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Non-monotonic potentials and vector analyzing powers of 6,7Li scattering by 12C, 26Mg,
58Ni, and 120Sn
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2011 EPL 94 62002
(http://iopscience.iop.org/0295-5075/94/6/62002)
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June 2011
EPL, 94 (2011) 62002
doi: 10.1209/0295-5075/94/62002
www.epljournal.org
Non-monotonic potentials and vector analyzing powers of
6,7
Li scattering by 12C, 26Mg, 58Ni, and 120Sn
A. K. Basak1(a) , M. M. Billah2 , M. J. Kobra1 , M. K. Sarkar1 , M. Mizanur Rahman1 , Pretam K. Das1 ,
S. Hossain3 , M. N. A. Abdullah4 , A. S. B. Tariq1 , M. A. Uddin1 , S. Bhattacharjee1 , I. Reichstein5
and F. B. Malik6,7
1
Department of Physics, University of Rajshahi - Rajshahi, Bangladesh
Department of Physics, Rajshahi University of Engineering & Technology - Rajshahi, Bangladesh
3
Department of Physics, Shahjalal University of Science & Technology - Sylhet, Bangladesh
4
Department of Physics, Jagannath University - Dhaka, Bangladesh
5
School of Computer Science, Carleton University - Ottawa, ON K1S 5B6, Canada
6
Department of Physics, Southern Illinois University - Carbondale, IL 62901, USA
7
Department of Physics, Washington University - St. Louis, MO 63130, USA
2
received 14 February 2011; accepted in final form 27 April 2011
published online 9 June 2011
PACS
PACS
PACS
25.70.Bc – Elastic and quasielastic scattering
24.10.Ht – Optical and diffraction models
24.70.+s – Polarization phenomena in reactions
Abstract – The data on the elastic scattering cross-section (CS) and vector analyzing power
(VAP ) of 6,7 Li incident on 12 C, 26 Mg, 58 Ni and 120 Sn nuclei are analyzed in terms of an
optical model (OM) potential, the real part of which is generated from a realistic two-nucleon
interaction using the energy-density functional (EDF) formalism. The EDF-generated real part
of the potential is non-monotonic (NM) in nature. This NM real potential part, without any
renormalization, along with an empirically determined imaginary part and spin-orbit potential,
embodying the underlying physics of projectile excitation, can successfully account for both CS
and VAP data in all four cases. This investigation, for the first time, using the simple OM analysis
accounts well for the opposite signs of the VAP data of elastically scattered 6,7 Li by 58 Ni at
Elab ≈ 20 MeV and by 120 Sn at Elab = 44 MeV. The ramification of successfully describing the
data by the EDF-generated potential to the equation of state of nuclear matter is discussed.
open access
c EPLA, 2011
Copyright Introduction. – An understanding of the interaction between lithium isotopes, the heaviest formed in
primordial nucleosynthesis, and other nuclei is likely to
be important for our understanding of stellar nucleosynthesis. Except in some recent studies [1–3], Li-nucleus
potentials are usually phenomenological Woods-Saxon
(WS) [4,5], squared WS (SWS) [6], or microscopic
double folding (DF) [6]. However, it is difficult to obtain
consistent global parameters of WS and SWS potentials
and they suffer from discrete [7] and continuous [8]
ambiguities. In the DF approach, potentials are generated
from the two-nucleon (NN) M3Y interaction [9] which
does not contain a tensor component that is critical to our
understanding of deuteron magnetic dipole and electric
quadrupole moments. Moreover, the DF method neglects
(a) E-mail:
akbasak@gmail.com
a proper consideration of the Pauli principle among the
nucleons. As a consequence, the DF potentials are able
to explain, using an empirical energy-dependent renormalization factor, the elastic scattering data of 6,7 Li [10]
only. Sakuragi and his group [11] demonstrated elegantly
that the renormalization may be done away with through
the generation of a repulsive dynamic polarization
potential (DPP) using a coupled discretized continuum
channels (CDCC) method [12] in conjunction with the
DF potential.
Alternatively, nucleus-nucleus potentials have been
derived [2] from a realistic NN potential using the
energy-density functional (EDF) [13] approach. As noted
in [1,2], the basic ingredients for deriving nucleus-nucleus
potentials using this method are: i) a realistic NN
potential, ii) experimental density distribution (DD)
functions for each of the colliding nuclei, and iii) local
62002-p1
A. K. Basak et al.
density approximation to determine the nucleonic mean
field that incorporates the Pauli principle. In the sudden
approximation version of this method, the functional
form of the nucleus-nucleus potential is non-monotonic
(NM) as a consequence of the Pauli principle. The latter
principle is not explicitly considered in other forms of
the EDF theory existing in the literature, such as those
based on the Skyrme type of the NN interaction [14]. The
coupled-channels (CC) and CDCC analyses in [3] demonstrated explicitly that the NM aspect of the real central
potential in conjunction with an optimum imaginary
part triggers the correct DPP leading to an adequately
effective SO potential to reproduce simultaneously the
CS and VAP data of the 6 Li-28 Si elastic scattering
without any need for renormalization or adjustment of
the parameters of the real part [3].
This motivated us to extend the applicability of the
optical model (OM) analyses of the 6 Li-28 Si elastic
scattering data in [1,2] to examine the 6,7 Li elastic
scattering by a wide mass range of targets, 12 C, 26 Mg,
58
Ni and 120 Sn, at different incident energies. The prime
focus of the present investigation is to address the
challenging situation of the opposite signs of available
VAP data of 6 Li and 7 Li elastic scattering on 58 Ni at
Elab ≈ 20 MeV [15,16] and on 120 Sn at 44 MeV [12,17] in
addition to accounting for their CS data. Only at our
selected energies, complete angular distributions of the
CS and VAP data of both the 6,7 Li elastic scattering are
available currently. The large difference in deformations
of 6 Li and 7 Li is likely to produce drastically different
DPP effects through their excitation process, as stressed
in the work of Nishioka et al. [18]. The very distinctive
dynamics in the excitation process of 6,7 Li may stimulate
a large difference in the correct SO potentials if the
nature of the central potential is chosen properly.
A successful application of potentials to determine
the elastic scattering CS as well as VAP data may
provide important information on the equation of state
(EOS) of nuclear and nucleonic matter as a function of
density, as noted in [19], including that at the saturation
density. This contrasts with the fact that the excitation
of breathing modes of nuclei [20] provides a knowledge of
incompressibility modulus K at the saturation density. K
at the saturation density for the EDF-generated potentials
herein and in [19] is 187 MeV. Both the studies use the
realistic NN potential of Gammel and Thaler [21].
Background. – Since the installation of the polarized
Li beams in Heidelberg [22] in the 1970s, studies of
the polarization effect in 6,7 Li-induced reactions have
drawn considerable attention. In particular, measured
VAP magnitudes of the 6,7 Li elastic scattering are found
large compared to the expected size of their spin-orbit
(SO) potentials. In the case of a composite projectile of
mass number AP , only a small fraction of the nucleons
do not form spin-zero pairs and the net SO interaction
[18]. Hence to
is expected to be proportional to A−1
P
6,7
describe the measured VAP, it is conjectured [22,23] that
the effective SO potential of 6,7 Li stems from the resultant of static and dynamic spin-dependent interactions.
The relative importance of these two spin-dependent
interactions and its energy dependence are subjects of
considerable interest [12,23]. The work of Basak et al. [3]
demonstrates that the effective SO potential, generated
through DPP alone, can account for the 22.8 MeV VAP
data of the 6 Li-28 Si elastic scattering well without having
a static SO term. This indicates that the static SO
potential is small at this energy.
Complete 6,7 Li potentials should include spindependent parts along with the central one. While
6
Li, with spin S(6 Li) = 1, has both first- and second-rank
SO terms; 7 Li with S(7 Li) = 3/2, bears up to third-rank
SO terms in the total potential. The works of refs. [18,24]
suggest that while the tensor analyzing power (T AP ) data
of the 6,7 Li elastic scattering on 58 Ni at Ecm (Li) = 12.7
and 18.1 MeV can be well accounted for by static tensor
potentials, the VAP data cannot be reproduced by static
SO potentials. In particular, the puzzling situation of
the iT11 data of the 6,7 Li elastic scattering on 58 Ni,
which are of opposite signs, is not expected to be
reproduced by static SO potentials. However, the iT11
data of the 6,7 Li-58 Ni systems could be reproduced in
the CC calculations [18,24] through the generation of
DPP in the projectile excitation process. In both the
works, the analyses have been accomplished using the
projectile-target potential derived in the cluster folding
(CF) model.
The need for using NM potentials is further substantiated in the investigation of Pakou [25], who demonstrated explicitly that a CDCC calculation induces DPP,
Upol (R) = Vpol (R) + iWpol (R), having a repulsive real part
which effectively renormalized the real part obtained in
the DF method. The total effective 6,7 Li-28 Si potential, U (R), is obtained by adding to the calculated DF
potential, Ubare (R), generated using the BDM3Y1 NN
interaction potential [26], an empirically determined DPP
Upol (R). The NM potential, U (R) = Ubare (R) + Upol (R),
thus obtained, can reproduce the 13 MeV CS data of the
6,7
Li elastic scattering by 28 Si. Without the inclusion of
Upol (R), the DF Ubare (R) potential needs a renormalization factor of 0.5 to 0.6 to fit the data. The effect of DPP,
arising from CDCC calculations, on the 22.8 MeV VAP
data of the 6 Li +28 Si elastic scattering has been shown
explicitly in [3].
The very distinctive dynamics in the excitation process
of 6,7 Li, as noted in [18], may stimulate a large difference in the correct SO potentials if the nature of the
central potential is chosen properly. It is, therefore, of
great interest to investigate whether the unadjusted
EDF-generated NM potentials, with empirically searched
imaginary and effective SO terms, in the simple OM
framework, succeed in describing particularly the opposite
signs of VAP data, observed for the 6,7 Li elastic scattering
by 58 Ni at Elab ≈ 20 MeV and by 120 Sn at 44 MeV in
62002-p2
Non-monotonic potentials and vector analyzing powers etc.
7
Li+12C
EDF
Parametrized
10
Parametrized
0
0
-10
-10
-20
-20
0
2
4
6
8
10
0
2
4
6
8
-30
10
10
10
26
Li+ Mg
7
26
Li+ Mg
EDF
EDF
Parametrized
Parametrized
0
0
-10
-10
-20
-20
-30
0
2
4
6
8
10
0
2
4
6
8
-30
10
10
10
6
58
Li+ Ni
7
58
Li+ Ni
EDF
EDF
Parametrized
0
Parametrized
0
-10
-10
-20
-20
-30
-30
0
2
4
6
8
10
0
2
4
Li+120Sn
Li+120Sn
6
8
10
7
6
0
EDF
0
EDF
Parametrized
VN (R) (MeV)
VN (R) (MeV)
6
addition to a wide range of CS and other VAP data in
terms of both target masses and incident energies.
VN (R) (MeV)
EDF
VN (R) (MeV)
VN (R) (MeV)
Binding energy
Calc.
Expt.
33.2
32.0
39.2
39.5
92.0
92.2
219.6
216.6
506.4
506.5
1023.2 1020.5
Optical potential parameters. – The real parts of
the central nuclear potentials for 6,7 Li are generated using
the DD functions in the EDF calculation, the details of
which are given in [2]. The energy-dependent part of the
mean field calculated using the NN potential [21] can
be represented by density-dependent polynomials [13,19]
with the coefficients, a1 = −0.2, a2 = 0.316, a3 = 1.646,
b1 = −741.28, b2 = 1179.89 and b3 = −467.54, which correspond to K ≈ 187 MeV [19] for symmetric nuclei. The nonhomogeneity parameter η = 8 [2] has been used for correction to the kinetic energy due to the finite size of the
nuclear matter and correlation effects, not included in the
mean field. The EDF calculations also involve the sudden
approximation ρ(r) = ρP (r) + ρT (r) with P and T referring to the projectile and target, respectively. The sources
of the DD functions are [27] for 6 Li; [28] for 7 Li; [29]
for 12 C; and [30] for 26 Mg, 58 Ni and 120 Sn. These DD
functions are transformed to the two-parameter Fermi
(2pF) function, ρ(r) = ρ0 [1 + exp((r − c)/z)]−1 , for use in
the EDF calculations. The reduced parameters along with
a comparison between the calculated and experimental
binding energies are given in table 1.
The EDF-derived real parts of the 6,7 Li NM potentials
are shown in solid dots in fig. 1, wherein these potentials
are parametrized in terms of solid lines with the simple
analytic expression
−1
R − R0
V (R) = −V0 1 + exp
a0
2 R − D1
+V1 exp −
.
(1)
R1
Li+12C
-30
VN (R) (MeV)
Li
Li
12
C
26
Mg
58
Ni
120
Sn
7
20
6
10
VN (R) (MeV)
6
2pF DD parameters
c
z
ρ0
1.333 0.577 0.2118
1.501 0.578 0.2006
2.294 0.434 0.1752
3.050 0.523 0.1695
4.154 0.553 0.1644
5.315 0.576 0.1710
20
Parametrized
-10
-10
-20
-20
-30
-30
-40
0
2
4
R (fm)
6
8
10
0
2
4
6
8
VN (R) (MeV)
Table 1: The parameters of the equivalent 2pF DD function for
the nuclei with the sources given in the text. c and z are in fm,
ρ0 in fm−3 (see text for definition of the parameters) and the
binding energy in MeV.
-40
10
R (fm)
Fig. 1: Parametrization in terms of the solid lines with the
analytical expression given in (1) fitting the nuclear part VN (R)
of the EDF-generated Li potentials (solid dots).
radius RC is added to obtain the total real part of the
potential. The parameters of the parametrized form of the
nucleus-nucleus potential, derived in the above-mentioned
way from the EDF-generated 6,7 Li potentials, are given in
table 2 along with the volume integrals JR /(AP AT ) (AT
being the target mass number) and RC .
The imaginary part of the 6,7 Li potential is taken
phenomenologically to be composed of volume and surface
terms as
2 R
W (R) = −W0 exp −
RW
2 R − DS
−WS exp −
.
(2)
RS
The effective SO part of 6,7 Li potentials is assumed to have
the standard WS form with only the real part as
VSO d
−1
[1 + exp((R − RSO )/aSO )] · I.
R dR
(3)
Here, and I are, respectively, the partial wave and spin
of 6,7 Li. The depth parameter VSO , and the geometry
parameters RSO and aSO have been adjusted for the best
possible fit to the iT11 data.
USO (R) = 2
The differences in the nuclear interior up to about 2.0
fm between the EDF-derived potentials (solid dots) and
the analytic lines for the cases of 58 Ni and 120 Sn are not
expected to have a significant impact on the angular distributions of the elastic scattering, which, at the considered
Analysis and results. – The optical model analyses
incident energies, are primarily determined by the surface
geometry of the nucleus-nucleus potential derived. The have been carried out using the code SFRESCO, which
Coulomb potential VC of a uniformly charged sphere with incorporates the coupled-channels code FRESCO 2.5 [31]
62002-p3
A. K. Basak et al.
Table 2: Analytic parameters in (1) of the EDF-generated NM projectile-target (system) potential along with the Coulomb
radius RC and the volume integrals JR /(AP AT ). The incident energy (ELi ) and depth parameters are in MeV, the geometry
parameters in fm and JR /(AP AT ) in MeV fm3 .
System
6
12
Li- C
Li-12 C
6
Li-26 Mg
7
Li-26 Mg
6
Li-58 Ni
7
Li-58 Ni
6
Li-120 Sn
7
Li-120 Sn
7
ELi
V0
R0
a0
V1
D1
R1
RC
JR /(6A)
20.0
21.10
44.0
44.0
20.0
20.3
44.0
44.0
28.71
29.81
43.87
41.11
38.62
35.52
39.21
38.71
4.17
4.27
4.830
4.95
6.00
6.10
7.29
7.39
0.691
0.705
0.780
0.783
0.792
0.804
0.780
0.776
31.06
32.38
28.82
27.36
16.50
13.31
16.88
16.19
0.138
0.093
0.811
0.687
2.275
2.167
3.173
3.211
2.211
2.158
2.923
2.679
2.208
2.081
2.308
2.283
7.8
8.0
8.5
8.6
9.6
9.7
10.7
10.9
124.1
123.2
120.8
115.4
100.1
87.10
84.81
73.75
Table 3: Empirical imaginary and SO parameters, respectively, in (2) and (3), of the projectile-target (system) potential along
with the total χ2 per point on combined fits to both CS and iT11 data. The incident energy (ELi ) and depth parameters are
in MeV, and the geometry parameters, in fm.
System
6
Li-12 C
Li-12 C
6
Li-26 Mg
7
Li-26 Mg
6
Li-58 Ni
7
Li-58 Ni
6
Li-120 Sn
7
Li-120 Sn
7
ELi
W0
RW
WS
DS
RS
VSO
RSO
aSO
χ2
20.0
21.10
44.0
44.0
20.0
20.3
44.0
44.0
3.257
13.50
35.48
31.12
84.86
200.0
95.32
300.9
6.10
4.595
4.279
4.527
4.078
4.067
4.433
4.372
24.96
8.918
7.029
6.00
169.42
200.0
6.831
12.458
3.663
4.419
4.574
5.332
5.748
6.232
8.084
7.300
0.310
0.241
0.500
0.54
0.15
0.200
0.787
0.327
+1.008
+1.800
−0.240
−0.272
+1.304
−1.111
+1.393
−0.458
4.121
3.434
3.851
3.570
5.617
6.193
5.460
7.917
0.980
0.450
1.10
0.900
1.091
0.90
0.938
0.613
5.93
8.17
13.0
5.81
4.08
1.97
0.70
0.52
coupled with the χ2 -minimization code MINUIT [32]. The
experimental CS and iT11 data are taken from [33] for
6
Li-12 C at the lab energy 20.0 MeV; [34] for 7 Li-12 C at
21.1 MeV; [35] for 6 Li-26 Mg and [36] for 7 Li-26 Mg both at
44.0 MeV; [15] for 6 Li-58 Ni at 20 MeV; [16] for 7 Li-58 Ni at
20.3 MeV; and [12] for 6 Li-120 Sn and [17] for 7 Li-120 Sn
both at 44.0 MeV. A systematic error of 15% has been
assumed for the experimental CS data normalized to the
Rutherford cross-sections (σ/σR ) for the angular points
without the error bars.
In the simultaneous analysis of the CS and iT11 data,
the EDF-generated parameters for the real part of the
6,7
Li-target potentials, given in table 2, have been held
fixed. In the first step of the analysis, the parameters of the
imaginary part of the 6,7 Li potentials have been obtained
from fitting the CS data only. In the second step, the
parameters of the SO potential have been optimized to
fit the iT11 data alone. Then the depths of the imaginary
and SO parts have been optimized by minimizing the χ2
in fitting the CS and VAP data simultaneously. Final
fits have been done visually after taking guidance from
the χ2 fits, since it is more important to reproduce the
features, e.g., positions of the peaks etc., of the angular
distributions than naively minimizing the χ2 only [37].
The final parameters for the imaginary and SO potentials
are noted in table 3. One can see that the effective SO
strengths are either attractive with positive sign of the
depth VSO or repulsive with negative depth.
The OM fits to the CS and VAP data of the 6,7 Li
elastic scattering are shown by solid lines in fig. 2 for
12
C and 26 Mg and those in fig. 3 are for 58 Ni and
120
Sn. Both the CS and VAP data are reproduced
simultaneously well with the unadjusted real parameters,
generated from the EDF calculations except for the CS
data for 7 Li +12 C. Interesting features for 58 Ni and 120 Sn
are that very similar CS distributions for the 6 Li and
7
Li elastic scattering and the opposite signs of their iT11
data are well accounted for by the imaginary and effective
SO potentials, the latter representing the combined effect
of the static and dynamic potentials. The latter, as
shown explicitly in the work of [3], is generated from the
projectile excitation processes and the imaginary potential
provided the real part of the 6,7 Li potential has the
appropriate NM nature.
Figure 4 shows the plot of the volume integral per
−1/3
which includes the
nucleon pair JR /(6AT ) vs. AT
integral value for the EDF-generated 6 Li-28 Si potential
from [1,2]. The plot is primarily linear with the rela−1/3
tionship JR /(6AT ) = 49.49(1 + 3.852AT ). Because of
the Pauli blocking effect, as noted in [5], the contribution to the volume integral is expected to be small from
the nuclear interior and the resulting effect leads to the
−1/3
proportionality of JR /(6AT ) to AT .
Discussion and conclusion. – The investigation
delineates, for the first time, that a physical process that
62002-p4
Non-monotonic potentials and vector analyzing powers etc.
Li + 12C, 20.0 MeV
0.2
6
Li + 58Ni, 20.0 MeV
0.6
0.4
0.2
10-1
0.0
6
Li + 58Ni, 20.0 MeV
100
0.0
10-1
iT11
Li + 12C, 20.0 MeV
σ/σR
6
6
iT11
σ/σR
101
0.8
100
-0.2
-0.4
10
Expt
-0.6
Expt
OM (EDF+SO)
OM (EDF+SO)
-0.8
20
40
60
7
12
80
100
120
140
20
40
7
12
60
80
100
120
140
20
Li + C, 21.1 MeV
Li + C, 21.1 MeV
0.0
σ/σR
-0.2
Expt
Expt
OM (EDF+SO)
100
120
140
160
20
40
60
80
100
120
140
Li + 58Ni, 20.3 MeV
0.0
10-1
10-2
-0.4
Expt
Expt
OM (EDF+SO)
OM (EDF+SO)
-0.6
20
40
60
6
26
80
100
120
140
20
40
60
80
100
120
140
20
100
Li + Mg, 44.0 MeV
Li +
σ/σR
0.6
Expt
Expt
0.4
OM (EDF+SO)
OM (EDF+SO)
0.2
10-2
0.0
10-3
60
80
100
20
7
40
60
80
-0.6
100
0.6
101
0.4
0
0.0
Expt
20
40
60
80
Angle Θc.m. (deg)
100
20
40
60
80
80
100
120
140
Li +
-0.2
160
0.1
Sn, 44.0 MeV
OM (EDF+SO)
60
80
10
20
40
60
80
0.1
7
Li +
120
Sn, 44.0 MeV
10-1
0.0
10-2
Expt
OM (EDF+SO)
10-3
-0.4
OM (EDF+SO)
60
120
Expt
40
-0.2
10-3
Sn, 44.0 MeV
Li +120Sn, 44.0 MeV
0.2
OM (EDF+SO)
40
-0.1
20
10-1
10-2
20
6
7
Expt
160
Expt
26
iT11
σ/σR
Li + 26Mg, 44.0 MeV
140
OM (EDF+SO)
Li + Mg, 44.0 MeV
7
120
0.0
10-2
σ/σR
40
100
100
-0.4
100
80
120
10-1
-0.2
20
60
6
26
iT11
σ/σR
Li + Mg, 44.0 MeV
10-1
40
101
0.8
6
-0.2
160
0.2
7
100
iT11
σ/σR
0.2
80
58
Li + Ni, 20.3 MeV
0.6
10-1
OM (EDF+SO)
60
7
0.4
10-2
40
101
0.8
100
iT11
Expt
OM (EDF+SO)
iT11
Expt
OM (EDF+SO)
-2
iT11
10
-2
Expt
OM (EDF+SO)
10-4
-0.6
100
-0.1
20
40
60
80
20
Angle Θc.m. (deg)
Angle Θc.m. (deg)
80
58
Ni and
140
JR/(6AT) for 6Li
130
JR/(6AT) MeV.fm
6
leads to the opposite signs of the VAP data of Li and
7
Li elastic scattering on 58 Ni at Elab ≈ 20 MeV and on
120
Sn at Elab = 44 MeV. These along with the respective
CS data can be well accounted for in the simple picture
of an OM analysis. Both the CS and VAP data of the
6,7
Li elastic scattering on 12 C, 26 Mg, 58 Ni and 120 Sn
are simultaneously reproduced using the EDF-generated
parameters of the real central potential, and empirical
imaginary and effective SO parts of the 6,7 Li potentials.
However, it can be seen in table 3 that in some cases, e.g.,
7
Li-26 Mg, 58 Ni, 120 Sn and 6 Li-26 Mg, the SO potentials
needed to fit the iT11 data are repulsive with negative
VSO . This is not surprising in the light of the findings
of [ 3,18,23], that demonstrate explicitly that the dynamic
part of the SO potential stems from the excitation process
of the projectile. The dynamic SO potential and its
influence on the predicted VAP depend on the number
of inelastic channels considered in the CDCC calculation
and hence on the absorption process. The present study
shows that after imaginary parts of the 6,7 Li potentials are
fixed using only the CS data, the VAP data are taken care
of by the effective SO potentials, arising from the static
and DPP potentials.
60
Fig. 3: Same as in fig. 2 for the elastic scattering on
120
Sn.
3
Fig. 2: The OM predicted σ/σR and iT11 (solid lines) for the
6,7
Li elastic scattering on 12 C and 26 Mg using the parameters
of the EDF-generated real central potential (table 2), and the
empirical imaginary and effective SO potentials (table 3), are
compared with the experimental data. The sources of the data
are given in the text.
40
Angle Θc.m. (deg)
120
110
100
EDF
Fit Curve
90
80
0.15
0.20
0.25
0.30
0.35
0.40
0.45
AT-1/3
−1/3
Fig. 4: Plot of the volume integral JR /(6AT ) vs. AT
for the
central real part of the 6 Li potential. The point for 28 Si is taken
from [1].
Aside from providing the underlying physical process
responsible for the opposite signature of the VAP data
of 6 Li and 7 Li, this analysis furnishes some insight to
density dependence of the EOS of the nuclear and nucleonic matter. This is because the nucleus-nucleus potential,
explaining the experimental data, has been derived by
the superposition of the observed DDs of the colliding
nuclei. The nuclear DD functions used by us also generate
proper binding energies of the nuclei considered herein. As
noted earlier, K at the saturation density for this EOS is
187 MeV, for the symmetric nuclear matter, which is at the
lower end of currently considered values [38]. Considering
62002-p5
A. K. Basak et al.
the yet unresolved uncertainties in the determination of
K [39], it would be interesting to study the sensitivity
of fits in this approach to the concerned values of K.
A preliminary investigation indicates that changing the
mean field slightly can still reproduce the elastic scattering CS and VAP data reasonably with a slightly different
value of K and a potential differing in the interior. Since
the elastic scattering data at the considered energies are
primarily determined by the geometry of the potential
near the surface region, which in turn is generated by
the overlap of densities of the two nuclei, we have some
information of EOS from very low to at least saturation
density. But K-sensitivity to the data is likely to become
more prominent at higher energies where the effects of the
nuclear interior become more prominent. Furthermore,
as noted in [40], the CS and VAP data of systems with
relatively low absorption at higher energies are needed to
extract information on EOS at higher densities.
In conclusion, as was the case in [1,3], the NM nature
of the real part of the central 6,7 Li-nucleus potential,
along with the dynamics of the projectile excitation, seems
to play a significant role in producing the proper DPP
responsible for generating the actual effective SO potential. Therefore, this approach merits further investigation
with the CS and VAP data at higher energies.
∗∗∗
We are indebted to Prof. K. Rusek of Heavy Ion
Laboratory, University of Warsaw for his helpful suggestions in the work. We would like to thank Prof. I. J.
Thompson, now at Lawrence Livermore Lab, for his
encouragements and helping us with the latest version
of his code FRESCO. A research grant from UGC of
Bangladesh and financial support to one of us (MJK)
from an MOSICT fellowship as well as another two
of us (MMR and PKD) from the M. A. Bary and
Feroza Bary scholarships, respectively, are also thankfully
acknowledged.
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