Extended Optical Model Analyses of Elastic Scattering and Fusion Cross Section Data for the 7Li+208Pb System at Near-Coulomb-Barrier Energies using the Folding Potential

Abstract

Simultaneous $\chi^{2}$ analyses previously made for elastic scattering and fusion cross section data for the $^{6}$Li+$^{208}$Pb system is extended to the $^{7}$Li+$^{208}$Pb system at near-Coulomb-barrier energies based on the extended optical model approach, in which the polarization potential is decomposed into direct reaction (DR) and fusion parts. Use is made of the double folding potential as a bare potential. It is found that the experimental elastic scattering and fusion data are well reproduced without introducing any normalization factor for the double folding potential and that both the DR and fusion parts of the polarization potential determined from the $\chi^{2}$ analyses satisfy separately the dispersion relation. Further, we find that the real part of the fusion portion of the polarization potential is attractive while that of the DR part is repulsive except at energies far below the Coulomb barrier energy. A comparison is made of the present results with those obtained from the Continuum Discretized Coupled Channel (CDCC) calculations and a previous study based on the conventional optical model with a double folding potential. We also compare the present results for the $^7$Li+$^{208}$Pb system with the analysis previously made for the $^{6}$Li+$^{208}$Pb system.

Full text

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arXiv:0706.0586v1 [nucl-th] 5 Jun 2007
Extended Optical Model Analyses of Elastic Scattering and
Fusion Cross Section Data for the7Li+208Pb System at
Near-Coulomb-Barrier Energies using the Folding Potential
W. Y. So and T. Udagawa
Department of Physics, University of Texas, Austin, Texas 78712
K. S. Kim
Department of Liberal Arts and Science,
Hankuk Aviation university, Koyang 412-791, Korea
S. W. Hong and B. T. Kim
Department of Physics and Institute of Basic Science,
Sungkyunkwan University, Suwon 440-746, Korea
Abstract
Simultaneous χ2analyses previously made for elastic scattering and fusion cross
section data for the6Li+208Pb system is extended to the7Li+208Pb system at near-
Coulomb-barrier energies based on the extended optical model approach, in which
the polarization potential is decomposed into direct reaction (DR) and fusion parts.
Use is made of the double folding potential as a bare potential. It is found that the
experimental elastic scattering and fusion data are well reproduced without intro-
ducing any normalization factor for the double folding potential and that both the

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DR and fusion parts of the polarization potential determined from the χ2analyses
satisfy separately the dispersion relation. Further, we find that the real part of the
fusion portion of the polarization potential is attractive while that of the DR part
is repulsive except at energies far below the Coulomb barrier energy. A comparison
is made of the present results with those obtained from the Continuum Discretized
Coupled Channel (CDCC) calculations and a previous study based on the conven-
tional optical model with a double folding potential. We also compare the present
results for the7Li+208Pb system with the analysis previously made for the6Li+208Pb
system.
PACS numbers : 24.10.-i, 25.70.Jj
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I.INTRODUCTION
It has been a long standing problem that one is forced to reduce the strength of
the folding potential by a factor N = 0.5 ∼ 0.6 in order to reproduce the elastic scat-
tering data [1, 2] for loosely bound projectiles such as6Li and9Be within the optical
model approach with a folding potential. This problem has been ascribed to the strong
breakup character of the projectiles; studies have been made of the effects of the breakup
on the elastic scattering, based on the coupled discretized continuum channel (CDCC)
method [3, 4]. These studies were very successful in reproducing the elastic scattering
data without introducing any arbitrary normalization factor and furthermore in under-
standing the physical origin of the factor N = 0.5 ∼ 0.6 needed to be introduced in
one-channel optical model calculations. The authors of Refs. [3, 4] projected their cou-
pled channel equations to a single elastic channel equation and deduced the polarization
potential arising from the coupling with the breakup channels. The resultant real part
of the polarization potential was then found to be repulsive at the surface region around
the strong absorption radius, Rsa. This shows that the reduction of the folding potential
by a factor of N = 0.5 ∼ 0.6 is to effectively take into account the effects of this repulsive
coupling with the breakup channels.
In our recent study [5], we explored this problem for the6Li +208Pb system in the
framework of the extended optical model [6, 7, 8], in which the optical potential con-
sists of the energy independent Hartree-Fock part and the energy dependent complex
polarization potential having two components, i.e., the direct reaction (DR) and fusion
parts, which we call the DR and fusion potentials, respectively. In Ref. [5], using such an
extended optical potential, we performed the simultaneous χ2analyses of the elastic scat-
tering and fusion cross section data, determining the two components of the polarization
potentials as functions of the incident energy. Our expectation was that the resulting
real part of the DR potential would become repulsive consistently with the results of the
CDCC calculations. Indeed the real DR polarization potential turned out to be repulsive.
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In addition, it was shown that both the DR and fusion potentials satisfy the dispersion
relation [9, 10] separately.
In this work, we extend the analysis made in Ref. [5] to the7Li+208Pb system. In this
system, such a normalization anomaly as observed in6Li+208Pb does not appear around
the Coulomb-barrier energies; the normalization factor N necessary for reproducing the
data is close to unity, N ≈ 1 (see Ref. [2]), in contrast to the factor N = 0.5 ∼ 0.6 for
6Li+208Pb.
In Sec. II of this article, we first discuss some characteristic features of elastic scat-
tering cross section data of7Li+208Pb [2] in comparison with those of6Li+208Pb. It will
be shown in the comparison that the DR cross section for7Li+208Pb is expected to be
significantly smaller than that for6Li+208Pb. In Sec. III, we first generate the so-called
semi-experimental DR cross section, σsemi-exp
D
, from the elastic scattering and fusion cross
section data [11, 12], following the method described in, e.g., Ref. [13]. (Note that use
is made of the fusion cross section data of7Li+209Bi, since the data are not available
for7Li+208Pb.) The data of σsemi-exp
D
are needed for separately determining the DR and
fusion potentials. χ2analyses of the elastic scattering, fusion, and semi-experimental DR
cross section data are then carried out in Sec. IV. In Sec. V, a comparison is made of
the present results with those obtained from the CDCC calculations [4] and a previous
study [2] based on the conventional optical model with a double folding potential. We
also show a comparison of the present result with the analysis previously made by us [5]
for the6Li+208Pb system. Section VI concludes the paper.
II.REVIEW OF EXPERIMENTAL CROSS SECTIONS
We begin with the discussion of some of the characteristic features of the elastic scat-
tering cross sections dσel/dσΩdata for7Li+208Pb in comparison with those for6Li+208Pb.
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Such features can best be seen in the ratio, PE, defined by
PE≡dσel
dσΩ/dσC
dσΩ
= dσel/dσC
(1)
as a function of the distance of the closest approach D (or the reduced distance d), where
dσC/dσΩis the Coulomb scattering cross section, while D (d) is related to the scattering
angle θ by
D = d(A1/3
1
+ A1/3
2 ) =1
2D0
?
1 +
1
sin(θ/2)
?
(2)
with
D0=Z1Z2e2
E
, (3)
D0being the distance of the closest approach in a head-on collision. Here (A1,Z1) and
(A2,Z2) are the mass and charge of the projectile and target ions, respectively, and
E ≡ Ec.m.is the incident energy in the center-of-mass system. PEas defined by Eq. (1)
will be referred to as the elastic probability.
In Figs. 1(a) and 1(b), we present the experimental values of PEfor incident energies
around the Coulomb barrier energy as a function of the reduced distance d for7Li+208Pb
and6Li+208Pb, respectively. As seen, the values of PE at different energies line up to
form a very narrow band. This is a characteristic feature seen in many of the heavy-ion
collisions, reflecting the semiclassical nature of these collisions. PEremains close to unity
until two ions approach each other within a distance dI, where PEbegins to fall off. The
distance dI is usually called the interaction distance, at which the nuclear interactions
between the colliding ions are switched on, so to speak. The values of dIare about 1.9 fm
for6Li+208Pb and 1.8 fm for7Li+208Pb.
As argued in Ref. [13], the fall off of the PE values in the region immediately next
to dIis due to DR. The fact that the dI-value (1.9 fm) for6Li+208Pb is larger than the
dI(1.8 fm) for7Li+208Pb shows DR starts to take place at larger distances for6Li+208Pb
than it does for7Li+208Pb. Also, it can be seen that the amount of decrease of the
PE value from unity in6Li+208Pb is significantly larger than in7Li+208Pb at 1.5 fm
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