# Thermal effects in hot and dilute homogeneous asymmetric nuclear matter

Vishal Parmar  Manoj K Sharma  School of Physics and Materials Science, Thapar
Institute of Engineering and Technology, Patiala-147004, India.
<EMAIL_ADDRESS>S K Patra  Institute of Physics, Bhubaneswar
751005, India
Homi Bhaba National Institute, Training School Complex, Anushakti Nagar,
Mumbai 400 085, India

###### Abstract

We present a comprehensive analysis of hot and dilute isospin-asymmetric
nuclear matter employing the temperature-dependent effective-relativistic
mean-field theory (E-RMF). The E-RMF is applied to study the effect of
$\delta$ and $\omega-\rho$ meson cross-coupling on the thermal properties of
asymmetric nuclear matter using two recently developed IOPB-I and G3 parameter
sets. These sets are known to reproduce the nuclear matter properties in
agreement with various experimental and observational constraints. We consider
the nuclear matter to be homogeneous and study the equation of state (EoS) for
densities, temperature and asymmetry which are relevant for astrophysical
simulations such as supernovae explosion. The effect of temperature is
investigated in reference to the density-dependent free symmetry energy and
its higher-order derivatives using the well known parabolic approximation. The
larger value of $\lambda_{\omega}$ cross-coupling in G3 in addition to the
$\delta$ meson coupling in G3 smoothen the free symmetry energy. Thermal
effects on various state variables are examined at fixed temperature and
isospin asymmetry by separating their T=0 and the finite-T expressions. The
thermal effects are mainly governed by effective mass with larger effective
mass estimating larger thermal contribution. The effect of temperature on
isothermal and isentropic incompressibility is discussed which is in harmony
with various available microscopic calculations. The liquid-gas phase
transition properties are examined in asymmetric matter with two conserved
charges in the context of different slope parameter and comparable symmetry
energy in IOPB-I and G3 set. The spinodal instability, binodal curve and
critical properties are found to be influenced by the slope parameter
$L_{sym}$. Finally, we consider a more realistic system with the inclusion of
electrons and analyse their effect on instability and adiabatic index of
isospin asymmetric nuclear matter.

###### pacs:

21.65.+f, 26.60.+c, 26.60.-c, 25.70.Pq

##  I  Introduction

Core-collapse supernova is nature’s one of the brightest optical display where
million-year life of a giant star ($M>8M\odot$) is put to an end violently and
abruptly within fractions of a second [1, 2]. The exact mechanism of collapse
explosion is still not well understood even after several decades of thorough
investigations. In recent years, such explosion has been studied using several
ab-initio core-collapse simulations where the hydrodynamics equations are
solved numerically [3, 4]. These simulations estimate that the explosion
energy of $\approx$ 10 51 erg is attained within the time scale of $\geq$ 1s
[5]. The temperature of the matter rises to 20 MeV and the density of the
bounce can vary up to two times the nuclear saturation density. The short time
scale of collapse does not allow the matter to reach $\beta$ equilibrium and
calculations are usually done at a fixed asymmetry
$\alpha=\frac{\rho_{n}-\rho_{p}}{\rho_{n}+\rho_{p}}\approx$ 0.4 [6, 7].

The determination of EoS for isospin-asymmetric nuclear matter (ANM) is
relevant in various areas of nuclear physics ranging from finite nuclei to
infinite matter. Not only the understanding of its ground state is important,
but its behaviour at finite temperature is equally significant. The finite
temperature behaviour of ANM is relevant in context to astrophysical events
such as neutron star mergers, gamma-ray bursts, proto-neutron stars, early
universe, etc [8]. Furthermore, the composition of matter inside the neutron
star impacts its transportation and cooling process which are governed by the
so-called direct URCA process [9]. With the recent detection of gravitational
wave (GW170817) which was accompanied by a gamma-ray-burst and electromagnetic
afterglow from the merger of a neutron star binary opened a new era of
astrophysics [10, 11]. In view of above, a systematic understanding of
asymmetric nuclear matter at finite temperature is highly desirable.

The nuclear matter which is predominantly governed by a residual short-range
strong force and long-range electromagnetic interaction shows various
structures which in turn depend upon the parameters such as density,
asymmetry, temperature etc. At low temperature or entropy, the matter is in
the non-homogeneous form below the subnuclear density ( $\rho<0.1$ fm-3). The
nuclear matter is a mixture of heavy nuclei and lights clusters in the
background of nucleon gas [12]. As the density increases, the nuclei become
deformed constituting a frustrating system usually known as pasta structures
because of the competition between nuclear and electromagnetic forces [13].
Above saturation density, the matter converts to a homogeneous mixture of
nucleons which may contain the exotic species such as kaon and pion
condensate, hyperons, quark phases etc. The leptons (electrons in low density
and electrons with muons at high density) are present in all these structures
for charge neutrality. At large-enough temperatures, the inhomogeneous nuclear
matter at low density again transforms to the homogeneous phase. In this work,
however, we assume the nuclear matter to be an ideal homogeneous mixture of
nucleons. Such investigation of ideal ANM system is significant to understand
the underlying qualitative behaviour. We employ the finite temperature
extension of the effective-relativistic mean-field theory (E-RMF) to study the
asymmetric nuclear matter at finite temperature. The recently developed E-RMF
parameters namely IOPB-I [14] and G3 [15] are used here to understand various
aspects of nuclear matter at finite temperature. These sets have comparable
saturation properties at zero temperature but differ in the number of
adjustable parameters i.e various self and cross couplings of $\sigma$,
$\omega$, $\rho$ and $\delta$ mesons. This provides us with an opportunity to
study the effect of these couplings on the finite temperature properties of
asymmetric nuclear matter.

The central motivation of this study is to perform a detailed analysis of the
EoS for dilute and hot homogeneous asymmetric nuclear matter within E-RMF
formalism. We aim to understand the nuclear matter properties like symmetry
energy $F_{sym}$, slope parameter ($L_{sym}$), skewness parameter ($Q_{sym}$),
curvature parameter ($K_{sym}$) as a function of temperature. These are
significant properties of asymmetric nuclear matter and are often used to
constraint the EoS around saturation density. Several finite temperature
effects such as thermal effects on various state variable, isothermal and
isentropic incompressibility are addressed. The results presented in this
study (below subnuclear density) serve only to differentiate between the
realistic inhomogeneous phase in a supernova and ideal homogeneous phase. This
is analogous to the van-der Walls equation of state and ideal-gas equation in
atomic physics. The phase transition property in asymmetric matter in
comparison with symmetric matter is discussed in context to incompressibility
$K$ and slope parameter $(L_{sym})$. With this study, we aim to verify the
trends available in various studies [6, 16, 17, 18] where effect of symmetry
energy and its derivative is discussed on the instability of ANM. Establishing
these trends is of primary importance as they serve as the bridge between
various nuclear matter properties which are not measured directly from the
experiments. In symmetric nuclear matter the trends are seen among the
properties at critical temperature [19]. The properties at ground state do not
necessarily dictate the critical properties of phase transition. However, for
asymmetric matter, the symmetry energy and its slope parameter decides the
energetic and therefore impacts the instabilities occurring in the system. In
a realistic case like supernovae it will affect the transportation and cooling
process whereas in neutron star crust, the core-crust boundary becomes the
variable of slope parameter $L_{sym}$.

The paper is organized as follows: In Section II, we give a brief formalism of
the E-RMF model at finite temperature. The thermal effects on various
thermodynamic functions and stability conditions are mentioned in this
section. In Section III, we summarise our results where we discuss the model
properties in Section III.1 and properties at finite temperature in Section
III.2. We discuss the liquid-gas phase transition in ANM in Section III.3. In
Section III.4, we discuss the influence of electrons on EoS. Finally, we
summarise our results in Sect. IV.

##  II  Formalism

The Lagrangian and the corresponding energy-density functional of the
effective field theory (EFT) motivated E-RMF is documented in literature [20,
14, 15, 19]. The advantage of E-RMF formalism is that one can ignore the basic
difficulties of the formalism, like renormalization and divergence of the
system. It takes care of several natural phenomena in ab initio manner which
otherwise are absent or are included in ad hoc manner in non-relativistic
formalisms. In the effective nuclear field theory, Lagrangian contains the
infinite number of terms. The Lagrangian is expanded in the power of meson
fields as a truncation scheme because the fields have a lower mass compared to
nucleon masses. The contribution of each term in the E-RMF Lagrangian can be
calculated by counting the power of expansion. Couplings present at a
particular order cannot be dropped arbitrarily without a proper symmetry
argument. The ambiguity in the expansion is checked by the inclusion of
naturalness constraints and naive dimensional analysis (NDA) . For
calibration, the coupling constants and mass of isoscalar-scalar $\sigma$
meson are fitted to reproduce the experimental values of saturation density
and ground state properties of some known nuclei. This method mocks the result
of two- loop contribution in mean-field theory [21]. The basic nucleon-meson
E-RMF which involve couplings of $\sigma$, $\omega$, $\rho$ and $\delta$
mesons and the photon with Dirac nucleon upto the fourth order is given as
[19]

$\displaystyle\begin{aligned}
\mathcal{E}(r)=&\psi^{\dagger}(r)\quantity{i\alpha\cdot\gradient+\beta[M-\Phi(r)-\tau_{3}D(r)]+W(r)+\frac{1}{2}\tau_{3}R(r)+\frac{1+\tau_{3}}{2}A(r)-\frac{i\beta\alpha}{2M}\quantity(f_{\omega}\gradient
W(r)+\frac{1}{2}f_{\rho}\tau_{3}\gradient R(r))}\psi(r)\\\
&+\quantity(\frac{1}{2}+\frac{k_{3}\Phi(r)}{3!M}+\frac{k_{4}}{4!}\frac{\Phi^{2}(r)}{M^{2}})\frac{m^{2}_{s}}{g^{2}_{s}}\Phi(r)^{2}-\frac{\zeta_{0}}{4!}\frac{1}{g^{2}_{\omega}}W(r)^{4}+\frac{1}{2g^{2}_{s}}\quantity\Big(1+\alpha_{1}\frac{\Phi(r)}{M})(\gradient\Phi(r))^{2}\\\
&-\frac{1}{2g^{2}_{\omega}}\quantity\Big(1+\alpha_{2}\frac{\Phi(r)}{M})(\gradient
W(r))^{2}-\frac{1}{2}\quantity\Big(1+\eta_{1}\frac{\Phi(r)}{M}+\frac{\eta_{2}}{2}\frac{\Phi^{2}(r)}{M^{2}})\frac{m^{2}_{\omega}}{g^{2}_{\omega}}W^{2}(r)-\frac{1}{2e^{2}}(\gradient
A^{2}(r))^{2}\\\ &-\frac{1}{2g^{2}_{\rho}}(\gradient
R(r))^{2}-\frac{1}{2}\quantity\Big(1+\eta_{\rho}\frac{\Phi(r)}{M})\frac{m^{2}_{\rho}}{g^{2}_{\rho}}R^{2}(r)-\Lambda_{\omega}(R^{2}(r)W^{2}(r))+\frac{1}{2g^{2}_{\delta}}(\gradient
D(r))^{2}+\frac{1}{2}\frac{m^{2}_{\delta}}{g^{2}_{\delta}}(D(r))^{2}.\end{aligned}$
(1)

Here $\Phi(r)$, W(r), R(r), D(r) and A(r) are the fields corresponding to
$\sigma$, $\omega$, $\rho$ and $\delta$ mesons and photon respectively. The
$g_{s}$, $g_{\omega}$, $g_{\rho}$, $g_{\delta}$ and $\frac{e^{2}}{4\pi}$ are
the corresponding coupling constants and $m_{s}$, $m_{\omega}$, $m_{\rho}$ and
$m_{\delta}$ are the corresponding masses. The zeroth component $T_{00}=H$ and
the third component $T_{ii}$ of energy-momentum tensor

$T_{\mu\nu}=\partial^{\nu}\phi(x))\frac{\partial\mathcal{E}}{\partial\partial_{\mu}\phi(x)}-\eta^{\nu\mu}\mathcal{E},$
(2)

yields the energy and pressure density respectively as

$\displaystyle\begin{aligned}
E=&\sum_{p,n}\frac{\gamma}{(2\pi)^{3}}\int\differential[3]{k}E^{*}_{p,n}(k)[n_{k}(T)+\bar{n}_{k}(T)]+\rho
W+\quantity\Big(\frac{1}{2}+\frac{k_{3}\Phi}{3!M}+\frac{k_{4}}{4!}\frac{\Phi^{2}}{M^{2}})\frac{m^{2}_{s}}{g^{2}_{s}}\Phi^{2}-\frac{1}{2}\quantity\Big(1+\eta_{1}\frac{\Phi}{M}+\frac{\eta_{2}}{2}\frac{\Phi^{2}}{M^{2}})\frac{m^{2}_{\omega}}{g^{2}_{\omega}}W^{2}\\\
&-\frac{\zeta_{0}}{4!}\frac{1}{g^{2}_{\omega}}W^{4}+\frac{1}{2}\rho_{3}R-\frac{1}{2}\quantity\Big(1+\eta_{\rho}\frac{\Phi}{M})\frac{m^{2}_{\rho}}{g^{2}_{\rho}}R^{2}-\Lambda_{\omega}(R^{2}W^{2})+\frac{1}{2}\frac{m^{2}_{\delta}}{g^{2}_{\delta}}(D)^{2}.\end{aligned}$
(3)

$\displaystyle\begin{aligned}
P=&\sum_{p,n}\frac{\gamma}{3(2\pi)^{3}}\int\differential[3]{k}\frac{k^{2}}{E^{*}_{p,n}(k)}[n_{k}(T)+\bar{n}_{k}(T)]-\quantity\Big(\frac{1}{2}+\frac{k_{3}\Phi}{3!M}+\frac{k_{4}}{4!}\frac{\Phi^{2}}{M^{2}})\frac{m^{2}_{s}}{g^{2}_{s}}\Phi^{2}&+\frac{1}{2}\quantity\Big(1+\eta_{1}\frac{\Phi}{M}+\frac{\eta_{2}}{2}\frac{\Phi^{2}}{M^{2}})\frac{m^{2}_{\omega}}{g^{2}_{\omega}}W^{2}\\\
&+\frac{\zeta_{0}}{4!}\frac{1}{g^{2}_{\omega}}W^{4}+\frac{1}{2}\quantity\Big(1+\eta_{\rho}\frac{\Phi}{M})\frac{m^{2}_{\rho}}{g^{2}_{\rho}}R^{2}+\Lambda_{\omega}(R^{2}W^{2})-\frac{1}{2}\frac{m^{2}_{\delta}}{g^{2}_{\delta}}(D)^{2}.\end{aligned}$
(4)

In above equations, $n_{k}(T)$ and $\bar{n}_{k}(T)$ are the baryons and
antibaryons occupation numbers respectively, which are defined by the Fermi
distribution function at finite temperature T as

$\displaystyle n_{k}(T)$ $\displaystyle=$
$\displaystyle\frac{1}{1+\exp\quantity(\frac{(E^{*}(k)-\nu)}{T})},$ (5a)
$\displaystyle\bar{n}_{k}(T)$ $\displaystyle=$
$\displaystyle\frac{1}{1+\exp\quantity(\frac{(E^{*}(k)+\nu)}{T})},$ (5b)

with E∗ as $\sqrt{k^{2}+{M^{*}}^{2}}$. Here the effective mass is written as
[22]

$\displaystyle M^{*}_{p}=M-\Phi(r)-D(r),$ (6a) $\displaystyle
M^{*}_{n}=M-\Phi(r)+D(r).$ (6b)

The effective chemical potential $\nu$ for proton and neutron are defined as

$\displaystyle\nu_{p}=\mu-W(r)+\frac{1}{2}R(r),$ (7a)
$\displaystyle\nu_{n}=\mu-W(r)-\frac{1}{2}R(r).$ (7b)

The entropy density ($S=s/\rho_{b}$) is given as

$s_{i}=-2\sum_{i}\int\frac{\differential^{3}k}{(2\pi)^{3}}[n_{k}\ln
n_{k}+(1-n_{k})\ln(1-n_{k})+(n_{k}\leftrightarrow\bar{n}_{k})]$ (8)

and the free energy is defined as

$F=E-TS$ (9)

The Free energy density can be written in the parabolic form of the asymmetry
parameter $(\alpha=\frac{\rho_{n}-\rho_{p}}{\rho_{n}+\rho_{p}})$ as [23, 24]

$F(\rho,\alpha,T)=F(\rho,\alpha=0,T)+F_{sym}(\rho,T)\alpha^{2}$ (10)

where $F_{sym}(\rho,T)\alpha^{2}$ is the free symmetry energy content per
nucleon of the system and $F(\rho,\alpha=0,T)$ is the free energy per nucleon
of symmetric ($\alpha=0$) nuclear matter. The free symmetric energy using the
empirical parabolic approximation then can be written as

$F_{sym}(\rho,T)=\frac{F(\rho,T,\alpha=1)}{\rho}-\frac{F(\rho,T,\alpha=0)}{\rho}.$
(11)

The free symmetric energy then can be expanded as a Taylor series around the
saturation density $\rho_{0}$ as

$\displaystyle
F_{sym}(\rho,T)=F_{sym}(\rho_{0},T)+L_{sym}\mathcal{x}+\frac{K_{sym}}{2!}\mathcal{x}^{2}+\frac{Q_{sym}}{3!}\mathcal{x}^{3}+$
(12) $\displaystyle O(\mathcal{x}^{4}),$

where $\mathcal{x}=\frac{\rho-\rho_{0}}{3\rho_{0}}$ and $L_{sym}$, $K_{sym}$
and $Q_{sym}$ are the slope parameter, curvature parameter and skewness
parameter which are written as

$\displaystyle L_{sym}$ $\displaystyle=$ $\displaystyle 3\rho\frac{\partial
F_{sym}(\rho,T)}{\partial\rho},$ (13) $\displaystyle K_{sym}$ $\displaystyle=$
$\displaystyle 9\rho^{2}\frac{\partial^{2}F_{sym}(\rho,T)}{\partial\rho^{2}},$
$\displaystyle Q_{sym}$ $\displaystyle=$ $\displaystyle
27\rho^{3}\frac{\partial^{3}F_{sym}(\rho,T)}{\partial\rho^{3}}.$

To infer the effects of finite temperature we focus on the thermal part of the
various state variables, that is, the difference between the T=0 and the
finite-T expressions for a given thermodynamic function $\mathcal{X}$ [25,
26],

$\mathcal{X}=\mathcal{X}(\rho,\alpha,T)-\mathcal{X}(\rho,\alpha,0)$ (14)

The thermal energy, thermal pressure, thermal free energy density and thermal
index are then written as

$\displaystyle E_{th}=E(\alpha,T)-E(\alpha,0)$ (15) $\displaystyle
P_{th}=P(\alpha,T)-P(\alpha,0)$ $\displaystyle F_{th}=F(\alpha,T)-E(\alpha,0)$
$\displaystyle\lambda_{th}=1+\frac{E_{th}}{P_{th}}$

Thermal contributions to the free symmetry energy is given by

$F_{sym,th}=F_{sym}(\alpha,T)-E_{sym}(\alpha,0).$ (16)

In asymmetric nuclear matter system, there are two conserved charges, baryon
number ($\rho_{b}=\rho_{p}+\rho_{n}$) and isospin number ($I_{3}=I_{p}+I_{n}$
). Therefore, one needs to treat it as a binary system. The system will be
stable against separation into two phases if the free energy of a single phase
is lower than the free energy in all multi-phase configurations.This is
formulated as

$\mathcal{F}(T,\rho_{i})<(1-\lambda)\mathcal{F}(T,\rho_{i}^{{}^{\prime}})+\lambda\mathcal{F}(T,\rho_{i}^{"}),$
(17)

with

$\rho_{i}=(1-\lambda)\rho_{i}^{{}^{\prime}}+\lambda\rho_{i}^{"},$ (18)

where the two phases are denoted by prime and double prime and $\lambda$ is
the volume fraction. In formal terms, stability implies that the free energy
density is a convex function of the densities. Convexity implies that the
stability against separation into two phases also guarantees stability against
separation into an arbitrary number of phases. In other terms, it is
necessarily true that the symmetric matrix [27]

$\mathcal{F}_{ij}=\Big{(}\frac{\partial^{2}\mathcal{F}}{\partial\rho_{i}\partial\rho_{j}}\Big{)}_{T}$
(19)

is positive. This results is mechanical and diffusive stability conditions as

$\frac{\partial
P}{\partial\rho_{b}}\bigg{|}_{T,\alpha}>0\quad\textrm{and}\quad\frac{\partial\mu_{p}}{\partial\alpha}\bigg{|}_{T,P}<0$
(20)

If one of the stability conditions is violated, a system with two phases is
energetically favorable. The phase coexistence is governed by the Gibbs
conditions

$\displaystyle\mu_{q}^{\prime}T,\rho_{b}^{\prime})$ $\displaystyle=$
$\displaystyle\mu_{q}^{\prime\prime}(T,\rho_{b}^{\prime\prime}),(q=n,p)$ (21)
$\displaystyle P^{\prime}(T,\rho_{b}^{\prime})$ $\displaystyle=$
$\displaystyle P^{\prime\prime}(T,\rho_{b}^{\prime\prime})$

At the critical points, the pressure, density and temperature are written as
$P_{c},\rho_{c}$ and $T_{c}$. For asymmetric nuclear matter, they are
calculated by finding an inflation point at chemical potential isobars as

$\frac{\partial\mu_{q}}{\partial\alpha}\bigg{|}_{T=T_{c}}=\frac{\partial^{2}\mu_{q}}{\partial\alpha^{2}}\bigg{|}_{T=T_{c}}=0$
(22)

Furthermore, one can define isothermal incompressibility of nuclear matter at
finite temperature T and asymmetry $\alpha$ as

$K^{T}(\alpha,T)=9\bigg{(}\rho_{b}^{2}\frac{\partial^{2}F}{\partial\rho_{b}^{2}}\bigg{)}\bigg{|}_{\rho_{b}^{T}(\alpha,T)}$
(23)

Here,$\rho_{b}^{T}$ is the density where free energy has its minimum. The
isentropic incompressibility at entropy S and asymmetry $\alpha$ which is an
important quantity in supernova collapse is written as [28]

$K^{S}(\alpha,S)=9\bigg{(}\rho_{b}^{2}\frac{\partial^{2}E}{\partial\rho_{b}^{2}}\bigg{)}\bigg{|}_{\rho_{b}^{T}(\alpha,S)}$
(24)

##  III  Results and Discussions

###  III.1  Model Properties

In this work, we consider two recent E-RMF parameteres, namely, IOPB-I [14]
and G3 [15]. The IOPB-I set contains the quartic-order cross-coupling
$R(r)^{2}W(r)^{2}$ $(\lambda_{\omega}\neq 0)$ and self-coupling of isoscalar-
vector $W(r)^{2}$ ($\zeta_{0}$). This set produces the infinite nuclear matter
properties at saturation and supersaturation density in consistent with the
empirical data. The maximum mass is found to be 2.15 M$\odot$ which satisfy
the current GW170817, observational constraint. The G3 set on the other hand
is the most comprehensive parameter that contains all the nucleons and tensor
coupling terms in addition to several self and cross coupling components. It
is known to estimate neutron-skin thickness over a wide range in harmony with
the experimental data. It estimates the maximum neutron star mass 2.03
M$\odot$ with canonical mass radius of 12.69 km which is a desirable feature
in context to observational analysis and finite nuclei experiments. The main
feature of the set G3 is that it includes the couplings of nucleons to the
$\delta$ and $\rho$ mesons and cross-couplings of $\sigma$$\omega$ and
$\sigma$$\rho$ mesons. The G3 set has positive scalar self couplings $k_{3}$
and $k_{4}$ and $\zeta_{0}$ nearly equal to 1. The G3 set and IOPB-I set also
differ in the value of $\lambda_{\omega}$. IOPB-I has relatively small
$\lambda_{\omega}$ as compared to G3. In Table 1, we present the bulk matter
properties of nuclear matter for the G3 and IOPB-I sets and the corresponding
empirical values. It is clear that these sets satisfy the well-accepted set of
laboratorial, theoretical, and observational constraints.

Figure 1:  EoS of nuclear matter below saturation density for pure nuclear
matter at T=0 MeV. The shaded magenta region corresponds to the microscopic
chiral EFT (NN + 3N) [29] calculations. Inner graph represents the difference
between the neutron matter energy and pressure and the average energy and
pressure with 1$\sigma$ calculation uncertainty area.

In Fig 1 we compare the neutron matter binding energy and pressure with the
microscopic calculations based on chiral effective field theory (EFT) with
realistic two and three-nucleon interactions [29]. Inner graph represents the
difference between the neutron matter energy and pressure and the average
energy and pressure normalised to the uncertainty of the microscopic
calculations($\sigma$=$\delta P$) . The uncertainties are represented by
orange band, and they indicate that the points that lie inside this band are
within the $1\sigma$ error limits. The G3 set satisfy nicely the microscopic
constraints whereas IOPB-I also fall within the $1\sigma$ error limit below
saturation density. These both parameter also satisfy the constraint form
collective flow data in heavy-ion collisions and Kaon experiment along with
the GW170817 gravitational wave constraints [14]. These features along with
the agreement of bulk matter properties with empirical data motivate us to use
them to study the E-RMF sets with and without the $\delta$ meson. $\delta$
meson couplings are a necessary feature in the dense asymmetric nuclear
matter. In this work we intend to investigate the effects of $\delta$ meson in
the dilute asymmetric matter in the finite temperature limit. We also try to
explore role of different self and cross couplings at the finite temperature
properties of asymmetric nuclear matter.

Table 1: Bulk matter properties of nuclear matter for the IOPB-I and G3 parameter and their corresponding empirical values. | IOPB-I | G3 | Empirical Value |
---|---|---|---|---
$\rho_{0}(fm^{-3})$ | 0.149 | 0.148 | 0.148/0.185 [30] |
$E_{0}$ (MeV) | -16.10 | -16.02 | -15.0/-17.0 [30] |
M*/M | 0.593 | 0.699 | 0.55/0.6 [31] |
$J$ (MeV) | 33.30 | 31.84 | 30.0/33.70 [32] |
$L$ (MeV) | 63.58 | 49.31 | 35.0/70.0 [32] |
$K_{sym}$ (MeV) | -37.09 | -106.07 | -174.0/-31.0 [33] |
$Q_{sym}$ (MeV) | 862.70 | 915.47 | -494/-10 [34] |
$K$ (MeV) | 222.65 | 243.96 | 220/260 [35] |

###  III.2  Finite temperature properties

Figure 2: Free symmetry energy $F_{sym.}$, slope parameter L, curvature
$K_{sym}$, and isovector skewness parameter $Q_{sym}$ as a function of density
at various temperature for IOPB-I (solid lines) and G3 (dashed lines) sets.

(a)  (b)

Figure 3: (a) Thermal energy, pressure, chemical potential and free energy
density for fixed T=10 MeV for various $\alpha=0,0.5,1$ (b) same as in (a) but
for various T = 5, 10, 15, 20 MeV at fixed $\alpha=0.5$. The solid lines are
for neutron chemical potential and the same colour dotted line represent the
proton chemical potential.

The nuclear symmetry energy (NSE) is one of a crucial property of asymmetric
nuclear matter governing several areas of nuclear matter calculations like
reaction dynamics, phase stability, cooling in the neutron star etc.
Temperature dependence of NSE on the other hand provides the much needed
knowledge on the dynamical evolution of neutron star and isoscaling analyses
of heavy-ion induced reactions. NSE is not a directly measurable quantity in
experiments and is extracted from the observables related to it. Despite
numerous theoretical and experimental efforts, it is still not a very precise
parameter even for cold nuclear matter. In Fig. 2, we show the variation of
Free nuclear symmetry energy (FNSE) (more relevant quantity in finite
temperature case, see Eq. 11 ) the slope parameter $L_{sym}$, the isovector
incompressibility $K_{sym}$, and the isovector skewness $Q_{sym}$ with the
density up to 2 times the saturation density. The Free NSE is scaled towards
higher magnitude due to decrease in entropy preserving its characteristic
shape at a higher temperature for both IOPB-I and G3 sets. At higher density,
the temperature range considered here does not affect FSNE much. The slope
parameter which has a direct correlation with neutron skin thickness, electric
dipole polarizabilities etc also follows the similar trend for both the sets.
NSE estimated from these sets are also consistent with the HIC Sn + Sn and IAS
data [24]. The low values of $F_{sym}$ and $L_{sym}$ are the result of cross-
couplings of $\rho$ meson with $\omega$ meson in IOPB-I set and coupling of
$\sigma$ meson with $\omega$ meson in G3 set which predicts even lower
$F_{sym}$ due to the presence of isovector scalar $\delta$ meson. The $\delta$
meson has the positive effect on binding energy and helps to estimate the
$F_{sym}$, $L_{sym}$ and $K_{sym}$ within the permissible limit [22]. The
sinusoidal variation of $K_{sym}$ with density is also shown. $K_{sym}$ is
constrained recently by combining the data from PSR J0030+0451 and GW170817
estimating the $K_{sym}$ = $102^{+71}_{-72}$ MeV within $1\sigma$ error [33] .
The IOPB-I and G3 both fall within this constraint. The Variation of $Q_{sym}$
with density is almost independent of temperature for the IOPB-I set and a
small variation is observed for G3 set. The $Q_{sym}$ is the least constrained
property in any experiment and several models predict it with a large
variation [21].

To study the finite temperature effect, we isolate the thermal part of a given
function according to equations 15. The subtraction scheme applies to only
those variable which depend on the kinetic energy density [26]. Fig 3 shows
the thermal effect on various state variables at a fixed temperature and
$\alpha$ for IOPB-I and G3 parameter set. The common observation is that i) at
the fixed temperature, the thermal energy decreases with density. The
difference due to asymmetry disappears at high densities and thermal effects
become weak and thermal energy gets density independent with asymptotically
tending to zero, ii) At very low density, the thermal energy and pressure have
linear T dependence as for a free Boltzmann gas (non-degenerate limit). This
linearity is changed when matter becomes increasingly degenerate, iii)
Temperature effects are more prominent in thermal pressure as compared to
thermal energy and iv) The thermal chemical potential becomes saturated after
saturation density. In reference [25, 26], the thermal effects are found to be
dominated by the behaviour of effective mass. These calculations were done for
Skyrme and APR forces where the effective mass is Landau effective mass
whereas in E-RMF the effective mass is Dirac effective mass. For the
consistency, one can establish the relationship between both Dirac and Landau
mass in relativistic model with constant couplings as
$M^{*}_{Landau,(n,p)}=\sqrt{k^{2}+M^{*}_{Dirac,(n,p)}}$ [36]. The qualitative
behaviour of both the mass for a given E-RMF set remain same with
$M_{Landau}>M_{Dirac}$. In view of this, we can then compare the Dirac mass of
E-RMF set with the Landau mass of non-relativistic models. The effective mass
mentioned hereafter will therefore be Dirac effective mass only.

In Fig 4 the density dependence of Dirac effective mass for PNM is shown in
the left panel. The effective mass for G3 decreases at a relatively slower
pace as compared to IOPB-I set. Due to the presence of $\delta$ meson, neutron
and proton mass gets split which is not the case for IOPB-I set due to the
absence of $\delta$ meson. In right panel, the effective mass at saturation
density is plotted for different values of $\alpha$ for G3 parameter set. This
$\delta$ meson mechanism on effective mass is an in important phenomenon in
studying the drip line nuclei which are of astrophysical interests [37] and is
analysed in experiments such as PREX [38]. The effective mass is the input for
the computation of energy, pressure and chemical potential which is determined
self consistently. The behaviour of effective mass therefore clearly dictates
the thermal pressure and thermal energy. The G3 set with larger effective mass
estimates greater thermal contribution on state variables as compared to the
IOPB-I set with smaller effective mass. This is in consistent with the Fermi-
liquid theory and non-relativistic calculations [25]. For IOPB-I set thermal
pressure increases and then decreases beyond the saturation density but for G3
it gets saturated at higher density. Furthermore, the quantitative difference
in thermal energy and pressure between IOPB-I and G3 set is due to the
difference in the self-coupling of isoscalar-scalar $\sigma$ meson which is
responsible for 3N interaction that plays an important role in determining the
thermal pressure and energy. These behaviour are analogous to chiral 2N and 3N
interaction, although with larger thermal contribution as compared to the
many-body self-consistent Green’s function method [39]. The decrease in
thermal pressure after reaching to its maximum is the combining effect of
incompressibility of EoS at zero temperature and how rapid is the finite
temperature pressure.

Figure 4:  The effective mass for IOPB-I and G3 set. The left panel shows the
mass splitting for PNM in G3 set (solid lines) as compared to the IOPB-I set
(dashed line). In the right panel, the effective mass at saturation density is
shown for the G3 set.

The chemical potential at fixed T=10 MeV has interesting behaviour. $\mu_{n}$
at $\alpha=1$ is crossed over by $\mu_{n}$ at $\alpha<1$ with increasing
density while that is not the case for $\mu_{p}$. Moreover, the crossings of
$\mu_{n}$ occur at a higher density at larger temperature. Chemical potential
becomes saturated at higher density because of the increasing degeneracy at
higher density. The different nature of chemical potential is again the
consequence of effective mass along with the self and cross-coupling of
$\sigma$ meson. The thermal free energy tends to zero with increasing density
like the chemical potential. Comparing IOPB-I and G3 sets for thermal
properties, it is seen that the parameter set G3 has additional $\delta$ meson
coupling whose contribution increases with density. This contribution directly
impacts the effective mass (see Eq. 6a) which in turn decides the behaviour of
various variable studied above. The $\delta$ meson along with the $\sigma$
meson therefore has the direct contribution in thermal properties of EoS.

Fig 5 shows the variation of the thermal index ($\Gamma$) with density for
IOPB-I and G3 set at fixed temperature and asymmetry. By comparing the case of
fixed temperature and $\alpha$ with those of thermal energy and pressure (Fig.
3) it is certain that $\Gamma$ depends mainly on i) the stiffness of pressure
ii) behaviour of effective mass with respect to density and iii) $\alpha$. For
$\rho\rightarrow 0$, $\Gamma$ approaches the nonrelativistic ideal gas index
$\frac{5}{3}$. $\Gamma$ is very sensitive to the asymmetry at a fixed
temperature which is opposite to the non-relativistic calculations where the
peak of $\Gamma$ is insensitive to asymmetry [26]. Furthermore, it is immune
to temperature change for fixed $\alpha$. For IOPB-I set the maximum $\Gamma$
is 2.1 for PNM while 1.97 for SNM. The G3 set reports these values to be 1.96
and 1.87 respectively. The G3 set with larger effective mass estimate the
lower pressure and therefore larger thermal index as compared to the IOPB-I
set with lower effective mass. These results of $\Gamma$ from the newly
developed E-RMF sets are in consistent with the dynamics of the neutron star
merger where $\Gamma$ is taken 1.5 and 2 indicating that these two sets can be
used for the calculations of such event [40]. However, it is to be noted that,
in the astrophysical simulation like binary star and proto-neutron star,
$\Gamma$ is taken as a constant while here it varies with the density. The
behaviour of $\Gamma$ is in agreement as with EFT theory [39].

Figure 5: Thermal index for IOPB-I (solid lines) and G3 set (dash line) for
fixed temperature in the left panel and fixed $\alpha$ =0.5 in the right
panel.

(a)  (b)

Figure 6: (a) Isothermal and (b) Isentropic incompressibility at $\alpha$ =
0.3 and 0.5 for IOPB-I and G3 set at saturation density.

Nuclear matter incompressibility along the isothermal ($K^{T}$) and isentropic
($K^{S}$) paths is shown in Fig 6 for IOPB-I and G3 set for two values of
$\alpha$ i.e. 0.3 and 0.5. These values are taken due to their relevance in a
core-collapse supernova. The incompressibility at the saturation in the cold
nuclear matter is governed‘ by self-coupling of $\sigma$ meson. The
incompressibility of both IOPB-I and G3 set fall within the accepted empirical
value as prescribed by the giant monopole resonance i.e. 240 $\pm$ 20 MeV. At
finite temperature, we define incompressibility within two channels. One being
the isothermal and other isentropic incompressibility defined according to eq.
23 and 24 respectively. The isentropic incompressibility is more relevant
quantity in context to supernova explosion as the time scale of collapse is
less than 1 second and the process is adiabatic instead of isothermal. It
prompts us to use energy instead of free energy ( see eq 24). The
incompressibility (both isothermal and isentropic) decreases quadratically
with temperature with G3 having higher magnitude at each temperature and
entropy. It also decreases with increasing asymmetry. We show the temperature
dependence of $K^{T,S}/K^{0}$ and $(\rho^{T,S}/\rho^{0})^{2}$ in context to
their relation with respective incompressibility ( see Eq. 23 and 24.) Their
behaviour remains almost similar irrespective of change in asymmetry. These
results satisfy the calculations carried out using microscopic approaches [41,
42] thereby suggesting that these newly developed parameters not only describe
finite nuclei and cold nuclear matter but can also be used in studying the
phenomenon at finite temperature such as proto-neutron star and supernova
explosion.

###  III.3  Liquid-gas phase transition

The asymmetric nuclear matter is a two-component system with two conserved
charges Q (B, $I_{3}$). In a two-component system, although the total charge
remains conserved, their ratio can be different in different phases. The
constraint on T, Q, and $\rho$ which determine the energetic of the system,
forces vapour pressure and chemical potential to change during the phase
transition. Apart from mechanical instability, the diffusive instability
(fluctuations on the charge concentration) appears and is more relevant to
describe the asymmetric matter. The phase transition in the asymmetric matter
is therefore described by the following three regions:

  1. 1.

Isothermal Spinodal (ITS): describe the mechanical instability given by
$\frac{\partial P}{\partial\rho_{b}}$. It defines the critical temperature in
the symmetric matter.

  2. 2.

Diffusive Spinodal (DS): describe the chemical instability. It essentially
means that energy is required to add extra protons in the system at a fixed
temperature and pressure. The The critical isobar $P_{c}$ is estimated by
finding a inflation point
$\frac{\partial\mu_{p}}{\partial\alpha}\Big{|}_{P_{c}}$= 0. The corresponding
T=$T_{c}$ and $\rho=\rho_{c}$ are called the critical temperature and density
respectively.

  3. 3.

Coexistence Curve (CE): Set of points where eq. 20 along with the Gibbs
conditions are satisfied. This curve may contain the critical points. Unlike
symmetric matter case, here CE or binodal is 2 dimensional.

Figure 7: EoS of nuclear matter at various $\alpha$ at T= 10 MeV along with
the ITS, DS and CE curves for IOPB-I and G3 sets.

The complexity of phase transition in the asymmetric nuclear matter is shown
in Fig 7. As one move from symmetric to asymmetric matter, a new behaviour
distinct to the two-component system is allowed. Asymmetry is held constant
during the phase transition which forces the system to change its chemical
potential and consequently the pressure (shown by dashed line in left panel).
Due to charge fluctuation during this phase transition, the diffusive
instability appears and plays more important role than mechanical instability
in describing the phase transformation. The right panel shows all three curves
i.e. ITS, DS and CE and it is visible that diffusive instability has larger
area as compared to mechanical instability.

(a)  (b)

Figure 8: The binodal surface on the P-$\alpha$ plane is shown on the left
(a). and spinodal boundary in $\rho_{n}$-$\rho_{p}$ plane and
$\alpha$-$\rho_{b}$ plane is shown on the right (b). The black dashed line on
the spinodal shows the critical points.

Binodal as per the Gibbs condition given in Eq. 21 at T=10 MeV is plotted in
Fig 8 by geometrical construction where a rectangle is drawn on the chemical
potential isobars of neutrons and protons [43]. It is characterised by the
point of equal construction (EC), The point of maximal asymmetry (MA) and the
critical point which determine the edge of instability area. In the phase
coexistence region, the proton fraction of two-phase changes (a unique feature
of two-component system) and the phase with higher asymmetry exhibits a lower
density or vice-versa. At the critical temperature of symmetric matter, all
the three points (EC, MA, CP) coincide and the surface becomes a point. The
vertical dashed magenta line indicates that during the phase transition,
$\alpha$ remain constant and both phases follow different paths i.e. liquid
follow the path A1-A2 while gas phase evolves from B1 to B2 during the
isothermal compression. Finally the system leave the instability at A2. This
condition is called stable condensation. On the other hand, when the system is
prepared with $\alpha>\alpha_{c}$ ($\alpha$ at CP), it operates in the gaseous
phase only and this unique phenomenon is called retrograde condensation. The
spinodal according to Eq. 20 is plotted on the right side of Fig. 8 on both
$\rho_{n}$-$\rho_{p}$, $\alpha$-$\rho_{b}$ plane. Fig .7 and 8 provide a
complete description of phase transition in asymmetric nuclear matter.

In symmetric nuclear matter, the compressibility is the deciding factor for
critical parameters of phase transition whereas, the phase transition in the
asymmetric matter is characterised by symmetry energy. This can be verified
from Eq. 12 where the contribution of iso-spin asymmetry is reflected from the
free symmetry energy $F_{sym}$ and its slope $L_{sym}$. $K_{sym}$ and
$Q_{sym}$ are the higher order derivative of FNSE in the Taylor series which
are still not well constrained. We have used two E-RMF sets IOPB-I and G3 to
account for the various EoS properties on the phase transition in the
asymmetric matter. The detailed analysis of phase transition on the symmetric
matter using the IOPB-I and G3 set is discussed in [19]. For ANM, the
asymmetry in density is introduced by $\rho$ meson and is dictated by cross
coupling $\lambda_{\omega}(R^{2}W^{2})$. The G3 and IOPB-I set has
$\lambda_{\omega}$= 0.038 and 0.024 respectively. The corresponding vales of
$J$ and $L$ at T=0 MeV are given in Table 1 whereas their finite temperature
dependence is shown in Fig 2. The G3 set has an additional mass asymmetry
introduced by $\delta$ meson. The $\delta$ meson allows one to vary the
$L_{sym}$ without altering the symmetry energy $F_{sym}$. At a given
temperature, the G3 set has larger coexistence area and large values of CP and
MA as compared to the IOPB-I set sue to the $\delta$ meson. A large
coexistence area favours highly asymmetric gas in coexistence with less
asymmetric dense fluid. This has a direct consequence for the core-crust
transition and crust structure of neutron star. Opposite to SNM, where
$\zeta_{0}$ plays the determining role, the value of $\lambda_{\omega}$ decide
the ANM which in turn affect the $L_{sym}$. A greater $\lambda_{\omega}$
usually gives smaller $L_{sym}$ and vice-versa.

Figure 9:  $\alpha$ as a function of temperature and corresponding $\rho_{c}$
for the IOPB-I and G3 set.

The spinodal for the G3 set also has a larger area at any given temperature as
compared to the IOPB-I set. One can observe the major variation among two sets
in the coexistence densities in the $\alpha-\rho_{b}$ plane. This means that
the densities where different structure in non-homogeneous phase occur will be
different. This property is again determined by the $L_{sym}$. The G3 set with
smaller $L_{sym}$ estimates the larger $\alpha$ and $\rho_{c}$ at any given
temperature. This is shown in Fig 9 where the dependence of $\alpha$ and
$\rho_{c}$ is shown on temperature. The $\alpha-T$ plots signify the
temperature at which the diffusive instability disappears (also called
critical temperature). This critical temperature is not similar to symmetric
matter where mechanical instability decides the phase transition but is
determined according to $\frac{\partial\mu_{p}}{\partial\alpha}|_{P,T}=0$ and
$\frac{\partial^{2}\mu_{p}}{\partial\alpha^{2}}|_{P,T}=0$. In the E-RMF sets
with constant couplings, the inflation point for proton and neutron coincides
having synchronous behaviour. This might not be the case with density-
dependent coupling sets [44, 45]. $\alpha$ decreases smoothly at low
temperatures but after $T>0.5T|_{\alpha=0}$, there is a steep fall in the
$\alpha$. The G3 set estimate larger $\alpha$ at a particular T due to its
smaller value of $L_{sym}$ and greater value of $\lambda_{\omega}$. This same
trend is observed in the $\rho_{c}$. These trends are similar to the
references [6, 16], where any one coupling in a parameter set were varied
keeping other fixed to obtain different $L_{sym}$. The agreement of those
trends while comparing two different parameter sets with almost same symmetry
energy indicated that the correlation between different properties of phase
transition still holds as in case of SNM [46]and these can be exploited to
constraint the EoS which do not take critical temperature into the account
[19, 47].

###  III.4  Effect of electrons

In a physical system, the electrons are present so that Coulomb energy do not
diverge. They are are included in EoS as a free non-interacting relativistic
Fermi gas described by [17]

Figure 10: The EoS with and without the electrons. Solid line represents the
nuclear matter without electrons and dashed line represents with electrons.

$L_{e}=\bar{\psi_{e}}[i\gamma_{\mu}\partial^{\mu}-m_{e}]\psi_{e},$ (25)

where $L_{e}$ is the Lagrangian, $m_{e}$ is the mass of electron. Since the
electrons only compensate the proton charge, we have
$\rho_{p}$=$\rho_{e}$=$\frac{1}{\pi^{2}}\int k^{2}dk(n_{ke}-\bar{n}_{ke})$.
Where $n_{ke}$ and $\bar{n}_{ke}$ are the Fermi integral for electrons and
positrons. Fig. 10 shows the effect of electrons on the EoS for IOPB-I and G3
parameter sets at T=0 MeV. The effect of electrons is dominant for the matter
with less asymmetry as the electron density becomes high to compensate for the
larger proton density. Electrons are taken as non-interacting particles and
therefore the underlying nature of a parameter set in unaltered. Electrons
have high Fermi energy and therefore, makes the system devoid of the
instability. Both the IOPB-I and G3 sets have no spinodal when electrons are
included for T= 5MeV. No spinodal means that stellar matter at $\beta$
equilibrium will be uniform at temperature above 5 MeV [12]. This is in
consistent with the various calculations of neutron star core-crust
transition.

To further understand the implication of electrons in the EoS, we study the
adiabatic index. In the processes like supernovae explosions and neutron
stars, the compression and rarefactions modes of vibration are adiabatic or
isentropic instead of isothermal [26] . The adiabatic index is related to the
stiffness of EoS and is given by

$\Gamma_{s}=\frac{\rho_{b}}{P}\frac{\partial P}{\partial\rho_{b}}\Big{|}_{s}.$
(26)

Figure 11: Adiabatic index $\Gamma_{s=0}$ for the G3 and IOPB-I set with
various asymmetry. Solid black line include only nucleons while red line
indicates the presence of electrons.

$\Gamma_{s=0}$ for the two models employed here is shown in Fig. 11. The solid
black curve represents the nucleon only while the red dashed curve include the
contribution from electrons. $\Gamma_{s=0}$ corresponding to nucleons goes
negative in some density regions showing the mechanical instability. For low
and high densities it varies asymptotically. The inclusion of electrons
restore the mechanical instability and value of $\Gamma_{s=0}$ increases
gradually around subsaturtaion density and become asymptotically constant at
low and high densities. These observations can be understood quantitatively by
examining the baryon and electron pressure as shown in Fig. 10. For
$\rho\rightarrow$ 0, $\Gamma_{s=0}$ tend to the 4/3 which is due to the
relativistic electrons and is an important requirement for the stability of
supernova simulation. As asymmetry rises, this value goes to 5/3 for pure
neutron matter. Although the underlying properties for $\Gamma_{s}$ are same
for both the model, but the position of instability and highest value of
$\Gamma_{s}$ in case of matter with electrons, is essentially determined by
the pressure due to baryon. The parameter set IOPB-I and G3 do not break the
causality condition on speed of sound [48]. The presence of electron in the
system also impacts the speed of sound $C_{s}^{2}=\frac{\partial p}{\partial
E}$. Addition of electrons do not yield the nonphysical region in low density
as is seen in the nuclear matter system without leptons. At higher density,
the electrons impart significant impacts on the more symmetric matter making
it smoother as compared to the asymmetric matter.

##  IV  Summary and Outlook

The primary aim of this work is to study the thermal properties of hot and
dilute isospin asymmetric nuclear matter within effective relativistic mean-
field (E-RMF) formalism. Although the thermodynamics of symmetric nuclear
matter is explored, the isospin effects are still not understood at finite
temperature. In this study, we consider the homogeneous nuclear matter and
analyse it using two E-RMF sets i.e. IOPB-I and G3 at different values of
temperature and isospin asymmetry because of their relevance in astrophysical
simulations. Both of these sets estimate the EoS at low density for pure
neutron matter in agreement with microscopic chiral EFT (NN + 3N)
calculations. Our motivation for selecting IOPB-I and G3 parameter sets lies
in the fact that these models have comparable symmetry energy at saturation
but differ in the value of slope parameter $L_{sym}$ which is an important
quantity for deciding the instabilities. Furthermore, the G3 set introduces
the mass asymmetry with the inclusion of $\delta$ meson which is responsible
for the different effective mass of neutrons and protons which make them
distinguishable. We study the temperature dependence of free nuclear symmetry
energy ($F_{sym}$) and its higher order derivative slope parameter
$(L_{sym,})$, skewness parameter ($Q_{sym}$) and curvature parameter
($K_{sym}$). The $F_{sym}$ increases with temperature at a given density due
to a decrease in entropy density. The higher-order derivative of $F_{sym}$
preserves the zero temperature behaviour with a slight change in magnitude
which shows that one can use the zero-temperature value of these parameters to
compare the relevant quantities at any given temperature.

To study the finite temperature effect, we separate the thermal component from
the zero temperature EoS. The thermal effects are sensitive to asymmetry at
low density. It is observed that the thermal effects in E-RMF formalism depend
mainly on the density dependence of effective mass. The effective mass is
calculated self consistently which depends on the $\sigma$ and $\delta$
mesons. A larger effective mass estimates larger thermal effects on the state
variables. This feature is also seen in non-relativistic formalisms. We have
calculated the isothermal thermal index ($\Gamma$) in view of its relevance in
supernovae. $\Gamma$ is very sensitive to the isospin asymmetry at a fixed
temperature which is opposite to the non-relativistic case. The isothermal and
isentropic incompressibility varies parabolically with temperature. The trends
of incompressibilities are in agreement with available microscopic
calculations. The phase transition is studied for the asymmetric nuclear
matter considering a two-component system with two conserved charges i.e.
Baryon number and isospin. The G3 set due to its low $L_{sym}$ estimates the
higher value of maximal asymmetry and critical pressure. The presence of
$\delta$ meson has a positive effect on binding energy and therefore
influences the boundary of spinodal. The critical density and asymmetry are
also larger for the G3 set, which can be attributed to its lower $L_{sym}$.
The Value of $L_{sym}$ is determined mainly by cross-coupling of $\rho$ and
$\omega$ meson and $\delta$ meson. One can say that a larger value of
$\lambda_{\omega}$ estimate the larger instability in asymmetric nuclear
matter. Critical asymmetry is a quadratic function of temperature and exhibits
different behaviour in the low and high-temperature range.

Finally, we study the effect of the electrons in EoS of nuclear matter and its
instability. Electrons due to their high Fermi energy make the system devoid
of instabilities. We study the adiabatic index ($\Gamma_{S=0}$) of matter with
and without the inclusion of electrons. The $\Gamma_{S}$ with electrons become
asymptotically constant at low and high densities with a small variation near
the saturation density. The density of this hump predominantly depends on the
baryon pressure. The electron being non-interacting particle do not alter the
underlying nature of the force parameter.

The present calculations can be extended to the case of inhomogeneous matter
where nuclei coexist with surrounding nucleon gas. Such studies are essential
to estimate an equation of state ranging from zero density to density of inner
core of neutron star.

## References

  * Couch [2017] S. M. Couch, Philos. Trans. Royal Soc. A 375, 20160271 (2017).
  * Roma  _et al._ [2019] V. Roma, J. Powell, I. S. Heng, and R. Frey, Phys. Rev. D 99, 063018 (2019).
  * Müller  _et al._ [2010] B. Müller, H.-T. Janka, and H. Dimmelmeier, ApJS 189, 104 (2010).
  * Stapleford  _et al._ [2020] C. J. Stapleford, C. Fröhlich, and J. P. Kneller, Phys. Rev. D 102, 081301 (2020).
  * Sawada and Maeda [2019] R. Sawada and K. Maeda, ApJ 886, 47 (2019).
  * Alam  _et al._ [2017] N. Alam, H. Pais, C. Providência, and B. K. Agrawal, Phys. Rev. C 95, 055808 (2017).
  * Nishizaki  _et al._ [1994] S. Nishizaki, T. Takatsuka, and J. Hiura, Prog. Theor. Phys 92, 93 (1994).
  * Liu [2010] M.-Q. Liu, Res. Astron. Astrophys. 11, 91 (2010).
  * Brown  _et al._ [2018] E. F. Brown, A. Cumming, F. J. Fattoyev, C. J. Horowitz, D. Page, and S. Reddy, Phys. Rev. Lett. 120, 182701 (2018).
  * Abbott  _et al._ [2017a] B. P. Abbott _et al._ (LIGO Scientific Collaboration and Virgo Collaboration), Phys. Rev. Lett. 119, 161101 (2017a).
  * Abbott  _et al._ [2017b] B. P. Abbott _et al._ , ApJ 848, L12 (2017b).
  * Avancini  _et al._ [2008] S. S. Avancini, D. P. Menezes, M. D. Alloy, J. R. Marinelli, M. M. Moraes, and C. Providência, Phys. Rev. C 78, 1 (2008).
  * Grams  _et al._ [2017] G. Grams, A. M. Santos, P. K. Panda, C. m. c. Providência, and D. P. Menezes, Phys. Rev. C 95, 055807 (2017).
  * Kumar  _et al._ [2018] B. Kumar, S. K. Patra, and B. K. Agrawal, Phys. Rev. C 97, 1 (2018), 1711.04940 .
  * Kumar  _et al._ [2017] B. Kumar, S. K. Singh, B. K. Agrawal, and S. K. Patra, Nucl. Phy. A 966, 197 (2017), 1705.02621 .
  * Sharma  _et al._ [2020] B. Sharma, S. Sathees, M. Meghaa, and T. Jha, Nucl. Phy. A 1002, 121974 (2020).
  * Avancini  _et al._ [2006] S. S. Avancini, L. Brito, P. Chomaz, D. P. Menezes, and C. Providência, Phys. Rev. C 74, 1 (2006).
  * Avancini  _et al._ [2004] S. S. Avancini, L. Brito, D. P. Menezes, and C. Providência, Phys. Rev. C 70, 1 (2004).
  * Parmar  _et al._ [2021] V. Parmar, M. K. Sharma, and S. K. Patra, J Phys G Nucl Part Phys 48, 025108 (2021).
  * Del Estal  _et al._ [2001] M. Del Estal, M. Centelles, X. Viñas, and S. K. Patra, Phys. Rev. C 63, 024314 (2001).
  * Kumar  _et al._ [2020] A. Kumar, H. Das, S. Biswal, B. Kumar, and S. Patra, The Eur. Phys. J. C 80, 1 (2020).
  * Singh  _et al._ [2014] S. K. Singh, S. K. Biswal, M. Bhuyan, and S. K. Patra, Phys. Rev. C 89, 044001 (2014).
  * Bednarek  _et al._ [2020] I. Bednarek, J. Sładkowski, and J. Syska, Symmetry 12, 10.3390/sym12060898 (2020).
  * Tsang  _et al._ [2009] M. B. Tsang, Y. Zhang, P. Danielewicz, M. Famiano, Z. Li, W. G. Lynch, and A. W. Steiner, Phys. Rev. Lett. 102, 122701 (2009).
  * Constantinou  _et al._ [2014] C. Constantinou, B. Muccioli, M. Prakash, and J. M. Lattimer, Phys. Rev. C 89, 065802 (2014).
  * Constantinou  _et al._ [2015] C. Constantinou, B. Muccioli, M. Prakash, and J. M. Lattimer, Phys. Rev. C 025801, 1 (2015).
  * Müller and Serot [1995] H. Müller and B. D. Serot, Phys. Rev. C 10.1103/PhysRevC.52.2072 (1995).
  * HADDAD [2003] S. HADDAD, Int. J. Mod. Phys. E 12, 125 (2003).
  * Hebeler  _et al._ [2013] K. Hebeler, J. M. Lattimer, C. J. Pethick, and A. Schwenk, ApJ 773, 11 (2013).
  * Bethe [1971] H. A. Bethe, Annu. Rev. Nucl. Sci. 21, 93 (1971).
  * Marketin  _et al._ [2007] T. Marketin, D. Vretenar, and P. Ring, Phys. Rev. C 75, 024304 (2007).
  * Danielewicz and Lee [2014] P. Danielewicz and J. Lee, Nucl. Phy. A 922, 1 (2014).
  * Zimmerman  _et al._ [2020] J. Zimmerman, Z. Carson, K. Schumacher, A. W. Steiner, and K. Yagi, (2020), arXiv:2002.03210 [astro-ph.HE] .
  * Cai and Chen [2017] B.-J. Cai and L.-W. Chen, Nuclear Science and Techniques 28, 185 (2017).
  * Garg and Colò [2018] U. Garg and G. Colò, Prog. Part. Nucl. Phys. 101, 55 (2018).
  * Chen  _et al._ [2007] L.-W. Chen, C. M. Ko, and B.-A. Li, Phys. Rev. C 76, 054316 (2007).
  * Horowitz and Piekarewicz [2001] C. J. Horowitz and J. Piekarewicz, Phys. Rev. Lett. 86, 5647 (2001).
  * Abrahamyan  _et al._ [2012] S. Abrahamyan _et al._ (PREX Collaboration), Phys. Rev. Lett. 108, 112502 (2012).
  * Carbone and Schwenk [2019] A. Carbone and A. Schwenk, Phys. Rev. C 100, 1 (2019), 1904.00924 .
  * Yasin  _et al._ [2020] H. Yasin, S. Schäfer, A. Arcones, and A. Schwenk, Phys. Rev. Lett. 124, 092701 (2020).
  * Modarres and Bordbar [1998] M. Modarres and G. H. Bordbar, Phys. Rev. C 58, 2781 (1998).
  * Bombaci  _et al._ [1993] I. Bombaci, T. Kuo, and U. Lombardo, Phys. Lett. B 311, 9 (1993).
  * Lattimer and Ravenhall [1978] J. M. Lattimer and D. Ravenhall, ApJ 223, 314 (1978).
  * Fedoseew and Lenske [2015] A. Fedoseew and H. Lenske, Phys. Rev. C 91, 034307 (2015).
  * Nikšić  _et al._ [2002] T. Nikšić, D. Vretenar, P. Finelli, and P. Ring, Phys. Rev. C 66, 024306 (2002).
  * Lourenço  _et al._ [2017] O. Lourenço, M. Dutra, and D. P. Menezes, Phys. Rev. C 95, 065212 (2017).
  * Yang  _et al._ [2019] S. Yang, B. N. Zhang, and B. Y. Sun, Phys. Rev. C 100, 054314 (2019).
  * Das  _et al._ [2020] H. C. Das, A. Kumar, B. Kumar, S. K. Biswal, T. Nakatsukasa, A. Li, and S. K. Patra, Mon. Notices Royal Astron. Soc. 495, 4893 (2020).