J. Nig. Soc. Phys. Sci. 5 (2023) 1392

Journal of the
Nigerian Society

of Physical
Sciences

Theoretical Study on 10C Elastic Scattering Cross Sections Using
Different Cluster Density Distributions and Different Potentials

Sunday D. Olorunfunmia,∗, A. Bahinib, Adenike S. Olatinwoa

aDepartment of Physics & Engineering Physics, Obafemi Awolowo University, Ile-Ife 220005, Osun State, Nigeria
biThemba Laboratory for Accelerator Based Sciences, Somerset West 7129, South Africa

Abstract

Elastic scattering cross sections are a fundamental aspect of nuclear physics research, and studying the cross sections of various nuclei can provide
important insights into the behavior of nuclei. In this study, the elastic scattering cross sections of 10C projectile by 27Al, 58Ni, and 208Pb target
nuclei are analyzed. The aim of this study is to investigate the cluster structure of 10C and the sensitivity of the elastic scattering cross sections
to different potentials. To achieve this objective, the double folding optical model and a simple cluster approach are used to analyze the cross
sections. The real part of the optical potential is obtained by folding two different effective interactions, Michigan-3-Yukawa (M3Y) and Jeukenne-
Lejeune-Mahaux (JLM), with four different cluster density distributions of the 10C nucleus: 6Be + α, 9B + p, 8Be + p + p, and α + α + p + p. The
imaginary part is taken to be a Woods-Saxon phenomenological form. The sensitivity of the elastic scattering cross sections to different potentials
is assessed by comparing the results obtained using different potentials. The cluster structure of 10C is validated by comparing the theoretical
results with experimental data. The results show that the cross sections are sensitive to the choice of potential used and that the cluster structure
of 10C is validated. The theoretical results show reasonable agreement with the experimental data.

DOI:10.46481/jnsps.2023.1392

Keywords: Elastic scattering, density distribution, Optical model, cluster model.

Article History :
Received: 11 February 2023
Received in revised form: 25 April 2023
Accepted for publication: 10 May 2023
Published: 21 May 2023

c© 2023 The Author(s). Published by the Nigerian Society of Physical Sciences under the terms of the Creative Commons Attribution 4.0 International license

(https://creativecommons.org/licenses/by/4.0). Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

Communicated by: O. J. Oluwadare

1. Introduction

There has been significant interest in studying reaction
mechanisms involving weakly bound neutron- and proton-rich
nuclei, especially because of their astrophysical importance or
applications [1, 2, 3, 4, 5, 6]. One of such weakly bound nuclei
that is of interest is the unstable proton-rich 10C, which exhibits
a three-cluster structure and can decay into 6Be + α, 9B + p, and

∗Corresponding author tel. no: +2349042713841
Email address: sundayolorunfunmi@gmail.com (Sunday D.

Olorunfunmi)

8Be + p + p channels with binding energies of 3.821, 4.006,
and 5.101 MeV, respectively [7, 8]. Curtis et. al., [9] studied
the break up reaction of 10C and concluded that the proton-rich
nucleus can also decay by α + α + p + p channel.

The elastic scattering angular distributions of 10C + 27Al
at incident energy of 29.1 MeV have been measured and the-
oretically analyzed using optical model potential constructed
from the combination of real São Paulo potential (SPP), imag-
inary Woods-Saxon potential and complex polarization poten-
tials [10]. The study concluded that the inclusion of the volume
and surface complex polarized potentials is needed in order to

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Olorunfunmi et al. / J. Nig. Soc. Phys. Sci. 5 (2023) 1392 2

successfully describe the data. These polarized potentials ac-
count for fusion and direct coupling. These distributions were
recently analyzed by Aygun [11] using double folding optical
potential with the real potential constructed by folding M3Y
effective interaction with relativistic mean-field density distri-
bution of 10C and Woods-Saxon (WS) imaginary potential. A
new set of global potential was obtained for the carbon isotope.

Yang et. al., [7] measured the quasi-elastic scattering an-
gular distributions of 10C + 208Pb at 226 and 256 MeV. Very
recently, the elastic scattering cross sections of the same reac-
tion at 66 MeV was measured by Linares et. al., [8]. The data
were analyzed and compared with the results of optical model
calculations performed using the SPP nucleus-nucleus interac-
tion. Again, the same data at 256 MeV was further analyzed in
Ref. [11] using optical-model based double folding potentials.

Recently, Guimaraes et al., [12] measured the elastic scat-
tering of 10C on 58Ni target nuclei at incident energy of 35.3
MeV with the purpose of studying the coupling effect in the
reaction. They analyzed the measured data using microscopic
approach within the framework of coupled-channels (CC) and
coupled-reaction channels (CRC) models. Results of the two
models could not describe the data satisfactorily. In order to
improve the description of the data they performed continuum-
discretized coupled-channels (CDCC) calculations. In CDCC
calculations the 10C nucleus was assumed to decay by two chan-
nels: 9B + p and 6Be + α. In the end they were able to achieve a
fair agreement between the theoretical calculations and exper-
imental data, but the need for a more realistic theoretical cal-
culations was emphasized. Consequent upon this, Aygun [13]
carried out a comprehensive theoretical analysis of this reaction
using different potentials and simple cluster model. The study
considered 6Be + α, 9B + p, and 8Be + p + p cluster configura-
tions for 10C, and found that the 6Be + α configuration describe
the data better then 9B + p, and 8Be + p + p cases. It was rec-
ommended that the cluster structure of 10C be evaluated in the
analysis of elastic scattering reactions of 10C with other target
nuclei.

In this present study, the elastic scattering cross section of
10C projectile nucleus from 27Al, 58Ni and 208Pb target nuclei
are calculated using the complex optical model potential with
folded real part and phenomenological Woods-Saxon imagi-
nary part. The real part is constructed by folding two differ-
ent effective nucleon-nucleon (NN) interactions M3Y and JLM
with four different cluster density distributions of 10C nucleus.
Here, we aim to study the structure effect in 10C + Nucleus reac-
tions via different choice of simple cluster density distributions
of 10C projectile nucleus and to investigate the sensitivity of 10C
+ Nucleus elastic scattering cross sections to different effective
NN interactions.

2. Theoretical Formalism

2.1. The Optical model potential

The theoretical calculations were performed using the
optical model of the form:

Table 1: The parameters of two-parameter Fermi (2pF) density distributions for
the 27Al, 58Ni, and 208Pb target nuclei.

Nucleus c (fm) z (fm) ρ0 (fm−3) Reference
27Al 2.840 0.569 0.2015 [14]
58Ni 4.094 0.540 0.1720 [15]
208Pb 6.620 0.551 0.1600 [14]

U(r) = V Coul(r) − V (r) − iW(r) , (1)

where V Coul(r) is the Coulomb potential, V (r) is the real poten-
tial and W(r) is the imaginary potential. The Coulomb potential
is defined as:

V Coul(r) =


1

4π�o
ZP ZT e2

r if r ≥ RCoul
1

4π�o
ZP ZT e2

2RCoul

(
3 − r

2

R2 Coul

)
if r ≤ RCoul

(2)

with

RCoul = 1.25(A
(1/3)
P + A

(1/3)
T ) , (3)

where ZP(T) and AP(T) are the proton and mass number of the
projectile (target) nuclei, respectively. In Refs. [16, 17], the
Coulomb potential was used in a theoretical investigation of the
half-life of certain nuclei. The real potential V (r) is obtained by
using the double folding potential given as:

V (−→r ) = NR

∫
d−→r1

∫
d−→r2ρP(

−→r )ρT(
−→r )υNN(

−→r12) , (4)

where −→r12 =
[
−→r − (−→r1 −

−→r2 )
]
, NR is the normalization constant,

υNN is the effective NN interaction, and ρP (ρT) is the density
distribution of the projectile (target). In this work, four different
cluster density distributions of the projectile nucleus are con-
sidered, and each is introduced in the following section. The
density distributions of target nuclei are obtained by using the
two-parameter Fermi (2pF) density

ρ(r) =
ρ0

1 + exp( r−cz )
, (5)

where ρ0 is the maximum density (central density) of the nu-
cleus and the Fermi-distribution parameters c and z describe the
half-density radius and the diffuseness, respectively. Their nu-
merical values are listed in Table 1. Two forms of effective NN
interaction υNN are considered, namely, M3Y and JLM. These
two interactions are presented in the next section. The imagi-
nary potential W(r) is taken in the Woods-Saxon form:

W(r) =
W0

1 + exp(
r−rI (A

1/3
P +A

1/3
T )

aI
)
, (6)

where W0, rI and aI represent the potential depth, the reduced
radius, and the diffuseness parameter, respectively.

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Olorunfunmi et al. / J. Nig. Soc. Phys. Sci. 5 (2023) 1392 3

Figure 1: Real folded potential for 10C + 27Al at 29.1 MeV, 10C + 58Ni at 35.3 MeV and 10C + 208Pb at 66 MeV, using M3Y and JLM effective interactions with
6Be + α, 9B + p, 8Be + p + p, and α + α + p + p cluster density distribution of 10C.

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Olorunfunmi et al. / J. Nig. Soc. Phys. Sci. 5 (2023) 1392 4

Figure 2: Elastic scattering angular distributions for 10C + 27Al at incident
energy of 29.1 MeV obtained using M3Y and JLM effective interactions with
6Be + α, 9B + p, 8Be + p + p, and α + α + p + p cluster density distribution
of the 10C. The experimental data are taken from Ref. [10].

2.2. Effective nucleon-nucleon interaction

The two different forms of effective NN interactions con-
sidered in the present study are fully described elsewhere
[18, 19, 20]. As such, only salient details are provided here.

2.2.1. M3Y interaction
The density-independent M3Y interaction is derived by

Bertsch et al., [19] and parameterized according to Satchler and
Love [20] as follows:

υM3YNN (r) = 7999
exp(−4r)

4r
− 2134

exp(−2.5r)
2.5r

− 276
[
1 − 0.005

ELab
AP

]
δ(r) , (7)

where ELab and AP are the laboratory energy and mass number
of the projectile, respectively. The first and the second terms
represent the direct part while the third term represents the ex-
change part of the interaction potential.

2.2.2. JLM interaction
The JLM potential derived by Jeukenne, Lejeune and Ma-

haux [18] was obtained in a Brueckner-Hartree-Fock (BHF)
approximation from the Reid soft-core NN interaction. The

Figure 3: Same as Figure 2 but for 10C + 58Ni at incident energy of 35.3 MeV.
The experimental data are taken from Ref. [12].

isoscalar component of the complex JLM interaction has the
form [18]:

υJLMNN (s,ρ, E) = g(s)v0(ρ, E) + ig(s)w0(ρ, E) , (8)

where g(s), v0 and w0 are the radial dependence factor, the real
component and the imaginary component of the effective inter-
action, respectively. In this study, only the real part of the JLM
effective interaction is considered and discussed, the imaginary
part is replaced with the Woods-Saxon potential (Eq. 6). The
density and energy dependence of the real part of JLM interac-
tion is parametrized as follows [18]:

v0 =
3∑

i, j=1

ai jρ
i−1 E j−1 . (9)

The values of the coefficient ai j are taken from Ref. [18]. In this
study, the local-density approximation (LDA) is considered us-
ing the arithmetic average approach as prescribed in Ref. [21]:

ρ = (ρp(r1)ρT (r2))
1/2 , (10)

where the local density is evaluated at each position of the in-
teracting nucleons. The radial dependence factor is taken to be
a single-Gaussian shape [22]

g(s) = (t
√
π)−3 exp(−s2/t2) (11)

with t = 1.2 fm.
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Olorunfunmi et al. / J. Nig. Soc. Phys. Sci. 5 (2023) 1392 5

Figure 4: Same as Figure 2 but for 10C + 208Pb at incident energy of 66 MeV.
The experimental data are taken from Ref. [8].

2.3. Cluster density distribution of 10C nucleus
The present work considers a simple cluster model, which

simply involves adding together the densities of the constituent
cluster nuclei. As an example, the density for 10C, is expressed
as, ρ10C = ρ6 Be + ρα, implying that the clusters are overlapping
at the same point inside the nucleus. Such an oversimplified
approach may still have some merit since in a more realistic
calculation, one would have to consider an overlap of the clus-
ter nuclei using the appropriate Jacobi coordinates, which, for
the case of three or four clusters can turn out to be very com-
plicated for the present investigation. The cluster model den-
sity distribution has been used successfully to analyze elastic
scattering cross sections of unstable projectile nuclei (see, e.g.,
Refs. [23, 24, 25, 26, 27, 28]. Four different forms of cluster
density distributions of the projectile nucleus are considered in
this study and each is presented in the following section.

2.3.1. 6 Be + α system
Firstly, the 10C nucleus is taken to be a cluster of 6Be and α

nuclei. Hence, the density distribution of 10C takes the form:

ρ10C = ρ6 Be + ρα . (12)

The São Paulo (SP) density distribution [29], is used for the
density of 6Be and parametrized as follows:

ρi6 Be(r) = ρ0i

(
1 + exp(

r − Ri
ai

)
)−1

, (i = n, p) , (13)

Figure 5: Same as Figure 2 but for 10C + 208Pb at ELab = 226 MeV . The
experimental data are taken from Ref. [7].

where

Rn = 1.49N
1/3
− 0.79, Rp = 1.81Z

1/3
− 1.12 , (14)

and

an = 0.47 + 0.00046N, ap = 0.47 − 0.00083Z . (15)

Here, Rn(Rp) and an(ap) represent the half-density radius and
surface thickness parameter of neutron (proton), while Z and N
are proton and neutron numbers, respectively. The α density is
taken to be [20]

ρα = 0.4229 exp(−0.7024r
2) . (16)

2.3.2. 9 B + p system
Another cluster density of 10C considered in this study is

given by

ρ10C = ρ9 B + ρp , (17)

where the density distribution of 9B is given in Eq. 13, and that
of proton is taken to be [30, 31]

ρp = (βπ)
−3 exp(−r2/β2) , (18)

where β is adjusted to reproduce the rms radius value of 10C.

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Olorunfunmi et al. / J. Nig. Soc. Phys. Sci. 5 (2023) 1392 6

Figure 6: Same as Figure 2 but for 10C + 208Pb at incident energy of 256 MeV.
The experimental data are taken from Ref. [7].

2.3.3. 8 Be + p + p system
The density distribution of 10C can also be evaluated as the

sum of 8Be, p and p densities

ρ10C = ρ8 Be + ρp + ρp , (19)

where the density distributions of 8Be and p are given in Eqs. 13
and 18, respectively.

2.3.4. α + α + p + p system
The last density distribution form of 10C considered here is

obtained from the addition of density distributions of α, α, p
and p

ρ10C = ρα + ρα + ρp + ρp , (20)

where the density distributions of α and p are given Eqs. 16
and 18, respectively. The approach of obtaining nuclear density
as a sum of the densities of the clusters has been used in Refs.
[23, 24].

3. Method of calculation

The first step in calculating elastic scattering cross section
is to obtain the complex total optical potential. In the present
study, the real part of the total optical potentials is calculated
via the double folding approach as expressed in Eq. 4 using

Figure 7: Normalization constant, NR versus incident energy, ELab obtained for
10C + 27Al at 29.1 MeV, 10C + 58Ni at 35.3 MeV and 10C + 208Pb at 66, 226
and 256 MeV, using M3Y and JLM effective interactions with 6Be + α, 9B + p,
8Be + p + p, and α + α + p + p cluster density distribution of 10C. The dashed
curves are to guide the eye.

M3Y and JLM effective N N interactions with 6Be + α, 9B + p,
8Be + p + p, and α + α + p + p cluster density distribution of
the 10C. The folded potentials are obtained using the computer
code DFPOT [32].

The elastic scattering cross sections of 10C are evaluated
with the computer code PTOLEMY [33, 34]. The code takes
as input the obtained folded potential to represent the real part
of the optical potential while the imaginary part of the potential
is taken in the usual phenomenological form as expressed in
Eq. 6. These potential are used to analyze experimental data of
10C + 27Al (at 29.1 MeV) [10], 10C + 58Ni (at 35.3 MeV) [12]
and 10C + 208Pb (at 66, 226 and 256 MeV) [7, 8]. In order to
reduce the number of fitting parameters, the imaginary reduced
radius, rI and diffuseness parameter aI are fixed at 1.3 and 0.4
fm, respectively.

Finally, in order to assess the quality of agreement between
the calculated results and experimental data, a search on NR and
WI was carried out using the usual reduced chi-square approach
[20]

χ2 = N−1
N∑

k=1

[
σcal(θk) −σex(θk)

∆σex(θk)

]2
, (21)

where σcal(θk) and σex(θk) are the calculated and experimental
cross sections, respectively, ∆σex(θk) is the experimental error,

6



Olorunfunmi et al. / J. Nig. Soc. Phys. Sci. 5 (2023) 1392 7

Table 2: Real normalization parameter, depth of the imaginary potential (WI), real and imaginary volume integrals (JR and JI), total reaction cross sections (σR) and
χ2/N values for the elastic 10C scattering on 27Al, 58Ni, and 208Pb target nuclei. The imaginary radius rI and diffuseness aI are fixed at 1.3 and 0.4 fm, respectively.

System ELab Potential Cluster NR WI JR JI σR χ2/N
(MeV) (MeV) (MeVfm3) (MeVfm3) (mb)

10C+27Al 29.1 M3Y 6Be+α 1.0 10.5 416.610 50.729 779.86 0.42
9B+p 1.0 33.5 416.365 161.856 904.88 0.94
8Be+p+p 1.0 30.5 416.783 147.355 894.90 0.91
α+α+p+p 1.3 10.5 416.610 50.729 775.29 0.37

JLM 6Be+α 1.0 55.5 345.483 268.139 967.45 0.56
9B+p 0.6 55.5 536.506 268.139 910.88 0.61
8Be+p+p 0.6 50.5 524.859 243.982 951.23 0.68
α+α+p+p 1.1 40.5 182.643 195.669 910.88 0.41

10C+58Ni 35.3 M3Y 6Be+α 0.7 52.5 415.575 186.713 469.23 2.33
9B+p 0.3 52.5 415.017 186.713 469.61 2.23
8Be+p+p 0.4 50.5 415.146 179.600 476.74 2.75
α+α+p+p 0.5 50.5 415.559 179.600 465.91 2.35

JLM 6Be+α 1.0 40.0 325.934 142.257 535.16 2.39
9B+p 1.0 40.0 500.371 142.257 535.15 2.42
8Be+p+p 1.0 40.0 489.754 142.257 535.16 2.44
α+α+p+p 1.0 40.0 176.553 142.257 535.16 2.42

10C+208Pb 66.0 M3Y 6Be+α 1.2 300.5 406.2207 595.5052 662.66 16.87
9B+p 1.0 250.5 402.3044 496.4195 672.49 20.50
8Be+p+p 1.0 250.5 402.7722 496.4195 664.92 19.79
α+α+p+p 1.2 300.5 407.7502 595.5052 648.46 15.80

JLM 6Be+α 1.0 450.5 353.997 892.76 708.24 15.56
9B+p 1.0 380.5 428.733 754.042 701.56 16.78
8Be+p+p 1.0 380.5 424.329 754.042 699.67 16.88
α+α+p+p 1.0 450.5 287.292 892.763 699.28 15.61

226.0 M3Y 6Be+α 1.0 30.0 384.199 59.451 3054.40 3.60
9B+p 1.0 30.5 380.620 60.422 3087.70 1.94
8Be+p+p 1.0 25.5 380.867 50.534 3046.90 2.39
α+α+p+p 1.0 40.5 385.806 80.26 3108.2 5.48

JLM 6Be+α 1.0 50.5 666.353 100.0765 3182.6 1.65
9B+p 0.7 40.5 211.937 80.259 3138.7 1.66
8Be+p+p 0.5 40.5 445.607 80.259 3138.1 1.98
α+α+p+p 1.1 50.5 1040.987 100.0766 3168.8 2.11

256.0 M3Y 6Be+α 1.0 30.0 384.203 59.451 3164.4 1.46
9B+p 0.6 30.0 381.022 59.451 3163.7 1.33
8Be+p+p 0.6 32.5 381.016 64.406 3184.6 1.36
α+α+p+p 1.0 30.5 386.112 60.442 3154.1 3.47

JLM 6Be+α 0.6 40.5 1103.551 80.259 3240.2 1.06
9B+p 0.4 40.5 401.396 80.259 3245.1 1.25
8Be+p+p 0.5 40.5 240.516 80.259 3238.0 1.22
α+α+p+p 0.8 40.5 1681.423 80.259 3234.5 1.65

and N is the number of data points. An average value of 10%
error is used as uncertainty on all the experimental data used in
this study.

The real (JR) and imaginary(JI) volume integrals are com-

puted using the expressions

JR(E) =
4π

AP AT

∫
V (r, E)r2dr , (22)

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Olorunfunmi et al. / J. Nig. Soc. Phys. Sci. 5 (2023) 1392 8

Table 3: Reduced reaction cross sections, σM3YRe and σ
JLM
Re , obtained in this work using M3Y and JLM potentials, respectively, compared with σ

SPP
Re obtained from

SPP in Refs.[7, 8, 10].

System ELab ERe Cluster σM3YRe σ
JLM
Re σ

SPP
Re

(MeV) (MeV) (mb) (mb) (mb)
10C+27Al 29.1 1.402 6Be+α 29.353 36.414 36.736 [10]

9B+p 34.058 34.285
8Be+p+p 33.683 35.803
α+α+p+p 29.181 34.285

10C+58Ni 35.3 1.079 6Be+α 12.925 14.741
9B+p 12.935 14.741
8Be+p+p 13.132 14.741
α+α+p+p 12.833 14.741

10C+208Pb 66.0 1.034 6Be+α 10.151 10.850 14.474 [8]
9B+p 10.302 10.747
8Be+p+p 10.186 10.718
α+α+p+p 9.934 10.712

226.0 3.541 6Be+α 46.791 48.755 48.685 [7]
9B+p 47.301 48.083
8Be+p+p 46.676 48.073
α+α+p+p 47.615 48.544

256.0 4.011 6Be+α 48.476 49.638 50.079 [7]
9B+p 48.466 49.713
8Be+p+p 48.786 49.604
α+α+p+p 48.319 49.550

and

JI(E) =
4π

AP AT

∫
W(r, E)r2dr , (23)

respectively.

4. Result and discussion

The real part of the optical potentials is calculated for the
reactions 10C + 27Al at 29.1 MeV, 10C + 58Ni at 35.3 MeV, and
10C + 208Pb at 66, 226 and 256 MeV using the double folding
model (Eq. 4) with M3Y and JLM effective N N interactions,
and four 10C cluster structure densities viz. 6Be + α, 9B + p,
8Be + p + p, and α + α + p + p. Typical calculated folding
potentials (with NR = 1) for the reactions 10C + 27Al at 29.1
MeV, 10C + 58Ni at 35.3 MeV and 10C + 208Pb at 66 MeV are
shown in Figure 1. It can be seen that the M3Y potentials ob-
tained differ primarily in depth and shape from the JLM poten-
tials. Furthermore, for the M3Y potential, the folding potentials
computed using the α + α + p + p configuration are observed
to be deeper than those obtained from the three other 10C clus-
ter density distributions. On the other hand, the use of the 9B
+ p cluster density results in potentials with a shallower depth
compared to the other cluster configurations. In contrast, for
the JLM potential, the 9B + p cluster configuration produces a
deeper potential compared to the other cluster configurations.

The elastic scattering cross sections of the reactions un-
der investigation are calculated using folded real potential and
Woods-Saxon imaginary potential with the four different forms
of cluster densities for 10C. The results of the calculations are
compared with appropriate experimental data and are shown in
Figures 2 - 6. The parameters that give good agreement with
experimental data, the real volume integral, JR, imaginary vol-
ume integral, JI and the reaction cross sections, σR for all the
reactions considered are presented in Table 2. It can be seen
from the figures, and values of NR and χ2/N in Table 2 that the
cross section obtained using M3Y and JLM interactions give al-
most the same quality of fit to experimental data. However, the
values of σR obtained for the JLM interaction are higher than
that of M3Y.

The elastic scattering cross sections of 10C + 27Al at inci-
dent energy 29.1 MeV are investigated for 6Be + α, 9B + p,
8Be + p + p, and α + α + p + p cluster densities of 10Ca, and
the results are shown in Figure 2. The results obtained using
M3Y N N interactions are shown in the top panel while the bot-
tom panel displays the results obtained with JLM interaction.
An excellent agreement between calculated results and data is
observed. Also, the results show similar shape for the different
cluster configurations.

Figure 3 shows the elastic scattering cross sections of 10C
+ 58Ni at 35.3 MeV obtained using the aforementioned set of
cluster densities for 10C and the two effective interactions M3Y
and JLM. Again, the theoretical results agree reasonably well

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Olorunfunmi et al. / J. Nig. Soc. Phys. Sci. 5 (2023) 1392 9

Figure 8: Reduced reaction cross sections, σRe, from the present work and
other results from Refs. [7, 8, 10] with respect to the reduced incident energy,
ERe. The curves are to guide the eye.

with the experimental data.
Finally, the theoretical results of elastic scattering cross sec-

tions of 10C from 208Pb at 66, 226 and 256 MeV using DF model
with the four different cluster densities the two effective inter-
actions are shown and compared with experimental data in Fig-
ures 4 - 6. Overall, good agreement is obtained between exper-
iment and theory, except at 66 MeV where theoretical results
underestimates experimental ones in the angular region 115◦

- 140◦. Similar discrepancy was reported for the same data in
Ref. [8]. The increase of the elastic cross sections at these back-
ward angles was attributed to coupling to excited states in the
projectile nucleus and target nucleus, and this was not captured
in the calculations presented in this study.

The renormalization factor NR is usually applied to folding
potential in order to assess the performance of the DF model
in describing a nuclear reaction [20]. The values of NR used
for the different cluster density configurations and the different
interactions considered in this study are presented in Table 2
and Figure 7. In general, it can be seen that the M3Y and JLM
interactions show the need for almost the same renormalization
factor NR. Also, the values of NR are mostly or close to unity,
except for the cases of 9B + p, and 8Be + p + p in 10C + 58Ni
(with M3Y) and 10C + 208Pb at 256 MeV (with JLM) where the
potentials are heavily reduced (NR = 0.3 to 0.5). In Figure 7,

Figure 9: Modulus of the scattering matrix |S L| versus the versus the orbital
angular momentum L obtained for 10C + 27Al at 29.1 MeV, 10C + 58Ni at 35.3
MeV and 10C + 208Pb at 226 MeV, using M3Y and JLM effective interactions
with 6Be + α, 9B + p, 8Be + p + p, and α + α + p + p cluster density distribu-
tion of the 10C.

the NR values obtained for both the M3Y and JLM interactions
are plotted versus the incident energy ELab. One observes from
this plot that for both interactions the 6Be + α cluster density
generally give NR value closer to unity than other cluster config-
urations. Also, from this figure, one sees that in the case of 10C
+ 58Ni reaction using M3Y potential, the potential is strongly
reduced almost for all the cluster configurations. This might be
due to the presence of other reaction mechanisms not consid-
ered in our calculations, such as the inclusion of the 2+ excited
state of the 10C nucleus, as mentioned in Ref. [12].

The real volume integrals JR values obtained for both the
M3Y and JLM interactions as well as the corresponding reac-
tion cross sections σR for each cluster configuration are pre-
sented in Table 2. One can see that the difference in the value of
the reaction cross section for the various cluster configurations
considered under the same potential is not more than 3%, ex-
cept for the reaction 10C + 27Al under the M3Y potential where
the difference is as high as 13%. In general, the cluster config-
uration 9B + p gives slightly higher value of σR compared to
the other three cluster configurations. Furthermore, in other to
compare the reaction cross sections of the different potentials
and densities with each other and with data from literature, a
reduction method is used. The reduced reaction cross section

9



Olorunfunmi et al. / J. Nig. Soc. Phys. Sci. 5 (2023) 1392 10

and reduced energy are given as follows [35]

σRe =
σR

(A1/3P + A
1/3
T )

2
, (24)

and

ERe = Ec.m. ×
A1/3P + A

1/3
T

ZPZT
, (25)

where Ec.m. is the incident energy in the center of mass frame,
and ZP(T) is the proton number of the projectile (target) nu-
cleus. The reduced energy and reaction cross section values for
the different potentials are listed in Table 3 and shown in Fig-
ure 8. The black filled circle represents the results obtained by
Yang et al., [7], Linares et al., [8] and Aguileral et al., [10], us-
ing SPP potential. It can be seen from the lower panel of Figure
8 that the results obtained with JLM potential agree reasonably
well with that of SPP potential for all the reactions, with the
exception of 10C+208Pb at 66 MeV where the SPP gives higher
reaction cross section ( see Table 3). In the upper panel of the
figure, it can be seen that the σRe obtained for M3Y potentials
are consistently lower than that reported for SPP. In general,
as shown in Table 3 and Figure 8, one observes that for the
M3Y effective interaction the cluster configuration 9B + p gives
slightly higher value of σRe compared to the other three cluster
configurations. For the case of JLM interaction, the cluster con-
figuration 6Be + α gives slightly higher value of σRe compared
to the other three cluster configurations.

The last parameter presented in Table 2 is the reduced chi-
square value χ2/N. It can be seen that, in general, the χ2/N
values are small, which is an indication of good agreement be-
tween the calculated and experimental cross sections.

Figure 9 shows the plot of the magnitudes of partial-wave
scattering (S-matrix) elements |S L| versus the orbital angular
momentum L, calculated for reactions 10C + 27Al at 29.1 MeV,
10C + 58Ni at 35.3 MeV and 10C + 208Pb at 226 MeV, using the
M3Y and JLM potentials. In this figure, we see that the value
of |S L| ≈ 0 at small L and increases rapidly as L becomes larger,
and finally reaches unity. The value of |S L| indicates the level
of absorption. For example, it is known that |S L| = 1 for elastic
scattering means no absorption. Furthermore, it has been sug-
gested that total absorption happens when the transmission co-
efficient (1-|S L|2) equals zero [36]. It can be seen from Figure 9
that the values of L obtained for both M3Y and JLM potentials,
increase with increasing target mass number. Furthermore, the
range of the values of L required for |S L| to rise from 0 to 1
is slightly higher for JLM than for M3Y potential. Lastly, it is
observed that |S L| does not show significant sensitivity to the
project cluster densities under the same potentials.

5. Conclusions

A systematic analysis of elastic scattering cross sections of
10C + 27Al, 10C + 58Ni, and 10C + 208Pb reactions has been
performed within the framework of the double-folding optical
model. This is with the view to investigating the nuclear struc-
ture of the 10C nuclei via the simple cluster model as well as

study the sensitivity of 10C elastic scattering cross sections to
different effective N N interactions. The real part of the opti-
cal potential is constructed by folding two different effective
NN interactions M3Y and JLM, with the density 10C. A cluster
model density distribution is assumed for 10C and four different
cluster configurations are considered, namely, 6Be + α, 9B + p,
8Be + p + p, and α + α + p + p. A phenomenological WS form
is used for imaginary part. For the reactions considered in this
study, the values of the reduced radius rI, and the diffuseness
parameter aI of the WS potential are fixed at 1.3 and 0.4 fm,
respectively, while the depth W0 is adjusted to fit the data.

A comparative study of the four cluster configurations for
10C shows that the results obtained with 6Be + α, 9B + p, 8Be
+ p + p and α + α + p + p cluster configurations describe
the experimental data quantitatively well. However, in terms of
NR, σR and χ2/N, the 6Be + α and 9B + p cluster configura-
tions yield better description of the experimental data than the
other cluster configurations. Also, it is clear that any cluster
density distribution for 10C can be compensated by the param-
eters W0 and NR . This further confirms the cluster structure of
10C nucleus.

In addition, a study of the effect of effective N N interactions
on the elastic scattering cross sections of the reactions consid-
ered in this study reveals that the JLM interactions is as good as
the popular M3Y interactions in terms of their agreements with
experimental data. However, the JLM potential gives higher σR
value than the M3Y potentials. The theoretical calculations
from both M3Y and JLM effective interactions with cluster den-
sity distributions of 10C (6Be + α, 9B + p, 8Be + p + p, and α
+ α + p + p) provide good description of the experimental data
for all the reactions considered in this work.

Furthermore, this study highlights the importance of con-
sidering the overlap of cluster nuclei using appropriate Jacobi
coordinates in a more realistic calculation. Future research can
focus on developing more sophisticated models to account for
the complex structure and dynamics of three or four-cluster nu-
clei and their overlap.

Overall, this study provides valuable insights into the reac-
tion dynamics of complex nuclear systems and can potentially
inform the design and development of future nuclear reactors
and other applications that require a detailed understanding of
nuclear reactions.

References

[1] L. Canto, P. Gomes, R. Donangelo, J. Lubian, and M. S. Hussein, “Recent
developments in fusion and direct reactions with weakly bound nuclei”,
Phys. Rep. 596 (2015) 1. https://doi.org/10.1016/j.physrep.2015.08.001

[2] N. Keeley, R. Raabe, N. Alamanos, and J. Sida, “Fusion and di-
rect reactions of halo nuclei at energies around the Coulomb
barrier”, Prog. in Part. and Nucl. Phys. 59, (2007) 579.
https://doi.org/10.1016/j.ppnp.2007.02.002

[3] D. Savran, T. Aumann, and A. Zilges, “Experimental studies of the
Pygmy Dipole Resonance”, Prog. in Part. and Nucl. Phys. 70 (2013) 210.
http://dx.doi.org/10.1016/j.ppnp.2013.02.003

[4] L. F. Canto, V. Guimarães, J. Lubián, and M. S. Hussein, “The total reac-
tion cross section of heavy-ion reactions induced by stable and unstable
exotic beams: the low-energy regime”, Eur. Phys. J. A 56 (2020) 281.
https://doi.org/10.1140/epja/s10050-020-00277-8

10



Olorunfunmi et al. / J. Nig. Soc. Phys. Sci. 5 (2023) 1392 11

[5] J. Wang et al., “7Be, 8B+208Pb Elastic scattering at above-
barrier energies”, J. of Phys.: Conf. Series 420 (2013) 012075.
https://doi.org/10.1088/1742-6596/420/1/012075

[6] T. Aumann and C. A. Bertulani, “Indirect methods in nuclear astrophysics
with relativistic radioactive beams”, Prog. in Part. and Nucl. Phys. 112
(2020) 103753. https://doi.org/10.1016/j.ppnp.2019.103753

[7] Y. Y. Yang et al., “Quasi-elastic scattering of 10,11C and
10B from a natPb target”, Phys. Rev. C 90 (2014) 014606.
https://doi.org/10.1103/PhysRevC.90.014606

[8] R. Linares et al., “Elastic scattering measurements for the 10C+208Pb
system at ELab = 66 MeV”, Phys. Rev. C 103 (2021) 044613.
https://doi.org/10.1103/PhysRevC.103.044613

[9] N. Curtis et al., “Breakup reaction study of the Brun-
nian nucleus 10C”, Phys. Rev. C 77 (2008) 021301(R).
http://dx.doi.org/10.1103/PhysRevC.77.021301

[10] E. F. Aguilera et al., “Elastic scattering of 10C+27Al”, IOP
Conf. Series: J. of Phys.: Conf. Series 876 (2017) 012001.
http://dx.doi.org/10.1088/1742-6596/876/1/012001

[11] M. Aygun, “Analysis with relativistic mean-field density distribu-
tion of elastic scattering cross-sections of carbon isotopes (10−14,16C)
by various target nuclei”, Pramana - J. Phys. 93 (2019) 72.
https://doi.org/10.1007/s12043-019-1835-y

[12] V. Guimarães et al., “Strong coupling effect in the elastic scattering of
the 10C+58Ni system near barrier”, Phys. Rev. C 100 (2019) 034603.
https://doi.org/10.1103/PhysRevC.100.034603

[13] M. Aygun, “Comprehensive Research of 10C nucleus using
different theoretical approaches”, Ukr. J. Phys. 66 (2021) 8.
https://doi.org/10.15407/ujpe66.8.653

[14] S. D. Olorunfunmi and A. Bahini, “Microscopic analysis of elas-
tic scattering angular distributions for five different density dis-
tribution of 9Be Nucleus”, Phys. Atom. Nuclei 84 (2021) 448.
https://doi.org/10.1134/S106377882104024

[15] M. Anwar, B. El-Naggar, and K. O. Behairy, “Microscopic analysis of the
8B + 58Ni elastic scattering at energies from 20.7 to 29.3 MeV”, J. Phys.
Soc. Jpn. 91 (2022) 014201. https://doi.org/10.7566/JPSJ.91.014201

[16] W. A. Yahya, “Alpha decay half-lives of 171−−189Hg isotopes
using modified Gamow–like model and temperature depen-
dent proximity potential”, J. Nig. Soc. Phys. Sci. 2 (2020) 250.
https://doi.org/10.46481/jnsps.2020.139

[17] S. Adams, E. Joseph, and G. Kamal, “Validation of Tritium Calibra-
tion Curve in CIEMAT/NIST Activity Measurement Using Non Lin-
ear Least Squared Fittings and Calculations of the Half-Life and De-
cay Constant of Potassium-40”, J. Nig. Soc. Phys. Sci. 4 (2022) 621.
https://doi.org/10.46481/jnsps.2022.621

[18] J. P. Jeukenne, A. Lejeunne, and C. Mahaux, “Optical-model potential in
finite nuclei from Reid’s hard core interaction”, Phys. Rev. C 16 (1977)
80. https://doi.org/10.1103/PhysRevC.16.80

[19] G. F. Bertsch, J. Borysowicz, H. McManus, and W. G. Love, “Interactions
for inelastic scattering derived from realistic potentials”, Nucl. Phys. A
284 (1977) 399. https://doi.org/10.1016/0375-9474(77)90392-X

[20] G. R. Satchler and W. G. Love, “Folding model potentials from real-
istic interactions for heavy-ion scattering”, Phys. Rep. 55 (1979) 183.
https://doi.org/10.1016/0370-1573(79)90081-4

[21] F. Carstoiu and M. Lassau, “Microscopic description of elastic scattering
and reaction cross sections of 6Li and 11Li”, Nucl. Phys. A 597, (1996)
269. https://doi.org/10.1016/0375-9474(95)00449-1

[22] O. M. Knyazov and E. F. Hefter, “An analytical folding po-
tential for deformed nuclei”, Z. Phys. A 301 (1981) 277.
https://doi.org/10.1007/BF01416304

[23] A. A. Ibraheem, M. El-Azab Farid, and A. S. Al-Hajjaji, “Analysis of
8B proton halo nucleus scattered from 12C and 58Ni at different ener-
gies”, Brazilian J. Phys. 48 (2018) 507. https://doi.org/10.1007/s13538-
018-0586-4

[24] M. Aygun and Z. Aygun, “A theoretical study on different cluster config-
urations of the 9Be nucleus by using a simple cluster model”, Nucl. Sci.
Tech. 28 (2017) 86. https://doi.org/10.1007/s41365-017-0239-2

[25] M. Aygun, “A comprehensive study on the internal structure and the den-
sity distribution of 12Be”, Rev. Mex. Fis. 62 (2016) 336.

[26] S. M. Lukyanov et al., “Some Insights into Cluster Structure of 9Be
from 3He + 9Be Reaction”, World J. Nucl. Sci. Technol. 5 (2015) 265.
http://dx.doi.org/10.4236/wjnst.2015.54026

[27] A. G. Camacho, P. R. S. Gomes, J. Lubian, and I. Padrón, “Simultaneous
optical model analysis of elastic scattering, fusion, and breakup for the
9Be + 144Sm system at near-barrier energies”, Phys. Rev. C 77 (2008)
054606. https://doi.org/10.1103/PhysRevC.77.054606

[28] Y. Sert, R. Yegin, and H. Doǧan, “A theoretical investigation of 9Be
+ 27Al reaction: phenomenological and microscopic model approxima-
tion”, Indian J. Phys. 89 (2015) 1093. https://doi.org/10.1007/s12648-
015-0685-9

[29] L. C. Chamon et al., “Toward a global description of the
nucleus-nucleus interaction”, Phys. Rev. C 66 (2002) 014610.
https://doi.org/10.1103/PhysRevC.66.014610

[30] A. K. Chaudhuri, “Density distribution of 11Li and proton elas-
tic scattering from 9Li and 11Li”, Phys. Rev. C 49 (1994) 1603.
https://doi.org/10.1103/PhysRevC.49.1603

[31] R. A. Rego, “Closed-form expressions for cross sections of exotic
nuclei”, Nucl. Phys. A 581 (1995) 119. https://doi.org/10.1016/0375-
9474(94)00424-L

[32] J. Cook, “DFPOT - A program for the calculation of dou-
ble folded potentials”, Commun. Comput. Phys. 25 (1982) 125.
https://doi.org/10.1016/0010-4655(82)90029-7

[33] M. Rhoades-Brown, M. H. Macfarlane, S. C. Pieper, “Tech-
niques for heavy-ion coupled-channels calculations. I. Long-
range Coulomb coupling”, Phys. Rev. C 21 (1980) 2417.
https://doi.org/10.1103/PhysRevC.21.2417

[34] M. H. Macfarlane, S. C. Pieper, “Ptolemy: a program for heavy-ion
direct-reaction calculations”, Argonne National Laboratory Report No.
ANL-76-11, (1978) (unpublished).

[35] P. R. S. Gomes, J. Lubian, I. Padron, and R. M. Anjos, “Uncertainties
in the comparison of fusion and reaction cross sections of different sys-
tems involving weakly bound nuclei”, Phys. Rev. C 71 (2005) 017601.
https://doi.org/10.1103/PhysRevC.71.017601

[36] M. C. Mermaz, “Phase shift analysis of heavy-ion elastic scatter-
ing measured at intermediate energies”, Z. Phys. A 321 (1985) 613.
https://doi.org/10.1007/BF01432438

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