††thanks: for correspondence<EMAIL_ADDRESS>
Superannuated from Inter-University Accelerator Centre, Aruna Asaf Ali Marg,
New Delhi-110067, India.

# Search for a viable nucleus-nucleus potential for heavy-ion nuclear
reactions

T. Nandi1#, D.K. Swami1, P.S. Damodara Gupta2,3, Yash Kumar4, S. Chakraborty1,
and H.C. Manjunatha5∗ 1∗1003 Regal, Mapsko Royal Ville, Sector 82,
Gurugram-122004, Haryana, India. 2Rajah Serfoji Govt. college affiliated to
Bharathidasan University, Thiruchirappalli-620 024 3Department of Physics,
Government First Grade College, Kolar-563101 Karnataka, India 4 Dipartimento
di Fisica "Galileo Galilei", Università di Padova, I-35131 Padova, Italy 5
Department of Physics, Government college for women, Kolar-563101, Karnataka,
India

###### Abstract

We have constructed an empirical formulae for the fusion and interaction
barriers using experimental values available till date. The fusion barriers so
obtained have been compared with different model predictions based on the
proximity, Woods-Saxon and double folding potentials along with several
empirical formulas, time dependent Hartree-Fock theories, and the experimental
results. The comparison allows us to find the best model, which is nothing but
the present empirical formula only. Most remarkably, the fusion barrier and
radius show excellent consonance with the experimental findings for the
reactions meant for synthesis of the superheavy elements also. Furthermore, it
is seen that substitution of the predicted fusion barrier and radius in
classic Wong formula [C. Wong, Phys. Rev. Lett. 31, 766 (1973)] for the total
fusion cross sections satisfies very well with the experiments. Similarly,
current interaction barrier predictions have also been compared well with a
few experimental results available and Bass potential model meant for the
interaction barrier predictions. Importantly, the present formulae for the
fusion as well as interaction barrier will have practical implications in
carrying out the physics research near the Coulomb barrier energies.
Furthermore, present fusion barrier and radius provide us a good nucleus-
nucleus potential useful for numerous theoretical applications.

##  I Introduction

The basic characteristics of nuclear reactions are usually described by an
interaction consisting of a repulsive Coulomb potential and a short range
attractive nuclear potential. The resultant potential can be expressed as a
function of the distance between the centres-of-mass of the target and
projectile nuclei. When a projectile ion approaches the target nucleus it
experiences a maximum force at a certain distance where the repulsive and
attractive forces cannot balance each other, the repulsive Coulomb force is
always higher. The projectile ion needs to overcome the barrier for coming
closer to the target nucleus. This barrier is referred to the fusion barrier
$(B_{fu})$, which is a basic parameter in describing the nuclear fusion
reactions and is simply defined as the maximum of the total potential without
the centrifugal term. The kinetic energy of the projectile ion must be
adequate to surmount this barrier in order to enter a pocket at the adjacent
to the barrier at the shorter distances, where the nuclei undergo the nuclear
fusion processes. Furthermore, the fusion barrier height and its width play a
crucial influence on the tunneling process during the sub-barrier fusion. The
situation is further complicated by the presence of multiple barriers. The
fusion barrier is determined by the excitation function measurement of the
nuclear fusion experiments [1], whereas it is estimated by many theoretical
models such as the Bass potential model [2, 3], proximity potential model [4],
double folding model [5], and semi-empirical models such as Christensen and
Winther (CW) model [6], Broglia and Winther (BW) model [7], Aage Winther (AW)
model [8], Denisov potential (DP) model [9], Siwek-Wilczyńska and Wilczyński
(SW) model [10], Skyrme energy density function (SEDF) model [11] and the Sao
Paulo optical potential (SPP)[12]. In contrast, the quasi-elastic (QEL)
processes, involving smaller energy transfer due to single nucleons or
clusters such as alpha particles than the fusion reactions, excite only
nuclear levels in either one of the participating nuclei or in both as soon as
the two bodies approach within the range of the nuclear forces. The position
where the resultant of the Coulomb and nuclear forces is still repulsive and
additional energy is required to get the two nuclei interacting over their
mutual potential barrier. This barrier is somewhat smaller than the fusion
barrier and is known as the interaction barrier $(B_{int})$, which is measured
by the excitation function studies of QEL scattering experiment [13].
Obviously, both the barriers are Coulomb barriers, but they are different from
each other. One is characterized by the fusion reaction when the two nuclei
fuse to form a compound nucleus and the other by the QEL scattering when the
two nuclei enter barely to the strong force regime [2, 3] keeping their
identities almost intact. However, many-a-time the distinction is overlooked,
for example, [13, 14], though the concept was introduced in seventies [2, 3].
Worth noting here that it is only the Bass who has segregated the appearance
of the Coulomb barrier in above mentioned, two different ways: one is the Bass
interaction model and the other is the Bass fusion model [2, 3].
Recently Sharma and Nandi [15] demonstrated the coexistence of the atomic and
nuclear phenomenon on the elastically scattered projectile ions while
approaching the Coulomb barrier. Here the projectile ion x-ray energies were
measured as a function of ion beam energies for three systems 12C(56Fe,56Fe),
12C(58Ni,58Ni) and 12C(63Cu,63Cu) and observed unusual resonance like
structures as the beam energy approaching the fusion barrier energy, according
to the Bass model [2, 3]. However, the resonance would have occurred near the
interaction barrier as the technique involved only the elastic phenomenon and
thus resembling the quasi-elastic (QEL) scattering experiment [13]. It implies
the resonance should have appeared adjacent to the interaction barrier. To
resolve this anomaly, we planned to examine both the fusion and interaction
barriers in a greater detail. Note that present attempt is not to make a
concrete theoretical model to describe various possible steps of a reaction
leading to the final products, but to find the best model available till date
so that the above anomaly be resolved. Besides the existing models, we have
used the experiments found in the literature to construct an empirical formula
for estimating the fusion barriers from the fusion excitation function
measurements alone and another for the interaction barriers from the QEL
scattering experiments only.
In the next step, we have compared the empirical model predictions for fusion
barrier with various models based on proximity type of potentials such as Bass
potential [2, 3] and Christensen and Winther (CW) [6] and Woods-Saxon type of
potentials such as Broglia and Winther (BW) [7], Aage Winther (AW) [8], Siwek-
Wilczyńska and Wilczyński (SW) [10], SEDF [11] models, and the Sao Paulo
optical potential (SPP) [12]. Further, the present interaction barrier formula
has been compared with the Bass interaction model [2, 3]. It is seen that this
work will be useful in various applications [1], for example, prediction of
$B_{fu}$ or $V_{B}$ for the formation of the superheavy elements [16] and that
of both $B_{fu}$ and $B_{int}$ for the significant physics research near the
Coulomb barriers [15].

##  II Determination of the barriers

Mean fusion barrier height may be obtained from the Gaussian fit of the
barrier distribution plot, which is defined as the second derivative of the
energy-weighted cross section $\frac{d^{2}(\sigma E)}{dE^{2}}$ (point
difference) versus beam energy in the center-of-mass frame [17, 18]. In many
articles only the excitation function is reported, we have converted it into a
barrier distribution plot to obtain the mean fusion barrier.
Similarly, the interaction barrier can be obtained from the QEL excitation
function studies. The QEL scattering is affected by the sum of elastic,
inelastic, and transfer processes, which is measured at backward angles of
nearly $180^{0}$, where the head-on collisions are dominant. The barrier
distribution is obtained by taking the first derivative, with respect to the
beam energy, of the QEL cross-section relative to the Rutherford cross
section, that is, $\frac{-d}{dE}(\frac{d\sigma_{QEL}}{d\sigma_{R}})$ [19].
This method has been examined in several intermediate-mass systems [20, 21].
One can notice that the QEL barrier distribution behaves similarly to the
fusion barrier distribution, although the former is less sensitive to the
nuclear structure effects.

##  III General background

Theoretically, the total nucleus-nucleus interaction potential $V_{T}(r)$
between the projectile and target nuclei, in general, is written as a function
of the distance $r$ between the two nuclei

$V_{T}(r)=V_{N}(r)+\frac{l(l+1)\hbar^{2}}{2\mu r^{2}}+V_{c}(r).$ (1)

where the first term $V_{N}(r)$ is the model dependent nuclear potential, the
second term is centrifugal potential so that
$\mu=\frac{A_{p}A_{t}}{A_{p}+A_{t}}$ is the reduced mass of the projectile
mass $A_{p}$ and the target nuclei mass $A_{t}$ in MeV/$c^{2}$ units and $l$
represents the angular momentum of the two body system. When we consider the
fusion barrier of the system, $l$ is set to zero which means the centrifugal
or the second term is zero. The third term is the Coulomb potential as given
by [22]

$V_{c}(r)=\frac{Z_{1}Z_{2}e^{2}}{4\pi\epsilon_{0}}\begin{cases}\frac{1}{r}&\quad\text{for
}r\geq R_{B}\\\
\frac{1}{2R_{B}}\Big{[}3-\Big{(}\frac{R_{B}}{r}\Big{)}^{2}\Big{]}&\quad\text{for}\;r<R_{B}.\end{cases}$
(2)

Here the fusion barrier radius $R_{B}$ = $R_{c}$($A_{p}^{\frac{1}{3}}$ \+
$A_{t}^{\frac{1}{3}}$), $R_{c}$ depends on the system as discussed below.
Putting the first term of equation (1) from any particular model, one can
solve the fusion barrier radius $R_{B}$ by using the conditions

$V_{T}(r=R_{B})=B_{fu}\;,\;\;\frac{dV_{T}(r)}{dr}\bigg{|}_{r=R_{B}}=0\;\;~{}and~{}\;\;\frac{d^{2}V_{T}(r)}{dr^{2}}\bigg{|}_{r=R_{B}}\leq
0.$ (3)

Also, $V_{T}(r=R_{int})$ = $B_{int}$, where $R_{int}$ is the interaction
barrier radius. Of course, $V_{N}(r)$ in equation (1) may need to be replaced
by another appropriate potential to obtain the interaction barrier; for
example, the Bass potential model [3] uses different potential forms to
estimate the fusion and interaction barriers.

##  IV Present empirical formula

According to the definition for the Coulomb potential given above, the shape
of the nuclear potential discussed below and representation of nuclear
distances, for example, one given above for the barrier radius, $B_{fu}$ and
$B_{int}$ may be written as a function of $Z_{p}$, $Z_{t}$, $A_{p}$ and
$A_{t}$. Hence, the experimentally obtained $B_{fu}$ from fusion excitation
function measurements and $B_{int}$ from QEL measurements can be plotted
against the Coulomb interaction parameter
$z=\frac{Z_{p}Z_{t}}{(A_{p}^{\frac{1}{3}}+A_{t}^{\frac{1}{3}})}$, as shown in
Fig.1(a) and Fig.1(b), respectively. The fusion experiments used for Fig. 1(a)
and the QEL experiments used for Fig. 1(b) are given in Table LABEL:t1 and 2,
respectively. Fusion data are available for $8\leq z\leq 286$, whereas QEL
data for $59\leq z\leq 313$. We can notice that both $B_{fu}$ vs $z$ and
$B_{int}$ vs $z$ are nonlinear. The whole range of $B_{fu}$-data has been
fitted by two non-linear functions of a sixth-degree polynomial to obtain the
reduced chi-square nearly equal to one. Whereas, the full range of
$B_{int}$-data is fitted with a single non-linear function.
The polynomial function that fits the $B_{fu}$ vs $z$ data extremely well is
as follows

$\displaystyle B_{fu}^{(1)}=$ $\displaystyle-1.2725+0.9106z+5.6932\times
10^{-4}z^{2}+2.335\times 10^{-5}z^{3}$ (4) $\displaystyle-4.4975\times
10^{-7}z^{4}+2.7836\times 10^{-9}z^{5}-5.2482\times 10^{-12}z^{6}\;\mbox{
for}\;8\leq z\leq 128$

and

$\displaystyle B_{fu}^{(2)}=$
$\displaystyle-34488.7618+1100.6666z-14.4066z^{2}+9.9275\times
10^{-2}z^{3}-3.7959\times 10^{-4}z^{4}$ (5) $\displaystyle+7.6357\times
10^{-7}z^{5}-6.3136\times 10^{-10}z^{6}\;\mbox{for}\;128\leq z\leq 286$

and the other function that fits the $B_{int}$ vs $z$ data is given by

$\displaystyle B_{int}=0.09295z^{3/2}-2.1601z+35.5879z^{1/2}-132.8943\;\mbox{
for}\;59\leq z\leq 313$ (6)

The $B_{fu}$ can be predicted using equation (4) for any system within $8\leq
z\leq 128$ and equation (5) for any system within $128\leq z\leq 286$. Such an
empirical formula was constructed in the past [23] as follows

$\displaystyle B_{WJS}=0.85247z+0.001361z^{2}-0.00000223z^{3}MeV.$ (7)

It was a polynomial function of z of degree 3, where as our empirical formula
contains up to a degree of 6.

To obtain $R_{B}$, we follow a method involving the reduced fusion barrier
position SB (the separation between the half-density surfaces of the two
nuclei) [24]

$\displaystyle S_{B}=R_{B}^{exp}-C_{1}-C_{2}$ (8)

Where the half-density radius of the matter distribution
$C_{i}=\left(R_{i}-\frac{b^{2}}{R_{i}}\right)$ (for i=1,2) [25], the sharp
radius $R_{i}=1.233A_{i}^{1/3}-0.98A_{i}^{-1/3}$ and the measure of the
diffuseness of the nuclear surface b=0.99 fm. $S_{B}$ is fitted with 4th order
polynomial as a function of z as

$\displaystyle S_{B}=-46.089+2.0478z-3.0962\times 10^{-2}z^{2}+1.9278\times
10^{-4}z^{3}-4.2492\times 10^{-7}z^{4}$ (9)

and the RB is obtained from the reduced fusion barrier position

$\displaystyle R_{B}=S_{B}+C_{1}+C_{2}$ (10)

##  V Nuclear potential models in literature

In this section we present different nucleus-nucleus potentials $V_{N}(r)$,
which can be used for obtaining the fusion and interaction to compare with the
available measured values.

###  V.1 Bass potential model

Bass potential model [2, 3] suggests that the Coulomb barrier for quasi-
elastic surface reaction is in general different from the Coulomb barrier for
fusion. The former results from the quasi-elastic processes, where not much
mass or energy transfer takes place, whereas maximum mass and energy can
transfer in the latter. Further, the quasi-elastic processes become
significant as the projectile and target nuclei approach to the range of
nuclear force, where the resultant of the Coulomb and nuclear forces is still
repulsive. Thus additional energy is required to get the nuclei within the
resultant attractive force, where the fusion can occur. According to the Bass
interaction model, the barriers responsible in the quasi-elastic reactions is
called the interaction barrier $B_{int}$ and it can be determined from the
elastic scattering experiments [13]. The other barrier is significant in
fusion reactions and is defined as the fusion barrier $B_{fu}$. The latter is
equal to the height of the potential barrier for zero angular momentum.
The total effective Bass potential consists of a Coulomb, nuclear and
centrifugal terms

$V_{l}(r)=\frac{Z_{p}Z_{t}e^{2}}{4\pi\epsilon_{0}r}+\frac{\hbar^{2}l^{2}}{2\mu
r^{2}}-a_{s}A_{p}^{\frac{1}{3}}A_{t}^{\frac{1}{3}}\frac{d}{R_{pt}}{e^{-(\frac{r-R_{pt}}{d})}}$
(11)

Where $d$ is the range of nuclear interaction. The influence of fragment
(projectile or target nuclei) properties on this potential can be expressed in
terms of the dimensionless parameters

$x=\frac{e^{2}}{r_{0}a_{s}4\pi\epsilon_{0}}\frac{Z_{p}Z_{t}}{A_{p}^{\frac{1}{3}}A_{t}^{\frac{1}{3}}(A_{p}^{\frac{1}{3}}+A_{t}^{\frac{1}{3}})}$
(12)

and

$y=\frac{\hbar^{2}}{2m_{0}r_{0}^{2}a_{s}}\frac{A_{p}+A_{t}}{A_{p}^{\frac{4}{3}}A_{t}^{\frac{4}{3}}(A_{p}^{\frac{1}{3}}+A_{t}^{\frac{1}{3}})^{2}}.$
(13)

Where $x$ is the ratio of the Coulomb force to the nuclear force and $yl^{2}$
is the ratio of the centrifugal force to the nuclear force at the point of
contact i.e., $r=R_{pt}=r_{0}(A_{p}^{\frac{1}{3}}+A_{t}^{\frac{1}{3}})$,
$r_{0}$=1.07 fm. Here, $a_{s}$ =17.23 MeV is the surface constant as used in
the liquid drop model of fission, $m_{0}$ the mass of a nucleon and other
notations have usual significance. Whereas $B_{fu}$ acts at $r$ = $R_{pt}$ \+
$d_{fu}$, $d_{fu}$ is the fusion distance. The $B_{int}$ is applicable for an
interaction distance between the two surfaces $(d_{int})$ or the centre to
centre distance, $R_{int}$ = $R_{pt}$ \+ $d_{int}$. The $d_{int}$ is always
longer than the $d_{fu}$, which can be approximately obtained from the
relation

$\frac{d_{fu}}{d}\approx-\frac{lnx}{(1-\frac{2d}{R_{pt}})}$ (14)

The $d_{fu}$ varies with the fragments in the nuclear interactions. The
barriers $B_{fu}$ and $B_{int}$ can be obtained from

$B_{fu}=\frac{Z_{p}Z_{t}e^{2}}{4\pi\epsilon_{0}R_{pt}}\Big{\\{}\frac{R_{pt}}{R_{pt}+d_{fu}}-\frac{1}{x}\frac{d}{R_{pt}}e^{\big{(}-\frac{d_{fu}}{d}\big{)}}\Big{\\}}$
(15)
$B_{int}=\frac{Z_{p}Z_{t}e^{2}}{R_{pt}+d_{int}}-2.90\frac{A_{p}^{\frac{1}{3}}A_{t}^{\frac{1}{3}}}{(A_{p}^{\frac{1}{3}}+A_{t}^{\frac{1}{3}})}$
(16) $d_{int}=2d=2\times 1.35=2.70fm$ (17)

Here, it is assumed in equation (17) that $d$ is independent of the mass of
the nuclei.

###  V.2 Christensen and Winther model

Christensen and Winther (CW) [6] use elastic scattering trajectory leading to
the rainbow pattern that is strongly connected to low-lying target excitation.
Hence, the phenomenon happens above the fusion barrier. They derived the
nucleus-nucleus interaction potential to represent the nuclear fusion on the
basis of semi classical arguments as given by

$V_{N}^{CW}(r)=-50{\overline{R}}{e^{\big{(}-\frac{r-R_{pt}}{a}\big{)}}}MeV$
(18)

where $R_{pt}$ = $R_{p}+R_{t}$, $\overline{R}=\dfrac{R_{p}R_{t}}{R_{p}+R_{t}}$
and $a$ is the diffuseness parameter ($a=0.63fm$). This form is similar to
that of the Bass model [2] with different sets of radius parameter.

$R_{i}=1.233A_{i}^{\frac{1}{3}}-0.98A_{i}^{-\frac{1}{3}}fm~{}(i=p,t)$ (19)

Here, the radius of the fusion barrier has the form

$R_{B}=1.07(A_{p}^{\frac{1}{3}}+A_{t}^{\frac{1}{3}})+2.72fm$ (20)

and the total nucleus-nucleus potential for $l=0$ is

$U^{CW}(r)=\dfrac{Z_{p}Z_{t}e^{2}}{4\pi\epsilon_{0}r}+V_{N}^{CW}(r)$ (21)

and thus, the fusion barrier can be obtained from $U^{CW}(r=R_{B}$).

###  V.3 Broglia and Winther model

Broglia and Winther (BW) [7] have refined the CW potential [6] in order to
make it compatible with the value of the maximum nuclear force of the
proximity potential [4]. This refined force is taken as the standard Woods-
Saxon potential given by

$V_{N}(r)=\dfrac{-V_{0}}{1+e^{\frac{r-R_{pt}}{a}}}MeV$ (22)

with $V_{0}$ = $16\pi a\gamma\frac{R_{p}R_{t}}{R_{p}+R_{t}},a=0.63fm$ and

$R_{pt}=R_{p}+R_{t}+0.29fm$ (23)

Here the nucleus radius $R_{i}$ is given by:

$R_{i}=1.233A_{i}^{\frac{1}{3}}-0.98A_{i}^{\frac{-1}{3}}fm(i=p,t)$ (24)

The surface energy coefficient ($\gamma$) has the form

$\gamma=\gamma_{0}\bigg{[}1-k_{s}\bigg{(}\frac{N_{p}-Z_{p}}{A_{p}}\bigg{)}\bigg{(}\frac{N_{t}-Z_{t}}{A_{t}}\bigg{)}\bigg{]}MeVfm^{2}$
(25)

Where $\gamma_{0}$ = $0.95MeVfm^{-2}$ and $k_{s}=1.8$. The total interaction
potential of the two heavy ions is

$U^{BW}(r)=\dfrac{Z_{p}Z_{t}e^{2}}{4\pi\epsilon_{0}r}+V_{N}^{BW}(r)$ (26)

and it displays a maximum, i.e., the fusion barrier and the barrier radius
$(R_{B})$ is the solution of the following equation

$\dfrac{dU^{BW}(r)}{dr}|_{r=R_{B}}=-\dfrac{Z_{p}Z_{t}e^{2}}{4\pi\epsilon_{0}R_{B}^{2}}+\dfrac{V_{0}\hskip
5.69054pte^{\frac{R_{B}-R_{pt}}{a}}}{{a\hskip
2.84526pt(1+e^{\frac{R_{B}-R_{pt}}{a}})}^{2}}=0$ (27)

and $U^{BW}(R_{B})$ is the fusion barrier.

###  V.4 Aage Winther model

Aage Winther (AW) [8] adjusted slightly the parameters of the Broglia and
Winther potential through an extensive comparison with the experimental data
for heavy-ion elastic scattering. The resulting values of $a$ and $R_{i}$ are
as follows:

$a=\Bigg{[}\dfrac{1}{1.17(1+0.53(A_{p}^{-\frac{1}{3}}+A_{t}^{-\frac{1}{3}}))}\Bigg{]}\hskip
5.69054ptfm$ (28)

and

$R_{i}=1.20A_{i}^{\frac{1}{3}}-0.09\hskip 5.69054ptfm\hskip 25.60747pt(i=p,t)$
(29)

and $R_{pt}$ of the BW model is written as $R_{pt}$ = $R_{p}+R_{t}$ only.

###  V.5 Siwek-Wilczyńska and Wilczyński model

Siwek-Wilczyńska and Wilczyński (SW) [10] have used a large number of
reactions to determine an effective nucleus-nucleus potential for reliable
prediction of the fusion barriers for the systems that are studied. In this
approach the nucleus-nucleus potential is taken also as the Woods-Saxon shape
is given in equation (22). Where $R_{pt}$ = $R_{p}+R_{t}$ and
$R_{i}=R_{c}A_{i}^{\frac{1}{3}}$ $(i=p,t)$, the radius parameter $R_{c}$ is
constant, $a$ is the diffuseness parameter and $V_{0}$ is the depth of the
potential. The $V_{0}$ is given by

$V_{0}=V_{0}^{{}^{\prime}}+S_{cn}.$ (30)

Where $S_{cn}$ is the shell correction energy [26] and

$V_{0}^{{}^{\prime}}=(M_{p}+M_{t}-M_{cn})c^{2}+C_{cn}-C_{p}-C_{t}$

$=Q_{fu}+C_{cn}-C_{p}-C_{t}.$ (31)

Here $Q_{fu}$ is the ground state $Q$ value for fusion and $C_{i}$ are the
intrinsic Coulomb energies [10] as given by

$C_{cn}-C_{p}-C_{t}=C_{0}$

$C_{0}=0.7054\Bigg{[}\dfrac{(Z_{p}+Z_{t})^{2}}{(A_{p}+A_{t})^{\frac{1}{3}}}-\dfrac{Z_{p}^{2}}{A_{p}^{\frac{1}{3}}}-\dfrac{Z_{t}^{2}}{A_{t}^{\frac{1}{3}}}\Bigg{]}\hskip
11.38109ptMeV.$ (32)

The Coulomb energy constant is taken from the standard liquid-drop-model fit
to nuclear masses [27]. Now, the equation (30) can be written as

$V_{0}=Q_{fu}+C_{0}+S_{cn}.$ (33)

For determination of the fusion barrier, one considers the nucleus-nucleus
potential in the region $R>R_{pt}$ as

$V(r)=V_{N}(r)+\dfrac{Z_{p}Z_{t}e^{2}}{4\pi\epsilon_{0}r}.$ (34)

For the region
$r<R_{p}+R_{t},e^{\big{[}\frac{r-R_{pt}}{a}\big{]}}\longrightarrow 0$, the
nucleus-nucleus potential takes the form

$V(r)=C_{0}-V_{0}=-Q_{fu}-S_{cn}.$ (35)

Equation (34) gives thus the fusion barrier. It has only two free parameters
$R_{c}$ and $a$ as $V_{0}$ is known from equation (33). These parameters are
obtained by fitting the barriers from equation (34). The experimental $B_{fu}$
values can be obtained where the measured fusion excitation functions are
fitted with the following expression

$\sigma_{fus}=\pi r_{\sigma}^{2}\dfrac{\omega}{E\surd
2\pi}[X\surd\pi(1+erf(X))+exp(-X^{2}].$ (36)

Where $X=\dfrac{E-B_{fu}}{\sqrt{2}\omega}$ and erf(X) is the Gaussian error
integral of the argument X as

$erf(X)=\dfrac{1}{\sqrt{\pi}}{\int_{0}^{X}e^{-t^{2}}dt}.$ (37)

The fitting gives three parameters the fusion barrier $B_{fu}$, the relative
distance corresponding to the position of the approximate barrier $r_{\sigma}$
and the width of the barrier $w$. However, the values of $R_{c}$ and $a$
depend on the Coulomb barrier parameter $z$, for example,
$R_{c}=1.25fm$ and $a=0.481fm$ for$z<70$
$R_{c}$ = $1.18fm$ and $a=0.675fm$ for $70<z<130$
$R_{c}=1.11fm$ and $a=0.895fm$ for $z>130$

###  V.6 Skyrme energy density function model

Skyrme energy density function (SEDF) model has been introduced by Wang et al.
[11, 28], where the total binding energy of a nucleus is represented as the
integral of the energy density function [29]

$E=\int{Hdr}.$ (38)

Here energy-density function $H$ has three parts: kinetic energy, Coulomb and
nuclear interactions and is generally defined as follows:

$H(r)=\frac{{\hbar}^{2}}{2m}\left[\tau_{p}\left(r\right)+\tau_{n}\left(r\right)\right]+H_{coulomb}\left(r\right)+H_{nuclear}\left(r\right)$
(39)

Where $\tau_{p}$ and $\tau_{n}$ are the kinetic energy density for proton and
neutron, respectively. The interaction potential $V_{B}\left(R\right)$ is
defined as

$V_{B}\left(R\right)=E_{tot}\left(R\right)-E_{p}-E_{t}$ (40)

where $E_{tot}\left(R\right)$ is the total energy of the interacting nuclear
system, $E_{p}$ and $E_{t}$ are the energies of the projectile and target,
respectively, at completely separated distance $R$. These energies can be
calculated by the following relation

$E_{tot}(R)=\int{H[\rho_{1p}(r)+\rho_{2p}(r-R),\rho_{1n}(r)+\rho_{2n}(r-R)]dr}$

$E_{p}\left(R\right)=\int{H\left[\rho_{1p}\left(r\right),\rho_{1n}\left(r\right)\right]dr}$

$E_{t}\left(R\right)=\int{H\left[\rho_{2p}\left(r\right),\rho_{2n}\left(r\right)\right]dr}$
(41)

The densities of the neutron $\rho_{n}$ and proton $\rho_{p}$ for projectile
and target can be described by the spherically symmetric Fermi function

$\rho{\left(r\right)}=\dfrac{{\rho}_{0}}{1+{e^{\frac{\left(r-c\right)}{a}}}}$
(42)

where $\rho_{0}$, $c$ and $a$ are the parameters of the densities of the
participating nuclei in the reactions, which are obtained by using the
density-variation approach and minimizing the total energy of a single nucleus
given by the SEDFM [11, 28].

Using the Skyrme energy density formalism, Zanganeh et al. [30] have
constructed a pocket formula for fusion barriers and positions in the range
8$\leq$ z $\leq$ 168 with respect to the charge and mass numbers of the
interacting nuclei as follows:

$V_{B}^{Par}=-0.01[(Z_{p}Z_{t})(A_{p}^{\frac{1}{3}}+A_{t}^{\frac{1}{3}})]+0.20(Z_{p}Z_{t})+0.60$
(43)
$R_{B}^{Par}=1.40[A_{p}^{\frac{1}{3}}+Z_{t}^{\frac{1}{3}}]-0.07(Z_{p}Z_{t})^{0.05}+1.40$
(44)

We have made use of the equation (43,44) for SEDFM predictions for various
reactions as shown in Table 2 and Table LABEL:t3. Since, the SEDFM is based on
the frozen density approximation, predicted values for each of the considered
fusion systems are a bit higher than the corresponding experimental data.

###  V.7 Sao Paulo optical potential (SPP)

This model [12] also takes a Woods-Saxon form for the nuclear potential as
given in equation (22). In the approximation of $\exp{(\frac{R_{B}-r}{a})}\gg
1$, the $R_{B}$ can be written as:

$R_{B}=r+0.65\ln{x}$ (45)

where $x=27.1\times\frac{A_{p}^{\frac{1}{3}}+A_{t}^{\frac{1}{3}}}{Z_{p}Z_{t}}$
is a positive dimensionless parameter. Note that the parameter $x$, which
appears in the argument of the logarithm of the above equation, can also be
written as $x=\exp({\frac{R_{B}-r}{a}})$ and is larger than one in the most
cases. The barrier potential $V_{B}$ is given by

$V_{B}=\frac{Z_{p}Z_{t}e^{2}}{4\pi\epsilon_{0}R_{B}}-\frac{15}{x+1}$ (46)

###  V.8 Moustabchir-Royer (MR) formula

Moustabchir and Royer [31] proposed two formulas for the fusion barrier height
and radius from a fitting procedure on generalised liquid drop model [32] data
on a large number of fusion reactions as follows:

$\displaystyle
V_{B}=-19.38+\frac{2.1388Z_{p}Z_{t}+59.427(A_{p}^{1/3}+A_{t}^{1/3})-27.07\ln\left(\frac{Z_{p}Z_{t}}{A_{p}^{1/3}+A_{t}^{1/3}}\right)}{(A_{p}^{1/3}+A_{t}^{1/3})(2.97-0.12\ln(Z_{p}Z_{t}))}$
(47) $\displaystyle
R_{B}=(A_{p}^{1/3}+A_{t}^{1/3})\left[1.908-0.0857\ln(Z_{p}Z_{t})+\frac{3.94}{Z_{p}Z_{t}}\right]$
(48)

The rms deviations were found to be 0.15 MeV and 0.08 fm in $V_{B}$ and
$R_{B}$, respectively.

##  VI Results and discussions

We have constructed the empirical formulae for the fusion and interaction
barriers using the experimental results available in the literature as
mentioned above. The fusion barriers $B_{fu}$ can be obtained from the
equations 4 and 5, and fusion barrier radius $R_{B}$ is from equation 10. The
$B_{fu}$ values obtained from various theoretical models for different systems
are compared with the experimental results in the rage of $9\leq z\leq 129$,
which are not used to construct the present fusion barrier formula, in Table
LABEL:t3. Similar comparisons have also been done for $R_{B}$ in Table
LABEL:t4 for the range $55\leq z\leq 170$. To check which model gives the best
agreement with the experimental results, the differences of both $B_{fu}$ and
$R_{B}$ values between the experiment and a specific model have been plotted
as a function of $z$ in Fig. 2 and 3, respectively. From the figure 2, it is
seen that the present empirical formula (Fig2(a)) is within $\pm 4\%$, most of
the SEDF data overestimates up to $7\%$ (Fig.2(b)), Bass model underestimates
for $z\leq 60$ and overestimates for $z\geq 60$ (Fig.2(c)), SW predicts within
$\pm 5\%$ (Fig.2(d)), similar comparison is found with CW data (Fig.2(e)), BW
data (Fig.2(f)) and AW data (Fig.2(g)), while SPP data shows about an
underestimation for $z$ up to 40 and overestimation up to 10% for $z>$40\. On
the other hand, Fig.3 gives a clear impression that the deviation between the
experiment and theory for the fusion barrier radius is much larger than the
fusion barrier. Whatsoever, we can find that the present empirical formula
(Fig3(a)) is within $\pm$20%, deviations of the SEDF data are within
$\begin{subarray}{c}+40\%\\\ -5\%\end{subarray}$ (Fig3(b)), Bass model
estimates within $\begin{subarray}{c}+40\%\\\ -6\%\end{subarray}$ (Fig.3(c)),
SW predicts within $\begin{subarray}{c}+50\%\\\ -0\%\end{subarray}$
(Fig.3(d)), similar deviation is found with CW (Fig.3(e)), BW data (Fig.3(f))
and AW data show about $\begin{subarray}{c}+40\%\\\ -20\%\end{subarray}$
(Fig.3(g)) deviations, while SPP data shows an overestimation of 40%. Further,
to have a quantitative evaluation on the model predictions we have obtained
the sum of the squared residuals ($SSR=\displaystyle\sum_{i=1}^{n}e_{i}^{2}$),
where $e_{i}$ is the $i^{th}$ residual or difference and n is the number of
data points. The mean squared errors ($\sigma_{\epsilon}^{2}=\frac{SSR}{n-2}$)
are shown in Table 5. The lowest mean squared error on both the fusion barrier
and radius is obtained for the the present model, thus the best available
formula till date. Note that the empirical formula of SJW [23] does not
predict fusion barrier radius, it is considered to be disqualified in this
test. The BW model is the second best and the AW model is the third best in
the region considered here.
We can see from the Fig. 1 that very few data are existing for $z>150$,
because the reaction mechanism at this region is complex and the data analysis
is a tricky affair. If the experimental apparatus is powerful enough in the
sense of measuring a number of physical parameters, right kind of data
analysis can then be possible. Often such apparatus is not available, thereby
appropriate analysis method cannot be straightforwardly chosen as discussed
below. Therefore, available experimental results are very limited. Whatsoever,
the available experimental results for the heavy ion reactions aiming to form
actinide and transactinide elements have been compared with the present
formula, WJS formula [23], W.D. Myers (WDM) model [25] and H.C. Manjumatha
(HCM) formula [24] in the Table 6. The mean square errors in the present
formula, WDM model, WJS formula, and HCM formula are found to be 2.26, 4.98,
4.42 and 5.72, respectively. Hence, the predictions from the present formula
is the best of all. An important point to note that excellent agreement is
found even with very precision experiments just conducted recently [33].
Hence, the present formula is an appropriate formula that may be very useful
for synthesizing the superheavy elements [discussed later]. This formula is
thus useful for the entire region of $8\leq z\leq 286$. Of course, for the
region of $0<z\leq 8$ where often nuclear astrophysics experiments [34] are
conducted, the BW model can be used.
Very recently, Ganiev and Nasirov [35] discussed predictive power of various
nucleus-nucleus potentials. The proximity potential [4] and its use in AW
model [8] give deviation more than 4 MeV from the experimental values of
$V_{B}$ for several fusion reactions. More importantly, the difference
increases for the reactions with massive nuclei including the mostly used Bass
potential [3, 2, 17, 36] of the proximity potential. For some fusion systems,
the results of CW [6] and BW model [37] are close to the experimental data,
but these models cannot describe the heavy-ion reactions such as
${86}Kr+^{208}Pb$. In contrast, predictions of a double-folding model (DFM)
[35] based on density-dependent Migdal potential [38] are closed to all the
experimental values including the heavy-ion reactions. We compare the present
results for the $V_{B}$ with this DFM model along with the DFM using M3YReid
interaction with zero-range exchange part [39] and the Paris CDM3Y3
interaction with finite-range exchange part [40] and experimental data
obtained from the cited references in Table 7. The mean squared error analysis
give a clear impression that predictions from our formula are the best of all
these models.
Synthesis of superheavy elements by heavy-ion reactions is a complex problem
because of quasi-fission etc. A fully equilibrated compound system needs to be
formed to produce the new elements. This depends on entrance channel
parameters [41], deformation parameters, orientation, internal excitation,
transfer and initial kinetic energy of the projectile [42], which can be
picked by the knowledge of theoretical $B_{fu}$. However, the barrier
distributions of many heavy-ion reactions used for synthesis of superheavy
nuclei shows double hump behavior, for example, less mass asymmetry cold
fusion reactions. Out of the two humps, the inner peak is the higher. While
the warm fusion reactions, due to higher mass asymmetry, exhibit no double
hump barriers [43]. At a given mass asymmetry up to about $\eta$ = 0.5 the
potential barrier exhibits a two hump shape, but for larger $\eta$ it displays
only one hump [44]. Hence, double hump problem appear only for the cold fusion
type of reactions only and the present predictions might represent the higher
potential barrier of the corresponding distribution for such reactions. Such
scenario has been discussed later by several authors time to time, for
example, a very recent article by Royer $et.al.$ [45]. Hence, a precisely
known value from this present formula will thus promote us to select the
kinetic energy judiciously. Recently, an reaction contact time experiment [46]
has made use of the WJS formula [23] (not the WDM model [25], although it is
quoted there). We showed just above that uncertainty in the WJS formula [23]
is much higher than the present formula. The fusion barriers of the reactions
aiming for the superheavy nucleus $=120$ have also been compared with
different formulae in Table 6.
Though the present empirical formula is not specific to a particular form of
the nucleus-nucleus potential, but the second as well as the third best models
are based on the Woods-Saxon potential (equation (22) and characterised by the
three potential parameters including the potential height parameter $V_{0}$,
the diffuseness parameter $a_{0}$ and the radius parameter $r_{0}$. The
present formula predicts the fusion barrier and radius pretty well, if we
substitute these values in the three equations 3 and solve for these three
parameters for a reaction, we can constitute the concerned nucleus-nucleus
potential quite well. Such an exercise is followed for several reactions and
the Woods-Saxon potential parameters are shown in Table 8 and compared with
the potential parameters used in the earlier analysis, where mostly the Akyuz
Winther parameters [47] have been used [48]. Sometimes the parameters are
altered to have an agreement with the fusion cross section measurements [49].
Even at times, modified Akyuz Winther parameters are chosen by fixing the
potential height at 100 MeV [50]. Note that in some models another parameter
is kept fixed, for instance, the $a_{0}$ value is fixed for the BW and AW
models to 0.63 fm. In contrast, we have kept all the parameters free while the
basic equations are solved (equation 3). The $V_{0},a_{0}$ and $r_{0}$ span
over 26-135 MeV, 0.92-1.76 fm, and 0.85-1.24 fm, respectively. In contrast,
earlier Akyuz Winther parameters for $V_{0},a_{0}$ and $r_{0}$ fall in the
range of 76-105 MeV, 0.67-1.06 fm and 1.06-1.19 fm, respectively.
Once we solve for the potential parameters $V_{0},a_{0}$ and $r_{0}$ by above
prescriptions and assume that the tail of the nuclear potential to be
exponential with diffuseness $a_{0}$, one can readily obtain $R_{B}$ from [51]

$\displaystyle V_{0}=\frac{Z_{1}Z_{2}e^{2}}{R_{B}}(1-\frac{a_{0}}{R_{B}}).$
(49)

We can obtain now another important parameter called the barrier width
$(\hbar\omega)$ from the relation given by Rowley $et~{}al.$ [18] as follows

$\displaystyle(\hbar\omega)^{2}=\frac{\hbar^{2}V^{\prime\prime}(R_{B})}{\mu}=\frac{Z_{1}Z_{2}e^{2}\hbar^{2}}{\mu
R_{B}^{2}}(\frac{1}{a_{0}}-\frac{2}{R_{B}}).$ (50)

Use of these barrier radius and width in the Wong’s phenomenological formula
[52] gives us the fusion cross section as follows

$\displaystyle\sigma_{fu}^{Wong}=\frac{\hbar\omega
R_{b}^{2}}{2E}ln[1+exp\frac{2\pi(E-V_{B})}{\hbar\omega}].$ (51)

We made an attempt to check whether the predicted fusion barrier and radius
from the present formula can reproduce the measured excitation fusion i.e.,
fusion cross section as a function of centre of mass energy of the projectile.
It is done by a direct substitution of $R_{B}$ and $\hbar\omega$ in the
classic Wong formula [52] given above for many reactions. As for example, the
Fig.4 shows a comparison for four reactions 19F+181Ta [49], 16O+208Pb,
16O+154Sm, and 58Ni+54Fe [50]. The comparison of the total fusion cross
section between the phenomenological formula [52] and experiment displays a
very good agreement for $E_{cm}>B_{fu}$ and a departure starts appearing at
low energies $E_{cm}<B_{fu}$, except for the reaction 16O+154Sm, because of
the strong channel coupling effects for some reactions in the sub-barrier
region [53]. The present scenario reiterates the fact that no single, energy-
independent potential can simulate the fusion cross sections at the sub-
barrier region, where the well-known influence of the coupling effects is
vital. A recent study [54] has discussed the sub-barrier fusion properties on
a different view point, which shows importance of the projectile mass and
surface energy coefficients in heavy-ion fusion at sub-barrier energies.
Further, it finds essence of fusion Q-value rule [55] for some fusion systems.
One point may be worth mentioning that owing to deformation in the projectile
and/or target nuclei, a distribution of barrier heights is observed in the
experiments. The fusion barrier height from the present empirical formula
represents an average value in the case of such distribution. Obviously, the
barrier width shows also an average value and its value for certain reactions
are compared with other models in Table 8.
In recent years, microscopic mechanisms which can impact the nucleus-nucleus
potential has been studied in the frame work of time dependent Hartree-Fock
(TDHF) approach [56] to provide a rather unique tool for describing nuclear
structure and nuclear reactions over the whole nuclear chart. Assuming the
densities of the target and projectile remain constant and equal to their
respective ground state densities, this leads to the so-called frozen density
TDHF (FD-TDHF) approximation. Washiyama and Lacroix [57] consider a different
approach based on a macroscopic reduction of the mean-field dynamics, called
dissipative-dynamics in TDHF (DD-TDHF). We have compared the DD-TDHF
predictions for the fusion barrier and barrier radius for several reactions
with our predictions and available experimental values in Table 9, and noticed
a good agreement. In recent years, Simenel and his co-workers [58, 59] and
Yilmaz $et.al.$ [60] have applied TDHF without any approximation to verify the
FD-TDHF and DD-TDHF predictions in several cases and found good accordance.
Mean squared error is given in the bottom row of the table, which implies that
the DD-TDHF predictions with low ECM are the best if we judge on both $V_{B}$
and $R_{B}$ simultaneously.
Let us discuss briefly the data analysis issue of the heavy ion reactions
here. When a projectile collides with a target nucleus near the fusion barrier
energy, besides the evaporation residues (ER) and binary fission [61], the
quasi-fission [62, 63] is another process that contributes considerably in the
reaction cross sections. Furthermore, quasi-fission occurs before the target
and projectile fuse into a compound nucleus not only for the heavy systems
$z_{t}.z_{p}>1600$ but also in much lighter systems $z_{t}.z_{p}\approx 800$
[64], which results in the hindrance of the formation of ER from the
equilibrated compound nucleus. The fusion-fission can also take place from an
incomplete fusion reaction [65, 66], in which only a part of the projectile
fuses with the target and the incompletely fused binary system equilibrates in
the compound nucleus. Often the quasi-fission products are considered as the
products of the deep-inelastic collisions [67] in the experimental data
analysis and their contribution is obviously not included in the capture cross
section, the estimation of the fusion probability from such analysis appears
unreliable. If the effect of quasi-fission and incomplete fusion-fission
reaction are not considered in the analysis the reaction cross sections will
certainly be erroneous. Such circumstances may of course distort the value of
the fusion barrier if the contribution of the quasi-fission and incomplete
fusion-fission reaction vary with the beam energy. However, a recent thorough
study shows that the cross section measured from the fission products
$\sigma_{fis}$ could be well reproduced by scaling the capture cross section
$\sigma_{cap}$ for a cold fusion by constant factors of 0.75 for 48Ca, 0.48
for 50Ti, and 0.22 for 54Cr projectiles on 208Pb target [33]. Hence, the
present empirical formula is suitable to realistically predict the fusion
barrier for any new reactions planned in search for the formation of the
superheavy elements.
Let us now discuss the interaction barriers, which can be obtained from
equation (6) and compared with the experiments and model predictions. The
experimental values available have been compared with the present empirical
formula in Table LABEL:t9. These experimental values compare far better with
the present formula than the Bass model. Mean squared error turns out to be
3.0 for the present formula, whereas that is 16.0 for the Bass model. Hence,
the present model of the interaction barriers is also recommended for future
applications for $59\leq z\leq 313$ and the Bass model for $0<z\leq 59$.
As discussed in the introduction, the motivation of the present work was based
on a work [15] that the observed resonance like structures with the projectile
x-ray energies appeared at a certain energy closure to the fusion barrier
energy even though the projectile x-ray production mechanism is close to the
elastic collision as examined by the Bass model [2, 3]. The scenario remains
unchanged with the refined fusion and interaction barrier energies for the
systems 12C(56Fe,56Fe), 12C(58Ni,58Ni) and 12C(63Cu,63Cu). The physical reason
for this anomaly is identified as a process called the nuclear orbiting
resonance (dinuclear complexes) [68]. Detail of this aspect is out of the
scope of this work and will be published elsewhere.

##  VII Conclusion

In this paper, using the experimental values available in the literature,
empirical formulae for the fusion and interaction barriers have been obtained.
The experimental values available for the fusion barrier radius give us an
option to find a formula for the fusion barrier radius also. The present study
is restricted to the fusion and interaction barriers for the reactions in the
regime $8\leq z\leq 286$ and $59\leq z\leq 313$, respectively. We have carried
out a comparative study of the fusion barriers as well as barrier radius
between the present empirical formula and various empirical, semi-empirical
models, and microscopic theories along with the experimental results.
According to a thorough comparison with the experimental values, the present
formula is found to be the best of all the models considered in this study for
the said regions. Further, to examine its predictability, the fusion barrier
and barrier radius have been used in the classic Wong fusion cross section
formula and the total fusion cross sections are found to be compared well with
the experimental values. The fusion barrier and barrier radius obtained from
the present formulae are used to solve for the potential parameters of the
Woods-Saxon potential using the three basic equations 3. The parameters so
obtained are quite different from the ones used in the earlier studies.
The present fusion barrier formula is showed an excellent accordance with the
experimental data in Table 6. This comparison includes mostly the heavy ion
reactions aiming to synthesize the superheavy elements. Further, the agreement
between the experimental and present interaction barrier formula is pretty
good. Hence the present fusion and interaction barrier formulae together can
be confidently used for planning experiments for the synthesis of the new
superheavy elements. Similarly, current interaction barriers have been
compared with the experimental interaction barriers quite well. Though we
revealed in this work that the fusion and interaction barrier from the present
work can be estimated very well, but it is an empirical formula. A refined and
complete theoretical model such as DD-TDHF [57] is found to be even better
than the present formula. More such theoretical calculations are highly
desirable to shed better light on the reaction mechanism of the heavy-ion
collisions. At the same time, a benchmark experiment is of high demand that
takes care of quasi-elastic events, deep inelastic collisions, capture
reactions, etc. for the heavy-ion reactions, in a proper manner.
Acknowledgements: We would like to acknowledge the illuminating discussions
with Subir Nath, Ambar Chatterjee, B.R. Behra and S. Kailas.

For corresponding: $\\#$<EMAIL_ADDRESS>and $*$
<EMAIL_ADDRESS>

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Table 1: The fusion barriers $(B_{fu})$ for the following two body systems have been used in Fig 1(a) to obtain the empirical formula for estimating the $B_{fu}$ for any system in the bound $8\leq z\leq 286$. System |  $z$ |  $B_{fu}(MeV)$
---|---|---
12C+15N |  8.83 |  6.80[69]
12C+16O |  9.98 |  7.50[70]
12C+26Mg |  13.71 |  11.5[71]
12C+30Si |  15.57 |  13.2[69]
16O+27Al |  18.84 |  43.6[72]
24Mg+26Mg |  24.63 |  20.8[73]
26Mg+32S |  31.28 |  27.5[74]
12C+92Zr |  35.27 |  32.3[75]
16O+72Ge |  38.32 |  35.4[76]
32S+40Ca |  48.52 |  43.3[30]
48Ca+48Ca |  55.03 |  51.7[30]
27Al+70Ge |  58.42 |  55.1[30]
32S+58Ni |  63.58 |  59.5[77]
40Ar+58Ni |  69.13 |  66.32[30]
37Cl+73Ge |  72.42 |  69.20[30]
40Ca+62Ni |  75.90 |  72.3[69]
32S+89Y |  81.68 |  77.8[78]
16O+238U |  84.43 |  80.8[30]
28Si+120Sn |  87.84 |  85.9[30]
48Ca+96Zr |  97.41 |  95.9[79]
40Ca+96Zr |  100.01 |  93.6[77]
40Ar+121Sb |  109.72 |  111[80]
40Ca+124Sn |  118.95 |  113[81]
28Si+198Pt |  123.18 |  121[82]
40Ar+154Sm |  127.11 |  121[83]
40Ar+165Ho |  135.43 |  141.4[30]
40Ca+192Os |  165.42 |  168.1[30]
84Kr+116Cd |  186.67 |  204[84]
74Ge+232Th |  278.45 |  310[84]
${}^{86}Kr+^{208}Pb$ |  285.52 |  299.00[9]

Table 2: The interaction barriers $(B_{int})$ for the following two body systems have been used in Fig 1 (b) to obtain the empirical formula for estimating the $B_{int}$ for any unknown systems in the range $59\leq z\leq 313$. System |  $z$ |  $B_{int(MeV)}$
---|---|---
12C+205Tl |  59.37 |  56.0[85]
12C+209Bi |  60.55 |  57.0[85]
12C+238U |  65.04 |  62.2[86]
14N+238U |  74.82 |  73.4[86]
16O+205Tl |  76.99 |  77.0[85]
16O+238U |  84.43 |  82.5[86]
20Ne+238U |  103.23 |  102[86]
40Ar+164Dy |  133.57 |  135[87]
40Ar+238U |  172.19 |  171[3]
48Ti+208Pb |  188.72 |  190.1[13]
54Cr+208Pb |  202.78 |  205.8[13]
56Fe+208Pb |  218.65 |  223[13]
58Ni+208Pb |  234.38 |  236[13]
70Zn+208Pb |  244.86 |  250.6[13]
84Kr+232Th |  307.86 |  332[3]
84Kr+238U |  313.14 |  333[88]

Table 3: Comparison of the fusion barrier $B_{fu}$ for different systems with experimental as well as various theoretical models. The reactions are listed in order of the increasing value of the $z$ parameter. The values for our model is taken from the equation (5).  ${System}$ |  $~{}~{}~{}z$ |  $Expt.[Ref.]$ |  $SEDF$ |  $Bass$ |  $SW$ |  $CW$ |  $BW$ |  $AW$ |  $SPP$ |  WJS |  MR |  $Pres.$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---
12C+14N |  8.93 |  7.00 [69] |  $7.03$ |  $5.19$ |  $7.14$ |  $7.06$ |  $7.11$ |  $7.04$ |  $6.73$ |  7.72 |  6.77 |  $7.47$ |
12C+18O |  9.77 |  7.45 [89] |  $7.84$ |  $5.93$ |  $7.85$ |  $7.80$ |  $7.86$ |  $7.78$ |  $7.53$ |  8.46 |  7.67 |  $7.69$ |
12C+17O |  $9.88$ |  $8.20$ [30] |  $7.97$ |  $5.97$ |  $7.86$ |  $7.89$ |  $7.94$ |  $7.86$ |  $7.59$ |  8.54 |  7.68 |  $7.78$ |
20Ne+20Ne |  $18.42$ |  $15.20$ [90] |  $16.02$ |  $13.15$ |  $15.57$ |  $15.59$ |  $15.69$ |  $15.61$ |  $15.46$ |  16.15 |  15.49 |  $15.78$ |
18O+28Si |  $19.79$ |  $16.90$ [30] |  $17.41$ |  $14.51$ |  $16.89$ |  $16.89$ |  $17.01$ |  $16.94$ |  $16.85$ |  17.39 |  17.01 |  $17.09$ |
16O+28Si |  $20.16$ |  $17.23$ [30] |  $17.75$ |  $14.76$ |  $17.34$ |  $17.23$ |  $17.34$ |  $17.26$ |  $17.14$ |  17.71 |  17.19 |  $17.43$ |
24Mg+${}^{24}Mg~{}$ |  $24.96$ |  $22.30$ [71] |  $22.47$ |  $19.16$ |  $21.63$ |  $21.69$ |  $21.82$ |  $21.76$ |  $21.79$ |  22.09 |  21.79 |  $22.01$ |
6Li+144Sm |  $26.35$ |  $24.65$ [30] |  $24.65$ |  $22.22$ |  $24.19$ |  $24.32$ |  $24.47$ |  $24.33$ |  $23.78$ |  23.36 |  24.68 |  $23.35$ |
7Li+159Tb |  $26.60$ |  $23.81$ [30] |  $25.00$ |  $22.58$ |  $24.44$ |  $24.45$ |  $24.73$ |  $24.64$ |  $24.14$ |  23.59 |  25.21 |  $23.59$ |
16O+40Ca |  $26.94$ |  $23.70$ [71] |  $24.52$ |  $21.18$ |  $23.69$ |  $23.68$ |  $23.81$ |  $23.75$ |  $23.78$ |  23.91 |  23.84 |  $23.91$ |
26Mg+30Si |  $27.68$ |  $24.80$ [30] |  $25.29$ |  $21.88$ |  $24.25$ |  $24.29$ |  $24.46$ |  $24.42$ |  $24.57$ |  24.59 |  24.70 |  $24.62$ |
14N+59Co |  $29.99$ |  $26.13$ [91] |  $27.75$ |  $24.45$ |  $26.68$ |  $26.79$ |  $26.95$ |  $26.90$ |  $26.95$ |  26.72 |  27.16 |  $26.86$ |
26Mg+34S |  $30.96$ |  $27.11$ [74] |  $28.62$ |  $25.06$ |  $27.64$ |  $27.42$ |  $27.61$ |  $27.59$ |  $24.07$ |  27.63 |  27.93 |  $27.81$ |
24Mg+34S |  $31.35$ |  $27.38$ [74] |  $28.95$ |  $25.39$ |  $27.69$ |  $27.80$ |  $27.96$ |  $27.92$ |  $24.37$ |  27.99 |  28.18 |  $28.19$ |
30Si+30Si |  $31.53$ |  $28.54$ [76] |  $29.19$ |  $25.62$ |  $27.93$ |  $27.97$ |  $28.16$ |  $28.14$ |  $28.42$ |  28.17 |  28.49 |  $28.37$ |
24Mg+32S |  $31.69$ |  $28.10$ [74] |  $29.23$ |  $25.65$ |  $28.01$ |  $28.09$ |  $28.25$ |  $28.21$ |  $24.61$ |  28.31 |  28.40 |  $28.51$ |
6Li+208Pb |  $31.77$ |  $30.10$ [30] |  $30.27$ |  $28.20$ |  $30.12$ |  $29.97$ |  $30.18$ |  $30.06$ |  $29.40$ |  28.39 |  30.54 |  $28.59$ |
28Si+30Si |  $31.90$ |  $29.13$ [73] |  $29.49$ |  $25.91$ |  $28.25$ |  $28.31$ |  $28.47$ |  $28.44$ |  $28.74$ |  28.51 |  28.72 |  $28.72$ |
28Si+28Si |  $32.27$ |  $28.89$ [73] |  $29.85$ |  $26.22$ |  $28.63$ |  $28.64$ |  $28.80$ |  $28.76$ |  $29.06$ |  28.85 |  28.97 |  $29.08$ |
20Ne+40Ca |  $32.60$ |  $28.60$ [30] |  $30.22$ |  $26.64$ |  $28.88$ |  $29.04$ |  $29.20$ |  $29.16$ |  29.16 |  $29.42$ |  29.37 |  $29.41$ |
24Mg+35Cl |  $33.14$ |  $30.70$ [30] |  $30.76$ |  $27.13$ |  $29.37$ |  $29.51$ |  $29.68$ |  $29.64$ |  29.66 |  $29.96$ |  29.90 |  $29.92$ |
16O+58Ni |  $35.05$ |  $31.67$ [30] |  $32.87$ |  $29.36$ |  $31.64$ |  $31.61$ |  $31.79$ |  $31.75$ |  31.45 |  $31.99$ |  32.04 |  $31.79$ |
12C+152Sm |  $48.78$ |  $46.39$ [30] |  $47.72$ |  $44.97$ |  $45.75$ |  $46.06$ |  $46.38$ |  $46.38$ |  44.56 |  $46.50$ |  46.98 |  $45.34$ |
18O+124Sn |  $52.58$ |  $49.30$ [30] |  $51.57$ |  $48.48$ |  $48.38$ |  $49.31$ |  $49.82$ |  $49.97$ |  $50.40$ |  48.26 |  50.72 |  $49.12$ |
16O+116Sn |  $54.08$ |  $50.94$ [30] |  $52.80$ |  $50.00$ |  $50.51$ |  $50.87$ |  $51.20$ |  $51.29$ |  $51.84$ |  49.72 |  51.91 |  $50.61$ |
30Si+64Ni |  $55.15$ |  $51.20$ [92] |  $53.90$ |  $50.25$ |  $50.85$ |  $51.20$ |  $51.59$ |  $51.71$ |  $52.82$ |  50.78 |  52.56 |  $51.69$ |
30Si+62Ni |  $55.48$ |  $52.20$ [92] |  $54.24$ |  $50.56$ |  $51.19$ |  $51.51$ |  $51.87$ |  $51.99$ |  $53.14$ |  51.11 |  52.82 |  $52.02$ |
28Si+64Ni |  $55.71$ |  $52.40$ [92] |  $54.35$ |  $50.82$ |  $51.29$ |  $51.76$ |  $52.08$ |  $52.18$ |  $53.36$ |  51.32 |  53.00 |  $52.24$ |
28Si+62Ni |  $56.04$ |  $52.89$ [92] |  $54.74$ |  $51.14$ |  $51.67$ |  $52.07$ |  $52.38$ |  $52.47$ |  $53.68$ |  51.65 |  53.27 |  $52.57$ |
40Ca+48Ca |  $56.70$ |  $52.00$ [30] |  $55.21$ |  $51.73$ |  $51.97$ |  $52.61$ |  $52.95$ |  $53.07$ |  $54.40$ |  52.31 |  54.01 |  $53.23$ |
28Si+58Ni |  $56.75$ |  $53.80$ [92] |  $55.46$ |  $51.82$ |  $52.39$ |  $52.71$ |  $53.00$ |  $53.08$ |  $54.37$ |  52.35 |  53.83 |  $53.28$ |
12C+204Pb |  $60.17$ |  $57.55$ [30] |  $59.84$ |  $57.89$ |  $57.88$ |  $57.92$ |  $58.33$ |  $58.50$ |  $58.53$ |  55.74 |  59.03 |  $56.71$ |
40Ca+48Ti |  $62.37$ |  $58.17$ [93] |  $61.38$ |  $57.82$ |  $57.56$ |  $58.22$ |  $58.56$ |  $58.68$ |  $60.32$ |  57.92 |  59.64 |  $58.89$ |
35Cl+54Fe |  $62.69$ |  $58.59$ [30] |  $61.89$ |  $58.20$ |  $58.08$ |  $58.56$ |  $58.90$ |  $59.02$ |  $60.65$ |  58.24 |  59.95 |  $59.21$ |
16O+144Sm |  $63.91$ |  $61.03$ [94] |  $63.62$ |  $61.08$ |  $60.60$ |  $61.00$ |  $61.39$ |  $61.56$ |  $62.25$ |  59.45 |  62.23 |  $60.42$ |
37Cl+64Ni |  $64.92$ |  $60.60$ [30] |  $64.32$ |  $60.80$ |  $60.32$ |  $60.90$ |  $61.36$ |  $61.56$ |  $63.12$ |  60.47 |  62.62 |  $61.43$ |
46Ti+46Ti |  $67.54$ |  $63.30$ [30] |  $66.99$ |  $63.50$ |  $62.66$ |  $63.39$ |  $63.76$ |  $63.92$ |  $65.81$ |  63.11 |  64.98 |  $64.05$ |
16O+186W |  $71.95$ |  $68.87$ [94] |  $72.26$ |  $70.53$ |  $68.66$ |  $69.48$ |  $69.99$ |  $70.27$ |  $70.91$ |  67.55 |  71.02 |  $68.44$ |
28Si+92Zr |  $74.16$ |  $70.93$ [75] |  $74.24$ |  $71.43$ |  $69.17$ |  $70.50$ |  $70.93$ |  $71.16$ |  $72.99$ |  69.81 |  72.27 |  $70.65$ |
40Ca+58Ni |  $76.81$ |  $73.36$ [69] |  $76.96$ |  $73.90$ |  $71.54$ |  $72.71$ |  $73.11$ |  $73.30$ |  $75.70$ |  72.51 |  74.52 |  $73.29$ |
16O+208Pb |  $77.68$ |  $74.90$ [95] |  $78.46$ |  $77.25$ |  $75.39$ |  $75.50$ |  $76.07$ |  $76.42$ |  $77.07$ |  7339 |  77.18 |  $74.15$ |
48Ti+58Ni |  $82.08$ |  $78.80$ [96] |  $82.74$ |  $79.86$ |  $76.76$ |  $78.09$ |  $78.57$ |  $78.83$ |  $81.42$ |  77.91 |  80.24 |  $78.53$ |
36S+90Zr |  $82.23$ |  $79.00$ [97] |  $83.17$ |  $80.39$ |  $77.48$ |  $78.55$ |  $79.18$ |  $79.51$ |  $81.68$ |  78.11 |  80.85 |  $78.68$ |
19F+197Au |  $83.77$ |  $81.61$ [30] |  $85.20$ |  $83.88$ |  $80.73$ |  $81.47$ |  $82.18$ |  $82.61$ |  $83.58$ |  79.66 |  83.50 |  $80.22$ |
35Cl+92Zr |  $87.34$ |  $82.94$ [75] |  $88.60$ |  $86.25$ |  $82.31$ |  $83.76$ |  $84.32$ |  $84.66$ |  83.35 |  $87.17$ |  86.09 |  $83.77$ |
35Cl+106Pd |  $97.70$ |  $94.30$ [30] |  $100.02$ |  $98.36$ |  $92.86$ |  $94.48$ |  $95.11$ |  $95.54$ |  $98.43$ |  94.21 |  97.13 |  $94.11$ |
32S+116Sn |  $99.36$ |  $97.36$ [30] |  $101.50$ |  $100.47$ |  $94.79$ |  $96.31$ |  $96.92$ |  $97.35$ |  $100.24$ |  95.95 |  98.93 |  $95.75$ |
58Ni+60Ni |  $100.69$ |  $96.00$ [30] |  $103.17$ |  $101.33$ |  $95.82$ |  $97.04$ |  $97.63$ |  $98.01$ |  $101.64$ |  97.36 |  99.93 |  $97.09$ |
40Ca+90Zr |  $101.25$ |  $96.88$ [30] |  $103.75$ |  $102.26$ |  $96.34$ |  $97.87$ |  $98.45$ |  $98.86$ |  $102.26$ |  97.94 |  100.63 |  $97.65$ |
58Ni+58Ni |  101.27 |  95.8 [98] |  $96.70$ |  $102.01$ |  $96.50$ |  $97.59$ |  $98.15$ |  $98.51$ |  $102.25$ |  97.97 |  100.36 |  $97.67$ |
40Ar+122Sn |  107.40 |  103.6 [99] |  $105.18$ |  $109.70$ |  $103.24$ |  $104.51$ |  $105.47$ |  $106.09$ |  $109.07$ |  104.51 |  107.86 |  $103.86$ |
40Ar+116Sn |  108.47 |  103.3 [83] |  $105.93$ |  $110.92$ |  $104.26$ |  $105.57$ |  $106.45$ |  $107.02$ |  $110.22$ |  105.63 |  108.84 |  $104.94$ |
40Ar+112Sn |  109.22 |  104.0 [83] |  $106.44$ |  $111.78$ |  $105.02$ |  $106.30$ |  $107.12$ |  $107.67$ |  $111.02$ |  106.44 |  109.52$105.71$ |  |
64Ni+74Ge |  109.29 |  103.2 [83] |  $106.34$ |  $111.37$ |  $104.46$ |  $106.00$ |  $106.92$ |  $107.50$ |  $111.09$ |  106.51 |  109.52 |  $105.77$ |
58Ni+74Ge |  111.04 |  106.8 [100] |  $107.50$ |  $113.49$ |  $105.97$ |  $107.76$ |  $108.49$ |  $109.01$ |  $112.99$ |  108.39 |  $111.11$ |  $107.56$ |
34S+168Er |  124.24 |  121.5 [21] |  $122.92$ |  $130.30$ |  $121.20$ |  $122.46$ |  $123.51$ |  $124.30$ |  $127.56$ |  122.63 |  126.11 |  $121.31$ |
40Ar+148Sm |  128.14 |  124.7 [83] |  $126.60$ |  $134.57$ |  $124.99$ |  $126.19$ |  $127.32$ |  $128.13$ |  $131.85$ |  126.88 |  130.25 |  $125.45$ |
40Ar+144Sm |  128.85 |  124.4 [83] |  $127.14$ |  $135.41$ |  $125.91$ |  $126.90$ |  $127.98$ |  $128.76$ |  $132.63$ |  127.66 |  130.92 |  $126.28$ |

Table 4: Comparison of the fusion barrier radius $R_{B}$ for different systems with experimental as well as various theoretical models. The reactions are listed in order of the increasing value of the $z$ parameter. The values for our model is taken from the equation (10). The superscript * indicates that the experimental radii for all the reactions are taken from the ones given in [10]. System |  $z$ |  $Expt.^{*}$ |  $SEDF$ |  $Bass$ |  $SW$ |  $CW$ |  $BW$ |  $AW$ |  $SPP$ |  MR |  $Pres.$ |
---|---|---|---|---|---|---|---|---|---|---|---|---
$11.30$ |  $10.50$ |  $10.46$ |  $10.37$ |  $10.28$ |  $11.09$ |  |  |  |  |  |  |
30Si+64Ni |  55.16 |  9.60 |  $9.96$ |  $9.82$ |  $10.60$ |  $10.32$ |  $10.21$ |  $10.16$ |  $10.05$ |  9.99 |  $8.79$ |
30Si+62Ni |  $55.48$ |  $9.70$ |  $9.91$ |  $9.75$ |  $10.50$ |  $10.28$ |  $10.15$ |  $10.10$ |  $9.99$ |  9.94 |  $8.77$ |
28Si+64Ni |  $55.71$ |  $7.60$ |  $9.87$ |  $9.69$ |  $10.50$ |  $10.25$ |  $10.11$ |  $10.03$ |  $9.95$ |  9.90 |  $8.75$ |
28Si+62Ni |  $56.04$ |  $7.70$ |  $9.81$ |  $9.62$ |  $10.40$ |  $10.20$ |  $10.03$ |  $9.97$ |  $9.88$ |  9.84 |  $8.72$ |
30Si+58Ni |  $56.18$ |  $8.80$ |  $9.78$ |  $9.60$ |  $10.40$ |  $10.19$ |  $10.02$ |  $9.94$ |  $9.86$ |  9.81 |  $8.72$ |
28Si+58Ni |  $56.75$ |  $8.10$ |  $9.68$ |  $9.47$ |  $10.20$ |  $10.11$ |  $9.89$ |  $9.84$ |  $9.76$ |  9.71 |  $8.67$ |
40Ca+44Ca |  $57.55$ |  $7.90$ |  $9.73$ |  $9.53$ |  $10.30$ |  $10.16$ |  $9.98$ |  $9.90$ |  $9.81$ |  9.76 |  $8.81$ |
40Ca+40Ca |  $58.48$ |  $9.50$ |  $9.58$ |  $9.33$ |  $10.20$ |  $10.04$ |  $9.79$ |  $9.73$ |  $9.64$ |  9.61 |  $8.72$ |
36S+64Ni |  $61.35$ |  $8.50$ |  $10.14$ |  $9.90$ |  $10.90$ |  $10.53$ |  $10.43$ |  $10.34$ |  $10.25$ |  10.18 |  $9.49$ |
34S+64Ni |  $61.88$ |  $8.50$ |  $10.05$ |  $9.79$ |  $10.70$ |  $10.47$ |  $10.33$ |  $10.25$ |  $10.16$ |  10.09 |  $9.42$ |
40Ca+50Ti |  $61.94$ |  $9.40$ |  $9.88$ |  $9.61$ |  $10.60$ |  $10.32$ |  $10.15$ |  $10.07$ |  $9.97$ |  9.91 |  $9.24$ |
40Ca+48Ti |  $62.37$ |  $9.40$ |  $9.81$ |  $9.52$ |  $10.40$ |  $10.27$ |  $10.07$ |  $10.00$ |  $9.90$ |  9.84 |  $9.19$ |
32S+64Ni |  $62.44$ |  $8.10$ |  $9.96$ |  $9.67$ |  $10.70$ |  $10.40$ |  $10.23$ |  $10.16$ |  $10.07$ |  10.00 |  $9.34$ |
36Si+58Ni |  $62.46$ |  $7.70$ |  $9.96$ |  $9.68$ |  $10.60$ |  $10.39$ |  $10.23$ |  $10.13$ |  $10.06$ |  10.09 |  $8.84$ |
40Ca+46Ti |  $62.83$ |  $9.40$ |  $9.74$ |  $9.42$ |  $10.40$ |  $10.21$ |  $9.98$ |  $9.92$ |  $9.82$ |  9.77 |  $9.13$ |
16O+154Sm |  $62.94$ |  $9.60$ |  $10.87$ |  $10.44$ |  $11.50$ |  $11.15$ |  $11.07$ |  $11.01$ |  $11.04$ |  10.91 |  $10.17$ |
17O+144Sm |  $63.49$ |  $10.80$ |  $10.78$ |  $10.35$ |  $11.40$ |  $11.08$ |  $10.98$ |  $10.91$ |  $10.94$ |  10.81 |  $10.10$ |
16O+148Sm |  $63.51$ |  $10.20$ |  $10.77$ |  $10.33$ |  $11.40$ |  $11.08$ |  $10.97$ |  $10.90$ |  $10.93$ |  10.82 |  $10.09$ |
32S+58Ni |  $63.59$ |  $8.30$ |  $9.78$ |  $9.44$ |  $10.40$ |  $10.26$ |  $10.02$ |  $9.95$ |  $9.87$ |  99.82 |  $9.19$ |
16O+144Sm |  $63.91$ |  $10.30$ |  $10.71$ |  $10.25$ |  $11.30$ |  $11.02$ |  $10.89$ |  $10.82$ |  $10.86$ |  10.47 |  $10.03$ |
16O+186W |  $71.95$ |  $10.60$ |  $11.22$ |  $10.63$ |  $11.70$ |  $11.52$ |  $8.73$ |  $8.50$ |  $11.43$ |  11.25 |  $10.46$ |
16O+208Pb |  $77.68$ |  $10.50$ |  $11.43$ |  $10.77$ |  $11.80$ |  $11.76$ |  $9.03$ |  $8.78$ |  $11.68$ |  11.47 |  $10.40$ |
36S+96Zr |  $81.21$ |  $11.00$ |  $10.66$ |  $10.16$ |  $11.30$ |  $11.15$ |  $8.40$ |  $10.91$ |  $10.87$ |  10.72 |  $9.55$ |
36S+90Zr |  $82.23$ |  $10.80$ |  $10.53$ |  $10.00$ |  $11.20$ |  $11.05$ |  $10.89$ |  $10.77$ |  $10.73$ |  10.59 |  $9.34$ |
36S+110Pd |  $90.94$ |  $8.20$ |  $10.83$ |  $10.23$ |  $11.50$ |  $11.38$ |  $8.72$ |  $8.45$ |  $11.09$ |  10.91 |  $9.05$ |
32S+110Pd |  $92.39$ |  $8.00$ |  $10.65$ |  $10.00$ |  $11.40$ |  $11.24$ |  $8.57$ |  $10.92$ |  $10.91$ |  10.7 |  $8.76$ |
64Ni+64Ni |  $98.00$ |  $7.80$ |  $10.64$ |  $10.01$ |  $11.50$ |  $11.28$ |  $8.65$ |  $10.94$ |  $10.92$ |  10.74 |  $8.44$ |
40Ca+96Zr |  $100.01$ |  $9.30$ |  $10.62$ |  $9.93$ |  $11.40$ |  $11.28$ |  $8.65$ |  $10.91$ |  $10.90$ |  10.72 |  $8.29$ |
58Ni+60Ni |  $100.70$ |  $7.50$ |  $10.34$ |  $9.64$ |  $11.00$ |  $11.05$ |  $10.77$ |  $10.62$ |  $10.60$ |  10.45 |  $7.98$ |
40Ca+90Zr |  $101.25$ |  $10.00$ |  $10.48$ |  $9.77$ |  $11.20$ |  $11.17$ |  $10.92$ |  $10.76$ |  $10.76$ |  10.59 |  $8.09$ |
40Ar+122Sn |  $107.40$ |  $9.80$ |  $11.03$ |  $10.33$ |  $11.80$ |  $11.69$ |  $9.15$ |  $8.87$ |  $11.38$ |  11.14 |  $8.43$ |
40Ar+116Sn |  $108.47$ |  $8.70$ |  $10.92$ |  $10.19$ |  $11.70$ |  $11.60$ |  $9.06$ |  $8.79$ |  $11.26$ |  11.03 |  $8.28$ |
40Ar+112Sn |  $109.22$ |  $8.90$ |  $10.84$ |  $10.10$ |  $11.60$ |  $11.54$ |  $8.99$ |  $8.73$ |  $11.1$ |  10.95 |  $8.18$ |
58Ni+74Ge |  $111.04$ |  $7.00$ |  $10.60$ |  $9.87$ |  $11.40$ |  $11.35$ |  $8.80$ |  $10.93$ |  $10.93$ |  10.73 |  $7.93$ |
40Ca+124Sn |  $118.95$ |  $9.60$ |  $10.96$ |  $10.17$ |  $11.80$ |  $11.72$ |  $9.23$ |  $8.95$ |  $11.35$ |  11.10 |  $8.27$ |
28Si+198Pt |  $123.18$ |  $9.80$ |  $11.50$ |  $10.64$ |  $12.30$ |  $12.21$ |  $9.79$ |  $9.49$ |  $11.96$ |  11.53 |  $9.56$ |
34S+168Er |  $124.24$ |  $10.30$ |  $11.35$ |  $10.53$ |  $12.20$ |  $12.09$ |  $9.68$ |  $9.38$ |  $11.81$ |  11.49 |  $8.78$ |
40Ar+148Sm |  $128.14$ |  $8.30$ |  $11.25$ |  $10.44$ |  $12.10$ |  $12.04$ |  $9.65$ |  $9.34$ |  $11.72$ |  11.41 |  $8.85$ |
40Ar+144Sm |  $128.85$ |  $8.30$ |  $11.19$ |  $10.37$ |  $12.00$ |  $11.99$ |  $9.60$ |  $9.30$ |  $11.65$ |  11.35 |  $8.81$ |
40Ca+192Os |  $165.42$ |  $10.70$ |  $11.54$ |  $10.62$ |  $12.20$ |  $12.55$ |  $10.35$ |  $10.04$ |  $12.21$ |  11.79 |  $10.38$ |
40Ca+194Pt |  $169.40$ |  $9.60$ |  $11.53$ |  $10.60$ |  $-$ |  $12.97$ |  $10.40$ |  $10.09$ |  $12.22$ |  11.79 |  $10.06$ |

Table 5: Comparison of mean squared error between the experimental value and a model prediction for the quantities Bfus and Rb.  Quantity | SEDF | Bass | SW | CW | BW | AW | SPP | WJS | MR |  | Present
---
Bfus | 7.63 | 15.14 | 0.82 | 0.84 | 1.61 | 2.41 | 11.41 | 3.76 | 6.88 | 0.81
Rb | 3.05 | 1.75 | 5.61 | 5.21 | 2.10 | 2.62 | 3.92 | - | 4.42 | 0.80

Table 6: Comparison of experimental fusion barrier (Bfu) with the present formula for different reactions in the superheavy region. The last four reactions are used for the reaction contact time experiments [46], where the WJS formula [23] has been used. We show here how much these values differ from the present formula. system |  z |  Expt. |  Present |  WDM[25] |  WJS[23] |  HCM[24]
---|---|---|---|---|---|---
${}^{32}S+^{232}Th$ |  154.52 |  155.73[9] |  157.53 |  153.24 |  155.99 |  158.04
${}^{48}Ca+^{208}Pb$ |  171.56 |  173.40$\pm$ 0.1[33] |  173.11 |  170.19 |  175.05 |  176.28
${}^{50}Ti+^{208}Pb$ |  187.00 |  192.60$\pm$ 0.1[33] |  191.82 |  187.62 |  192.42 |  194.04
${}^{48}Ti+^{208}Pb$ |  188.71 |  190.10[9] |  193.02 |  188.18 |  194.35 |  195.10
${}^{54}Cr+^{208}Pb$ |  202.79 |  207.30$\pm$ 0.3[33] |  207.89 |  203.61 |  210.22 |  210.68
${}^{56}Fe+^{208}Pb$ |  218.64 |  223.00[9] |  221.73 |  220.71 |  228.14 |  228.67
${}^{64}Ni+^{208}Pb$ |  231.33 |  236.00[9] |  233.74 |  233.58 |  242.43 |  243.25
${}^{70}Zn+^{208}Pb$ |  244.86 |  250.00[9] |  252.41 |  248.55 |  257.61 |  259.02
${}^{86}Kr+^{208}Pb$ |  285.52 |  299.00[9] |  299.58 |  292.12 |  302.44 |  308.39
${}^{86}Kr+^{208}Pb$ |  285.52 |  303.3[101] |  299.58 |  292.12 |  302.44 |  308.52
${}^{50}Ti+^{249}Cf$ |  216.14 |  - |  219.62 |  218.31 |  225.32 |  226.41
${}^{54}Cr+^{248}Cm$ |  228.97 |  - |  231.23 |  232.58 |  239.78 |  241.05
${}^{58}Fe+^{244}Pu$ |  241.51 |  - |  247.01 |  245.75 |  253.85 |  255.53
${}^{64}Ni+^{238}U$ |  252.62 |  - |  266.84 |  258.22 |  266.25 |  268.54
Table 7: Comparison of the fusion barrier heights (second column) ($V_{B}^{exp}$) (unit is MeV) with various theoretical results calculated by the DFM $V^{DFM}_{B}$ (third column), the DFM potential using the M3YReid interaction with zero-range exchange part ($V_{B}^{\delta R}$) (fourth column) and using the Paris CDM3Y3 interaction with finite-range exchange part ($V_{B}^{Pf}$) (fifth column), respectively. All these DFM values have been taken from Table 3 of [35]. In last column and last row, we give the present values ($V_{B}^{Pres.}$) and mean squared error of each set of theoretical values, respectively.  Proj.+Targ. | z | V${}_{B}^{exp.}$ | V${}_{B}^{DFM}$ | V${}_{B}^{R\delta}$ | V${}_{B}^{Pf}$ | V${}_{B}^{Pres.}$
---|---|---|---|---|---|---
${}^{16}O+^{92}Zr$[102] | 45.49 | 42.00 | 43.50 | 42.26 | 41.14 | 42.08
${}^{12}C+^{204}Pb$[103] | 60.17 | 57.60 | 57.90 | 58.61 | 57.30 | 56.70
${}^{16}O+^{148}Sm$[94] | 63.51 | 59.80 | 61.10 | 61.19 | 59.61 | 60.03
${}^{17}O+^{144}Sm$[94] | 63.49 | 60.60 | 60.80 | 61.10 | 59.53 | 60.00
${}^{16}O+^{144}Sm$[94] | 63.91 | 61.00 | 61.40 | 61.53 | 59.94 | 60.42
${}^{28}Si+^{92}Zr$[103] | 74.16 | 70.90 | 71.80 | 71.46 | 69.59 | 70.65
${}^{16}O+^{208}Pb$[95] | 77.68 | 74.55 | 75.30 | 77.08 | 75.40 | 74.15
${}^{36}S+^{96}Zr$[97] | 81.21 | 76.70 | 77.80 | 77.65 | 75.45 | 77.67
${}^{34}S+^{89}Y$[78] | 80.99 | 76.90 | 78.50 | 77.55 | 75.42 | 77.45
${}^{32}S+^{89}Y$[78] | 81.68 | 77.80 | 79.50 | 78.21 | 76.04 | 78.14
${}^{36}S+^{90}Zr$[97] | 82.23 | 78.00 | 79.10 | 78.26 | 76.01 | 78.68
${}^{19}F+^{197}Au$[94] | 83.77 | 80.80 | 81.10 | 83.83 | 81.90 | 80.22
${}^{35}Cl+^{92}Zr$[103] | 87.34 | 82.90 | 84.50 | 83.57 | 81.15 | 83.77
${}^{19}F+^{208}Pb$ [104] | 85.88 | 83.00 | 83.50 | 85.26 | 83.25 | 82.31
${}^{40}Ca+^{96}Zr$[98] | 100.01 | 94.60 | 96.50 | 97.01 | 94.32 | 96.41
${}^{40}Ca+^{90}Zr$[98] | 101.25 | 96.90 | 98.10 | 97.78 | 95.01 | 97.65
${}^{28}Si+^{144}Sm$[105] | 104.86 | 104.00 | 102.20 | 104.37 | 101.72 | 101.28
${}^{40}Ca+^{124}Sn$[81] | 118.95 | 113.10 | 114.90 | 117.89 | 114.96 | 115.73
${}^{28}Si+^{208}Pb$[106] | 128.10 | 128.10 | 126.10 | 130.90 | 128.08 | 126.34
Mean squared error | 1.88 | 3.77 | 1.94 | 1.56
Table 8: Comparison of the Woods-Saxon potential parameters $V_{0}$ (MeV), $r_{0}$ (fm), $a_{0}$ (fm) and $\hbar\omega$ from the present empirical formula and other models. The value marked with a superscript (∗) indicates Akyuz Winther (AkW) parameters [47] are altered to have agreement with measurements. Modified Akyuz Winther (MAkW) parameters are obtained by fixing the $V_{0}$ = 100 MeV and then calculating the values of $a_{0}$ and $r_{0}$ from the AkW model. The width $\hbar\omega$ is obtained from equation (48). Systems | Present work | Other Models
---|---|---
$V_{0}$ | $a_{0}$ | $r_{0}$ | $\hbar\omega$ | $V_{0}$ | $a_{0}$ | $r_{0}$ | $\hbar\omega$ | Model | Ref.
19F+181Ta | 45.30 | 1.17 | 1.15 | 3.79 | 104.5 | 0.700 | 1.120 | 5.21 | AkW∗ | [49]
16O+154Sm | 26.40 | 0.92 | 1.24 | 4.19 | 100.0 | 1.060 | 1.019 | 3.85 | MAkW | [50]
19F+208Pb | 67.73 | 1.45 | 1.06 | 3.51 | 100.0 | 1.060 | 1.059 | 4.34 | MAkW | [50]
64Ni+64Ni | 135.07 | 1.76 | 0.85 | 2.61 | 75.98 | 0.676 | 1.190 | 5.06 | AkW | [48]
36S+90Zr | 53.72 | 1.18 | 1.12 | 3.27 | 100.0 | 0.970 | 1.070 | 3.74 | MAkW | [50]

Table 9: Comparison of the present fusion barrier $B_{fus}^{Pres.}=V_{B}^{Pres.}$ Energy (in MeV) and radii (in fm) of the Coulomb barrier extracted from the DD-TDHF method [57]. $V^{DD}_{B}$ (high $E_{c.m.}$) refers to the barrier deduced for $E_{c.m.}>V^{FD}_{B}$, while $V^{DD}_{B}$ (low $E_{c.m}$.) corresponds to the lowest Coulomb barrier deduced from TDHF using $E_{c.m.}<V^{FD}_{B}$ and available experimental values. The reactions are listed in order of the increasing value of the $z$ parameter. Mean squared error (MSE) are given at the bottom row. See text for details. Reaction | V${}_{B}^{FD}$ |  | V${}_{B}^{DD}$
---
(high Ec.m)
| V${}_{B}^{DD}$
---
(low Ec.m)
V${}_{B}^{Pres.}$ | V${}_{B}^{exp}$ | R${}_{B}^{FD}$ |  | R${}_{B}^{DD}$
---
(high Ec.m)
| R${}_{B}^{DD}$
---
(low Ec.m)
R${}_{B}^{Pres.}$ | R${}_{B}^{exp}$
${}^{16}O+^{16}O$ | 10.2 | 10.13 | 10.12 | 10.41 | 10.61[69] | 8.4 | 8.46 | 8.52 |  | 7.91[69]
${}^{16}O+^{40}Ca$ | 23.5 | 23.36 | 23.07 | 23.91 | 23.06[69] | 9.2 | 9.18 | 9.50 |  | 9.21[69]
${}^{16}O+^{48}Ca$ | 23.0 | 22.77 | 22.48 | 23.01 |  | 9.4 | 9.50 | 9.75 |  |
${}^{40}Ca+^{40}Ca$ | 54.7 | 54.54 | 53.35 | 55.00 | 52.80[107] | 9.8 | 9.82 | 10.32 | 8.72 |
${}^{40}Ca+^{48}Ca$ | 53.4 | 53.24 | 52.13 | 53.53 | 52.00[108] | 10.1 | 10.09 | 10.56 | 8.88 | 9.99[108]
${}^{48}Ca+^{48}Ca$ | 52.4 | 52.13 | 50.97 | 51.56 | 51.49[108] | 10.3 | 10.38 | 10.82 | 9.02 | 10.16[108]
${}^{16}O+^{208}Pb$ | 76.0 | 75.91 | 74.51 | 74.15 | 74.52[108] | 11.8 | 11.74 | 12.14 | 10.41 | 11.31[108]
${}^{48}Ca+^{90}Zr$ | 99.8 | 99.98 | 97.71 | 97.65 | 96.88[108] | 10.8 | 10.63 | 11.27 | 8.55 | 10.53[108]
MSE | 3.51 | 3.37 | 0.34 | 1.74 | - | 0.17 | 0.13 | 0.99 | 3.63 | -

Table 10: Comparison of the interaction barrier $B_{int}$ for different systems between the present empirical model, Bass interaction model [2] and available experimental values. The reactions are listed in order of the increasing value of the $z$ parameter. Mean squared errors for Bass model and present formula are 15.22 and 1.74, respectively. System |  $z$ |  $Bass$ |  $Pres.$ |  $Exptl.$ |  System |  $z$ |  $Bass$ |  $Pres.$ |  $Exptl.$
---|---|---|---|---|---|---|---|---|---
32S+24Mg |  $31.69$ |  $25.72$ |  $15.56$ |  |  16O+150Nd |  $61.29$ |  $57.41$ |  $57.90$ |
32S+27Al |  $33.69$ |  $27.71$ |  $19.06$ |  |  16O+148Nd |  $61.46$ |  $57.57$ |  $58.13$ |
18O+64Ni |  $33.83$ |  $28.37$ |  $19.31$ |  |  12C+238U |  $65.04$ |  $62.63$ |  $62.38$ |  $62.2$[86]
18O+62Ni |  $34.05$ |  $28.55$ |  $19.68$ |  |  35Cl+64Ni |  $65.46$ |  $60.18$ |  $62.87$ |
18O+60Ni |  $34.27$ |  $28.72$ |  $20.06$ |  |  35Cl+62Ni |  $65.85$ |  $60.49$ |  $63.31$ |  $63.30$[48]
16O+64Ni |  $34.36$ |  $28.85$ |  $20.20$ |  |  35Cl+60Ni |  $66.24$ |  $60.81$ |  $63.77$ |
18O+58Ni |  $34.51$ |  $28.91$ |  $20.45$ |  |  35Cl+58Ni |  $66.65$ |  $61.13$ |  $64.24$ |  $64.2$[48]
16O+62Ni |  $34.58$ |  $29.03$ |  $20.58$ |  |  14N+238U |  $74.82$ |  $72.83$ |  $73.47$ |  $73.4$[86]
16O+60Ni |  $34.81$ |  $29.21$ |  $20.97$ |  |  16O+205Tl |  $76.99$ |  $74.60$ |  $75.85$ |  $77.0$[85]
18O+65Cu |  $34.93$ |  $29.47$ |  $21.17$ |  |  16O+238U |  $84.43$ |  $82.92$ |  $83.84$ |  $82.5$[86]
18O+64Cu |  $35.05$ |  $29.39$ |  $21.35$ |  |  35Cl+90Zr |  $87.71$ |  $83.57$ |  $87.29$ |
18O+63Cu |  $35.15$ |  $29.64$ |  $21.54$ |  |  40Ar+110Pd |  $100.84$ |  $98.01$ |  $100.77$ |
35Cl+27Al |  $35.24$ |  $29.28$ |  $21.69$ |  |  35Cl+124Sn |  $102.93$ |  $100.38$ |  $102.89$ |
16O+65Cu |  $35.47$ |  $29.95$ |  $22.07$ |  |  20Ne+238U |  $103.24$ |  $102.81$ |  $103.19$ |  $102.0$[86]
18O+70Zn |  $35.59$ |  $30.21$ |  $22.28$ |  |  35Cl+116Sn |  $104.31$ |  $101.52$ |  $104.28$ |
16O+63Cu |  $35.69$ |  $30.13$ |  $22.44$ |  |  40Ar+164Dy |  $133.57$ |  $133.93$ |  $133.37$ |  $135.0$[87]
18O+68Zn |  $35.81$ |  $30.30$ |  $22.63$ |  |  40Ar+197Au |  $153.92$ |  $156.45$ |  $153.64$ |
18O+66Zn |  $36.02$ |  $30.55$ |  $22.98$ |  |  40Ar+208Pb |  $157.95$ |  $161.08$ |  $157.69$ |
16O+70Zn |  $36.14$ |  $30.71$ |  $23.17$ |  |  40Ar+238U |  $172.19$ |  $177.18$ |  $172.17$ |  $171.0$[3]
18O+64Zn |  $36.25$ |  $30.73$ |  $23.35$ |  |  48Ti+208Pb |  $188.72$ |  $194.41$ |  $189.32$ |  $190.1$[13]
16O+68Zn |  $36.36$ |  $30.89$ |  $23.52$ |  |  54Cr+208Pb |  $202.79$ |  $209.91$ |  $204.27$ |  $205.8$[13]
16O+66Zn |  $36.58$ |  $31.05$ |  $23.89$ |  |  54Cr+207Pb |  $202.99$ |  $210.07$ |  $204.48$ |
16O+64Zn |  $36.81$ |  $31.23$ |  $24.26$ |  |  52Cr+208Pb |  $203.78$ |  $210.79$ |  $205.34$ |
12C+152Sm |  $48.78$ |  $44.68$ |  $41.95$ |  |  56Fe+208Pb |  $218.64$ |  $227.02$ |  $221.55$ |  $223.0$[13]
35Cl+48Ti |  $54.16$ |  $48.39$ |  $49.06$ |  |  58Ni+208Pb |  $234.38$ |  $244.03$ |  $239.19$ |  $236.0$[13]
16O+134Ba |  $58.66$ |  $54.44$ |  $54.72$ |  |  70Zn+208Pb |  $244.87$ |  $256.33$ |  $251.22$ |  $250.6$[13]
12C+205Tl |  $59.37$ |  $56.29$ |  $55.59$ |  $56.0$[85] |  84Kr+232Th |  $307.86$ |  $326.78$ |  $328.62$ |  $332.0$[3]
12C+209Bi |  $60.55$ |  $57.57$ |  $57.04$ |  $57.0$[85] |  84Kr+238U |  $313.14$ |  $332.81$ |  $-$ |  $333.0$[88]

Figure 1: Fusion barrier, interaction barrier, fusion barrier radius and
percentage of deviation plots: (a) the fusion barrier Bfu (MeV), as obtained
from the fusion excitation function experiments, (b) the interaction barrier
Bint (MeV) from the quasi-elastic scattering experiments, (c) the fusion
barrier radius RB (fm), (d) percentage of deviation of present value from the
experimental Bint are plotted against the dimensionless parameter z. The
percentage of deviation for Bfu and RB are shown in figures 2 and 3.

Figure 2: Comparison of percentage of deviation of different theoretical
models from the experimental fusion barriers as a function of $z$.

Figure 3: Comparison of percentage of deviation of different models from the
experimental barrier radius as a function of $z$

Figure 4: Comparison of total fusion cross section between theory and
experiment as a function of $E_{cm}$ for the reactions (a) 19F+181Ta [49] and
(b) 16O+154Sm [50](c) ${}^{16}O+^{208}Pb$[95] (d)${}^{58}Ni+^{54}Fe$. The
dashed vertical line indicates the fusion barrier for the corresponding
reaction.