# Systematic study of surface properties for Ne, Na, Mg, Al and Si isotopes in
a coherent density fluctuation model using the relativistic mean field
formalism

Jeet Amrit Pattnaik1<EMAIL_ADDRESS>R. N. Panda1
<EMAIL_ADDRESS>M. Bhuyan2,3<EMAIL_ADDRESS>S. K. Patra4,5
<EMAIL_ADDRESS>1Department of Physics, Siksha ${}^{\prime}O^{\prime}$
Anusandhan, Deemed to be University, Bhubaneswar-751030, India  2Center of
theoretical and Computational Physics, Department of Physics, University of
Malaya, Kuala Lumpur, 50603, Malaysia  3Institute of Research and Development,
Duy Tan University, Da Nang 550000, Vietnam  4Institute of Physics, Sachivalya
Marg, Bhubaneswar-751005, India  5Homi Bhabha National Institute, Training
School Complex, Anushakti Nagar, Mumbai 400094, India

###### Abstract

We have systematically studied the surface properties, such as symmetric
energy, neutron pressure, and symmetry energy curvature coefficient for Ne,
Na, Mg, Al, and Si nuclei from the proton to neutron drip-lines. The Coherent
Density Fluctuation Model (CDFM) is used to estimate these quantities taking
the relativistic mean-field densities as inputs. The Brückner energy density
functional is taken for the nuclear matter binding energy and local density
approximation is applied for its conversion to coordinate space. The symmetry
energy again decomposed to the volume and surface components within the liquid
drop model formalism to the volume and surface parts separately. Before
calculating the surface properties of finite nuclei, the calculated bulk
properties are compared with the experimental data, whenever available. The
NL3* parameter set with the BCS pairing approach in an axially deformed frame-
work is used to take care of the pairing correlation when needed. The deformed
density is converted to its spherical equivalent with a two Gaussian fitting,
which is used as an input for the calculation of weight function in the CDFM
approximation. With the help of the symmetric energy for the recently isotopes
29F, 28Ne, 29,30Na and 31,35,36Mg are considered to be within the island of
inversion emphasized [Phys. Lett. B 772, 529 (2017)]. Although we get large
symmetric energies corresponding to a few neutron numbers for this isotopic
chain as expected, an irregular trend appears for all these considered nuclei.
The possible reason behind this abnormal behavior of symmetry energy for these
lighter mass nuclei is also included in the discussion, which gives a
direction for future analysis.

###### pacs:

21.10.−k,21.10.Dr,21.10.Ft,21.10.Gv,21.60.−n,21.30.−x

##  I Introduction

The surface property of a nucleus is one of the pivotal quantities to
determine the structural information of finite and infinite nuclear matter at
isospin asymmetry. The symmetric $S$ energy determines the nuclear matter
equation of state (EoS), which is connected with the isospin asymmetry. This
$S$ controls the properties of the neutron star (NS), such as its mass and
radius li2008 . Consequently, the Gravitational Waves (GW), the parameter of
NS merger, etc. are automatically governed by the surface properties. Because
of the interrelation of the nuclear matter symmetric energy with the neutron-
skin thickness of finite nuclei, one can study the NS structure knowing the
symmetric energy of the nuclear matter EoS li2019 . Similarly, there is a
strong bond between the nuclear matter surface properties with the
corresponding quantities of finite nuclei. There has been many significant
efforts both in nuclear physics and astrophysics over the last two decades,
and much progress in this regard is constraining both the magnitude of the
symmetry energy $S(\rho_{0})$ and the slope parameter $L_{sym}^{A}$ =
$3\rho_{0}(dS(\rho)/d\rho)_{\rho_{0}}$ at the saturation density $\rho_{0}$ of
nuclear matter baran2005 ; li2008 ; tsang2012 ; horowitz2014 ; baldo2016 ;
xu2020 .

Recently, a large number of works have been reported, which connect the
surface properties of finite nuclei with the nuclear matter observables anto18
; anto2 ; anto3 ; anto4 ; gad11 ; gad12 ; bhu18 ; bhu20 . To calculate these
properties, a large number of methods are adopted such as Coherent Density
Fluctuation Model anto1 ; anto2 ; anto3 ; anto4 ; gad12 ; bhu18  and Liquid
Drop Approximation dani03 . Recently, the Coherent Density Fluctuation Model
(CDFM) is gaining momentum for the evaluation of such properties. Although the
original idea is published antozphys1980  long back, substantial work is
waiting yet. Some of the relations of symmetric energy and related quantities
have a strong correlation with the structures of the finite nuclei. We have
reported in an isotopic series that the symmetric energy shows a peak at the
corresponding magic number bhu18 ; bhu20 ; manpreet . Due to the advancement
of the radioactive ion beam facility, the study of highly asymmetric isospin
neutron/proton-rich nuclei open up a new area of research in Nuclear Physics.
Some of the neutron number for certain proton combination behaves like a magic
number. For example, neutron number N= 14, 16 and 32/34 act like close shell
combination for neutron-rich Ne and Mg isotopes tanihata . The nuclei Ne, Na,
Mg, Al, and Si show unexpected characteristics near the neutron-rich region.
Some of the isotopic region termed as island of inversion. In this region of
the mass table, the experimental binding energy deviates considerably from the
shell model calculations. Long back Patra and Praharaj patra91  shown that
these isotopes exhibit a large deformation in their ground states, which is
later on verified by Tanihata et al. tanihata  in their experimental study. In
this paper, we would like to perform a systematic study of surface properties
of these nuclei starting from the proton deficient region to the neutron-drip
line and look for various structures.

A large number of theoretical formalisms are available in the literature for
the structural analysis of finite nuclei and EoS of infinite nuclear matter.
Among those well-known approaches, the Skyrme Hartree-Fock and relativistic
mean-field (RMF) theory are the most successful frame-work for such study. The
RMF approach has an advantage over the non-relativistic counterpart because of
the inclusion of the spin-orbit interaction through the relativistic
equations. This approach also reproduces the binding energy, quadruple
deformation parameter $\beta_{2}$, and other related properties of finite
nuclei throughout the mass table. With a few parameters of meson-nucleon
couplings and their respective masses, the method successfully reproduces the
experimental quantities for almost all known nuclei. This method also explains
the nuclear EoS and NS properties quite well. Because of the satisfactory
reproduction of experimental quantities by using RMF formalism, in the present
paper, we use the formalism to get the bulk properties of finite nuclei
including densities. These densities are used in the CDFM to generate surface
properties of finite nuclei.

The paper is arranged as follows: The relativistic mean-field model and the
fitting procedure of energy density functional ${\cal{E}(\rho)}$ to an
analytical expression in coordinate space with the help of the Brückner
prescription bruk68 ; bruk69  is discussed in Sec. II. Also, the Coherent
Density Fluctuation Model is briefly outlined. The Sec. III is dedicated to
the discussion of our results derived from the calculations. A summary and
concluding remarks are presented in Sec. IV.

##  II Relativistic mean field (RMF) Model

In this section, we briefly describe the formalism of the well documented
relativistic mean-field theory. The standard nonlinear RMF Lagrangian density
is built up by the interactions of isoscalar-scalar $\sigma$, isoscalar-vector
$\omega$, and isovector-vector $\rho$ mesons with nucleons. In this version of
the Lagrangian, the self-interaction of the $\sigma-$meson is also considered.
Recently, the extended versions of the relativistic mean-field formalism based
on the effective field theory motivated relativistic mean-field formalism are
very popular frun96 ; frun97 . In these models, all possible meson-nucleon and
their self-interactions are considered. The standard RMF formalism is quite
successful for finite nuclei not only for $\beta-$stable but also predicts
reasonably the properties of drip-lines and super-heavy nuclei kumar18 ;
kumar17 . Thus, we have considered this model in the present work. The RMF
Lagrangian is discussed in detail in Refs. kumar18 ; kumar17 ; frun96 ; frun97
. During the last few decades, the pertinence of this decorum to nuclear
astrophysics is in limelight. It foresees the structure of NS and reveals the
tidal deformability adequately malik2018 . The recently reported experimental
data of Gravitational Wave GW170817  measurements constraint the RMF nuclear
EoS in a precise manner. Thus, the RMF Lagrangian gives enough confidence to
adopt for further implementations to different nuclear environments. For a
nucleon-meson interacting system the energy density functional is granted as
kumar18 :

$\displaystyle{\cal E}({r})$ $\displaystyle=$
$\displaystyle\sum_{\alpha=p,n}\varphi_{\alpha}^{\dagger}({r})\Bigg{\\{}-i\mbox{\boldmath$\alpha$}\\!\cdot\\!\mbox{\boldmath$\nabla$}+\beta\bigg{[}M-\Phi(r)\bigg{]}+W({r})+\frac{1}{2}\tau_{3}R({r})+\frac{1+\tau_{3}}{2}A({r})\bigg{)}\Bigg{\\}}\varphi_{\alpha}(r)$
(1) $\displaystyle\
+\left(\frac{1}{2}+\frac{\kappa_{3}}{3!}\frac{\Phi({r})}{M}+\frac{\kappa_{4}}{4!}\frac{\Phi^{2}({r})}{M^{2}}\right)\frac{m_{s}^{2}}{g_{s}^{2}}\Phi^{2}({r})-\frac{1}{2}\frac{m_{\omega}^{2}}{g_{\omega}^{2}}W^{2}({r})-\frac{1}{2}\frac{m_{\rho}^{2}}{g_{\rho}^{2}}R^{2}({r})\;,$

Here, $\Phi$, $W$ and $R$ are the re-considered fields for $\sigma$, $\omega$
and $\rho$ mesons written as $\Phi=g_{s}\sigma_{0}$, $W=g_{\omega}\omega_{0}$
and $R=g_{\rho}\rho$ respectively. $M$, $m_{\sigma}$, $m_{\omega}$ and
$m_{\rho}$ are the nucleon masses $\omega$ , $\sigma$ and $\rho$ mesons,
sequentially. From Eq. (1), we obtain our energy density ${\cal{E}}_{nucl.}$
kumar18 ; kumar17  by taking that the exchange of mesons establish an uniform
field, where the oscillations done by nucleons in a periodic motion said to be
simple harmonic . From the effective-RMF energy density, the equation of
motions (EoS) for the mesons and the nucleons are procured by using the Euler-
Lagrange equation. A bunch of coupled differential equations are retained and
settled accordingly kumar18 . The scalar and vector densities,

$\displaystyle\rho_{s}(r)$ $\displaystyle=$
$\displaystyle\sum_{\alpha}\varphi_{\alpha}^{\dagger}({r})\beta\varphi_{\alpha},$
(2) $\displaystyle\rho_{v}(r)$ $\displaystyle=$
$\displaystyle\sum_{\alpha}\varphi_{\alpha}^{\dagger}({r})\tau_{3}\varphi_{\alpha},$
(3)

are figured out from the converged elucidations within the spherical
harmonics.

###  II.1 Spherical Equivalent Density using two Gaussian Fitting

We have obtained the axially deformed density from RMF model for considered
parameter sets and converted it into it’s spherical equivalent density by
following the below steps,

  * •

Initially we have the deformed density $\rho(r_{\bot},z)$ which is converted
to one dimensional as $\bar{\rho}(\omega)$ where $\omega=\sqrt{x^{2}+y^{2}}$
by performing the z-integration over the whole space as done in patra2009 ;
panda2014 ; sharma2016 :

$\vec{\rho}(\omega)=\int_{-\infty}^{\infty}\rho(\sqrt{\omega^{2}+z^{2}})dz\;.$
(4)

  * •

In the second step, the z-integrated density $\bar{\rho}(\omega)$ is fitted to
a two Gaussian function expressed as:

$\rho(r)=\sum_{i=1}^{2}c_{i}exp[-a_{i}r^{2}]\;,$ (5)

where the co-efficient $c_{i}$ and range $a_{i}$ are given with initial values
for respective nuclei.

Then we have used the spherical equivalent density in place of deformed
density for further calculations. The spherical equivalent densities are
normalized to mass number of the nucleus.

To carry a detailed study of the nuclear bulk properties of open-shell nuclei,
the contribution of pairing correlation becomes a major factor. In the present
calculations, we will be using BCS approach for the calculation of pairing
correlation. Here, the RMF+BCS formalism with NL3* parameter set lala09  is
used for the finite nuclei. Within this approach, we evaluate the binding
energy, quadruple deformation parameter $\beta_{2}$, root mean square radii
and the axially deformed densities for both neutrons and protons distribution.
The vector density $\rho_{v}(r)$ is farther handled within the CDFM to locate
the weight function $|F(x)|^{2}$, which is an extensive quantity to reckon the
symmetry energy ($S^{A}$), neutron pressure ($P^{A}$) and symmetry energy
curvature coefficient ($K_{sym}^{A}$) for Ne, Na, Mg, Al and Si nuclei in the
isotopic chains.

###  II.2 Nuclear Matter Parameter

The expression for the energy density of infinite and isotropic nuclear matter
(NM) are obtained from the Brückner functional defined as bruk68 ; bruk69 :

$\displaystyle{\cal{E}}(\rho)_{nucl.}$ $\displaystyle=$ $\displaystyle
AV_{0}(x)+V_{C}-V_{Cx},$ (6)

where

$\displaystyle V_{0}(x)$ $\displaystyle=$ $\displaystyle
37.53\Big{[}(1+\delta)^{5/3}+(1-\delta)^{5/3}]\rho_{0}(x)^{2/3}$ (7)
$\displaystyle+b_{1}\rho_{0}(x)+b_{2}\rho_{0}(x)^{4/3}+b_{3}\rho_{0}(x)^{5/3}$
$\displaystyle+\delta^{2}[b_{4}\rho_{0}(x)+b_{5}\rho_{0}(x)^{4/3}+b_{6}\rho_{0}(x)^{5/3}\Big{]}.$

Here, $b_{1}=-741.28$, $b_{2}=1179.89$, $b_{3}=-467.54$, $b_{4}=148.26$,
$b_{5}=372.84$, $b_{6}=-769.57$. In each Flucton there are protons having
Coulomb energy $V_{C}=\frac{3}{5}\frac{Z^{2}e^{2}}{x}$ and Coulomb exchange
energy $V_{Cx}=0.7386Ze^{2}(3Z/4\pi x^{3})^{1/3}$.

The important part of the present calculation is to convert the NM quantities
Eq. (6) from momentum ($\rho-$) space to coordinate ($r-$) space in local
density approximation (LDA). The NM parameters $S^{NM}$, $P^{NM}$,
$L_{sym}^{NM}$ and $K_{sym}^{NM}$ are obtained from the well defined relations
kumar18 ; Chen2014 ; bruk68 :

$\displaystyle S^{NM}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\frac{\partial^{2}({\cal
E}/\rho)}{\partial\alpha^{2}}\Big{|}_{\alpha=0},$ (8) $\displaystyle
L_{sym}^{NM}$ $\displaystyle=$ $\displaystyle 3\rho\frac{\partial
S(\rho)}{\partial\rho}\Big{|}_{\rho=\rho_{0}}=\frac{3P^{NM}}{\rho_{0}},$ (9)
$\displaystyle K_{sym}^{NM}$ $\displaystyle=$ $\displaystyle
9\rho^{2}\frac{\partial^{2}S(\rho)}{\partial\rho^{2}}\Big{|}_{\rho=\rho_{0}}.$
(10)

Here,

$\displaystyle S^{NM}$ $\displaystyle=$ $\displaystyle
41.7\rho_{0}(x)^{2/3}+b_{4}\rho_{0}(x)+b_{5}\rho_{0}(x)^{4/3}$ (11)
$\displaystyle+b_{6}\rho_{0}(x)^{5/3},$ $\displaystyle P^{NM}$
$\displaystyle=$ $\displaystyle
27.8\rho_{0}(x)^{5/3}+b_{4}\rho_{0}(x)^{2}+\frac{4}{3}b_{5}\rho_{0}(x)^{7/3}$
(12) $\displaystyle+\frac{5}{3}b_{6}\rho_{0}(x)^{8/3},$ $\displaystyle
K_{sym}^{NM}$ $\displaystyle=$
$\displaystyle-83.4\rho_{0}(x)^{2/3}+4b_{5}\rho_{0}(x)^{4/3}$ (13)
$\displaystyle+10b_{6}\rho_{0}(x)^{5/3},$

are the nuclear matter quantities at local density. The nuclear densities of
Ne, Na, Mg, Al and Si nuclei are calculated using RMF formalism. These
densities are used as input in CDFM (described in the following sub-section)
to calculate our weight function, which is a key quantity acting as a overpass
among the NM parameters within ($\rho-$) space and in ($r-$) space of finite
nuclei (manipulating LDA). To contest with the coordinate ($r-$) and momentum
($\rho-$) space simultaneously, we can build up the total density of a
nucleus. The number of Fluctons in an infinite manner can be superimposed to
form the total density of the nucleus, following the approach of CDFM
discussed below.

###  II.3 Coherent Density Fluctuation Model

The Coherent Density Fluctuations Model (CDFM) is firstly prescribed by
Antonov et al antozphys1980 , which plays a vital role to take care of the
fluctuation of momentum and coordinate. This method can be easily used to
interpret the surface properties of finite nuclei. In this CDFM formalism we
can use NM tools $S^{NM}$, $P^{NM}$ and $K_{sym}^{NM}$ from Eqs. (8)$-$(13) to
obtain their values for a finite nucleus anto1 ; anto2 ; anto3 ; antozphys1980
. Within this model, the density $\rho$ (r, r′) of a finite nucleus can be
rewritten as the coherent superposition of infinite number of one-body density
matrix (OBDM) $\rho_{x}$ (r, r′) for spherical parts of NM coined as Fluctons
bhu18 ; gad11 ,

$\rho_{x}({\bf r})=\rho_{0}(x)\,\Theta(x-|{\bf r}|),$ (14)

where $\rho_{o}(x)=\frac{3A}{4\pi x^{3}}$. The generator coordinate x is the
radius of a sphere consisting of Fermi gas having all the A nucleons
distributed uniformly over it. It is appropriate to administer for such a
system the OBDM disclosed as below bhu18 ; anto2 ; gad11 ; gad12 ,

$\rho({\bf r},{\bf r^{\prime}})=\int_{0}^{\infty}dx|F(x)|^{2}\rho_{x}({\bf
r},{\bf r^{\prime}}),$ (15)

where $|F(x)|^{2}$ is called as weight function (WF). The coherent
superposition of OBDM $\rho_{x}({\bf r},{\bf r^{\prime}})$ is given below as:

$\displaystyle\rho_{x}({\bf r},{\bf r^{\prime}})$ $\displaystyle=$
$\displaystyle 3\rho_{0}(x)\frac{J_{1}\left(k_{f}(x)|{\bf r}-{\bf
r^{\prime}}|\right)}{\left(k_{f}(x)|{\bf r}-{\bf r^{\prime}}|\right)}$ (16)
$\displaystyle\times\Theta\left(x-\frac{|{\bf r}+{\bf
r^{\prime}}|}{2}\right),$

where J1 is said to be the spherical Bessel function kind of first order and
$k_{f}$ is the Fermi momentum of nucleons inside the Flucton having radius $x$
and $k_{f}(x)=(3\pi^{2}/2\rho_{0}(x))^{1/3}=\gamma/x$, ($\gamma\approx
1.52A^{1/3}$). The Wigner distribution function of the OBDM of Eq. (16) is
given by,

$\displaystyle W({\bf r},{\bf k})=\int_{0}^{\infty}dx\,|F(x)|^{2}\,W_{x}({\bf
r},{\bf k}).$ (17)

Here, $W_{x}({\bf r},{\bf k})=\frac{4}{8\pi^{3}}\Theta(x-|{\bf
r}|)\Theta(k_{F}(x)-|{\bf k}|)$. The density $\rho$ (r) in terms of weight
function within the CDFM access is:

$\displaystyle\rho(r)$ $\displaystyle=$ $\displaystyle\int d{\bf k}W({\bf
r},{\bf k})$ (18)
$\displaystyle=\int_{0}^{\infty}dx\,|F(x)|^{2}\,\frac{3A}{4\pi
x^{3}}\Theta(x-|{\bf r}|),$

which is normalized to A, i.e., $\int\rho({\bf r})d{\bf r}=A$. In the
$\delta$-function limit, the Hill-Wheeler integral equation, that is the
differential equation for the WF in the generator coordinate is retrieved
anto1 . The $|F(x)|^{2}$ for a provided density $\rho$ (r) is described as
here

$|F(x)|^{2}=-\left(\frac{1}{\rho_{0}(x)}\frac{d\rho(r)}{dr}\right)_{r=x},$
(19)

with $\int_{0}^{\infty}dx|F(x)|^{2}=1$. A detailed genealogy can be found in
Refs. bhu18 ; anto1 ; anto2 ; gad11 ; gad12 . The finite nuclear symmetry
energy $S^{A}$ neutron pressure $P^{A}$ and surface curvature coefficients
$K_{sym}^{A}$ are calculated by weighting the corresponding quantities for
infinite NM within the CDFM, as given below anto4 ; gad11 ; gad12 ; fuch95 ;
anto17

$\displaystyle S^{A}=\int_{0}^{\infty}dx\,|F(x)|^{2}\,S^{NM}(\rho(x)),$ (20)
$\displaystyle P^{A}=\int_{0}^{\infty}dx\,|F(x)|^{2}\,P^{NM}(\rho(x)),$ (21)
$\displaystyle K_{sym}^{A}=\int_{0}^{\infty}dx\,|F(x)|^{2}\
K_{sym}^{NM}(\rho(x)).$ (22)

The $S^{A}$, $P^{A}$ and $K_{sym}^{A}$ in Eqs. ($\ref{s0}-$22) are the surface
weighted average of the corresponding NM quantities in the LDA limit for
finite nuclei.

###  II.4 Volume and surface symmetric energy in Danielewicz’s liquid drop
prescription

The nuclear binding energy E(A, Z) in the liquid drop model incorporating the
volume symmetry energy parameter $S_{V}$ and modified surface symmetry energy
parameter $S_{S}$ is written as steiner2005 ; myers :

$\displaystyle
E(A,Z)=-B.A+E_{S}A^{2/3}+S_{V}A\frac{(1-2Z/A)^{2}}{1+S_{S}A^{-1/3}/S_{V}}$
$\displaystyle+E_{C}\frac{Z^{2}}{A^{1/3}}+E_{dif}\frac{Z^{2}}{A}+E_{ex}\frac{Z^{4/3}}{A^{1/3}}+a\Delta
A^{-1/2}.$ (23)

In Eq. (23), B= binding energy per particle of symmetric NM at saturation.
$E_{S}$, $E_{C}$, $E_{dif}$, and $E_{ex}$ are the coefficients of the surface
energy of symmetric matter, the Coulomb energy, the diffuseness correction,
and the exchange correction to the Coulomb energy, conjointly. The pairing
corrections are delivered by the uttermost term, which is crucial for open-
shell nuclei. The symmetry energy is rewritten as [3rd term of Eq. (23) in the
form S = $(N-Z)^{2}$/A, where

$\displaystyle
S=\frac{S_{V}}{1+\frac{S_{S}}{S_{V}}A^{-1/3}}=\frac{S_{V}}{1+A^{-1/3}/\kappa}.$
(24)

From the above Eq. (24), the individual components of $S_{V}$ and $S_{S}$ can
be written as:

$\displaystyle S_{V}=S\left(1+\frac{1}{\kappa A^{1/3}}\right)$ (25)

and

$\displaystyle S_{S}=\frac{S}{\kappa}\left(1+\frac{1}{\kappa A^{1/3}}\right).$
(26)

The symmetry energy and its volume and surface components are calculated
within the CDFM formalism bhu18 ; anto2 ; antozphys1980 . Following Refs.
dani03 ; dani04 ; dani06 , an approximate expression for the ratio
$\kappa\equiv\frac{S^{V}}{S^{S}}$ can be written within the CDFM,

$\displaystyle\kappa=\frac{3}{R\rho_{0}}\int_{0}^{\infty}dx|F(x)|^{2}x\rho_{0}(x)\left[\left(\frac{S(\rho_{0})}{S(\rho(x)}\right)-1\right].$
(27)

Here S($\rho_{0}$) is the nuclear symmetry energy at equilibrium nuclear
matter density $\rho_{0}$. Employing the density dependence of symmetry energy
dani03 :

$\displaystyle
S[\rho(x)]=S^{V}\left(\frac{\rho(x)}{\rho_{0}}\right)^{\gamma},$ (28)

There exist various estimations for the value of the parameter $\gamma$. In
present work, we use $\gamma$ = 0.3 in reference to anto18 . Using above eq.
and S($\rho_{0}$) = $S_{V}$, the Eqs. (20) and (27) can be re-written as
follows:

$\displaystyle
S=S(\rho_{0})\int_{0}^{\infty}dx|F(x)|^{2}\left(\frac{\rho(x)}{\rho_{0}}\right)^{\gamma},$
(29)

and

$\displaystyle\kappa=\frac{3}{R\rho_{0}}\int_{0}^{\infty}dx|F(x)|^{2}x\rho_{0}(x)\left(\left(\frac{\rho_{0}}{\rho(x)}\right)^{\gamma}-1\right).$
(30)

Figure 1: (Color online) (a) The total binding energy B.E. (b) Binding energy
per particle B.E./A and (c) Two neutron separation energy $S_{2n}$ as a
function of neutron number N for Ne, Na, Mg, Al and Si isotopes, respectively.
Figure 2: (color online) (a) The charge distribution radius and (b) the
quadruple deformation parameter $\beta_{2}$ as a function of neutron number N
for Ne, Na, Mg, Al, and Si isotopes.

##  III Results and discussions

The surface properties, like symmetric energy, neutron pressure, and symmetry
energy curvature coefficient of finite nuclei are studied using the coherent
density fluctuation model by taking the RMF density as input in the
calculations. Before going to the explanation of the surface properties of
finite nuclei, we highlight herewith the bulk properties (binding energy,
deformation, and nuclear charge radius) of the considered nuclei such as Ne,
Na, Mg, Al, and Si. The results are discussed through Table 1 and Figs. 1-8.
In Table 1 , the values of the coefficients of the Gaussian function [Eq. 5]
$c_{1}$, $c_{2}$ and $a_{1}$, $a_{2}$ are given, which are used to get the
spherical equivalent densities of the Ne, Na, Mg, Al and Si isotopes. The
obtained spherical equivalent densities are further used to calculate the
weight function $|F(x)|^{2}$ [Eq. 22]. Then the weight function $|F(x)|^{2}$
is used to calculate the nuclear matter parameters, such as symmetric energy
$S^{A}$, neutron pressure $P^{A}$ and symmetry energy curvature coefficient
$K_{sym}^{A}$ for finite nuclei. The predicted results are given in Figs. 1-8
in the following subsections.

###  III.1 Binding energy and two neutron-separation energy

To estimate the surface properties such as symmetry energy, neutron pressure,
and symmetry energy curvature coefficient of finite nuclei, one needs to have
an understanding of the ground state bulk properties. The binding energy (BE),
root mean square (rms), charge radius ($R_{ch}$) and quadruple deformation
parameter ($\beta_{2}$) for Ne, Na, Mg, Al, and Si isotopes starting from
proton-rich to the expected neutron drip-line are obtained by using RMF model
and NL3∗ parameter set. The calculated results with available experimental
data for the above-considered isotopes are shown in Figures 1 and 2. From the
overall observation of binding energy for all the considered isotopic chains,
we note a good agreement between our results and the experimental data raman ;
angeli . A careful analysis of Fig. 1 shows that the expected drip-lines for
Ne, Na, Mg, Al, and Si isotopic chain are at mass number (A) = 34, 39, 40, 47,
and 48 respectively.

Figure 3: (color online) The neutron skin thickness $\triangle{r_{np}}$ is
presented as a function of neutron number N for Ne, Na, Mg, Al and Si nuclei.
Figure 4: (color online) The spherical equivalent density (solid line) for
some of the selected nuclei are shown. The corresponding weight function
(dashed line) are also displayed.

The two neutron separation energy $S_{2n}$ can be determined using the
relation $S_{2n}(N,Z)$= BE(N,Z) - BE(N-2,Z). Where both the $BE^{\prime}s$ are
evaluated using RMF(NL3*) parameter for theory and experimental values are
taken from the Ref. wang2017  for comparison. The estimated $S_{2n}$ values
along with the available experimental data are presented in the lower panel of
Fig. 1 for the 18-34Ne, 20-39Na, 22-40Mg, 24-47Al and 26-48Si isotopic series.
One can notice from the figure, $S_{2n}$ decreases smoothly with an increase
in neutron number except some sharp discontinuities (i.e kinks) at some
particular neutron number $N_{magic}$ indicating the shell closure property.
It is worthy here to mention that the energy requires to remove two neutrons
from a nucleus with $N_{magic}$+2 is much less than the nucleus with
$N_{magic}$ breaking the regular pattern of the $S_{2n}$. In the lower panel
of Fig. 2, we have displayed the total quadruple deformation parameter
$\beta_{2}$ for the considered isotopic series. It is clear from the figure
that there are shape transitions at neutron numbers (N) = 14 for Ne, 20 for Na
and Mg, 28 for Si isotopic series.

To find the ground state solution of a given nucleus, we obtain the conversed
solutions by taking various initial quadruple deformation. From all those
solutions, we have considered the maximum binding energy and corresponding
quantities as the ground state of the given nucleus. The radial components,
which are having odd under time-reversal symmetry and parity must be
neglected. In the present paper, we have implemented the blocking method which
rectifies the time-reversal symmetry. Normally we need to block different
states around the Fermi level to determine the lowest energy states for the
odd nucleus as per odd-odd, odd-even, and even-odd nuclei, the time-reversal
symmetry violates. In order to take care of the time-reversal violation into
account, we used the simple blocking procedure as discussed in Ref. patra . In
many of the cases, we get almost nearly equal energies among various states
indicating the shape co-existence nature of the isotopes. As we are following
the criteria of taking the highest BE result, there are some low-laying
excited states (degenerate states), which may be the reason behind the
irregular plots. From those obtained values we got our ground state, excited
states and degenerate states (almost same magnitude of binding energy with the
different state).

In Fig. 2, we have displayed the charge distribution radius $R_{ch}$ (upper
panel) and quadruple deformation parameter $\beta_{2}$ (lower panel) for the
isotopic chain of Ne, Na, Mg, Al, and Si nuclei. The experimental data raman ;
angeli  are also compared, wherever available. Here in the lower mass region,
the charge radius $R_{ch}$ decreases and then increases with the mass number
for all the isotopic chain. From the quadruple deformation parameter
($\beta_{2}$), we find shape transition appears at N = 14, 20, 20, 29, and 28
for Ne, Na, Mg, Al, and Si isotopes, respectively. It is to be noted that the
irregular nature of the results mostly due to the shape co-existence nature of
the isotopes. A low-lying excited state is very often noticed in these
isotopic chains.

###  III.2 The neutron skin thickness

In Fig. 3, we have presented the neutron skin thickness $\triangle
r_{np}=r_{n}–r_{p}$ as a function of neutron number. The $\triangle r_{np}$ is
an important quantity, which is connected with the surface property of the
nucleus in terms of isospin asymmetry. It has a direct relation with the
nuclear equation of state (EoS) which controls the structure of the neutron
star. Thus, it has an important role in nuclear astrophysics. We have shown it
for Ne, Na, Mg, Al, and Si isotopic series. As we see the neutron skin
thickness increases with neutron number, i.e., the presence of more neutrons
enhanced the nuclear radius of the nucleus. The neutron skin thickness in the
nucleus is formed by a combination of volume and surface contributions. The
volume part explains an increase in the local mean field of the surface of the
neutron to the proton individual and the surface part in which the surface
width of the neutron increases as compared to one of the individual protons.
Both the contributions increase with neutron excess vinas2012  in an isotopic
series. The linear increase in neutron skin thickness indicates the
surface/volume saturation, which also tells the increase in symmetric energy.

###  III.3 The spherical equivalent density and weight function

In Fig. 4, we have plotted the spherical equivalent densities (solid) and
corresponding weight functions $|F(x)|^{2}$ (dashed) as a function of radius.
We have converted the deformed densities into their spherical equivalents
through two Gaussian fittings by using Eq. 5. The spherical equivalent density
for any nucleus can be easily obtained by using the values of $c_{1}$, $a_{1}$
and $c_{2}$, $a_{2}$ given in Table 1. For a careful inspection of Table 1, it
is noticed that for all the nuclei, the values for $c_{1}$ are found to be
negative and $c_{2}$ with a positive values. However, the values of $a_{1}$
and $a_{2}$ are almost comparable to each other with positive signs. More
detail analysis of the fitting and also technical details can be found in
Refs. patra2009 ; panda2014 ; sharma2016 .

Table 1: The axially deformed RMF density obtained from the NL3∗ parameter set are converted to its spherical equivalent using two Gaussian functionals. The coefficients $c_{1}$, $a_{1}$ and $c_{2}$, $a_{2}$ are listed for Ne, Na, Mg, Al, and Si isotopes. Nucleus | $c_{1}$ | $a_{1}$ | $c_{2}$ | $a_{2}$ | Nucleus | $c_{1}$ | $a_{1}$ | $c_{2}$ | $a_{2}$
---|---|---|---|---|---|---|---|---|---
19Ne | -3.16481 | 0.331586 | 3.47687 | 0.311655 |  20Ne | -3.04522 | 0.333589 | 3.41182 | 0.313962
21Ne | -3.59429 | 0.337037 | 3.95496 | 0.317189 |  22Ne | -4.08107 | 0.339088 | 4.4364 | 0.319048
23Ne | -3.83833 | 0.316714 | 4.17055 | 0.297795 |  24Ne | -3.61795 | 0.42193 | 3.99949 | 0.373673
25Ne | -1.61846 | 0.409952 | 2.0865 | 0.337241 |  26Ne | -1.44537 | 0.400685 | 1.90112 | 0.322851
27Ne | -0.856237 | 0.394508 | 1.30403 | 0.291229 |  28Ne | -0.805704 | 0.365513 | 1.24565 | 0.274331
29Ne | -2.09689 | 0.306084 | 2.50567 | 0.272813 |  30Ne | -3.85888 | 0.276814 | 4.23455 | 0.260330
31Ne | -2.18123 | 0.208782 | 2.44085 | 0.194627 |  32Ne | -1.55223 | 0.202165 | 1.8495 | 0.187297
33Ne | -1.07001 | 0.210559 | 1.37341 | 0.186965 |  34Ne | -0.284251 | 0.253762 | 0.588779 | 0.171028
20Na | -3.30838 | 0.33129 | 3.61359 | 0.309985 |  21Na | -3.36721 | 0.336049 | 3.72673 | 0.315269
22Na | -4.20169 | 0.339772 | 4.55721 | 0.320112 |  23Na | -4.93147 | 0.342127 | 5.28423 | 0.323103
24Na | -1.59925 | 0.3403 | 1.92878 | 0.289744 |  25Na | -5.01558 | 0.302793 | 5.32092 | 0.287093
26Na | -4.65003 | 0.297148 | 4.94264 | 0.280147 |  27Na | -5.93258 | 0.398734 | 6.37662 | 0.369386
28Na | -2.01555 | 0.388844 | 2.46011 | 0.324078 |  29Na | -1.57596 | 0.361364 | 2.01826 | 0.297191
30Na | -3.8188 | 0.315588 | 4.2291 | 0.291141 |  31Na | -5.99753 | 0.27771 | 6.3129 | 0.263971
32Na | -2.84109 | 0.216843 | 3.11021 | 0.203505 |  33Na | -2.05982 | 0.217242 | 2.3596 | 0.200594
34Na | -0.815459 | 0.235539 | 1.12203 | 0.193029 |  35Na | -0.440892 | 0.262169 | 0.748431 | 0.182956
36Na | -0.425172 | 0.257764 | 0.729539 | 0.178082 |  37Na | -0.426312 | 0.252979 | 0.725345 | 0.174057
38Na | -0.560764 | 0.236309 | 0.85135 | 0.174332 |  39Na | -0.767664 | 0.221727 | 1.04873 | 0.174823
22Mg | -3.5102 | 0.337634 | 3.86364 | 0.315119 |  23Mg | -4.68847 | 0.341192 | 5.03992 | 0.32147
24Mg | -5.58442 | 0.345115 | 5.93632 | 0.326273 |  25Mg | -5.70374 | 0.32602 | 6.03196 | 0.308936
26Mg | -5.91305 | 0.308559 | 6.21652 | 0.293159 |  27Mg | -6.04836 | 0.305345 | 6.33983 | 0.289534
28Mg | -6.28349 | 0.303037 | 6.56421 | 0.287016 |  29Mg | -2.13344 | 0.407667 | 2.57616 | 0.332949
30Mg | -3.19583 | 0.260324 | 3.48083 | 0.240888 |  31Mg | -4.19839 | 0.228173 | 4.44081 | 0.216497
32Mg | -2.65606 | 0.288664 | 2.97542 | 0.256755 |  33Mg | -2.40436 | 0.311802 | 2.77146 | 0.270189
34Mg | -1.27359 | 0.233869 | 1.56893 | 0.201356 |  35Mg | -2.06917 | 0.231846 | 2.37601 | 0.209307
36Mg | -0.808364 | 0.255092 | 1.1172 | 0.197461 |  37Mg | -0.700842 | 0.25534 | 1.00767 | 0.191015
38Mg | -0.779685 | 0.247165 | 1.07905 | 0.188717 |  39Mg | -0.944354 | 0.236374 | 1.23856 | 0.187856
40Mg | -1.6618 | 0.220155 | 1.94604 | 0.191281 |  |  |  |  |
24Al | -1.45476 | 0.340458 | 1.78352 | 0.286481 |  25Al | -5.33616 | 0.325916 | 5.6647 | 0.30783
26Al | -8.96115 | 0.414522 | 9.43319 | 0.393303 |  27Al | -4.25196 | 0.384866 | 4.61596 | 0.346254
28Al | -9.60618 | 0.387342 | 10.0265 | 0.368222 |  29Al | -10.1518 | 0.387245 | 10.5648 | 0.367966
30Al | -8.57763 | 0.358694 | 8.99408 | 0.340275 |  31Al | -3.93913 | 0.363267 | 4.39077 | 0.326436
32Al | -2.62455 | 0.320816 | 3.00351 | 0.280775 |  33Al | -1.90801 | 0.223972 | 2.13351 | 0.199603
34Al | -1.76454 | 0.22558 | 2.02282 | 0.200253 |  35Al | -4.40195 | 0.215828 | 4.66863 | 0.205308
36Al | -4.45984 | 0.219293 | 4.74574 | 0.208377 |  37Al | -4.54495 | 0.220995 | 4.83685 | 0.209645
38Al | -4.7981 | 0.217558 | 5.07953 | 0.206567 |  39Al | -4.69265 | 0.214125 | 4.97399 | 0.203054
40Al | -3.60492 | 0.225556 | 3.88433 | 0.208409 |  41Al | -3.50796 | 0.222836 | 3.78546 | 0.205275
42Al | -2.99132 | 0.248402 | 3.32657 | 0.222474 |  43Al | -3.05889 | 0.253024 | 3.39251 | 0.22516
44Al | -3.02871 | 0.249435 | 3.36787 | 0.221941 |  45Al | -2.9738 | 0.260693 | 3.31665 | 0.228223
46Al | -3.1057 | 0.245182 | 3.43189 | 0.217474 |  47Al | -0.319467 | 0.289022 | 0.63011 | 0.153346
26Si | -7.91999 | 0.414491 | 8.39401 | 0.390892 |  27Si | -4.17883 | 0.381022 | 4.53972 | 0.342842
28Si | -3.26289 | 0.341632 | 3.63435 | 0.30699 |  29Si | -3.778 | 0.363259 | 4.1903 | 0.326487
30Si | -6.2779 | 0.265629 | 6.52777 | 0.253178 |  31Si | -2.90607 | 0.336128 | 3.28085 | 0.294991
32Si | -2.94072 | 0.303651 | 3.28119 | 0.270526 |  33Si | -1.99993 | 0.20584 | 2.19687 | 0.186214
34Si | -2.06805 | 0.212795 | 2.27827 | 0.191571 |  35Si | -2.00258 | 0.217443 | 2.24468 | 0.195173
36Si | -2.08563 | 0.206827 | 2.29686 | 0.185987 |  37Si | -2.02117 | 0.221807 | 2.28665 | 0.197815
38Si | -2.03642 | 0.226282 | 2.31387 | 0.200671 |  39Si | -2.14661 | 0.225337 | 2.41648 | 0.199744
40Si | -3.18035 | 0.30413 | 3.59945 | 0.266638 |  41Si | -2.98979 | 0.250094 | 3.33998 | 0.225143
42Si | -2.72873 | 0.236071 | 3.07376 | 0.21272 |  43Si | -2.60821 | 0.230403 | 2.95853 | 0.20749
44Si | -2.87398 | 0.234379 | 3.22239 | 0.211086 |  45Si | -6.43192 | 0.225856 | 6.78428 | 0.215118
46Si | -3.39586 | 0.244309 | 3.75371 | 0.219991 |  47Si | -3.19051 | 0.231893 | 3.5316 | 0.208824
48Si | -0.357803 | 0.322425 | 0.671824 | 0.159255 |  |  |  |  |
Figure 5: (color online) The symmetric energy is shown as a function of
neutron number N for Ne, Na, Mg, Al and Si nuclei. Figure 6: (color online)
Here we have represented neutron pressure as a function of neutron number N
for Ne, Na, Mg, Al and Si nuclei.

We have shown the equivalent density as a function of radial coordinate for
20,29Ne, 23,31Na, 24,39Mg and 28,45Si isotopes as representative cases. It is
to be noted that the density plots in Fig. 4 reflect the normalized equivalent
density of the deformed nucleus. In each panel, we have shown the equivalent
densities for the isotopes from the $\beta-$stable and neutron drip-line
region for each atomic nucleus, which provides the information relative
changes of the density with-respect-to isospin asymmetry. For example, panel
(a), (b), (c), and (d) are assigned for the isotopes of 20,29Ne, 23,31Na,
24,30Mg, and 28,45Si, respectively. From the spherical equivalent density, we
obtained the weight function $|F(x)|^{2}$ of the nucleus. This is one of the
most important ingredients to determine the surface properties of the nucleus.
In each specific representation, although the small difference in the surface
region of the densities reflects a wide difference in the weight functions. As
a consequence, one can expect a very different surface property for
$\beta-$stable and drip-line nuclei.

Figure 7: (color online) The symmetry energy curvature coefficients are given
for the Ne, Na, Mg, Al and Si nuclei isotopic series. Figure 8: (color online)
The volume ($S_{V}$), surface ($S_{s}$) and total surface energy (S) are
shown. The $\kappa-$factor is also given as a function of neutron number N for
Ne, Na, Mg, Al and Si nuclei.

###  III.4 Nuclear surface properties

There are several prescriptions to analyze nuclear surface properties of
finite nuclei in terms of isospin dependent nuclear matter quantities such as
symmetry energy and its derivatives. In the present work, we have mainly
highlighted the CDFM approach of Brückner’s functional and the liquid drop
approximation of Danielewicz dani03 . Furthermore, the symmetry energy is
separated into it’s volume and surface components, which are mainly
approximated in connection with the nuclear matter’s properties at the
saturation. For example, Eq. (29) shows that the saturation symmetric energy
$S(\rho_{0})$ and nuclear matter density are needed to calculate the symmetric
energy of a finite nucleus. On the other hand, for the same quantity, we need
in the nuclear matter Brückner’s functional as of Eqs. (20)$-$(22). The
calculated nuclear surface properties such as symmetric energy, neutron
pressures, and symmetry energy curvature are given in Fig. 5, 6, and 7,
respectively as a function of the neutron number. In each figure, we have
given the corresponding calculated quantities for the isotopic chain of
considered nuclei Ne, Na, Mg, Al, and Si. From the Fig. 5, it can be noticed
that the symmetry energy for all the isotopes bounded within a range of 23-25
$MeV$. It is worth mentioning that the exotic nuclei having large neutron-to-
proton asymmetry respond to the nature of nuclear symmetry energy. Further,
the appropriate information of symmetry energy from finite nuclei will be
added to a wide range of nuclear phenomena starting from the study of nuclear
structure, dynamics of heavy-ion reactions to the high isospin asymmetry
system such as neutron star matter.

Observing the characteristics of symmetry energy $S^{A}$, and surface symmetry
curvature $K_{sym}^{A}$ in Figs. 5, and 7, in general, distorted peaks appear
at N = 14, 16, 20, and 28, for some of the cases. Similarly, we also notice a
fall in the neutron pressure $P^{A}$ at the same neutron numbers in Fig. 6. We
found the value of $S^{A}$ within the range of 24-25 $MeV$ at the neutron
number N = 20 for all the isotopic series of Ne, Na, Mg, Al, and Si nuclei. In
all the figures, the trend is not smooth, this is either from the transform of
deformed density into the spherical equivalent one by using the two Gaussian
fittings (discussed in sub-section III.3) or by neglecting the shape degrees
of freedom. Although there is an anomalous trend appear in all the isotopic
chain, a careful inspections show that the magnitude of the peak and/or depth
are assuredly at the neutron magic N =20 and also for 28 along with a few more
neutron numbers, namely N = 14, and 16. These larger (lower) magnitude of
$S^{A}$, $P^{A}$ and $K_{sym}^{A}$ provides the signature of shell/sub-shell
closures over the isotopic chain. It is to be noted that, one can obtain the
$K_{sym}^{A}$ by adopting the Thomas-Fermi approach along with relativistic
mean-field theory within the leptodermus expansion for finite nuclei
patra2002curv . Here the neutron pressure and symmetry energy curvature are
distributed within a range of -7 to -1 $MeV/fm^{3}$ and -550 to -250 $MeV$
respectively. In some of our earlier study bhu18 ; abdul19 ; manpreet , we
notice a relation of $S^{A}$ with the shell closure nature of magic number,
i.e., we get a maximum of $S^{A}$ at the magic number. This correlation is
seen only for light and medium mass nuclei. While searching such relation in
the considered mass region although, we noticed such type of peaks at the
presently reported magic number plb2020Han  within the Brückner’s functional,
but it does not show a regular pattern, as we are getting in medium mass
region manpreet . For further analysis of the surface properties of symmetry
energy, we subdivided the $S$ values into its surface and volume components as
shown in Fig. 8. To calculate these quantities, we have used the prescription
of P. Danielewicz dani03 ; dani04 ; dani06 ; dani09 . The estimation of $S$
with Eq. (29) is quite different than that of (20) dani03 ; dani04 ; dani06 ;
dani09 . As a result, we predict nearly two different values of $S$ in both
two approaches, which can be seen clearly in Figs. (5) and (8). In this Fig. 8
we have produced the volume, the surface, and the total surface energy. The
parameter $\kappa=S_{V}/S_{S}$, i.e., the ratio of the volume to the surface
part of the symmetric energy is also given in the figure. We observed that the
symmetry energy related to volume is more than the surface contribution and
their ratio $\kappa$ decreases with increasing of N. As the symmetric energy
is pretty much related to N-Z asymmetry, hence it is crucial to examine the
surface and volume part. At the surface of the nucleus, the nuclear saturation
does not acquire, and also due to the unequal population of the number density
of the nucleons, the surface symmetry energy get more crucial, detail
explanation can be found in Ref. feenberg1947 .

##  IV Concluding Remarks

In our outline, the properties of the surface of a nucleus, such as symmetric
energy, neutron pressure, and the symmetry energy curvature coefficient as a
function of the neutron number are investigated for Ne, Na, Mg, Al, and Si
nuclei from the proton to neutron drip-lines. The Coherent Density Fluctuation
Model (CDFM) is used to evaluate these quantities within the framework of the
relativistic mean-field formalism. Here, in CDFM the nuclear/nucleonic density
does not follow any sharp edge surface; thus, the diffuseness parameter is
also not neglected. The fluctuation of the flucton attains infinitesimal
size/dimension and is distributed over the range of nuclear density
distribution from the relativistic mean-field model. Hence, there is no
possibility of sloppy surface contribution from the density in terms of weight
function. And further, we have examined the CDFM account the surface effect
substantially compared to Liquid-Drop-Approximation subrat  and reference
therein. We have discussed the bulk properties of these isotopic chain, which
are essential for determining the symmetry energy and its co-efficient. We
evaluated the binding energy, the binding energy per particle, charge radius,
nuclear quadruple moment deformation parameter, and the neutron-skin
thickness. To locate the recently reported magic structure of some of the
isotopes in the considered region, we analyzed the two-neutron separation
energy. The correlation of the magic number with the surface symmetric
properties of the isotopes are studied. We noticed some relation in the
reported isotopes, which is recently a fact. As expected, the neutron-skin
thickness increases monotonously with neutron numbers in an isotopic series.
On the other hand, the BE/A increases initially and then decreases with N. The
shape of the nucleus in an isotopic chain changes and does not show any
regular pattern. This is because a larger number of nuclei exhibit shape
coexistence.

We converted the deformed densities to their spherical equivalent with a two
Gaussian fitting. The coefficients $a_{1}$ and $a_{2}$ are in general
decrease, but $c_{1}$ increases in an zig-zag manner. The values of $c_{2}$
however decrease with neutron number. In practice, when we use the Gaussian
fitting for the axial density distribution to obtain the spherical one, there
is an enhancement of central density with a long tail. As a result, the
surface covers a wide range of the radius, which is the limitation of the
Gaussian fitting procedure. In some of the previous studies by our
collaborators, various fitting techniques were used to convert axial deform
density to spherical form and their effect on the surface properties of finite
nuclei quantitatively kishore20  and reference therein.

In our present work, we reported here two types of symmetry energies, one due
to liquid drop approximation and the other due to LDA by Brückner method.
Although we can not measure the symmetric energy often directly, we can
extract the values from other nuclear observable, indirectly which are related
to it bhu18 . But the question is why we are not getting the same values or
the same trends of symmetry energy for two different approximations? Although
the dimensions are the same in both approaches the expressions are different
and the fact is that these nuclei are small in size. Nuclei having large size
show almost the same values of symmetry energy in both approximations as we
have shown in our previous work quddus2020 . Here we get the maximum value of
symmetry energy in the liquid drop approach which is $33.11$ and $25.52$ $MeV$
in Brückner functional adoption for NL3* parameter set.

In the earlier studies, it is shown that the symmetric energy and/or neutron
pressure is connected with the shell/sub-shell closure in the emergence of a
peak and/or fall over an isotopic chain gad11 ; gad12 ; bhu18 ; quddus2020 .
Hence the symmetry energy and its co-efficient (neutron pressure) can be used
as an observable for predicting the neutron and/or proton magic at the drip-
line region. Although we get considerable magnitude in the symmetric energy
for these neutron numbers (N=14, 16 and 32/34), but a broke-down/ distorted
trend in the isotopic chain of all the considered nuclei. The abnormal trend
in these mass regions can be correlated with the possible reasons:

  * •

The considered nuclei in the present study are lighter mass nuclei, hence
quite sensitive to the neutron-to-proton ratio. Analogous to the neutron
proton ratio, the neutron-to-proton difference is very small, hence the
isospin dependency of the system not effective, as a result, the symmetry
energy for these mass nuclei is not followed the systematic trend.

  * •

The considered mass nuclei are found to be highly deformed in their ground
state, which reflects in their densities. Following the limitation of the
calculation, the spherical equivalent densities using a slender approximation
are implemented into the CDFM to obtain the weight function. It is worth
mentioning that, the density in terms of weight function is one of the central
ingredients for obtaining the symmetry energy kishore20  and its components.

  * •

As we mentioned above, the considered isotopes are small in size, hence it is
likely to get clustering configuration with sharp surface instead of a smooth
surface as that of nuclei of mass A $\geq$ 40 arima72 ; aremugumpatra05 ;
bhu13 ; nora08 , it has been shown that the smaller nuclei in the mass table
show cluster properties unlike developing a smooth surface as that of medium
or heavy mass nuclei.

Therefore we suggest, while doing these further calculations in the lower mass
region, one must need to take care of these points.

Acknowledgement: One of the authors (JAP) is thankful to the Institute of
Physics, Bhubaneswar for providing the computer facilities during the work.
This work is partly reinforced by SERB, Department of Science and Technology,
Govt. of India, Project No. CRG/2019/002691. MB acknowledges the support from
FAPESP Project No. 2017/05660-0, FOSTECT Project No. FOSTECT.2019B.04, and the
CNPq - Brasil.

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# Threshold and frequency properties of a cold ytterbium laser

Dmitriy Sholokhov<EMAIL_ADDRESS>Experimentalphysik, Universität des Saarlandes, D-66123 Saarbrücken, Germany
Saran Shaju  Experimentalphysik, Universität des Saarlandes, D-66123
Saarbrücken, Germany  Ke Li  Experimentalphysik, Universität des Saarlandes,
D-66123 Saarbrücken, Germany  Simon B. Jäger  Physics Department and Research
Center OPTIMAS, University of Kaiserslautern-Landau, D-67663 Kaiserslautern,
Germany  Jürgen Eschner<EMAIL_ADDRESS>Experimentalphysik, Universität des Saarlandes, D-66123 Saarbrücken, Germany

###### Abstract

We investigate properties of the lasing action observed on the
1S${}_{0}\,\leftrightarrow\,^{3}$P1 intercombination transition of ytterbium
atoms that are laser-cooled and -trapped inside a high-finesse cavity. The
dressing of the atomic states on the 1S${}_{0}\,\leftrightarrow\,^{1}$P1
transition by the magneto-optical trap (MOT) laser light allows the coupled
atom-cavity system to lase, via a two-photon transition, on the same line on
which it is pumped. The observation and basic description of this phenomenon
was presented earlier by Gothe et al. [1]. In the current work, we focus on a
detailed analysis of the lasing threshold and frequency properties and perform
a comparison to our theoretical models.

cold atoms, lasing, ytterbium, superradiant laser, collective coupling, cavity
QED

##  I Introduction

The conventional laser, based on population inversion and cavity-stimulated
emission between two atomic eigenstates, has been extended to more complex
lasing mechanisms such as the cold atom [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12],
the inversionless [13, 14, 15, 16, 17] and the super-radiant [18, 19, 20, 21,
22, 23, 24] laser. These approaches aim to discover, study, and apply new
light-matter interactions [25, 26, 27, 28, 29, 30, 31, 32], for example to
better understand collective dynamics and matter phases in cold atom gases.
New advances in this direction will also help to overcome thermal-noise-
limited laser linewidth [33, 34, 23], providing new tools for the atomic clock
world, such as gravimeters [35], dark matter analysers [36], or fundamental
constants testers [37].

For such purposes, it is particularly promising to realize ultra-narrow
tunable lasers, which are based on optical transitions in cold atomic systems
[38, 18, 21]. Atoms with a narrow transition are thus placed inside of a
rapidly decaying cavity, in the so-called ”bad-cavity” regime [39, 40, 41]. In
this regime the laser linewidth is less affected by noise in the cavity
degrees of freedom since the coherence is mostly stored in the atomic degrees
of freedom. Using the forbidden transitions of cooled and trapped atoms would
allow one to achieve lasing linewidth on the $1\text{\,}\mathrm{mHz}$ scale
[39, 38, 18, 42].

Lasing mechanisms in cold atomic systems have been studied with rubidium [6,
19], calcium [43, 44], strontium [38, 18, 21, 45, 46], and ytterbium [1].
Recent progress towards a super-radiant laser has been obtained with strontium
atoms [47, 48, 49, 50, 51, 31, 52, 53]. Cold ytterbium (Yb) is an equally
promising platform, in particular with respect to its ultra-narrow clock
transition ${}^{1}\text{S}_{0}\,\leftrightarrow\,^{3}\text{P}_{0}$
($\sim$10\text{\,}\mathrm{mHz}$$ linewidth). In related work, we recently
demonstrated a cold-atom laser in continuous-wave operation on the semi-
forbidden ${}^{1}\text{S}_{0}\,\leftrightarrow\,^{3}\text{P}_{1}$ transition
($\sim$182\text{\,}\mathrm{kHz}$$ linewidth) [1].

In that work we identified a laser gain mechanism based on a two-photon
process that involves the light used for trapping and cooling the atoms in a
magneto-optical trap (MOT). The two-photon transition effectively couples the
3P1 and 1P1 atomic states through simultaneous photon emission into the cavity
and photon absorption from the MOT light. Population inversion is established
because of the large difference in the 1P1 and 3P1 lifetimes. The cavity with
a decay time on the order of the 3P1 lifetime keeps the system in between the
good- and bad-cavity regimes and supports the described process.

While we established the basic mechanism, the detailed parameter regimes
required for the lasing have been far less understood. This was partially due
to the absence of a qualitative and quantitative theory that can predict the
lasing threshold and behavior. In this paper, we close these gaps and provide
a detailed analysis of the lasing threshold and frequency, both as functions
of the parameters of the exciting (pump) light, the MOT light, and the cavity.
With two different approaches based on optical Bloch equations [12], we model
steady-state and dynamical properties of the laser, and we compare the results
to our experimental findings, finding excellent qualitative and good
quantitative agreement.

In the next Section II, we introduce our experimental setup. We continue by
describing the performed measurements and their results in Sec. III. In Sec.
IV and V we present an analysis of threshold and frequency properties of the
cold ytterbium laser and compare them with our models. We conclude and give an
outlook in Sec. VI.

##  II Experiment

Figure 1: (a) Energy levels and transitions of atomic ${}^{174}\mathrm{Yb}$
with relevant parameters. Cavity parameters are also indicated. (b) Schematic
of magneto optical trap geometry, cavity interaction, and detection pathways.

Our laser is based on a magneto-optically trapped cloud of 174Yb atoms inside
a high-finesse cavity. Atoms are evaporated from an oven operating at
$\qty{500}{}$ and collimated. While travelling towards the trapping region,
they are decelerated from $\sim\qty{300}{\per}$ to $\sim\qty{1}{\per}$ by a
Zeeman slower. The MOT operates on the
$\Gamma_{\textbf{b}}=$29.1\text{\,}\mathrm{MHz}$$-wide
1S${}_{0}\,\leftrightarrow\,^{1}$P1 transition at
$\lambda_{\textbf{b}}=$399\text{\,}\mathrm{nm}$$ wavelength (blue), with a
typical detuning of $\Delta_{\text{MOT}}\simeq-\qty{30}{MHz}$ and with
$38\,\mathrm{G/cm}$ axial magnetic field gradient. [Note that throughout the
paper, frequencies are given in technical units. The corresponding angular
frequencies are used in the theoretical analyses.] It traps and cools the
atoms in a $\sim\qty{1}{mm}$-size spatial region around the center of our
optical cavity, see Fig. 1. The number of trapped atoms is controlled by a
servo-driven mechanical shutter that regulates the atomic-beam flux. We trap
up to $10^{7}$ atoms in the MOT at 5 to $\qty{10}{mK}$ temperature.

The $4.78\text{\,}\mathrm{cm}$-long cavity with
$87\text{\,}\mathrm{\SIUnitSymbolMicro m}$ waist radius is resonant with the
$\Gamma_{\textbf{g}}=\qty{182.4}{\kilo}$-wide
1S${}_{0}\,\leftrightarrow\,^{3}$P1 intercombination transition at
$\lambda_{\textbf{g}}=$556\text{\,}\mathrm{nm}$$ wavelength (green). It has an
energy decay rate $\kappa=$70\text{\,}\mathrm{kHz}$$ (finesse $\mathcal{F}=$
45000) and a vacuum Rabi frequency $g_{0}=$66\text{\,}\mathrm{kHz}$$
(cooperativity per atom $C_{1}\simeq 0.34$). Taking into account the ratio
between the cloud size and the cavity waist, we estimate that $N=10^{4}$ to
$10^{5}$ of atoms interact with the TEM00 cavity mode, placing the system well
within the collective strong-coupling regime $C=NC_{1}\gg 1$ [54]. The cavity
is rotated by $45^{\circ}$ with respect to the in-plane MOT beams, as
schematically depicted in Fig. 1(b). It is frequency-locked by means of the
TEM15,0 cavity mode detuned by $-420\text{\,}\mathrm{MHz}$ from TEM00.

The trapped atomic cloud is additionally laser-excited (pumped) on the green
1S${}_{0}\,\leftrightarrow\,^{3}$P1 line by six beams spatially overlapped
with the blue MOT beams, as depicted in Fig. 1(b). For appropriate
experimental parameters, the light that the atoms emit into the cavity builds
up to a large intra-cavity field. About $T_{\text{m}}\simeq\qty{1.5}{ppm}$ per
mirror of the intra-cavity power leaks out and is observed as laser output
[1].

We characterize the output with different techniques, see Fig. 1(b): most
importantly, a heterodyne measurement is performed by mixing the cavity TEM00
mode output with a 30MHz-detuned reference beam. The beat note is detected by
a fast photoreceiver (New Focus) and measured on a spectrum analyzer (Rohde
$\&$ Schwarz). The local oscillator of the heterodyne measurement and the
excitation light are derived from the same green laser using different
acousto-optical modulators (AOMs), hence the spectrum analyzer reveals any
frequency shift of the laser output relative to the pump. Additionally, the
structure of the cavity mode is detected by a CCD-camera (Imaging Source), and
the fluorescence on the green and blue transitions and the power emitted from
the cavity are measured by photomultiplier tubes (PMTs, Hamamatsu); these
measurements are presented in the supplementary material.

It should be noted that the six-beam configuration of the green pump light
creates an additional trapping and cooling effect (”green MOT”), co-acting or
counter-acting to the blue MOT, depending on its detuning relative to the
green resonance. When co-acting at negative detuning, the green MOT leads to
lower cloud temperatures, and hence higher atomic densities. On the other
hand, at positive detuning, the green light is heating the cloud, thus
reducing the trapping and cooling action of the blue MOT and depleting the
number of atoms interacting with the cavity mode. In a comprehensive model,
this mechanical effect would have to be taken into account; here we use a
simplified treatment that neglects this contribution.

##  III Measurement

The raw data that we record to characterize the lasing properties are obtained
from the heterodyne measurement of the TEM00 mode emission for a wide range of
cavity detunings $\Delta_{\text{cavity}}$, green pump and blue MOT detunings,
$\Delta_{\text{pump}}$ and $\Delta_{\text{MOT}}$, and pump and MOT powers,
$P_{\text{pump}}$ and $P_{\text{MOT}}$. Detunings are always referenced to the
respective atomic line centers.

For a given set of parameters, we record the spectrum of the beat signal from
the photodiode on one side of the cavity with the rf spectrum analyser.
Simultaneously, we record the cavity output power and the mode structure
observed with a CCD camera on the opposite side of the cavity, see Fig. 2(b).
The atomic fluorescence signals on the blue and green transitions are measured
with an oscilloscope. For a given value of $\Delta_{\text{cavity}}$, we step
$\Delta_{\text{pump}}$, while $\Delta_{\text{MOT}}$ as well as
$P_{\text{pump}}$ and $P_{\text{MOT}}$ are kept constant. The measured
heterodyne beat spectra are transformed into two types of heatmaps (color
maps): one representing the cavity output power, as obtained from the peak
area of the heterodyne signal, the other representing the output frequency, as
obtained from the peak position. Both maps are then rendered, with pump
detuning as X-axis and cavity detuning as Y-axis. These maps are generated for
each set of values of $\Delta_{\text{MOT}}$, $P_{\text{pump}}$ and
$P_{\text{MOT}}$. Details of the data processing will be provided as
supplementary information. The camera data are saved as video recordings for
additional analyses.

##  IV Threshold

As a first step we illustrate the typical lasing threshold behaviour of our
system. In Fig. 2, we display the continuous-wave laser output power as a
function of pump power $P_{\text{pump}}$, for various values of
$P_{\text{MOT}}$ as indicated in the inset. The detunings
$\Delta_{\text{pump}}=0$, $\Delta_{\text{MOT}}=-$30\text{\,}\mathrm{MHz}$$,
and $\Delta_{\text{cavity}}=-$30\text{\,}\mathrm{MHz}$$ are fixed. We observe
typical laser characteristics with a threshold and with output power that
increases approximately linearly above threshold. The threshold is in the
range of $P_{\text{pump}}\simeq\qty{2}{mW}$ for all displayed situations. A
systematic dependence of the threshold on the detunings is discussed in the
following in Fig. 3. There, the pump power is fixed to
$P_{\text{pump}}=\qty{5.7}{mW}$, for which all curves in Fig. 2 are well above
threshold.

Figure 2: Output power of the cold Yb laser as a function of the green pump
power for several values of blue MOT power.

We now explore the lasing behavior as function of the various detunings.
Figure 3(a) shows the measured laser output power, as color scale maps, vs.
the detunings $\Delta_{\text{pump}}$ and $\Delta_{\text{cavity}}$. The MOT
power and detuning values are indicated at the sub-plots. We observe distinct
regions of lasing centered around
$\Delta_{\text{pump}}\sim$0\text{\,}\mathrm{MHz}$$ and
$\Delta_{\text{cavity}}\sim\Delta_{\text{MOT}}$. In our earlier work [1], we
observed up to three regions of lasing emission, corresponding to
$\sigma^{\pm}$ and $\pi$ transitions, that were separated due to an offset
magnetic field in the MOT. Here we see only one lasing region, as a
consequence of better alignment of the atomic cloud to the center of the MOT,
where the magnetic field is zero.

Figure 3: Laser output power and threshold. (a) Measured laser power, color
coded from dark blue (zero) to dark red (maximum). (b) Simulated laser
threshold shown as contours; outside the contours, the model predicts no
lasing, inside it predicts laser output. In both (a) and (b), each sub-plot is
measured resp. calculated as a function of $\Delta_{\text{pump}}$ (along X)
and $\Delta_{\text{cavity}}$ (along Y); the other variables are
$\Delta_{\text{MOT}}$ (rows) and $P_{\text{MOT}}$ (columns). MOT power values
are given in $\mathrm{mW}$ for experimental data in (a); for the simulation in
(b), the Rabi frequencies in $\mathrm{MHz}$ are displayed, which are in good
agreement with experimentally calibrated ones.

In the maps of Fig. 3(a) we observe a systematic shift of the lasing regions
along the $\Delta_{\text{pump}}$ axis, when $P_{\text{MOT}}$ is varied, and
along the $\Delta_{\text{cavity}}$ axis, when $\Delta_{\text{MOT}}$ changes.
That dependence is a consequence of the MOT light field dressing the
${}^{1}\text{S}_{0}$ atomic ground state: the state splits into the lower
dressed state, from which the atoms are pumped to ${}^{3}\text{P}_{1}$, and
the upper dressed state, which is the intermediate state in the two-photon
lasing process, as depicted in Fig. 1(a). Hence the observed systematics
confirm and elaborate the gain mechanism proposed in [1]: an atom is first
pumped from the lower dressed ${}^{1}\text{S}_{0}$ state to
${}^{3}\text{P}_{1}$; it emits a green photon into the cavity mode and absorbs
a blue MOT photon, transiting through the upper dressed state; spontaneous
decay from ${}^{1}\text{P}_{1}$ back to the ground state closes the cycle.

We perform a simulation of our system to describe our experimental results in
terms of the lasing threshold. To that end, we calculate the steady state of a
three-level Bloch equation model that treats the pump as incoherent drive,
while the atom-cavity interaction is treated as coherent, with a collective
Rabi frequency $\Omega_{\textbf{cavity}}=g_{0}\sqrt{N}$. The simulation shown
in Fig. 3(b) was performed for an atom number $N={75000}$. The other
parameters are varied according to the measurement shown in Fig. 3(a); the
green Rabi frequency is fixed to
$\Omega_{\text{pump}}=$1.5\text{\,}\mathrm{MHz}$$ which corresponds, according
to our calibration, to $P_{\text{pump}}\simeq\qty{5.7}{mW}$. We extract the
threshold values from the simulation by equating the calculated gain rate with
the cavity loss rate, and we find the contours displayed in Fig. 3(b). The
calculated contours agree well with the observed boundaries of the lasing
regions shown in Fig. 3(a). Inside the contours the model predicts lasing, but
in view of the strong simplifications, we do not expect it to provide a
reliable value for the lasing power, hence we do not attempt a comparison to
the measurement in that regime.

Apart from the threshold contours, two important features of the experimental
results are reproduced by the model: (i) the shift of the lasing regions
towards larger pump frequencies, when the MOT power increases, which we
attribute to the MOT light-induced Stark shift of the ground state; and (ii)
the shift of the regions when the MOT frequency is varied, such that the
center is always near $\Delta_{\text{cavity}}\simeq\Delta_{\text{MOT}}$, which
is a signature of the two-photon gain mechanism.

At the same time, one sees a discrepancy between the ranges of pump detuning
for which lasing occurs in the experiment and in the simulation, visible as a
systematic difference of $\sim+$2\text{\,}\mathrm{MHz}$$ (see the horizontal
axes of Fig. 3(a) and (b)). This is partially attributed to an inaccuracy in
the laser lock calibration, partially its origin is not yet understood. An
obvious source of discrepancies between model and experiment is that the
number of atoms interacting with the cavity is assumed constant in the
simulations, whereas it varies in the measurement, even within one sub-plot of
Fig. 3(a). The reason is the already mentioned frequency-dependent impact of
the pump light on the trapping (the green MOT), which varies from cooling and
compressing the atomic cloud to heating and diluting it, as the pump light is
tuned across the (Stark-shifted) resonance. One indication for this effect is
the sudden transition from lasing to no-lasing on the positive pump frequency
side of the lasing regions.

##  V Frequency

When analysing the frequency of the laser light, obtained via the heterodyne
measurement, we observe a systematic redshift with respect to the empty-cavity
frequency, the latter set by the cavity lock. We associate this shift with the
change of the refractive index inside the cavity due to the presence of the
atoms. The amount of redshift depends on the pump and cavity detunings and
varies in a rather complex manner over the lasing regions (Fig. 4(a)). While
an intuitive understanding is hard to attain, we carried out a simulation
based on mean-field theory [12] and find that it reproduces a similar detuning
dependence. The model dynamically calculates the cavity field, and a Fourier
transform provides its frequency relative to the empty cavity.

A comparison of measured and simulated frequency shifts is displayed in the
maps of Fig. 4, where (a) shows the experimental data for
$\Delta_{\text{MOT}}=$-30\text{\,}\mathrm{MHz}$$,
$P_{\text{MOT}}=$20\text{\,}\mathrm{mW}$$, and
$P_{\text{pump}}=$5.7\text{\,}\mathrm{mW}$$, and (d) shows the model result
for the corresponding parameters and an atom number $N={75000}$. The Rabi
frequencies of the MOT and pump light are
$\Omega_{\text{MOT}}=$19\text{\,}\mathrm{MHz}$$ and
$\Omega_{\text{pump}}=$1.5\text{\,}\mathrm{MHz}$$, respectively, in good
agreement with an independent calibration of the experimental values. Despite
the simplifications of the model, it reproduces the main features of the
experimental results to a good degree, in particular the absolute value of the
frequency shift, up to $-1.5\text{\,}\mathrm{MHz}$ and its dependence on
$\Delta_{\text{cavity}}$, shown by the vertical cuts in Figs. 4(c) and (f).

We note that this model does not take into account atomic motion nor the
cooling and trapping action of the green and blue light, as possible
contributions to the frequency shift. Corresponding extension of the theory
will allow us to investigate the finer details and obtain a deeper
understanding of the phenomena.

Figure 4: Frequency shift map of the cavity output, measured (a) and simulated
(d). The vertical and horizontal cuts, taken along the red lines, depict
examples for the frequency shift dependencies on $\Delta_{\text{cavity}}$, (c)
and (f), and on $\Delta_{\text{pump}}$, (b) and (e). See text for experimental
and model parameters.

##  VI Conclusion

We have systematically studied the laser emission from cold 174Yb atoms in a
high-finesse cavity, on the semi-forbidden 3P1 to 1S0 transition. Using
heterodyne measurement of the laser output, we record the power and the
frequency of the emission as functions of the detunings and powers of the
involved light fields. We map the parameter regions where lasing occurs and
study their systematics, in particular their shifts with the frequency and
power of the trapping (MOT) light. The behaviour confirms and elaborates our
proposed mechanism for the laser gain [1], a two-photon process that couples
the 3P1 state to the 1P1 state. The intermediate level of the process emerges
from the 1S0 ground state by its dressing with the MOT light.

For a comparison to model calculations, we have focused on the laser threshold
and the laser output frequency, calculated from a Bloch equation model and a
mean-field dynamical model, respectively. Despite significant simplifications,
the models reproduce main observed features well, thus corroborating our
understanding of the system. Some particular discrepancies are also obvious,
which will be subject to further investigation.

##  VII Acknowledgements

We gratefully acknowledge funding by the Deutsche Forschungsgemeinschaft (DFG,
German Research Foundation) through the Collaborative Research Center TRR-306
”QuCoLiMa”, sub-project B04. S.B.J. acknowledges support from the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) through projects A4
and A5 in TRR-185 “OSCAR”. The authors acknowledge helpful discussions with J.
Cooper, F. Fama, G. Harmon, M. Holland, N. Kukharchyk, G. Morigi, J. Reilly,
and S. Schäffer.

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