exotic nuclei structure and reaction noyaux exotiques ... - IPN - IN2P3

Spectral analysis in the case of a complex potential.
IPNO Participation: R.J. Lombard
Collaboration : R. Mezhoud, Faculty of Sciences, Boumerdes University, Boumerdes, Algeria, and F.
-Z. Ighezou, Laboratoire de Physique Théorique, USTHB, Bab Ezzouar, Alger, Algeria
Nous avons étendu au cas des potentiels complexes, une méthode développée pour résoudre le problème
inverse pour des potentiels locaux à partir des états liés. Nous présentons un premier exemple entièrement
soluble basé sur le potentiel de Kratzer complexe. Ce modèle permet de tester la méthode et de mettre en
évidence ses limitations. La méthode est appliquée ensuite à deux atomes hadroniques présentant des
caractéristiques très différentes : Π - – 28 Si and K - - 208 Pb.
The present work is dealing with the inverse problem
from bound states in the case of a complex
potential. At first glance, this domain is thought as
the natural extension of techniques developed for
real potentials. The method relies on a connection
between the moments of the ground state density
and the energies of the yrast levels. It is well suited
for systems with a large (infinite) number of bound
states. Hadronic atoms, for instance, could be a
good field of application.
As the question was studied thoroughly, limitations
appeared. First, the method does not determine
the shape of the imaginary part of the potential,
though it has an influence on the real part of the
eigenvalues and on the width. Assuming the imaginary
component to be known, either from scattering
experiment or from a sound model, the present
method could determine the real part of the potential.
It turns out, however, that the effective potential
governing the radial wave function is dependent
on the state through the repulsive contribution
generated by the imaginary part. This could be
redhibitory, since the method developed for the
real case implies local (state independent) potentials.
Taken as an illustrative example, the case of the
complex Kratzer potential shows, however, that
this difficulty can be overcome, taking into account
the shift in energy produced by the state dependence.
This open an opportunity, and, in the special
case of the Kratzer potential, it leads to the exact
solution. In other words, in this case the method
we have developed is able to recover the real part
of the potential.
The present work has been completed by the
study of two hadronic atoms : Π - – 28 Si and
K - - 208 Pb. In the first case use is made of the experimental
data, in the second one, the calcula-
tions of Baca et al have been taken as pseudo
data. Although the conditions are quite different,
the two examples yield similar answers. The
ground state density is well determined outside the
nuclear volume. At short distances, roughly speaking
from the origin to the nuclear surface, the present
method lacks of sensitivity. This is due to the
strong repulsion, which reflects a repulsive nuclear
interaction in the case of pionic atoms, and a
strong state dependence in the kaonic case. Consequently,
the real part of the nuclear potential can
only be reached at a qualitative level. For the
kaonic case, the surface shape is obtained, while
the inside part get the wrong sign. Note that only a
tiny fraction of the density is responsible for this
shortcoming.
The fact that with approximative nuclear potentials
the spectra are reproduced to better than 1 \%,
clearly indicates that the eigenvalues are not sensitive
to details of the potential over the nuclear
volume. It means that a model independent determination
of the real part of the nuclear potential is
barely possible.
However, the good determination of the ground
state density, except at short distances, suggests
that the approach could be improved, in order to
get a better description inside the nuclear volume.
This is possible, though not obvious. It would require
to reproduce the eigenvalues of the yrast
levels to better than 0.1 \%. While a great accuracy
is possible with pseudo data resulting from calculations,
such a situation is difficult to reach experimentally.
References:
A. Baca, C. Garcia-Recio and J. Nieves, Nucl.
Phys. A 673 (2000) 335.
95