# Simultaneous study of scattering and fusion hindrance near Coulomb barrier
in $F+Pb$ systems

Kamala Kanta Jena  P. G. Department of Physics, Bhadrak Autonomous College,
Bhadrak-756100, India  P. G. Department of Physics and Ballistics, Fakir Mohan
University, Balasore-756019, India  Bidhubhusan Sahu  School of Applied
Sciences, KIIT Deemed to be University, Bhubaneswar-751024, India  Jajati K.
Nayak  Variable Energy Cyclotron Centre, 1/AF, Bidhan Nagar, Kolkata-700064,
India  Variable Energy Cyclotron Centre, 1/AF, Bidhan Nagar, Kolkata-700064,
India  Raj Preethi P  Department of Sciences, Amrita Vishwa Vidyapeetham,
Ettimadai, Coimbatore-641105, India  B. K. Sharma  Department of Sciences,
Amrita Vishwa Vidyapeetham, Ettimadai, Coimbatore-641105, India  Santosh Kumar
Agarwalla  P. G. Department of Physics and Ballistics, Fakir Mohan University,
Balasore-756019, India

###### Abstract

A phenomenological optical potential is used to study the elastic angular
distributions for the system ${}^{19}F+^{208}Pb$ close to the Coulomb barrier.
This potential is constructed by taking into account the flexible potential
developed by Ginocchio. The fluctuations in the real and imaginary parts of
the optical model potential follow the trends of the threshold anomaly. The
set of optical potential parameters needed to analyze the fusion cross
sections of the same system are obtained through analysis of the scattering
cross sections. Theoretical fusion cross-sections and results from four
different experimental groups well agree for a range of energies. Several
Fluorine (F) isotopes are used as projectiles in this study of fusion cross-
sections by slightly altering the radial parameter. It was found that the
fusion process occurs unfettered in the ${}^{19}F+^{208}Pb$ system below the
Coulomb barrier but is seriously hindered in the case of its isotopic
projectiles.

Keywords: Optical potential, elastic scattering, threshold anomaly, fusion
cross-section
PACS: 25.70.– z,25.70.Jj, 25.70.Bc

##  1 INTRODUCTION

To examine the various nuclear characteristics, the analysis of experimental
data from nucleus-nucleus scatterings using an optical model has been proven
successful. In optical model analysis, phenomenological nuclear potentials
like Woods-Saxon (WS), Gaussian, modified WS, and many more are employed. In
heavy-ion elastic scatterings, the fluctuation in the real and imaginary
components of an optical potential is seen as a crucial characteristic near
the Coulomb barrier. Threshold anomaly (TA) is an interesting phenomenon, in
the case of systems having heavy projectiles, in which the real component of
the potential is practically constant at higher energies but rapidly increases
as the incident energy gets closer to the Coulomb barrier. When the incident
energy is below the barrier, it slowly starts to decline after reaching its
maximum at the barrier. Around the barrier, thus, the variation takes a bell-
shape. The imaginary part, on the other hand, shows nearly a constant
magnitude at higher energies but decreases to a low value [1, 2, 3, 4, 5, 6,
7, 8, 9, 10] around the barrier in the same vicinity. In other words, when the
collision energy rises above the top of the Coulomb barrier, the strength of
imaginary potential rises rapidly and then its value becomes nearly constant.
The maximum value of the real part can be two fold of the constant value it
assumes at higher energies [11]. This anomalous variation is caused by the
coupling of several elastic and quasi-elastic response channels. This is
explicable by the dispersion relation developed by Byron and Fuller [12] using
the causality principle. This study demonstrates how the optical potential,
which has an imaginary component that is noticeably small, corresponds with TA
occurrences and explains the fusion cross-section. In our discussions, we take
into consideration a semi-classical heavy-ion elastic collision system,
${}^{19}F+^{208}Pb$, whose experimental results may be interpreted in terms of
an optical model by employing a complex potential with the appropriate
parameterization. Lin et al. [13] carried out experimental measurements and
theoretical analyses of the system. The angular distributions were observed
with a fluorine beam (19F) at six energies ranging from 80.6 MeV to 93.5 MeV
in the center-of-mass frame. We analyze the outcomes for the same energy range
to broaden our investigation of elastic scattering with a focus on TA. For TA
analysis, mostly spherical nuclei have been investigated [14]. We work on one
that is deformed. A sizable static ${}^{19}F$ nucleus with deformation is
present in the system ${}^{19}F+^{208}Pb$ [15]. To explore the nuclear
characteristics, 19F has also been utilized as a projectile [16, 17, 18, 19,
20] for decades in several elastic scatterings with various targets and
incidence energies. All these facts motivate our team to choose the projectile
19F and realize the versatility of our potential.

We use a phenomenological optical potential [21, 22] based on a short-ranged,
smooth, and analytically solvable asymmetric potential developed by Ginocchio
[23] that possesses the versatility to control the volume and surface regions
of the potential. The parameters dealt in the optical potential are
significantly less in number. The experimental results of ${}^{16}O+^{28}Si$
and ${}^{12}C+^{24}Mg$ systems were fairly explained by G. S. Mallick et al.
[21] over a wide range of energy by using this potential. The interesting
feature of our potential is the neck structure near the Coulomb barrier. This
non-trivial feature helps us match theoretical data with experimental data.
The potential agrees with the presence of ‘threshold anomaly’ because of the
fast rise of the imaginary part with the rapid fall of the real part of the
potential as the incident energy rises above the Coulomb barrier. The optical
potentials used by most of the researchers deal with large imaginary parts.
The absorption of a major share of partial waves cannot be avoided in cases
having large imaginary parts. The imaginary parts remain below 12% of their
corresponding real parts in Ref.[13] for incident energies 80.6 – 85.2 MeV but
exceed 29% for 87.9 – 93.5 MeV. The fact that high imaginary parts
substantially destroy the resonance states generated by the volume part of the
effective potential cannot be ruled out. Hence, a small imaginary part may be
more convincing. In this work, we use the potential where the imaginary part
is very small as compared to its real part.

We extend the applicability of our optical potential in fusion as well. We use
the potential to analyze fusion cross-section data obtained from four
different experiments performed by D. J. Hinde et al. [24], B. B. Back et al.
[25], K. E. Rehm et al. [26] and Zhang Huanqiao et al. [27] for the same
collision system ${}^{19}F+^{208}Pb$ but over different energy ranges. The
analysis of fusion cross-sections involves mostly the same set of parameters
used for the analysis of elastic scattering cross-sections.

As far as fusion hindrance at sub-barrier energies for drip-line nuclei is
considered, the periodic table expanded (to now include 118 elements), and
super heavy elements (SHE) were made available to humankind due to conceptual
and experimental developments in physics. According to conventional content,
elements with more than 104 protons shouldn’t exist since the element would
undergo spontaneous fission if the fission barrier had been zero. Yet again,
the stabilization of these elements and the formation of SHEs with distinct
features are caused by quantum shell effects. Even though the fusion process
between massive nuclei has been well studied thus far, the fusion probability
between massive nuclei is dependent on the charge product $Z_{P}Z_{T}$ of the
projectile and the target. This is because when the charge product grows, the
Coulomb repulsion between them grows, decreasing the likelihood of fusion.
Nevertheless, nuclear processes involving heavy ions are utilized to produce
SHEs. The formation of SHEs involves the use of both cold fusion and hot
fusion nuclear processes.

The doubly magic nucleus ${}^{208}Pb$ is employed as a target in cold fusion
reactions together with the suitable projectile. The one-dimensional barrier
penetration model, which takes into consideration the coupling of inelastic
excitations, has been noted to accurately represent the fusion cross section
for charge products smaller than 1800. On the other hand, in contrast to the
model’s calculated results, the fusion cross section is hampered when the
charge product is greater than 1800 [28]. But in addition to the charge
product, other factors that affect nuclear fusion between heavy nuclei include
the nuclear structures of the projectile and target. According to reports, the
fusion probability is significantly influenced by the number of valence
nucleons outside of a major shell closure [28, 29]. In the fusion processes,
${}^{130}Xe+^{86}Kr$ and ${}^{136}Xe+^{86}Kr$, where the nucleus ${}^{136}Xe$
has a closed neutron shell N=82 and the neutron number of the nucleus
${}^{130}Xe$ is 76, six neutrons fewer than the closed shell, the evaporation
residue cross sections were determined by Oganessian et al. [30] in 1987. They
discovered that, in the vicinity of the Coulomb barrier, the measured
evaporation residue cross sections for the fusion process ${}^{136}Xe+^{86}Kr$
are about two to three orders of magnitude greater than those for the fusion
reaction ${}^{130}Xe+^{86}Kr$. The enhancement of the evaporation residue
cross sections near the Coulomb barrier region between the double closed shell
nuclei ${}^{208}Pb$ and ${}^{48}Ca$ is also pointed out by Oganessian et al.
[31] in 2001. The dependence of fusion on the nuclear shell structure was
investigated by K. Satou et al. [32] in 2002 for the two reaction systems
${}^{82}Se+^{138}Ba$ and ${}^{82}Se+^{134}Ba$, where the nucleus ${}^{138}Ba$
has a closed neutron shell N=82 while the nucleus ${}^{134}Ba$ has a neutron
number N=78; four neutrons less than the closed shell. The fusion reaction
${}^{82}Se+^{138}Ba$ takes place without hindrance, but ${}^{82}Se+^{134}Ba$
fusion is significantly hindered, as is typically observed in major reaction
systems with the charge product $Z_{P}Z_{T}$ $\geq$ 1800 of the projectile and
target. These results suggest that a crucial part of the low-energy fusion
process involves the shell structure. We analyze the isotopic dependence in
the ${}^{19-23}F+^{208}Pb$ systems to realize how the nuclear shell structure
affects the fusion process.

The paper is organized to discuss the formulation of the optical model based
on Ginocchio potential in section $\bf{2}$. Section $\bf{3}$ explains the
application of our optical potential to the elastic scattering of tightly
bound projectile ${}^{19}F$ by ${}^{208}Pb$ target at energies near the
Coulomb barrier and TA phenomena thereof. The fusion cross-sections for this
system are also presented and compared with the data from the various
experiments. Finally, the summary and conclusions are presented in section
$\bf{4}$.

Figure 1: The plot of real part $V_{n}(r)$ of optical potential for the system
${}^{19}F+^{208}Pb$ at incident energy $E_{c.m.}$ = 87.9 MeV. Values of
parameters are taken to be $R_{0}$ = 12.5 fm, $B_{0}$ = 118 MeV, $B_{1}$ =
7.0, $B_{2}$ = 0.6 and $V_{B}$ = 2.6 MeV. The curve shows a neck formation at
$R_{0}$.

[t]

Figure 2: Imaginary part $W_{n}(r)$ of optical potential for the system
${}^{19}F+^{208}Pb$ at incident energy $E_{c.m}$= 87.9 MeV. Values of
parameters are $R_{0W}$ = 12.8 fm, $W_{0}$= 1.5 MeV, $V_{BW}$ = 0.6 MeV,
$W_{1}$ = 1.4 and $W_{2}$= 0.001.

##  2 FORMULATION OF THEORY

The phenomenological optical potential used here is based on a short-ranged,
smooth, and analytically solvable asymmetric potential developed by Ginocchio
[23] and used by others [21, 22, 33, 34, 35] which possesses the versatility
to control the volume and surface regions. A potential to describe nucleus-
nucleus interaction usually consists of Coulomb potential $V_{C}(r)$ due to
electric charges of two nuclei and nuclear potential $V_{N}(r)$. Taking the
centrifugal force into account, the effective potential $V_{eff}(r)$) for
nucleus-nucleus collision with reduced mass $\mu$ and orbital quantum number
$l$ can be described by Eq.1 in which the last term represents the potential
owing to centrifugal force.

$V_{eff}(r)=V_{N}(r)+V_{C}(r)+\frac{l(l+1){\hbar}^{2}}{2\mu r^{2}}$ (1)

The nuclear part $V_{N}(r)$ is an optical potential. It is taken from Ref.[23]
by considering the value of parameter $\lambda$=1, where $\lambda$ parameter
is responsible for the flatness of the potential. Nuclear potential $V_{N}(r)$
is the most important part of the effective potential which is not uniquely
described to date. But $V_{N}(r)$, as has been argued in many articles, takes
a complex form to describe the experimental observations. We also consider
$V_{N}(r)$ to be complex and represent as, $V_{N}(r)$ = $V_{n}(r)+iW_{n}(r)$.
The variable $\mu$ represents the reduced mass of the projectile-target system
and is defined as $\mu=\frac{m_{P}\times m_{T}}{m_{P}+m_{T}}$, where $m_{P}$
and $m_{T}$ being the masses of projectile and target respectively.

Following the potential developed in Ref.[23], which is further simplified in
Ref.[22], we consider the real part $V_{n}(r)$ of the potential as given in
Eq.2 by putting $\lambda$=1.

$V_{n}(r)=\left\\{\begin{array}[]{ll}\frac{-V_{B}}{B_{1}}\left[B_{0}+(B_{1}-B_{0})(1-y_{1}^{2})\right]&\textrm{if}~{}~{}0<r<R_{0}\\\
&\\\
\frac{-V_{B}}{B_{2}}\left[B_{2}(1-y_{2}^{2})\right]&\textrm{if}~{}~{}r\geq
R_{0}\\\ \end{array}\right.$ (2)

On substitutions of $y=tanh\rho_{n}$, $\rho_{n}=(r-R_{0})b_{n}$, and
$V_{B}=V_{01}B_{1}$ = $V_{02}B_{2}$, we find

$V_{n}(r)=\left\\{\begin{array}[]{ll}-V_{01}\left[B_{0}+\frac{(B_{1}-B_{0})}{cosh^{2}\rho_{1}}\right]&\textrm{if}~{}~{}0<r<R_{0}\\\
&\\\ -V_{02}\left[\frac{B_{2}}{cosh^{2}\rho_{2}}\right]&\textrm{if}~{}~{}r\geq
R_{0}\\\ \end{array}\right.$ (3)

Here, $V_{B}$ is the height of the barrier. The slope parameter $b_{n}$ is
given by, $b_{n}=\frac{\sqrt{2\mu V_{B}}}{\hbar^{2}B_{n}}$ in which n = 1 or
2. The radial distance $R_{0}$ in the surface region is close to the radial
position of the effective S-wave barrier potential. The depth of potential at
origin and $R_{0}$ are controlled by the parameters $B_{0}$ and $V_{B}$
respectively. Slope parameter $b_{n}$ on either side of $R_{0}$ depends on
$B_{n}$ and $V_{B}$. Parameters $B_{0}$ and $B_{1}$ specify the potential for
$r\leq R_{0}$. $V_{01}$ specifies the strength in that region, and is given
by,

$V_{01}=\frac{V_{B}}{\lambda_{1}^{2}B_{1}+\frac{1-\lambda_{1}^{2}}{2}}=\frac{V_{B}}{B_{1}}$
for $\lambda_{1}$ =1. The parameter $\lambda_{1}$ controls the flatness of the
potential in the region for $r\leq R_{0}$. Similarly, the parameters $B_{2}$
and $V_{B}$ specify the potential for the region $r>R_{0}$, where $R_{0}$ is
sum of radii of two interacting nuclei, i.e.,
$R_{0}{=r_{0}(A_{P}^{\frac{1}{3}}+A_{T}^{\frac{1}{3}})}$= $R_{1}$ \+ $R_{2}$.
The quantities $\rho_{1}$ and $\rho_{2}$ are the transformed distance
variables and given by,
$\rho_{n}=(r-R_{0})\frac{\sqrt{2mV_{B}}}{\hbar^{2}B_{n}}=(r-R_{0})b_{n}$. With
the above consideration, the real part $V_{n}$(r) of the optical potential for
the collision system ${}^{19}F+^{208}Pb$ is depicted in Fig.1. The potential
has two regions; volume and surface. Two parts of the potential corresponding
to the volume region and surface region are connected at $r=R_{0}$ satisfying
the analytic continuity. Unlike monotonous fall with ‘r’ in a nuclear
potential of a standard Woods-Saxon form, our optical potential shows a neck-
formation near $r=R_{0}$. The optical potential consists of two analytically
solvable regions, namely, the volume region and the surface region. The
regions are smoothly joined near r = $R_{0}$ forming a neck-like structure. We
refer the location to an analytic junction [34, 35], where the two regions of
the potential meet each other. As the name suggests, the junction is
analytically solvable and the Schrodinger equation can be solved there. The
structure appears unusual, but our consideration ensures indifference in two
parts of potential and keeps the respective derivatives (concerning ‘r’) the
same at the meeting point satisfying analytic continuity. This new feature
helps us suitably explain the differential scattering cross-sections and
fusion cross-sections over a wide range of energies. The feature also enables
us to apprehend the effects of frictional forces, resonance in the formation
of a composite binuclear system and transfer of one or cluster of nucleons
from the target to the projectile and/or vice versa in this configuration,
when the bombarding nuclei touch each other in the surface region around
$r=R_{0}$.

The form of the imaginary part $W_{n}(r)$ is similar to that of the real part,
but its strength differs. The imaginary part in strength is weaker than that
of the real part, i.e., the real part with a larger value is very deep and the
imaginary part with a smaller value is comparatively weak. With substitution
$V_{0nW}=\frac{V_{BW}}{W_{n}}$, the imaginary part is given by Eq.4 and its
behaviour is plotted in Fig.2 with a suitable set of parameters.

$W_{n}(r)=\left\\{\begin{array}[]{ll}-V_{01W}\left[W_{0}+\frac{(W_{1}-W_{0})}{cosh^{2}\rho_{1}}\right]&\textrm{if}~{}~{}0<r<R_{0W}\\\
&\\\
-V_{02W}\left[\frac{W_{2}}{cosh^{2}\rho_{2}}\right]&\textrm{if}~{}~{}r\geq
R_{0W}\\\ \end{array}\right.$ (4)

The parameter $W_{0}$ represents the depth of imaginary potential at the
origin and $V_{BW}$ controls the depth of potential at $R_{0W}$. The other two
parameters, namely, $W_{1}$ and $W_{2}$ are slope parameters. Parameter
$W_{1}$ specifies the potential for $r\leq R_{0W}$, whereas, $W_{2}$ specifies
the potential for $r>R_{0W}$. We use a set of these parameters to represent
imaginary part in Fig.2. The Coulomb potential for the projectile nucleus and
target nucleus interacting system is given by Eq.5 as follows.

$V_{C}(r)=\left\\{\begin{array}[]{ll}\frac{Z_{p}Z_{T}e^{2}}{2R_{C}^{3}}(3R_{C}^{2}-r^{2})&\textrm{if}~{}~{}r<R_{C}\\\
&\\\ \frac{Z_{p}Z_{T}e^{2}}{r}&\textrm{if}~{}~{}r>R_{C}\\\ \end{array}\right.$
(5)

Here, $R_{C}=r_{C}(A_{P}^{1/3}+A_{T}^{1/3})$;$A_{P}$ and $A_{T}$ being the
mass numbers of projectile and target nuclei respectively. $Z_{P}$ and $Z_{T}$
are the atomic numbers of those nuclei. The value of the Coulomb radius
parameter $r_{C}$ is taken to be 1.33 fm. Neglecting the centrifugal term with
orbital quantum number $l$ = 0, Eq.1 describes effective potential as :

$V_{eff}(r)=V_{n}(r)+iW_{n}(r)+V_{C}(r)$ (6)

The real part of the effective potential $V_{eff}(r)$ is depicted in Fig.3
with a set of parameters considered earlier for the real part in Fig.1. With
the above effective potential for the various partial waves ($l$), we solve
the following Schrodinger equation to obtain the total scattering amplitude
$f(\theta)$.

$\left[\frac{-\hbar^{2}}{2\mu}\nabla^{2}+V_{eff}(r)\right]\psi(\vec{r})=E\psi(\vec{r})$
(7)

Total scattering amplitude $f(\theta)$ is expressed as the sum of Coulomb
scattering amplitudes $f_{C}(\theta)$ and nuclear scattering amplitude
$f_{N}(\theta)$ respectively thus,

$f(\theta)=f_{C}(\theta)+f_{N}(\theta)$ (8)

The amplitudes $f_{N}(\theta)$ and $f_{C}(\theta)$ have expansions as follows.

$f_{N}(\theta)=\frac{1}{2ik}\sum_{l}(2l+1)e^{2i\sigma_{l}}\left(e^{2i\bar{\delta_{l}}-1}\right)P_{l}(cos\theta)$
(9)
$f_{C}(\theta)=\frac{1}{2ik}\sum_{l}(2l+1)\left(e^{2i\bar{\delta_{l}}-1}\right)P_{l}(cos\theta)$
(10)

Here k is the magnitude of wave vector $\overrightarrow{k}$, $\sigma_{l}$ is
the Coulomb phase shift due to scattering and $\bar{\delta_{l}}$ is the
nuclear phase shift. The ratio of the measured elastic scattering cross-
section to Rutherford’s scattering cross-section is given by

$\frac{d\sigma_{el}}{d\sigma_{Ruth}}=|{\frac{f(\theta)}{f_{C}(\theta)}}|^{2}$
(11)

For $l^{th}$ partial wave and its S-matrix $S_{l}$, the elastic scattering
cross-section $\sigma_{el}$ and the reaction cross-section $\sigma_{r}$ are
given by Eq.12 and Eq.13 respectively as follows.

$\sigma_{el}=\frac{\pi}{k^{2}}(2l+1)|1-S_{l}|^{2}$ (12)
$\sigma_{r}=\frac{\pi}{k^{2}}(2l+1)T_{l}(E)=\frac{\pi}{k^{2}}(2l+1)(1-|S_{l}|^{2})$
(13)

Here, $T_{l}(E)=(1-|S_{l}|^{2}$ . This is known as the transmission
coefficient for the orbital angular momentum $l$. The fusion cross section is
given by, $\sigma_{fus}=\frac{\pi}{k^{2}}(2l+1)P_{l}^{F}$ , where $P_{l}^{F}$
is the fusion (absorption) probability. The wave function, in the case of
fusion, is expected to be absorbed completely inside the barrier; hence the
fusion probability can be assumed to be close to the probability that the
incident current reaches the point of total absorption. Therefore, fusion
probability, i.e.,$P_{l}^{F}\sim T_{l}(E)=(1-|S_{l}|^{2}$ . Thus, we have,
$\sigma_{r}=\sigma_{fus}$. Based on the above theory and potential, the
results of elastic scattering and fusion are discussed.

Figure 3: The plot of real part of effective potential Veff(r) for $l$= 0 of
the system ${}^{19}F+^{208}Pb$ at energy 87.9 MeV in the centre-of-mass frame.
Values of parameters are $R_{0}$= 12.5 fm, $B_{0}$= 118 MeV, $B_{1}$= 7.0,
$B_{2}$= 0.6 and $V_{B}$ = 2.6 MeV. The curve retains the neck structure at
$R_{0}$.

Figure 4: Comparison of the calculated angular distribution of elastic
scattering cross-sections with experimental data for six incident energies of
the system ${}^{19}F+^{208}Pb$. The dark circles (green) represent
experimental data and the solid curve (red) represents theoretical
calculations. The experimental data for Ref.[13] are downloaded from the site
http: //nrv.jinr.ru. The values of the six independent parameters are found
$R_{0}$= 12.5 fm, $R_{0W}$ = 12.8 fm, $B_{2}$= 0.6, $B_{0}$= 118 MeV,
$W_{1}$=1.4 and $W_{0}$=1.5 MeV for all incident energies.

##  3 RESULTS

While analyzing angular distribution cross-sections of the elastic scattering
${}^{19}F+^{208}Pb$, C. J. Lin et al. [13] have used Woods-Saxon based optical
potential, in which imaginary potential depths are approximately 29 - 60$\%$
or more of the real part to get the best fit of data for energy span 87.9 -
94.0 MeV. The minimum value of the real part differs by 21$\%$ from the
maximum value, and the difference is more than 96$\%$ in the case of imaginary
parts for the same range of incident energy. Scattered values may raise
uncertainty to conclude threshold anomaly and higher imaginary values
concerning real parts may suppress resonance states of the system generated by
the effective potential. We use a substantially small imaginary part to real
part ratio to measure angular distribution cross-sections at different
incident energies. The success story of the potential generated by using
Ginocchio potential [21, 22, 23] catalyzes us to explain the experimental data
in this optical model analysis. The variations in real and imaginary parts
near the Coulomb barriers are studied to realize threshold anomaly. The
results are presented in the following sub-sections.

###  3.1 Analysis of scattering cross-sections

The laboratory energies for elastic scattering of collimated ${}^{19}F$ beam
by ${}^{208}Pb$ target are taken at 88, 91, 93, 96, 98, and 102 MeV, which are
equivalent to 80.6, 83.4, 85.2, 87.9, 89.8, and 94.0 MeV respectively in the
center of mass frame. The system ${}^{19}F+^{208}Pb$ has the Coulomb barrier
at about $E_{c.m.}$= 84 MeV. Energy-dependent parameters (both real and
imaginary) of the optical potential are presented in Table-I for the best fit.
The calculated angular distribution of elastic cross-sections is compared with
experimental values in Fig.4 for the given range of incident energies. The
experimental values digitized with GSYS-2.4 are obtained from the source http
://nrv.jinr.ru.

Table 1: Energy-dependent parameters of the Optical Potential $E_{CM}$ | $V_{B}$ | $W_{2}$ | $V_{BW}$ | $B_{1}$
---|---|---|---|---
(MeV) | (MeV) |  | (MeV) |
80.6 | 2.2 | 2.5 | 0.06 | 2.5
83.4 | 2.4 | 2.2 | 0.1 | 6.0
85.2 | 2.5 | 1.2 | 0.3 | 6.0
87.9 | 2.6 | 0.001 | 0.6 | 7.0
89.8 | 2.3 | 2.5 | 0.7 | 2.0
94.0 | 2.2 | 3.0 | 1.2 | 1.0

Six number of parameters show energy-independence. The radial distance in the
surface region $R_{0}$ is kept constant with energy. The values of the
independent parameters are $R_{0}$ = 12.5 fm, $R_{0W}$ = 12.8 fm, $B_{2}$ =
0.6, $B_{0}$= 118 MeV, $W_{0}$=1.5 MeV and $W_{1}$=1.4 while matching
theoretical results with experimental outcomes for the entire range of
energies. Four parameters, namely, $B_{1}$, $V_{B}$, $W_{2}$, and $V_{BW}$
mentioned in Table-I vary with collision energies. We keep the Coulomb radius
parameter $r_{C}$ at 1.33 fm following the literature referred [13]. The
theoretical data (solid line curve in red colour) are compared with the
experimental data (dark circles filled with green colour) obtained from Ref.
[13]. Theoretical calculations fairly agree with the experimental values.

It is worth mentioning that the best fit takes a smaller imaginary part in
comparison to the corresponding real part of the potential which ensures less
suppression of resonance states generated by the effective potential. The real
parts of the potential are taken the same, i.e., 118.0 MeV for all the six
incident energies. The imaginary parts are also kept small and the same, i.e.,
1.5 MeV in comparison to their counterpart’s real potential. Thus, the ratios
of imaginary parts to real parts remain the same, i.e., 0.0127 for all
colliding energies. Such a small imaginary-to-real ratio will help not to
substantially destroy the resonance states generated by the volume part of the
effective potential. The need for a small imaginary part in an optical
potential is explained with supporting graphs in sub-section 3.4.

Figure 5: Variation of real part ($V_{B}$) and imaginary part ($V_{BW}$) of
the optical potential around the Coulomb barrier. The blue dash-line with red
star-symbols represents the variation of real parts with incident energies,
whereas, the blue dash-line with dark red circles represents simultaneous
variation of imaginary parts. The variation shows the presence of TA in the
vicinity of the Coulomb barrier.

###  3.2 Phenomenon of threshold anomaly

While reproducing the experimental results using theoretical calculations of
the potential, we find variations in real and imaginary parts of the potential
near the Coulomb barrier. The variations are described in Fig.5. When we
proceed from the lower energy of the collision, the real part first increases
and then decreases in the vicinity of the Coulomb barrier, and ultimately
saturates around 2.2 MeV at higher energies away from the barrier. On the
other hand, the imaginary part remains almost constant at 1.2 MeV at higher
energies but decreases in the vicinity of the Coulomb barrier as we move from
higher to lower values of incident energy. The variation in the real part
follows a bell-shaped dash line (upper plot), whereas, the variation in the
corresponding imaginary part follows an L-shaped dash line (lower plot). The
bell-shaped and L-shaped curves are obtained by the dispersion relations in
Ref.[13], which describes threshold anomaly. Thus, theoretical calculations
fairly agree with the TA phenomenon described in the reference.

###  3.3 Analysis of fusion cross-sections

Along with the angular distribution, the present analysis is extended with the
potential to check the fusion cross-sections. The experimental data of the
fusion cross-section ($\sigma_{fus}$) with ${}^{19}F+^{208}Pb$ system are
available by different groups from independent experiments. B. B. Back et al.
[25] measured the fission fragment angular distributions in 1985 by using
Argonne Superconducting LINAC in the energy range 100.8 to 174.0 MeV in the
center-of-mass frame. The fission cross sections and angular distributions
were measured in 1990 by Zhang Huanqiao et al. [27] with HI-13 tandem Van de
Graff accelerator at bombarding energies from 75.51 to 137.5 MeV. The fusion-
fission cross-sections were also measured in 1998 by K. E. Rehm et al. [26] by
using ATLAS superconducting linear accelerator for the reaction at energies
from 77.68 to 99.6 MeV. 1n 1999, D. J. Hinde et al. [24] were able to measure
fission fragment cross sections and angular anisotropies to high accuracy for
the reaction by using 14UD tandem electrostatic accelerator and LINAC in the
energy range 76.0 to 144.7 MeV. The experimentally measured fusion cross-
section values digitized with GSYS-2.4 are downloaded from the website
http://nrv.jinr.ru.

To explain the data of the fusion cross-section, we use the same parameters
except for $R_{0W}$ and $W_{2}$ of the optical potential obtained while
matching theoretical calculations with the corresponding angular distributions
in the scatterings of the system at the incident energy of 94 MeV in the
center-of-mass frame. The value of $R_{0W}$ has been altered from 12.8 MeV to
8.2 MeV and that of $W_{2}$ from 3.0 to 0.8 to explain fusion data. Thus, the
set of parameters to reproduce fusion cross-sections is $R_{0}$=12.5 fm,
$R_{0W}$ =8.2 fm, $B_{1}$=1.0, $B_{2}$=0.6, $V_{B}$=2.2 MeV, $V_{BW}$=1.2 MeV,
$B_{0}$= 118 MeV, $W_{0}$= 1.5 MeV, $W_{1}$=1.4 and $W_{2}$= 0.8. The set of
parameters fairly explains the experimental values of four independent
experiments performed for different energy ranges. The results are shown in
Fig.6 with fusion cross-sections (expressed in mb) as the function of the
center of mass energy (taken in MeV). It is challenging to find a unique
potential that can address both of these phenomena simultaneously. $R_{0W}$ in
case of fusion is considered to be 8.2 fm, which is less than the Coulomb
barrier position ($R_{B}$ = 12.8 fm) keeping the imaginary $W_{0}$ unchanged
at 1.5 MeV to observe the structure phenomena. The value of $R_{0W}$
($<R_{B}$) confirms that fusion takes place only after the barrier has been
fully penetrated. The other parameter $W_{2}$ was reduced from 3.0 to 0.8 to
observe the fusion phenomena. Finally, we get the required fusion cross-
section which is plotted in Fig.6. The structure effect is also visible here
because ${}^{208}Pb$ is a double magic and shell closure nucleus and
${}^{19}F$ is relatively stable projectile. The binding energy per nucleon
value of ${}^{19}F$ is higher compared to other isotopes such as ${}^{21}F$
and ${}^{23}F$. We do not find hindrances below the Coulomb barrier because of
the shell closure of both the projectile and target. As we move toward the
neutron drip-line we see the hindrance phenomena as described in Fig.7.

Figure 6: Plot of fusion cross-sections as a function of colliding energy in
centre-of-mass frame. The solid curve (blue) represents theoretical values of
fusion cross-sections. The experimental values obtained from Back et al. are
represented by rhombus, Hinde et al. by red circles, Rehm et al. by triangles
and Zhang et al. by green circles. The experimental data are downloaded from
the site http://nrv.jinr.ru. Set of parameters taken are $R_{0}$=12.5 fm,
$R_{0W}$ =8.2 fm, $B_{1}$=1.0, $B_{2}$=0.6, $V_{B}$=2.2 MeV, $V_{BW}$=1.2 MeV,
$B_{0}$= 118 MeV, $W_{0}$= 1.5 MeV, $W_{1}$=1.4 and $W_{2}$= 0.8.  Figure 7:
Plots of fusion cross-sections with different isotopes of Fluorine projectile.
The figure shows the structure effect due presence of double magic and shell
closure nucleus.

It is commonly known that a projectile’s binding energy has a significant
impact on how easily it fragments into smaller parts, which has an impact on
the fusion of weakly bound projectiles. It is difficult to calculate the
dynamic polarization potential theoretically, thus up to now, TA has only been
studied experimentally. Most of the examined systems are spherical or very
close to spherical shape. The implications of nuclear structure on well-
deformed systems have received very little attention in research. C. J. Lin et
al. [13] used the system ${}^{19}F+^{208}Pb$ to investigate the role of
deformed nuclei in fusion and TA responses. The projectile nucleus ${}^{19}F$
possesses [36] quite large static deformations ($\beta_{2}$ = 0.44 and
$\beta_{4}$ = 0.14). The results of the fusion reaction [13] for the system
have been carefully examined. We analyze the cross-sections of fusion
reactions for the fluorine isotopes with the double shell closure of
${}^{208}Pb$. The nucleus of a nearby projectile with double shell closure is
${}^{16}O$. In the case of ${}^{19}F$, one half-filled valence proton and two
valence neutrons are present, and the same is evident for ${}^{21}F$ and
${}^{23}F$ nuclei. As we approach the drip-line nuclei, the two-neutron
separation energy ($S_{2n}$) of fluorine continues to decrease. The magnitudes
of $S_{2n}$ for the isotopes ${}^{19}F$, ${}^{21}F$ and ${}^{23}F$ are 19.582
MeV, 14.742 MeV, and 12.81 MeV respectively [37]. Moreover, the corresponding
binding energies per particle (B.E./A) are 7.779 MeV, 7.738 MeV, and 7.622 MeV
respectively. Again, Z=82 and N=126 in the target nucleus are double magic
numbers. The isotopes of Fluorine (F) could split into ${}^{16}O$ and neutron
clusters due to the valence neutrons’ low binding energy.

Again, it is observed that the presence of magic shells in the entrance
channel increases the probability of fusion [38], because magic nuclei are
difficult to excite, which lowers energy dissipation and facilitates the
creation of a more compact di-nuclear system. So, when we go away from the
shell closure, the 2n, 4n, and 6n evaporation residue cross-sections may
indeed be enhanced close to the Coulomb barrier, which might account for the
hindrance of the fusion cross-section shown in Fig.7. Nevertheless, it is
beyond the scope of our approach to demonstrate the entire reaction cross-
section, including quasi-fission and evaporation residues. We have just
displayed the fusion cross-section here. Because of quasi-fission, the fusion
probability could well be significantly suppressed below the Coulomb barrier,
which might be the cause of the fusion hindrance [39]. Hence, the fusion of
weakly bound projectiles is affected by the breakup channel coupling. Fig.7
illustrates how the breakup channel coupling affects ${}^{19-23}F+^{208}Pb$
systems depending on the valence nucleon separation energy. As per
expectation, the breakup cross-section grows as the binding and separation
energies fall, making the breakup of a weakly bound nucleus more feasible. As
the structure of interacting nuclei influences the mechanism of the fusion and
other processes leading to the absorption of particles from the elastic
channel then it is natural to expect that the Optical Model parameters should
vary from one system of colliding nuclei to another one. The parameter $R_{0}$
is altered. So the potential barrier and hence the fusion cross section
increase. We demonstrate the effects of only charge and mass on the fusion
phenomenon. Nuclear shell structures, deformation, and orientation are
additional factors that influence the fusion processes in addition to charge
and mass [28, 29, 30].

###  3.4 Need for a small imaginary part in the optical potential

The scattering process in a nuclear collision is sensitive to the nature of
the potential on the surface region. On the contrary, the fusion process is an
interior activity. It is quite difficult to find a unique nuclear potential
that can take care of both phenomena.

It is a common assumption that fusion takes place only after the barrier has
been fully penetrated [40, 41]. Based on this concept that the fusion of two
nuclei occurs in the region interior to the radial position ($R_{B}$) of the
Coulomb barrier, the region $0<r<R_{B}$ is expected to account for the
experimental data of fusion cross section as the total reaction cross section
includes the cross section for different reaction channels of which the fusion
channel is predominant in the low-energy collision activities. The values of
fusion radius and the Coulomb radius used in the heavy ion collision system
${}^{19}F+^{208}Pb$ agree with the fact that fusion is an interior phenomenon,
whereas the surface phenomenon is attributed to scattering and other
peripheral, less absorptive direct reaction processes.

In this study, we identify two crucial aspects of our potential: (i) the real
component with a larger magnitude, and (ii) the imaginary part with a smaller
value. Thus, in contrast to light ion systems, this potential has a less
absorptive character. Because of its less absorptive nature, standing waves
are formed in the nuclear well, which allows shape resonances to survive in
the collision process. As a result, these resonances produce the oscillatory
structures in the fusion (total reaction) cross-section, $\sigma_{fus}$
($\sigma_{r}$) as a function of colliding energy $E_{c.m.}$ Although the
resonances exist, it is very difficult to detect the resonances experimentally
through direct observations [42].

Figure 8: Comparison of reaction cross section curve at $W_{0}$=1.5 MeV with
reaction cross section curve at $W_{0}$=15 MeV keeping other parameters
unaltered. Resonances are shown in the case of a lower $W_{0}$ value (blue
curve).  Figure 9: Reaction cross-section curves for the system
${}^{19}F+^{208}Pb$ against incident energy. Resonances are shown in case of
lower $W_{0}$ value, i.e., $W_{0}$=1.5 MeV keeping other parameters unaltered
except the values of $l$. The amplitude of oscillation increases with the
increase of $l$-values of partial waves.

In the potential scattering theory, these resonances are manifested clearly as
maxima in the results of reaction cross-section ($\sigma_{r}$) at the
respective resonance energies [43, 44]. This small value of the imaginary part
further indicates that the fusion only occurs when the barrier has been
completely penetrated [41, 45, 46]. Due to the potential’s smaller absorption
capacity, standing waves in the nuclear well might occur, which would allow
shape resonance states (which have not been experimentally detected) [42] to
survive the collision process. As a result, these resonances take on the role
of being the cause of the oscillatory structure in the barrier distribution,
$D(E_{c.m.})$ findings as a function of $E_{c.m.}$ [45, 46]. When the
potential is made more absorbed by considering a bigger imaginary part
$W_{0}$, the width of the resonance caused by the real part of the potential
widens. Consequently, larger width leads to the extinction of the
corresponding resonance in the collision process. In this study, we have
considered a deep real potential associated with a relatively weak imaginary
strength $W_{0}$. As explained above, the resonances are visible in the form
of peaks in the partial wave trajectories for a smaller value of $W_{0}$, but
the oscillation in $\sigma_{r}$ vanishes for a larger value of $W_{0}$, which
is shown explicitly by taking $W_{0}$=1.5 MeV and 15 MeV in Fig.8. The
cumulative effect of all these resonances is primarily responsible for the
oscillation in reaction cross-section ($\sigma_{r}$). The fusion radius is
found to be more than the Coulomb radius when a larger value of $W_{0}$ is
considered in the case of heavy ion collisions.

The amplitudes of oscillation increase with increasing $l$-values for a
particular low imaginary potential. This is verified by changing the variation
of resonance structures for different $l$’s with a particularly low value of
$W_{0}$ as shown in Fig.9. The plots in Fig.9 explain how the amplitudes of
oscillation increase with increasing $l$-values for a particular low imaginary
potential. Here the imaginary depth is kept low at $W_{0}$=1.5 MeV and the
oscillations with increasing amplitudes are shown for different values of $l$,
i.e., $l$=10, 20, 30, 50.

##  4 CONCLUSIONS

We use the optical potential taken into consideration in the paradigm of
Ginocchio potential to explain the angular distributions of elastic scattering
of the system ${}^{19}F+^{208}Pb$ for the center of mass energies $E_{c.m.}$=
80.6, 83.4, 85.2, 87.9, 89.8, and 94.0 MeV. At a nucleus’ surface, the
potential has a particular deformation effect. We calculate the fusion cross-
sections for the same system and compare these values with various independent
findings from four distinct experiments carried out by the researchers in Ref.
[24, 25, 26, 27]. A theoretical calculation provides a good explanation for
the data showing threshold anomaly close to the system’s Coulomb barrier. To
ensure that resonance states are not too suppressed, the imaginary components
of the potential employed in the current study are kept relatively modest in
comparison to the real parts. Due to the shell closure of both interacting
nuclei, no hindrance phenomenon is seen in the system ${}^{19}F+^{208}Pb$.

As shown in Fig.7, a weakly bound nucleus significantly impacts the fusion due
to the increased possibility of dissociation. So, it may be argued that
although the Coulomb repulsion is stronger, the existence of neutron shell
closure and breakup probability in the entrance channel favours fusion
hindrance just below the Coulomb barrier. More studies are needed to
understand the dynamics in the sub-barrier area and to identify additional
influencing elements that may further favour or hinder the likelihood of
fusion, such as deformation of the colliding partners, projectile direction
upon striking the target, isospin asymmetry of the colliding partners and
shell energy [47] etc. So it has not only the kinematical origin governed by
the atomic mass and charge number, but it could be due to shell structures,
deformation and shell energy [31, 32, 47]. The adaptability of the potential
with less energy-dependent parameters encourages further analysis of more
pertinent systems.

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