Simultaneous Optical Model Analyses of Elastic Scattering, Breakup, and Fusion Cross Section Data for the $^{6}$He + $^{209}$Bi System at Near-Coulomb-Barrier Energies

Abstract

Based on an approach recently proposed by us, simultaneous $\chi^{2}$-analyses are performed for elastic scattering, direct reaction (DR) and fusion cross sections data for the $^{6}$He+$^{209}$Bi system at near-Coulomb-barrier energies to determine the parameters of the polarization potential consisting of DR and fusion parts. We show that the data are well reproduced by the resultant potential, which also satisfies the proper dispersion relation. A discussion is given of the nature of the threshold anomaly seen in the potential.

Full text

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arXiv:nucl-th/0111061v1 22 Nov 2001
Simultaneous Optical Model Analyses of Elastic Scattering,
Breakup, and Fusion Cross Section Data for the
6He +209Bi System at Near-Coulomb-Barrier Energies
B. T. Kim, W. Y. So, and S. W. Hong
Department of Physics and Institute of Basic Science,
Sungkyunkwan University, Suwon 440-746, Korea
T. Udagawa
Department of Physics, University of Texas, Austin, Texas 78712
Abstract
Based on an approach recently proposed by us, simultaneous χ2-analyses are per-
formed for elastic scattering, direct reaction (DR) and fusion cross sections data for the
6He+209Bi system at near-Coulomb-barrier energies to determine the parameters of the
polarization potential consisting of DR and fusion parts. We show that the data are well
reproduced by the resultant potential, which also satisfies the proper dispersion relation.
A discussion is given of the nature of the threshold anomaly seen in the potential.
24.10.-i, 25.70.Jj
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A great deal of effort has recently been focused on studies of the so-called threshold
anomaly [1,2] (rapid energy variation in the strength of the optical potential) in heavy
ion scattering induced, particularly, by very loosely bound projectiles such as6He [3],
6Li [4–6], and9Be [7,8]. The experimental results accumulated so far indicate that the
imaginary part of the optical potential, W(r;E), extracted by analysis of the elastic
scattering data, does not show such an anomaly as is observed in the potentials for
normal, tightly bound projectiles. For tightly bound projectiles, W(r;E) at around the
strong absorption radius r = Rsais found to decrease rapidly as the incident energy E
falls below the Coulomb-barrier energy Ec, and eventually vanishes at some threshold
energy E0. Contrary to this, for loosely bound projectiles W(Rsa;E) remains large at
energies even below Ec[3,5,6,8].
The reason for W(Rsa;E) being so large at low energies has been ascribed to the
weak binding of the extra neutrons to the core nucleus, leading to breakup. In fact,
the breakup cross sections have been measured for these projectiles [9–11], confirming
that they are indeed large, even larger than the fusion cross sections at E ∼ Ec. It
was argued [1] that since the energy dependence of the polarization potential due to the
breakup must be weak, one might not be able to observe a noticeable energy variation
in the resultant potential when the breakup cross section is larger than the fusion cross
section as for loosely bound projectiles.
The threshold anomaly of W(r;E) observed for tightly bound projectiles may be
ascribed to the coupling of the elastic and fusion channels [12]. This is substantiated by
the fact that the threshold energy E0of W(r;E) (i.e., the energy where W(r,E0) = 0)
agrees very well with that of the fusion cross section σF, or more precisely the threshold
energy of S(E) ≡√EσF[13]. It is thus natural that if the breakup cross section is larger
than the fusion cross section, and if one is concerned only with the total W(r;E), the
rapid change in the fusion cross section and the anomaly would not show up clearly in
the total W(r;E).
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Insight into this problem may be obtained if one decomposes the total W(r;E) into
the direct reaction (DR) and fusion parts, WD(r;E) and WF(r;E), respectively, and
determines them separately [14]. The aim of the present study is to make such a deter-
mination of WD(r;E) and WF(r;E) by performing simultaneous χ2-analyses of elastic
scattering, DR (breakup), and fusion cross section data. We take the6He+209Bi system,
for which data are available not only for elastic scattering [3], but also for breakup [9]
and for fusion [15]. Following Ref. [3], we identify the breakup cross section with the DR
cross section. Optical model analyses of the elastic scattering and total reaction cross
section data have already been presented in Refs. [3,16]. The present analysis is thus an
extension of the previous studies.
The optical potential U we use has the following form;
U = UC(r) − [V0(r) + V (r;E) + iW(r;E)], (1)
where UC(r) is the Coulomb potential, whose radius parameter is fixed at a standard value
of rc=1.25 fm, and V0(r) is the energy independent Hartree-Fock part of the potential,
while V (r;E) and W(r;E) are, respectively, real and imaginary parts of the so-called
polarization potential [17] that originates from couplings to reaction channels. W(r;E) is
assumed to have a volume-type fusion and a surface-derivative-type DR part. Explicitly,
V0(r) and W(r;E) are given, respectively, by
V0(r) = V0f(X0)(2)
and
W(r;E) = WF(r;E) + WD(r;E) = WF(E)f(XF) + 4WD(E)aDdf(XD)
dRD
, (3)
where f(Xi) = [1 + exp(Xi)]−1, with Xi= (r − Ri)/ai(i = 0, D and F), is the usual
Woods-Saxon function. The real part of the polarization potential is also assumed to have
DR and fusion parts; V (r;E) = VF(r;E) + VD(r;E). Each real part may be generated
from the corresponding imaginary potential by using the dispersion relation [1];
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Vi(r;E) = Vi(r;Es) +E − Es
π
P
?∞
0
dE′
Wi(r;E′)
(E′− Es)(E′− E),(4)
where P stands for the principal value and Vi(r;Es) is the value of the potential at a
reference energy E = Es. Later, we will use Eq. (4) to generate the final real polarization
potentials VF(r;E) and VD(r;E), after WF(r;E) and WD(r;E) have been fixed from
χ2-analyses. For V0(r), we simply use the potential determined for the α+209Bi system
at E=22 MeV [18], assuming that all the unusual features of the scattering may be
described by the polarization part of the potential, particularly by the DR part. The
parameters used for V0(r) are V0=100.4 MeV, r0=1.106 fm, and a0=0.54 fm.
The unusual behavior of the elastic scattering and DR data for loosely bound projec-
tiles can most dramatically be seen in plots of the ratios of the elastic differential cross
section (dσE/dΩ), and the DR cross section (dσD/dΩ), to the Rutherford scattering cross
section (dσc/dΩ), i.e.,
Pi≡dσi
dΩ/dσc
dΩ= (dσi
dσc),(i = E or D),(5)
as a function of the distance of closest approach D (or the reduced distance d) [19,20]
that is related to the scattering angle θ by
D = d(A1/3
1
+ A1/3
2 ) =1
2D0
?
1 +
1
sin(θ/2)
?
with D0=Z1Z2e2
E
. (6)
Here D0 is the distance of closest approach in a head-on collision (s-wave). Further,
(A1,Z1) and (A2,Z2) are the mass and charge of the projectile and target ions, respec-
tively, and E is the incident energy in the center-of-mass system.
In Fig. 1, we present such plots for two incident energies of E=18.5 and 21.9 MeV [9].
As seen, PE is close to unity for large d, but starts to decrease at an unusually large
distance of d =2.2 fm (≡ dI, interaction distance). This value is much larger than the
usual value of dI≈1.6 fm for normal, tightly bound projectiles. On the other hand, it
is remarkable that the sum, PE+ PD, remains close to unity until d becomes as small
as ≈ 1.7 fm, implying that the absorption in the elastic channel up to this distance, and
the unusual character of the scattering data, is due to the breakup.
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Since the theoretical cross sections are not very sensitive to the real polarization
potential, we tentatively treat it in a rather crude way in carrying out χ2-analyses;
we simply assume Vi(r;E) has the same radial shape as the imaginary part Wi(r;E):
Vi(r;E) = Vi(E)(Wi(r;E)/Wi(E)), Vi(E) being the strength of the real potential. We
then carry out χ2-analyses treating WF(E) and rD as adjustable parameters, keeping
all other parameters fixed as VF=3.0 MeV, rF=1.40 fm, aF=0.55 fm, VD=0.25 MeV,
WD=0.40 MeV and aD=1.25 fm. The necessity of varying aD or rD as a function of
E has been shown in previous studies [3,16], and in the present work we take rD as
a variable parameter to study as a function of E. In the χ2-analyses, data for elastic
scattering, angle-integrated total DR, and fusion cross sections at E =14.3, 15.8, 17.3,
18.6, and 21.4 MeV are employed.
The values of WF(E) and rD(E) fixed from the χ2-analyses are presented in Fig. 2
by the open and the solid circles, respectively. Each set of circles can be well represented
by (in MeV and fm, respectively, for WF(E) and rD(E))
WF(E) =















0, E ≤15.4
, 15.4< E ≤18.5
, 18.5≤ E
1.25(E − 15.4)
4.0
(7)
and
rD(E) =















1.73, E ≤14.0
, 14.0< E ≤21.4
, 21.4≤ E.
1.73 − 0.03(E − 14.0)
1.508
(8)
Note that the threshold energy E0=15.4 MeV, at which WF(E) = 0, is set equal to that
of the linear representation of quantity S(E) =√EσF ∝ (E − E0) discussed earlier.
Kolata et al. [15] found the value to be 15.4 MeV, which is used in Eq. (7). At this
moment, we have no experimental information on rD-values below 14.0 MeV and above
21.4 MeV. Thus, in Eq. (8), we tentatively set rD(E) to be a constant as 1.73 fm for
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