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

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.

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

where

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 :

Here,

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.

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 ,

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 ,

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:

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,

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:

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

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

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.

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 :

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

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

and

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,

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

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:

and

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.

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.

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.

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 .

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.

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 .