Correlated continuum wave functions for three particles with Coulomb interactions
G. Gasaneo,* F. D. Colavecchia, and C. R. Garibotti
Centro Ato ´mico Bariloche and Consejo Nacional de Investigaciones Cientı ´ficas y Te ´cnicas, 8400 San Carlos de Bariloche,
Rı ´o Negro, Argentina
J. E. Miraglia and P. Macri
Instituto de Astronomı ´a y Fı ´sica del Espacio, Consejo Nacional de Investigaciones Cientı ´ficas y Te ´cnicas, Casilla de Correo 67,
Sucursal 28, 1428 Buenos Aires, Argentina
Received 29 August 1996
We present an approximate solution of the Schro ¨dinger equation for the three-body Coulomb problem. We
write the Hamiltonian in parabolic curvilinear coordinates and study the possible separation of the wave
equation as a system of coupled partial differential equations. When two of the particles are heavier than the
others, we write an approximate wave equation that incorporates some terms of the Hamiltonian that before
had been considered as a perturbation. Its solution can be expressed in terms of a confluent hypergeometric
function of two variables. We show that the proposed wave function includes a correlation between the motion
of the light particle relative to the heavy particles and verifies the correct asymptotic behavior when all
particles are far from each other. Finally, we discuss the possible uses of this function in the calculation of
transition matrices and differential cross sections in ionizing collisions. S1050-29479700504-0
PACS numbers: 34.50.Fa, 34.10.+x, 03.65.Nk
INTRODUCTION
The full three-body Coulomb problem has particular im-
portance in many areas of physics, especially in atomic col-
lisions. The initial and the final channel of ion-atom or
electron-atom collisions can be considered as three-body
Coulomb states in a first approximation when we assume that
only one electron of the target atom is active. The character-
istics of these processes are described by the transition ma-
trix in the post or prior form T
if
=
f
| V
f
|
i
+
=
f
-
| V
i
|
i
, respectively. The exact chan-
nel functions
i
+
(
f
-
) are not known in the three-body
case and then they should be replaced by proper approxima-
tions. These approximate wave functions also determine the
channel potential V
f
( V
i
) 1. From an experimental point of
view, the spectra of electrons emitted in the collisions reveal
the main features of these processes. Nowadays many double
differential cross sections DDCSs, in terms of the energy
and direction of the emitted electrons are available for a va-
riety of processes ion-atom or electron-atom ionization,
charge transfer, excitation, etc.. Recently, measurements of
differential cross sections that take into account the momenta
of the recoil atom have been reported 2. Triply differential
cross sections TDCSs, in which the momentum of the pro-
jectile is considered, are known in some particular geom-
etries of electron-atom ionization 3. However, they remain
an open question for ion-atom collisions.
The choice of approximate wave functions has been found
to be critical when comparing theory with experimental re-
sults. Initial attempts to describe total cross sections TCSs
were carried out decades ago. The wave functions included
in that theory were simply plane waves for the three outgo-
ing particles and their results described the TCS very roughly
in fast ion-atom collisions. The main drawback of this ap-
proach is that it does not take into account the long-range
behavior of the Coulomb potential among the particles. Fur-
ther approximate wave functions were based on the exact
solution of the two-body Coulomb problem that can be writ-
ten in terms of the confluent hypergeometric function
1
F
1
.
The first Born approximation FBA relies on the assumption
that the ejected electron completely screens the target poten-
tial, including a final state described by the free wave func-
tion of the projectile leaving the collision region, while the
electron interacts with the target through the Coulomb poten-
tial 4. In this way, the final wave function is a product of a
plane wave and the solution of the two-body Coulomb prob-
lem electron target. This approximation has been useful in
describing the single differential cross sections in the high-
impact-energy regime, but fails to reproduce the well-known
electron capture to the continuum peak that appears in the
double differential cross sections. This effect can only be
understood with the introduction of the projectile-electron
interaction, which is treated perturbatively in the FBA 5,6.
If we focus our attention on ion-atom ionization pro-
cesses, there are many theories that incorporate the
projectile-electron interaction at the final channel, but with
different approximate initial states. The continuum distorted
wave CDW eikonal initial state EIS approximation 7
and the impulse approximation IA8 include the electron-
target and the electron-projectile interactions in the final
channel and the projectile-target interaction is introduced in
an eikonal way. The Coulomb projectile-target interaction is
included on an equal basis in the multiple scattering MS
approximation 9. All these theories show qualitative agree-
ment with DDCS in fast bare ion-atom ionizations. In spite
of the relative success of these theories in the description of
the overall features of the DDCS, many discrepancies remain
unexplained.
*Permanent address: Departamento de Fı ´sica, Universidad Nacio-
nal del Sur, Avenida Alem 1253, 8000 Bahı ´a Blanca, Argentina.
PHYSICAL REVIEW A APRIL 1997 VOLUME 55, NUMBER 4
55 1050-2947/97/554/280912/$10.00 2809 © 1997 The American Physical Society