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method applicable to high-energy electrons was proposed by Lisitsa and Savel'ev
( 1987 ). The scope of that study was limited by the condition (5). The theory
was extended to 1 and f 1 by Klinskikh and Rapoport ( 1987 ), where
a general expression was obtained for the n -photon differential cross section. In
the alternative approach to the same problem used by Golovinskii ( 1988 ), the
semiclassical wavefunction was found for an electron scattered by the finite-
range potential in an electromagnetic field. Moreover, the resulting expression for
the potential scattering cross section allows transitions to the limit of the Born
approximation and the case of classical scattering.
It should be noted here that the phenomena described by the time-independent
theory of radiative collisions developed in several studies (Ivanov et al. 1988 ,
1995 , 1997b , c , 1999 ; Vartazaryan et al. 1989 ; Golubkov et al. 1993 , 1999a , b ;
Golubkov and Ivanov 1993 , 1994 , 1997 ) occur in the near-threshold energy region
of E p ! (! is the oscillation frequency of the molecular ion), where the Born
approximation and semiclassical description are strongly inapplicable. According
to (5), the electron quiver amplitude satisfies the relationship
! f
e n 2 ;
a f Š
where the electron wavelength e is comparable with the dimensions of the Rydberg
XY ** complex. Because of the multichannel Rydberg electron dynamics, electron
energy is not conserved during the complex formation stage. The ponderomotive
energy (cycle-averaged quiver energy) of the slow electron in a laser focal region
is E D h f i
2 . 4! f (Collins and Csanak 1991 ). If the energy and momentum
of the incident electron are E p E, p e p h f i ı ! f , and the transit time
L f =p through a beam with size L f is much longer than the laser pulse duration
l , then the limitation on the field strength becomes
h f i L f ! f = l :
The conditions depicted by Eqs. 2.6 , 2.7 , 2.8 , 2.9 , 2.10 ,and 2.11 allow
ignoring the ponderomotive interaction, which would be significant for relatively
fast electrons (Kibble 1966 ; Corkum et al. 1989 , 1992 ;Reiss 1990 ; Salamin and
Faisal 1997 ; Salamin 1997 ). Indeed, as a result of the analytical properties of
the Coulomb wave functions, the corresponding ponderomotive contributions to
matrix elements, resulting from the electric field inhomogeneity across the laser
focal region, are independent of E p and the scattering channel characteristics. The
resulting general shift of all energy levels in the “electron C molecular ion” system
can easily be eliminated by introducing a phase factor. For example, in the case of
a typical laboratory laser with ! f 10 1 , L f 10 8 ,and l 10 7 , it takes place
when the incident electron energy and the average field strength are E p
10 1 ,
and h f i 1, respectively.
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