# Properties of hot finite nuclei and associated correlations with infinite
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

This work aim to study the various thermal characteristics of nuclei in view
of the saturation and critical behavior of infinite nuclear matter. The free
energy of a nucleus is parametrized using the density and temperature-
dependent liquid-drop model and interaction among nucleons is worked out
within the effective relativistic mean-field theory (E-RMF). The effective
mass (m,∗) and critical temperature of infinite symmetric nuclear matter
($T_{c}$) of a given E-RMF parameter force play a seminal role in the
estimation of thermal properties. A larger (m∗) and $T_{c}$ of the E-RMF set
estimate larger excitation energy, level density, and limiting temperature
$(T_{l})$ for a given nucleus. The limiting temperature of a nucleus also
depends on the behavior of the nuclear gas surrounding the nucleus, making the
equation of state (EoS) at subsaturation densities an important input. A stiff
EoS in the subsaturation region estimates a higher pressure of the nuclear gas
making it less stable. Since the $T_{c}$ plays an important part in these
calculations, we perform a Pearson correlation statistical study of fifteen
E-RMF parameter sets, satisfying the relevant constraint on EoS. Effective
mass seems to govern the thermal characteristics of infinite as well as finite
nuclear matter in the framework of E-RMF theory.

††preprint: APS/123-QED

##  I  Introduction

One of the astonishing universality in the laws of nature is the resemblance
between the nuclear and the molecular force. Molecular force is of van der
Waals type and nuclear force behaves similarly, albeit on the different energy
scale. Therefore one may arrive at the notion that nuclear matter should
undergo a liquid-gas phase transition (LGPT) like a classical liquid drop.
This phenomenon of LGPT in both infinite nuclear matter and finite nuclei is
an important feature of heavy-ion-induced reactions (HIR) [1, 2, 3]. In these
reactions, the participating hot nuclei undergo multi-fragmentation after the
initial dynamic stage of compression upon reaching sub-saturation density
($\approx 0.2\rho_{0}$) [4]. In this sub-saturation density region, the
properties of nuclei are modified [5, 6, 7] which are very essential for the
understanding of thermodynamics of hot nuclei, and the medium in which they
are created. The knowledge of the nuclear matter in the sub-saturation region
is also important in context to core-collapse supernovae [8], neutron star
crust and giant astrophysical explosions where nuclear matter minimizes its
energy by forming clusters at temperature $\approx 4$ MeV [9].

The $\gamma$ ray emission is the dominating process in the nucleus at low
excitation energy, where nuclear levels are well resolved. As excitation
energy increases slightly, the nuclear energy levels are substantially
modified. The single-particle energy states become degenerate and nuclear
shells start melting leading to a spherical nucleus after a temperature
usually known as shell melting temperature T${}_{m}\approx 1-2$ MeV [10].
Further increase in temperature leads to nucleon emission, which is generally
studied within the framework of nuclear statistical equilibrium. On further
heating, the nucleon evaporation turns violent, and at a certain limiting
temperature, Tl, a new decay channel known as multi-fragmentation becomes
dominant. This Tl was found to be $\approx 5.6$ MeV for the mass region A
$\approx$ 90 in ALADIN experiment [11]. Nuclear multi-fragmentation occurs in
the region of spinodal or phase instability boundary in the nuclear matter
phase diagram [12]. The nucleus, which resembles a hot liquid drop, expands
because of thermal pressure and moves to the spinodal region where it is
surrounded by a nucleon gas. As the spinodal is the region of instability, the
nucleus explodes violently and the process is known as multi-fragmentation at
freeze-out volume $\approx 7V_{0}$ [13]

There have been several qualitative attempts to study the limiting temperature
of nuclei in terms of Coulomb instability, where the EoS of infinite matter is
taken from various theoretical frameworks such as Skyrme effective NN
interaction [14, 15], microscopic EoS such as Friedman and Pandharipande,
finite temperature relativistic Dirac-Brueckner, chiral perturbation theory
[16, 17, 18], EoS considering the degeneracy of the Fermi system [19]
relativistic calculations using quantum hadrodynamics and Thomas-Fermi
approach [20, 21, 22], Gogny interactions [23], chiral symmetry model [24].
Some calculations have been carried out by analyzing the plateau in caloric
curve obtained from various experimental observations [25]. These calculations
give a qualitative picture of $T_{l}$ and it is seen that $T_{l}$ is model
dependent and hence needs to be investigated for appropriate outcome.

To understand the properties of LGPT in nuclei and most importantly the
temperature at which the nucleus undergoes multi-fragmentation and loses its
entity, we use one of the most successful effective relativistic mean-field
(E-RMF) theory [26]. The E-RMF is the effective theory of hadrons as per
quantum chromodynamics (QCD), which successfully explains the nuclear matter
properties from finite nuclei to the neutron star and gives valuable inputs in
the supernovae simulations. The E-RMF formulation calculates the volume energy
of infinite nuclear matter on which the finite size corrections: surface,
symmetry, Coulomb are added to evaluate the properties of a realistic nucleus.
The idea behind using the E-RMF framework for the bulk volume energy part is
that the nuclear drop is usually surrounded by a nucleon gas in complete
thermodynamic equilibrium. To calculate the properties of such a system, one
usually needs to solve the Gibbs conditions [27] where it is expected that the
same equation of state (EoS) are used for the gaseous as well as the liquid
phase.

The aim of present study is twofold: First, we investigate the properties of
hot isolated nuclear drop by studying the variation of thermodynamic variables
such as excitation energy, entropy, level density, fissility etc. We compare
them with available experimental or microscopic theoretical calculations [28,
10]. The second and important part of this work is the qualitative analysis of
the limiting temperature of a hot nucleus. In HIRs, nuclei can be heated to
their limiting temperature which provides an opportunity to investigate the
collective motion of nucleons, and their highly chaotic and disordered
behavior at high excitation energy. We use E-RMF parameter sets namely
FSUGarnet, G3, IOPB-I, and most successful NL3 [29] for the volume energy of a
nucleus. The temperature-dependent surface energy term depends on the $T_{c}$
which is calculated for these individual E-RMF parameter sets. In the analysis
of critical properties of infinite nuclear matter using these E-RMF sets in
[12], we found that the $T_{c}$ is not a well-constrained quantity and the
majority of E-RMF sets that satisfy the relevant observational and
experimental constraints on EoS underestimates it. Since the experimental
value of $T_{c}$ is calculated by extrapolating the data from multi-
fragmentation reaction data on finite nuclei, it is interesting to see the
variation of $T_{l}$ of finite nuclei using different E-RMF forces. To further
generalize the relationship between various saturation properties of infinite
nuclear matter, its critical properties, and the limiting properties of a hot
nucleus, we have used fifteen parameter sets that lie within the allowed
incompressibility range and satisfy other constraints [30]. An effort is made
to establish correlations among these properties.

The paper is organized as follows: In Section II, we have discussed the
formalism to calculate the energy of a finite nucleus from infinite nuclear
matter . In subsequent subsections II.3, and II.4 we discuss the formalism for
the excitation energy, fissility parameter, and limiting temperature along
with the lifetime of the hot nuclear liquid drop. In section III, we have
discussed results related to various properties of hot nucleus. Finally, we
summarise our results in Section IV.

##  II  Theoretical formalism

###  II.1  From infinite matter to finite nuclei

We consider a nucleus to be a liquid drop and resort to the conventional
liquid-drop model to define the free energy of the drop with given mass number
A, proton number Z, and neutron number N as

$\displaystyle F_{A}(\rho,T)=$
$\displaystyle\mathcal{F}_{v}(\rho,T)A+\mathcal{F}_{corr}(\rho,T),$ (1)

Where $\mathcal{F}_{v}(\rho,T)$ is the free energy of infinite symmetric
nuclear (SNM) matter calculated within the effective-relativistic mean-field
theory (E-RMF) corresponding to the volume and $\mathcal{F}_{corr}$ is the
finite size correction due to surface, symmetry, and Coulomb effects and is
written as

$\displaystyle\begin{aligned} \mathcal{F}_{corr}(\rho,T)=&f_{surf}(\rho,T)4\pi
R^{2}+f_{sym}(\rho,T)\frac{(N-Z)^{2}}{A}\\\ &+f_{Col}.\end{aligned}$ (2)

Here $R$ is the radius of the drop and is defined as

$R=\quantity(\frac{3A}{4\pi\rho(T)})^{1/3}.$ (3)

The coefficient of free surface energy (FSE) ($f_{surf}(\rho,T)$) is a crucial
parameter that introduces the surface and is assumed to be factorized and
density-dependent [31]. This is written as

$\displaystyle\begin{aligned}
f_{surf}(\rho,T)=\alpha_{surf}(\rho_{0},T=0)\mathcal{D}(\rho)\mathcal{Y}(T).\end{aligned}$
(4)

Here, $\alpha_{s}(\rho_{0},T)$ is the surface energy coefficient at saturation
density ($\rho_{0}$) of infinite SNM. As the density of liquid evolves, the
surface energy should change. Therefore the density dependence is taken from
and is written as [32]

$\mathcal{D}(\rho)=1-\frac{\mathcal{K_{\rho}}}{2}\quantity(\frac{\rho-\rho_{0}}{\rho_{0}})^{2}$
(5)

The temperature dependence of the coefficient of FSE is another significant
parameter that ensures that the surface tension vanishes above a certain
temperature $T_{c}$. In this work, we use two parametrizations of the
temperature dependence of surface energy which are widely used in the
calculation of multi-fragmentation in nuclei and structure of neutron star
crust. The first expression is taken from [33] which takes into account the
plane sharp interface between liquid and gaseous phase of nuclear matter in
equilibrium. It is written as

$\displaystyle\mathcal{Y}(T)$ $\displaystyle=$
$\displaystyle\quantity(\frac{T_{c}^{2}-T^{2}}{T_{c}^{2}+T^{2}})^{\frac{5}{4}}.$
(6)

The second expression is derived based from the semiclassical modified Seyler-
Blanchard interaction and takes the form [34] as

$\displaystyle\mathcal{Y}(T)$ $\displaystyle=$
$\displaystyle\quantity(1+1.5\frac{T}{T_{c}})\quantity(1-\frac{T}{T_{c}})^{\frac{3}{2}}.$
(7)

In these expressions, $T_{c}$ is the critical temperature of liquid-gas phase
transition in infinite SNM. $\alpha_{s}(\rho_{0},T)$ is taken as 1.15 MeV fm-2
and $\mathcal{K_{\rho}}$ is a dimensionless constant taken to be 5.0 as
prescribed in [35].

The coefficient of free symmetry energy (FSYE) ($f_{sym}(\rho,T)$) which
depend on the mass number of liquid drop is written as

$f_{sym}(\rho,T)=\alpha_{sym}(\rho,T=0)\mathcal{G}(T)\quantity(\frac{\rho}{\rho_{0}})^{\Gamma}.$
(8)

Here, $\alpha_{sym}(\rho,T=0)$ is further defined as

$\alpha_{sym}(\rho,T=0)=\frac{J}{1+\mathcal{C}A^{-1/3}},$ (9)

where $J$ is the symmetry energy of cold SNM and is taken as $31$ MeV and
$\mathcal{C}=2.4$. The dependence of $f_{sym}(\rho,T)$ on the temperature is
ensured using the function $\mathcal{G}(T)$ in line with the infinite matter
calculations that suggest that free FSYE increases with temperature [27]. It
is taken in a schematic form as [36]

$\mathcal{G}(T)=(1+\mathcal{X}_{1}T+\mathcal{X}_{2}T^{2}+\mathcal{X}_{4}T^{4}),$
(10)

where $\mathcal{X}_{1}=-0.00848$, $\mathcal{X}_{2}=0.00201$,
$\mathcal{X}_{4}=0.0000147$ with dimensions as relevant power of unit of
temperature. The density dependence is ensured with the $\Gamma=0.69$ in
congruence with the experimental observations [37]. The free Coulomb energy
FCE which is otherwise absent in the infinite matter is responsible for the
Coulomb instability of the liquid drop. It is taken as [38]

$f_{Col}=\frac{3}{5}\frac{Z^{2}e^{2}}{R}\quantity(1-\frac{5}{2}\quantity(\frac{b}{R})^{2}),$
(11)

where b is the surface thickness which is also a temperature-dependent
quantity taken as

$b\approx 0.72(1+0.009T^{2}).$ (12)

The ratio $\frac{b}{R}$ increases with temperature resulting in the reduction
of Coulomb free energy in addition to that arising from the expansion of bulk
matter. We do not include the exchange term in Coulomb free energy due to its
low contribution. In the construction of the liquid drop, we do not include
other finite-size effects such as pairing and shell corrections because they
become insignificant for temperature $>1-2$ MeV due to shell melting.

###  II.2  E-RMF at zero and finite temperature

The relativistic mean-field model (RMF) treats nucleons as Dirac particles
that interact in the relativistic covariant way by exchanging virtual mesons
namely, isoscalar-scalar $\sigma$ meson, isoscalar-vector $\omega$ meson,
isovector-vector $\rho$ meson, and isovector-vector $\delta$ mesons. Further
modification in the RMF model leads to effective relativistic mean-field
formalism (E-RMF) which has the advantage that one can ignore the
renormalization and divergence of the system. In E-RMF, the Lagrangian
contains an infinite number of terms consistent with the underlying QCD
Symmetries. The ratio of meson fields to the nucleon mass is used for the
expansion and truncation scheme. Taking recourse to the naturalness and naive
dimensional analysis (NDA), it is possible to truncate the Lagrangian at the
given level of accuracy. The detailed formalism and theoretical background of
E-RMF can be found in [27, 29, 10, 12, 39, 40] and here we present a general
outline of the formalism. The typical E-RMF Lagrangian fro infinite nuclear
matter is written as

$\displaystyle\mathcal{E}=\psi^{\dagger}(i\alpha.\gradient+\beta[M-\Phi(r)-\tau_{3}D(r)]+W(r)+\frac{1}{2}\tau_{3}R(r)$
(13)
$\displaystyle+\frac{1+\tau_{3}}{2}A(r))\psi+\quantity(\frac{1}{2}+\frac{k_{3}\Phi(r)}{3!M}+\frac{k_{4}}{4!}\frac{\Phi^{2}(r)}{M^{2}})\frac{m^{2}_{s}}{g^{2}_{s}}\Phi(r)^{2}$
$\displaystyle-\frac{\zeta_{0}}{4!}\frac{1}{g^{2}_{\omega}}W(r)^{4}\\-\frac{1}{2}\quantity\Big(1+\eta_{1}\frac{\Phi(r)}{M}+\frac{\eta_{2}}{2}\frac{\Phi^{2}(r)}{M^{2}})\frac{m^{2}_{\omega}}{g^{2}_{\omega}}W^{2}(r)$
$\displaystyle-\frac{1}{2}\quantity\Big(1+\eta_{\rho}\frac{\Phi(r)}{M})\frac{m^{2}_{\rho}}{g^{2}_{\rho}}R^{2}(r)-\Lambda_{\omega}(R^{2}(r)W^{2}(r))$
$\displaystyle+\frac{1}{2}\frac{m^{2}_{\delta}}{g^{2}_{\delta}}(D(r))^{2}.$

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 and the third component
of energy-momentum tensor yields the energy and pressure density [29, 12]. For
cold matter i.e. T=0 case, the complete field equations and related density,
energy and pressure integrals are well given in [29, 41] . At $T\neq 0$, the
energy and pressure for finite temperature can be written by using the concept
of canonical thermodynamic potential $\Omega$ which are also documented in
[12, 27]. The Dirac effective mass which is calculated self consistently is
written as

$\displaystyle M^{*}_{n/p}=M-\Phi(r)\pm D(r).$ (14)

###  II.3  Excitation energy, level density and fissility parameter

The binding energy $E(T)$ of a liquid-drop with given A and Z can be found by
minimizing Eq. 1 to obtain the density of a nucleus at a given temperature.
The excitation energy then attain a simple form as $E^{*}(T)=E(T)-E(T=0)$,
which essentially signifies the difference of binding energy of ground level
to that at any given temperature. Here the energy can be determined from the
relation

$E(T)=\mathcal{F}(T)+TS.$ (15)

The inter-relationship between temperature, excitation energy, and entropy
which determine the level density parameter (a) is written as [42]

$E^{*}=aT^{2},\hskip 14.22636ptS=2aT,\hskip 14.22636ptS^{2}=4aE^{*}.$ (16)

In a heavy nucleus, The competition between Coulomb and surface energy
determines the fissility of the nucleus: the larger the ratio, the smaller is
the fission barrier. The fissility parameter is given by dimensionless
parameter $x(T)$ which is defined as [38]

$x(T)=\frac{\mathcal{F}^{0}_{Col}}{2\mathcal{F}^{0}_{s}},$ (17)

here superscript signifies the spherical drop. We then define the fission
barrier or potential energy of deformed drop in terms of standard liquid-drop
conventions as

$\mathcal{B}_{f}(T)=((B_{s}-1)+2x(T)(B_{c}-1)).$ (18)

Here, $B_{s}$ and $B_{c}$ are the surface and Coulomb energy at saddle point
in the units of surface and Coulomb free energy respectively. Values of
$B_{s}$ and $B_{c}$ can be determined from [43] where these values are
tabulated against fissility parameter $x(T)$.

###  II.4  Limiting temperature

The most important aspect of the thermodynamics of a finite nucleus is its
multi fragmentation which can be explained in terms of liquid-gas phase
transition. We consider the nucleus to be a spherical drop of liquid
surrounded by a gas of nucleons under the assumptions that the hot nucleus at
a temperature T is surrounded by homogeneous gas of symmetric nuclear matter
in a complete mechanical and chemical thermodynamic equilibrium with no
exchange of particle. A sharply defined surface separates the liquid and
gaseous phase and there is no interaction between nucleons in the gaseous and
liquid phase so that the gas remains unchanged and without Coulomb effect.
These approximations then lead us to the following modified phase equilibrium
condition similar to the infinite matter case.

$\displaystyle P_{0}^{g}(\rho^{g},T)=P^{l}_{0}(\rho^{l},T)+\delta P^{l},$
(19a)
$\displaystyle\mu_{p0}^{g}(\rho^{g},T)=\mu_{p0}^{l}(\rho^{l},T)+\delta\mu_{p}^{l}.$
(19b)

Here, 0 in the subscript refers to the bulk matter conditions, and $\delta
P^{l}$ and $\delta\mu_{p}^{l}$ are the pressure and chemical potential
corrections which are given as [44]

$\displaystyle\delta
P^{l}=-\rho^{2}\big{(}\frac{\partial\mathcal{F}_{corr}}{\partial\rho}\big{)}\big{|}_{T,N,Z},$
(20a)
$\displaystyle\delta\mu_{p}^{l}=\big{(}\frac{\partial\mathcal{F}_{corr}}{\partial
Z}\big{)}\big{|}_{T,N,\rho}.$ (20b)

Where, $\mathcal{F}_{corr}$ is defined in Eq 1. The expressions for other
thermodynamical quantities such as critical temperature (Tl), flash
temperature (Tf), etc can be found in [27, 12] and they are used similarly in
this work. The external nucleon gas also defines the stability of a hot
nuclear liquid drop. In this context, we define the lifetime of a hot drop by
using the concept of statistical average and assuming neutron emission to be
the dominant process along with neglecting the energy dependency of the cross-
section as [28] as

$\frac{1}{\tau}=4\pi\gamma\frac{1}{h^{3}}2m(kT)^{2}\sigma\exp{\frac{\mu_{n}}{kT}},$
(21)

where $\gamma$ is the spin degeneracy and $\sigma$ is taken to be geometric
cross section.

##  III  Results and Discussion

In this section, we present the results of our calculation of a hot nucleus.
We use FSUGarnet, G3, IOPB-I, and NL3 E-RMF [29] parameter set . These E-RMF
forces are known to reproduce the properties of finite nuclei as well as
infinite nuclear matter [12, 27, 10, 39, 29]. They also satisfy the relevant
constraint on EoS such as incompressibility, symmetry energy, slope parameter,
etc., and observational constraints like Flow and Kaon experiments [10]. In
[12, 27] we have discussed in detail the critical properties of SNM using
these parameters and, here we extend those to the finite nuclei case. In our
calculations, for a fixed nuclear system and E-RMF parameter set, we use two
parametrization i.e. Eq. 6 and Eq. 7 for surface energy. These
parameterizations are widely used in the astrophysical [45] and statistical
calculations [46] and are used here for comparison . We compare the results in
reference to the properties of nuclei at finite temperature and consequently
study the role of critical temperature of infinite matter. This section is
divided into three subsection. We discuss the caloric curve and related
aspects in section III.1 and limiting temperature in section III.2. In section
III.3 we establish the correlation among various zero and finite temperature
properties.

###  III.1  Excitation energy, level density and fissility

We begin with the discussion of the caloric curve which is the relation
between excitation energy and temperature for the three isolated spherical
nuclei i.e. 56Fe, 90Zr, 208 Pb and 236U which is formed when thermally fissile
235U absorb a thermal neutron. In experiments, the temperature of the nucleus
is not measured directly and it is calculated using excitation energy which
can be obtained using resonance or energy of evaporation residue. Above
mentioned nuclei are most studied nuclear systems and their microscopic
calculations are available in literature. Fig 1 shows the caloric curve for
these nuclei using the four E-RMF sets FSUGarnet, IOPB-I, G3, and NL3.

Figure 1: The excitation energy of 56Fe, 90Zr, 208Pb and 236U as a function of
temperature for FSUGarnet, IOPB-I, G3 and NL3 sets. The solid lines represents
calculation from Eq. 6 and dashed lines are from Eq. 7. The theoretical data
in black cross is taken from [47], plus [28] from and star from [10]. The
experimental values for A $\approx$ 200 are taken from [48].

The estimations of theoretical caloric curves from the E-RMF are in reasonable
agreement with microscopic calculations [47, 28, 10]. The experimental value
for mass A $\approx$ 200 extracted from [48] also align with our calculations
for T $<5$ MeV. The deviation at higher temperature and excitation energy may
be associated with the production of heavier particles in the multi-
fragmentation process which may change the energy of the system. The behavior
of different parameter sets is tightly constrained and the spread of curves
becomes narrower as one moves from 56Fe to 208Pb. The effect of different
parametrization of surface energy from Eq.6 and 7 is also visible. Eq. 7
derived from the semi-classical Seyler-Blanchard interaction estimate a
steeper slope for caloric curve as compared to the Eq 6 based on thermodynamic
equilibrium of sharp interface between liquid and gaseous phase. It is because
the Eq. 6 estimates relatively lower surface energy at any given temperature.

For a particular nucleus, the G3 set with the largest effective mass
($m^{*}/m$=0.699) estimates the steepest caloric curve while the FSUGarnet
with the smallest ($m^{*}/m$=0.578) corresponds to the softest caloric curve.
The effective mass in E-RMF formalism is determined from the strength of
scalar field because of NN interaction. The G3 set due to small scalar self
couplings $k_{3},k_{4}$ and scalar-vector cross couplings $\eta_{1},\eta_{2}$
estimate the softest scalar field while the FSUGarnet yields the stiffest
scalar field. The scalar field consequently determine the mechanical
properties of the system and therefore, the effective mass becomes a crucial
saturation property at finite temperature. The effective mass which is
obtained self consistently also determine the chemical potential and kinetic
energy of nucleons which are essential input for the thermal properties
calculations. Furthermore the G3 set estimate the softest repulsive
contribution arising from the vector self coupling $\zeta_{0}$. The combine
effect of scalar and vector field determine the critical temperature. The
parameter set G3 and FSUGarnet estimate the largest and smallest $T_{c}$ among
these four sets (see table 2). Therefore, in finite nuclei, the thermal
contribution of energy essentially depends on the combined effect of effective
mass, $T_{c}$ and the zero-temperature EoS. It may be noted that the
saturation properties are not unique and different combination of mesons
coupling can yield the similar nuclear matter properties. Therefore, it is
relevant to analyse the finite temperature properties of the nuclear matter in
terms of saturation properties and not the coupling constants.

Figure 2: The Relation between square of entropy and excitation energy for the
systems as in Fig 1.

In the Fermi gas model, the point of minimum entropy in the transition state
nucleus corresponds to its minimum excitation energy (E∗) [49]. Therefore we
show the relation of the square of entropy and E∗ in Figure 2 for the systems
considered in Fig. 1. The square of entropy increases monotonically with the
E∗ signifying a disordered and chaotic nucleus. The disorder increases with
mass number implying a more violent multi-fragmentation process once the
nucleus reaches its limiting temperature Tl. Eq. 7 estimates higher entropy at
a given E∗ as compared to 6. For a particular nucleus, the spread of different
E-RMF sets increases with E∗. This effect can be attributed to the effective
mass and $T_{c}$ of a particular E-RMF parameter. In our model, we have not
considered the shell correction which deviates the straight-line behaviour of
this curve at low temperature, where shell structure is still intact. These
shells melt at around $E^{*}\approx 40$ MeV or $T\approx 1-2$MeV [50]. After
this temperature, the nucleus is highly disordered where nucleons are
constantly trying to push out from the nuclear boundary which is ensured by
the surface as in Eq. 4. The behaviour of $S^{2}$ is in agreement with results
in [47, 10].

Table 1: The level density parameters obtained using different expression of Eq. 16 for the FSUGarnet, IOPB-I, G3 and NL3 parameter set. Element | Forces |  $a$ (MeV-1) Using Eq. 7
---|---|---
|  | ${E^{*}}/{T^{2}}$ | $S^{2}/4E^{*}$ | $S/2T$
56Fe | NL3 | 4.695 | 4.931 | 4.789
FSUGarnet | 4.582 | 4.323 | 4.357
IOPB-I | 5.033 | 4.789 | 4.808
G3 | 5.149 | 4.942 | 4.963
90Zn | NL3 | 7.267 | 7.740 | 7.491
FSUGarnet | 7.102 | 7.185 | 7.072
IOPB-I | 7.8123 | 7.857 | 7.758
G3 | 7.872 | 8.065 | 7.930
208Pb | NL3 | 15.683 | 17.030 | 16.394
FSUGarnet | 15.752 | 16.725 | 16.233
IOPB-I | 16.998 | 18.040 | 17.531
G3 | 17.126 | 18.191 | 17.683
236U | NL3 | 17.64 | 19.19 | 18.4632
FSUGarnet | 17.761 | 18.946 | 18.353
IOPB-I | 19.196 | 20.400 | 19.818
G3 | 19.296 | 20.532 | 19.949

The caloric curve gives us the opportunity to study the level density
parameter (a) which plays a crucial role to understand the particle spectra
and nuclear fission. Level density signifies the available excited state level
at a given energy. In order to study the level density we use Eq. 16 and fit
them for the value of $a$ with R-squared value $>$ 0.99. The level density
parameters obtained using different expression of Eq. 16 are listed in Table
1. The level density calculated from all the three equations in Eqs. 16 are
comparable. A larger effective mass and Tc corresponds to the larger level
density as in the case of G3. These calculations are performed using Eq. 6. On
the other hand Eq. 7 estimates lower magnitude of level density although the
trend remain same. The value of level density lie within the empirical
relations $A/11.93$ from [51] and A/14.75 from [52]. Nuclear level density can
also be studied in terms of temperature where one can take the relevant ratio
in a straightforward manner for eg. $a=E^{*}/T^{2}$ at a particular
temperature. The G3 set with largest effective mass yield the largest
temperature-dependent level density. The above analysis of thermal properties
advocates the importance of effective mass over other saturation properties.

Now we shell discuss the temperature dependence of fissility and fission
barrier. Fissility characterizes the stability of a charged nuclear drop
against fission. In general, when Coulomb free energy $\mathcal{F}_{col}$
becomes twice the surface free energy $\mathcal{F}_{surf}$, the spherical
liquid drop become critical towards spheroidal deformation and split into two
equal parts. This feature is excessively used in the equilibrium condition
determining the structure of neutron star crust or supernovae explosion (see
Eq. 44 in [45]). One thing to note here is that, similar to a classical liquid
drop, on increasing the temperature, the nuclear liquid drop becomes more
spherical [10] i.e shell structure becomes trivial and deformations in the
nucleus vanish. Therefore, a drop can not undergo spontaneous fission only by
the temperature and one always needs external disturbance like a thermal
neutron in case of 235U. Although, at a certain maximum temperature $T_{l}$,
the nucleus will undergo multi-fragmentation process.

Figure 3: The fissility parameter $x(T)/x(T=0)$ as a function of
$\frac{T}{T_{c}}$ for ${}^{236}U$ using parameter sets FSUGarnet, IOPB-I, G3
and NL3 on the left panel. Right panel show the Liquid-drop fission barrier
for ${}^{236}U$. Solid and dashes lines have the same meaning as in Fig 1.

We show in Fig. 3, the variation of fissility as a function of $T/T_{c}$ using
Eq. 17 with different forces and both the parametrization of temperature
dependence of surface energy i.e. Eq. 6 and 7. The fissility for 236U
increases exponentially with temperature suggesting that the surface energy
decreases much faster on increasing the temperature. Eq. 7 has steeper slope
than Eq 6 which is again the result of lower surface energy in case of Eq. 6.
The fission barrier decreases with temperature and almost vanishes for
$T/T_{c}$=0.4 for all the forces. G3 parameter set estimates the largest
barrier and FSUGarnet the lowest which may be due to their effective mass. The
effective mass controls the mechanical properties and consequently determine
the equilibrium density of the nuclear liquid drop. One may notice in in fig 3
the dominant effect of $T_{c}$ as these quantities do not include the volume
term (see Eq. 17). The FSUGarnet and IOPB-I shows the similar trend with
almost similar $T_{c}$. G3 parameter set estimates the softest fissility and
largest fission barrier followed by the NL3 set as their value of $T_{c}$ are
15.3 and 13.75 respectively. The vanishing points of liquid-drop fission
barrier are aligned with their respective value of $T_{c}$ (see Table 2).

###  III.2  Limiting temperature

Figure 4: The limiting temperature Tl, the ratio of $\frac{T_{l}}{T_{c}}$,
limiting excitation energy per nucleon and life time of nuclear liquid drop at
the limiting temperature as a function of mass number A for the nuclei on
$\beta$-stability line. The temperature dependent expression used here is Eq
6. Experimental points in solid square are taken from [53] for Tl which are
calculated using double isotope yield ratio and thermal bremsstrahlung
measurements and from [25] for excitation energy. The points represented in
upper triangle are taken from the fisher droplet model derived from [54].

Determination of the temperature at which a hot nucleus drop will undergo
multi-fragmentation by loosing its entity, is one of a challenging problem in
nuclear physics. Experimentally it is difficult to estimate $T_{l}$ and other
related properties such as specific heat for a particular nucleus as there are
large number of nucleons involved. Although, theoretically we can study these
properties by applying appropriate constraints. In that context, we consider a
simplistic approach to determine the $T_{l}$ of a nucleus. We employ our
assumption stated in II.4 and solve Eqs. II.4. These Equations will not have
any solution for a given T, $\rho_{v}$ and $\rho_{l}$ for temperature greater
than $T_{l}$ signifying that the nucleus can no longer exists.

In fig. 4 we show the variation of limiting temperature $T_{l}$,
$T_{l}/T_{c}$, limiting excitation energy ($E^{*}(T_{l})/A$) and the life time
($\tau$) of nucleus at limiting temperature as a function of mass number for
the nuclei along $\beta$ stability line where the atomic number can be written
as

$Z=0.5A-0.3\crossproduct 10^{-2}A^{\frac{5}{3}}.$ (22)

The value of $T_{l}$ decreases exponentially with increasing mass number as
the Coulomb energy rises due to larger Z. At lower Z, $T_{l}$ decreases at
faster pace because the Coulomb component dominates the surface and symmetry
energy of liquid drop. At a higher mass number, the situation becomes a little
different. There is competition between Coulomb, surface, and symmetry terms.
On moving from low to higher mass number along the $\beta$ stability line, the
$Z/A$ ratio decreases. The decrease in the $Z/A$ ratio weakens the A
dependence causing $T_{l}$ to increases. On the other hand, the symmetry and
surface energy increase with the increase in mass number which tries to bring
down the $T_{l}$. For comparison we show points determined from
phenomenological analysis [53, 25, 54] for the Tl and $E^{*}(T_{l})$. The
results from E-RMF forces are within reasonable agreement.

Figure 5: EoS, Chemical potential and effective mass at low density at T=5 MeV
for the FSUGarnet, IOPB-I, G3 and NL3 parameter sets.

The value of $T_{l}$ for a particular nucleus and a particular EoS depends on
the $T_{c}$ of infinite nuclear matter and the low density ( $\rho_{0}<0.01$)
variation of EoS which determine the properties of surrounding gaseous phase.
Since the finite-size corrections are employed externally, they are the same
for every EoS. To understand the effect of EoS, we plot in Fig. 5 the EoS,
chemical potential ($\mu$), and effective mass ($M^{*}$) calculated using the
FSUGarnet, IOPB-I, G3, and NL3 parameter sets for the density range
significant for nuclear vapor surrounding the hot nucleus. Chemical potential
is a function of temperature-dependent effective mass which consequently
determines the chemical equilibrium between nuclear gas and nuclear drop. The
IOPB-I and FSUGarnet have similar ground state saturation properties and they
have similar behavior at T=5 MeV. The incompatibilities of the NL3 and G3 sets
are 271.38 and 243.96 MeV respectively but their behavior is opposite in the
low-density regime. G3 set estimates the maximum value of pressure, and
effective mass at any given density. This is the reason G3 set have larger
value of Tc than the NL3 set. This trend in Fig 5 for different EoS, validates
the variation of $T_{l}$ in Fig 4, where the magnitude of Tl explicitly
depends upon the low density EoS. In other word, to understand the effect of
EoS on the $T_{l}$ one has to take into account the $T_{c}$ and low density
behaviour of EoS instead of incompressibility at saturation.

Further the ratio $T_{l}/T_{c}$ signify the finite size effect where the
limiting temperature decreases sharply as compared to the critical temperature
of infinite symmetric matter. It reduces up to the 0.3$T_{c}$ for heavy
nuclei. Furthermore, there is still model dependency in the $T_{l}/T_{c}$ .
The larger effective mass yields smaller $T_{l}/T_{c}$ which is clear from the
fact the FSUGarnet and G3 estimate the largest and smallest $T_{l}/T_{c}$.
Limiting excitation energy per nucleon is calculated at $T_{l}$ and our
calculations from E-RMF forces agree with the phenomenological calculation
[25]. We have performed these calculations using Eq. 6 as there were no
significant difference between the values of $T_{l}$ calculated from Eq. 6 and
7. However, the Eq. 7 estimates the larger excitation energy for a given
nucleus as compared to the Eq. 6. The Eq. 6 and 7 are frequently used in
various calculations such as statistical equilibrium analysis and supernovae
matter. In that context, these equations correctly estimate the finite nucleus
observables with slight difference in magnitude. Eq. 6 has a slight edge as it
is consistent with the surface energy estimated from thermal Hartree-Fock
approximation [38]. Our calculations show better agreement with experimental
and theoretical values when using Eq. 6 as well. However, the judicious use of
these can be made depending on the problem such as supernova where the thermal
energy plays a very important role.

(a)

(b)

Figure 6: (a) Limiting temperature for fix atomic number Z=82 as a function of
mass number calculated from the expression 7 (b) Same as in (a) but for fix
neutron number N=126.

To further understand the behaviour of $T_{l}$, we calculate the lifetime of
hot nucleus $\tau$ using Eq. 21. As we have not considered the temperature
dependence of neutron-capture cross-section, these values will slightly
underestimate the lifetime but the trend will remain the same. The radius R
which is the input for Eq. 21 is determined after solving the coexistence Eqs.
II.4 for a particular nucleus. We have seen that the nuclear gas surrounding
the nuclear liquid plays a significant role in determining the Tl. In terms of
lifetime, a larger pressure and smaller density corresponds to a less stable
liquid drop and therefore, lower lifetime. The IOPB-I set that estimate the
loweer Tl for a given nucleus yields the higher lifetime. We see that the
lifetime $\tau$ is of the order of $10^{-22}$S at $T_{l}$ for all the nuclei
on the $\beta$ stability line. Nuclei at the lower mass range are slightly
more stable than heavy nuclei. This time scale is just enough for a nucleus to
allow thermalization. This also states the fact that at $T_{l}$ the nucleus is
highly unstable and will undergo violent multi-fragmentation which has the
time scale of $10^{-22}$S [4, 55].

In Fig. 6a, the variation of $T_{l}$ is shown for a fixed atomic number Z=82
and Fig. 6b demonstrate the behaviour for a fixed neutron number N=126. For a
fixed atomic number, the $T_{l}$ rises $\approx$ 1.5 MeV when we move from
A=178 to A=220 or from $Z/A$ = 0.46 to 0.37. The increase in $T_{l}$ with a
decrease in $Z/A$ ratio is because Coulomb free energy reduces as the radius
of nuclear liquid drop increases as a function of charge number. The surface
energy then dominates over the Coulomb energy which helps in preserving the
surface of the drop at a much higher temperature. This trend is confirmed with
the non-relativistic Hartree-Fock calculation where the solution becomes
unstable after a certain temperature [56]. When we keep the neutron number
fixed, there is an interesting binodal type trend in the values of $T_{l}$
with increasing mass number. The $T_{l}$ increases with increasing Z and
reaches its maximum at A $\approx$ 170\. It then decreases at a faster rate on
further increasing the value of Z. This effect is the result of competition
between Coulomb and surface energy at lower and higher mass region. This shape
of the graph then signifies that one can make nuclei in the unconventional
regimes, which might not be stable at zero temperature but can exist at some
higher temperature. In Figs. 6a and 6b, the trends of EoS are similar to the
ones obtained at low density regime.

###  III.3  Correlations

In the analysis of a hot nucleus and its limiting temperature, we saw that the
critical temperature $T_{c}$ of infinite nuclear matter affects the
observables through Eq. 6 and 7. They also depend on the properties of EoS
such as effective mass and low density behaviour of a particular EoS. The
$T_{c}$ which is basically an inflation point on critical isotherm, is one of
the most uncertain parameter in nuclear matter studies. The value of $T_{c}$
is an important factor in calculation of finite nuclei as well as supernovae
matter and neutron star crust [45]. Hence it becomes important to relate the
$T_{c}$ of a particular EoS to its saturation properties. In Refs. [12, 27] we
have studied the thermodynamics of liquid gas phase transition in infinite
nuclear matter using the E-RMF parameter sets used in this study. It has been
observed that the critical temperature $T_{c}$ is not a well constrained
quantity. It requires a comprehensive statistical analysis of nuclear
properties at critical points and saturation properties of cold nuclear matter
as their analytical relationship is difficult to establish. For this, we take
fifteen E-RMF parameter sets satisfying relevant constraints [30, 29, 27, 10,
40] on EoS and first of all calculate the properties at critical point of
liquid-gas phase transition in infinite matter.

Table 2: The zero temperature incompressibility K, binding energy $e_{0}$, saturation density $\rho_{0}$, effective mass M∗ and critical temperature $T_{c}$, pressure $P_{c}$, density $\rho_{c}$ along with flash temperature $T_{f}$, density $\rho_{f}$, incompressibility $C_{f}$ and effective mass at $T_{c}$ for infinite symmetric nuclear matter using the several forces.  Parameter | K | $e_{0}$ | $\rho_{0}$ | $m^{*}/m$ | $T_{c}$ | $P_{c}$ | $\rho_{c}$ | $T_{f}$ | $\rho_{f}$ | Cf | $m^{*}_{c}/m$
---|---|---|---|---|---|---|---|---|---|---|---
| MeV | MeV | fm-3 |  | MeV | MeV fm-3 | fm-3 | MeV | fm-3 |  |
G2 [26] | 215.00 | -16.10 | 0.153 | 0.664 | 14.30 | 0.181 | 0.0432 | 11.80 | 0.080 | 0.293 | 0.879
IOPB-I [29] | 222.65 | -16.10 | 0.149 | 0.593 | 13.75 | 0.167 | 0.0424 | 11.20 | 0.071 | 0.286 | 0.864
Big Apple [57] | 227.00 | -16.34 | 0.155 | 0.608 | 14.20 | 0.186 | 0.0441 | 11.45 | 0.073 | 0.297 | 0.876
BKA22 [40] | 227.00 | -16.10 | 0.148 | 0.610 | 13.90 | 0.178 | 0.0442 | 11.33 | 0.072 | 0.290 | 0.855
BKA24 [40] | 228.00 | -16.10 | 0.148 | 0.600 | 13.85 | 0.177 | 0.0450 | 11.31 | 0.073 | 0.284 | 0.845
FSUGarnet [29] | 229.50 | -16.23 | 0.153 | 0.578 | 13.80 | 0.171 | 0.0430 | 11.30 | 0.071 | 0.288 | 0.850
FSUGold [58] | 230.00 | -16.28 | 0.148 | 0.600 | 14.80 | 0.205 | 0.0460 | 11.90 | 0.074 | 0.301 | 0.844
IUFSU [59] | 231.31 | -16.40 | 0.155 | 0.610 | 14.49 | 0.196 | 0.0457 | 11.73 | 0.074 | 0.296 | 0.862
FSUGold2 [60] | 238.00 | -16.28 | 0.151 | 0.593 | 14.20 | 0.187 | 0.0450 | 11.51 | 0.073 | 0.293 | 0.855
BKA20 [40] | 240.00 | -16.10 | 0.146 | 0.640 | 15.00 | 0.209 | 0.0458 | 11.91 | 0.073 | 0.304 | 0.868
G3 [29] | 243.96 | -16.02 | 0.148 | 0.699 | 15.30 | 0.218 | 0.0490 | 12.10 | 0.075 | 0.291 | 0.879
NL3* [61] | 258.27 | -16.31 | 0.150 | 0.590 | 14.60 | 0.202 | 0.0466 | 11.70 | 0.075 | 0.297 | 0.861
Z27v1 [62] | 271.00 | -16.24 | 0.148 | 0.800 | 18.03 | 0.304 | 0.0515 | 13.70 | 0.077 | 0.327 | 0.914
NL3 [29] | 271.38 | -16.29 | 0.148 | 0.595 | 14.60 | 0.202 | 0.0460 | 11.80 | 0.070 | 0.301 | 0.846
TM1 [63] | 281.10 | -16.26 | 0.145 | 0.630 | 15.60 | 0.236 | 0.0486 | 12.09 | 0.076 | 0.311 | 0.862
Exp/Emp | 240 [30] | -16 [27] | 0.166 [27] | 0.63 [64] | 17.9 [65] | 0.31[65] | 0.06 [65] | - | - | 0.288 [12] | -
|  $\pm$ 20 |  $\pm$ 1 |  $\pm$ 0.019 |  $\pm$ 0.05 |  $\pm$ 0.40 |  $\pm$ 0.07 |  $\pm$ 0.01 | - | - | - | -
Figure 7: The Pearson correlation matrix for the critical parameter for
infinite symmetric nuclear matter and some cold nuclear matter properties. The
number of stars in a circle represents the p-value given at the bottom. The
strength of correlation is colour mapped.

In table 2, we present the saturation properties of cold nuclear matter i.e.
incompressibility (K), binding energy ($e_{0}$), saturation density
($\rho_{0}$), effective mass (M∗) and critical temperature ($T_{c}$), pressure
($P_{c}$), density ($\rho_{c}$) along with flash temperature ($T_{f}$),
density ($\rho_{f}$), incompressibility factor ($C_{f}$) and effective mass at
($T_{c}$) for infinite symmetric nuclear matter using different force
parameters. For further details on these quantities please see Ref. [12]. We
have selected variety of forces with different meson couplings, which include
up to the quartic order scalar and vector terms in order to have a generalised
analysis of E-RMF forces. The E-RMF sets satisfying the allowed
incompressibility range and other observational constraints underestimate the
critical values of temperature, density and pressure when compared to
experimental data [65].

We then calculate the Pearson Correlation matrix [66] for variables calculated
in table 2 and the results are shown in Fig. 7. The colour coded correlation
matrix also show the statistical significance in form of p-value [66] for
different confidence interval i.e. 95%, 99% and 99.9%. The binding energy
($e_{0}$) and saturation density ($\rho_{0}$) of cold infinite nuclear matter
have very weak strength of correlation with the critical properties at finite
temperature. This is against the natural intuition that binding energy of
infinite matter should impact the $T_{c}$.

The incompressibility on the other hand show positive correlation with
critical properties i.e. $T_{c}$, $p_{c}$, $\rho_{c}$ and $T_{f}$. Although
this correlation do not exceed the value of 0.77. Therefore, we can conclude
that the saturation properties of cold nuclear matter do not significantly
impact the value of critical parameter individually. The reason for this can
be the fact that saturation properties are calculated at saturation density
$\rho_{0}\approx$ 0.16 fm-3, whereas, the nuclear matter convert from liquid
to gaseous phase at $\approx$ 0.25-0.3 $\rho_{0}$. The behaviour of EoS in
this density region is not always as per the properties at saturation, as
noticed in Fig 5. One exception is the effective mass which shows a strong
positive correlation with critical properties. This is in line with our
analytical analysis of infinite nuclear matter that finite temperature
properties in E-RMF formalism are governed by the effective mass. This
behaviour is consistent with the non-relativistic formalisms as well, although
the definition of effective mass is different in both the cases [27].

From Table 2 we see that the parameter sets G3 and Z27v1 have relatively high
effective mass and a high value of $T_{c}$. A high positive correlation
between $m^{*}$ and $T_{c}$ in Fig 7 suggests the same. Therefore, one way to
construct a model at par with experimental findings is to exploit this
property of effective mass. This fact was also considered in [67]. However,
the prescribed range of effective mass 0.58 $\leq$ m∗/m $\leq$ 0.68 in
agreement with spin-orbit splitting experiments [64] should be kept in mind.
The Z27v1 set does not satisfy this constraint and it was also not considered
in [30], from where the constraints on EoS are taken for this study.
Therefore, no standard RMF and E-RMF parameter sets, that satisfy all the
available constraints can reproduce the experimental value of the critical
parameter for infinite nuclear matter and hence needs more analysis especially
on the low-density regime of EoS. Moreover, the effective mass dependence of
thermal properties will also be useful in the microscopic calculations, where
the concept of $T_{c}$ is not explicitly used for the surface energy
calculation.

The low correlation means that the variables are acting as independent
parameters. This is also justified as the properties like K, $\rho_{0}$,
$e_{0}$, and $m^{*}$ are the inherent characteristic properties of an EoS. The
critical temperature therefore can be understood as a result of competition
between various nuclear matter observables. To demonstrate this, we construct
a very simple multiple linear regression (MLR) fit of following form.

$T_{c}=\beta_{0}+\beta_{1}K+\beta_{2}e_{0}+\beta_{3}\rho_{0}^{(1/3)}+\beta_{4}m^{*},$
(23)

where, all the variables are in MeV except $\rho_{0}$ which is in MeV3 and
coefficients have relevant dimensions with $\beta_{0,1,2,3,4}$=-11.5033,
0.00201, -4.32248, -0.52433, 0.01795. These coefficients are statistically
significant as well for 95 % confidence interval. In Fig. 8, we show the
result of Eq. 23 against the actual $T_{c}$ from table 2. The regression
equation estimate the $T_{c}$ excellently with R-square=0.987. The fitted
regression equation suggest that the binding energy and saturation density has
opposite variation with $T_{c}$. The regression equation 23 is better than the
empirical relations suggested in [68] based on Lattimer–Swesty and Natowitz
predictions. This is because the greater degree of freedom are considered in
this equation. However, this will yield a strange value of $T_{c}$ when all
the saturation properties tend to zero. This equation gives an useful insight
in the form of free coefficient $\beta_{0}$ which suggests that there is a
missing link between our current understanding of critical temperature and its
relationship with the saturation properties. The $\beta_{0}$ becomes
inevitable as the equation then gives a bad fitting.

Figure 8: Actual value of $T_{c}$ from different forces and regression fit
values calculated from Eq. 23.

Unlike saturation and critical properties, the critical parameters are
strongly correlated with each other, except the flash density $\rho_{f}$. The
flash density seems to be model independent with standard deviation = 0.0025.
It is to note that these correlations are for the E-RMF sets considered in
Table 2 and are not universal. However, the selected parameter sets have a
wide range of meson couplings and nuclear matter saturation properties.
Moreover, we have presented our results with the statistical significance of
Pearson correlation to make it as general as possible.

Table 3: Limiting temperature (MeV), chemical potential (MeV), pressure (MeV fm-3), gas density ($fm^{-3}$), liquid density ($fm^{-3}$), radius ($fm$) and lifetime ($\tau\crossproduct\exp{-22}$ Sec) of 208Pb nucleus for several forces. Parameter | $T_{l}$ | $\mu$ | P | $\rho_{v}$ | $\rho_{l}$ | R | $\tau$
---|---|---|---|---|---|---|---
G2 | 5.4 | -8.55 | 0.0162 | 0.0075 | 0.147 | 6.964 | 1.49
IOPB-I | 5.88 | -9.61 | 0.02 | 0.0084 | 0.143 | 7.028 | 1.29
Big Apple | 5.37 | -8.36 | 0.0191 | 0.0075 | 0.148 | 6.948 | 1.47
BKA22 | 5.46 | -8.65 | 0.0197 | 0.0076 | 0.142 | 7.045 | 1.42
BKA24 | 5.51 | -8.73 | 0.0196 | 0.0075 | 0.142 | 7.045 | 1.40
FSUGarnet | 5.9 | -9.48 | 0.024 | 0.0082 | 0.148 | 6.948 | 1.28
FSUGold | 5.92 | -9.21 | 0.0239 | 0.0085 | 0.143 | 7.028 | 1.18
IUFSU | 5.69 | -8.97 | 0.0224 | 0.0081 | 0.149 | 6.933 | 1.34
FSUGold2 | 5.59 | -8.88 | 0.0207 | 0.0078 | 0.145 | 6.996 | 1.38
BKA20 | 5.85 | -9.03 | 0.0238 | 0.0085 | 0.140 | 7.078 | 1.18
G3 | 5.9 | -9.22 | 0.0245 | 0.0087 | 0.141 | 7.061 | 1.19
NL3* | 5.74 | -9.08 | 0.022 | 0.0082 | 0.144 | 7.012 | 1.30
Z27v1 | 6.95 | -10.49 | 0.0369 | 0.0110 | 0.14 | 7.078 | 0.80
NL3 | 5.88 | -9.17 | 0.0213 | 0.0084 | 0.144 | 7.012 | 1.21
TM1 | 5.85 | -8.63 | 0.025 | 0.0086 | 0.138 | 7.112 | 1.09

After establishing the relationship between critical properties and saturation
properties of cold nuclear matter, we extend these correlation to limiting
properties. In Table 3, we present the values of $T_{l}$, chemical potential
$\mu$ , pressure (P), gas density ($\rho_{g}$), liquid density ($\rho_{l}$),
radius (R) and lifetime ($\tau$) of 208Pb nucleus for the forces considered in
Table 2. To establish the relation of different properties we calculate the
correlation matrix for limiting properties of 208Pb nucleus, critical
properties of infinite nuclear matter $T_{c}$ and saturation properties of
cold nuclear matter.

Figure 9:  The Pearson correlation matrix for the critical parameter for
infinite symmetric nuclear matter, some cold nuclear matter properties and
limiting properties for the 208Pb.

Once again, the binding energy and saturation density of cold nuclear matter
is weakly correlated with the limiting properties. The incompressibility shows
a weak correlation with the limiting properties which is in agreement with the
analysis of the low density behaviour of EoS. However, it is correlated
negatively with the lifetime of nucleus. This is justified as the stiff EoS
corresponds to the the larger pressure, which in turn make the nucleus less
stable surrounded in a nucleon gas. The effective mass is strongly correlated
with the limiting properties. A strong correlation between Tc and effective
mass then suggest that the limiting properties of a nucleus essentially depend
on the Tc and M∗ of the model applied. This statement has a far reaching
implication as the majority of the calculations employing statistical
thermodynamics as well as compress liquid-drop model (CLDM) in astrophysical
applications heavily depend on the value of Tc for surface energy. Also in the
microscopic calculations where the surface energy is determined using the
derivative of mean-fields, effective mass plays the determining role. On the
other hand, the limiting properties for 208Pb i.e. limiting temperature (MeV),
chemical potential (MeV), pressure (MeV fm-3), gas density ($fm^{-3}$), liquid
density ($fm^{-3}$) and radius ($fm$) are tightly correlated. A higher $T_{l}$
means that the chemical potential will be smaller and the equilibrium pressure
and gas density will be larger.

##  IV  Summary and Outlook

In summary, we use the effective relativistic mean-field theory (E-RMF) to
analyze the thermal properties of hot nuclei. The free energy of a nucleus is
estimated by using temperature and density-dependent parameters of the liquid-
drop model. We parametrize the surface free energy using two approaches based
on the sharp interface of the liquid-gaseous phase and the semi-classical
Seyler-Blanchard interaction. The later parametrization estimates relatively
stiff behavior of excitation energy, entropy, and fissility parameter. The
estimations of these properties are in reasonable agreement with the available
theoretical microscopic calculations and experimental observations.

It has been observed that the thermal properties of the finite nuclear system
are influenced strongly by the effective mass and critical temperature
($T_{c}$) of the E-RMF parameter sets employed. A larger effective mass
corresponds to the higher excitation energy, level density, limiting
temperature, and limiting excitation energy. The limiting temperature also
depends on the behavior of EoS at subsaturation densities which helps to
calculate the properties of surrounding nuclear gas in equilibrium with the
hot nucleus. A stiff EoS at subsaturation density corresponds to the larger
limiting temperature. The temperature-dependent liquid-drop fission barrier is
also influenced by the $T_{c}$. A larger $T_{c}$ estimates a larger
temperature where the barrier vanishes.

Finally we have performed a detailed Correlation matrix analysis to account
for the large deviations in the value of critical parameters among various
E-RMF sets. The effective mass shows a strong positive correlation with the
critical parameters namely ($T_{c}$, $\rho_{c}$, $P_{c}$) and limiting
temperature of the nucleus, which is consistent with the analytical analysis.
The binding energy and saturation density act as independent parameters which
prompts us to establish a simple multiple linear regression (MLR) between the
$T_{c}$ and saturation properties of cold nuclear matter. Our MLR equation
fits the original $T_{c}$ and gives useful relationship between saturation
properties and critical temperature.

The present calculations can be extended to various astrophysical problems. A
similar situation is encountered in supernovae explosion and neutron star
crust, where the nuclei are surrounded in a nuclear and relativistic electron
gas. The model dependence can also be studied within statistical multi-
fragmentation calculations. Furthermore, a comprehensive analysis is required
to address the anomaly in the magnitude of the critical temperature of nuclear
matter by employing the low-density correction in the EoS.

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