Download Resonant Effects in Collisions of Relativistic Electrons in the Field of

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
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(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
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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
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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
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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,
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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
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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)
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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
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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 )
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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
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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
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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
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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
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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
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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
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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
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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
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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).
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LASER PHYSICS
Vol. 6
No. 5
1996