New relativistic effective interaction for finite nuclei, infinite nuclear matter and neutron stars
We carry out the study for finite nuclei, infinite nuclear matter and neutron star properties with the newly developed relativistic force named as the Institute Of Physics Bhubaneswar-I(IOPB-I). Using this force, we calculate the binding energies, charge radii and neutron-skin thickness for some selected nuclei. From the ground state properties of superheavy nuclei (Z=120), it is noticed that considerable shell gaps appear at neutron numbers N=172, 184 and 198, manifesting the magicity at these numbers. The low-density behavior of the equation of state for pure neutron matter is compatible with other microscopic models. Along with the nuclear symmetry energy, its slope and curvature parameters at the saturation density are consistent with those extracted from various experimental data. We calculate the neutron star properties with the equation of state composed of nucleons and leptons in which are in good agreement with the X-ray observations by Steiner and Nättilä. Based on the recent observation GW170817 with a quasi-universal relation, L. Rezzolla et. al. have set a limit for the maximum mass that can be supported against gravity by a nonrotating neutron star is in the range . We find that the maximum mass of the neutron star for the IOPB-I parametrization is 2.15. The radius and tidal deformability of a canonical neutron star mass 1.4 are 13.2 km and 3.910 g cm s respectively.
pacs:26.60.+c, 26.60.Kp, 95.85.Sz
In the present scenario nuclear physics and nuclear astrophysics are well described within the self-consistent effective mean field models furnstahl97 (). These effective theories are not only successful to describe the properties of finite nuclei but also explain the nuclear matter at supra normal densities aru05 (). Recently, a large number of nuclear phenomenas have been predicted near the nuclear drip-lines within the relativistic and non-relativistic formalisms estal01 (); chab98 (); sand20 (). Consequently, several experiments are planed in various laboratories to probe deeper side of the unknown nuclear territories, i.e., the neutron and proton drip-lines. Among the effective theories, the relativistic mean field (RMF) model is one of the most successful self-consistent formalism, which has currently drawn attention for theoretical studies of such systems.
Although, the construction of the energy density functional for the RMF model is different than those for the non-relativistic models, such as Skyrme doba84 (); doba96 () and Gogny interactions dech80 (), the obtained results for finite nuclei are in general very close to each other. The same accuracy in prediction is also valid for the properties of the neutron stars. At higher densities, the relativistic effects are accounted appropriately within the RMF model walecka74 (). In the RMF model the interactions among nucleons are described through the exchange of mesons. These mesons are collectively taken as effective fields and denoted by classical numbers, which are the quantum mechanical expectation values. In brief, the RMF formalism is the relativistic Hartree or Hartree Fock approximations to the one boson exchange (OBE) theory of nuclear interactions. In OBE theory, the nucleons interact with each other by exchange of isovector , , and mesons and isoscalars like , , and mesons. The , and mesons are pseudo-scalar in nature and do not obey the ground state parity symmetry. In mean-field level, they do not contribute to the ground state properties of even nuclei.
The first and simple successful relativistic Lagrangian is formed by taking only the , and mesons contribution into account without any non-linear term to the Lagrangian density. This model predicts an unreasonably large incompressibility of MeV for the infinite nuclear matter at saturation walecka74 (). In order to lower the value of to an acceptable range, the self-coupling terms in sigma meson are included by Boguta and Bodmer boguta77 (). Based on this Lagrangian density, a large number of parameter sets, such as NL1 pg89 (), NL2 pg89 (), NL-SH sharma93 (), NL3 lala97 () and NL3 nl3s () were calibrated. The addition of meson self-couplings improved the quality of finite nuclei properties and incompressibility remarkably. However, the equation of states at supra-normal densities were quite stiff. Thus, the addition of vector meson self-coupling is introduced into the Lagrangian density and different parameter sets are constructed bodmer91 (); gmuca92 (); sugahara94 (). These parameter sets are able to explain the finite nuclei and nuclear matter properties to a great extent, but the existence of the Coester-band as well as the 3-body effects need to be addressed. Subsequently, nuclear physicists also change their way of thinking and introduced different strategies to improve the result by designing the density-dependent coupling constants and effective field theory motivated relativistic mean field (E-RMF) model furnstahl97 (); vretenar00 ().
Further, motivated by the effective field theory, Furnstahl et. al. furnstahl97 () used all possible couplings up to fourth order of the expansion, exploiting the naive dimensional analysis (NDA) and naturalness, obtained the G1 and G2 parameter sets. In the Lagrangian density, they considered only the contributions of the isoscalar-isovector cross-coupling, which has a greater implication on neutron radius and equation of state (EoS) of asymmetric nuclear matter pika05 (). Later on it is realized that the contributions of mesons are also needed to explain certain properties of nuclear phenomena in extreme conditions kubis97 (); G3 (). Though the contributions of the mesons to the bulk properties are nominal in the normal nuclear matter, but effects are significant for highly asymmetric dense nuclear matter. The meson splits the effective masses of proton and neutron, which influences the production of and in the heavy ion collision (HIC) ferini05 (). Also, it increases the proton fraction in the stable matter and modifies the transport properties of neutron star and heavy ion reactions chiu64 (); bahc65 (); lati91 (). The source terms for both the and mesons contain isospin density, but their origins are different. The meson arises from the asymmetry in the number density and the evolution of the meson is from the mass asymmetry of the nucleons. The inclusion of mesons could influence the certain physical observables like neutron-skin thickness, isotopic shift, two neutron separation energy , symmetry energy , giant dipole resonance (GDR) and effective mass of the nucleons, which are correlated with the isovector channel of the interaction. The density dependence of symmetry energy is strongly correlated with the neutron-skin thickness in heavy nuclei, but till now experiments have not fixed the accurate value of neutron radius, which is under consideration for verification in the parity-violating electron-nucleus scattering experiments horowitz01b (); vretenar00 ().
Recently, the detection of gravitational waves from binary neutron star GW170817 is a major breakthrough in astrophysics which is detected for the first time by the advanced Laser Interferometer Gravitational- wave Observatory (aLIGO) and advanced VIRGO detectors BNS (). This detection has certainly posed to be a valuable guidance to study the matter under the most extreme conditions. Inspiraling and coalescing objects of a binary neutron star result in gravitational waves. Due to the merger, a compact remnant is remaining whose nature is decided by two factors i.e (i) the masses of the inspiraling objects and (ii) the equation of state of the neutron star matter. For final state, the formation of either a neutron star or a black hole depends on the masses and stability of the objects. The chirp mass is measured very precisely from data analysis of GW170817 and it is found to be 1.188 for the 90 credible intervals. It is suggested that total mass should be 2.74 for low-spin priors and 2.82 for high-spin priors BNS (). Moreover, the maximum mass of nonspinning neutron stars(NSs) as a function of radius are observed with the highly precise measurements of . From the observations of gravitational waves, we can extract information regarding the radii or tidal deformability of the nonspinning and spinning NSs flan (); tanja (); tanja1 (). Once we succeed in getting this information, it is easy to get the neutron star matter equation of state latt07 (); bharat ().
In present paper, we constructed a new parameter set IOPB-I using the simulated annealing method (SAM) Agrawal05 (); Agrawal06 (); kumar06 () and explore the generic prediction of properties of finite nuclei, nuclear matter and neutron stars within the E-RMF formalism. Our new parameter set yields the considerable shell gap appearing at neutron numbers N=172, 184 and 198 showing the magicity of these numbers. The behavior of the density-dependent symmetry energy of nuclear matter at low and high densities are examined in detail. The effect of the core EoS on the mass, radius and tidal deformability of an NS are evaluated using the static perturbation of a Tolman-Oppenheimer-Volkoff solution.
The paper is organized as follows: In Sec. II, we outline the effective field theory motivated relativistic mean field (E-RMF) Lagrangian. We outline briefly the equations of motion for finite nuclei and equation of states (EoS) for infinite nuclear matter. In Sec. III, we discuss the strategy of the parameter fitting using the simulated annealing method (SAM). After getting the new parameter set IOPB-I, the results on binding energy, two neutron separation energy, neutron-skin thickness for finite nuclei are discussed in Sec. IVA. In sub-Sec. IVB and IVC, the EoS for symmetric and asymmetric matters are presented. The mass-radius and tidal deformability of the neutron star obtained by the new parameter set is also discussed in this section. Finally, the summary and concluding remarks are given in Sec. V.
Conventions: We have taken the value of throughout the manuscript.
ii.1 Energy density functional and equations of motion
In this section, we outline briefly the E-RMF Lagrangian furnstahl97 (). The beauty of effective Lagrangian is that, one can ignore the basic difficulties of the formalism, like renormalization and divergence of the system. The model can be used directly by fitting the coupling constants and some masses of the mesons. The E-RMF Lagrangian has an infinite number of terms with all types of self and cross couplings. It is necessary to develop a truncation procedure for practical use. Generally, the meson fields constructed in the Lagrangian are smaller than the mass of the nucleon. Their ratio could be used as a truncation scheme as it is done in Refs. furnstahl97 (); muller96 (); serot97 (); estal01 () along with the NDA and naturalness properties. The basic nucleon-meson E-RMF Lagrangian (with meson, ) up to 4 order with exchange mesons like , , meson and photon is given as furnstahl97 (); singh14 ():
where , , , and are the fields, , , , , are the coupling constants and , , and are the masses for , , , mesons and photon, respectively.
Now, our aim is to solve the field equations for the baryons and mesons (nucleon, , , , ) using the variational principle. We obtained the mesons equation of motion using the equation The single particle energy for the nucleons is obtained by using the Lagrange multiplier , which is the energy eigenvalue of the Dirac equation constraining the normalization condition furn87 (). The Dirac equation for the wave function becomes
The mean-field equations for , , , and are given by
where the baryon, scalar, isovector, proton and tensor densities are
Here is the nucleon’s Fermi momentum and the summation is over all the occupied states. The nucleons and mesons are composite particles and their vacuum polarization effects have been neglected. Hence, the negative-energy states do not contribute to the densities and current pg89 (). In the fitting process, the coupling constants of the effective Lagrangian are determined from a set of experimental data which takes into account the large part of the vacuum polarization effects in the no-sea approximation. It is clear that the no-sea approximation is essential to determine the stationary solutions of the relativistic mean-field equations which describe the ground-state properties of the nucleus. The Dirac sea holds the negative energy eigenvectors of the Dirac Hamiltonian which is different for different nuclei. Thus, it depends on the specific solution of the set of nonlinear RMF equations. The Dirac spinors can be expanded in terms of vacuum solutions which form a complete set of plane wave functions in spinor space. This set will be complete when the states with negative energies are the part of the positive energy states and create the Dirac sea of the vacuum.
The effective mass of proton and neutron are written as
The vector potential is
The set of coupled differential equations are solved self-consistently to describe the ground state properties of finite nuclei. In the fitting procedure, we used the experimental data of binding energy (BE) and charge radius for a set of spherical nuclei (O, Ca, Ca, Ni, Zr, Sn and Pb). The total binding energy is obtained by
where is the sum of the single particle energies of the nucleons and , , , , are the contributions of the respective mesons and Coulomb fields. The pairing and the center of mass motion MeV energies are also taken into account elliott55 (); negele70 (); estal01 ().
The pairing correlation plays a distinct role in open-shell nuclei greiner72 (); ring80 (). The effect of pairing correlation is markedly seen with the increase in mass number A. Moreover, it helps in understanding the deformation of medium and heavy nuclei. It has a lean effect on both bulk and single particle properties of lighter mass nuclei because of the availability of limited pairs near the Fermi surface. We take the case of T=1 channel of pairing correlation i.e, pairing between proton- proton and neutron-neutron. The pairs of nucleons are invariant under time reversal symmetry when the pairing interaction has non-zero matrix elements:
where and (with and ) are the quantum states.
A nucleon of quantum states pairs with another nucleon having same value with quantum states , since it is the time reversal partner of the other. In both nuclear and atomic domains, the ideology of BCS pairing is the same. The even-odd mass staggering of isotopes was the first evidence of its kind for the pairing energy. Considering the mean-field formalism the violation of the particle number is seen only due to the pairing correlation. We find terms such as (density) in the RMF Lagrangian density but we put an embargo on terms of the form or since it violates the particle number conservation. Thus, we affirm that BCS calculations have been carried out by constant gap or constant force approach externally in the RMF model gambhir90 (); reinhard86 (); toki94 (). In our work, we consider seniority type interaction as a tool by taking a constant value of G for pairs of the active pair shell.
The above approach does not go well for nuclei away from the stability line because in the present case, with the increase in the number of neutrons or protons the corresponding Fermi level goes to zero and the number of available levels above it minimizes. To complement this situation we see that the particle-hole and pair excitations reach the continuum. In Ref. doba84 () we notice that if we make the BCS calculation using the quasiparticle state as in HFB calculation, then the BCS binding energies are coming out to be very close to the HFB, but rms radii (i.e the single-particle wave functions) greatly depend on the size of the box where the calculation is done. This is because of the unphysical neutron (proton) gas in the continuum where wavefunctions are not confined in a region. The above shortcomings of the BCS approach can be improved by means of the so-called quasibound states, i.e, states bound because of their own centrifugal barrier (centrifugal-plus-Coulomb barrier for protons) estal01 (); meyer98 (); liotta20 (). Our calculations are done by confining the available space to one harmonic oscillator shell each above and below the Fermi level to exclude the unrealistic pairing of highly excited states in the continuum estal01 ().
ii.2 Nuclear Matter Properties
1. Energy and pressure density
In a static, infinite, uniform and isotropic nuclear matter, all the gradients of the fields in Eqs.(4)-(8) vanish. By the definition of infinite nuclear matter, the electromagnetic interaction is also neglected. The expressions for energy density and pressure for such a system is obtained from the energy-momentum tensor walecka86 ():
The zeroth component of the energy-momentum tensor gives the energy density and the third component compute the pressure of the system singh14 ():
where = is the energy and is the momentum of the nucleon. In the context of density functional theory, it is possible to parametrize the exchange and correlation effects through local potentials (Kohn–Sham potentials), as long as those contributions are small enough Ko65 (). The Hartree values control the dynamics in the relativistic Dirac-Brückner-Hartree-Fock (DBHF) calculations. Therefore, the local meson fields in the RMF formalism can be interpreted as Kohn-Sham potentials and in this sense equations (3-8) include effects beyond the Hartree approach through the non-linear couplings furnstahl97 ().
2. Symmetry Energy
The binding energy per nucleon = can be written in the parabolic form of asymmetry parameter :
where is energy density of the symmetric nuclear matter (SNM) ( = 0) and is defined as the symmetry energy of the system:
The isospin asymmetry arises due to the difference in densities and masses of the neutron and proton. The density type isospin asymmetry is taken care by meson (isovector-vector meson) and mass asymmetry by meson (isovector - scalar meson). The general expression for symmetry energy is a combined expression of and mesons, which is defined as matsui81 (); kubis97 (); estal01 (); roca11 ():
The last function is from the discreteness of the Fermi momentum. This momentum is quite large in nuclear matter and can be treated as a continuum and continuous system. The function is defined as:
In the limit of continuum, the function . The whole symmetry energy () arises from and mesons is given as:
where is the Fermi energy and is the Fermi momentum. The mass of the -meson modified, because of the cross-coupling of fields and is given by
The cross-coupling of isoscalar-isovector mesons () modifies the density dependence of without affecting the saturation properties of the symmetric nuclear matter (SNM) singh13 (); horowitz01b (); horowitz01a (). In the numerical calculation, the coefficient of symmetry energy is obtained by the energy difference of symmetric and pure neutron matter at saturation. In our calculation, we have taken the isovector channel into account to make the new parameters, which incorporate the currently existing experimental observations and predictions are done keeping in mind some future aspects of the model. The symmetry energy can be expanded as a Taylor series around the saturation density as:
where is the symmetry energy at saturation and . The coefficients , , and are defined as:
Similarly, we obtain the asymmetric nuclear matter (ANM) incompressiblity as and is given by chen09 ()
where in SNM.
Here, is the slope and represents the curvature of at saturation density. A large number of investigations have been made to fix the values of , and singh13 (); tsang12 (); dutra12 (); xu10 (); newton11 (); steiner12 (); fattoyev12 (). The density dependence of symmetry energy is a key quantity to control the properties of both finite nuclei and infinite nuclear matter roca09 (). Currently, the available information on symmetry energy MeV and its slope MeV at saturation density are obtained by various astrophysical observations li13 (). Till date, the precise values of and the neutron radii for finite nuclei are not known experimentally, it is essential to discuss the behavior of the symmetry energy as a function of density in our new parameter set.
Iii Parameter Fitting
The simulated annealing method (SAM) is used to determine the parameters used in the Lagrangian density kirkpatrick83 (); press92 (). The SAM is useful in the global minimization technique, i.e., it gives accurate results when there exists a global minimum within several local minima. Usually, this procedure is used in a system when the number of parameters are more than the observables kirkpatrick84 (); ingber89 (); cohen94 (). In this simulation method, the system stabilizes, when the temperature (a variable which controls the energy of the system) goes down Agrawal05 (); Agrawal06 (); kumar06 (). Initially, the nuclear system is put at a high temperature (highly unstable) and then allowed to cool down slowly so that it is stabilized in a very smooth way and finally reaches the frozen temperature (stable or systematic system). The variation of should be very small near the stable state. The values of the considered systems are minimized (least-square fit), which is governed by the model parameters . The general expression of the can be given as:
Here, and are the number of experimental data points and the fitting parameters, respectively. The experimental and theoretical values of the observables are denoted by and , respectively. The ’s are the adopted errors dob14 (). The adopted errors are composed of three components, namely, the experimental, numerical and theoretical errors dob14 (). As the name suggests, the experimental errors are associated with the measurements, numerical and the theoretical errors are associated with the numerics and the shortcomings of the nuclear model employed, respectively. In principle, there exists some arbitrariness in choosing the values of which is partially responsible for the proliferation of the mean-field models. Only guidance available from the statistical analysis is that the chi-square per degree of freedom (Eq. 37) should be close to the unity. In the present calculation, we have used some selected fit data for binding energy and root mean square radius of charge distribution for some selected nuclei and the associated adopted errors on them klup09 ().
In our calculations, we have built a new parameter set IOPB-I and analyzed their effects for finite and infinite nuclear systems. Thus, we performed an overall fit with 8 parameters, where the nucleons as well as the masses of the two vector mesons in free space are fixed at their experimental values i.e., MeV, MeV and MeV. The effective nucleon mass can be used as a nuclear matter constraint at the saturation density along with other empirical values like incompressibility, binding energy per nucleon and asymmetric parameter . While fitting the parameter, the value of effective nucleon mass , nuclear matter incompressibility and symmetry energy coefficient are constrained within 0.50-0.90, 220-260 MeV and 28-36 MeV, respectively. The minimum is obtained by simulated annealing method Agrawal05 (); Agrawal06 (); kumar06 () to fix the final parameters. The newly developed IOPB-I set along with NL3 lala97 (), FSUGarnet chai15 () and G3 G3 () are given for comparison in Table 1. The calculated results of the binding energy and charge radius are compared with the known experimental data audi12 (); angeli13 (). It is to be noted that in the original E-RMF parametrization, only five spherical nuclei were taken into consideration while fitting the parameters with the binding energy, charge radius and single particle energy furnstahl97 (). However, here, eight spherical nuclei are used for the fitting as listed in Table 2.
Iv Results and Discussions
In this section we discuss our calculated results for finite nuclei, infinite nuclear matter and neutron stars. For finite nuclei, binding energy, rms radii for neutron and proton distributions, two neutron separation energy and neutron-skin thickness are analyzed. Similarly, for infinite nuclear matter system, the binding energy per particle for symmetric and asymmetric nuclear matter including pure neutron matter at both sub-saturation and supra-saturation densities are compared with other theoretical results and experimental data. The parameter set IOPB-I is also applied to study the structure of neutron star using equilibrium and charge neutrality conditions.
iv.1 Finite Nuclei
(i) Binding energies, charge radii and neutron-skin thickness
We have used eight spherical nuclei to fit the experimental ground-state binding energies and charge radii using SAM. The calculated results are listed in Table 2 and compared with other theoretical models as well as experimental data audi12 (); angeli13 (). It can be seen that the NL3 lala97 (), FSUGarnet chai15 () and G3 G3 () successfully reproduce the energies and charge radii as well. Although, the ”mean-field models are not expected to work well for the light nuclei ” but the results deviate only marginally for the ground state properties for light nuclei patra93 (). We noticed that both the binding energy and charge radius of O are well produced by IOPB-I. However, the charge radii of Ca slightly underestimate the data. We would like to emphasize that it is an open problem to mean-field models to predict the evolution of charge radii of Ca (see Fig. 3 in Ref. gar16 ()).
The excess of neutrons give rise to a neutron-skin thickness. The neutron-skin thickness is defined as
with and being the rms radii for the neutron and proton distributions, respectively. The , strongly correlated with the slope of the symmetry energy pg10 (); brown00 (); maza11 (), can probe isovector part of the nuclear interaction. However, there is a large uncertainty in the experimental measurement of neutron distribution radius of the finite nuclei. The current values of neutron radius and neutron-skin thickness of Pb are 5.78 and 0.33 fm, respectively abra12 (). This error bar is too large to provide significant constraints on the density-dependent of symmetry energy. It is expected that PREX-II result will give us the neutron radius of Pb within accuracy. The inclusion of some isovector dependent terms in the Lagrangian density is needed which would provide the freedom to refit the coupling constants within the experimental data without compromising the quality of fit. The addition of cross-coupling into the Lagrangian density controls the neutron-skin thickness of Pb as well as for other nuclei. In Fig. 1, we show neutron-skin thickness for Ca to U nuclei as a function of proton-neutron asymmetry . The calculated results of for NL3, FSUGarnet, G3 and IOPB-I parameter sets are compared with the corresponding experimental data thick (). Experiments have been done with antiprotons at CERN and the are extracted for 26 stable nuclei ranging from Ca to U as displayed in the figure along with the error bars. The trend of the data points show approximately linear dependence of neutron-skin thickness on the relative neutron excess of nucleus that can be fitted by thick (); xavier14 ():
The values of obtained with IOPB-I for some of the nuclei slightly deviate from the shaded region as can be seen from Fig. 1. This is because, IOPB-I has a smaller strength of cross-coupling as compared to FSUGarnet. Recently, F. Fattoyev et. al. constraint the upper limit of fm for Pb nucleus with the help of GW170817 observation data fatt17 (). The calculated values of neutron-skin thickness for the Pb nucleus are 0.283, 0.162, 0.180 and 0.221 fm for the NL3, FSUGarnet, G3 and IOPB-I parameter sets respectively. The proton elastic scattering experiment has recently measured neutron-skin thickness fm for Pb zeni10 (). Thus values of for IOPB-I are consistent with recent prediction of neutron-skin thickness.
(ii) Two-neutron separation energy
The large shell gap in single particle energy levels is an indication of the magic number. This is responsible for the extra stability for the magic nuclei. The extra stability for a particular nucleon number can be understood from the sudden fall in the two-neutron separation energy . The can be estimated by the difference in ground state binding energies of two isotopes i.e.,
In Fig. 2, we display results for the as a function of neutron numbers for Ca, Ni, Zr, Sn, Pb and Z=120 isotopic chains. The calculated results are compared with the finite range droplet model (FRDM) moller () and latest experimental data audi12 (). From the figure, it is clear that there is an evolution of magicity as one moves from the valley of stability to the drip-line. In all cases, the values decrease gradually with increase in neutron number. The experimental manifestation of large shell gaps at neutron number N = 20, 28(Ca), 28(Ni), 50(Zr), 82(Sn) and 126(Pb) are reasonably well reproduced by the four relativistic sets. Figure 2 shows that the experimental of Ca are in good agreement with the prediction of NL3 set. It is interesting to note that all sets predict the sub-shell closure at N=40 for Ni isotopes. Furthermore, the two-neutron separation energy for the isotopic chain of nuclei with Z=120 is also displayed in Fig. 2. For the isotopic chain of Z=120, no experimental information exit. The only comparison can be made with theoretical models such as the FRDM audi12 (). At N=172, 184 and 198 sharp falls in separation energy is seen for all forces, which have been predicted by various theoretical models in the superheavy mass region rutz (); gupta (); patra99 (); mehta (). It is to be noted that the isotopes with Z=120 are shown to be spherical in their ground state mehta (). In a detail calculation, Bhuyan and Patra using both RMF and Skyrme-Hartree-Fock formalisms, predicted that Z=120 could be the next magic number after Z=82 in the superheavy region bunu12 (). Thus, the deformation effects may not affect the results for Z=120. Therefore, a future mass measurement of 120 would confirm a key test for the theory, as well as direct information about the closed-shell behavior at N=172, 184 and 198.
iv.2 Infinite Nuclear Matter
The nuclear incompressibility determines the extent to which the nuclear matter can be compressed. This plays important role in the nuclear equation of state (EoS). Currently, the accepted value of MeV has been determined from isoscalar giant monopole resonance (ISGMR) for Zr and Pb nuclei colo14 (); piek14 (). For our parameter set IOPB-I, we get MeV. The density-dependent symmetry energy is determined from Eq.(32) using IOPB-I along with three adopted models. The calculated results of the symmetry energy coefficient (), the slope of symmetry energy () and other saturation properties are listed in Table 3. We find that in case of IOPB-I, MeV and MeV. These values are compatible with MeV and MeV obtained by various terrestrial experimental informations and astrophysical observations li13 ().
Another important constraint