordinates, for instance in acounter-propagating plane wave. Then, by knowing the initial trajectory weknow the final one (up to a lateral translation). Let us nevertheless test theBK formula in axial channeling, where the potential depends on transversecoordinates. Conservation of energy and longitudinal momentum gives( ω/ (cid:2) ( γγ ′ ) − + θ (cid:3) = ǫ T − ǫ ′ T . (20)whereas, for a periodic trajectory of period L , the BK formula predicts( γ/γ ′ ) ( ω/ (cid:0) γ − + θ + h v i (cid:1) = 2 νπ/L , (21) ν being the harmonic number. Let us assume that the transverse motionis quasi-classical and all channeled trajectories are periodic. Then ν = n − n ′ , where n ≫ Lǫ h v i = 2 nπ . For a ω ≪ ǫ and ν ≪ n , we have ǫ T − ǫ ′ T = 2 νπ/L − ω h v i (22)(the last term comes from applying variational principle to the transverseHamiltonian V ( r T ) + p / (2 ǫ ) when ǫ changes). Then (20) and (21) agree.However they give different results when ω ∼ ǫ or ν ∼ n . Note that ν ≪ n usually implies ω ≪ ǫ , therefore we can conclude that the BK spectral linesare shifted only for the high harmonics. But it does not matter, since thelatter build a quasi-continuum.The BK formula can be applied to linear and nonlinear Compton scat-tering, and to ordinary BR by factorizing the scattering amplitude and theradiation amplitude. This allows, in simulations [11], to treat simultaneouslythe Channeling Radiation and the incoherent Bremsstrahlung.11 Impact parameter description
Radiation as a tunneling effect.
In vacuum, the reaction e − ( p ) → γ ( k ) + e ′− ( p ′ ) is kinematically forbidden if we consider it as a local processbetween classical particles. One cannot satisfy p = p ′ + k , p = p ′ = m , and k = 0 together. In an external field, we can keep the particle description,assuming a nonlocal process, where the e ′− trajectory is at distance δ r = − ω ǫ ′ R ( γ − + θ ) , (23)from the e − one, R being the e − position measured from the curvature centre.This electron side slipping [31] is of the order of | δ r | ∼ ω/ ( m ω c ) < ∼
400 fermi,where ω c = γ /R is the SR cutoff. In the the ~ → b = − ( ǫ ′ /ω ) δ r = R ( γ − + θ ) / . (24)Then, the ( e ′− + γ ) center-of-energy prolongates the intial trajectories forsome time (see Fig.1, right). For θ = 0 the photon trajectory is tangent tothe light cylinder of radius R/v ≃ R (1 + γ − / b can be considered as a tunnelingeffect. For a circular electron orbit, the photon has a angular momentum ωR/ ( v cos ψ ), where ψ is defined below Eq.(10). It must cross a centrifugalbarrier which ends at the radius R/ ( v cos ψ ). The above-cutoff behaviour ∼ exp (cid:8) − ( ω/ω c ) (1 + γ ψ ) / (cid:9) (25)of SR can be simply derived from the standard tunneling factor. A similartunneling effect takes place in the crossed process γ → e + e − in an externalfield [33]. The impact relative impact parameter (see Fig.1, right) is b ( e + ) − b ( e − ) = m F T / (2 ω F ) , (26)where ± F T is the tranverse force acting on e ± .12 Summary
We have presented, relying on significant equations, a few phenomenologicalaspects of radiation by relativistic electrons in external field or in matter :annular lobes with infrared-divergence, constructive or destructive longitudi-nal coherence effects, release of the equivalent photons, semi-bare electrons,shadowing, quantum recoil and spin effects in “artificial” strong fields, elec-tron side-slipping, photon impact parameter and tunnelling mechanism.We did not broach many other important topics, for instance kinematicalversus dynamical approaches of PXR, density effect, ionization energy loss,coherent emission from a whole electron bunch, stimulated emission, etc.They would deserve several other reviews.
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