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CHINESE JOURNAL OF PHYSICS
VOL. 53, NO. 4
August 2015
Extended Double Folding Model Analysis of the Elastic Scattering of 6 Li
Using the No-Core Full Configuration Density Distribution
M. Aygun
Department of Physics, Bitlis Eren University, Bitlis, Turkey
(Received December 9, 2014; Revised March 13, 2015)
A large amount of experimental data on the scattering of 6 Li from various nuclei with A
ranging from 12 C to 208 Pb at ELab = 12.3–210 MeV is considered. The analysis of the data
is performed by using the double folding model with an effective nucleon-nucleon interaction
by means of the known densities of 6 Li and the target nuclei within the framework of the
optical model. The free parameters are very few, however the agreement of the theoretical
calculations with the data is highly appreciable.
DOI: 10.6122/CJP.20150323
PACS numbers: 21.60.De, 25.60.Bx
I. INTRODUCTION
6 Li
is one of the weakly bounded nuclei intensively studied in nuclear physics. In
this context, a lot of study have been carried out both theoretically and experimentally on
the interaction of a 6 Li nucleus with different target nuclei. A comprehensive survey of the
literature information about these interactions investigated in the present study is given in
the following.
The elastic scattering angular distributions of the 6 Li + 12 C system at varied incident energies were measured and analyzed by the help of the Woods-Saxon and double
folding type potentials and the cluster folding model [1–5]. Bindal et al. [6] and Vineyard
et al. [7, 8] measured the elastic and inelastic scattering of 6 Li scattered from 12 C at diverse energies. They fitted the experimental data using the optical model (OM) and the
distorted wave Born approximation (DWBA). El-Azab Farid and Hassanain [9–11] investigated the differential cross-section of elastic and inelastic scattering by using the double
folding model. The different experimental and theoretical studies for the 6 Li + 16 O reaction
were presented by [12–14]. They reported the theoretical results obtained using different
methods at various energies. For the 6 Li + 28 Si interaction the elastic scattering data were
measured and examined via the optical model and double folding model [15–17]. Hossain
et al. [18] investigated the differential cross-sections of the 6 Li + 28 Si system with a potential microscopically derived from the energy density functional (EDF) theory. Also, a
simultaneous analysis of the elastic scattering, fusion, and total reaction cross-sections for
the 6 Li + 28 Si system was performed [19].
Elastic and inelastic scattering of the 6 Li on 40 Ca were measured at different incident
energies [20–23]. The 6 Li + 70 Ge reaction was investigated to obtain the elastic and inelastic scattering angular distributions at 28 and 44 MeV [24, 25]. The experimental data
were analyzed by means of the optical model, the coupled-channels, and the distorted wave
approximation. The interaction of 6 Li with 90 Zr has had different experimental studies
http://PSROC.phys.ntu.edu.tw/cjp
080301-1
c 2015 THE PHYSICAL SOCIETY
⃝
OF THE REPUBLIC OF CHINA
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EXTENDED DOUBLE FOLDING MODEL ANALYSIS . . .
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performed at various energies [26–28]. In these studies, the experimental data were analyzed by using the optical model, the double folding model, and the distorted wave Born
approximation.
Elastic scattering of the 6 Li + 144 Sm system was reported by Figueira et al. [29].
They examined the data within the framework of the optical model. For the 6 Li+197 Au
reaction, Fulmer et al. [30] measured the elastic scattering data and analyzed the experimental data with the help of the Woods-Saxon and double folding potentials. Finally,
elastic and inelastic scattering of 6 Li + 208 Pb were measured and investigated with different approaches [16, 30–32]. Consequently, a large body of experimental data over a wide
energy range has been accumulated for both the elastic and inelastic scattering of 6 Li from
different target nuclei.
Recently, the no-core full configuration (NCFC) calculations for the Lithium isotopes,
6 Li, 7 Li, and 8 Li have been performed by Cockrell et al. [33]. NCFC suggests a solution
for the many-body problem to examine microscopically the features of the nuclei. One
dimensional and three dimensional translationally invariant one body density distributions
for various ground and excited states of 6 Li, 7 Li, and 8 Li have been obtained. Aygun [34, 35]
has used the NCFC density distribution for the 6 Li + 58 Ni and 6 Li + 209 Bi systems and
has obtained good agrement results with the experimental data. However, a simultaneous
theoretical analysis for the same geometry from light to heavy target nuclei has not been
performed. To provide the elastic scattering angular distributions and the reaction cross
sections which are widely used in nuclear physics (e.g., coupled channels, cluster model,
transfer reaction) and astrophysics for any energy and reaction, the determination of global
optical models is needed. The difficulties about this case of 6 Li have been reported in
previous studies [10, 11]. Therefore, we think that the extended double folding model
analysis via the NCFC density distribution of 6 Li for target nuclei ranging from 12 C to
208 Pb in the energy range of 12.3 ≤ E
Lab ≤ 210 MeV will be very interesting and important.
In the next section, we present the theoretical model used in our calculations and
the results of these calculations are presented in Section III. Section IV is devoted to our
summary and conclusions.
II. DOUBLE FOLDING MODEL CALCULATIONS
In this part, we investigate the elastic scattering angular distributions of 6 Li scattered
from 12 C, 16 O, 28 Si, 40 Ca, 70 Ge, 90 Zr, 144 Sm, 197 Au, and 208 Pb target nuclei at ELab = 12.3–
210 MeV via the double folding model based on the optical model. In this context, the
complex nuclear potential can be written as following form:
VNuclear (r) = V (r) +
| {z }
RealP art
iW (r).
| {z }
(1)
ImaginaryP art
To determine the real part of the VNuclear (r), we use the double folding model by
the nuclear matter distributions of both the projectile and target nuclei together with an
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M. AYGUN
080301-3
effective nucleon-nucleon interaction potential (νN N ). Thus, the double folding potential is
given as the following form:
∫
∫
−
→
−
→
→
−
−
→
→
→
VDouble Folding ( r ) = d r 1 d−
r 2 ρP ( →
r 1 )ρT (→
r 2 )νN N (−
r −−
r1+−
r 2 ),
(2)
→
−
where ρP (−
r 1 ) and ρT (→
r 2 ) are the nuclear matter density of the projectile and target
nuclei, respectively. For the 6 Li nucleus, we use the NCFC density distribution which is
taken from [33]. In Fig. 1, this density distribution of 6 Li is shown. In our folding model
calculations, we have chosen the Gaussian form for the 12 C ground state matter density
distribution,
ρ(r1 ) = ρ0 (1 + wr12 ) exp (−βr12 ),
(3)
where ρ0 = 0.1644 fm−3 , w = 0.4988 fm−2 , and β = 0.3741 fm−2 [11, 36]. The density
distributions of the other target nuclei have been taken from the Hartree-Fock-Bogolubov
(HFB) method based on the BSk2 Skyrme force [37]. We have used the M3Y nucleonnucleon realistic interaction, which is given by [5]
νN N (r) = νD (r) + Jˆ00 (E)δ(r),
(4)
where νD (r) is the direct part of the M3Y interaction and Jˆ00 (E) is the exchange term.
νD (r) and Jˆ00 (E), respectively, can be expressed as
νD (r) = 7999
exp(−4r)
exp(−2.5r)
− 2134
,
4r
2.5r
Jˆ00 (E) ≃ −276 [1 − 0.005 (ELab /Ap )]MeV fm3 .
(5)
(6)
To obtain the imaginary part of the nuclear potential, a Woods-Saxon type potential
of the following form has been used:
W (r) = −W0 f (r, Rw , aw ),
f (r, Rw , aw ) =
1/3
1
,
w
1 + exp( r−R
aw )
1/3
(7)
(8)
where Rw = rw (AP + AT ) and AP and AT are mass numbers of the projectile and target
nuclei, respectively. The code Fresco [38] has been used for the calculations.
In the determination of the global potential parameters for all the reactions, we have
performed test calculations to see the variation of the parameters, which are the depth
(W0 ), the radius (rw ) and the diffusion parameter (aw ), at different energies. Then we
have used a rw = 1.32 fm value for all the systems in order to minimize the number of the
free parameters. In a similar manner, aw has been fixed as 0.90 fm. We have investigated
the W0 values so as to obtain a good agreement between the theoretical results and the
experimental data. All the parameters of the imaginary potential used in the calculations
have been shown in Table I.
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EXTENDED DOUBLE FOLDING MODEL ANALYSIS . . .
VOL. 53
FIG. 1: The proton and neutron density distributions of 6 Li within the NCFC.
III. RESULTS AND DISCUSSIONS
We have analyzed the elastic scattering angular distribution of 6 Li by different target
nuclei over a wide energy range within the framework of the double folding model. The
microscopic real potential has been produced by using the NCFC density distribution of
6 Li. The imaginary potential parameters used in our calculations are given in Table I.
The 6 Li + 12 C system has been investigated at ELab = 12.3, 30, and 210 MeV. The
results have been shown in Fig. 2. In a general sense, the results are in good agreement with
the data. However, at 30 MeV, some experimental points show an inconsistency with the
theoretical results, because of using the same geometry for all systems and the oscillations.
The elastic scattering angular distributions of the 6 Li + 16 O system have been obtained
at ELab = 20 and 32 MeV. The theoretical results shown in Fig. 3 catch the phase of
the experimental data. The 6 Li + 28 Si system at ELab = 13, 20, and 32 MeV has been
investigated and the theoretical results for the elastic scattering have been obtained. As
seen from Fig. 4, the theoretical results with the experimental data are quite consistent with
each other. Both the phase and amplitude give the behavior of the data. The theoretical
results obtained for the 6 Li + 40 Ca system at ELab = 20, 30, 32, and 210 MeV have been
shown in Fig. 5. It has been noticed there is a harmony between the experimental and
theoretical results. However, at 210 MeV, the results are not very good for some points of
the experimental data. The elastic scattering results of the 6 Li + 70 Ge system at ELab =
28 and 44 MeV have been obtained and are given in Fig. 6. The results are almost perfect
in comparison with the data. The 6 Li + 90 Zr reaction at ELab = 34, 60, and 210 MeV has
VOL. 53
M. AYGUN
080301-5
TABLE I: The optical model parameters, the real and imaginary volume integrals, cross-sections,
and χ2 /N values obtained for the NCFC density distribution of all the reactions investigated by
using the double folding model.
6
6
6
6
6
ELab
W
rw
aw
Jv
Jw
σ
χ2 /N
−
MeV
MeV
fm
fm
MeVfm3
MeVfm3
mb
−
12.3
6.80
1.32
0.90
406.5
80.1
1287.4
0.64
30.0
7.35
1.32
0.90
402.6
86.6
1524.5
1.11
210.0
23.0
1.32
0.90
362.5
271.1
1743.5
3.51
20.0
5.55
1.32
0.90
404.7
56.5
1448.8
1.72
32.0
5.85
1.32
0.90
402.0
59.5
1535.2
43.5
13.0
5.60
1.32
0.90
406.9
43.8
1014.4
0.76
20.0
8.50
1.32
0.90
405.3
66.5
1555.6
0.27
32.0
10.0
1.32
0.90
402.6
78.3
1856.2
0.13
20.0
10.0
1.32
0.90
405.7
67.3
1484.7
0.04
30.0
11.0
1.32
0.90
403.5
74.0
1891.9
0.03
32.0
11.5
1.32
0.90
403.0
77.4
1950.6
0.07
210.0
15.7
1.32
0.90
363.2
105.6
2265.4
23.4
28.0
9.30
1.32
0.90
404.7
50.4
1704.9
0.01
44.0
9.70
1.32
0.90
401.1
52.6
2196.1
0.13
34.0
11.0
1.32
0.90
403.6
54.7
1934.7
0.07
60.0
14.0
1.32
0.90
397.8
69.6
2646.6
0.18
210.0
24.0
1.32
0.90
364.2
119.3
3252.8
3.72
23.0
5.80
1.32
0.90
406.8
24.8
152.7
0.01
35.1
5.90
1.32
0.90
404.1
25.2
1360.0
0.11
42.3
9.60
1.32
0.90
402.5
41.1
1957.1
0.17
88.0
11.50
1.32
0.90
392.7
45.0
3212.0
0.06
29.0
6.30
1.32
0.90
406.0
24.2
240.6
0.01
88.0
13.30
1.32
0.90
392.7
51.2
3335.7
0.08
210.0
20.80
1.32
0.90
365.3
80.2
4208.1
1.84
Li +
Li +
Li +
12
16
28
40
Li +
70
Li +
6
6
System
Li +
Si
Ge
90
Zr
Li +
Li +
197
Li +
208
6
O
Ca
144
6
C
Sm
Au
Pb
been analyzed. The theoretical results plotted in Fig. 7 show very good agreement with the
experimental data. The other system analyzed in our study is 6 Li + 144 Sm. The theoretical
results for this system have been acquired at ELab = 23, 35.1, and 42.3 MeV. As seen from
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EXTENDED DOUBLE FOLDING MODEL ANALYSIS . . .
VOL. 53
Fig. 8, the double folding model results coincide well with the experimental results. The
elastic scattering angular distribution of 6 Li + 197 Au reaction has been investigated at ELab
= 88 MeV and the results have been presented in Fig. 9. If one examines the compliance
with the data of the theoretical results, one can see that the double folding results are in
very good agreement with the data. Finally, we have analyzed the elastic scattering angular
distribution of the 6 Li + 208 Pb system at ELab = 29, 88, and 210 MeV. We have showed the
theoretical results in comparison with the experimental data in Fig. 10. We have observed
that the double folding results successfully describe the experimental data. Thus, it can be
said that the double folding model for the NCFC density distribution of the 6 Li nucleus is
valid for a heavy target nucleus system. A similar result has been reported by [35]. Thus,
the present results confirm the previous study.
0
10
-1
10
Exp.
Double Folding Model
-2
10
12.3 MeV
-3
10 0
10
20
30
40
50
60
70
80
90
100
0
σ/σR
10
-1
10
-2
30 MeV
10
-3
10
0
20
40
80
60
100
120
140
160
180
6
10
3
10
210 MeV
0
10
-3
10
-6
10
-9
10 0
10
20
30
40
θc.m.(deg)
50
60
70
80
FIG. 2: The elastic scattering angular distributions for the NCFC density distribution of the 6 Li
+ 12 C reaction at ELab = 12.3, 30, and 210 MeV in comparison with the experimental data. The
experimental data have been taken from [4, 7].
In the calculations of the present work, we have used the double folding model.
This model includes the normalization factor (NR ), which is changed to obtain the good
agreement results with the experimental data. The value NR = 1.0 means the success of
the model used in the calculations [39]. In the previous studies [9–11, 39], it has been
reported that the normalization value for the light nucleus such as 6 Li deviates from 1.0 to
0.5. As a result of this, we have not changed the normalization constant in order to see the
validity of the density distribution used for the 6 Li nucleus over a wide energy and target
nucleus range. As seen from the figures, in a general sense, the results are in agreement
with the experimental data. We do not claim perfect results for each energy and target
VOL. 53
M. AYGUN
080301-7
FIG. 3: The elastic scattering angular distributions for the NCFC density distribution of the 6 Li +
16
O reaction at ELab = 20 and 32 MeV in comparison with the experimental data. The experimental
data have been taken from [12, 13].
13 MeV
0
10
-1
10
Exp.
Double Folding Model
-2
10
-3
10
0
20
40
80
60
100
120
140
160
180
0
10
20 MeV
σ/σR
-1
10
-2
10
-3
10
-4
10
0
10
20
30
40
50
60
70
80
90
100
110
120
0
10
32 MeV
-1
10
-2
10
-3
10
-4
10
0
10
20
30
40
θc.m.(deg)
50
60
70
80
FIG. 4: The elastic scattering angular distributions for the NCFC density distribution of the 6 Li
+ 28 Si reaction at ELab = 13, 20, and 32 MeV in comparison with the experimental data. The
experimental data have been taken from [12, 13, 16].
080301-8
EXTENDED DOUBLE FOLDING MODEL ANALYSIS . . .
0
VOL. 53
20 MeV
10
-1
10
Exp.
Double Folding Model
-2
10
-3
10 0
10
20
30
40
50
70
60
80
90
100
0
10
30 MeV
-2
10
-4
σ/σR
10
0
10
20
30
40
50
70
60
80
90
100
0
10
-1
10
-2
10
-3
10
-4
10
6
32 MeV
0
10
20
30
40
50
70
60
80
10
4
10
210 MeV
2
10
0
10
-2
10
0
10
20
30
θc.m.(deg)
40
50
60
FIG. 5: The elastic scattering angular distributions for the NCFC density distribution of the 6 Li +
40
Ca reaction at ELab = 20, 30, 32, and 210 MeV in comparison with the experimental data. The
experimental data have been taken from [12, 13, 16, 22].
FIG. 6: The elastic scattering angular distributions for the NCFC density distribution of the 6 Li +
70
Ge reaction at ELab = 28 and 44 MeV in comparison with the experimental data. The experimental
data have been taken from [24, 25].
VOL. 53
M. AYGUN
080301-9
FIG. 7: The elastic scattering angular distributions for the NCFC density distribution of 6 Li +
90
Zr reaction at ELab = 34, 60, and 210 MeV in comparison with the experimental data. The
experimental data have been taken from [16, 26, 27].
nucleus. In addition to this, we cannot obtain good results for some nuclei and energy by
means of test calculations in the same geometry. This is consistent with earlier results in
the literature [9–11, 39]. Our focus is to find a good agreement of results with the data
without renormalization for the NCFC density distribution. Consequently, a good harmony
between the theoretical results and the experimental data for the reactions and the energies
investigated in our study has been obtained. As a result of this case, we evaluate that the
NCFC accepts as active all the nucleons in the investigated nucleus. Thus, the NCFC
approaches the system more microscopically [34].
We have calculated the χ2 /N values for all the systems at each energy and have given
the results in Table I. We have observed that the χ2 /N values are sufficiently small except
for 6 Li + 16 O at 32 MeV and 6 Li + 40 Ca at 210 MeV. It should noticed that the NR values
are kept fixed as 1.0 in defining the χ2 /N values.
In our study we have given the cross-sections for all the reactions investigated in
Table I. The similar behavior seen for each of the reactions is increasing with the energy
of ELab . In Fig. 11, we have shown as comparative the variation of the cross-section with
1/3
AT for the 6 Li + 16 O, 6 Li + 32 Si, and 6 Li + 40 Ca reactions at 32 MeV and for the 6 Li +
12 C, 6 Li + 40 Ca, 6 Li + 90 Zr, and 6 Li + 208 Pb reactions at 210 MeV. We have seen that the
1/3
cross-sections show a dependence on AT of the target nucleus. Similar results have been
080301-10
EXTENDED DOUBLE FOLDING MODEL ANALYSIS . . .
VOL. 53
23 MeV
0
10
Exp.
Double Folding Model
0
20
40
80
60
100
120
140
160
0
35.1 MeV
10
σ/σR
180
-1
10
-2
10
-3
10
0
10
20
30
40
50
60
70
80
90
100
0
10
110
120
42.3 MeV
-1
10
-2
10
-3
10
0
10
20
30
40
50
60
θc.m. (deg)
70
80
90
100
FIG. 8: The elastic scattering angular distributions for the NCFC density distribution of the 6 Li +
144
Sm reaction at ELab = 23, 35.1, and 42.3 MeV in comparison with the experimental data. The
experimental data have been taken from [29].
found in a previous study [16].
The shapes of the real and the imaginary potentials used in the elastic scattering
calculations of the 6 Li projectile scattered from different target nuclei at 32 and 210 MeV
energies are shown in Figs. 12 and 13. At 32 MeV, it has been observed that the real
potential of 6 Li+16 O goes to zero faster than the other potentials. If we examine the
imaginary potentials, we can say that the 6 Li+40 Ca potential is deeper than the other
potentials. At 210 MeV, it can be said that the real potential of 6 Li+12 C goes to zero
faster than the other potentials. However, the imaginary potential of the 6 Li+90 Zr system
is deeper than the other potentials.
Finally, we have calculated the volume integrals of the real and imaginary potentials
of all the systems at each energy and have given the results in Table I. As seen from the
results, an energy dependence for the real and imaginary potentials is seen. That is, while
the real volume integrals increase with incident energy of the projectile, the real volume
integrals decrease with the energy.
VOL. 53
M. AYGUN
080301-11
FIG. 9: The elastic scattering angular distributions for the NCFC density distribution of the 6 Li
+ 197 Au reaction at ELab = 88 MeV in comparison with the experimental data. The experimental
data have been taken from [30].
0
29 MeV
10
Exp.
Double Folding Model
0
20
40
80
60
100
120
140
160
180
0
10
88 MeV
σ/σR
-2
10
-4
10
-6
10
0
10
20
30
40
50
70
60
80
9
10
6
10
210 MeV
3
10
0
10
-3
10
-6
10 0
10
20
30
θc.m.(deg)
40
50
60
FIG. 10: The elastic scattering angular distributions for the NCFC density distribution of the 6 Li
+ 208 Pb reaction at ELab = 29, 88, and 210 MeV in comparison with the experimental data. The
experimental data have been taken from [16, 30, 32].
080301-12
EXTENDED DOUBLE FOLDING MODEL ANALYSIS . . .
VOL. 53
5000
4500
E=32 MeV
4000
E=210 MeV
Linear Fit
mb
3500
3000
2500
2000
1500
1000
1,0
1,5
2,0
2,5
3,0
3,5
4,0
4,5
5,0
5,5
6,0
6,5
7,0
1/3
A
T
1/3
FIG. 11: The cross-sections calculated as a function of AT for the 6 Li + 16 O,
at 32 MeV and the 6 Li + 12 C, 40 Ca, 90 Zr, and 208 Pb systems at 210 MeV.
28
Si,
40
Ca systems
FIG. 12: The shapes of the real and the imaginary potentials of the nuclear potential of 6 Li which
interacts with different target nuclei at 32 MeV.
VOL. 53
M. AYGUN
080301-13
FIG. 13: The shapes of the real and the imaginary potentials of the nuclear potential of 6 Li which
interacts with different target nuclei at 210 MeV.
IV. SUMMARY AND CONCLUSIONS
The elastic scattering angular distributions of the 6 Li nucleus scattered from the 12 C,
and 208 Pb target nuclei have been investigated.
The theoretical results have been obtained by the double folding model based on the optical
model. The global optical model parameters have been acquired as given in Table I. Also,
the real and imaginary volume integrals and the cross-sections for the systems investigated
have been acquired. The obtained results have been given in a table and have been presented
in figures. The shapes of the real and the imaginary potentials used in the elastic scattering
calculations of the 6 Li projectile at 32 and 210 MeV have been shown in figures. It has been
seen that the theoretical results obtained for the NCFC density distribution of 6 Li are in
good agreement with the experimental data. Consequently, this study will be useful within
the scope of both the ab-initio density distribution of 6 Li and the global optical potential
parameters by means of different target nucleus reactions.
16 O, 28 Si, 40 Ca, 70 Ge, 90 Zr, 144 Sm, 197 Au,
Acknowledgements
The author would like to thank the referee for valuable comments about the
manuscript.
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EXTENDED DOUBLE FOLDING MODEL ANALYSIS . . .
VOL. 53
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