v_{eff}(r)=\frac{g_{\omega}^{2}}{4\pi}\frac{e^{-m_{\omega}r}}{r}+\frac{g_{\rho}^{2}}{4\pi}\frac{e^{-m_{\rho}r}}{r}-\frac{g_{\sigma}^{2}}{4\pi}\frac{e^{-m_{\sigma}r}}{r}$
(6.7)
$\displaystyle+\frac{g_{2}^{2}}{4\pi}{r}{e^{-2m_{\sigma}r}}+\frac{g_{3}^{2}}{4\pi}\frac{e^{-3m_{\sigma}r}}{r}-\frac{\xi^{2}}{4\pi}\frac{e^{-3m_{\omega}r}}{r}$
$\displaystyle+J_{00}(E)\delta(r).$

Here $m_{\sigma}$, $m_{\rho}$, $m_{\omega}$ are the masses of sigma, rho, and
omega mesons respectively, whereas the zero range pseudo potential $J_{00}(E)$
is given in Eqs.(6.5).

Table 6.1:  Model parameters for the Lagrangian FSUGold[90], NL3[49] and TM1[87]. | FSUGold | NL3 | TM1
---|---|---|---
$M$ (MeV) | 939 | 939 | 938
$m_{\sigma}$ (MeV) | 491.500 | 508.194 | 511.198
$m_{\omega}$ (MeV) | 782.500 | 782.501 | 783.000
$m_{\rho}$ (MeV) | 763.000 | 763.000 | 770.000
$g_{\sigma}$ | 10.592 | 10.2170 | 10.0290
$g_{\omega}$ | 14.298 | 12.8680 | 12.6140
$g_{\rho}$ | 11.767 | 4.4740 | 4.6320
$g_{2}$ (fm-1) | -4.2380 | -10.4310 | -7.2330
$g_{3}$ | -49.8050 | -28.8850 | 0.6180
$\xi$ | 2.0460 | - | 71.3070
$\Lambda$ | 0.0300 | - | -

Using NL3 parameters from Table 6.1, Eqs. (6.7) becomes[186]

$\displaystyle
v_{eff}(r)=10395\frac{e^{-3.97r}}{4r}+1257\frac{e^{-3.87r}}{4r}-6554\frac{e^{-2.58r}}{4r}$
(6.8)
$\displaystyle+6830{r}\frac{e^{-5.15r}}{4}+52384\frac{e^{-7.73r}}{4r}+J_{00}(E)\delta(r).$

The authors of Ref. [186] denoted this $NN$ interaction potential as
NR3Y(NL3). Further, putting parameter sets from TM1 (Table 6.1), one can
obtain $v_{eff}$ for NR3Y(TM1).

Since the DDM3Y folded potential described above do not include any spin-orbit
term, the spin-orbit potential from the Scheerbaum prescription[218] has been
coupled with the phenomenological complex potential depths $\lambda_{vso}$ and
$\lambda_{wso}$ . The spin-orbit potential is given by

$U^{so}_{n(p)}(r)=(\lambda_{vso}+i\lambda_{wso})\frac{1}{r}\frac{d}{dr}(\frac{2}{3}\rho_{p(n)}+\frac{1}{3}\rho_{n(p)}).$
(6.9)

The depths are functions of energy, given by

$\lambda_{vso}=130\exp(-0.013E)+40,$

and

$\lambda_{wso}=-0.2(E-20),$

where $E$ is in MeV. These standard values have been used in the present work.
However, in case of nonlinear $NN$ folded potentials from RMF (NR3Y(NL3),
NR3Y(TM1)), one need not require to add spin-orbit term from outside, as it is
contained within the RMF[186].

Finally reaction cross-sections and astrophysical reaction rates are
calculated in the Hauser-Feshbach formalism using the computer package
TALYS1.2[221].

###  6.3 Results and discussions

For simplicity, this section is divided in three subsections. In the Sec.
6.3.1, results from RMF calculations are given. We will concentrate on the
reaction cross-sections and astrophysical S factors in the Sec. 6.3.2.
Furthermore, results for reaction rates for astrophysically important nuclei
are provided. Sec. 6.3.3 is devoted to the effects of different $NN$
potentials in this mass region.

####  6.3.1 RMF calculations

Table 6.2:  Calculated binding energy per nucleon (B.E/A) [219] and charge radii $r_{ch}$ [220] for some selected $p$ nuclei compared with experimental values. | B.E./A(MeV) |  $r_{ch}$(fm)
---|---|---
| FSUGold | TM1 | NL3 | Exp | FSUGold | TM1 | NL3 | Exp
102Pd | 8.480 | 8.537 | 8.572 | 8.580 | 4.460 | 4.476 | 4.483 | 4.483
106Cd | 8.494 | 8.518 | 8.532 | 8.539 | 4.525 | 4.535 | 4.535 | 4.538
108Cd | 8.498 | 8.529 | 8.537 | 8.550 | 4.537 | 4.549 | 4.552 | 4.558
113In | 8.507 | 8.461 | 8.523 | 8.523 | 4.480 | 4.575 | 4.588 | 4.601
112Sn | 8.514 | 8.520 | 8.502 | 8.514 | 4.595 | 4.598 | 4.594 | 4.594
114Sn | 8.534 | 8.526 | 8.490 | 8.523 | 4.636 | 4.611 | 4.662 | 4.610
115Sn | 8.530 | 8.527 | 8.494 | 8.514 | 4.607 | 4.611 | 4.617 | 4.615
120Te | 8.461 | 8.461 | 8.460 | 8.477 | 4.682 | 4.688 | 4.735 | 4.704

In some earlier works [207, 208, 209, 210, 211], FSUGold has been proved to be
successful in reproducing experimentally obtained binding energy, charge
radius and charge density data in the A$\sim$55-100 region. Again in 1997, NL3
parameter set had been introduced by Lalazissis et.al[49] with a aim to
provide a better description not only for the properties of stable nuclei but
also for those far from the $\beta$ stability line and during last two
decades, this parameter set successfully reproduces binding energy, charge
radius etc. for various elements throughout the periodic table[49, 222]. In
order to confirm the applicability of RMF calculations in A$\sim$100-120
region, in Table 6.2, we compare nuclear binding energy per nucleon and charge
radii of $p$ nuclei in the concerned mass region with different sets of
parameters of RMF formalism with existing experimental data[219, 220]. We find
that, in most cases, our calculations with different sets of parameters match
quite well with the experimental data. In Fig. 6.1 charge density from our
calculations are compared with existing electron scattering data[203] for Pd
isotopes and here also, the agreement is well enough to confirm the
credibility of RMF models in this mass region.

####  6.3.2 Astrophysical S factor and reaction rates

In the present case, our calculations, being more microscopic, are more
restricting. In general, phenomenological models are usually fine-tuned for
nuclei near the stability valley, but not very successful in describing
elements near the proton and neutron rich regions. Microscopic models, in
contrary, can be extended to the drip line regions and therefore, this method
can be used to study the reaction rates of nuclei involved in $p$ process
nucleosynthesis network ($\sim$ 2000 nuclei are present in the total $p$
network). However, only a few number of stable $p$ nuclides are available in
nature that can be accessed by the experiment and therefore we are restricted
to those nuclei for the purpose of comparison.

Let us first take the case of DDM3Y folded potential. As a first test of the
optical model potential, we have calculated elastic proton scattering at low
energies where experimental data are available. As the elastic scattering
process involves the same incoming and outgoing channel for the optical model,
it is expected to provide the easiest way to constrain various parameters
involved in the calculation. Here we are mainly interested in the energy
region between 2-8 MeV as the astrophysical important Gamow window lies within
this energy range in the concerned mass region. However, scattering
experiments are very difficult at such low energies, because the cross
sections are extremely small, and hence no experimental data are available.
Therefore we have compared the cross sections from our calculations with the
lowest energy experimental data available in the literature.

Figure 6.2:  Experimental and calculated cross sections for elastic proton
scattering at 9.7 MeV proton energy.

In Fig. 6.2, we present the result of our calculation with DDM3Y folded
potential for 120Sn with available experimental data[223]. To fit the
experimental data, at first, the folded DDM3Y potential is multiplied by
factors 0.3 and 0.7 to obtain the real and imaginary parts of the optical
potential, respectively. However better fits for individual reactions can be
possible by varying different parameters. But if the present calculation has
to be extended to an unknown mass region, this approach is clearly inadequate.
Therefore, we have refrained from fitting individual reactions. A detailed
description of these normalizing constants are available in references[207,
208, 209, 210, 211].

Yet, the astrophysical reaction rates depend on the proper choice of the level
density and the E1 gamma strength. Therefore, we have calculated all of our
results with microscopic level densities in Hartree-Fock (HF) and Hartree-
Fock-Bogoliubov (HFB) methods, calculated for TALYS database by Goriley and
Hilaire[221, 224] on the basis of Hartree-Fock calculations[225] . We have
also compared our results using phenomenological level densities from a
constant-temperature Fermi gas model, a back-shifted Fermi gas model, and a
generalized super-fluid model from TALYS. All these model parameters can be
availed from TALYS database. We find that the cross sections are very
sensitive to the level density parameters. We therefore analyzed, in most of
the cases, the HF level densities fit the experimental data better in this
mass region. Again, for E1 gamma strength functions, results derived from HF +
BCS and HFB calculations, available in the TALYS database, are employed. In
this case also, the results for HF+BCS calculations describe the experimental
data reasonably well and we present our results for that approach only.

We now calculate some $(p,\gamma)$ cross sections relevant to $p$ nuclei in
A$\sim$100-120 region, where experimental data are available. At such low
energies, reaction cross-section varies rapidly making comparison between
theory and experiment rather difficult. Therefore the usual practice in low-
energy nuclear reaction is to compare another key observable, viz. the S
factor. It can be expressed as[208]

$S(E)=E\sigma(E)e^{2\pi\eta},$ (6.10)

where E is the energy in center of mass frame in keV, which factorises out the
pre-exponential low energy dependence of reaction cross-section $\sigma(E)$,
and $\eta$ indicates the Sommerfeld parameter with

$2\pi\eta=31.29Z_{p}Z_{t}\sqrt{\frac{\mu}{E}}.$ (6.11)

The factor exp$(2\pi\eta)$ is inversely proportional to the transmission
probability through the Coulomb barrier with zero angular momentum(s-wave) and
therefore removes exponential low energy dependence of $\sigma(E)$. Here
$\sigma(E)$ is in barn, $Z_{p}$ and $Z_{t}$ are the charge numbers of the
projectile and the target, respectively and $\mu$ is the reduced mass (in amu)
of the composite system. This S factor varies much slowly than reaction cross-
sections and for this reason, we calculate this quantity and compare it with
experimentally obtained values.

Figure 6.3:  S factors from two different microscopic potentials are compared
with experimental measurements for 102Pd. Here “Exp-1”is the experimental data
from reference[226], “Exp-2”from reference[227] and “DDM3Y”is for the
DDM3Y-folded potential.

Figure 6.4:  S factors extracted from theory compared with experimental
measurements for 106,108Cd. Here “Exp”is the experimental data from
reference[228].

Figure 6.5:  S factors extracted from theory compared with experimental
measurements for for 112Sn.

In Figs. 6.3-6.5 we present the results of some of our calculations with
folded DDM3Y potential for Pd, Cd and Sn isotopes, respectively, along with
the corresponding experimental results. The experimental values for 102Pd are
from Ref. [226](red point) and [227](blue cross), 106,108Cd from Gy. Gyürky
et.al[228] and 112Sn from Ref. [229].In case of 102Pd in Fig. 6.3, theoretical
prediction is in a good agreement, mainly in the low energy regime, with the
experimental data from Ref. [227] but under estimates the data obtained from
the Ref. [226]. In Ref. [226], an activation technique was used in which gamma
rays from decays of the reaction products were detected off-line by two hyper-
pure germanium detectors in a low background environment, whereas in Ref.
[227], cross-section measurements have been carried out at the cyclotron and
Van de Graaff accelerator by irradiation of thin sample layers and subsequent
counting of the induced activity. However, we can not comment on the
individual merits of these experiments.

In case of 106,108Cd (in Fig. 6.4), one can find that the agreement of theory
with experimental values are good enough, however there is a slight over
estimation for 108Cd in the low energy regime. In case of 112Sn in Fig. 6.5,
our calculation follows the experiment in a fairly good fashion.

The success of this microscopic optical potential (DDM3Y interaction folded
with FSUGold density) in reproducing S-factor data for the above $p$ nuclei
leads us to calculate reaction rates of some astrophysically important
reactions. In Fig. 6.6, we compare $(p,\gamma)$ reaction rates for some
important $p$ nuclei with NONSMOKER rates[206] obtained from statistical model
calculation with a global approach. Again in Fig. 6.7, reaction rates for
charge exchange reactions $(p,n)$ for some nuclei, however not astrophysically
significant enough, in this mass region are compared with existing NONSMOKER
calculations. One can see that the present calculation is very similar to the
NONSMOKER values in almost all cases. Therefore, it is expected that all the
results can also be reproduced with commonly used NONSMOKER rates.

Figure 6.6:  Astrophysical reaction rates for $(p,\gamma)$ reactions of some
important $p$ nuclei compared with NONSMOKER rates[206]. Here Green Continuous
line: Present calculation, Red Dotted line: NONSMOKER calculation.

Figure 6.7:  Astrophysical reaction rates for $(p,n)$ reactions compared with
NONSMOKER rates[206]. Here Green Continuous line: Present calculation, Red
Dotted line: NONSMOKER calculation.

Figure 6.8:  S factors extracted from our calculations compared with
experimental measurements for 120Te. Here “Exp”is the experimental data from
reference[230]. For other details, see the text.

In the remaining part, we mainly concentrate on the effects of optical
potentials obtained by folding nonlinear interactions from RMF (NR3Y). In Fig.
6.8, S factors for 120Te obtained from NR3Y(NL3) and NR3Y (TM1) potentials are
compared with the experimental data taken from Ref. [230]. The S- factor with
DDM3Y interaction folded with FSUGold density is also given for comparison.

One can see that our calculation with folded DDM3Y potential shows a very nice
agreement with experimental values throughout the energy range. In contrary,
in case of NR3Y(NL3) folded potential, there is a wide deviation of the theory
with experimental data after 6 MeV whereas the TM1 folded potential NR3Y(TM1)
shows a decrease in S factor value around 6 MeV energy unlike the NR3Y(NL3)
case.

The rapid drop of S factor values with increasing energy actually takes place
due to the increasing contribution of higher angular momentum channels
(l$>$0). Therefore, if the center of mass energy Ec.m. becomes larger than the
Coulomb barrier for a specific set of nucleon-nucleus reaction
(E${}_{c.m.}>$Ec), as a result the S factor will decrease rapidly with the
growth of energy (Ec.m.)[231]. In the next subsection, we illustrate this
physics in detail and show how this phenomenon is associated with different
form of potentials.

####  6.3.3 Optical potentials and effects of non-linearity

Figure 6.9: (Color online) Effective interaction potential for ${}^{120}Te$.

We now interpret the above results (for example, see Figs. 6.8) with help of
the microscopic potentials obtained from different $N-N$ interactions. In Fig.
6.9, the effective $NN$ interaction potentials (in MeV) are plotted with the
radius r (fm) for 120Te. The DDM3Y interaction, being dependent on the
density, is different for different elements of the periodic table, whereas in
contrary, other interactions remain unaltered for different elements. In Fig.
6.9 different forms of $NN$ interactions are given. We find that the curves
from DDM3Y and NR3Y(NL3) interactions generated from two different formalisms
show almost similar trend.

Figure 6.10: (Color online) Real central part of folded potentials and Coulomb
potential for 7 MeV proton(Lab) incident on 120Te

A graphical representation of microscopic potentials for 120Te after folding
the interactions is represented in Fig. 6.10. Here the real central part of
the optical potential is plotted with the radius. In the figure, one can see
that the DDM3Y folded potential provides an attractive potential whereas in
case of NR3Y(NL3) folded potential, the repulsive part overpowers the
attractive part, as well as the Coulomb part of the potential. As a result,
the resultant repulsive barrier becomes greater than the Coulomb barrier
almost upto a range for a nuclear reaction to occur. Therefore the
penetrability of the higher angular momentum channels get reduced and as a
obvious consequence, the desirable sharp drop in S factor (Fig. 6.8) has not
been achieved. In case of TM1 folded potential, we can see that the effective
contribution of the optical potential is attractive in nature similar to the
DDM3Y potential and therefore, the Coulomb energy serves as the only repulsive
barrier. As a result the penetration probability for higher angular momentum
channels becomes higher than that of the NR3Y(NL3) case. This reason is
replicated as a drop of S factor values at higher energies in Fig. 6.8. In
case of the imaginary part of the potential, the curves follow exactly the
similar trend as that of the real part, i.e., apart from NR3Y(NL3) potential,
rest of them gives attractive contribution. One can explain the above scenario
from the numerical value of the nonlinear coupling constant $g_{3}$ of TM1
parameter set, as given in Table 6.1, which is much less than that of the NL3
parameter set. Therefore it can be understood that with decreasing values of
the nonlinear coupling constants $g_{2}$ and $g_{3}$, the repulsive component
of the optical potential also gets reduced and one point is attained when only
the effect of Coulomb barrier remains as a dominating repulsive contributor
and we will get patterns like TM1, DDM3Y as shown in Fig. 6.10 and we can find
the expected drop of S factor at higher energies due to the opening of higher
angular momentum channels. So from the above observations we can comment that
there should be an upper cut-off for the coupling constants values of the
nonlinear components.

###  6.4 Summary

To summarize, cross section for low energy $(p,\gamma)$ reactions for a number
of $p$ nuclei in A$\sim$100-120 region have been calculated using microscopic
optical model potential with the Hauser Feshbach reaction code TALYS. Mainly,
microscopic potential is obtained by folding DDM3Y interaction with densities
from RMF approach. Astrophysical reaction rates for $(p,\gamma)$ and $(p,n)$
reactions are compared with standard NONSMOKER results. Finally, the effect of
microscopic optical potential obtained by folding nonlinear NR3Y(NL3) and
NR3Y(TM1) interactions with corresponding RMF densities are employed to fit
the experimental S factor data for 120Te. The reason of the deviation of
theoretical prediction with nonlinear NR3Y(NL3) potential from experiment at
higher energies has been discussed and finally we made a comment on magnitude
of the coupling terms of the nonlinear components that an upper cut-off for
$g_{2}$ and $g_{3}$ should be fixed to get proper repulsive component of the
$NN$ interaction.

##  Chapter 7 Effects of hyperon on both static and rotating neutron star

In last few chapters, we discussed the structure of finite nuclei by using
relativistic as well as non-relativistic formalisms. Along with the finite
nuclear system, an infinite nuclear system also plays very crucial role in
understanding the nucleon-nucleon interaction and hence the nuclear structure.
In this chapter, we have discussed the structure of the neutron star, which is
a perfect example of the infinite nuclear matter system. Particularly , we
study the effects of isovector-scalar ($\delta$)-meson on neutron and hyperon
stars. Influence of $\delta$-meson on both static and rotating stars are
discussed. The $\delta$-meson in a neutron star consisting of protons,
neutrons, and electrons makes the equations of state stiffer at a higher
density and consequently increases the maximum mass of the star. But induction
of $\delta$-meson in the hyperon star decreases the maximum mass. This is due
to the early evolution of hyperons in the presence of $\delta$-meson.

###  7.1 Introduction

Neutron star is a venerable candidate to discuss the physics at high density.
We can not create such a high density in a terrestrial laboratory, so a
neutron star is and the only object, which can provide much information on
high-density nature of the matter[232, 233]. But it is not an easy task to
deal with the neutron star for it’s complex nature, as all the four
fundamental forces (strong, weak, gravitational and electromagnetic) are
active. High gravitational field makes mandatory to use general theory of
relativity for the study of neutron star structure. Equations of states (EOS)
are the sole ingredient that must be supplied to the equation of stellar
structure, Tolman-Oppenheimer-Volkoff (TOV) equation, whose output is the
mass-radius profile of the dense neutron star. In this case, the nuclear EOS
plays an intimate role in deciding the mass-radius of a neutron star. Its
indispensable importance attracts the attention of physicists to have an
anatomy of the interactions Lagrangian. As the name suggests, a neutron star
is not completely made up neutrons, a small fraction of protons and electrons
are also present, which is the consequence of the $\beta-$equilibrium and
charge neutrality condition[234]. Also, the presence of exotic degrees of
freedom like hyperons and kaons can not be ignored in such high dense matter.
It is one among the most asymmetric and dense nuclear object in nature.

From last three decades [235, 236], the relativistic mean field (RMF)
approximation, generalized by Walecka [8] and later on developed by Boguta and
Bodmer [99] is one amongst the most reliable theory to deal with the infinite
nuclear matter and finite nuclei. The original RMF formalism starts with an
effective Lagrangian, whose degrees of freedom are nucleons, $\sigma-,$
$\omega-$, $\rho-$ and $\pi-$mesons. To reproduce proper experimental
observable, it is extended to the self-interaction of $\sigma-$meson.
Recently, all other self- and crossed interactions including the baryon octet
are also introduced keeping in view the extra-ordinary condition of the
system, such as highly asymmetric system or extremely high- density medium
[237]. Since the RMF formalism is an effective nucleons-mesons model, the
coupling constants for both nucleon-meson and hyperon-meson are fitted to
reproduce the properties of selected nuclei and infinite nuclear matter
properties [8, 99, 238, 239]. In this case, it is improper to use the
parameters obtained from the free nucleon-nucleon scattering data. The
parameters, with proper relativistic kinematics and with the mesons and their
properties are already known or fixed from the properties of a small number of
finite nuclei, the method gives excellent results not only for spherical
nuclei but also well-known deformed cases. The same force parametrization can
be used both for $\beta-$stable and $\beta-$unstable nuclei through-out the
periodic table [240, 241, 193, 49].

The importance of the self- and crossed- interactions are significant for some
specific properties of nuclei/nuclear-matter in certain conditions. For
example, self-interaction of $\sigma$-meson takes care the reduction of
nuclear matter incompressibility $K_{\infty}$ from an unacceptable high value
of $K_{\infty}\sim 600$ MeV to a reasonable number of $\sim 270$ MeV [99,
141], while the self-interaction of vector meson $\omega$ soften the equations
of states [193, 90]. Thus, it is imperative to include all the mesons and
their possible interactions with nucleons and hyperons, self- and crossed
terms in the effective Lagrangian density. However, it is not necessary to do
so, because of the symmetry reason and their heavy masses [242]. For example,
to keep the spin-isospin and parity symmetry in the ground state, the
contribution of $\pi-$meson is ignored [243] and also the effect of heavier
mesons are neglected for their negligible contribution. Taking into this
argument, in many versions of the RMF formalism, the inclusion of isovector-
scalar ($\delta$) meson is neglected due to its small contribution. But
recently it is seen [244, 245, 246, 247, 248, 249] that the endowment of the
$\delta$-meson goes on increasing with density and asymmetry of the nuclear
system. Thus, it will be impossible for us to justify the abandon of
$\delta-$meson both conceptually and practically, while considering the high
asymmetry and dense nuclear systems, like the neutron star and relativistic
heavy ion collision. Recent observation of neutron star like PSR J1614-2230
with mass of (1.97$\pm$0.04)$M_{\odot}$ [201] and the PSR J0348+0432 with mass
of (2.01$\pm$0.04)$M_{\odot}$ [250] re-open the challenge in the dense matter
physics. The heavy mass of PSR J0348+0432 (M=2.01$\pm$0.04$M_{\odot}$) forces
the nuclear theorists to re-think the composition and interaction inside the
neutron star. Therefore, it is important to establish the effects of the
$\delta$-meson and all possible interactions of other mesons for such compact
and asymmetry system.

The paper is organized as follows: In Sec. 7.2, we have outlined a brief
theoretical formalism. The necessary steps of the RMF model and the inclusion
of $\delta-$meson is explained. The results and discussions are devoted in
Sec. 7.3. Here, we have attempted to explain the effects of $\delta$-meson on
the nuclear matter system like hyperon and proton-neutron stars. This analysis
is done for both static and rotating neutron and neutron-hyperon stars. In
this calculations, the E-RMF Lagrangian (G2 parameter set) is used to take
care of all possible self- and crossed interactions [251]. On top of the G2
Lagrangian, the $\delta-$meson interaction is added to take care of the
isovector channel. The concluding remarks are given in section 7.4.

###  7.2 Theoretical formalism

From last one decade a lot of work have been done to emphasize the role of
$\delta-$meson on both finite and infinite nuclear matter [252, 253, 254,
255]. It is seen that the contribution of $\delta$-meson to the symmetry
energy is negative [256]. To fix the symmetry energy around the empirical
value ($\sim$30 MeV ) we need a large coupling constant of the $\rho-$meson
($g_{\rho}$) value in the absence of the $g_{\delta}$. The proton and neutron
effective masses split due to inclusion of $\delta$-meson and consequently it
affects the transport properties of neutron star[246]. The addition of
$\delta$-meson not only modify the property of infinite nuclear matter, but
also enhances the spin-orbit splitting in the finite nuclei[252]. A lot of
mysteries are present in the effects of $\delta$-meson till date. The
motivation of the present chapter is to study such information. It is to be
noted that both the $\rho-$ and $\delta-$mesons correspond to the isospin
asymmetry, and a careful precaution is essential while fixing the
$\delta$-meson coupling in the interaction.

The effective field theory and naturalness of the parameter are described in
[251, 257, 258, 259, 260]. The Lagrangian is consistent with underlying
symmetries of the QCD. The G2 parameter is motivated by E-RMF theory. The
terms of the Lagrangian are taken into account up to $4^{th}$ order in meson-
baryon coupling. For the study of isovector channel, we have introduced the
isovector-scalar $\delta$-meson. The baryon-meson interaction is given by
[237]:

$\displaystyle{\cal L}$ $\displaystyle=$
$\displaystyle\sum_{B}\overline{\psi}_{B}\left(i\gamma^{\mu}D_{\mu}-m_{B}+g_{\sigma
B}\sigma+g_{\delta
B}\delta.\tau\right)\psi_{B}+\frac{1}{2}\partial_{\mu}\sigma\partial_{\mu}\sigma$
(7.1) $\displaystyle-
m_{\sigma}^{2}\sigma^{2}\left(\frac{1}{2}+\frac{\kappa_{3}}{3!}\frac{g_{\sigma}\sigma}{m_{B}}+\frac{\kappa_{4}}{4!}\frac{g_{\sigma}^{2}\sigma^{2}}{m_{B}^{2}}\right)-\frac{1}{4}\Omega_{\mu\nu}\Omega^{\mu\nu}$
$\displaystyle+\frac{1}{2}m_{\omega}^{2}\omega_{\mu}\omega^{\mu}\left(1+\eta_{1}\frac{g_{\sigma}\sigma}{m_{B}}+\frac{\eta_{2}}{2}\frac{g_{\sigma}^{2}\sigma^{2}}{m_{B}^{2}}\right)-\frac{1}{4}R_{\mu\nu}^{a}R^{\mu\nu
a}$
$\displaystyle+\frac{1}{2}m_{\rho}^{2}\rho_{\mu}^{a}\rho^{a\mu}\left(1+\eta_{\rho}\frac{g_{\sigma}\sigma}{m_{B}}\right)+\frac{1}{2}\partial_{\mu}\delta.\partial_{\mu}\delta-
m_{\delta}^{2}\delta^{2}+\frac{1}{4!}\zeta_{0}\left(g_{\omega}\omega_{\mu}\omega^{\mu}\right)^{2}$
$\displaystyle+\sum_{l}\overline{\psi}_{l}\left(i\gamma^{\mu}\partial_{\mu}-m_{l}\right)\psi_{l}.$

The co-variant derivative $D_{\mu}$ is defined as:

$\displaystyle
D_{\mu}=\partial_{\mu}+ig_{\omega}\omega_{\mu}+ig_{\rho}I_{3}\tau^{a}\rho_{\mu}^{a},$
(7.2)

where $R_{\mu\nu}^{a}$ and $\Omega_{\mu\nu}$ are field tensors and defined as
follow

$\displaystyle
R_{\mu\nu}^{a}=\partial_{\mu}\rho_{\nu}^{a}-\partial_{\nu}\rho_{\mu}^{a}+g_{\rho}\epsilon_{abc}\rho_{\mu}^{b}\rho_{\nu}^{c},$
(7.3)
$\displaystyle\Omega_{\mu\nu}=\partial_{\mu}\omega_{\nu}-\partial_{\nu}\omega_{\mu}.$
(7.4)

Here, $\sigma$, $\omega$ , $\rho$ and $\delta$ are the sigma, omega, rho and
delta meson fields, respectively and in real calculation, we ignore the non-
abelian term from the $\rho-$field. All symbols are carrying their own usual
meaning [237, 244].

The Lagrangian equation for different mesons are given by [237]:

$\displaystyle
m_{\sigma}^{2}\left(\sigma_{0}+\frac{g_{\sigma}\kappa_{3}\sigma_{0}}{2m_{B}}+\frac{\kappa_{4}g_{\sigma}^{2}\sigma_{0}^{2}}{6m_{B}^{2}}\right)\sigma_{0}-\frac{1}{2}m_{\rho}^{2}\eta_{\rho}\frac{g_{\sigma}\rho_{03}^{2}}{m_{B}}$
$\displaystyle-\frac{1}{2}m_{\omega}^{2}\left(\eta_{1}\frac{g_{\sigma}}{m_{B}}+\eta_{2}\frac{g_{\sigma}^{2}\sigma_{0}}{m_{B}^{2}}\right)\omega_{0}^{2}=\sum
g_{\sigma}\rho^{s}_{B},$ (7.5) $\displaystyle
m_{\omega}^{2}\left(1+\eta_{1}\frac{g_{\sigma}\sigma_{0}}{m_{B}}+\frac{\eta_{2}}{2}\frac{g_{\sigma}^{2}\sigma_{0}^{2}}{m_{B}^{2}}\right)\omega_{0}+\frac{1}{6}\zeta_{0}g_{\omega}^{2}\omega_{0}^{3}=\sum
g_{\omega}\rho_{B},$ (7.6) $\displaystyle
m_{\rho}^{2}\left(1+\eta_{\rho}\frac{g_{\sigma}\sigma_{0}}{m_{B}}\right)=\frac{1}{2}\sum
g_{\rho}\rho_{B3},$ (7.7) $\displaystyle m_{\delta}^{2}{\delta^{3}}$
$\displaystyle=$ $\displaystyle g_{\delta}^{2}\rho^{s}_{3B},$ (7.8)

with $\rho^{s}_{3B}=\rho^{s}_{p}-\rho^{s}_{n}$, $\rho^{s}_{p}$ and
$\rho^{s}_{n}$ are scalar densities for the proton and neutron, respectively.
The total scalar density is expressed as the sum of the proton and neutron
densities $\rho^{s}_{B}=\rho^{s}_{p}+\rho^{s}_{n}$, which is given by

$\displaystyle\rho^{s}_{i}=\frac{2}{(2\pi)^{3}}\int_{0}^{k_{i}}\frac{M_{i}^{*}d^{3}k}{E_{i}^{*}},i=p,n$
(7.9)

and the vector (baryon) density

$\displaystyle\rho_{B}=\frac{2}{(2\pi)^{3}}\int_{0}^{k_{i}}d^{3}k,$ (7.10)

where, $E_{i}^{*}=(k_{i}^{2}+M_{i}^{*2})^{1/2}$ is the effective energy,
$k_{i}$ is the Fermi momentum of the baryons. $M_{p}^{*}$ and $M_{n}^{*}$ are
the proton and neutron effective masses written as

$\displaystyle M_{p}^{*}=M_{p}-g_{\sigma}\sigma_{0}-g_{\delta}{\delta^{3}}$
(7.11) $\displaystyle
M_{n}^{*}=M_{n}-g_{\sigma}\sigma_{0}+g_{\delta}{\delta^{3}},$ (7.12)

which are solved self-consistently. $I_{3}$ is the third component of isospin
projection and $B$ stands for baryon octet. The energy and pressure density
depends on the effective mass $M_{B}^{*}$ of the system, which first needed to
solve these self-consistent equations and obtained the fields for mesons.
Using the Einstein’s energy-momentum tensor, the total energy and pressure
density are given as [237]:

$\displaystyle\cal{E}$ $\displaystyle=$
$\displaystyle\sum_{B}\frac{2}{(2\pi)^{3}}\int_{0}^{k_{B}}d^{3}kE_{B}^{*}(k)+\frac{1}{8}\zeta_{0}g_{\omega}^{2}\omega_{0}^{4}+m_{\sigma}^{2}\sigma_{0}^{2}\left(\frac{1}{2}+\frac{\kappa_{3}}{3!}\frac{g_{\sigma}\sigma_{0}}{m_{B}}+\frac{\kappa_{4}}{4!}\frac{g_{\sigma}^{2}\sigma_{0}^{2}}{m_{B}^{2}}\right)$
(7.13)
$\displaystyle+\frac{1}{2}m_{\omega}^{2}\omega_{0}^{2}\left(1+\eta_{1}\frac{g_{\sigma}\sigma_{0}}{m_{B}}+\frac{\eta_{2}}{2}\frac{g_{\sigma}^{2}\sigma_{0}^{2}}{m_{B}^{2}}\right)+\frac{1}{2}m_{\rho}^{2}\rho_{03}^{2}\left(1+\eta_{\rho}\frac{g_{\sigma}\sigma_{0}}{m_{B}}\right)$
$\displaystyle+\frac{1}{2}\frac{m_{\delta}^{2}}{g_{\delta}^{2}}(\delta^{3})^{2}+\sum_{l}\varepsilon_{l},$

and

$\displaystyle\cal{P}$ $\displaystyle=$
$\displaystyle\sum_{B}\frac{2}{3(2\pi)^{3}}\int_{0}^{k_{B}}d^{3}kE_{B}^{*}(k)+\frac{1}{8}\zeta_{0}g_{\omega}^{2}\omega_{0}^{4}-m_{\sigma}^{2}\sigma_{0}^{2}\left(\frac{1}{2}+\frac{\kappa_{3}}{3!}\frac{g_{\sigma}\sigma_{0}}{m_{B}}+\frac{\kappa_{4}}{4!}\frac{g_{\sigma}^{2}\sigma_{0}^{2}}{m_{B}^{2}}\right)$
(7.14)
$\displaystyle+\frac{1}{2}m_{\omega}^{2}\omega_{0}^{2}\left(1+\eta_{1}\frac{g_{\sigma}\sigma_{0}}{m_{B}}+\frac{\eta_{2}}{2}\frac{g_{\sigma}^{2}\sigma_{0}^{2}}{m_{B}^{2}}\right)+\frac{1}{2}m_{\rho}^{2}\rho_{03}^{2}\left(1+\eta_{\rho}\frac{g_{\sigma}\sigma_{0}}{m_{B}}\right)$
$\displaystyle-\frac{1}{2}\frac{m_{\delta}^{2}}{g_{\delta}^{2}}(\delta^{3})^{2}+\sum_{l}P_{l},$

where $P_{l}$ and $\varepsilon_{l}$ are lepton’s pressure and energy density,
respectively.

###  7.3 Results and discussions

Before going to the discussions of our results, we give a brief description of
the parameter fitting procedure for gρ and gδ. As it is commonly known, the
symmetric energy $E_{sym}$, is an important quantity to select the equation of
states. This value of $E_{sym}$ determines the structure of both static and
rotating neutron stars. On the other hand, an arbitrary combination of gρ and
gδ with a fixed value of Es can affect the ground state properties of finite
nuclei. Thus, to have a clear picture on the effect of gδ on hyperon star
structure, we have chosen two different prescriptions for the selection of gδ
in our present calculations. (1) In the first method, we have constructed
various sets of gρ and gδ keeping Es fixed. Here, all the other parameters of
G2 set are remained unchanged. The G2 set and the combination of gρ and gδ are
displayed in Table 7.1. (2) In the second procedure, we have chosen the gρ, gδ
pairs keeping the binding energy constant (experimental binding energy) for
finite nuclei. The values of these gρ and gδ are given in Table 7.3 with other
properties of infinite nuclear matter. It is worthy to re-emphasized here that
we are not looking for a full-fledged parameter set including the
$\delta-$meson coupling, but our aim in this paper is to study the effects of
$\delta-$meson coupling on hyperon star and the production of baryon octet.
Therefore, after splitting the gρ coupling constant into two parts (gρ, gδ)
using the first prescription, the results on hyperon star along with the
neutron star structures both for static and rotating cases under
$\beta-$equilibrium condition are discussed in the subsequent subsections
7.3.2, 7.3.3, 7.3.4, 7.3.5 and 7.3.6. In Sec. IV, we follow the second
procedure to get the (gρ, gδ) pairs and applied these to some selective cases.

####  7.3.1 Parametrization of $g_{\rho}$ and $g_{\delta}$ with constant
symmetry energy

It is important to fix $g_{\delta}$ value to see the effects of the
$\delta$-meson. The isovector channels in RMF theory come to exist through
both the $\rho-$ and $\delta-$mesons couplings. While considering the effects
of the $\delta$-meson, we have to take the $\rho$-meson into account. Since
both the isovector channels are related to isospin, one can not optimize the
$g_{\delta}$ coupling independently. Here, we have followed a more reliable
procedure by fixing the symmetry energy $E_{sym}$ with adjusting
simultaneously different values of $g_{\rho}$ and $g_{\delta}$ [246]. In
general, for most of the non-relativistic formalism, the symmetry energy
$E_{sym}$ is around 30-33 MeV. However, in some specific parametrization like
GS4, $E_{sym}=$12.83 MeV and for PRC45 set it is 51.01 MeV [261, 262]. On the
other hand, in non-linear, density-dependent and point-coupling relativistic
mean field forces, the $E_{sym}$ varies from 26.1 to 44.0 MeV. Here we have
used the well known G2 parametrization, which has a moderate symmetry energy
$E_{sym}=36.4$ MeV. It is to be noted that the symmetry energy plays a crucial
role both in finite nuclei and in the equation of state, which include the
neutron distribution radius in the nucleus and the mass and radius of a
neutron star, respectively. For a smaller value of $E_{sym}$, both the
relativistic and non-relativistic forces predict a smaller neutron star mass
contrary to the recent observation of about 2M⊙. A detail variation of
symmetry energies for Skyrme effective interaction and non-linear relativistic
mean field formalism is available in Refs. [261, 262]. Recently, a large
number of papers have been devoted to $E_{sym}$ for a definite value, but till
it is under active discussions.

As it is mentioned earlier, we have added $g_{\delta}$ on top of the G2
parameter set. Thus, the symmetry energy of G2 parameter $E_{sym}=36.4$ MeV is
kept constant at the time of re-shuffling $g_{\rho}$ and $g_{\delta}$. It is
to be noted that, we do not want to change the value of $E_{sym}$ of the
original G2 parameter set with the addition of $\delta$-meson. The G2
parameters and the $g_{\delta}$ and $g_{\delta}$ combinations are displayed in
Table 7.1. The nuclear matter properties are also listed in the table (middle
panel).

Figure 7.1: Variation of $g_{\rho}$ and $g_{\delta}$ at a constant value of
symmetry energy $E_{sym}=36.4$ MeV for both nuclear and neutron matter. Table
7.1: The parameters for G2 set are in the upper panel of the Table. The
nuclear matter saturation properties are in the middle panel and various
$g_{\rho}$ and $g_{\delta}$ combinations are in the lower panel, keeping
symmetry energy Esym = 36.4 MeV fixed.

$m_{n}$ = 939.0 MeV |  $m_{\sigma}$ = 520.206 MeV |  $m_{\omega}$ = 782.0 MeV |  $m_{\rho}$ = 770.0 MeV |  $m_{\delta}$ = 980.0 MeV |  $\Lambda$= 0.0 |  $\zeta_{0}$ = 2.642 |  $\eta_{\rho}$ = 0.39 |
---|---|---|---|---|---|---|---|---
$g_{\sigma}$ = 10.5088 |  $g_{\omega}$ = 12.7864 |  $g_{\rho}$ = 9.5108 |  $g_{\delta}$ = 0.0 |  $k_{3}$ = 3.2376 |  $k_{4}$ = 0.6939 |  $\eta_{1}$ = 0.65 |  $\eta_{2}$ = 0.11 |
$\rho_{0}$ = 0.153$fm^{-3}$ | E/A = -16.07 MeV |  $K_{\infty}$ = 215 MeV |  $E_{sym}$ = 36.4 MeV |  $m^{*}_{n}/m_{n}$ = 0.664 |  |  |  |
$(g_{\rho},g_{\delta})$ | (9.510, 0.0) | (9.612, 1.0) | (9.973, 2.0) | (10.550, 3.0) | (11.307, 4.0) | (12.212, 5.0) | (13.234, 6.0) | (14.349, 7.0)

Table 7.2: Binding energy (MeV) and charge radius (fm) are calculated with
various combination of $g_{\rho}$ and $g_{\delta}$ in G2+$\delta$. The results
are compared with experimental data [263].

($g_{\rho}$,$g_{\delta}$) | (9.510, 0.0) | (9.612, 1.0) | (9.973, 2.0) | (10.550, 3.0) | (11.307, 4.0) | (12.212, 5.0) | (13.234, 6.0) | (14.349, 7.0) |
---|---|---|---|---|---|---|---|---|---
Nucleus | Theory | Expt.
16O (BE) | 127.2 | 127.2 | 127.2 | 127.2 | 127.2 | 127.2 | 127.2 | 127.2 | 127.6
rch | 2.718 | 2.718 | 2.718 | 2.718 | 2.718 | 2.717 | 2.717 | 2.716 | 2.699
40 Ca (BE) | 341.1 | 341.1 | 341.1 | 341.1 | 341.1 | 341.1 | 341.1 | 341.1 | 342.0
rch | 3.453 | 3.453 | 3.453 | 3.453 | 3.452 | 3.451 | 3.450 | 3.449 | 3.4776
48Ca (BE) | 416.0 | 415.8 | 415.2 | 414.1 | 412.6 | 410.7 | 408.4 | 405.7 | 416.0
rch | 3.440 | 3.439 | 3.438 | 3.437 | 3.435 | 3.432 | 3.430 | 3.427 | 3.477
56Ni (BE) | 480.4 | 480.3 | 480.3 | 480.3 | 480.3 | 480.3 | 480.3 | 480.2 | 484.0
rch | 3.730 | 3.730 | 3.730 | 3.730 | 3.730 | 3.730 | 3.730 | 3.724 |
58Ni (BE) | 497.2 | 497.2 | 497.1 | 497.0 | 496.9 | 496.7 | 496.5 | 496.3 | 506.5
rch | 3.765 | 3.765 | 3.763 | 3.762 | 3.761 | 3.758 | 3.756 | 3.753 | 3.775
90Zr (BE) | 781.6 | 781.2 | 780.6 | 779.4 | 777.9 | 775.9 | 773.6 | 770.8 | 783.9
rch | 4.238 | 4.238 | 4.237 | 4.235 | 4.233 | 4.230 | 4.228 | 4.225 | 4.269
116Sn (BE) | 981.2 | 980.7 | 979.4 | 977.2 | 974.1 | 970.2 | 965.6 | 960.3 | 988.7
rch | 4.604 | 4.603 | 4.601 | 4.598 | 4.594 | 4.589 | 4.584 | 4.579 | 4.625
118Sn (BE) | 997.6 | 997.1 | 995.4 | 992.7 | 989.0 | 984.3 | 978.7 | 972.2 | 1004.9
rch | 4.620 | 4.619 | 4.617 | 4.613 | 4.610 | 4.604 | 4.599 | 4.594 | 4.639
120Sn (BE) | 1013.9 | 1013.2 | 1011.2 | 1008.0 | 1003.5 | 998.0 | 991.3 | 983.8 | 1020.5
rch | 4.627 | 4.626 | 4.624 | 4.620 | 4.616 | 4.610 | 4.605 | 4.600 | 4.652
208Pb (BE) | 1633.3 | 1631.4 | 1625.7 | 1616.2 | 1603.0 | 1586.4 | 1566.4 | 1543.5 | 1636.4
rch | 5.499 | 5.498 | 5.497 | 5.494 | 5.492 | 5.489 | 5.487 | 5.485 | 5.501

Table 7.3:  Mass and radius of the neutron star are calculated at different
values of $g_{\rho}$ and $g_{\delta}$ keeping binding energy of 208Pb
(1633.296 MeV) constant. The calculated results of $E_{sym}$, $L_{sym}$ and
$K_{sym}$ are for nuclear matter at different combinations of ($g_{\rho}$,
$g_{\delta}$) pairs.

$(g_{\rho},g_{\delta})$ | $\frac{M}{M_{\odot}}$ | Radius (Km) | Esym (MeV) | Lsym (MeV) | Ksym (MeV)
---|---|---|---|---|---
(9.510, 0.0) | 1.980 | 11.230 | 36.4 | 101.0 | -7.58
(9.588, 1.746) | 1.993 | 11.246 | 35.3 | 98.3 | -0.60
(9.896, 3.543) | 1.997 | 11.262 | 31.7 | 90.2 | 20.90
(10.518, 5.742) | 2.004 | 11.294 | 23.8 | 72.5 | 67.07
(11.774, 8.834) | 2.018 | 11.510 | 6.35 | 30.6 | 169.03

For a particular value of $E_{sym}=36.4$ MeV, the variation of $g_{\rho}$ and
$g_{\delta}$ are plotted in Fig. 7.1. From Fig. 7.1, it is clear that as the
$g_{\delta}$ increases the $g_{\rho}$ value also increases, almost linearly,
to fix the symmetry energy unchanged. This implies that $\rho-$ and
$\delta$-mesons have opposite effect on $E_{sym}$ contribution, i.e., the
$\delta$-meson has negative contribution of the symmetry energy contrary to
the positive contribution of $\rho$-meson.

We feel that it is instructive to check the finite nuclear properties with
these combinations of $g_{\rho}$ and $g_{\delta}$. We have tabulated the
binding energy and charge radius of some spherical nucleus in Table 7.2. From
the table, it is clear that binding energy for asymmetric nucleus goes on
decreasing with increasing $\delta-$meson and decreasing $\rho-$meson
couplings. However, it is well understood that the scalar $\delta-$meson gives
a positive contribution to the binding energy. Thus, the binding energy of
asymmetric nuclei should go on increasing with gδ contrary to the observation
seen in Table 7.2. This happens, because of the simultaneous change of
($g_{\rho}$, $g_{\delta}$) pair to keep the constant symmetry energy, i.e., gρ
is decreasing and $g_{\delta}$ is increasing. As a result, the contribution of
$\rho-$ meson, which is negative to the binding energy dominates over the
$\delta-$meson effect on binding energy. But in case of symmetric nucleus,
like 16O etc. the effects of both $\rho-$ and $\delta-$mesons are absent due
to iso-spin symmetry. A further inspection of Table 7.2 reveals a slight
change in binding energy and charge radius even for symmetric nuclei because
of the slight different in density distribution of protons and neutrons,
although it is small.

####  7.3.2 Fields of $\sigma,\omega,\rho$ and $\delta$ mesons

The fields of the meson play a crucial role to construct the nuclear
potential, which is the deciding factor for all type of calculations in the
relativistic mean field model. In Fig. 7.2, we have plotted various meson
fields included in the present calculations, such as $\sigma$, $\omega$,
$\rho$ and $\delta$ with $g_{\delta}$ on top of G2 parameter set
$(G2+g_{\delta})$.

Figure 7.2: Various meson fields are obtained from the RMF theory with
$G2+g_{\delta}$ and NL3 parameter sets. The $\sigma-$ meson field $V_{\sigma}$
and $\omega-$meson field $V_{\omega}$ from $G2+g_{\delta}$ calculations are
compared with the results of DBHF theory [264] and NL3 set.

It is obvious that $V_{\sigma}$ and $V_{\omega}$ are opposite to each other,
which is also reflected in the figure. This means, the positive value of
$V_{\omega}$ gives a strong repulsion, which is compensated by the strongly
attractive potential of the $\sigma-$meson field $V_{\sigma}$. The nature of
the curves for $V_{\sigma}$ and $V_{\omega}$ are almost similar except the
sign. The magnitude of $V_{\sigma}$ and $V_{\omega}$ looks almost equal.
However, in real (it is not clearly visible in the curve, because of the
scale), the value of $V_{\sigma}$ is slightly larger than $V_{\omega}$, which
keeps the overall nuclear potential strongly attractive. The attractive
$V_{\sigma}$ and repulsive $V_{\omega}$ potentials combinely give the
saturation properties of the nuclear force. It is worthy to mention that the
contributions of self-interaction terms are taken care both in $V_{\sigma}$
and $V_{\omega}$, which are the key quantities to solve the Coester band
problem [186] and the explanation of quark-gluon-plasma (QGP) formation within
the relativistic mean field formalism [265]. The self-interaction of the
$\sigma-$meson gives a repulsive force at long range part of the nuclear
potential, which is equivalent to the 3-body interaction and responsible for
the saturation properties of nuclear force. The calculated results of
$V_{\sigma}$ and $V_{\omega}$ are compared with the results obtained from DBHF
theory with Bonn-A potential[264] and NL3 [49] force.

Fig. 7.2 clearly shows that in the low density region (density $\rho_{B}$
$\sim 2\rho_{0}$) both RMF and DBHF theories well matched. But as it increases
beyond density $\rho_{B}$ $\sim 2\rho_{0}$ ($\rho_{0}$ is the nuclear
saturation density) both the calculations deviate from each other. The
possible reason may be the fitting procedure of parameters in Bonn-A potential
is up to $2-3$ times of saturation density $\rho_{0}$, beyond that the DBHF
data are simple extrapolation of the DBHF theory. Again, the $V_{\omega}$ and
$V_{\sigma}$ fields of NL3 are very different from $G2+\delta$ results. The
$V_{\omega}$ for NL3 follows a linear path contrary to the results of
$G2+\delta$ and Bonn-A. This could be due to the absence of self- and crossed
couplings in NL3 set. The contribution of both $\rho-$ and $\delta-$ mesons
correspond to the isovector channel. The $\delta-$meson gives different
effective masses for proton and neutron, because of their opposite iso-spin of
the third component. The nuclear potential generated by the $\rho-$ and
$\delta-$mesons are also shown in Fig 7.2. We noticed that although their
contributions are small, but non-negligible. These non-zero values of
$V_{\rho}$ and $V_{\delta}$ to the nuclear potential has a significant
consequence, mostly in compact dense object like neutron or hyperon stars,
which will be discussed later in this paper.

####  7.3.3 Energy per particle and pressure density

The energy and pressure densities as a function of baryonic density $\rho_{B}$
are known as equations of states (EOS). These quantities are the key
ingredients to describe the structure of neutron/hyperon stars. To see the
sensitivity of the EOS, we have plotted energy per particle ($E/\rho_{B}-M$)
as a function of density for pure neutron matter in Fig 7.3. Each curve
corresponds to a particular combination of $g_{\delta}$ and $g_{\rho}$ (taken
from Table 7.1), which reproduce the symmetry energy $E_{sym}=36.4$ MeV
without destabilizing other parameters of G2 set. The green line represents
for $g_{\delta}=0$, i.e., with pure G2 parameter set. Both the binding energy
per particle as well as the pressure density increase with the value of
$g_{\delta}$. This process continues till the value of $g_{\delta}$ reaches,
at which $E/\rho_{B}-M$ equals the nuclear matter binding energy per particle.
An unphysical situation arises beyond this value of $g_{\delta}$ because the
binding energy of the neutron matter will be greater than $E/\rho_{B}-M$ for
the symmetric nuclear matter. In the case of G2+$\delta$ parametrization, this
limiting value of $g_{\delta}$ reaches at $g_{\delta}$= 0.7, after which we do
not get a convergence solution in our calculations.

Figure 7.3: (Color online) Variation of binding energy per particle with
density at various $g_{\rho}$ and $g_{\delta}$.  Figure 7.4:  Variation of
energy per particle (panel (a)) and pressure density (panel (b)) with
$\rho_{B}/\rho_{0}$ at different values of $g_{\rho}$ and $g_{\delta}$. The
enlarged version of energy per particle for sub-saturation region is in panel
(c). The results of other theoretical models like Baldo-Maieron [267], DDHF
[268], Firedman [266] and AFDMC [269] are also given for comparison.

In Fig 7.4 we have plotted the variation of energy and pressure densities as a
function of ${\rho_{B}/\rho_{0}}$ for different combinations of $g_{\rho}$ and
$g_{\delta}$. The enlarge version of energy density in the sub-saturation
region is shown in panel (c) of the figure. Similar to other parameter sets of
RMF formalism, the G2+$\delta$ set also deviates from the experimental data.
It is to be recalled here that special attentions are needed to construct
nucleon-nucleon interaction to fit the data at sub-saturation density. For
example, the potentials of Friedman and Pandharipande [266], Baldo-Maieron
[267], DDHF [268] and AFDMC [269] are designed to fit the data in this region.
The three-body effect also can not be ignored in this sub-saturation region of
the density [270]. Although, the non-linear interactions fulfilled this demand
to some extent [186, 265, 165], like Coester band problem [197], till some
further modification of the couplings are needed. In this regard, the
relativistic mean field calculations with density dependent meson-nucleon
coupling [271] and constraining the RMF models of the nuclear matter equation
of state at low densities [272] are some of the attempts. The mean field
approximation is also a major limitation in the region of sub-saturation
density. This is because, the assumption of classical meson field is not a
proper approximation in this region to reproduce precisely the data. In higher
density region, most of the RMF forces reproduces the experimental data quite
well and the predictive power of these forces for finite nuclei is in
excellent agreement both for $\beta-$stable and $\beta-$unstable nuclei. The
energy and pressure densities with G2 set reproduce the experimental data
satisfactorily [273]. The variation of pressure density as a function of
$\rho_{B}$ is shown in panel (b) of Fig. 4, which passes inside the stiff flow
data at higher density [274]. Also, the $\delta-$meson coupling has
significant effect in supersaturation density than the sub-saturation region.
All the EOS with different $g_{\rho}$ and $g_{\delta}$ remain inside the stiff
flow data (Fig. 4, panel (b)). In the present investigation, we are more
concerned for highly dense neutron and hyperon stars, which are considered to
be super-saturated nuclear objects.

####  7.3.4 Stellar properties of static and rotating neutron stars

The $\beta$-equilibrium and charge neutrality are two important conditions to
justify the structural composition of the neutron/hyperon stars. Both these
conditions force the stars to have $\sim$90$\%$ of neutron and $\sim$10$\%$
proton. With the inclusion of baryons, the $\beta-$equilibrium conditions
between chemical potentials for different particles:

$\displaystyle\mu_{p}=\mu_{\Sigma^{+}}=\mu_{n}-\mu_{e},$
$\displaystyle\mu_{n}=\mu_{\Sigma^{0}}=\mu_{\Xi^{0}}=\mu_{n},$
$\displaystyle\mu_{\Sigma^{-}}=\mu_{\Xi^{-}}=\mu_{n}+\mu_{e},$
$\displaystyle\mu_{\mu}=\mu_{e},$

and the charge neutrality condition is satisfied by

$\displaystyle
n_{p}+n_{\Sigma^{+}}=n_{e}+n_{\mu^{-}}+n_{\Sigma^{-}}+n_{\Xi^{-}}.$ (7.16)

To calculate the mass and radius profile of the static (non-rotating) and
spherical neutron star, we solve the general relativity Tolmann-Oppenheimer-
Volkov (TOV)[275] equations which are written as:

$\displaystyle\frac{dP(r)}{dr}=-\frac{G}{c^{2}}\frac{[{\cal
E}(r)+P(r)][M(r)+\frac{4\pi
r^{3}P(r)}{c^{2}}]}{r^{2}(1-\frac{2GM(r)}{c^{2}r})}$ (7.17)

and

$\displaystyle\frac{dM(r)}{dr}=\frac{4\pi r^{2}{\cal E}(r)}{c^{2}},$ (7.18)

with G as the gravitational constant, $\cal E$$(r)$ as the energy density,
$P(r)$ as the pressure density and $M(r)$ as the gravitational mass inside
radius $r$. We have used c=1. For a given EOS, these equations can be
integrated from the origin as an initial value problem for a given choice of
the central density $\cal E$c($r$). The value of r (= R) at which the pressure
vanish defines the surface of the star. In order to understand the effect of
$\delta-$meson coupling on neutron star structure, we must also look, what
happens to massive objects as they rotate and how this affects the space-time
around them.

Figure 7.5: (Color online) The mass-radius profile for static star with
different parametrization like G2[251], NL3[49], NL3*[93], NL-SH[241],
FSU[90]and FSU2[279]. (a) is for proton-neutron star and (b) is for the
hyperon star. The maximum mass M and the corresponding radius obtained by
various parameter sets are given in the parenthesis.

For this, we use the code written by Stergioulas[276] based on Komastu,
Eriguchi, and Hachisu (KEH) method (fast rotation)[277, 278] to construct
mass-radius of the uniform rotating star. One should note that the maximum
mass of a static star is less than the rotating stars. Because, when the
massive objects rotate they flatten at their poles. The forces of rotation,
namely the effective centrifugal force, pulls the mass farthest from the
center further out, creating the equatorial bulge. This pull away from the
center will, in part, counteract gravity, allowing the star to be able to
support more mass than its non-rotating star.

We know that the core of neutron stars contain hyperons at very high density
($\sim$7-8 $\rho{{}_{0}}$) matter. As it is mentioned before, with the
presence of baryons, the EOS becomes softer and stellar properties will
change. The maximum mass of hyperon star decreases about 10-20$\%$ depending
on the choice of the meson-hyperon coupling constants. The hyperon couplings
are expressed as the ratio between the meson-hyperon and meson-nucleon
couplings as:

$\displaystyle\chi_{\sigma}=\frac{g_{Y\sigma}}{g_{N\sigma}},\chi_{\omega}=\frac{g_{Y\omega}}{g_{N\omega}},\chi_{\rho}=\frac{g_{Y\rho}}{g_{N\rho}},\chi_{\delta}=\frac{g_{Y\delta}}{g_{N\delta}}.$
(7.19)

In the present calculations, we have taken
$\chi_{\sigma}=\chi_{\rho}=\chi_{\delta}$ = 0.6104 and $\chi_{\omega}$ =
0.6666[280]. One can find similar calculations for stellar mass in Refs. [281,
282, 283]. Now we present the star properties like mass and radius in Figs.
7.5 and 7.6. In Fig. 7.5 we have plotted the mass-radius profile for the
proton-neutron star as well as for the hyperon star using a wide variation of
parameter sets starting from the old parameter like NL-SH[241] to the new set
of FSU2 [279]. The mass-radius profile varies to a great extend over the
choice of the parameter. For example, in FSU parameter set [90], the maximum
possible mass of the proton-neutron star is $\sim$ 1.75 $M_{\odot}$, while the
maximum possible mass for the NL3 set [49] is $\sim$ 2.8 $M_{\odot}$. These
results are shown in the left panel of the Fig. 7.5, while right panel shows
the same things for the hyperon star (the maximum mass and the corresponding
radius for different forces are given in the parenthesis).

####  7.3.5 Effects of $\delta-$meson on static and rotating stars

The main aim of this paper is to understand the effects of $\delta$-meson on
neutron stars both with and without hyperons. Fig. 7.6 represent the mass-
radius profiles for non-rotating and rotating stars taking into account the
presence of with and without hyperons. These profiles are shown for various
combinations of $g_{\rho}$ and $g_{\delta}$ (see Table 7.1), which we have
obtained by fitting the symmetry energy $E_{sym}$ of pure nuclear matter.

Figure 7.6: (Color online) The mass-radius profile of the static and rotating
proton-neutron and hyperon stars with various combination of $g_{\delta}$ and
$g_{\rho}$ in G2+$\delta$. (a,c) is for proton-neutron star and (b,d) is for
the hyperon star. (a,b) for static and (c,d) for rotating cases. Figure 7.7:
(Color online) Keplerian frequency of the rotating proton-neutron and hyperon
stars with various combination of $g_{\delta}$ and $g_{\rho}$ in G2+$\delta$.
(a) is for proton-neutron star and (b) is for the hyperon star. The results
are obtained from RNS code [276].

Analyzing the graphs, we notice a slight change in the maximum mass with gδ
value. That means, the mass of the star goes on decreasing with an increase
value of the $\delta$-meson coupling in hyperon star. A further inspection of
the results reveals that, although the $\delta$-meson coupling has a nominal
effects on the maximum mass of the proto-neutron stars, we get an asymptotic
increase in the mass. This asymptotic nature of the curves is more prominent
in presence of hyperons inside the stars. Similar phenomena are also observed
in case of rotating stars.

The empirical formula for the relation between maximum frequency $f_{max}$
with mass of the neutron star for a given EOS is given as [284, 285]

$f_{max}\approx
1.22KHz\sqrt{M_{max}^{static}/M_{\odot}}({R_{max}^{static}/10Km})^{-3/2},$
(7.20)

where $M_{max}^{static}$=maximum static mass and $R_{max}^{static}$=maximum
allowed radius for a neutron star. In actual, the neutron stars have a wide
range of frequencies due to the fluid of the stars oscillate in various modes
[286]. Among them, the most important modes are the first pressure mode
($p_{I}$-mode) and the fundamental mode of the fluid oscillation (f-mode). The
empirical formula for the frequencies of these two modes are
$f_{f}=(0.79\pm{0.09})+(33\pm 2)\sqrt{M/R^{3}}$ and
$f_{p}=1/M(-1.5\pm{0.8})+(79\pm{4}){M/R}$, respectively, where M and $f$ in Km
and KHz. The above two relations are obtained by using a wide sample of EOSs
[286].

In the present calculations, we assume the frequency of the rotating neutron
star is within the Keplerian frequency limit. At this limit the spin frequency
of the neutron star is equal to the orbital frequency $f_{or}$ (along a
circular path on the equator of the NS) [284]. If the orbital frequency
$f_{or}>f_{K}$ (Kepler frequency), then the hydrostatic equilibrium of the NS
does not hold good. To make it clear, the Kepler frequency as a function of NS
mass is shown in Fig. 7 with and without considering hyperon into account. The
results of Fig. 7 is obtained from the RNS code and the expression for the
Keplarian frequency, i.e. the maximum frequency obtained from the general
theory of relativity can be found in Refs. [287, 276]. In this figure, the
variation of $f_{K}$ is shown as a function of $M/M_{\odot}$ with various
combinations of $g_{\rho}$ and $g_{\delta}$ which we have already fixed (see
Table 1). We noticed a finite effect of $g_{\rho}$ and $g_{\delta}$ variation
on the mass and Keplerian frequency of the pure neutron and hyperon stars.

For a quite some time pulsar B1937+21 with frequency 642 Hz was considered as
the fasted spinning NS. However, Hessels et. al.[288] found even more faster
spinning NS pulsar J1748-2446ad at frequency 716.356 Hz. This NS has a mass of
0.14$M_{\odot}$ companion. It is difficult to obtain 0.14$M_{\odot}$ from the
equation of state at supra-nuclear densities. Our calculations suggest that if
pulsar has a mass less than 1.4$M_{\odot}$ than the larger density slope of
the symmetry energy at saturation would be excluded. If we consider the
neutron star mass to be greater than 2.0$M_{\odot}$ and hyperons are present
in it, then the star mass will be 1.6$M_{\odot}$, within the pulsar XTE
J1739–285 NS[289].

Here, we analyzed the effects of $\delta-$meson coupling on neutron and
hyperon stars. We observed that, the mass of the star decreases when hyperons
are included in the calculations, as a result, the maximum mass of the star
with G2+$\delta$ set become much less than 2M⊙, the latest observation of a
massive neutron star. In summary, the following possibilities are in order:

(i) Since the mass obtained without hyperon for static case is $\sim
2M_{\odot}$, in this situation one do not need to reduce the mass any more by
adding hyperons into it. This can be justify by assuming that in massive
neutron star, there is no hyperon. The absence of hyperons in massive neutron
star may not be a convincing explanation, because of the highly dense matter
in the core of the NS, which favor the production of hyperon. (ii) The
rotation of a NS increases the maximum allowed mass. On the other hand, the
inclusion of hyperon deceases the mass. In the present case, even if consider
the rotation of the star, it is not sufficient to get the maximum mass$\sim$
2$M_{\odot}$ (see Fig. 6(d)). (iii) The 3rd possibility is the effect of
$\delta-$meson coupling, which may increase the mass of the hyperon star after
its insertion into the model. Although, its effect is finite, it is not
sufficient to increase the hyperon mass to two solar unit. Thus, the addition
of $\delta-$meson may not be sufficient to explain the heavier mass of the NS.
(iv) Probably, the fourth possibility may be the most acceptable explanation
in which we suggest for the modification of the EOS, such that after the
addition of hyperon, the mass of the static neutron-hyperon star will be $\sim
2M_{\odot}$. In particular, the hyperon-meson coupling should be re-
investigated to get a proper coupling constants, which allowed the maximum
mass$\sim 2M_{\odot}$ with hyperon. Work in this direction is in
progress[290].

####  7.3.6 Effects of $\delta-$meson on baryon production

Finally, we want to see the effects of $\delta-$meson coupling on the particle
production for the whole baryonic family at various densities in nuclear
matter system.

Figure 7.8: (Color online) Yield of strange particles as a function of
density. The upper panel (a) is with G2 parameter set (without taking
$\delta-$meson coupling) and the lower panel (b) is with $\delta-$meson
coupling.

The Fermi energy of both proton and neutron increases with density for their
Fermionic nature. After a certain density, the Fermi energy of the nucleon
exceeded the rest mass energy of the nucleon ($\sim$1000 MeV), and strange
particles ($\Sigma,\Lambda,\Xi$) are produced. As a result, the equations of
state of the star becomes soft and gives a smaller star mass compare to the
neutron star containing only protons, neutrons and electrons. The decrease in
star mass in the presence of whole baryon octet can be understood from the
analysis of Fig. 7.8. From the figure, it is clear that $\delta$-meson has a
great impact on the production of hyperons. The inclusion of $\delta-$meson
accelerate the strange particle production. For example, the evolution of
$\Sigma^{-}$ takes place at density $\rho_{B}=1.75\rho_{0}$ in absence of
$\delta-$meson. However, it produces at $\rho_{B}=1.67\rho_{0}$ when
$\delta-$meson is there in the system. Similarly, analyzing the evolution of
other baryons, we notice that although the early production of baryons in the
presence of $\delta-$meson is not in a definite proportion to each other, in
each case the yield is faster. A significant shifting towards lower density is
maximum for heaviest hyperon ($\Xi^{0}$) and minimum for nucleon (see Fig.
7.8). For example, $\Xi^{-}$ evolves at $\rho_{B}$ = 6.5 $\rho_{0}$ for a
non-$\delta$ system and $\rho_{B}$$\sim$5.0 $\rho_{0}$ for medium when
$\delta-$meson is included. Thus, the $\delta-$coupling has a sizable impact
on the production of hyperons like $\Xi^{-},\Xi^{0}$ and $\Sigma^{+}$.

Figure 7.9: (Color online) Mass and radius profile of hyperon star with
G2+$\delta$ parameter set, but with different meson-hyperon coupling of
Ref.[281].

####  7.3.7 Fitting of $g_{\rho}$ and $g_{\delta}$ with fixed binding energy
and charge radius

In previous sub-sections we have seen the effects of ($g_{\rho}$,
$g_{\delta}$) pair with a constant symmetry energy on the maximum mass and
radius of the neutron and hyperon stars. The effects of the ($g_{\rho}$,
$g_{\delta}$) pairs are not prominent on the star structure in this method. On
the other hand, it affects the bulk properties, like binding energy and root
mean square radius considerably for asymmetric finite nucleus. In Table 7.2,
we have given the mass and charge radius for some of the selected nuclei.
Although, all the combination of $g_{\rho}$ and $g_{\delta}$ are fixed at a
constant symmetry energy, the binding of 208Pb differ by 90 MeV in the first
and last combination of $g_{\rho}$ and $g_{\delta}$. In this sub-section, we
would like to change the strategy to select the ($g_{\rho}$, $g_{\delta}$)
pairs. Here, we have followed the second procedure as we have discussed in the
previous sub-section, i.e., we find the values of $g_{\rho}$ and $g_{\delta}$
by adjusting the binding energy and charge radius of 208Pb. Once we get the
($g_{\rho}$, $g_{\delta}$), we used the pair for the calculations of other
nuclei of Table 7.2. Surprisingly, the outcome of binding energy and charge
radius matches pretty well with the original calculations. The $g_{\rho}$ and
$g_{\delta}$ combination along with the corresponding mass and radius of a
neutron star is given in the Table 7.3. From the table it is clear that these
combinations are also not affecting much to the maximum mass and radius of the
neutron star. However, the $E_{sym}$, $L_{sym}$ and $K_{sym}$ calculated from
the corresponding ($g_{\rho}$, $g_{\delta}$) combinations for nuclear matter
changes a lot (See Table 7.3). We used the hyperon-meson coupling constants of
Ref. [281] to evaluate the hyperon star structure. The calculated results for
static and rotating hyperon star are plotted in Fig. 7.9. The maximum mass
increases and the radius decreases slightly with the addition of
$\delta-$meson to the star system.

###  7.4 Summary and Conclusions

In summary, using the effective field theory approach, we discussed the
effects of isovector scalar meson on hyperon star. The inclusion of
$\delta$-meson with G2 parameter set, we have investigated the static and
rotating stellar properties of neutron star with hyperons. We fitted the
parameters and see the variation of gρ and gδ at a constant symmetry energy
for both the nuclear and neutron matter. We also used these (gρ, gδ) pairs to
finite nuclei and find a large change in binding energy for asymmetric nuclei.
Then we re-fitted the (gρ, gδ) pairs keeping binding energy and charge radius
fixed for 208Pb and tested the effects for some selected nuclei and able to
reproduced the data similar to the original G2 set. With the help of
G2+$\delta$ model, for static and rotating stars without hyperon core, we get
the maximum mass of $\sim$2$M_{\odot}$ and $\sim$2.4$M_{\odot}$, respectively.
This prediction of masses is in agreement with the recent observation of
$M\sim$$2M_{\odot}$ of the stars. However, with hyperon core the maximum mass
obtained are $\sim$1.4$M_{\odot}$ and $\sim$1.6$M_{\odot}$ for static and
rotating hyperon stars, respectively. In addition, we have also calculated the
production of whole baryon octet with variation in density. We find that the
particle fraction changes a lot in the presence of $\delta-$meson coupling.
When there is $\delta-$meson in the system the evolution of baryons are faster
compare to a non-$\delta$ system. This effect is significant for heavier
masses and minimum for lighter baryon. Hence, one can conclude that the yield
of baryon/hyperons depends very much on the mesons couplings. One important
information is drawn from the present calculations is that the effect of gδ is
just opposite to the effect of gρ. As a consequence, many long standing
anomaly, such as the comparable radii of 40Ca and 48Ca be resolved by
adjusting the (gρ, gδ) pairs properly. Keeping in view the importance of
$\delta-$meson coupling and the reverse nature of gρ and gδ, it is necessary
to get a new parameter set including proper values of $g_{\delta}$ and
$g_{\rho}$, and the work is under progress.

##  Chapter 8 Summary and Conclusions

The relativistic mean field formalism provides a common platform to study both
the finite and infinite nuclear matter systems. Its relativistic nature
defined a pathway from the less dense finite nuclear system to highly dense
neutron star. We studied various aspects of finite nuclear structure like
magicity of proton and neutron number and collective excitation like giant
resonance and the nucleon-nucleon interaction . For the infinite nuclear
system, we applied RMF to study the mass and radius of the neutron star. In
brief, we have applied RMF formalism to study both finite and infinite nuclear
systems.

Recently the advent of radioactive ion beam (RIB) facility inspires the
nuclear physicists to look forward to the structure of drip-lines and super-
heavy nuclei more seriously. Nuclei away from the $\beta$-stability lines are
far different from the nuclei on the $\beta$-stability line. In Ch. 2, we have
calculated BE, $S_{2n}$ energy, single particle levels, pairing gaps and
chemical potential, in the isotopic chain of Z = 82, 114, 120 and 126. All our
calculations are done in the framework of non-relativistic SEI and
relativistic RMF interactions. We have compared our results with FRDM and
other theoretical predictions. Overall, the discussions and analysis of all
possible evidence of shell closure property with SEI interaction and RMF show
that one can take Z = 120 and N = 182 or 184 as the next magic combination
beyond Z = 82 and N = 126, which is consistent with other theoretical models.

Not only the single particle properties, which are discussed in first part of
the thesis but also the collective excitation plays an important role in
nuclear structure physics. In Ch.3, we have calculated the excitation energy
of isoscalar giant monopole resonance and incompressibility for O, Ca, Ni, Sn,
Pb, Z=114, and Z=120 isotopic series starting from proton to neutron drip
lines. We used four successful parameter sets, NL1, NL3, NL3*, and FSUGold,
with a wide range of nuclear matter incompressibility starting from 211.7 MeV
to 271.76 MeV to see the dependency of the ISGMR on $K_{\infty}$. Also, we
have analyzed the predictions of ISGMR with these forces, which originate from
various interactions and found that whatever may be the parameter set, the
differences in excitation energy ISGMR predicted by them are found to be
marginal in the super heavy region. A recently developed scaling approach in a
relativistic mean field theory is used. A simple, but accurately constrained
approximation is also performed to evaluate the isoscalar giant monopole
excitation energy. From the scaling and constrained ISGMR excitation energies,
we have evaluated the resonance width $\Sigma$ for the whole isotopic series.
This is obtained by taking the root mean square difference of ${E^{s}}_{x}$
and ${E^{c}}_{x}$ . The value of ${E^{s}}_{x}$is always higher than the
constrained result ${E^{c}}_{x}$ . In a sum rule approach, the ${E^{s}}_{x}$
can be compared with the higher and ${E^{c}}_{x}$ as the lower limit of the
resonance width. In general, we found an increasing trend of resonance width
$\Sigma$ for both the light and super heavy regions near the proton and
neutron drip lines. The magnitude of $\Sigma$ is predicted to be minimum in
the vicinity of N=Z or in the neighborhood of a double closed nucleus and it
is maximum for highly asymmetric system. In the present thesis, we have also
estimated the incompressibility of finite nuclei. For some specific cases, the
incompressibility is compared with the nuclear matter incompressibility and
found a linear variation among them. It is also concluded that the nucleus
becomes less compressible with the increase of neutron or proton number in an
isotopic chain. Thus neutron-rich matter, like neutron star as well as drip-
line nuclei, are less compressible than normal nuclei. In the case of exotic
(drip-line nuclei) system, the nucleus is incompressible, although it
possesses a normal density.

The recent experiment on isoscalar monopole excitation in the isotopic chain
of the Sn isotopes indicates a new problem in the medium heavy mass range
(A$\sim$ 100). In the second part of Ch. 3, we analyzed the predictive power
of various force parameters, like NL1, NL2, NL3, Nl-SH and FSUG in the
framework of RTF and RETF approaches for giant monopole excitation energy of
Sn-isotopes. Then the calculation is extended to some other relevant nuclei in
the mass range A$\sim$100\. The analysis shows that relativistic Thomas-Fermi
and extended relativistic Thomas-Fermi approximation give comparable results
with pairing+MEM prediction. It can be exactly reproduced the experimental
data for Sn isotopes, when the incompressibility of the force parameter is
within 210-230 MeV, however, fails to reproduce the GMR data for other nuclei
within the same accuracy. We have qualitatively analyzed the difference in GMR
energies ($\triangle E_{GMR}$=RETF-RTF) using RETF and RTF formalisms in
various force parameters. The FSUGold parameter set shows different behavior
from all other forces. Also, we extended our calculations for monopole
excitation energy of Sn isotopes with a force parametrization having softer
symmetry energy (NL3 + $\Lambda_{V}$ ). The excitation energy decreases with
the increase of proton-neutron asymmetry agreeing with the experimental trend.
In conclusion, after all, from these thorough analyses, it seems that the
softening of Sn isotopes is an open problem in nuclear theory and more work in
this direction is needed. In Ch. 4 we discussed about the new constrained
caiculation developed by us. This new method is based on the Taylor series
expansion. Although this method is very simple, it gives reasonable results
which are comparable with other microscopic calculation like RRPA. For
simplicity, we applied the constrained method to Thomas-Fermi approximation.
But this method can be extended to extended Thomas-Fermi approach without
loosing any generality.

Nucleon-Nucleon interaction has a very crucial role in the nuclear structure
as well as the other branch of nuclear physics. So it is important to study
more about the nucleon-nucleon interaction and its effects on various
properties of the nuclear system. In Ch. 5, we extensively discussed a new
approach R3Y, for the N-N interaction, which is first suggested by Patra et
at. We have added a new self-interacting $\omega$ meson contribution to this
new formalism and checked its contribution to finite and infinite nuclear
matter system. In Ch. 6, we used the density dependent M3Y and R3Y to study
the p-$\gamma$ reaction for the proton-rich nuclei. We have folded the M3Y and
R3Y interaction with RMF densities to obtain the cross-section of p-$\gamma$
reaction. From the cross-section of these reaction, we can study the
astrophysical- S factor, which is important in the study of the r-process of
nuclear synthesis.

In Ch. 7, using the effective field theory approach, we discussed the effects
of isovector scalar meson on hyperon star with the inclusion of $\delta$-meson
with G2 parameter set and then investigated the static and rotating stellar
properties of neutron star with hyperons. We fitted the parameters and see the
variation of gρ and gδ at a constant symmetry energy for both the nuclear and
neutron matter. We also used these (gρ, $g_{\delta}$) pairs to finite nuclei
and find a large change in binding energy for asymmetric nuclei. Then we re-
fitted the (gρ, gδ) pairs keeping binding energy and charge radius fixed for
208Pb and tested the effects for some selected nuclei and able to reproduce
the data similar to the original G2 set. With the help of the G2+$\delta$
model, for static and rotating stars without hyperon core, we get the maximum
mass of $\sim$2$M_{\odot}$ and $\sim$2.4$M_{\odot}$, respectively. This
prediction of masses is in agreement with the recent observation of
$M\sim$$2M_{\odot}$ of the stars. However, with hyperon core the maximum
masses obtained are $\sim$1.4$M_{\odot}$ and $\sim$1.6$M_{\odot}$ for static
and rotating hyperon stars, respectively. In addition, we have also calculated
the production of whole baryon octet with variation in density. We find that
the particle fraction changes a lot in the presence of $\delta-$meson
coupling. When there is $\delta-$meson in the system the evolution of baryons
is faster compared to a non-$\delta$ system. This effect is significant for
heavier masses and minimum for lighter baryon. Hence, one can conclude that
the yield of baryon/hyperons depends very much on the mesons couplings. One
important information is drawn from the present calculations is that the
effect of gδ is just opposite to the effect of gρ. As a consequence, many
long-standing anomalies, such as the comparable radii of 40Ca and 48Ca can be
resolved by adjusting the (gρ, gδ) pairs properly. Keeping in view the
importance of $\delta-$meson coupling and the reverse nature of gρ and gδ with
each other, it is necessary to get a new parameter set including proper values
of $g_{\delta}$ and $g_{\rho}$, and the work is in progress.

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