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M. J. CONDENSED MATTER VOLUME 12, NUMBER 3 DECEMBER 2010
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M. J. CONDENSED MATTER VOLUME 12, NUMBER 3 DECEMBER 2010
Coulomb Scattering Of An Electron In Strong Laser Fields
S.Taj1, B. Manaut2, Y.Attaourti1, S.Elhandi 1 and L.Oufni3
1
Laboratoire de Physique des Hautes Energies et d'Astrophysique, Faculté des Sciences Semlalia, Université Cadi
Ayyad Marrakech, BP : 2390, Morocco
2
Laboratoire Interdisciplinaire de Recherche en Science et Technique (LIRST), Faculté Polydisciplinaire Université
Sultan Moulay Slimane Béni Mellal, BP : 523, Morocco.
3
Université Sultan Moulay Slimane, Faculté des Sciences et Techniques, Département de Physique, LPMMERM,
BP : 523, 23000 Béni Mellal, Morocco.
Email: [email protected]
Abstract : In this work, we review and correct the first Born differential cross section for the process of Mott
scattering of a DiracVolkov electron, namely, the expression (26) derived by Szymanowski et al [Physical Review
A56, 3846 (1997)]. In particular, we disagree with the expression of they obtained and we give the exact
coefficients multiplying the various Bessel functions appearing in the scattering differential cross section. Comparison
of our numerical calculations with those of Szymanowski et al. shows qualitative and quantitative differences when
the incoming total electron energy and the electric field strength are increased particularly in the direction of the laser
propagation. Such corrections are very important since the relativistic electronic dressing of any DiracVolkov
charged particle gives rise to these coefficients that multiply the various Bessel functions and the relativistic study of
other processes (such as excitation, ionization, etc....) depends strongly of the correctness and reliability of the
calculations for this process of Mott Scattering in presence of a laser field. Our work has been accepted [Y. Attaourti,
B. Manaut, Physical Review A68, 067401 (2003)] but only as a comment. In this paper, we give the full details of the
calculations as well as the clear explanation of the large discrepancies that their results could cause when working in
the ultra relativistic regime and using a very strong laser field corresponding to an electric field in atomic
units.
PACS number(s): 34.80.Qb, 12.20.Ds
I. Introduction [1], where is the phase stemming from the
In a pioneering and very often cited paper, expression of the circularly polarized electromagnetic
Szymanowski et al. [1] have studied the Mott field. The claim of [1] that they vanish is not true.
scattering process in a strong laser field. The main These terms do not depend on the chosen description
purpose of their work was to show that the of the circular polarization in cartesian components.
modifications of the Mott scattering differential cross On the other hand, we perform the calculations with
section for the scattering of an electron by the some details and throughout this work, we use atomic
Coulomb potential of a nucleus in the presence of a units where denotes the electron
strong laser field can yield interesting importance and mass. The abbreviation DCS stands for the differential
the signatures of the relativistic effects. Their spin cross section.
dependent relativistic description of Mott scattering The organization of this paper is as follows: in Section
permits to distinguish between kinematics and spin 2, we give the expression of the matrix transition
orbit coupling effects. They have compared the results amplitude as well as the formal expression of
of a calculation of the first Born differential cross scattering DCS. In section 3, we give some estimates
section for the Coulomb scattering of the Dirac of the numerical significance of our corrections. In
Volkov electrons dressed by a circularly polarized particular, we compare numerically the DiracVolkov
laser field to the first Born cross section for the DCS we have obtained with the corresponding DCS of
Coulomb scattering of spinless KleinGordon particles [1]. We end by a brief conclusion in Section 4.
and also to the non relativistic Schrö dingerVolkov
treatment. The aim of our work is to provide the
correct expression for the firstBorn differential cross
sections corresponding to the Coulomb scattering of II. Theory
the DiracVolkov electrons. On the one hand, we show The interaction of the dressed electrons with the
that the terms proportional to are missing in central Coulomb field
12 271 © 2010 The Moroccan Statistical Physical and Condensed Matter Society
272 Coulomb Scattering Of An Electron In Strong Laser Fields 12
(1) where the three coefficients , and are
respectively given by
is considered as a firstorder perturbation. This is well
justified if , where is the nuclear charge of
C 0 0 2 k 0 a 2 kc ( p i ) c ( p f )
the nucleus considered and is the finestructure C 1 c ( p i ) 0 ka1 c ( p f ) a1 k 0 (8)
constant. We evaluate the transition matrix element for C 2 c ( p i ) 0 ka 2 c ( p f ) a 2 k 0
the transition ( )
We now invoke the wellknown identities involving
(2) ordinary Bessel functions
We consider the quantity
(9)
With
(10)
With
After some algebraic calculations, and using textbook
We have by A. G. Grozin [6] which is full of worked examples
in various fields of physics particularly in QED. We
give the final result for the unpolarized DCS for the
Mott scattering of a DiracVolkov electron:
Where is such that
(5)
Whereas the quantities and are given by
(11)
The coefficients , , and are respectively given
and the phase is such that by
With and .
Therefore, the transition matrix element becomes
(12)
12 S.Taj, B. Manaut, Y.Attaourti, S.Elhandi and L.Oufni 273
(13)
(14)
Fig 1 (a): Envelope of the non relativistic differential crosssection
(15) scaled in unit of as a function of energy transfer
scaled in units of the laser photon energy $w$ for an
Where
electric field strength of . and a relativistic parameter
, (b). Envelope of the relativistic differential cross
section scaled in unit of as a function of
III. Results and discussion
the laser photon energy for the same parameters. The envelope
For the description of the scattering geometry, we
for and are almost identical.
work in a coordinate system in which This
means that the direction of the laser propagation is
We have compared our DiracVolkov DCS and the
along the axis. To avoid any confusion, we will DiracVolkov DCS (26) of [1] and We turn to a
compare the DiracVolkov DCS (26) of [1] with the qualitative and quantitative discussion of the physical
corresponding DCS (11) we have obtained in the same process. We shall comment and analyze the results
coordinate system. obtained in [1] in the light of those we have obtained
bearing in mind that we can hardly escape rephrasing
the physical insights and explanations contained in [1].
Our disagreement is quantitative since we have shown
in the first part of this work that the expression (26) of
[1] contains errors and a missing term proportional to
. So, our primary task is to assess the
importance of this errors and missing term and to what
extent they modify the quantitative and qualitative
contents of [1].
274 Coulomb Scattering Of An Electron In Strong Laser Fields 12
III.1 The non relativisticlow electric field The visual cutoff occurs for and .
strength regime The difference between our results and that
In this regime, we choose as in [1] for the of [1] is just a matter of convention. We now analyze
the angular distributions. We have summed as in [1] ±
relativistic parameter and for the 100 peaks around the elastic one in order to draw the
electric field strength. This relativistic parameter angular dependence of the DCS. In Figure 6. Of [1],
corresponds to an incoming electron kinetic energy the accumulated DCS is shown for an electric field
. We plot in the upper part strength . The computer code we have
(a) of Figure 1 the non relativistic DCS given by written calculates the DiracVolkov DCS (11) of our
Eq.(34) of [1] and in the lower part (b) of the same work, the DiracVolkov DCS (26) of [1], the spinless
figure, the generalized DiracVolkov DCS given either particle DCS and the non relativistic DCS. At least, in
by Eq.(26) of [1] or Eq.(11) of our work as a function the non relativistic regime, our results and that of [1]
of the final electron energy scaled to the photon agree very well and are both close to the results for a
energy. The scattering angle is large enough so that an spinless particle. We give in Figure 3 the angular
important number of photons can be exchanged in the distribution of the various DCSs.
course of the collision. In this lowintensity regime,
the envelope of the non relativistic DCS is
qualitatively different from the envelope for the Dirac
Volkov and KleinGordon DCSs. The observed cutoffs
occur at and for the non
relativistic DCS and and
both for the DiracVolkov and KleinGordon DCSs
since the argument that appears in the ordinary Bessel
functions is the same for both DCSs. So the comments
made in [1] concerning the interpretation of the
envelope obtained do not apply for the DiracVolkov
and KleinGordon cases. While the spectrum of Figure
(1.a) of our work (which is identical to that of Figure
(1.a) of [1]) exhibits an overall asymmetric envelope
with peaks of negative energy transfer higher than
peaks of positive energy transfer, this asymmetry is Fig. 2: Summed differential cross sections of ±100 peaks
less pronounced in the case of the DiracVolkov and
spinless particle DCSs. This emphasized asymmetry in around the elastic one as a function of the angle for a
the non relativistic case can easily be traced back by a relativistic parameter and an electric field strength
close look at Eq.(34) of [1]. Indeed, the non relativistic The solid line denotes the result for DiracVolkov
DCS depends on ( depends only weakly on ) electrons, the long dashed one sketches the values for
so the asymmetry can only come from the dependence and the short dashed is the result for spinless
of the modulus of the final momentum on the particles
number of the transferred photons $s$ according to
Apart from minor differences, all three calculations
Eq.(25) of [1]. As mentioned in [1], we have an
exhibit maxima for and a giggling
enhancement of negative over positiveenergy transfer
crosssection. The DCSs fall of abruptly beyond the oscillatory behaviour (as in [1]) and minima slightly
points where the argument of the Bessel functions shifted from ( at Let aside the order of
equal to the order. For the DiracVolkov and Klein magnitude, we have in our case, three DCSs that are
Gordon DCSs, this cutoff occurs ( up to machine close to each other and not as differentiated as shown
precision) numerically for and an in Figure 6. of [1]. So, this adds to the controversy.
argument of the ordinary Bessel functions almost Even if we use the expression for the DiracVolkov
constant and equal to 380.016. However Figure (1.b) DCS given by Eq.(26) of [1], we have a different
figure for the non relativistic regime. If we now
shows a visual cutoff for . For the non
increase the electric field strength from
relativistic DCS, the numerical cutoff occurs (again up
to machine precision) for and . to the agreement remains good
12 S.Taj, B. Manaut, Y.Attaourti, S.Elhandi and L.Oufni 275
between the three relativistic calculations. There is still parameter and for an electric field
a maximum at $ while the minima are shifted strength and , our results for
towards To give an idea the small differences the DiracVolkov DCS and the corresponding results
between our result and the result of [1] for the Dirac of [1] do not agree at all. In the lower part (b) of
Volkov DCS, we have plotted in the upper part (a) of Figure (4), there is an over estimation varying from
Figure 4, the ratio of the DCS given by Eq.(26) of [1] 22.5 % to 30 % with some peaks giving an over
to the DCS given by Eq.(11) of our work as a function estimation of up to 45 % for the DCS (26) of [1]
of the angle . The ratio is defined compared to the corresponding DCS (11}) of this
work. All these peaks are nearly multiples or
by
submultiples of an angle close to
(16)
Fig 4 : (a) Summed differential cross sections of peaks
around the elastic one as a function of the angle for a
Fig 3 : Ratio of the two DiracVolkov DCS for relativistic parameter and an electric field strength
and . (b): Ratio of the two Dirac . The solid line denotes the result for the Dirac
Volkov DCS for and Volkov electrons, the long dashed one sketches the values for
. and the short dashed is the result for spinless
The deviations from the expected value 1 are shown particles. (b): Ratio of the two DiracVolkov DCSs for a
and have the same shape as the corresponding DCS. relativistic parameter and an electric field strength
However, for increasing electric field strength, the
values for this ratio are not close to 1. For a relativistic
276 Coulomb Scattering Of An Electron In Strong Laser Fields 12
III2. Relativisticstrong electric field
strength regime
For the relativistic regime, we have chosen the
parameters of [1] which corresponds to an
incoming electron total energy or a
. The electric field strength is now
. In this regime, dressing effects are
important. The envelope of the energy distribution of
the scattered electrons is similar to the one displayed
in the lower part (b) of Figure 1 However, there is a
more important asymmetry than in the non relativistic
regime with was to be expected. The corresponding
cutoffs are for
and for ,
and The three relativistic calculations Fig 5 : (a): Summed differential cross section of peaks
lead to angular distributions peaked in the direction of around the elastic one as a function of the angle for a
the laser propagation . The two DiracVolkov relativistic parameter and an elastic field strength
DCSs (solid line and long dashed line) are slightly . Th solid line denotes the result for DiracVolkov
different only in the vicinity of the two minima electrons, the long dashed one sketches the values for
located at In this regime, the shape of the and the short dashed is the result for spinless
ratio is similar to that of the corresponding DCSs. particles. (b): Ratio of the two DiracVolkov DCSs for the same
This ratio is equal to 1 for but there is values of the relevant parameters, and
now an overall amplitude of around the
expected value 1. If we increase the electric field
In the upper part(a) of Figure 5, we give the two
strength from $ to DiracVolkov DCSs and in the lower part (b) of the
and keep the same value of the relativistic parameter same figure we give the ratio of the two DCSs for
the difference between our DiracVolkov
the relativistic parameter and an electric
results and the corresponding results of [1]$ becomes
important. field strength For the angles
, the ratio is while for the
peak in the direction of the laser propagation,
the ratio is
III3. Relativistic and high electric field
strength regime
To study this regime, we use the same parameters as in
[1], that is a relativistic parameter or an
incoming electron kinetic energy
$ The electric field
strength is . In the upper part (a) of
Figure 6, we show the various DCSs. For angles $
, the agreement between our results and
the results of [1] is good but deteriorates for small
values of . For , the result of our work gives
a value ( scaled in a.u )
while de corresponding result found using Eq.(26) of
12 S.Taj, B. Manaut, Y.Attaourti, S.Elhandi and L.Oufni 277
[1] is . Our results (solid line) polarized electrons in a strong laser field [12]. For the
are always smaller than the results for spinless difficult process of ionization of atomic hydrogen by
particles while those obtained using Eq.(26) of [1] are electron impact, we published an article concerning the
greater than for small angles around the importance of the relativistic electronic dressing in
laserassisted ionization of atomic hydrogen by
direction of the laser propagation. In the lower part (b) electron impact [13]. All these works relied heavily on
of Figure 5, we show the ratio R defined by Eq.(16). the corrections that we made in this work.
For , this ratio is
VI. References
V. Conclusion [1] C. Zsymanowski, V. Véniard, R. Taïeb, A. Maquet
In this work, we derived the correct expression of the and C.H. Keitel, Phys. Rev. A, 56 3846, (1997).
first Born differential cross section for the scattering of [2] V. G. Bagrov and D. M. Gitman, Exact solutions of
the DiracVolkov electron by a Coulomb potential of a relativistic Wave Equations (Kluwer Academic
nucleus in the presence of a strong laser field. We have Publishers, Dordrecht, 1990)
given the correct relativistic generalization of the [3] D. M. Volkov, Z. Phys, 94 250,(1935).
Bunkin and Fedorov treatment [7] that is valid for an [4] A. C. Hearn, Reduce User's and Contributed
arbitrary geometry. We are adamant that the core of Packages Manual, Version 3.7 (KonradZuseZentrum
the whole controversy stems from the fact that in [1], für Informationstechnik, Berlin, 1999).
the vector of our work has not been properly dealt [5] V. Berestetzkii, E. M. Lifshitz and L. P. Pitaevskii,
Quantum Electrodynamics, 2nd ed. (Pergamon Press,
with while it is the common method to use when a
Oxford, 1982).
trace contains a matrix. Any standard QED
[6] A. G. Grozin, Using Reduce in High Energy
textbook introduces this very elementary method. Physics (Cambridge University Press, 1997).
Comparison of our numerical calculations [9] with [7] F.V. Bunkin and M.V. Fedorov, Zh Eksp. teor. Fiz.
those of Szymanowski et al. [1] shows qualitative and 49 1215, (1965)[Sov. Phys. JETP 22, 844 (1966)].
quantitative differences when the incoming total [8] Y. Attaourti, B. Manaut, eprint hepph/0207200.
electron energy and the electric field strength are [9] Y. Attaourti, B. Manaut, Phys. Rev. A, 68,
increased particularly in the direction of the laser 067401, (2003).
propagation. The difference between our results and [10] Y. Attaourti, B. Manaut and A. Makoute, Phys.
those of [1] can only be traced back to the mistakes Rev. A, 69 063407, (2004).
and the omitted term in Eq.(26) of [1]. The corrections [11] Y. Attaourti, B. Manaut and S. Taj, Phys. Rev. A,
that we made allowed us to study other processes that 70 023404, (2004).
were published in Physical Review A., namely an first [12] B. Manaut, S. Taj and Y. Attaourti, Phys. Rev. A,
article concerning the relativistic electronic dressing in 71 043401, (2005).
laser assisted electronhydrogen elastic collisions [10], [13] Y. Attaourti and S. Taj, Phys. Rev. A, 69, 063411,
another concerning the process of Mott scattering in an (2005).
elliptically polarized laser field [11] as well as a third
work dealing with the process of Mott scattering of
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