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J.
Phys. A: Math. Gen.
14
(1981) 2305-2315. Printed in Great Britain
e-e scattering
in
the presence
of
an external field
J
Bergout,
S
Varr6-t and M V FedorovS
t
Central Research Institute for Physics, H-1525 Budapest, POB 49, Hungary
fP
N
Lebedev Physical Institute
of
the Academy
of
Sciences of the USSR, Moscow,
Leninsky prospect 53, USSR
Received 30 October 1980
Abstract.
A non-relativistic treatment is given of electron-electron scattering in the
presence of a laser field. The field is accounted for by the external field approximation and is
represented by a circularly polarised monochromatic plane-wave field.
A
simple analytic
expression is derived for the transition amplitude, which is shown to exhibit internal
resonances as well as intensity-dependent shifts. The former is the non-relativistic limit of
the resonant M~ller scattering predicted previously by Oleinik. The latter, however,
appears in a higher order of
u/c
and is consequently negligible for very slow electrons. The
differential cross section of the scattering is also given where the effect of the spin and
symmetry is taken into account explicitly. The width of resonances is introduced
phenomenologically but its connection with previous methods is established. Consideration
is also given to the experimental conditions under which the effects may become observable.
1.
Introduction
In
connection with the rapid development of high-power lasers, in the last two decades
considerable theoretical effort has been devoted to the description of fundamental
EM
processes in the presence of an intense field. Of particular interest are the electromag-
netic processes of.free charged particles in such an environment. The theory developed
so
far (e.g. Mitter 1975, Karapetyan and Fedorov 1978) predicts two different types of
corrections whose physical origin is also entirely different. Namely, intensity-depen-
dent resonances and intensity-dependent shifts are the main phenomena occurring in
external field problems. The occurrence of resonances is connected with induced
processes at integer multiples of the frequency of the external field, whereas the
occurrence of intensity-dependent (energy and momentum) shifts is a classical effect
connected with the average kinetic energy and momentum of a classical particle
oscillating in an external field. Well known examples of these phenomena are the
nonlinear bremsstrahlung (Bunkin and Fedorov 1965) and nonlinear Compton scat-
tering (Brown and Kibble 1964) in the presence of an intense external field. Especially
interesting and much less commonly investigated are, in this context, the resonances
and shifts predicted for Mpller scattering. The problem was first investigated by Oleinik
(1967a, b). His results, however, were not given in a form suitable for direct
comparison with experiments. Very recently, the problem was reinvestigated using a
different calculational technique developed especially to treat intense field problems
(BOs
et
a1
1979a,
b).
The formalism was still rather complicated and allowed the
calculations for non-relativistic energies only. In view of this, we think that the present
status of the theory requires a consistent, non-relativistic treatment
of
this problem.
0305-4470/81/092305
+
11$01.50
@
1981 The Institute of Physics 2305
2306
J
Bergou,
S
Varrd and
M
V
Fedorov
The calculations are based
on
the formalism where the laser field is taken into
account exactly and all the other interactions are treated by perturbation theory
(Bergou 1980). The laser field is represented as an external (classical) plane-wave field,
In
order to simplify the calculations, we introduce a few assumptions which are
essentially very similar to the assumptions of Bos
et
a1
(1979a), namely:
(1)
the external field is circularly polarised;
(2)
consideration is given to non-relativistic unpolarised electrons;
(3)
the e-e interaction takes place via the static Coulomb potential;
(4)
there is
no
transverse-photon exchange;
(5)
interaction with the spin momentum is neglected.
The last four assumptions indicate that we use a consistent non-relativistic descrip-
tion from the very beginning.
As
a result, we obtain a simple analytical expression for
the transition amplitude in
§
2.
The kinematics of the process can easily be deduced
from the delta-function parts of the transition amplitude.
In
§
3
we transform the
transition amplitude to a form where the internal resonances appear explicitly and we
then derive the cross section of the scattering. We also discuss how the inclusion of the
spin and symmetry modifies the preceding results.
In
the last part of the paper
(§
4)
we
briefly summarise the main results, give a physical interpretation to the internal
resonances and discuss their connection with experimental possibilities of observation.
2.
The transition amplitude
In the following we shall investigate non-relativistic electron-electron scattering in the
presence of an external field. For an intense (laser) field the one-mode, external field
approximation is reasonably good provided that the following two requirements are
satisfied (Bergou and Varr6 1981):
(i)
the number of photons in the mode is large;
(ii)
the change of the photon number (depletion) in a given process
is
much less than
the initial number of photons. For a non-relativistic approximation this change should
actually be less than the ratio of electron rest energy to photon energy
(2
mc2/hw)
which, in the case of optical frequencies, is still a large number (105-106).
We start from the Schrodinger equation of the problem:
iha+/at
=
H+
(2.1)
where
+
is the two-electron wavefunction and the Hamiltonian
H
can be written in the
form
where
H=Ho+
V(r)
(2.2)
and
V(r)
=
e2
e-"'/r,
r
=
Ixl
-
x2j.
(2.4)
Ho
corresponds to the free electrons interacting with the external field. The external
field itself is represented by the
A(v)
vector potential
(2.5)
A(v1.2)
=
a(e1
cos
~1,2+ez
sin
v1,2),
~1,2
=
Ut
-
kx1,2,
e-e scattering in the presence
of
an external field 2307
which describes a circularly polarised plane wave. The electron-electron interaction is
represented by the static Coulomb potential
(2.4).
The shielding factor
(Y
is introduced
here merely to simplify the following calculations. Its physical significance will be
discussed in
0
4.
First, it should be noted that in dipole approximation
(k
=
0),
the Schrodinger
equation
(2.1)
can be decoupled in centre-of-mass and relative coordinates
$(xi,
X2,
t)
=
Qi(R,
f)Q2(r,
t),
R
=;
(x1
+xz),
r=xl-x2,
(2.6)
to yield
(2.7)
Here
3
and
8
are the respective operators of total and relative momenta,
M
=
2m
is the
total mass,
m,
=
m/2
is the reduced mass. We can easily recognise that equation
(2.7)
corresponds to the motion of a free particle (with total charge
2e
and mass
2m)
in an
external field, whereas equation
(2.8)
is the Schrodinger equation of a particle moving
in a background potential field
V(r).
Since now the scattering problem is entirely
separated from the external field problem, we immediately see that the external field
affects the scattering only
if
we go beyond dipole approximation (Brehme
1971,
Bergou
1976).
Let us now return to the full Hamiltonian
(2.2)
which is beyond dipole approxima-
tion. According to the spirit of intense field calculations, we shall treat
V(r)
as a
perturbation and use the solution of the
HO
part as a basis set for perturbation theory.
Since
Ho
is the sum of two one-electron Hamiltonians, the unperturbed solution can be
taken in the form of the product of two one-electron wavefunctions:
(2.9)
+(O)(Xl,
x2,
t)
=
$(O)(Xl,
t)+‘0’(x2,
t).
The exact solution for
+(O)(X,
t)-which is given in terms of Mathieu functions and is
therefore rather complicated-was found recently (Bergou and Varr6
1980).
However, for all practical purposes, the following approximative solution is satisfactory
(Bergou and Varr6
1980,
Ehlotzky
1978):
4“)
=
$p(x,
t)
=
(2~h)-~”
exp [(i/h)(px
-Et)],
(2.10)
where
E
=p2/2m
and
w’
=
w(1-
kp/mw)
is the Doppler-shifted frequency.
The required validity condition of the above solution is the following. One may
expand both
(2.10)
and the exact solution into Fourier series of
einv.
The coefficients of
this Fourier series satisfy a complicated recurrence relation in the case of the exact
solution, which can be shown to reduce to the simpler recurrence relation for the
coefficients of the Fourier series of
(2.10)
if
nhw/2mc2f(a)
K
1,
2 2
112
where
f(a)
=
[(
1
-
kp/mw)2
-
(ea/mc
)
]
is a slowly varying function
of
amplitude
a.
For a large range of the U-values
f(a)
-
1,
and the validity condition is well fulfilled in
optical interactions.
2308
J
Bergou,
S
Varrd and M V Fedorov
Performing the integration in (2.10), we obtain
$p(~,
t)
=
(2.rrh)-3/2 exp [(i/h)(p'x
--Et)]
exp[i(ea/mchw)(elp'sin v-eg' cos
v)]
where (2.11)
(2.11a)
We see that
p'
and
B
depend nonlinearly on the original momentum
p
and contain an
intensity-dependent shift.
p'
reduces to
p
in the case of very slow electrons.
With the help of first-order perturbation theory, we obtain for the transition
amplitude due to V(r) the expression
Tfi=
-i
jm
dtl d3x1 d3x2
+?*
(XI,
XZ,
t)V(r)$?)(xI, x2,
t)
(2.12)
h
-m
where
do)
=
+Pli
(XI,
t)+PZi
(X2,
t),
$i"'
=
$Plf(X1,
t)
$PZf(X2,
t)
(2.12~)
Using expression (2.1
1)
for +p(~,
t)
in (2.12), we have the following explicit form of
are initial and final wavefunctions, respectively.
Tfi
:
X
exp[iz (el
Ap;
sin
VI
-
e2Ap; cos
v1
+
elApk sin
v2
-
e2Ap; cos
v2)]
(2.13)
where
(2.13~)
ea
mchw'
z=-
A#
=
bi
-p'f,
Ap'=pl
-pi,
AB=Bi-Ef.
If we now introduce centre-of-mass and relative coordinates,
R
=
4(X,
+X2),
r=x1-x2,
and centre-of-mass and relative momenta,
P=p1+p2,
P
=
2.m
-P2),
then
Tfi
takes the form
Tfi=---
6
Im
dtl
d3R
d3r V(r)
exp{(i/h)[A~R+Apr-(A&~+A&)t]}.
h
(2Th)
-m
x
exp {i[ -zl sin
t
kr
cos
(ut-
kR
-9)
+z2 cos
1
kr
sin
(ut-
kR
-@)I}
(2.14)
where
elAP'
=
APl
cos
@,
elAp'
=
ApY cos
Q,
2zApL =ti,
ZAP:
=
z2,
e2AP'
=
AP:
sin
a,
e2Ap'
=
Ap: sin
Q,
(2.14~)
e-e scattering in the presence
of
an external field
after having made use of the trigonometric relations
cos
sin
[
wt
-
kR
f
ikr}
=
'""I
sin
wt
-
kR] cosikr
f
cos
sin(
wt
-
kR}
sinikr.
Using the expansions
eiz
sin
0
=
c
Jl(z)
eile,
,is
cos
0
-
-
C
imJm(z)
e"'.
I
m
in (2.14), the time and
R
integrations can easily be carried out, yielding
2309
(2.15)
where
n
=
1
+
m.
In the extreme non-relativistic case (scattering
of
very
slow
electrons) v/c
<<
1
and
P
=
P'
=
P.
Since
z2
is proportional to the transverse component
of
AP'
=
P! -PI,
according to the conservation of the transverse components
of
the centre-of-mass
momentum expressed by the delta function, we have
22
=
0
and JIt-dO)
=
6,~.
Consequently, in this case
Tfi
can be represented as a single sum over n-photon emitted
and absorbed parts,
Tfi
=
Tji"'
It
where
(2.16)
(2.17)
and
tji")
=
(2~h)-~ exp [-in
(cp
+
$7r)]
d3r V(r)JIt(zl sin
d
kr) exp -Apr
.
(2.18)
I
(d
)
From here we can deduce the following conservation laws.
(1)
Transverse components of
P
are conserved.
(2) The longitudinal component
of
P
is shifted by an amount which depends on the
momentum itself and is proportional to the light intensity. Furthermore, in the process
of
scattering it may acquire an integer number
of
the photon momentum
hk.
(3) The energy is also shifted by an amount depending on the momentum and being
proportional to the light intensity. Furthermore, during scattering it may change by an
integer number of the photon energy
hw.
The momentum shift
SP
is connected with the energy shift SE by the relation
(6PI
=
ISEl/c and is therefore small even at high intensities. This justifies our process of
neglecting the difference between
#
and
P'
but keeping when we made the transition
from (2.15) to (2.17)-(2.18).
The energy conservation law can be brought to a somewhat different form where the
kinematics of the process is more apparent. The initial and final energies can be
2310
J
Bergou,
S
Varro' and
M
V
Fedorov
expressed with the help of the centre-of-mass and relative momenta as follows (to first
order in
v/c):
P:f p:f e2a2 e2a2 kPi,f
4m
m
2mc 2mc
mu
Ei,~=Eii,f+EZi,f=-+-+2---2++-,
Conservation of energy requires now
(2.19)
The uniform shift
e2a2/2mc2
cancels from the difference, and only the non-uniform
part of the shift
(e2a2/2mc2) kPi,f/mw
gives a non-vanishing contribution. We can now
make use of the conservation law
of
the centre-of-mass momentum
Pf= Pi-nhk
in
(2.19),
yielding
(2.20)
where
VB
=ckPi/mw
is the longitudinal (parallel to
k)
component of the initial
centre-of-mass velocity.
In
other words, the system takes
on
momentum as a whole (momentum change
appears in the centre-of-mass motion). The coresponding Doppler shift and recoil
together with an additional intensity-dependent energy shift appear in the relative
motion. Equation
(2.17)
indicates that absorption and emission of real photons are also
connected with the relative motion (for the process under consideration this means the
internal degrees of freedom). To close this section, let us mention that in dipole
approximation
kr
=
0,
J,(O)
=
and equation
(2.18)
gives the usual fieldless scatter-
ing amplitude as expected from equations
(2.7)
and
(2.8).
3.
Scattering cross section
3.1.
Inclusion
of
spin
and symmetry
Since the electrons are fermions, the two-electron wavefunction must be antisymmetric
under the exchange of the two particles. Accordingly, we have to modify the theory
developed in the preceding section in order to include this symmetry property in the
treatment. First, we note that the Hamiltonian
(2.2)
does not affect the spin variables,
and hence the total wavefunction is a product of the spin-dependent and the coor-
dinate-dependent terms. The wavefunction describing the spin state of the two particles
may be either symmetric (triplet state) or antisymmetric (singlet state)
on
the exchange
of the particles. The corresponding orbital part is either antisymmetric (triplet state) or
symmetric (singlet state). The orbital part of the transition amplitude
Tfi
will then be
either antisymmetric
(Tf,J
or symmetric
(Tfi,J.
Since the effect of the exchange
0:
the
particles is equivalent to changing the sign of the relative momentum in the final state,
and therefore does not affect the centre-of-mass part
of
the transition amplitude, we
have for the symmetric case
(3.1)
e-e scattering in the presence
of
an external
field
2311
where
t::
=
(2~h)-~
[
d3r V(r){exp [(i/h)p'-'r]J,(z'-)sin
1
kr) exp
[
-in(q~(-)+h)]
+e~p[(i/h)p'~)r]J,(z'~'sin
ikr) exp[ -in(p(+)+kr)]}
(3.2)
and
Pi
*
Pf,
z(*)
=
(ea/mcho)lpiL*ppfll,
(3.3)
p(*)
=
p'*)
=
tan-'
(e2p(*)/elp(*)),
$2
=
tP)(e)+ tji")(r
-
e)
Equation
(3.2)
can be written in the form
(3.4)
where tt'(8) is the amplitude of transition due to the V(r)Jn(zl sin kr)
ePiq
effective
potential as given by equation
(2.18)
and
8
is the scattering angle (the angle between
pf
and
pi).
Similarly, for the antisymmetric case we have
where now
tt:
=
tp(8)-tP)(r--O),
(3.6)
Quite generally, there are three symmetric and one antisymmetric spin states for the
two-electron system. For unpolarised electrons each state is equally probable. There-
fore, in the cross section the weight of the antisymmetric combination
(3.6)
is three
times larger than the weight of the symmetric combination
(3.2),
that is
(3.7)
The cross section dv/dfl is a sum of incoherent nth-order cross sections and
(3.7)
holds
for each of them separately. In the following we shall confine ourselves to the study
of
the (unsymmetrised)
$'(e)
scattering amplitude
(2.18),
since symmetrisation is essen-
tially equivalent with the inclusion of
tji"'(r
-
8)
in the treatment, and therefore effects
connected with symmetry can be obtained by changing
8
to
r
-
8.
3.2.
Internal resonances
To
perform a numerical calculation for direct comparison with experiments, we can
most conveniently use equation
(2.18).
In
order to obtain a better insight into the
dynamics of the physical processes involved, and to obtain a qualitative understanding
of
them, we transform the transition amplitude to a form where the internal resonances
are manifest. We start from equation
(2.18).
Let us first introduce the Fourier
representation of V(r) as
V(r)
=
[
V(q) e-iqr d3q, V(q)
=
4~e~/(q~+a')(2r)~. (3.8)
2312
J
Bergou,
S.
Varrd and
M
VFedorov
Then we use the integral representation for the Bessel function
Jn(x)
(Abramowitz and
Stegun 1964):
1"
2T
-"
J,
(x)
=
-
exp(
-
in6
+
ixsin6) do.
In our case z1
sin
1
kr
=
x.
Inserting this expression for
x
in the exponential and
expanding exp (izl sin 6 sin
3
kr) in Fourier series of exp(imkr/2), we obtain for
tg'
from (2.18)
$)
=(2~A)-~exp[-in(cp+2~)]C
d6e-'"'Jm(zlsin 6)
"
m
5-w
+
imkk
-
(3.9)
Integration over
r
yields the Oirac delta function S(Ap
-
Aq
+$mhk) and the integra-
tion over q yields then V(q
=
kmk
+
Aplh). The only remaining integral (integration
over 6) yields (Gradshtein and Ryahik 1971)
We can summarise the results as follows. The transition amplitude
T,
can be represen-
ted by a sum of n-photon emitted
(n
>
0)
and absorbed
(n
<
0)
parts
Tg)
(2.16)
n
where
Tfi"'
(
- - -
2.rriS(Pf
-Pi
+
ntik)S(&
-Bi
+
n
Aw)tg'.
(2.17)
is given either by equation (2.18)-which is more convenient for practical
Here
numerical calculations-or by
(3.10)
(m+n=even)
The scattering amplitude given in this form explicitly exhibits a resonance structure.
Apart from the Coulomb resonance (Ap
=
0),
additional new resonances may occur
if the scattering geometry is properly chosen. From (3.10) we might expect that the
resonances occur whenever 2Ap
+
mtik
=
0.
This is not quite the case, however, since
then
z1
=
0
and from (2.18) we see that the amplitude reduces to that
of
the Coulomb
scattering. Nevertheless, the condition of these 'geometrical resonances' is very close to
the condition stated above. The geometrical resonances will be discussed further in a
separate paper (Bergou et
a1
1980).
Introducing a new summation index
i(m
-
n)
=
r in (3.10), we obtain an equivalent
but simpler form for
tg)
(we avoid the use of ostensibly fractional indices)
tg)
-
-iw
16~e'
J,
GZ
1)
J,+
GZ
)
A
?
[2Ap
+
(2r
+
n)hkI2+ (2ha)*'
-e
(3.10~)
e-e scattering
in
the presence
of
an
external field
2313
3.3.
Cross section
The cross section can be calculated in the usual manner from the square of the scattering
amplitude
ts.
For unpolarised electrons we average over initial and sum over final spin
states. Since the Hamiltonian (2.2) does not affect the spin state, this procedure actually
leads to the weighted sum of symmetric and antisymmetric cross sections. Further-
more, similar to other processes in the presence of an external field, the differential
cross section du is a sum of incoherent contributions du'") corresponding to
n
-photon
absorption
(n
<
0)
and emission
(n
>
0):
da
=
1
da'")
n
where, according to equation (3.7),
da(")
=
a
daF)
fa
dff?).
In our system of normalisation and with the help
of
equations
(3.4)
and (3.6)
and
Using the explicit expressions (3.4) and (3.6), we obtain finally
(3.11)
This expression clearly has the advantage over the ones given by
Bos
et
a1
(1979a) and
Oleinik (1967a) that the average over polarisation is explicitly carried out and the
problem is reduced to its essence, i.e. to the manifestation of the internal resonances. In
addition,
tr)
is given by two different but equivalent expressions, equations (2.18) and
(3.10). The first one is more convenient for performing numerical analysis, while the
second one explicitly exhibits a resonance structure. Far from resonances, the depen-
dence of the denominator on the summation index can be neglected, and the summation
can be carried out to yield a closed form expression for the non-resonant scattering
cross section.
4.
Discussion and summary
In the previous sections we have derived explicit expressions for the transition ampli-
tude andscattering cross section of the non-relativistic electron-electron scattering in
the presence of an external electromagnetic field (laser). This scattering is the non-
relativistic limit of the Maller scattering in an external field (Oleinik 1967a, b,
Bos
et
a1
1979a, b). We have found the previously predicted intensity-dependent shifts and
resonances to persist in the non-relativistic case as well. However, shifts appear in
higher order in
v/c
than resonances. In lowest order in
v/c
there is still an intensity-
dependent shift of the kinetic energy and momentum of the free electron, but it is
2314
J
Bergou,
S
Varro' and
M
V
Fedorov
uniform (i.e. independent of the state of the electron, which is characterised by
E
and
p).
This uniform shift therefore cancels out in the transition amplitude which depends
on
differences of energy and momentum. Only the next-order term (which is non-
uniform, i.e.
p
dependent) gives a non-vanishing contribution to the shift. But since the
momentum shift
SP
is related to the energy shift
SE
via the relation
ISPI=
(l/c)lSEI,
the
non-vanishing contribution to the momentum shift is in fact already second order in
l/c,
and in non-relativistic approximation it can safely be neglected. Thus only the
energy shift leads to observable consequences.
The advantage of the consistent non-relativistic treatment is the possibility of
carrying out explicitly the averaging over spin polarisation. We have obtained two
different forms for the transition amplitude, namely equations
(2.18)
and
(3.10).
Equation
(3.10)
more explicitly shows the resonant structure, and its connection with
previous calculations is straightforward. On the other hand, equation
(2.18)
has a few
advantages. First, it is more suitable for performing numerical calculations. Second, it
allows an interesting interpretation for the occurrence of resonances. Apart from a
phase factor, in first Born approximation the scattering can be interpreted as being
caused by an effective potential
Vefl(r)
=
V(r)J,(zl
sin
ikr).
It is clear at first sight that
all effects of the external field result from the non-dipole character of the effective
potential, since in the
k
=
0
case the scattering reduces to the Coulomb scattering. For a
certain electron separation
r
this effective potential may become attractive
(J,,
<
0),
which tends to cause the electrons to form pairs. It is known that to a bound state there
corresponds a resonance in the scattering amplitude. Therefore the resonances in the
scattering cross section might be regarded as the manifestation of the attractive effective
potential between electrons in the presence of an external field. From
(3.10)
it is clear
that in the case of a resonance the absorption and emission of virtual photons might be
regarded as real processes (energy-momentum conservation is satisfied). The effect
bears some analogy with the formation of electron pairs in superconductors through the
electron-phonon interaction. However, for a resonance one must simultaneously
satisfy conservation of energy-momentum in the virtual photon exchange (labelled by
m
in
(3.10))
and conservation of the total energy-momentum (as described by the
argument of the delta functions in
(2.17)).
At this point it should be mentioned that in our treatment we introduced
phenomenologically a damping factor in the Coulomb potential. Far from resonances,
the damping can be neglected and we are left with the usual Coulomb potential for the
electron-electron interaction. In the vicinity
of
resonances, the role of the radiative
corrections becomes significant. As was discussed
by
Bos
eta1
(1979a,
b), two types of
radiative corrections have to be taken into account: vacuum polarisation for the photon
and self-energy correction for the electron. For slow electrons and optical photons the
second of these is dominant. From their discussion it also follows that the distances
between resonances are much larger than their width given by the radiative corrections.
In beam experiments two other types of corrections have also to be taken into account.
First, the finite width of the laser line; second, the finite width of the momentum
distribution of the incoming electrons, which leads to further broadening effects. It
seems to be hard to overcome this latter difficulty in the case of non-relativistic
electrons. Nevertheless, one can assume an ideal experimental situation and neglect
the broadening due to these effects. In this case the resonant scattering occurs at very
small scattering angles (near the Coulomb resonance), and one needs high angular
resolution, but the effect is in principle observable. We have to mention that in this case
not only the position but also the width of the resonance is mainly determined by the
e-e scattering
in
the presence
of
an
external field
2315
scattering geometry which again justifies calling them geometrical resonances (Bergou
eta1
1980).
We finally note that in homogeneous but time-varying fields
(k
=
0)
the effect
disappears and the external field has no influence at all
on
the scattering. Thisis not the
case, however, in the scattering of oppositely charged particles (e.g. bremsstrahlung).
In that case the particles oscillate with opposite phases and this oscillatory motion
enters the relative part of the equation
of
motion. One then obtains in the
k
=
0
limit a
non-vanishing contribution which is the leading term, and
k
#
0
gives only small
corrections. Thus the main effect of the external field is to distinguish between the
relative signs of the scattered particles.
Acknowledgments
We wish to express our gratitude to Professor Gy Farkas and Dr
Z
Gy Horvdth for
valuable discussions on the experimental aspects of the problem. They also proposed,
independently, the idea of an attractive effective electron-electron interaction in an
intense electromagnetic environment (private communication); we wish to thank them
for bringing this to our knowledge and for their continuous interest in the problem.
Note added
in
proof.
At first sight resonances occur whenever the denominator
of
(3.10a) vanishes. At these
points, however, the numerator also vanishes (see the definition of
zl
in (2.14a).
As
a consequence, the true
location of resonances is slightly shifted from the zero
of
the denominator and the width and height of the
resonances remain always finite even without introducing radiative corrections.
A
closer analysis reveals also
that resonance conditions can be met in the elastic
(n
=
0)
channel
only.
This point together with further
details
of
the cross sections is left, however,
to
a separate publication (Bergou
et
al
1980).
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