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Laser Physics, Vol. 6, No. 5, 1996, pp. 837–858. Original Text Copyright © 1996 by Astro, Ltd. English Translation Copyright © 1996 by åÄàä ç‡Û͇ /Interperiodica Publishing (Russia). REVIEWS Resonant Effects in Collisions of Relativistic Electrons in the Field of a Light Wave S. P. Roshchupkin Sumy State University, Sumy, 244030 Ukraine Received May 12, 1996 Abstract—This review is devoted to the theoretical consideration of one of the fundamental problems of quantum electrodynamics of strong fields—resonances related to a virtual intermediate particle that falls within the mass shell in processes of the second order in the fine-structure constant in an external electromagnetic field. It is demonstrated that resonant cross sections may be several orders of magnitude greater than the cross sections of the corresponding processes in the absence of the external field. CONTENTS 1. INTRODUCTION 2. RESONANT ELECTRON–ELECTRON SCATTERING IN THE FIELD OF AN ELLIPTICALLY POLARIZED LIGHT WAVE 2.1. The Amplitude of Electron–Electron Scattering in the Field of a Plane Light Wave 2.2. Poles of the e––e– Scattering Amplitude 2.3. Resonant Differential Cross Section 3. RESONANT BREMSSTRAHLUNG OF AN ELECTRON SCATTERED BY A NUCLEUS IN THE FIELD OF A LIGHT WAVE 3.1. The Amplitude of Electron–Nucleus SB in a Light Field 3.2. Poles of the SB Amplitude 3.3. Resonant Differential SB Cross Section 3.4. The Range of Relativistic Energies 3.5. The Range of Nonrelativistic Electron Energies 3.6. The Range of Ultrarelativistic Energies of Electrons Moving within a Narrow Cone with a Photon from the Wave 3.7. The Range of Ultrarelativistic Energies of Electrons Moving within a Narrow Cone with a Spontaneous Photon 4. RESONANT EFFECTS IN THE PHOTOCREATION OF ELECTRON–POSITRON PAIRS ON A NUCLEUS IN A LIGHT FIELD 5. CONCLUSION REFERENCES 1. INTRODUCTION A characteristic feature of electrodynamic processes of higher orders in the fine-structure constant in a laser field is associated with the fact that such processes may occur under resonant conditions. This may be due to lower order processes, such as spontaneous emission or one-photon creation and annihilation of electron–positron pairs, that may be allowed in the field of a light wave. Therefore, within a certain range of the energy and momentum, a particle in an intermediate state may fall within the mass shell. Then, the considered higher order process is effectively reduced to two 837 840 840 842 845 846 847 848 849 850 852 853 853 855 857 858 (or more than two) sequential lower order processes. The appearance of resonances in a laser field is one of the fundamental problems of quantum electrodynamics of strong fields, which has attracted attention of physicists since the mid-1960s. However, although nearly three decades have passed, the number of publications on these problems does not exceed two tens [1–17]. This is due to the fact that analysis of electrodynamic processes of the second and higher orders in the finestructure constant is complicated by computational difficulties and a cumbersome form of the results. Even when these studies were generalized (see the monographs [18, 19]), it was done in a rather fragmentary 837 838 ROSHCHUPKIN form and in connection with particular problems only. Therefore, it is of importance to generalize the results of studies devoted to resonances in stimulated bremsstrahlung emission and absorption (SBEA) and spontaneous bremsstrahlung (SB) of electrons in a light field. We emphasize that the range of resonances is a substantially relativistic domain. For this reason, in this review, we will be mainly concentrated on the studies that use a general relativistic approach. Nonlinear effects in the processes of interaction of electrons with the field of a wave are governed by a classical relativistic-invariant parameter [20] η = eF / mω,1 (1.1) where e and m are the charge and the mass of an electron, respectively, and F and ω are the strength and the frequency of the electric field in the wave. Multiphoton SBEA and SB processes that involve an electron scattered by a nucleus or an electron in a plane-wave field are also characterized by quantum parameters— Bunkin–Fedorov parameter γ of the multiplicity of a multiphoton process [21, 22] and parameter β [excluding a circularly polarized wave, when β = 0; see expression (2.24) for β–] [23], γ = η(mv / ω), β = η2(m2v / Eω). (1.2) Here, E and v = | p | / E are the energy and the velocity of an electron. In what follows, we assume that the frequencies of the external field satisfy the condition ωⰆ m v ⁄ 2 , if v Ⰶ 1 m, if E ⲏ m. 2 (1.3) Within the range of optical frequencies (ω ~ 1015 s–1), parameters η, γ, and β become on the order of unity in the fields F ~ (1010–1011), (104–105)v, and (107– 108)v (V/cm), respectively. In the latter expressions, the electron velocities are bounded below by the condition of the Born approximation in the interaction of electrons with each other and with the field of the nucleus, v Ⰷ α = 1/137 (the charge of the nucleus is Z ~ 1). Usually, in calculating the SBEA and SB amplitudes, we can expand the solutions to the Dirac equation in the field of a plane wave (Volkov functions) [24] as Fourier series. The resulting expressions are integrated with respect to spatial and temporal variables. Within the framework of such an approach, the amplitude of a process can be represented as a sum of partial amplitudes (with emission or absorption of a definite number of photons from the wave). In a general relativistic case, these partial amplitudes for an elliptically polarized wave are expressed in terms of functions Lr (2.26) (r = ±1, ± 2, … stand for the number of emitted or absorbed photons from the wave) [16, 23], which specify multiphoton processes by means of parameters γ and β (1.2). A resonant behavior of the cross sections of electron–electron SBEA and electron–nucleus SB pro1 Hereafter, we use the relativistic system of units: ប = c = 1. cesses in the presence of the field of a plane electromagnetic wave can be accounted for in the following manner [1, 2]. The Fourier transform of the Green function of an electron in the field of a plane wave has poles where 2 Ẽ + rω = ± ( p̃ + rk ) + m * ; r = 0, ± 1, ± 2, … . 2 (1.4) Here, p̃ = ( Ẽ , p̃ ) and m* are the four-quasimomentum and the effective mass of an electron in the field of the wave [see expressions (2.5) and (2.6)] [20], respectively. The plus and minus signs in (1.4) correspond to the electron and positron states, respectively. According to the conventional interpretation of the poles of the Green function [25], the quantities Epr ≡ Ẽ + rω and Pr ≡ p̃ + rk can be considered as the energy and the momentum of a quasiparticle that corresponds to a system consisting of an electron and a plane electromagnetic wave [26]. Consequently, although the potential of the external field depends on time, we can introduce a discrete energy spectrum of the system under study, and this spectrum consists of an infinite number of levels (1.4). Therefore, the physical nature of resonances in the system under consideration is the same as in resonant transitions in a discrete spectrum. The interaction with a quantized electromagnetic field causes transitions between states with different quasienergies, i.e., the wave packet spreads in the space of quasienergies. Formally, we can take into account this spreading if we define the quasienergy Ẽ as a complex quantity, Ẽ = ˜ – iΓ. Then, 1/Γ determines the lifetime of a state E' with a definite quasienergy. Rigorously, the divergence of the cross section of scattering in the resonance range indicates that expansion into a perturbation series is inapplicable in the situation under study. Correct calculation of the cross section of scattering requires an approach that would fall beyond the framework of the perturbation theory. Specifically, we can perform summation of a principal sequence of Feynman diagrams. In practice, such summation is reduced to a consideration of radiative corrections to the energies of particles involved in the process under investigation. This procedure leads to a finite width of a resonance [1, 4, 5, 10–16]. In Section 2, we consider resonant scattering of an electron by an electron in the field of a light wave. Within the framework of the Born approximation in electron interaction, we derive a general relativistic expression for the amplitude of e––e– scattering in the field of an elliptically polarized wave with an arbitrary intensity. It is demonstrated that this amplitude can be represented as a sum of partial amplitudes with emission and absorption of a definite number of photons of the wave by both electrons. We thoroughly study the poles of the scattering amplitude related to the Green function of an intermediate photon that falls within the LASER PHYSICS Vol. 6 No. 5 1996 RESONANT EFFECTS IN COLLISIONS OF RELATIVISTIC ELECTRONS mass shell. We demonstrate that, under resonant conditions, the Bunkin–Fedorov quantum parameter γ has no influence on the scattering process. Multiphoton emission and absorption of wave photons by electrons is governed by a classical relativistic-invariant parameter η. Under these conditions, the process of resonant electron–electron scattering can be effectively divided into two processes similar to Compton scattering of a wave by an electron: the first electron absorbs a certain number of photons from the wave and emits a real intermediate photon, which is absorbed by another electron with emission of a definite number of photons of the wave. For wave intensities η Ⰶ 1, the resonances are shown to occur, in the frame of reference related to the center of inertia, in forward and backward scattering of electrons by small angles θ ~ ω/|p| Ⰶ 1. The wave vectors of photons involved in this process (photons of the wave and intermediate photons) are directed along the generatrices of a cone whose axis is directed along the initial relative momentum of electrons. For this case, we derive a relativistic expression for the resonant differential cross section of electron scattering within a solid angle dΩ for an elliptically polarized wave. A resonant singularity in this expression is eliminated by the imaginary part of the electron energy, which is determined by the total probability (per unit time) of the Compton scattering of a light wave by an electron. It is demonstrated that the ratio of the resonant cross section dσres /dΩ to the Möller cross section dσMöl /dΩ (in the absence of the external field) is of the same order of magnitude as Rres ~ α–2(m/E)4. It can be seen from this relation that the ratio of the resonant cross section to the Möller cross section of free electrons reaches its maximum in the range of nonrelativistic electrons, where the resonant cross section may be up to four orders of magnitude greater than the Möller cross section. As the energy grows, the ratio of these cross sections decreases. For ultrarelativistic electrons, this ratio tends to zero. In Section 3 of this review, we consider SB for a relativistic electron scattered by a nucleus in a light field. Within the framework of the Born approximation in the interaction of an electron with the field of the nucleus, we derive a general relativistic expression for the amplitude of SB due to the scattering of an electron by a nucleus in the field of an elliptically polarized wave with an arbitrary intensity. We study in detail the poles of the SB amplitude related to the Green function of an electron that falls within the mass shell in a plane-wave field. It is demonstrated that, under resonant conditions, the process of resonant electron–nucleus SB can be effectively divided into two processes of the first order in the fine-structure constant: Compton scattering of the light wave by an electron (where the multiplicity of the multiphoton process is determined by the classical parameter η) and scattering of a real intermediate electron (positron) by a nucleus in the field of the wave (where the multiplicity of the multiphoton process is determined by the quantum parameter γ). Note that resLASER PHYSICS Vol. 6 No. 5 1996 839 onances occur only when photons (of the spontaneous and external field) propagate nonparallel to each other. In the sequel, we restricted our consideration of resonances to the range of intensities where η Ⰶ 1. It is demonstrated that, within a sufficiently broad range of electron energies and scattering angles (excluding an ultrarelativistic electron with an energy E ~ m2 /ω that moves within a narrow cone related to a spontaneously emitted photon), the resonant frequency ω' of the spontaneously emitted photon is multiple of frequencies ωi,f (3.19) for the direct and exchange amplitudes, respectively, ω' = rωi, f , where r = 1, 2, 3, … corresponds to absorption of 1, 2, 3, … photons from the external field. Under these conditions, a process that occurs with absorption of a single photon from the wave is characterized by the maximum probability. In such a process, an electron spontaneously emits a photon due to absorption of a single photon from the external field. However, the scattering of a real intermediate electron by a nucleus in a field with η Ⰶ 1 is generally associated with multiphoton stimulated bremsstrahlung. Note that the frequency ωi, f has four characteristic ranges of values: in the nonrelativistic situation, the resonant frequency is ω' = ω; in the ultrarelativistic limit of energies, when an electron moves within a narrow cone related to a photon of the external field, ω' Ⰶ ω, and ω' Ⰷ ω when an electron moves within a narrow cone with a spontaneous photon; otherwise, ω' ~ ω. Resonant amplitudes (direct and exchange ones) interfere with each other when an electron is scattered by a small angle, θ ~ ω/|p| Ⰶ 1. We derive a general relativistic expression for resonant electron–nucleus SB when the emission angle of the spontaneous photon is detected simultaneously with the ejection angle of an electron scattered by a large angle (in the absence of the interference between the direct and exchange amplitudes). A resonant singularity in this expression is eliminated by the imaginary part of the mass of the intermediate electron, which is determined by the total probability (per unit time) of the Compton scattering of a light wave by the intermediate electron. The derived resonant cross section was successively considered within four characteristic spectral ranges. The performed analysis allowed us to derive simple expressions for the resonant cross section in units of the conventional (in the absence of the external field) electron–nucleus SB cross section. The derived expressions provide the following estimates: Rres ~ π2α–1v –3 ~ 103v –3 for nonrelativistic electrons (v Ⰶ 1 and ω' = ω), Rres ~ π2α–1(m/|p|)2 for relativistic electron energies (ω' ~ ω), Rres ~ π2α–1(m/E)2 for ultrarelativistic electron energies when an electron moves within a narrow cone related to a photon of the wave (ω' Ⰶ ω), and Rres ~ πα–1 ln–1(E/m) for ultrarelativistic electron energies when an electron moves within a narrow cone with a spontaneous photon (ω' Ⰷ ω). Hence, the ratio of the resonant electron–nucleus SB cross section in the field of a light wave to the conventional SB cross section (in the absence of the external field) reaches its maximum in the range of nonrelativistic 840 ROSHCHUPKIN electron energies, where the resonant electron–nucleus SB cross section may be up to seven orders of magnitude greater than the conventional SB cross section. In Section 4, we consider resonant photocreation of electron–positron pairs on a nucleus in a light field. We derive a general relativistic expression for the amplitude of this process in the field of an elliptically polarized wave with an arbitrary intensity. We analyze resonances related to the Green function of an intermediate electron (positron) that falls within the mass shell. It is demonstrated that, in the absence of the interference between the direct and exchange amplitudes, resonant photocreation of electron–positron pairs on a nucleus in the field of a light wave can be effectively reduced to two sequential processes of lower orders in the finestructure constant: creation of an electron–positron pair by an incident photon in the wave field and scattering of a real intermediate electron (positron) by a nucleus in the field of the wave. In the sequel, we restricted our consideration to the range of intensities where η Ⰶ 1. For this range of intensities, we demonstrated that resonances may occur only in the ultrarelativistic situation, when the energy of an incident γ-quantum is higher than a certain threshold energy, ω' ≥ ω *' ~ m2 /ω, and when at least one of the particles in this pair falls within a narrow cone related to the incident γ-quantum. The interference of resonant amplitudes (direct and exchange ones) occurs when both particles in the pair move within a narrow cone with the incident γ-quantum. Near the threshold, the energies of the emerging ultrarelativistic electron and positron are close to each other (E– = E+ = ω *' /2). Far from the threshold, these energies considerably differ from each other. We derive the expression for the resonant cross section of photocreation of electron–positron pairs on a nucleus in the field of a light wave with an intensity η Ⰶ 1 in the absence of the interference between the direct and exchange amplitudes. It is demonstrated that the resonant cross section is mainly determined by the terms implying that the creation of a pair by an incident γ-quantum occurs through absorption of one photon from the external field and the scattering of an intermediate particle by a nucleus is, generally, of multiphoton nature. Within the framework of the logarithmic approximation, we demonstrate that the resonant cross section of pair photocreation on a nucleus when one of the particles in the pair is scattered by a large angle with respect to the momentum of the incident γ-quantum may be an order of magnitude greater than the conventional cross section of pair photocreation on a nucleus (in the absence of the external field). In Conclusion, we summarize the main results concerning resonant processes that occur in the field of a light wave, including electron–electron scattering, electron–nucleus spontaneous bremsstrahlung, and photocreation of electron–positron pairs on a nucleus. 2. RESONANT ELECTRON–ELECTRON SCATTERING IN THE FIELD OF AN ELLIPTICALLY POLARIZED LIGHT WAVE Resonant scattering of an electron by an electron in the field of a linearly polarized wave was first considered by Oleinik [1, 2], who demonstrated the fundamental possibility of such processes and estimated the resonant cross section for such a process at a single point within the nonrelativistic range. Bös et al. [5, 6] applied numerical methods to simulate specific resonant cross sections. Because of technical difficulties, these simulations were performed only for nonrelativistic electrons within the range of intensities of the external field η2 ⱗ 1. Roshchupkin et al. [10, 14, 16, 17] carried out a systematical study of resonances in the general relativistic situation and derived an expression for the resonant cross section in the field of an elliptically polarized wave within the range of intensities η Ⰶ 1. Note that, under resonant conditions, electrons involved in the scattering process exchange a real rather than a virtual photon. As a result, electron–electron scattering becomes a cascade process, i.e., it can be effectively divided into two processes of the first order in the fine-structure constant: one of the electrons absorbs a certain number of photons from the external field and emits a real intermediate photon, which is absorbed by another electron with emission of a definite number of photons of the wave (two processes similar to the Compton scattering of a light wave by an electron). Therefore, under resonant conditions, the quantum parameter γ has no influence on electron– electron scattering, and nonlinear effects are governed by the classical parameter η. Kazantsev et al. [27] considered the interaction of classical electrons in the field of a circularly polarized electromagnetic wave. Zavtrak et al. [28] investigated e––e– scattering in the field of a plane wave in the limiting case of high frequencies. The authors of these studies demonstrated that, in principle, the potential of electron interaction in the field of a plane wave may reverse its sign. 2.1. The Amplitude of Electron–Electron Scattering in the Field of a Plane Light Wave Let us choose the four-potential of the external elliptically polarized light wave in the following form: A(ϕ) = ( F ⁄ ω )(e 1 cos ϕ + δe 2 sin ϕ), ϕ = kx = ωt – kx. (2.1) Here, δ is the ellipticity parameter of the wave (δ = 0 corresponds to the linear polarization, and δ = ±1 corresponds to the circular polarization), e1,2 = (0, e1,2), and k = ωn = ω(1, n) are the polarization four-vectors of the 2 photon momentum in the external field (n2 = 0, e 1, 2 = –1, LASER PHYSICS Vol. 6 No. 5 1996 RESONANT EFFECTS IN COLLISIONS OF RELATIVISTIC ELECTRONS and e1,2k = 0). The amplitude of electron–electron scattering with an exchange of a virtual photon in the field of wave (2.1) is defined by the following expression (the Feynman diagrams of the process are shown in Fig. 1): 2 ∫ 4 4 S = ie d x 1 d x2 G µν( x 1 – x 2) µ p2' p1' k1 p2 p1' (2.2) k2 ν ( 2', 1' ) } . µ exp ( – iφ p ) ψ p( x A) = -----------------------D r( p)exp { – i( p̃ – rk)x }u p ˜ 2E r = –∞ (2.3) for amplitude in the following form: 2(2π) exp ( iφ ) S = ie ----------------------------------Ẽ 1 Ẽ 2 Ẽ 1' Ẽ 2' 5 ∑ Here, up is the Dirac bispinor, and the phase φp and the four-quasimomenta p̃ = ( Ẽ , p̃ ) [20] are written as eF(e 2 p) - , κ p = np = E – np; φ p = δ -----------------2 ω κp 2 ∞ (2.5) ∞ ∑ ∑ 1/2 (4) × δ [ p˜ 1' + p˜ 2' – p˜ 1 – p˜ 2 + ( r + r' ) k ] , φ = φ 1' + φ 2' – φ 1 – φ 2 , ∞ µ G1 = s 1' µ (2.11) D r + s( p 1), (2.12) ∞ G 2µ = The Fourier coefficients in (2.3) are given by ∑ D ( p )γ s = –∞ (2.6) ∑ D ( p )γ s 2' µ D r' + s( p 2). s = –∞ π 1 D r( p) = ------ F p(ϕ) exp ( – irϕ ) dϕ, 2π (2.7) en̂ Â(ϕ) F p(ϕ) = 1 + ----------------- exp { iS p' (ϕ) }, 2κ p (2.8) ∫ ( 2', 1' ) ] -– [ ( 1', 2' ) 2 m. (2.10) where Note that p̃ = m * , where m* is the effective mass of an electron in the field of a plane wave, 2 2 1 m * = 1 + ---(1 + δ )η 2 µ ( u 1' G 1 u 1 ) ( u 2' G 2µ u 2 ) × --------------------------------------------------------------------------2 [ p̃ 1' – p̃ 1 + p̃ 2 – p̃ 2' + ( r – r' )k ] r = – ∞ r' = – ∞ (2.4) 2 + ± 2 2 m p̃ = p + 2β p k, β p = ( 1 ± δ )η ------------- . 8ωκ p p1 Fig. 1. Scattering of an electron by an electron in the field of a plane electromagnetic wave. The solid lines correspond to the wave functions of an electron in the field of the wave (the Volkov functions), and the inner line shows the Green function of a free photon. (a) Direct diagram and (b) exchange diagram. ∞ 2 (b) p2 Here, γ̃ (µ = 0, 1, 2, 3) are the Dirac matrices; Gµν is the Green function of a free electron; pj (Ej, pj) are the four-momenta of electrons before (j = 1, 2) and after ( j = 1', 2') scattering; and ψp(x|A) are the wave functions of electrons in the field of a plane wave (Volkov functions) [24], which can be conveniently expanded as Fourier series, p1 p2' (a) × { [ ψ 1'( x 1 A)γ̃ ψ 1( x 1 A) ] [ ψ 2'( x 2 A)γ̃ ψ 2( x 2 A) ] } – { ( 1', 2' ) 841 –π Introducing the total number of photons l = r + r' emitted and absorbed by electrons in the process of scattering and performing summation in (2.12), we can write the scattering amplitude in the final form, ∞ S = ∑S (l) , (2.13) l = –∞ eF – - [ ( e 1 p ) sin ϕ + δ ( e 2 p ) cos ϕ ] – β p sin 2ϕ. S p' (ϕ) = ----------2 ω κp (2.9) where the partial amplitude corresponding to emission (l > 0) and absorption (l < 0) of l photons of the wave by both electrons is As usually [24], hats above notations in (2.8) stand for µ µ scalar products, n̂ = nµ γ̃ and  = Aµ γ̃ . Substituting (2.3) into (2.2) and integrating over the four-coordinates of electrons, we can write the soughtLASER PHYSICS Vol. 6 No. 5 1996 S (l) 2 ( 2π ) exp ( iφ ) = ie --------------------------------2 Ẽ 1 Ẽ 2 Ẽ 1' Ẽ 2' (1) (4) 5 × M δ ( p̃ 1' + p̃ 2' – p̃ 1 – p̃ 2 + lk). (2.14) 842 ROSHCHUPKIN Here, we introduced the notations M (l) ∞ = µ ( u 1' G 1 u 1 )(u 2' G 2µ u 2) ----------------------------------------------2 k 1 r = –∞ ∑ (2.15) ( 2', 1' ) ] , --- – [ ( 1', 2' ) k 1 = p̃ 1 – p̃ 1' – rk = p̃ 2' – p̃ 2 + ( l – r )k, µ G1 µ = a Lr + µ b– Lr – 1 µ b+ Lr + 1 + eral relativistic case, the amplitude of electron scattering in the field of a wave with an arbitrary intensity is expressed in terms of functions Lr (2.21), which determine the probability of emission and absorption of photons from the wave. The functions Lr considerably depend on the polarization of the wave. Expanding exponentials as series in integer-order Bessel functions Js and performing simple algebraic transformations, we can write the functions Lr in the following form: (2.16) L r(χ 1, γ 1, β 1 ) – µ + c ( L r + 2 + L r – 2), (2.17) ∞ = exp ( – irχ 1 ) 2 ε̂ ± = ê 1 ± iδê 2 , (2.19) 2 µ 2 2 m µ ˆ c = – ( 1 – δ )η --------------n n. 8κ 1 κ 2 (2.20) L r(χ 1, γ 1, 0) = exp ( – irχ 1 )J r(γ 1). L r ≡ L r(χ 1, γ 1, β 1 ) – (2.21) π 1 – = ------ exp { i [ γ 1 sin ( ϕ – χ 1 ) + β 1 sin 2ϕ – rϕ ] } dϕ. 2π ∫ –π in (2.21), which depend on Parameters χ1, γ1, and the electron momenta and parameters of the wave, are defined as tan χ 1 = δ ( e 2 g 1 ) ⁄ ( e 1 g 1 ), g 1 = p 1' ⁄ κ 1' – p 1 ⁄ κ 1 , (2.22) 2 m 2 2 (2.23) γ 1 = η ---- ( e 1 g 1 ) + δ (e 2 g 1) , ω 2 ± ± 2 2m ± 1 1 β 1 = β p1' – β p1 = ( 1 ± δ )η ------- ------ – ----- . 8ω κ 1' κ 1 ∞ J r(γ 1, β 1 ) ≡ L r(0, γ 1, β 1 ) = ⊥ g 1 + g 1 , we can represent the quantum parameter γ (2.23) in the following form: cos τ 1 + δ sin τ 1 , τ 1 = 2 2 – ∑J – r – 2s(γ 1)J s(β 1 ). s = –∞ (2.28) The derived expression for the amplitude of electron– electron scattering in the field of an elliptically polarized electromagnetic wave with an arbitrary intensity is rather cumbersome, which is due to the fact that the scattering amplitude involves both resonant and nonresonant parts. In what follows, we will consider resonant scattering. (2.24) Taking into account that zeroth components of the polarization four-vectors of the light wave are equal to zero and expanding the vector g1 into components parallel and perpendicular to the polarization plane, g1 = 2 (2.27) For a linearly polarized wave (δ = 0), the phase is χ1 = 0 [see (2.22)], and the functions Lr (2.26) are expressed in terms of the generalized Bessel functions [29], – – β1 m || γ 1 = η ---- g 1 ω (2.26) For a circularly polarized wave (δ2 = 1) and for an elliptically polarized wave considered in the dipole approximation in the interaction of electrons with the electric – field of the wave, the quantum parameter is β 1 = 0 [see (2.24)], and expression (2.26) for the functions Lr can be considerably simplified, The functions Lr in (2.17) are written as || – r – 2s(γ 1)J s(β 1 ). (2.18) µ µ γ̃ n̂ε ε̂ ± n̂γ̃ 1 = --- ηm ------------+ -------------± , 4 κ 1' κ1 1 s = –∞ µ µ 2 2 m µ a = γ + ( 1 + δ )η --------------n n̂, 4κ 1 κ 2 µ b± ∑ exp ( 2isχ )J || ∠(e 1, g 1 ). (2.25) 2.2. Poles of the e––e– Scattering Amplitude The scattering amplitude described by (2.14) and (2.15) displays poles when a virtual intermediate photon becomes a real one. This implies that the fourmomenta squared of intermediate photons vanish either for the direct or exchange scattering amplitudes, 2 2 2 2 k 1 = ( Ẽ 1' – Ẽ 1 + rω ) – ( p̃ 1' – p̃ 1 + rk ) = 0, (2.29a) 2 k 2 = ( Ẽ 2' – Ẽ 1 + rω ) – ( p̃ 2' – p̃ 1 + rk ) = 0. (2.29b) 2 µ The operator G2µ can be obtained from the operator G 1 [see (2.17–(2.24)] after the following replacements: p1 p2, p1' p2', γ µ γµ, and r (l – r). It can be seen from expressions (2.14)–(2.17) that, in the gen- For given parameters of the wave and electron energies, resonant conditions (2.29a) and (2.29b) allow us to determine resonant scattering angles. Let us analyze first the conditions for the appearance of resonances in LASER PHYSICS Vol. 6 No. 5 1996 RESONANT EFFECTS IN COLLISIONS OF RELATIVISTIC ELECTRONS the direct scattering amplitude (in this case, the inner line in Fig. 1, which corresponds to an intermediate photon is cut, see Fig. 2). Using expression (2.16) for the four-momentum of an intermediate photon k1, we can write the laws of energy–momentum conservation in resonant scattering in the form of two equalities for the first and second electrons, 2 p̃ 1 – rk = p̃ 1' + k 1 , (2.30) p̃ 2 + k 1 = p̃ 2' + r'k. (2.31) 2 2 2 E 1 – np 1 -, ω 1 = r ω ----------------------------------------------------------( n 1 p̃ 1 ) + r ω ( 1 – cos θ 1 ) (2.32) E 2 – np 2 -. ω 1 = r'ω -----------------------------------------------------( n 1 p̃ 2 ) – r'ω(1 – cos ω) (2.33) Here, k1 = ω1n1 = w1(1, n1) and θ1 is the angle between the momenta of the intermediate photon and the photon of the external field [see (2.49)]. Note that, in the frame of reference where, on the average, an electron remains at rest ( p̃ 1 = 0 and Ẽ 1 = m * ), expression (2.32) yields the well-known formula for the Compton effect if we replace four-momenta of the electron by four-quasimomenta and substitute a four-vector |r|k for the fourmomentum of the incident photon, rω ω 1 = r ω 1 + --------- ( 1 – cos θ 1 ) m* –1 . (2.34) Therefore, the frequencies ω1 of the intermediate photon can be considered as harmonics of the frequency of the external field. Let us express parameters χ1, γ1, and β 1 (2.22)– (2.24), which determine the functions Lr (2.21) through the parameters of the intermediate photon. Employing expressions (2.30)–(2.33), we can easily derive the following relationships: – r ω(nn 1) 1 1 1 1 r'ω(nn 1) - , ------ = ----- – ------------------------ = ----- + --------------------- . (2.35) κ 1 κ 1(n p̃ 1) κ 2' κ 1' κ 2 κ 2 ( n p̃ 2 ) Taking these expressions into account and using (2.16) to eliminate the four-quasimomentum p̃ 1' , we finally Vol. 6 No. 5 p1' k12 = 0 p2 p1 Fig. 2. Resonant electron–electron scattering in the field of a plane electromagnetic wave. The solid lines correspond to the wave functions of an electron in the field of the wave (the Volkov functions), and the dotted lines represent a real intermediate photon. 2 2 Since p̃ 1 = p̃ 1' = p̃ 2 = p̃ 2' = m * and k2 = k 1 = 0, equalities (2.30) and (2.31) are satisfied only for r ≤ –1 and r' ≥ 1. Hence, resonant electron–electron scattering in the field of a light wave can be effectively reduced to two processes similar to Compton scattering of a wave by an electron: the first electron absorbs |r| photons from the wave and emits a real photon k1; next, the second electron absorbs the intermediate photon k1 and emits r' photons of the wave. Using expressions (2.16), (2.30), and (2.31), we can easily find the frequency of the intermediate photon emitted by the first electron and absorbed by the second electron, LASER PHYSICS p2' 843 1996 arrive at the following expressions for the sought-for parameters: tan χ 1 = δ(e 2 ζ 1) ⁄ ( e 1 ζ 1 ) , ζ 1 = n 1 – ( nn 1 ) p 1 ⁄ κ 1 , (2.36) m 2 2 2 γ 1 = r η ------------- ( e 1 ζ 1 ) + δ ( e 2 ζ 1 ) , ( n p̃ 1 ) (2.37) 2 – 2 2 m (nn 1) β 1 = ( 1 – δ ) r η ----------------------. 8κ 1(n 1 p̃ 1) (2.38) Note that, in deriving parameters χ2, γ2, and β 2 , which determine the functions Ll – r in the operator G2µ, we should make the following replacements in appropriate expressions (2.36)–(2.38): p1 p2 and |r| r'. As one might expect [16, 20], the derived expressions – (2.36)–(2.38) indicate that parameters γ1 and β 1 become classical (the Planck constant in the denominator is canceled). These parameters are governed by a classical relativistic-invariant parameter η. In other words, the quantum parameter of the multiplicity of a multiphoton process (the Bunkin–Fedorov parameter) has no influence on resonant electron–electron scattering. For η Ⰶ 1, a process with r' = |r| = 1 and l = r + r' = 0 has the maximum probability (the domain where we can apply the perturbation theory with respect to the external field), whereas for η ⲏ 1, multiphoton processes become important. Note also that, if the direction of motion of the intermediate photon coincides with the direction of motion of the photon from the external – field (nn1 = 1 – cosθ1 = 0), then we have γ1 = β 1 = 0 [see (2.36)–(2.38)] and resonances vanish. Therefore, in what follows, we assume that nn1 ≠ 0. – Now, let us analyze the conditions when resonances appear. Equating expressions (2.32) and (2.33), we derive an expression that allows us to determine the direction of emission of an intermediate photon for given initial electron energies and fixed parameters of the wave, r κ1 r'κ 2 ------------------------------------------- = -----------------------------------------. ( n 1 p̃ 1 ) + r ω(nn 1) ( n 1 p̃ 2 ) – r'ω(nn 1) (2.39) 844 ROSHCHUPKIN Taking (2.35) into account, we can represent the resonant condition (2.29a), which determines resonant scattering angles, in the following form: ( E 1' – E 1 – r * ω ) = ( p 1' – p 1 – r * k ) . 2 2 (2.40) Here, we use the notation 2 m (nn 1) r * = r 1 – ( 1 + δ )η ----------------------. 4κ 1(n 1 p̃ 1) 2 2 (2.41) To perform subsequent analysis, we will use the frame of reference related to the center of inertia of electrons, p 1, 2 = ( E i, ± p i ); p 1', 2' 1 = --- P f ± p f , 2 (2.42) E 1' + E 2' – 2E i + l * ω = 0, P f + l * k = 0 , 2 2 Ef − + l *(kp f ) + ( l * ω ⁄ 2 ) , (2.43) (2.44) 1 2 2 2 r r' l * = l + --- ( 1 + δ )η (nn 1)m -------------------- – -------------------- , 4 κ 1(n 1 p̃ 1) κ 2(n 1 p̃ 2) (2.45) κ 1 = E i – np i ; κ 2 = E i + np i . (2.46) Using the derived expressions, we can rewrite the resonant conditions (2.39) and (2.40) in the frame of reference related to the center of inertia of electrons. Performing simple calculations, we find r (1 – v i cos θ i)(1 + v i cos θ 1i) – r' ( 1 + v i cos θ i )(1 – v i cos θ 1i) = 2(1 – cos θ 1)[r' r + r'β 1 (1 + v i cos θ i) + – r + β 2 (1 (2.47) ω – v i cos θ i)] ----- , Ei ( E 1' – E i – r * ω ) = [ p f – p i – ( r * + l * ⁄ 2 )k ] . (2.48) 2 (2.51) r' = r = 1, l = r + r' = 0. Then, the resonant condition (2.47) can be written as θ i + θ 1i θ i – θ 1i - sin ---------------sin --------------- 2 2 (2.52) cos θ i m 2θ 2 2 ω - ----- . = sin -----1 ---- + ( 1 + δ )η ------------------------------------2 2 E 2 pi 2(1 – v i cos θ i) i 2 where Ei,f and pi,f are the relative energies and momenta of electrons before and after scattering. Thus, the laws of energy and momentum conservation [see the argument of the delta function in (2.14)] are written as E 1', 2' = This inequality implies that we can apply the perturbation theory with respect to the external field [because – r γ1 Ⰶ 1, β 1 Ⰶ γ1, and Lr ~ γ 1 , see (2.37), (2.38), and (2.26)]. Under these conditions, the maximum probability is reached for a process where the first electron absorbs and the second electron emits one photon of the wave, 2 Here, vi, f = pi, f /Ei, f (pi, f = |pi, f |); E1' and β 1, 2 are defined by expressions (2.44) and (2.24) with allowance for (2.46), respectively; and the angles θi, θ1i, and θ1 are given by + θ i, f = ∠(k, p i, f ), θ 1i = ∠(k 1, p i), θ 1 = ∠(k, k 1). (2.49) It is rather difficult to analyze resonant conditions (2.47) and (2.48) for an arbitrary intensity of the external field. Therefore, we will consider a situation when the wave intensity is such that η Ⰶ 1. (2.50) Since the expression in square brackets is much less than unity, we find that, under resonant conditions, the angle between the initial relative momentum of electrons pi and the wave vector k1 of the intermediate photon is close to the angle between the initial relative momentum of electrons and the wave vector k of photons from the external field (|θ1i – θi | Ⰶ 1). In other words, the wave vectors of intermediate photons and photons from the external field are directed along the generatrices of a cone whose axis is directed along the initial relative momentum of electrons, and the angle between the momenta of photons is θ1 ~ 1. Taking this circumstance into account, we find from (2.32) that, in this approximation, the frequency of the intermediate photon coincides with the frequency of the photon from the wave of the external field (ω1 = ω). To determine the scattering angles of electrons in the frame of reference related to the center of inertia under conditions (2.50), we can employ relationship (2.48). In the case under consideration, this relationship can be written as ( p f – pi – k ) = ω . 2 2 (2.53) We can easily demonstrate that, within the range of optical frequencies, this equality can be satisfied only when electrons are scattered by small angles in the frame of reference related to the center of inertia, θ = θ res ≡ 2 sin ( θ 1 ⁄ 2 )(ω ⁄ p i) Ⰶ 1, θ = ∠(p i, p f ). (2.54) Note that the exchange amplitude features a resonance 2 ( k 2 = 0) corresponding to backward scattering of electrons in the frame of reference related to the center of inertia, π – θ = θ res ∼ ω ⁄ p i Ⰶ 1 . (2.55) Consequently, when the intensity of the external wave meets condition (2.50), resonances of the direct and exchange amplitudes lie in different kinematic LASER PHYSICS Vol. 6 No. 5 1996 RESONANT EFFECTS IN COLLISIONS OF RELATIVISTIC ELECTRONS domains. In the following section, we will calculate the resonant cross section for the direct amplitude. 2.3. Resonant Differential Cross Section Let us determine the resonant cross section of electron scattering in the case when the direct amplitude displays a resonance (see Fig. 2), i.e., when an electron is scattered by a small angle θ = θres ~ ω/pi Ⰶ 1 in the frame of reference related to the center of inertia. In this case, we can keep only the first term in the amplitude described by (2.14) and (2.15). Then, taking into ± account that we can neglect parameters β 1, 2 (2.38) in (2.17) and (2.26), we can represent the functions Lr (2.26) in the form (2.27). Expanding the Bessel functions as power series in γ1,2 Ⰶ 1 and taking into consideration that |r| = 1 and l = 0, we finally arrive at the followµ ing expressions for the operators G 1 and G2µ in (2.17): γ µ m µ µ µ G 1 = – -----1 exp ( iγ 1 )γ̃ + η -------- ( n ε̂ + – ε + n̂ ) , (2.56) 2 2κ 1 γ m G 2µ = -----2 exp ( – iγ 2 )γ̃ µ + η -------- ( n µ ε̂ – – ε –µ n̂ ) . (2.57) 2 2κ 2 To eliminate a resonant singularity in the field of a wave with intensity that meets condition (2.50), we can apply the Breit–Wigner procedure. In the field of a wave, electron energies become complex [4–6, 9, 10, 16], Ei – iΓi; Ef Ef + iΓf , (2.58) Ei where the widths Γi, f are determined by the total probabilities (per unit time) of the Compton scattering of a photon from the external field by electrons with relative momenta pi, f, respectively, 1 2 2 Γ i, f = --- ( 1 + δ )αη ( 1 – v i, f cos θ i, f )ω . 6 (2.59) Performing transformation (2.58), we can rewrite expression (2.29a) in the following form: k 1 = ( E f – Ei – ω ) – ( p f – pi – k ) 2 2 2 (2.60) p i [ θ(θ – θ res) + iR 0 ] , 2 where ω 2 2 2 R 0 = --- ( 1 + δ )αη ( 1 – v i cos θ i ) ------- pi 3 2 2 ω ∼ αη ------- Ⰶ 1. pi 2 (2.61) Setting Ẽ 1' = Ẽ 1 = Ẽ 2' = Ẽ 2 = Ei, we can use expressions (2.14) and (2.15) for the amplitude with allowance for (2.56), (2.57), and (2.60) to find the resonant differential cross section appropriately averaged and summed in accordance with a standard technique [24]. LASER PHYSICS Vol. 6 No. 5 1996 845 Performing the corresponding transformations, we finally derive the following expression for the resonant differential cross section of electron scattering into an elementary solid angle dΩ: ( ηm ⁄ 2 p i ) ( m ⁄ E i ) dσ res 2 -. ---------- = r e f 0 ----------------------------------------------2 2 2 dΩ θ (θ – θ ) + R 4 res 2 (2.62) 0 Here, re is the classical electron radius, and the function f0 is determined by electron energies, scattering angles, and polarization of the wave. This function has a rather cumbersome form [16]. In the main kinematic domain, this function is on the order of unity. By analyzing (2.62), we can clearly see the resonant structure of the scattering cross section. For scattering angles θ – θ res Ⰶ R 0 ⁄ θ res ∼ αη ( ω ⁄ p i ) Ⰶ 1 , 2 (2.63) the cross section displays a sharp peak, dσres / r e ~ 0.1α–2(m / ω)4(m / Ei)2 ⱗ 1023. Under these conditions, the ratio of the resonant differential cross section (2.62) to the conventional (in the absence of the external field) Möller differential cross section [24] of scattering by small angles θres (2.54) is given by 2 4 dσ res –2 m R res = ------------ = f 1 α ----- sin ( θ 1 ⁄ 2 ) , Ei dσ Möl (2.64) where f1 ~ 1. Hence, we can see that the ratio of the resonant cross section to the conventional Möller cross section reaches its maximum in the range of nonrelativistic electron energies (when the intermediate photon is emitted in the direction opposite to the wave vector of the laser wave). In this case, the resonant cross section is four orders of magnitude greater than the Möller cross section. As the electron energy grows, the ratio of these cross sections decreases. For ultrarelativistic energies Ei / m ~ α ~ 10, the resonant cross section becomes of the same order of magnitude as the Möller cross section. Figures 3 and 4 illustrate the influence of the polarization of the laser wave on the resonant cross section (2.62) and show the dependence of the resonant scattering angle (2.54) on the relative electron velocity. As can be seen from Fig. 3, the polarization of the laser wave exerts the most considerable influence on the resonant cross section within the range of relativistic electron energies. Note that, for a linearly polarized wave, the resonant cross section may be several times greater than that for a circularly polarized wave (e.g., for Ei = 0.85 MeV, we have dσres(δ = 0)/ dσres(δ2 = 1) = 6.5). As can be seen from Fig. 4, within the range of nonrelativistic electron energies, the resonant scattering angle reaches its maximum and equals several hundredths of a degree. In this context, we should note that numerical simulations performed by Bös et al. [5] in the range of fields η2 < 10–2 did not reveal any increase in the resonant cross section away from resonances. The authors of [5] obtained such a result because, in simulations, –1 846 ROSHCHUPKIN ativistic electrons. Specifically, within the range of optical frequencies, for η = 0.1 and vi = 0.1, the reso- logRres 5 nant cross section can be estimated as σres ~ 1012 r e . 2 4 3 2 1 0 0.2 0.4 0.6 0.8 1.0 v Fig. 3. The ratio of the resonant differential cross section of e––e– scattering to the cross section of Möller scattering (in the absence of the external field) as a function of the relative velocity of electrons in the logarithmic scale (2.64) [16] for preset directions of the relative momentum (θi = 60° and ϕi = 95°) and the intermediate photon (θ1 = 106° and ϕ1 = 135°). The solid curve corresponds to a linearly polarized laser wave and the dashed curve corresponds to a circularly polarized laser wave. The laser frequency is ω = 4 eV and the strength of the laser field satisfies the condition F Ⰶ 3 × 1010 V/cm. Resonant angle × 10–2, deg 3 3. RESONANT BREMSSTRAHLUNG OF AN ELECTRON SCATTERED BY A NUCLEUS IN THE FIELD OF A LIGHT WAVE 2 1 0 We should take into account that, in experiments, a laser field may have a finite spectral width Γω , which may considerably influence the magnitude of the resonant cross section. Therefore, the Compton width should be at least no less than the spectral width, Γi > Γω . Hence, we find the condition that restricts the applicability of lasers with very short pulses, Γω /ω < αη2. In particular, for η = 10–1, we have Γω /ω < 10–4, i.e., the estimates of the resonant cross section are valid for nanosecond pulsed lasers (Γω /ω ~ 10–6). Note also that the results obtained by Bergou et al. [7] are incorrect in the resonant domain because the denominator of the scattering amplitude [formula (3.10) in this paper] does not involve the term (mបω)2. As a result, the denominator vanishes simultaneously with the numerator, and the poles disappear. As mentioned above, this incorrectness is associated with the use of a nonrelativistic approach to the problem, namely, with the use of the Coulomb potential, which cannot yield an additional term in the denominator of the scattering amplitude (this term can be obtained only when we consider the Green function of a free photon, i.e., within the framework of the general relativistic approach). 0.2 0.4 0.6 0.8 1.0 Electron velocity, v Fig. 4. Dependence of the resonant scattering angle on the relative velocity of electrons (2.54) for preset directions of the relative momentum (θi = 60° and ϕi = 95°) and the intermediate photon (θ1 = 106° and ϕ1 = 135°) in a laser field with frequency ω = 4 eV and strength F Ⰶ 3 × 1010 V/cm. they chose scattering angles θ ~ 0.1°. No resonances can occur within this range because θres ~ 0.01°. Let us estimate the contribution of a resonance to the integral cross section. Representing the elementary solid angle in the form dΩ = θresdθdϕ and performing relevant integration procedures in (2.62), we find that the integral cross section can be estimated as 3 2 m –1 2 2 σ res ∼ α η ----------------- r e . ω p i E i (2.65) Hence, we can see that, within the resonant range, the integral cross section reaches its maximum for nonrel- Let us consider spontaneous bremsstrahlung of an electron scattered by a nucleus in the external field of a light wave, when an electron decelerated by a nucleus not only absorbs or emits photons of the external field but also emits a single photon of an arbitrary frequency (see Fig. 5). The specific feature of this process is that it can occur under resonant conditions. This possibility is associated with the Green function of the intermediate electron that falls within the mass shell. A detailed analysis of electron–nucleus SB in the field of a light wave for arbitrary parameters η and γ and arbitrary electron energies is complicated by considerable computational difficulties and the necessity to investigate cumbersome expressions. Therefore, we analyzed this problem for various particular values of parameters η and γ. Note that Borisov and Zhukovskii [30] considered nonresonant electron–nucleus SB in a plane-wave field when an ultrarelativistic electron was scattered by small angles close to the direction of the momentum of a spontaneous photon for arbitrary values of the parameter (1.1) (the authors of this paper assumed that the transferred momentum satisfies the condition |q| ~ ω, and, consequently, the problem is, in fact, characterized by a single classical Bunkin– Fedorov parameter). From the viewpoint of intensities of electromagnetic radiation that can be achieved in LASER PHYSICS Vol. 6 No. 5 1996 RESONANT EFFECTS IN COLLISIONS OF RELATIVISTIC ELECTRONS modern laser sources in a stable mode [31], the range of parameters η and γ [see (1.1) and (1.2)] where η Ⰶ 1, γ ⲏ 1 3.1. The Amplitude of Electron–Nucleus SB in a Light Field The SB amplitude of an electron scattered by a nucleus Ze in the field of a plane wave (2.1) is given by the following expression (Feynman diagrams of the process are shown in Fig. 5): × 2 ∫ ∫ dx dx ψ (x 4 1 Cl { γ̃ 0 A 0 ( x 2)G( x 2 x 1 4 2 f 2 A) ˆ ( x 1, k') A) A' A 0 ( x) = Ze ⁄ x , Cl (3.4) LASER PHYSICS No. 5 1996 (b) pf Fig. 5. Spontaneous bremsstrahlung accompanying the scattering of an electron by a nucleus in the field of a plane electromagnetic wave. The inner lines show the Green function of an electron in the field of a plane wave, the incoming and outgoing solid lines correspond to the wave function of an electron in the field of the wave (the Volkov functions), and the dashed lines represent a spontaneous photon k' and a pseudophoton q of a Coulomb center. where e 2κ p ω ᑣ p(x) = 1 + ------------- k̂ Â(kx) exp [ iS(x) ], (3.6) kx e 2 e S( x) = – ( px ) – ---------- dϕ p A(ϕ) – --- A (ϕ) . 2 κ pω ∫ 0 Here, κp is defined by expression (2.4). Expanding the Volkov functions and the Green function of an electron in the plane-wave field as Fourier series and performing the relevant integration procedures, we derive an expression for the sought-for amplitude in the momentum representation, S fi = ∑S (l) fi , (3.7) 5/2 3 δ(q 0) 8π Ze (l) (l) - , (3.8) S fi = – i ------------------------ exp ( iφ fi ) [ u f M u i ] ----------3 q 2ω'Ẽ i Ẽ f ∞ q̂ i + m * ( r ) q̂ + m * ( r + l ) (r) f ----------------------------------- M2 + F . F i f 2 2 2 2 qi – m* q f – m* r = –∞ (3.9) Here, we use the notations (l) ∑ = (r + l) M1 0 (3.5) (r + l) M1 = a L r + l(χ 1, γ 1, β 1 ) + b – L r + l – 1 0 – 0 0 (3.10) + b + L r + L + 1 + c ( L r + l – 2 + L r + l + 2), (r) Vol. 6 q qf M 4 d p p̂ + m -2 ᑣ p( x 1), -------------4 ᑣ p( x 2) ----------------2 p –m ( 2π ) pi Ze l = –∞ (3.3) 2π * ------ε µ exp ( ik'x ) , ω' (a) where the partial amplitude with emission and absorption of l photons of the wave is written as where ε µ* and k' = ω'n' = ω'(1, n') are the polarization four-vector and the four-momentum of the spontaneous photon, respectively; pi and pf are the four-momenta of the electron before and after scattering, respectively; ψp(x|A) is the Volkov function [24] [see (2.3)]; and G(x2x1 |A) is the Green function of an electron in the field of the wave (2.1), which is written as [34–38] ∫ qi pf k' (3.2) Here, we introduced the notation G( x 2 x 1 A) = q ∞ ˆ ( x 2, k')G( x 2 x 1 A)γ̃ 0 A 0Cl( x 1) }ψ i( x 1 A). + A' ˆ = γ̃ µ A µ' ; A µ' ( x, k') = A' k' (3.1) is of greatest interest. Within this range, nonlinear multiphoton processes are characterized by the quantum parameter γ. Resonant spontaneous bremsstrahlung of a nonrelativistic electron scattered by a nucleus in the plane-wave field was studied by Lebedev [3]. Nonresonant nonrelativistic electron–nucleus SB was investigated by Karapetyan and Fedorov [32]. The authors of these studies employed the Born approximation with respect to the interaction of a nonrelativistic electron with a nucleus. In other words, sufficiently fast electrons with Ze2 Ⰶ v Ⰶ 1 were taken into consideration. Paper [33] was devoted to the opposite limiting case, when the electron–nucleus interaction can be considered in the quasi-classical approximation and the interaction of an electron with the wave field can be taken into account in the first order of the perturbation theory. Borisov et al. [4] considered resonant SB that accompanies collisions of ultrarelativistic electrons within the range defined by (3.1) for large transferred momenta. In [12, 15, 17], I systematically studied resonances in the range (3.1). S fi = – ie Ze 847 Fi – = ( aε∗ )L –r(χ i, γ i, β i ) + ( b – ε∗ )L – r – 1 + ( b + ε∗ )L – r + l + ( cε∗ )( L – r – 2 + L – r + 2). (3.11) 848 ROSHCHUPKIN In these expressions, matrices a, b±, and c are defined by relationships (2.18)–(2.20); the functions Lr + l and L–r are given by expressions (2.21) and (2.26); and their arguments χ1, γ1, and β 1 and χi, γi, and β i are determined in accordance with (2.22)–(2.24). To find (r + l) amplitudes M 1 , we should perform the replacements p1 qi and p1' pf in expressions (2.18)–(2.24). – – (r) To determine F i , we should make the replacements pi and p1' qi in the same expressions. In the p1 above-derived formulas, m is the effective mass of an * electron in the field of a plane wave (2.6), and the fourmomenta qi and qf and the transferred four-momentum q = (q0, q) are given by q i = p̃ i – k' + rk, q f = p̃ f + k' – rk (3.12) q = p̃ f – p̃ i + k' – lk , (3.13) and respectively. In the above expressions, the four-quasimomenta p̃ i, f are defined by (2.5). Formula (3.8) involves the phase φf,i = φf – φi, where the phases φf,i are given by (r + l) (r) (2.4). To determine the amplitudes M 2 and F f , we qi , qi pi and should make replacements pf qi pf , p i qf in expressions (3.10) and (3.11), respectively. 3.2. Poles of the SB Amplitude As mentioned above, the resonant behavior of the amplitude described by (3.8) and (3.9) is due to a quasidiscrete structure of the system consisting of an electron and a plane electromagnetic wave. As a result, by virtue of the energy–momentum conservation in elementary processes involved in the considered phenomenon, the four-momentum of an intermediate electron lies on the mass surface. Under these conditions, the following equalities are satisfied for the first or second term in (3.9) (see also Figs. 5 and 6): q j = m * , j = i, f . (3.14) It is convenient to write expressions (3.12) and (3.13), 2 2 Ze k' q qi2 = m2* pf pi Fig. 6. Resonant spontaneous bremsstrahlung related to the scattering of an electron by a nucleus in the field of a plane electromagnetic wave. The inner line of the diagram in Fig. 5a is cut up, which corresponds to a real intermediate electron. which specify qi,f and q in a resonance, in terms of the amplitudes of the processes shown in Figs. 5a and 5b, p̃ i + rk = q i + k', q = p̃ f – q i + ( r + l )k (3.15a) and q f + rk = p̃ f + k', q = q f – p̃ f + ( r + l )k . (3.15b) 2 2 2 2 2 2 Since p̃ i = q i = m * or p̃ f = q f = m * , and k2 = k'2 = 0, equalities (3.15a) and (3.15b) can be satisfied only when r ≥ 1. Hence, taking into account expressions (3.10) and (3.11) for the amplitudes, we infer (see also Fig. 6) that, with allowance for the four-momen(r) tum conservation [see the first equality in (3.15a)], F i represents the amplitude of the process where an electron with a four-momentum pi absorbs r photons from the wave and emits a photon with a four-momentum k'. Such a process was considered by Nikishov, Ritus, et al. (see review [20]). With allowance for the transferred four-momentum q [see the second equality in (3.15a)], (r + l) the quantity M 1 is the amplitude of scattering of an intermediate electron with a four-momentum qi by a nucleus in the field of a light wave with absorption or emission of |r + l| photons of the wave. In the nonrelativistic limiting case, this process was studied by Bunkin and Fedorov [39]. Denisov and Fedorov [40] investigated this process in the general relativistic situation. Similarly to the above-considered case, for the diagram shown in Fig. 5b, with allowance for the sec(r + l) ond equality in (3.15b), M 2 is the amplitude of scattering of an electron with a four-momentum pf by a nucleus in the wave field. Simultaneously, with allow(r) ance for the first term in (3.15b), F f is the amplitude of the process where an electron with a four-momentum qf emits a photon with a four-momentum k' in the wave field. Consequently, in the absence of the interference between the direct and exchange amplitudes, the process of resonant electron–nucleus SB in the field of a light wave can be effectively reduced to two sequential processes of the first order in the fine-structure constant: emission of a photon with a four-momentum k' by an electron in a light wave and scattering of an electron by a nucleus in the field of the wave (see Fig. 6). It can be easily verified that, if the spontaneous photon propagates in the same direction as the photon from the external field, condition (3.14) cannot be satisfied simultaneously with the first equality in (3.15a) or (3.15b). Therefore, resonances may occur only when photons propagate nonparallel to each other. Taking (3.14) into account, we can use (3.15a) and (3.15b) to find the frequency of the spontaneous photon in a resonance (the resonant frequency) for the direct and exchange amplitudes (corresponding to diagrams shown in Figs. 5a and 5b, respectively), (n p j) ' ≡ ω 'j = rω -------------, ω res j = i, f . (3.16) ( n'q j ) LASER PHYSICS Vol. 6 No. 5 1996 RESONANT EFFECTS IN COLLISIONS OF RELATIVISTIC ELECTRONS The interference of resonant amplitudes (corresponding to the direct and exchange diagrams) implies the equality of their resonant frequencies, ω i' = ω 'f . With allowance for (3.16), we derive ( n p i )(n'q f ) = ( n p f )(n'q i) . (3.17) For arbitrary wave intensities, expressions (3.8) and (3.9) for the amplitude are rather cumbersome. Therefore, in what follows, we restrict our consideration to resonant electron–nucleus SB within the range of intensities specified by (3.1), where multiphoton processes are characterized by the quantum parameter γ. Within the range (3.1), resonant frequencies (3.16) can be conveniently written in the following form: ω 'j = rω j ( 1 + d j ) , (3.18) κ ω + ω j = ω -----j , d j = ( nn' )(β j ± r) ----- , κ 'j κ 'j (3.19) –1 where κ j = E j – np j , κ 'j = E j – n'p j , j = i, f . (3.20) Here, the quantities β i, f are defined by expression (2.5). As can be seen from (3.19), within a rather broad range of electron energies and scattering angles, we have |dj | Ⰶ 1 (except for an ultrarelativistic electron with an energy of m2 /ω moving within a narrow cone close to the direction of the momentum of the spontaneous photon). Therefore, resonances are mainly observed when the frequency of the spontaneous photon is multiple of ωj (3.19). Let us introduce a positive integer r that stands for the number of a resonance, i.e., r = 1, 2, 3, etc. correspond to the first, second, third, etc. resonances. As can be seen from (3.19), we can separate four characteristic domains of the frequency ωj: in the nonrelativistic case, ωj ≅ ω; in the limiting case of ultrarelativistic energies when an electron moves within a narrow cone related to the photon from the external field, ωj Ⰶ ω; for an ultrarelativistic electron moving within a narrow cone with the spontaneous photon, ωj Ⰷ ω; otherwise, ωj ∼ ω. Below, we will consider resonant frequencies in greater detail. Within the range specified by (3.1), condition (3.17) of interference between the direct and exchange resonant amplitudes is written as + ( v f – v i )(n – n') + ( v f × v i )(n' × n) rω(κ i + κ f ) -. = ( n'n ) -------------------------Ei E f (3.21) Here, vj = pj /Ej is the electron velocity before ( j = i) and after (j = f ) scattering. The quantity involved in the right-hand side of (3.21) is small as compared with LASER PHYSICS Vol. 6 No. 5 1996 849 unity. Therefore, this equality is satisfied if the directions of motion of photons (the spontaneous photon and the photon from the external field) or electrons (before and after scattering) are close to each other. Since resonances vanish when the direction of motion of the spontaneous photon is close to the direction of motion of the photon from the external field, it can be seen from (3.21) that resonant amplitudes corresponding to the processes shown in Figs. 5a and 5b interfere when an electron is scattered by a small angle θ ~ (1 – nvi)(ω/ viEi) Ⰶ 1. 3.3. Resonant Differential SB Cross Section Within the range specified by (3.1), we can simplify expressions (3.8)–(3.11) for the amplitudes. In particular, we can neglect the terms proportional to the param(r + l) eter η in expression (3.10) for the amplitude M 1, 2 (for an ultrarelativistic electron that moves within a narrow cone with the photon from the external field, we require that the inequality η Ⰶ m/Ei should be satis(r) fied). For the amplitude F i, f (3.11), we should keep the terms proportional to the zeroth and first powers of the parameter η, whereas the terms proportional to the second power of η can be neglected, which is due to the fact that, in a resonance, the order of magnitude of arguments of the Bessel functions is determined by η [see (2.26)]. Taking this circumstance into account, we derive the following expressions for the amplitudes (r + l) (r) M1 and F i : (r + l) M1 (r) Fi 0 = γ̃ L r + l(χ 1, γ 1, β 1 ) , – (3.22) = ( aε∗ )L –r(χ 1, γ 1, 0) + ( b + ε∗ )L 1 – r(χ i, γ i, 0). (3.23) Here, the functions Lr + l and L–r are defined by expressions (2.26) and (2.27), respectively. Let us determine the resonant cross section in the absence of interference between the direct and exchange amplitudes, i.e., when an electron is scattered by a large angle θ Ⰷ (1 – nvi)(ω/ viEi). (3.24) Assume, for example, that the resonant condition is satisfied for the direct amplitude (see Fig. 6), i.e., the frequency of the spontaneous photon coincides with the resonant frequency ω i' [see (3.18) and (3.19)]. Then, the sum in (3.9) involves only the first term with fixed r. The resonant singularity is eliminated by the imaginary part of the electron mass, i.e., m µi = m – iΓi, where q i0 -W . Γ i = -----2m (3.25) Here, qi0 is the zeroth component of the four-momentum qi (3.12) and W is the total probability of the Compton scattering of a light wave by an intermediate 850 ROSHCHUPKIN electron with a four-momentum qi. Now, we can apply a conventional procedure [24] to find the differential cross section from the amplitude determined by (3.8), (3.9), (3.22), and (3.23). Performing the relevant procedures of averaging and summation in polarizations of the photon and initial and final electrons, we derive the general relativistic expression for the resonant differential cross section of electron–nucleus SB in the field of a light wave when the electron is scattered by a large angle (3.24), ( l, r ) dσ res Ei κi qi u 1 dΩ k' (r) (r + l) - ----------------------------------- dW dσ S . = -----2 ------------------2 2 ( nn' ) p (1 + u) i π qi – µi (3.26) 2 Here, (l + r) dσ S 2 2 2 2 pf m 2 - ( m + 2E f q i0 – p f q i ) L r + l = 2Z r e -------------4 qi q (3.27) × δ [ Ẽ f – q̃ i0 + ( r + l )ω ]dE f dΩ is the differential cross section of scattering of an intermediate electron with a four-momentum qi by a nucleus in the field of the wave, and dW (r) αm = ---------4E i a multiquantum process. Note that, if the wave intensity is subject to requirements more stringent than (3.1) that can be written, depending on the electron energy, as η Ⰶ vi, if vi Ⰶ 1, 2 is the probability that an electron with a four-momentum pi absorbs r photons from the external field and emits a photon with a four-momentum k'. For a circularly polarized wave, the argument of the Bessel function in (3.28) can be represented in the following form: (3.29) Here, invariant parameters u and ur are given by (3.30) The argument of the Bessel functions in (3.27) can be represented as mq γ 1 = α 1 γ , γ = η ---------- , ωE i 2 2 ∞ ∑ (1 + l) dσ S l = –∞ 2 2 2 2 2 m = dσ S(q i) = Z r e (4E i – q ) -----4- dΩ, q (3.33) where dσS(qi) is the differential cross section of elastic scattering of an intermediate electron by a nucleus into an elementary solid angle dΩ and q = pf – pi [24]. Within the range (3.1), the width Γi in (3.25) is written as ( nq i ) 1 2 2 -ω , Γ i = --- ( 1 + δ )αη f (q i) ----------6 m (3.34) f (q i) = σ c(q i) ⁄ σ T ≤ 1 . (3.35) where (3.28) 2 2 du 2 2 2 1 u × 4J r (γ i) + ---(1 + δ )η J r – 1 (γ i) 2 + ------------ -------------------2 2 1 + u ( 1 + u ) ( k pi ) ( kk' ) -. u = ------------ , u r = 2r ----------2 ( kq i ) m (3.32) ωE i ⁄ m , if E i ⲏ m, then |Lr + l | 2 = J r + l (γ 1) , and the cross section (3.27) can be integrated in the energy of final electrons and summed over all possible l. This procedure yields 2 u u γ i = 2rη ---- 1 – ---- . ur u r ω ⁄ p i , if v i Ⰶ 1 η Ⰶ (3.31) where α1 ~ 1. Since γi ~ η Ⰶ 1 [see (3.29)], within the range of fields specified by (3.1), the first resonance, i.e., the resonance with r = 1, provides the main contribution to the resonant cross section. This implies that the Compton scattering of a light wave by an initial electron is mainly due to the absorption of one photon from the external field. Since γ1 ~ γ ⲏ 1 [see (3.31)], the scattering of an intermediate electron by a nucleus in the wave field (3.1) under these conditions is generally Here, σc(qi) is the total cross section of the Compton effect that involves a photon from the external field and an intermediate electron, and σT is the cross section of Thomson scattering [24]. Since the resonant cross section within the range of fields specified by (3.1) is mainly determined by the first resonance (r = 1), in what follows, we will analyze this situation for a circularly polarized external field. 3.4. The Range of Relativistic Energies In this section, we will consider neither the nonrelativistic limit of electron energies nor ultrarelativistic electrons moving within a narrow cone with a spontaneous photon or a photon from the external field. Except for these restrictions, we will not impose any specific requirements on electron energies and scattering angles. Then, as it follows from (3.19), |di | Ⰶ 1. Therefore, the resonant frequency ω i' (3.18) in this case is on the order of the frequency of the external field. Depending on the emission angle of the spontaneous photon with respect to the direction of the momentum of the initial electron, the resonant frequency falls within the interval κi κi -. ω -----------------≤ ω i' ≤ ω ----------------Ei + pi Ei – pi (3.36) This frequency reaches its minimum and maximum (see Fig. 7) when the spontaneous photon is emitted LASER PHYSICS Vol. 6 No. 5 1996 RESONANT EFFECTS IN COLLISIONS OF RELATIVISTIC ELECTRONS along the direction of motion of the initial electron and in the opposite direction, respectively. Invariant parameters (3.30) are written as ωκ i ω' - , u ≅ ( nn' ) ----- Ⰶ 1 . u r = 2r -------2 κi m 851 ω'res /ω 3 (3.37) 2 Taking these relationships into account, we can represent the resonant denominator in the following form: 1 2 2 2 2 ' ) + C 12 ], q i – µ i = ( 2ω' p i ) [ ( cos θ i' – cos θ res (3.38) 0 where θ i', f = ∠(k', p i, f ), θ i, f = ∠(k, p i, f ), (3.39) E i – ( ω ⁄ ω' )κ i mΓ ' = --------------------------------- , C 1 = ------------i- . cos θ res pi ω' p i (3.40) For resonant angles that are not too close to 0 and π, we can expand cos θ i' in (3.38) as a Taylor series near the resonant angle with an accuracy up to the term of the ' . The solid angle that correfirst order in t = θ i' – θ res sponds to the emission of the spontaneous photon is ' dϕdt. Then, performing sumwritten as dΩk' = sin θ res mation over all possible l [see (3.33)], we derive the following expression for the resonant cross section (3.26) for a circularly polarized wave and r = 1: Ei κi 1 dϕd(t ⁄ y) (1) (1) - dW dσ S(q i). dσ res = --------2 ------------------------2 ----------------------------4π 1 + ( t ⁄ y ) ( nn' ) p i Γ i m (3.41) ' ) ~ αη2 Ⰶ 1. Since the Here, y = mΓi /(ω'|pi |sin θ res angular width of the resonance is very small, we can integrate (3.41) with respect to the azimuthal angle dϕ and with respect to d(t/y) within the limits from –∞ to +∞ (we extend the integration limits to the infinity because of a fast convergence of the integral). Finally, we derive Ei κi (1) (1) dσ res = -------------------------------dW dσ S(q i) , ε(nn')m p i Γ i (3.42) where dW (1) 2 m 2u u = αη ( nn' ) ------------- 1 – ------ 1 – ----- dω' . (3.43) u1 2E i κ i u 1 2 Here, ε = 2 corresponds to resonant angles of emission of the spontaneous photon that are not close to 0 and π, ' = 0, π (in the latter case, we and ε = 4 corresponds to θ res should expand cos θ i' as a Taylor series with an accuracy up to t2, represent the solid angle as dΩk' = (1/2)dϕdt2, and perform similar integration procedures). The derived expressions (3.42) and (3.43) for the resonant cross section are valid within the range of field intensiLASER PHYSICS Vol. 6 No. 5 1996 40 80 120 160 200 Scattering angle θi', deg Fig. 7. The ratio of the resonant frequency of the spontaneous photon to the frequency of a photon from the external field (3.19) as a function of the scattering angle of the spontaneous photon with respect to the electron momentum in the initial state (θi = 60° and ϕi = 140°) for a circularly polarized laser wave with frequency ω = 4 eV and strength F Ⰶ 3 × 1010 V/cm. The electron energy is (solid line) Ei = 0.59 and (dashed line) 0.85 MeV. logRres 5 4 3 2 1 0 0.2 0.4 0.6 0.8 1.0 Electron velocity, v Fig. 8. The ratio of the resonant differential cross section of electron–nucleus SB to the cross section of bremsstrahlung in the absence of the external field (3.46) as a function of the electron velocity for preset orientations of the electron momentum in the initial (θi = 60° and ϕi = 140°) and final (θf = 120° and ϕf = 60°) states and fixed orientation of the spontaneous photon (θ' = 30° and ϕ' = 40°) in a circularly polarized laser wave with frequency ω = 4 eV and strength F Ⰶ 3 × 1010 V/cm. ties specified by (3.1) and (3.32) when an electron is scattered by a large angle θ Ⰷ ω/|pi | [see (3.24)]. The frequency and the emission angle of the spontaneous photon with respect to the momentum of the initial electron are unambiguously related to each other by expression (3.40), where the frequency of the spontaneous photon is chosen from the interval (3.36). Note that, in the case under consideration, the conventional cross section dσ of electron–nucleus bremsstrahlung (in the * absence of an external field) [24] can be factorized as a product of the cross section dσS(pi) of electron–nucleus 852 ROSHCHUPKIN elastic scattering [see (3.33)] and the probability dWγ to emit a photon, dσ * = dσ S dW γ , (3.44) 2 α 2 2 m dω' dW γ = --------2 q – ( n'q ) ----------- ----------------dΩ k' , κ i' κ 'f ω'κ i' κ 'f 4π (3.45) q = p f – pi . Let us consider the ratio of the resonant cross section (3.42) to the conventional cross section of electron– nucleus bremsstrahlung (in the absence of an external field) (3.44) with allowance for the resonant relation (3.36) between the frequency and the emission angle of the spontaneous photon, (1) R res 2 dσ res 2 –1 m = ------------------------ = f 2 π α ------- , pi dσ * ⁄ dΩ k' (3.46) where the function f2 ~ 1 has a rather cumbersome form [17]. Figure 8 displays Rres (3.46) as a function of the initial velocity of the electron. As can be seen from (3.46) and Fig. 8, within the range of relativistic electron energies, the resonant differential cross section of electron–nucleus SB when the ejection angle of the scattered electron is detected simultaneously with the emission angle of the spontaneous photon may be four orders of magnitude higher than the corresponding cross section in the absence of the external field. Within the range of ultrarelativistic electron energies, this ratio 0. drastically decreases, Rres ~ (m/Ei)2 3.5. The Range of Nonrelativistic Electron Energies In this section, we assume that the energies of the initial and final electrons are small as compared with the speed of light, Zα ~ vi, f Ⰶ 1. As it follows from (3.18) and (3.19), resonant frequencies for nonrelativistic electrons are given by the expression 1 2 2 ω i', f = rω 1 + v i, f (n' – n) – --- ( 1 + δ )(nn')η ≅ rω. 4 (3.47) Thus, resonances occur when the frequency of the spontaneous photon is multiple of the frequency of the external field. In the case under consideration, the condition of interference between the direct and exchange resonant amplitudes (3.21) is written as ω (3.48) ( v f – v i )(n – n') = 2r(nn') ---- Ⰶ 1 . m Consequently, such an interference is manifested when an electron is scattered by a small angle θ ~ ω/mvi Ⰶ 1. The resonant cross section corresponding to the scattering of a nonrelativistic electron by a large angle θ Ⰷ ω/mvi is determined from the nonrelativistic limit of expression (3.26), where we should abandon summation in l. Finally, we derive 1 (1) ( l, 1 ) (1 + l) , (3.49) dσ res = ------------------------- dW dσ S 2(nn') v i Γ i where dW (1) 2u 1 2 u = --- αη ( nn' ) 1 – ------ 1 – ----- dω' , u 2 u 1 1 (3.50) 4 2 2 2m = ( 2Z ) r e -----4- ρ l + 1 J l + 1 (γ 1)dΩ . q Here, we introduced the notations ω' u ----- = ( nn' ) ------- , 2ω u1 (l + 1) dσ S m 2 2 2 γ 1 = η ---- [ e 1(v f – v i) ] + δ [ e 2(v f – v i) ] , ω (3.51) (3.52) (3.53) 2 p 2 η 1 ρ l + 1 = -------f- = – ---(1 + δ ) ----- cos θ f v i 4 pi (3.54) ω' + ( l + 1 )ω 1 2 η 2 η 2 1 - + ---(1 + δ ) ----- cos θ i – ---(1 + δ ) ----- cos θ f . + 1 – 2 -----------------------------2 v v 8 2 i i mv i 2 The resonant frequency of the spontaneous photon depends on the emission angle of this photon with respect to the momentum of the initial electron and lies within a narrow interval ' ≤ ω [1 + 2 v i sin ( θ i ⁄ 2 ) ]. ω [1 – 2 v i cos ( θ i ⁄ 2 )] ≤ ω res (3.55) 2 2 Note that Lebedev [3] determined the total cross section of electron–nucleus SB in the plane-wave field under the assumption that this cross section is mainly determined by Compton resonances. In the limiting 2 case of a strong field, η Ⰷ vi, Karapetyan and Fedorov [32] revealed resonances at the frequencies of the spontaneous photon multiple of the frequency of the external field that are not related to the Green function of the intermediate electron falling within the mass shell. Taking into account that, in the nonrelativistic limit, the width Γi (3.34) is given by Γi = αη2ω/3, we can write the ratio of the resonant cross section (3.49) to the corresponding conventional nonrelativistic cross section of electron–nucleus bremsstrahlung (in the absence of an external field) as LASER PHYSICS Vol. 6 No. 5 1996 RESONANT EFFECTS IN COLLISIONS OF RELATIVISTIC ELECTRONS (l) R res = f 3π α v 2 –1 (l + 1) (q i) – 3 dσ S i -------------------------, dσ S( p i) (3.56) 853 Rres × 106 8 where the function f3 ~ 1 is given by 3 [ 1 – ( 1/2 ) sin θ' ] -. f 3 = -------------------------------------------------------------------------2 2 4 sin ( θ ⁄ 2 ) – ( cos θ 'f – cos θ i' ) 2 (3.57) 4 Provided that conditions (3.32) are satisfied (the nonrelativistic limit), we can perform summation in expression (3.56) over all l, which yields R res = f 3 π α v 2 –1 –3 i . (3.58) Figure 9 shows the dependence of Rres (3.58) on the polar emission angle of the spontaneous photon for a nonrelativistic electron with the energy Ei = 2.5 keV. As can be seen from Fig. 9 and expression (3.58), for electron energies of several kiloelectronvolts (recall that, within the framework of the Born approximation, electron velocities are bounded below, v Ⰷ α = 1/137), the resonant differential SB cross section when the emission angle of the spontaneous photon is detected simultaneously with the ejection angle of an electron scattered by a large angle may be seven orders of magnitude greater than the corresponding cross section of bremsstrahlung in the absence of the external field. 3.6. The Range of Ultrarelativistic Energies of Electrons Moving within a Narrow Cone with a Photon from the Wave In this section, we will consider an ultrarelativistic electron that moves (in the initial or final state) within a narrow cone related to a photon from the external field. Therefore, the quantities κi, f (3.20) in expressions (3.18) and (3.19) can be written in the following form: 2 E i, f m 2 -. κ i, f = ( 1 + δ i, f ) ------------ , δ i, f = θ i, f -------2E i, f m (3.59) Taking these relations into account and using (3.18) and (3.19), we find that the resonant frequencies are much lower than the frequency of the external field and are given by ( 1 + δ i, f ) m 2 - --------- ω Ⰶ ω. ω i', f = rω i, f , ω i, f = --------------------2(nn') E i, f 2 (3.60) As it follows from (3.60), condition of interference between the direct and exchange resonant amplitudes implies that δi = δf , or θi ≅ θf , i.e., the initial and final electrons form equal angles with the momentum of the photon from the external field and are located on different sides of the momentum of this photon. As can be 3 seen from (3.21), θi ~ ωm2 / E i Ⰶ 1. If the initial ultrarelativistic electron moves within a narrow cone with a photon from the external field and experiences LASER PHYSICS Vol. 6 No. 5 1996 0 40 80 120 160 200 Scattering angle of photons, θ', deg Fig. 9. The ratio of the resonant differential cross section of SB due to the scattering of a nonrelativistic electron with an energy Ei = 2.5 keV by a nucleus to the cross section of bremsstrahlung in the absence of the external field (3.58) as a function of the polar emission angle of the spontaneous photon for preset orientations of the electron momentum in the initial (θi = 60° and ϕi = 140°) and final (θf = 120° and ϕf = 60°) states in a circularly polarized laser wave with frequency ω = 4 eV and strength F Ⰶ 3 × 1010 V/cm. The azimuthal angle corresponding to the emission of the spontaneous photon is (solid line) ϕ' = 160° and (dashed line) 130°. scattering by a large angle θi Ⰷ ωm2 / E i , expression (3.26) and equalities (3.59) yield 3 (1) dσ res ( 1 + δ i )m (1) - dW dσ S(q i) . = -----------------------4(nn')E i Γ i 2 (3.61) Here, the resonant frequency of the spontaneous photon is given by expression (3.60) with r = 1, and the emission angle of the spontaneous photon is not close to the direction of motion of the initial electron. The ratio of the resonant cross section (3.61) to the conventional cross section of electron–nucleus bremsstrahlung can be derived from (3.46) with allowance for (3.59), 2 2 –1 m R res = f 4 π α ----- , E i (3.62) where f4 ~ 1. Hence, for the electron energy Ei = 5 MeV, the resonant cross section when the emission angle of the spontaneous photon is detected simultaneously with the ejection angle of an electron scattered by a large angle may be an order of magnitude greater than the conventional cross section of bremsstrahlung in the absence of the external field. As the electron energy grows, the resonant cross section drastically decreases. 3.7. The Range of Ultrarelativistic Energies of Electrons Moving within a Narrow Cone with a Spontaneous Photon Suppose that an ultrarelativistic electron (an initial or a final one) moves within a narrow cone with a spon- 854 ROSHCHUPKIN taneous photon. Then, the quantities κ i', f (3.20) can be written in the form analogous to (3.59) where E i, f -. δ i', f = θ i', f -------(3.63) δi, f m Here, depending on the electron energy, we deal with one of two situations. Provided that m Ⰶ Ei, f Ⰶ m2 /ω, the resonant frequencies fall within the interval ω Ⰶ ω i', f Ⰶ Ei, f and are given by Taking (3.26)–(3.28) and (3.67) into account (for a circularly polarized wave and r = 1), writing the solid 2 2 angle as dΩk' = (m2 /2 E i )dϕd δ i' , and performing inte2 gration with respect to the azimuthal angle and d δ i' within the limits from zero to +∞, we derive the following expression for the resonant cross section: q i0 (1) (1) -dW dσ S(q i) . dσ res = Y ( xτ) -------mΓ i 2 2(nn') E i, f - --------- ω . ω i', f = rω i, f , ω i, f = --------------------2 ( 1 + δ i', f ) m (3.64) 2 2 m (3.66) E f < ( 1 + δ 'f ) --------------------- . 2(nn')rω As it follows from energy considerations, the condition ∆i ≥ 1 should be also satisfied for the initial electron. Therefore, resonances do not occur for energies Ei, f Ⰷ m2 /ω. It can be easily seen that the direct and exchange resonant amplitudes can interfere with each other only when the energies of the initial and final electrons satisfy the condition Ei, f ⱗ m2 /ω and when the initial and final electrons move within a narrow cone with a spontaneous photon so that δ i' = δ 'f . Provided that the initial ultrarelativistic electron moves within a narrow cone with a spontaneous photon and experiences scattering by a large angle θ Ⰷ Ei /ω, we can use formulas (3.26)–(3.28) to find the resonant cross section. In this case, it is convenient to represent the resonant denominator in the following form: 2 2 2 u 4 2 2 2 q i – µ i = m [ ( x – δ i' ) + τ ] -------------------2 , (1 + u) where u ( 1 + u )Γ i 2(1 + u)Γ x = -----1 + ---------------------- – 1, τ = ------------------------i . 2 u um um Here, ∞ If the energy of an electron (the initial or the final one) is Ei,f ~ m2 /ω, then the resonant frequencies ω i', f ~ Ei, f are given by Ei Ei - , ω 'f = -------------ω i' = ------------∆i + 1 ∆f – 1 (3.65) 2 ( 1 + δ i', f ) m 2 ∆ i, f = ---------------------- ---------------- . 2(nn') rωE i, f Hence, the final electron may fall in resonance only when the energy does not exceed a certain limit, (3.67) 2 (3.68) Here, the invariant parameters u and u1 (3.30) are given by ωE ω' u ≅ ---------------- , u 1 = 2(nn') --------2-i . (3.69) E i – ω' m (3.70) 2 dδ i' 1 x 1 1 Y ( xτ) = --- -------------------------------- = --- + --- arctan -- (3.71) 2 2 π π τ 2 2 0 ( x – δ i' ) + τ ∫ is a smoothed step function. In the regions far from the resonant point, |u1 – u| Ⰷ 2(1 + u)(Γi /m) and at the resonance point u1 = u, this function takes the following limiting values [4]: 1, if u < u 1 Y ( xτ) = 0.5, if u = u 1 (3.72) τu ⁄ π(u – u 1) , if u > u 1 . The probability involved in (3.70) is given by dW (1) 4u m u u du = αη -------- 2 + ------------ – ------ 1 – ----- -------------------2 . 4E i u 1 ( 1 + u ) 1 + u u1 (3.73) 2 2 2 In deriving formula (3.70), we took into account that the argument of the Bessel function in (3.28) is γi ~ η Ⰶ 1 and performed summation over all l. Let us consider the ratio of the resonant cross section (3.70) to the conventional cross section of electron–nucleus bremsstrahlung in the case when an ultrarelativistic electron moves within a narrow cone with the photon created in bremsstrahlung and experiences scattering by a large angle dσa. Baier, Fadin, and Khoze [41] derived the following expression for this ratio: (1) E dW ( 1 ) dσ res - = Y ( xτ) --------i - ------------------- . R res = ----------mΓ i dW pi(k') dσ a (3.74) Here, dW pi(k') is the probability that an electron with a four-momentum pi emits a photon with a four-momentum k' [41]. For electron energies m Ⰶ Ei Ⰶ m2 /ω, expression (3.74) can be written as 3 –1 –1 E R res = --- πα ln -----i . m 4 (3.75) Note that Borisov, Zhukovskii, et al. [4] derived an expression for resonant bremsstrahlung that accompanies the scattering of an ultrarelativistic electron by an LASER PHYSICS Vol. 6 No. 5 1996 RESONANT EFFECTS IN COLLISIONS OF RELATIVISTIC ELECTRONS electron in the field of a wave with a moderate intensity. The expression derived in [4] is similar to (3.75). Let us estimate the ratio of the cross sections under study. For the electron energy Ei = 5 × 104 MeV, formula (3.75) gives Rres ≈ 28. 4. RESONANT EFFECTS IN THE PHOTOCREATION OF ELECTRON–POSITRON PAIRS ON A NUCLEUS IN A LIGHT FIELD In this section, we will consider resonances that occur in the photocreation of electron–positron pairs on a nucleus in the field of a plane electromagnetic wave. The creation of an electron–positron pair by a photon and bremsstrahlung due to the interaction of an electron and a nucleus in the field of a wave form two cross channels of the same reaction. Therefore, we can find the amplitude of pair photocreation on a nucleus in the field of a wave using expressions (3.7)–(3.11) with the replacements p–, pi pf –p+, k' –k' (4.1) and qi q–, qf –q+. (4.2) Here, p– and p+ are the four-momenta of the electron and the positron, respectively, and q– and q+ are the four-momenta of the intermediate electron and positron (see Fig. 10). Taking (3.11), (3.12), (4.1), and (4.2) into account, we can derive the following expressions for the four-momenta of the intermediate electron and positron and the transferred four-momentum: q – = k' + rk – p̃ + , q + = k' + rk – p̃ – , (4.3) q = p̃ – + p̃ + – k' + lk . (4.4) Resonant features of the amplitude corresponding to the photocreation of electron–positron pairs on a nucleus in the field of a wave are due to the fact that the intermediate electron or the intermediate positron (see diagrams shown in Figs. 10a and 10b) may fall within the mass shell, i.e., may become real, 2 2 2 2 q– = m* , q+ = m* . (4.5) It is convenient to write the expressions that determine the four-momenta of the intermediate electron and positron and the transferred four-momentum for the resonant amplitudes corresponding to the processes shown in Figs. 10a and 10b in the following form: k' + rk = q – + p̃ + , q = p̃ – – q – + ( l + r )k , (4.6a) k' + rk = p̃ – + q + , q = p̃ + – q + + ( l + r )k . (4.6b) 2 2 2 2 2 2 Since p̃ + = q – = m * or p̃ – = q + = m * , and k2 = k'2 = 0, the first equalities in (4.6a) and (4.6b) can be satisfied LASER PHYSICS Vol. 6 No. 5 1996 855 Ze q k' q– (a) p– k' – p+ Ze q q+ p– (b) – p+ Fig. 10. Photocreation of electron–positron pairs on a nucleus in the field of a plane electromagnetic wave. The solid lines represent the wave functions of the electron and the positron in the field of the wave, the inner lines show the Green function of the electron (positron) in the field of the wave, and the dashed lines correspond to an incident γ-quantum k' and a pseudophoton q of a Coulomb center. only for r ≥ 1. Hence, taking into account expressions (3.8)–(3.11) for the amplitude and replacements (4.1) (r + l) (r + l) and (4.2), we infer that M 1 and M 2 with the transferred momentum described by (4.6a) and (4.6b) are the amplitudes of scattering of the intermediate electron q– and the intermediate positron q+, respectively, by a nucleus in the field of the wave [40]. At the same time, with allowance for the conservation of the four-momentum, which is expressed by (4.6a) and (r) (r) (4.6b), F i and F f are the amplitudes of pair creation by a photon k' in the field of the wave. This process was considered by many researchers (e.g., see review [20]). Consequently, in the absence of interference between the amplitudes of the processes shown in Figs. 10a and 10b, resonant photocreation of electron–positron pairs on a nucleus in the field of a light wave can be effectively divided into two processes of the first order in the finestructure constant: creation of electron–positron pairs by a photon k' in the wave field and scattering of an intermediate particle by a nucleus in the field of the wave. Using (4.6a) and (4.6b), we can find the resonant frequency of the incident photon for the amplitudes of the processes shown in Figs. 10a and 10b, ( n p+ ) -, ω' = ω +' ≡ rω ------------( n'q – ) (4.7a) ( n p– ) -. ω' = ω –' ≡ rω ------------( n'q + ) (4.7b) Thus, the condition of interference between the resonant amplitudes can be written as ( n p – )(n'q –) = ( n p + )(n'q +) . (4.8) In what follows, we restrict our consideration to the range of intensities of the external field specified by (3.1). Within this range, conservation of the energy [the zeroth 856 ROSHCHUPKIN Ze Then, the resonant cross section corresponding to the photocreation of electron–positron pairs on a nucleus in the field of a wave can be represented as k' q q2 = m2* 2 – p+ p– ( r, l ) dσ res Fig. 11. Resonant photocreation of electron–positron pairs on a nucleus in the field of a plane wave. The inner line of the diagram in Fig. 10a is cut up, which corresponds to a real intermediate electron. component of the four-vector (4.4)] is expressed by the following relationship: ω' ≅ E – + E + . (4.9) Thus, as can be seen from (4.7a) and (4.7b), within the range of fields specified by (3.1), resonances may occur only for an ultrarelativistic positron p+ (diagram in Fig. 10a) and an ultrarelativistic electron p– (diagram in Fig. 10b) moving within a narrow cone with the incident γ-quantum k'. Under these conditions, the resonant frequencies are given by 2 δ ±' ) m 2 (1 + E± - , W ± = -------------------- ------, ω ±' = ------------------------1 – W ± ⁄ E± 2(nn') rω (4.10) (4.11) Hence, resonances may occur only when the energies of the positron and the electron are higher than a certain threshold W±, E± > W± ~ m2 /ω. Applying the law of energy conservation in the form of (4.9) and (4.10), we find that the resonant amplitudes interfere with each other if the electron p– and the positron p+ move within a narrow cone with the incident γ-quantum k' so that δ –' = δ +' and θ –' ~ ω/E–. If the interference is absent and the amplitude of the process shown in Fig. 10a provides a dominant contribution (see Fig. 11), i.e., the positron p+ moves within a narrow cone with the incident γ-quantum k', whereas the electron p– moves outside this cone ( θ –' Ⰷ ω/E–), we can find the resonant cross section using expression (3.26). For this purpose, we should make replacements (4.1) and (4.2) in this expression, reverse the sign, and change the density of final states in accordance with ( E i ⁄ p i )d p f d k' 3 3 d 3p–d 3p+. (4.12) In the case under consideration, it would be convenient to replace invariant parameters u and ur (3.30) by invariant parameters z+ and zr , ( k p+ ) E+ ( kk' ) -. - ≅ ---------------- , z r = 2r ---------z + = -----------2 ( kq – ) ω' – E + m (4.14) ω'z + (r) (r + l) -dW (q –). × ------------------------pair dσ S 2 2 m (1 + z +) (r + l) Here, dσ S (q –) is the cross section of the scattering of the intermediate electron by the nucleus in the field (r) of the wave [see (3.27)] and dW pair is the probability that the γ-quantum k' creates an electron–positron pair (q–p+) in the field of the wave, 2 m 2 (r) dW pair = α -------- 4J r (α +)--4ω' (4.15) ( 1 + z + ) dz + 1 2 2 2 – ---(1 + δ )η J r – 1 (α +) 2 – -------------------- ---------------------2 , 2 z+ ( 1 + z + ) 2 2z + Γ – z+ Γ– z+ zr -. x + = --------------------2 + ------------------------2 – 1, y + = ---------------------( 1 + z + )m ( 1 + z + ) ( 1 + z + )m (4.16) For a circularly polarized external field, the argument of the Bessel function in (4.15) is written as 2 where δ ±' = θ ±' (E ± ⁄ m), θ ±' = ∠(k', p ±) Ⰶ 1 . dϕdδ +' 1 = --------2 -------------------------------------------2π [ ( x – dδ ' 2 ) 2 + y 2 ] + + + (4.13) ( 1 + z+ ) 2 z+ zr – ( 1 + z+ ) . α + = 2rη -----------------z+ zr (4.17) Summing (4.14) over all possible l, integrating the 2 resulting expression in dϕ and d δ + , and setting r = 1, we find the resonant cross section corresponding to the photocreation of electron–positron pairs in the field of a circularly polarized wave, ω' (1) (1) dσ res = Y ( x + y +) -------------------------------dW pair dσ S(q –) . (4.18) 2mΓ –(1 + z +) Here, the function Y(x+y+) is given by (3.71). At the resonance (z1 = (1 + z+)2 /z+), this function is equal to 0.5. The width Γ– in (4.18) is defined by the expression q –0 - ( W + W k' ) , Γ – = -----2m k (4.19) where q–0 is the zeroth component of the four-vector q– and Wk and Wk' are the total probabilities of the Compton effect involving the intermediate electron and the photons k and k', respectively. Since Wk' Ⰶ Wk, the width is given by [see (3.34)] 1 2 ( nq – ) -ω . Γ – = --- f (q –)αη -----------3 m LASER PHYSICS Vol. 6 (4.20) No. 5 1996 RESONANT EFFECTS IN COLLISIONS OF RELATIVISTIC ELECTRONS Note that, in a particular case of counterpropagating photons k and k', formula (4.18) is reduced to the expression derived by Borisov et al. [9]. Applying the law of energy conservation (4.9), we can easily find the resonant positron energy, ω *' 1 E + = --- 1 ± 1 – ------- ω' . 2 ω' (4.21) Here, ω *' is the threshold frequency of the incident γquantum, which is equal to 2 2m ω *' = ------------------------------- , θ' = ∠(k, k') . ( 1 – cos θ' )ω (4.22) Note that we can determine the electron energy by reversing the sign in front of the square root in (4.21). As can be seen from (4.21), near the threshold (ω' ≈ ω *' ), the energies of the created electron and positron are equal to each other (E+ = E– ≅ ω *' /2). If the frequency of the incident γ-quantum is high (ω' Ⰷ ω *' ), then the energies of the electron and positron considerably differ from each other (E+ = ω' – ω *' /4 ≈ ω' and E– ≈ ω *' /4). Baier et al. [41] considered a process γ + Ze e+ + e– + Ze that occurs within the same kinematic domain in the absence of a wave field. The authors of this study demonstrated that the amplitudes of the processes shown in Figs. 10a and 10b feature poles within different ranges of angles of pair ejection. Therefore, these amplitudes do not interfere with each other. The cross section can be factorized as dσpair = dWk'(p+, q–)dσS(q–), (4.23) where q– = k' – p+ and dWk'(p+, q–) is the probability that a γ-quantum k' creates an electron–positron pair (p+q–). Let us express the resonant cross section (4.18) in terms of the conventional cross section (4.23), (1) (1) R res dW pair dσ res ω' -. - = Y ( x + y +) ------------------------------- ---------------------------= -----------2mΓ ( 1 + z ) dW dσ pair – + k'( p +, q –) (4.24) Taking into account formulas (4.15) and (4.20) and the expression for dWk', we can write the ratio (4.24) in the logarithmic approximation [41] as R res E 3 = --- π α f (q –) ln -----+8 m –1 . (4.25) Let us estimate the ratio of the cross sections (4.25). For the laser frequency ω = 4 eV, the threshold frequency of the incident γ-quantum is ω *' = 250 GeV. Therefore, for ω' = 106 GeV, formula (4.25) gives Rres ≈ 12. LASER PHYSICS Vol. 6 No. 5 1996 857 5. CONCLUSION We performed analysis of resonant effects in an external light field related to the scattering of an electron by an electron, spontaneous bremsstrahlung of an electron scattered by a nucleus, and photocreation of electron– positron pairs on a nucleus. This analysis demonstrates that, for the intensities of light fields widely used in physical experiments [I ~ (1012–1019) W/cm2], a virtual intermediate particle may fall within the mass shell (i.e., may become a real particle), and a process of the second order in the fine-structure constant can be effectively divided into two first-order processes. Note that resonant cross sections may be considerably greater than the corresponding cross sections in the absence of the external field. Specifically, within the range of strengths of light fields (5.1) F Ⰶ (1010–1011) V/cm, resonant electron–electron scattering occurs in forward (for the direct amplitude) or backward (for the exchange amplitude) small-angle (θ ~ ω/|pi | ⱗ 0.01°) scattering in the frame of reference related to the center of inertia. In such a situation, the emission angle of the intermediate photon is correlated with the emission angle of the photon from the external field. The wave vectors of these photons are directed along the generatrices of a cone whose axis is directed along the initial relative momentum of electrons. Under these conditions, the ratio of the resonant cross section to the conventional Möller cross section is inversely proportional to the fourth power of the electron energy. For nonrelativistic electrons, the resonant cross section may be up to four orders of magnitude greater than the conventional Möller cross section. Depending on the energy of electrons and scattering angles, resonant spontaneous bremsstrahlung due to the scattering of an electron by a nucleus in a light field (5.1) features a characteristic spectrum. Specifically, for nonrelativistic electrons, the frequency of the spontaneous photon coincides with the frequency of the wave field. For ultrarelativistic energies of an electron moving within a narrow cone related to a photon from the light wave, the frequencies of the emission spectrum are much lower than the frequency of the external field. Conversely, if an electron moves within a narrow cone with a spontaneous photon, the frequencies of the emission spectrum are much higher than the frequency of the external field. In other cases, the frequencies of the emission spectrum are of the same order as the frequency of the wave field. We emphasize that the resonant cross section of electron–nucleus SB when the emission angle of the spontaneous photons is detected simultaneously with the ejection angle of an electron scattered by a large angle considerably depends on the energy and the scattering angle of the electron. This cross section may be much greater than the conventional (in the absence of the external field) cross section of electron–nucleus SB. The ratio of the resonant cross section to the conventional cross section reaches its 858 ROSHCHUPKIN maximum in the range of nonrelativistic electrons (where the resonant cross section may be up to seven orders of magnitude greater than the conventional cross section). The resonant photocreation of electron–positron pairs on a nucleus in the presence of a light field (5.1) occurs when the threshold energy of the incident γ-quantum is much higher than twice the rest energy of the created pair. Therefore, the emerging electron and positron have ultrarelativistic energies. Under these conditions, the resonant cross section of photocreation can be factorized as a product of the probability that the incident γ-quantum creates a pair in the field of the wave and the cross section of scattering of an intermediate electron by a nucleus in the light field. The resonant cross section may be an order of magnitude greater than the conventional cross section corresponding to the photocreation of pairs on a nucleus (in the absence of the external field). REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. Oleinik, V.P., 1967, Zh. Eksp. Teor. Fiz., 52, 1049. Oleinik, V.P., 1967, Zh. Eksp. Teor. Fiz., 53, 1997. Lebedev, I.V., 1972, Opt. Spektrosk., 32, 120. Borisov, A.V., Zhukovskii, V.Ch., and Eminov, P.A., 1980, Zh. Eksp. Teor. Fiz., 78, 530. Bös, J., Brock, W., Mitter, H., and Schott, Th., 1979, J. Phys. A, 12, 715. Bös, J., Brock, W., Mitter, H., and Schott, Th., 1979, J. Phys. A, 12, 2573. Bergou, J., Varroó, S., and Fedorov, M.V., 1981, J. Phys. A, 14, 2305. Belousov, I.V., 1977, Opt. Commun., 20, 205. Borisov, A.V., Zhukovskii, V.Ch., Nasirov, A.K., and Eminov, P.A., 1981, Izv. Vyssh. Uchebn. Zaved., Fiz., 24, 12. Kazakov, A.E. and Roshchupkin, S.P., 1983, Preprint FIAN, no. 18, Moscow, USSR. Kazakov, A.E. and Roshchupkin, S.P., 1983, Preprint FIAN, no. 115, Moscow, USSR. Roshchupkin, S.P., 1983, Izv. Vyssh. Uchebn. Zaved., Fiz., no. 4, 18. Roshchupkin, S.P., 1983, Izv. Vyssh. Uchebn. Zaved., Fiz., no. 8, 12. Roshchupkin, S.P., 1984, Opt. Spektrosk., 56, 36. Roshchupkin, S.P., 1985, Yad. Fiz., 41, 1244. Roshchupkin, S.P., 1994, Laser Phys., 4, 139. 17. Roshchupkin, S.P., 1994, Stimulated and Spontaneous Emission in Collisions of Relativistic Electrons in a Strong Light Field, Doctoral Dissertation (Moscow: Moscow Engineering Physics Inst.). 18. Oleinik, V.P. and Belousov, I.V., 1983, Problems of Quantum Electrodynamics of Vacuum, Dispersive Media, and Strong Fields (Chisinau) (in Russian). 19. Ternov, I.M., Zhukovskii, V.Ch., and Borisov, A.V., 1989, Quantum Processes in a Strong External Field (Moscow: Mosk. Gos. Univ.) (in Russian). 20. Nikishov, A.I. and Ritus, V.I., 1979, Trudy FIAN, 111. 21. Bunkin, F.V. and Fedorov, M.V., 1965, Zh. Eksp. Teor. Fiz., 49, 1215. 22. Fedorov, M.V., 1991, An Electron in a Strong Light Field (Moscow: Nauka) (in Russian). 23. Roshchupkin, S.P., 1994, Zh. Eksp. Teor. Fiz., 106, 102. 24. Berestetskii, V.B., Lifshitz, E.M., and Pitaevskii, L.P., 1980, Quantum Electrodynamics (Moscow: Nauka) (in Russian). 25. Abrikosov, A.A., Gor’kov, L.P., and Dzyaloshinskii, I.E., 1962, Methods of Quantum Field Theory in Statistical Physics (Moscow: Fizmatgiz) (in Russian). 26. Yakovlev, V.P., 1966, Zh. Eksp. Teor. Fiz., 51, 617. 27. Kazantsev, A.P. and Sokolov, V.P., 1984, Zh. Eksp. Teor. Fiz., 86, 896. 28. Zavtrak, S.T. and Komarov, L.I., 1990, Teor. Mat. Fiz., 84, 431. 29. Reiss, H.R., 1980, Phys. Rev. A, 22, 1786. 30. Borisov, A.V. and Zhukovskii, V.Ch., 1976, Zh. Eksp. Teor. Fiz., 70, 477. 31. 1991, Itogi Nauki Tekh., Ser. Sovrem. Probl. Lazernoi Fiz., vol. 4, Akhmanov, S.A., Ed. 32. Karapetyan, R.V. and Fedorov, M.V., 1978, Zh. Eksp. Teor. Fiz., 75, 816. 33. Krainov, V.P. and Roshchupkin, S.P., 1983, Zh. Eksp. Teor. Fiz., 84, 1302. 34. Schwinger, J., 1951, Phys. Rev., 82, 664. 35. Brown, L.S. and Kibblie, T.W.B., 1964, Phys. Rev. A, 133, 705. 36. Eberly, J.H. and Reiss, H.R., 1966, Phys. Rev., 145, 1035. 37. Reiss, H.R. and Eberly, J.H., 1966, Phys. Rev., 151, 1058. 38. Yakovlev, V.P., 1966, Zh. Eksp. Teor. Fiz., 51, 617. 39. Bunkin, F.V. and Fedorov, M.V., 1965, Zh. Eksp. Teor. Fiz., 49, 1215. 40. Denisov, M.M. and Fedorov, M.V., 1967, Zh. Eksp. Teor. Fiz., 53, 1340. 41. Baier, V.N., Fadin, V.S., and Khoze, V.A., 1973, Nucl. Phys. B, 65, 381. LASER PHYSICS Vol. 6 No. 5 1996