PACS: 25.60.Pj; 21.60.Ev; 24.10.Eq.

1.Introduction

In the past few decades, the large number of theoretical and experimental efforts has been
concentrated to investigate the role of the nuclear structure degrees of freedom of
collision partners on fusion process. The fusion reactions, wherein the colliding
nuclei leads to the formation of a compound nucleus either by overcoming or by
quantum mechanical tunneling through the Coulomb barrier, have emerged as one of the
most sensitive nuclear spectroscopic tool to examine the role of the nuclear
structure of the participating nuclei as well as their nuclear interactions ^{1}^{,}^{2}^{,}^{3}. Despite of the lots of investigations done so far, the
dynamics of fusion reactions still shows unexpected facets and attracts researcher
to explore many unexplored features. Many theoretical and experimental evidences
showed that the sub-barrier fusion excitation function data of various fusing
systems is dramatically enhanced over the predictions of the one-dimensional barrier
penetration model. Such fusion enhancement at near and sub-barrier energies has an
intimate link with the nuclear structure degrees of freedom such as permanent
deformation (deformed nuclei), vibration of nuclear surface (spherical nuclei),
rotations of nuclei during collision, neck formation and nucleon (multi-nucleon)
transfer reactions ^{1}^{,}^{2}^{,}^{3}^{,}^{4}. The coupling of such dominant intrinsic channels to the
relative motion of the collision partners effectively reduces the interaction
barrier between the colliding systems and consequently results in an anomalously
large fusion excitation functions at below barrier energies. Within the coupled
channel approach, the effect of inclusion of intrinsic channels associated with the
colliding systems is to replace the nominal Coulomb barrier by a distribution of
barriers of different height and weight. In barrier distribution, the presence
fusion barriers whose heights are smaller than the Coulomb barrier allow the passage
of flux from entrance channel to fusion channel and hence coupled channel
calculations are capable of reproducing the observed fusion dynamics ^{5}^{-}^{6}.

In theoretical description, the influences of the relevant intrinsic channels are incorporated
through the nucleus-nucleus potential. In this respect, the choice of nuclear
potential is very crucial in order to judge the success of the theoretical approach.
In heavy ion fusion reactions, the optimum choice of the nucleus-nucleus potential,
which consists of the Coulomb repulsive interaction, centrifugal term and short
range attractive nuclear potential, strongly affects the magnitude of fusion
excitation functions and hence an accurate knowledge with regard the nuclear
potential greatly simplifies the problem of understanding of the basic mechanism
involved in the nuclear reaction dynamics ^{ 7
}^{,}^{8}^{,}^{9}^{,}^{10}^{,}^{11}^{,}^{12}^{,}^{13}^{,}^{14}^{,}^{15}^{,}^{16}^{,}^{17}^{,}^{18}^{,}^{19}^{,}^{20}. The nuclear potential of the Woods-Saxon form, wherein
the depth, range and diffuseness parameters are interrelated, is often used to
preview the various features of the heavy ion collisions. For this potential, the
different sets of potential parameters are associated with different nuclear
interactions between the collision partners. In case of elastic scattering analysis,
a diffuseness of *a* = 0.65 *fm* is the most suitable
for good description of the data. In contrast to this, a much larger value of the
diffuseness parameter ranging from *a* = 0.75 *fm* to
*a* = 1.5 *fm* has been extracted from the
systematics of the fusion reactions ^{ 21
}^{,}^{22}^{,}^{
23}. This diffuseness anomaly, which might preview the various
static and dynamical physical effects, is one of the interesting and challenging
issue of heavy ion collisions. For heavy ion reactions, the recent analysis^{16}^{,}^{24}^{,}^{25}^{,}^{26}^{,}^{27}^{,}^{28} suggested that the energy dependence in nucleus-nucleus potential
is another essential feature of nuclear potential. Such energy dependence is also
pointed out in double folding potential wherein it originates from the
nucleon-nucleon interactions as well as the non-local quantum effects. The non-local
quantum effects are directly linked with the exchange of nucleons between the
colliding systems and consequently generate a velocity dependent nuclear
potential^{16}^{,}^{24}. The energy dependence of the
local equivalent potential is related to the finite range of Pauli nonlocality which
in turn manifests the exchange of nucleons during nuclear interactions ^{16}^{,}^{24}. It is quite interesting to
note that the energy dependence in nucleus-nucleus potential is also reflected from
the microscopic time-dependent Hartree-Fock theory ^{25}^{,}^{26}^{,}^{27}^{,}^{ 28
}. In Ref. 25 to 26, it has been shown that in the domain of the
Coulomb barrier, the nuclear potential becomes energy dependent and such energy
dependence occurs due to coordinate-dependent mass and the involvement channel
coupling effects associated with the collision partners. In this sense, the energy
dependent nuclear potential may give better explanation of the many uncharted
features of nuclear interactions. To include nuclear structure effects as well as
the energy dependence in nucleus-nucleus potential, the earlier work undertook
several efforts by introducing the energy dependence in the Woods-Saxon potential
via its diffuseness parameter ^{29}^{,}^{30}^{,}^{31}.

In this work, the fusion dynamics of ^{32}^{,}^{33}^{,}^{34}^{,}^{35}^{,}^{36}^{,}^{37}^{,}^{38}^{,}^{39}^{,}^{40}^{,}^{41}^{,}^{42} is analyzed within the view of the static Woods-Saxon potential
and the energy dependent Woods-Saxon potential model (EDWSP model) ^{4}^{,}^{8}^{-}^{9}^{,}^{13}^{,}^{17}^{-}^{18}^{,}^{29}^{,}^{30}^{,}^{31} along with the Wong’s approximation ^{43}. As far as the colliding systems
are concerned, the projectiles exhibit dominance of the different nuclear structure
degrees of freedom and consequently results in the different energy dependence of
the sub-barrier fusion cross-sections. The lighter projectile (^{ 44}, overcome these deviations. In
contrast, the energy dependence in nucleus-nucleus potential introduces various
kinds of barrier modifications and consequently reduces the fusion barrier which in
turn responsible for the predictions of larger sub-barrier fusion excitation
functions with reference to the simple one dimensional barrier penetration model.
The brief description of the method of calculation is given in Sec. 2. The results
are discussed in detail in Sec. 3 while the conclusions drawn are presented in Sec.
4.

2.Theoretical Formalism

2.1. Single channel description

The total fusion cross-section within the framework of partial wave analysis is defined as

Hill and Wheeler proposed an expression for tunneling probability (^{ 45 }. In parabolic approximations, the effective
interaction potential between the collision partners has been replaced by a
parabola and the tunneling probability through this barrier can be estimated by
using the following expression.

This parabolic approximation was further simplified by Wong using the following assumptions
for barrier position, barrier curvature and barrier height ^{ 43}.

with, *V _{B}* is the Coulomb barrier which corresponds to

*ℓ*= 0.

Using Eq. (2) to Eq. (5) into Eq. (1), the fusion cross-section can be written as

By incorporating the contributions from the infinite number of partial waves to fusion
process, one can change the summation over *ℓ* into integral with
respect to *ℓ* in Eq.
(6). By solving the integral one obtains the following expression of
the one dimensional Wong formula ^{ 43
}.

In earlier work, the EDWSP model ^{
4}^{,}^{
8}^{-}^{
9}^{,}^{
13}^{,}^{17}^{-}^{18}^{,}^{29}^{,}^{30}^{,}^{
31} successfully explores the fusion dynamics of various
heavy ion fusion reactions wherein the inelastic surface excitations and nucleon
(multi-nucleon) transfer channels are the most relevant intrinsic channels. This
work examines the fusion mechanism of different projectiles (^{44} wherein the static Woods-Saxon potential model has
been used to entertain the influence of nuclear structure degrees of freedom of
the fusing systems. In this sense, the optimum form of the static Woods-Saxon
potential is defined as

with *V*_{0}’ is depth and ‘*a*’ is diffuseness parameter of the Woods-Saxon potential. In EDWSP model, the depth of real part of the Woods-Saxon potential is defined as

where

and

are the isospin asymmetry of fusing pairs.

In collision dynamics, the large number of static and dynamical physical effects occurs in
the surface region of nuclear potential or tail region of the Coulomb barrier
and consequently changes the parameters of nuclear potential. For instance, the
variation of N/Z ratio of the colliding pairs, variation of surface energy and
nucleon densities during nuclear interactions, the channel coupling effects like
permanent deformation and low lying inelastic surface excitations of the
colliding systems, nucleon (multi-nucleon) transfer channels, neck formation,
dissipation of kinetic energy of the relative motion of the collision partners
to their internal structure degrees or other static and dynamical physical
effects generally occur in the tail region of the Coulomb barrier. These
physical effects induce modifications in the values of the potential parameters
and henceforth, results in the requirement of the different set of potential
parameters for the different type of the nuclear interactions. In fusion
dynamics, the diffuseness parameter of the static Woods-Saxon potential strongly
alters the energy dependence of low energy fusion cross-section at near and
below barrier energies and there is large number of experimental evidences
wherein an abnormally large value of the diffuseness parameter is needed to
explore the sub-barrier fusion data. The recently observed steep fall of fusion
excitation function data at deep sub-barrier energy region in many medium mass
nuclei, which is termed as fusion hindrance, can only be explained if one uses
an abnormally large diffuseness parameter ^{2}^{-}^{
3}. In addition, the nuclear structure effects present in
surface region produce fluctuation in the strength of nuclear potential and this
kind of fluctuation of nuclear strength is associated with the variation of the
diffuseness parameter. It is worth noting here that the different channel
coupling effects and non-local quantum effects which originate from the
underlying nucleon-nucleon interactions induce the energy dependence in
nucleus-nucleus potential. Therefore, to include all the above mentioned
physical effects, the energy dependence in the Woods-Saxon potential was
introduced via its diffuseness parameter. The energy dependent diffuseness
parameter is defined as

The range parameter (*r*_{0}) is an adjustable parameter and its value
is optimized in order to vary the diffuseness parameter required to address the
observed fusion dynamics of fusing system under consideration. In addition, the
value of the range parameter (*r*_{0}) strongly depends
on the nuclear struc- ture of the participating nuclei and the type of dominance
of nuclear structure degrees of freedom and hence the different set values of
the range parameter (*r*_{0}) are required to explain the
fusion dynamics of the different fusing systems. The potential parameters
(*r*_{0}, *a* and
*V*_{0}) of the EDWSP model are interrelated and the
change in one parameter automatically brings the corresponding adjustment in the
values of other parameters. In the present model, the value of
*V*_{0} depends on the surface energy and isospin
term of the interacting nuclei and the other two parameters
(*r*_{0} and *a*) are linked through
the Eq. (10). Therefore, the
values of the diffuseness parameter is directly related with the range parameter
(*r*_{0}) which in turn geometrically defines the
radii of the fusing systems (^{44}^{,}^{ 46}^{,}^{47}^{,}^{48 }. The values of the range
parameter used in the EDWSP model calculations for the chosen reactions are
consistent with the commonly adopted values of the range parameter
(*r*_{0} = 0.90 *fm* to
*r*
_{0} =1.35 *fm*), which are generally used in literature
within the context of the different theoretical models for different colliding
systems ^{ 1}^{-}^{3}^{,}^{5}^{-}^{6}^{,}^{ 48}^{-}^{49}.

2.2Coupled channel description

The coupled channel method that provides an adequate description of the fusion dynamics of
various heavy on fusion reactions at near and sub-barrier energies is the most
fundamental approach. In this method, the influences of intrinsic channels
associated with the fusing systems are properly incorporated ^{44}^{,}^{46}^{,}^{47}^{,}^{48}. In coupled channel approach,
the following set of the coupled channel equation is solved numerically.

where, *μ* is defined as the reduced mass of the colliding systems.
*E*_{cm} and *ε*_{n} represent the bombarding energy in the center of mass frame and the
excitation energy of the *n*^{th} channel respectively. *V*_{nm} , which consists of the Coulomb and nuclear components, is the matrix
elements of the coupling Hamiltonian. The realistic coupled channel calculations
are obtained within the view of the coupled channel code CCFULL ^{44}. In code CCFULL, the coupled
channel equations are solved numerically by imposing the no-Coriolis or rotating
frame approximation and ingoing wave boundary conditions (IWBC). The no-Coriolis
or rotating frame approximation has been entertained for reducing the number of
the coupled channel equations ^{44}^{,}^{46}^{,}^{47}^{,}^{48}. The ingoing wave boundary conditions (IWBC),
which are quite sensitive to the existence of the potential pocket of the
interaction fusion barrier, are well applicable for heavy ion collisions.
According to IWBC, there are only incoming waves at the minimum position of the
Coulomb pocket inside the barrier and there are only outgoing waves at infinity
for all channels except the entrance channel. By incorporating the influence of
the dominant intrinsic channels, the fusion cross-section can be written as

where, *P*_{J} (*E*) is the total
transmission coefficient corresponding to the angular momentum
*J*. The rotational coupling with a pure rotor and
vibrational coupling in the harmonic limit are considered in the coupled channel
approach. The rotational (

Where, *R*_{T} is defined as _{λ} is the deformation parameter and *n*-phonon state *m*-phonon state

and

respectively. The Coulomb coupling matrix elements are computed by the linear coupling approximation and are defined as

and

for the rotational and vibrational couplings respectively.

3.Results and Discussion

The present paper systematically analyzed the fusion dynamics of various heavy ion fusion reactions within the context of the static Woods-Saxon potential and energy dependent Woods-Saxon potential model along with Wong’s approximation. The influences of nuclear structure degrees of freedom of the fusing pairs are investigated using the coupled channel calculations. In this work, the fusing systems are selected in such a way that the different projectiles: doubly magic (^{+} and 3^{-} vibrational states of the colliding systems as required in the coupled channel calculations are listed in Table I. The barrier characteristics such as barrier height, barrier position and barrier curvature of various colliding pairs used in the EDWSP model calculations are listed in Table II. The potential parameter like range, depth and diffuseness parameters as used in the EDWSP model calculations for the chosen reactions are given in Table III.

The details of the coupled channel calculations for the fusion dynamics of

The energy dependence in the Woods-Saxon potential modifies the barrier characteristics of the
interaction barrier between the colliding systems which in turn results in a
spectrum of the variable fusion barriers as shown in Fig. 2. The spectrum of the energy dependent fusion barrier is shown for
the *a* = 0.95 *fm* for the *a* = 0.96 *fm* for the *FB* = 53.20 MeV and for *FB* = 66.20 MeV). This fusion barrier is
smaller than the Coulomb barrier by an amount of 1.22 MeV for

At above barrier energies, the fusion cross-sections are less sensitive towards nuclear structure as well as the channel coupling effects and consequently saturate at above barrier energies. This physical effect is properly modeled in the present approach wherein the magnitude of the diffuseness parameter gets saturated to its lowest value (*a* = 0.85 *fm*) at above barrier energies. At well above the barrier, the highest fusion barrier for the *FB* = 54.75 MeV and for the *FB* = 68.15 MeV). This fusion barrier is still smaller than the corresponding value of the Coulomb barrier as given in Table II. Therefore, the EDWSP model based calculation and the coupled channel calculation reasonably explored the fusion dynamics of the chosen reactions in quantitative as well as the qualitative way and henceforth, indicates that these theoretical methods produce analogous modifications in the barrier characteristics (barrier height, barrier position, barrier curvature) of the interaction fusion barrier between the colliding systems.

Morton *et al*. ^{50} suggested
the weak influence of the collective vibrations of the ^{+} vibrational state of the target enhances
the magnitude of the sub-barrier fusion excitation functions but unable to bring the
required order of magnitude of the sub-barrier fusion enhancement. This confirms the
possible influences of the more intrinsic channels. The couplings to one phonon
2^{+} and 3^{-} vibrational states of the target along with
their mutual couplings considerably improve the theoretical results. However, the
target degrees of freedom are not sufficient to properly explain the data and the
additions of the projectile excitations are necessarily required to obtain the
consistent fits with the experimental data. To overcome small discrepancies between
theoretical predictions and the below barrier fusion data, the projectile
excitations have been included in the coupled channel calculations. Therefore, the
couplings to one phonon 2^{+} vibrational state of the projectile, one
phonon 2^{+} and 3^{-} vibrational states of the target along with
their mutual couplings quantitatively address the observed fusion dynamics of ^{+} and 3^{-} vibrational states of the
target along with their mutual couplings recovers the discrepancies between the
theoretical calculations based on static Woods-Saxon potential along with one
dimensional Wong formula and the experimental fusion data as depicted from Fig. 3d. On the other hand, the energy dependence
in nucleus-nucleus potential lowers the fusion barrier between the colliding pairs
and hence reasonably addresses the sub-barrier fusion enhancement of

^{36}^{,}^{37}^{,}^{38}^{,}^{50}^{-}^{51}. The measurements on quadrupole moment and transition
probability (^{36}^{,}^{37}^{,}^{38}^{,}^{52}^{-}^{55}. Furthermore, several authors based on equivalent spheres
model have shown that the consideration of the oblate deformed shape for the
projectile (^{36}. The couplings to these intrinsic
channels strongly alter the energy dependence of the fusion cross-sections at below
barrier energies. All these odd-spin states are added as quadrature, which produces
dominant effects and hence entertained in coupled channel calculations. The
couplings to one phonon 2^{+} or one phonon 3^{-} vibrational state
of the target nucleus alone significantly enhances the magnitude of sub-barrier
fusion excitation functions with respect to no coupling calculations but unable to
recover the required order of magnitude of the observed fusion enhancement at
sub-barrier energies. This suggested that more intrinsic channels must be included
in the coupled channel calculations. The inclusion of the odd-spin states as a
quadrature in projectile and one phonon 2^{+} and 3^{-} vibrational
states along with their mutual couplings in target nucleus bring the observed fusion
enhancement of

The theoretical results of the fusion dynamics of ^{38} and hence significantly enhance the magnitude of the
sub-barrier fusion excitation functions. In the fusion of ^{+} or one phonon 3^{-}
vibrational state of the target nucleus alone is insufficient to account the
experimental data at sub-barrier energies. This demands the couplings to more
intrinsic channels for the complete description of the fusion data. The inclusion of
the one phonon 2^{+} and 3^{-} vibrational states in target as well
as odd-A spin states in projectile along with their mutual couplings reasonably
reproduces the observed fusion enhancement of

In Fig. 4, a comparison of the fusion excitation function
data of

The different kinds of channel coupling effects display their signature on the fusion
excitation functions at sub-barrier energies while such physical effects have
negligible influence on the above barrier fusion data. Therefore, the one
dimensional barrier penetration model should provide a good description of the
fusion data at above barrier energies. In this sense, a comparison of above barrier
fusion data and the predictions of the present model for the fusion of

4. Conclusions

The present work analyzed the role of collective excitations of the fusing systems on the fusion mechanism of