Elastic differential cross section and critical point for positron-hydrogen collisions
Arijit Ghoshal
1
and Puspajit Mandal
2
1
Department of Mathematics, Suri Vidyasagar College Suri 731 101 West Bengal, India
2
Department of Mathematics, Visva-Bharati University Santiniketan 731 235 West Bengal, India
Received 22 April 2005; published 14 October 2005
A detailed study is made of the elastic differential cross section in positron-hydrogen collisions using the
accurate Schwinger variational results for partial waves up to l = 19 of the preceding paper Ghoshal and
Mandal, Phys. Rev. A, 72, 042709 2005. Dramatic behavior of the differential cross section is evident in the
surface plots, suggesting interference of partial-wave contributions at small and large scattering angles for the
appearance of rich structure. It is found that the behavior of the critical point basically characterizes the nature
of the differential cross section. Further interferences of the scattered waves for different angular momentum
states produced by the interaction between the incoming positron and the target hydrogen atom determine such
behavior and consequently build up the nature of the differential cross section. These findings reveal the
remarkable fact that positron-hydrogen scattering almost does not occur in the direction around 87.3° at an
incident energy of about 3.4 eV, corresponding to incident positron momentum of 0.5 atomic units.
DOI: 10.1103/PhysRevA.72.042710 PACS numbers: 34.80.Gs, 34.70.e, 36.10.Dr
I. INTRODUCTION
In the preceding paper, paper I 1, the authors have ap-
plied the Schwinger variational method to investigate elastic
positron-hydrogen collision. With the help of the correlated
dipole-polarized basis, accurate results of the phase shifts
have been reported for incident positron momentum in the
range 0.1–3.5 atomic units a.u.. These results are in confor-
mity with the available theoretical calculations using both
variational and nonvariational methods.
The theoretical study of differential cross section is of
utmost importance in collision phenomena not only because
the results of experiment are expressed by means of this
quantity, but also because this quantity characterizes the col-
lision process. The nature of the differential cross section is
basically characterized by the behavior of the critical points.
These points are defined to be those incident energies or
scattering angles for which the differential cross section as-
sumes its minimum value. A careful observation of the exis-
tence and behavior of the critical points unfolds some dra-
matic properties of the differential cross section in positron-
hydrogen collisions.
In spite of having a vital role in determining the detailed
nature of the differential cross section, the study of critical
points has not drawn the serious attention of workers. One of
the reasons for this seems to be that accurate results of the
scattering parameters are limited to a few partial waves.
However, using empirical relationships satisfied by the phase
shifts, Wadhera et al. 2 predicted the existence of critical
points for positron–rare-gas-atoms collisions with Ar, Kr,
and Xe due to low-energy positron diffraction. An analysis
by Buhring 3 in the case of electron-atom collisions has
shown that elastically scattered electrons are fully polarized
when the scattering angle and the incident energy correspond
to the critical values. The origin of this polarization effect
has been found to be the spin-orbit coupling which arises in
a many-electron atomic system.
Of late, using accurate variational results Kar et al. 4
were able to demonstrate in an elegant way the existence and
behavior of critical angles in elastic positron-hydrogen colli-
sion in the low-energy region with the help of surface plots.
In this paper our main objective is to make a detailed
study of the elastic differential cross section for positron-
hydrogen collisions using the accurate Schwinger variational
results of the preceding paper 1. Use of surface plots is
made to have a concrete idea about the existence and behav-
ior of the critical points which determine the nature of the
differential cross section. In the process we give a detailed
analysis of the occurrence of a primary minimum and sec-
ondary maximum in the differential cross section by display-
ing striking features in their structures.
The plan of the paper is as follows. In Sec. II we discuss
our approach for obtaining the elastic differential cross sec-
tion in positron-hydrogen collisions. Section III is devoted to
the discussions of the results as obtained by the present cal-
culations. Finally in Sec. IV we make our concluding re-
marks.
II. THEORY
The elastic differential cross section for positron-
hydrogen scattering is obtained as
d
d
= f k
f
, k
i
2
units of a
0
2
= f
r
k
f
, k
i
2
+ f
i
k
f
, k
i
2
units of a
0
2
, 1
where f
r
k
f
, k
i
and f
i
k
f
, k
i
are, respectively, the real and
imaginary parts of the scattering amplitude f k
f
, k
i
. In order
to obtain convergence in the results, it is required to take a
large number of partial-wave contributions into account. This
is accomplished by making use of our accurate Schwinger
variational results for a maximum of L = 19 partial waves and
approximating the next higher partial-wave amplitudes with
the help of the effective-range formula of O’Malley et al. 5.
With this approximation f
r
k
f
, k
i
and f
i
k
f
, k
i
take the forms
PHYSICAL REVIEW A 72, 042710 2005
1050-2947/2005/724/0427109/$23.00 ©2005 The American Physical Society 042710-1