An optical potential method was applied to the calculation of positron scattering from the noble gases in order to determine the effect of open excitation channels on the shape of differential scattering cross sections

Bartschat, K.

Mceachran, R. P.

Stauffer, A. D.

English

NASA Technical Reports Server

N90-18971
NUMERICAL GREEN'S FUNCTIONS IN OPTICAL POTENTIAL CALCULATIONS
FOR POSITRON SCATTERING FROM ARGON AND NEON
K. Bartschat t, 1_. P. McEachran$ and A. D. Stauffer$
tDepartment of Physics and Astronomy, Drake University,
Des Moines, Iowa 50311, U.S.A.
_Physics Department, York University, Toronto, Canada M3J 1P3
ABSTRACT
We have applied an optical potential method to the
calculation of positron scattering from the noble gases in
order to determine the effect of open excitation channels
on the shape of differential scattering cross sections.
THEORY
In positron-atom scattering the usual close-coupling
expansion for the total wavefunction in terms of the sum
of products of the bound-state wavefunctions of the tar-
get atom and the one-projectile scattering wavefunctions
leads to the following set of integro-dlfferential equations
for the radial parts F_ of the scattering wavefunctions:
-,_,ld_2 l_(l,r 2+ 1) + F,(,)
= (I)
2
Here the potential terms _j are given by
N 1
ViJ(r)'-Z6_J-_'-_<"_'r {'k-_'I ['j> (2)
k=1
and the @_ are the bound-state wavefunctions of the N-
electron target.
In practice only a finite number of bound target states
can be included in the dose-coupling expansion. Hence,
we approximate the effect of the higher discrete target
states as well as the ionization continuum by means of an
optical potential. We divide the space of scattering func-
tions into P- and Q-spaces. We choose for the P-space
the elastic channel only and thus the Q-space contains
all inelastic channels. In the Q-space we neglect all cou-
plings between different channels but retain the couplings
between the P- and Q-spaces. Thus our method requires
the solution of the inhomogeneous differential equation
In either case the solution to equation (3) can be writ-
ten as
/0 °°
FdCr) = -2 dr' Gi(r, r') V_o(r' ) Fo(r' ) (4)
where the Green's function G i(% r') is given by
1
C,(r,/) = V,/"(t'r<) [g,,(k,r>) + i/,,(k,r>)] (5)
The functions .fz and gz are the regular and irregular
solutions of equation (3) with the right-hand-side put to
zero.
Upon substitution of equation (4) into the P-space
form of equation (1) we obtain
/a2_r2 l(l r _+1) 2 Voo(r) + k 2) Fo(r )
= - V_)_t(r) Fo(r ) (6)
where the optical potential is given by
Fo(,.)
= 4 dr'Vo,(r ) G,(r, r') V_o(r' ) Fo(r' ) (7)
The real part of the optical potential represents polariza-
tion while the imaginary part represents absorption due
to the inelastic channels.
When the V_i axe ignored, f_ and gz are the Pdccati-
Bessel functions. 1 However, if we retain the diagonal po-
tentials in equation (3), then fz and g_ have to be found
by a numerical solution of the homogeneous differential
equation.
2v,,(,)+ F,(,)
: 2 V_o(r ) F0(r ) (3)
for the radial functions F_ belonging to the Q-space. Here
F 0 is the P-space (elastic) channel wavefunction. In our
previous work 1 we ignored the diagonal term V_¢ above
and solved equation (3) by means of the free-particle
Green's function involving the standard Pdccati-Bessel
functions.
RESULTS
We have extended our previous work on argon 1 by
using the numerical Green's functions in the optical po-
tential. We have also carried out similar calculations for
positron scattering from neon.
The overall effect of retaining the diagonal potentials
and hence using numerical Green's functions is quite
small, i.e. of the order of a few percent of the differ-
ential cross sections at all angles.
95
PRECEDING PAGE BLANK NOT FILMED
https://ntrs.nasa.gov/search.jsp?R=19900009655 2020-03-19T22:57:50+00:00Z
yThe results of using optical potentials for positron-
neon scattering yields cross sections whose behaviour is
very similar to that of argon. In particular, the dis-
tinct minimum in the differential cross section which we
obtained in our previous polarized-orbital calculations 2
is no longer present when the optical potential is used.
Below we present some typical results for positron scat-
tering from argon and neon.
I
10 ,,,............. , .............. , .............. , ..............
\
,r4
I
vl
N
0
b
Figure 1 illustrates the differential cross section for
positron scattering from argon at 30 eV. The polarized-
orbital calculation 2 shows a deep minimum at 21 ° while
the two optical potential calculations, which differ only
slightly, do not exhibit any such behaviour. The normal-
ized experimental data s clearly favour the latter calcula-
tions.
,1 i • , | = i • • • i = , , 1 ! ! i i t i | i | | • , • , I ! I i i i i | i • i i i • • i i i • • i i • • • i • • i i
0 30 60 90 120
0 (aeg)
FIG. 1. Positron scattering from argon at 30 eV. (_), ten-state optical potential using free:
wave Green's functions1; (- - -), ten-state optical potential using numerical Green's functions;
(---), polarized-orbital approximaticn2;., experimental data normalized at 60 °3
I
Figure 2 illustrates similar results for positron scat- shape of the polarized-orbital calculations. 2
tering from neon at 20 eV. Finally in figure 3 we show
ACKNOWLEDGMENTS
positron scattering from argon at 8.5 eV. At this energy This research was supp0rted by the Natural Sciences
only the elastic channel is open. Here the experimental
and Engineering Research Council of Canada and by thedata clearly show a minimum and agree well with the
Deutsche Porschungsgemeinschaft.
K. Floeder, P. H6ner, W. Raith, A. Schwab, G. Sinapius
and G. Spicher, Phys. Rev. Left. 60, 2363, 1988.
4 W. E. Kauppila, S. 3. Smith, C. K. Kwan and T. S.
Stein, Proc. of the Workshop on Annihilation in Gases
and Galaxies, to be published.
z K. Bartschat, R. P. McEachran and A. D. Stauffer, J.
Phys. B. 21, 2789, 1088.
_ R. P. McEachran and A. D. Stauffer, Proc. of the Third
Int. Workshop on Positron (Electron )-Gas Scattering,
ed. W. E. Kauppila, T. S. Stein and 3. W. Wadehra
(Singapore: World Scientific) p. 122, (1980).
h
96
=
=
0.8
0.7
0.6
+-i
I
0.5
N
Q
ml 0.4
0.3
b
"1=1
0.2
0.1
0.0
_i ii iii i i i i i iIi i i l i i ii i i I I IIII |T I I I IIII I III III III IIIIII II IIIIIII IIIIII
:
i //-" "-'::'I-_. ___..
s'"
ss*
0
ill 11 . iiii lll I ill III I llllll_lll III II lllJll I + I III II II III II[ _II IIIII I
30 60 90 120
9 (deg)
FIG. 2. Positron scattering from neon at 20 eV. (--), fourteen-state optical potential using
free-wave Green's functions; (- - -), fourteen-state optical potential using numerical Green's
functions; (- - -), polarized-orbital approximation2; *, experimental data normalized at 90.04
l
m
N
'x3
b
'13
iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii
I0
]e\"_ /
0 30 60 90
0 (deg)
FIG. 3. Positronscatteringfrom argonat 8.5eV. (---),polarized-orbitalapproximation2;
% experimental data norma_zed at 45 °a
i
2O
97
rz_ x


Numerical Green's functions in optical potential calculations for positron scattering from argon and neon

Abstract

Similar works

Full text

Available Versions

NASA Technical Reports Server