Symmetry Energy Coefficient Research Articles

Study of the Symmetry Energy and the Nuclear Equation of State for 13O -13B and 13N-13C Mirror Nuclei

In parallel with the shell model using the harmonic oscillator's single-particle wave functions, the Hartree-Fock approximation was also used to calculate the neutron skin thickness, the mirror charge radii, and the differences in proton radii for 13O-13B and 13N-13C mirror nuclei. The calculations were done for both mirror nuclei in the psdpn model space. Depending on the type of potential used, the calculated values of skin thickness are affected. The symmetry energy and the symmetry energy's slope at nuclear saturation density were also determined, and the ratio of the density to the saturation density of nuclear matter and the symmetry energy has a nearly linear correlation. The mirror energy displacement was calculated, and the findings corresponded well with the available experimental data for the binding energies of the studied mirror nuclei. The measured values of the symmetry energy coefficient for the pair of mirror nuclei agreed with the computed ones, and this coefficient's value rises exponentially as the difference in charge radius increases.

The validity of the [formula omitted] lattice calculations for the finite asymmetric [formula omitted] matter using the central potentials

The lattice version of the lowest order constrained variational (LOCV) approach at the three-body cluster approximation is applied to the finite asymmetric nucleonic matter (ASM) in the periodic boxes (PBLOCV). The PBLOCVASM three- to two-body cluster energy ratios (R), using the central Bethe homework and (AV′8)c potentials, are obtained for small cubic PBs. The smallness of the ratios R could be considered evidence for the validity of the lowest order approximation. It is demonstrated that the truncation of the energy cluster expansion at the lowest order is confirmed for the Bethe homework interaction case. Also, the finite size effects and their impact on the symmetry energy coefficient are investigated. It is observed that using simple potentials and small finite nucleonic systems, one cannot reproduce the experimental prediction of the symmetry energy coefficient, even by including the three-body cluster energy contribution.

The isovector density-dependence of the M3Y nucleon-nucleon interactions and related soft and hard equations of state

Towards a precise equation of state (EOS) of nuclear matter (NM) based on a realistic interaction, we use non-relativistic Brueckner-Hartree-Fock (BHF) scheme to deduce the isovector (IV) density dependence (DD) of M3Y-Paris and Reid nucleon-nucleon interactions, via comparison with previous BHF calculations of nucleon optical potential. The single particle rearrangement correction is taken into account. With analytical parameterization of the isoscalar DD, the IV DD is parameterized (IVF1) for a broad range of EOSs, from soft EOS of saturation incompressibility K0 = 150 MeV to stiff one with K0 = 300 MeV. These Paris (Reid) EOSs indicate symmetry energy coefficient of 30.06±0.02 MeV (31.76±0.06 MeV), with corresponding density-slope sticks around L = 49.7 MeV (51.9 MeV). We found that the IVF1 DD indicates larger (smaller) saturation binding-energy (density) of asymmetric NM, and more high-density repulsion increasing with density and isospin asymmetry, as well as higher pressure, than those based on the IVF0 DD that scaled in terms of the isoscalar DD. Also, the IVF1 confirms the mostly indicated hard symmetry energy at supra-saturation density, on the contrary of the IVF0 that indicates soft one. Increasing the stiffness of the EOS increases the NM energy, though it decreases the symmetry energy. The more neutron-rich NM gets less bound, over less bound-density range. Only the fine ranges of Paris (K0 = 240 - 250 MeV) and Reid (K0 = 230 - 240 MeV) EOSs maintain simultaneous consistency with earlier experimental constraints and accurate calculations on NM energy, symmetry energy, and pressure.

Nuclear shell structure in a finite-temperature relativistic framework

The shell evolution of neutron-rich nuclei with temperature is studied in a beyond-mean-field framework rooted in the meson-nucleon Lagrangian. The temperature-dependent Dyson equation with the dynamical kernel taking into account the particle-vibration coupling (PVC) is solved for the fermionic propagators in the basis of the thermal relativistic mean-field Dirac spinors. The calculations are performed for $^{68-78}$Ni in a broad range of temperatures $0 \leq T \leq 4$ MeV. The special focus is put on the fragmentation pattern of the single-particle states, which is further investigated within toy models in strongly truncated model spaces. Such models allow for quantifying the sensitivity of the fragmentation to the phonon frequencies, the PVC strength and to the mean-field level density. The model studies provide insights into the temperature evolution of the PVC mechanism in real nuclear systems under the conditions which may occur in astrophysical environments.

Nuclear Symmetry Energy and Hyperonic Stars in the QMC Model

The nuclear symmetry energy, together with the other saturation properties of symmetric nuclear matter, plays an important role in low energy nuclear structure of terrestrial systems, as well as astrophysical objects. In particular, its density dependence, both in sub- and supra-saturation regions in high density matter in neutron stars, is of utmost significance and has been a subject of active research for decades, usually within a mean-field framework. We report results obtained using the latest version of Quark-Meson-Coupling Model (QMC-A) with just three variable parameters, the baryon-meson coupling constants in free space. It is shown that these parameters can be determined directly using nuclear matter (NM) properties at saturation; two parameters of symmetric nuclear matter (SNM), the baryon number density and the energy per particle, and the symmetry energy coefficient of asymmetric nuclear matter (ANM). The effects of uncertainties in the these parameters and propagation of these uncertainties through the calculation of properties of dense hyperonic matter and cold neutron stars are demonstrated. This approach leads to new limits on both the NM parameters and the QMC coupling constants. The results, which exploit the unique features of the QMC model, are discussed and future prospects are outlined.

Systematic study for the surface properties of neutron stars

[Background] In our earlier work {\bf [Phys. Rev. C 104, 055804 (2021)]}, we studied the surface properties of a neutron star, assuming it as a huge finite nucleus containing protons, neutrons, electrons, and muons. For the first time, we reported these results of a neutron star for a few representative masses. In the present paper, we give a detailed study of these quantities to draw definite conclusions. [Method]To carry forward our earlier idea, the energy density functional of the momentum space of neutron star matter is converted to the coordinate space in a local density approximation. This functional is again used to derive the neutron star surface properties within the coherent density fluctuation model using the weight function obtained from the density profile of the neutron star using the recently developed G3 and widely used NL3 and IU-FSU parameter sets in the context of relativistic mean-field formalism. [Results] The systematic surface properties of the neutron star, such as incompressibility, symmetry energy, slope parameter, and curvature coefficient, is calculated. The volume and surface components of the total symmetry energy are decomposed with the help of the $\kappa$ factor obtained from the volume to surface ratio of the symmetry energies in the liquid drop limit of Danielewicz. The magnitude of the computed surface quantities increases with the neutron star's mass. [Conclusion] The incompressibility $K^{\rm star}$, symmetry energy $S^{\rm star}$, slope parameter $L_{\rm sym}^{\rm star}$ and curvature coefficient $K_{\rm sym}^{\rm star}$ of the neutron stars with different mass are analyzed and found to be model dependent. NL3 is the stiffest equation of state endue us with the higher magnitude of surface quantities as compared to the G3 and IU-FSU forces.

Uncertainty analysis for the nuclear liquid drop model and implications for the symmetry energy coefficients

Background: Despite its age, the nuclear liquid drop (LD) model, plus the microscopic corrections, still plays an important role in nuclear mass studies. Especially, the LD model readily correlate the finite nucleus and the nuclear matter through the symmetry energy term.Purpose: To systematically analyze the model uncertainty of the LD mass formula and check the corresponding symmetry energy coefficients.Method: The Monte Carlo bootstrap approach, based on the nonparametric sampling, is applied to determine the statistical uncertainties of the parameter set in two popular LD formulas. The dependence between these parameters is also quantified via the correlation coefficients.Results: The least required proportion of the experimental mass data is fixed for the fitting process of the LD formula. After the statistical deviation is determined for each parameter in the LD formula, the model uncertainty is evaluated as illustrated for one heavy isotopic chain. The Pearson coefficients between each two parameters involved in the LD mass formula are tabulated and figured plus the detailed discussions on the surface and volume symmetry energy coefficients.Conclusion: The uncertainties of the fitted parameters and the LD model itself are smooth, leading to a relatively stable extrapolation. It is necessary to include the Wigner energy term in the LD model when yielding the reasonable symmetry energy coefficients.

Refining the nuclear mass model via theαdecay energy

Besides the direct comparison with the experimental mass values, the $\ensuremath{\alpha}$ decay energy $({Q}_{\ensuremath{\alpha}})$, as one type of the mass difference, can be considered as another factor to judge the accuracy and validity of modern nuclear mass models. In the present study, the Duflo-Zuker (DZ) mass model is employed to match the measured ${Q}_{\ensuremath{\alpha}}$ value besides the mass data. This procedure can not only provide us a new way to evaluate and predict the $\ensuremath{\alpha}$ decay energy, but also raise one more constraint on the theoretical mass scheme. It is found that all the experimental $\ensuremath{\alpha}$ decay energies, from the ground state (g.s.) to g.s. transition of even-even nuclei above $A=100$, can be well reproduced in a quite improved accuracy. Moreover, the symmetry term, involved in the macroscopic part of the DZ model, is refined further to regulate the behavior of the ambiguous symmetry energy coefficient in the nuclear equation of state. Through the relationship between the slope parameter of the symmetry energy and the neutron skin thickness, the neutron thickness $\mathrm{\ensuremath{\Delta}}{R}_{np}$ of $^{208}\mathrm{Pb}$ is determined in the vicinity of 0.201 fm, which is compatible to the very recent measurement.

Constraining the nonanalytic terms in the isospin-asymmetry expansion of the nuclear equation of state

We examine the properties of the isospin-asymmetry expansion of the nuclear equation of state from chiral two- and three-body forces. We focus on extracting the high-order symmetry energy coefficients that consist of both normal terms (occurring with even powers of the isospin asymmetry) as well as terms involving the logarithm of the isospin asymmetry that are formally nonanalytic around the expansion point of isospin-symmetric nuclear matter. These coefficients are extracted from numerically precise perturbation theory calculations of the equation of state coupled with a new set of finite difference formulas that achieve stability by explicitly removing the effects of higher-order terms in the expansion. We consider contributions to the symmetry energy coefficients from both two- and three-body interactions. It is found that the coefficients of the logarithmic terms are generically larger in magnitude than those of the normal terms from second-order perturbation theory diagrams, but overall the normal terms give larger contributions to the ground state energy. The high-order isospin-asymmetry terms are especially relevant at large densities where they affect the proton fraction in beta-equilibrium matter.

Constraining nuclear matter parameters from correlation systematics: a mean-field perspective

The nuclear matter parameters define the nuclear equation of state (EoS), they appear as coefficients of expansion around the saturation density of symmetric and asymmetric nuclear matter. We review their correlations with several properties of finite nuclei and of neutron stars within mean-field frameworks. The lower order nuclear matter parameters such as the binding energy per nucleon, incompressibility and the symmetry energy coefficients are found to be constrained in narrow limits through their strong ties with selective properties of finite nuclei. From the correlations of nuclear matter parameters with neutron star observables, we further review how precision knowledge of the radii and tidal deformability of neutron stars in the mass range $1 - 2 M_\odot$ may help cast them in narrower bounds. The higher order parameters such as the density slope and the curvature of the symmetry energy or the skewness of the symmetric nuclear matter EoS are, however, plagued with larger uncertainty. From inter-correlation of these higher order nuclear matter parameters with lower order ones, we explore how they can be brought to more harmonious bounds.

Trajectory in 2D plot of isoscalar and isovector densities of Ca48 and Pb208 , and symmetry energy

We propose a new two-dimensional (2D) plotting method of isoscalar-isovector (IS-IV) densities of $^{48}\mathrm{Ca}$ and $^{208}\mathrm{Pb}$ to extract not only empirical symmetry energy coefficients $J, L$, and ${K}_{\mathrm{sym}}$ but also the asymmetric isospin term ${K}_{\ensuremath{\tau}}$ in the nuclear incompressibility. We adopted several modern relativistic and nonrelativistic energy density functional (EDF) for the theoretical calculations of IS and IV densities to compare with experimental neutron and proton density distributions determined by recent proton elastic scatterings. We found very strong correlations between the trajectory of the 2D density at the density $\ensuremath{\rho}=0.1\phantom{\rule{4pt}{0ex}}{\mathrm{fm}}^{\ensuremath{-}3}$ and the symmetry energy coefficients $J, L$, and also ${K}_{\ensuremath{\tau}}$. For ${K}_{\mathrm{sym}}$, the Skyrme EDF SAMi-J family shows a strong correlation in the trajectory of 2D density, while the RMF Lagrangian DDME-J family shows a peculiar quadratic curve and does not provide the experimental value. The symmetry energy term ${K}_{\ensuremath{\tau}}$ of nuclear incompressibility is also discussed in term of the slope of 2D trajectory.

Isospin dependence of the nuclear binding energy * *Supported by the National Key R&DProgram of China (2018YFA0404403),and the National Natural Science Foundation of China(12075169, 11605122, 11875188)

In this paper, we study the symmetry energy and the Wigner energy in the binding energy formula for atomic nuclei. We simultaneously extract the symmetry energy and Wigner energy coefficients using the double difference of "experimental" symmetry-Wigner energies, based on the binding energy data of nuclei with . Our study of the triple difference formula and the "experimental" symmetry-Wigner energy suggests that the macroscopic isospin dependence of binding energies is explained well by the symmetry energy and the Wigner energy, and further consideration of the term in the binding energy formula does not substantially improve the calculation result.

Study of the dependence of alpha decay half-life on the surface symmetry energy

In this study, the effect of the surface symmetry energy on the neutron skin thickness and division of it into the bulk and surface parts are investigated by determination of the symmetry energy coefficient [Formula: see text] of finite nuclei. We demonstrate the importance of the isospin asymmetry distribution in the symmetry energy coefficient of finite nuclei at the surface region. We attempt to find out how different surface symmetry energies may affect alpha decay half-life. The Skyrme interactions are used to describe the neutron and proton density distributions and to calculate the symmetry energy coefficient [Formula: see text] of four nuclei and the surface symmetry energy. The chosen Skyrme interactions can produce the binding energy and root-mean-square charge radii of both mother and daughter nuclei. We single out the spherical isotones of [Formula: see text] named [Formula: see text]Pb, [Formula: see text]Po, [Formula: see text]Rn and [Formula: see text]Ra for daughter nuclei and explore the dependence of the bulk and surface contributions on the surface symmetry energy. The half-life of mother nuclei, i.e., [Formula: see text]Po, [Formula: see text]Rn, [Formula: see text]Ra and [Formula: see text]Th, is employed to investigate the extent to which it is affected by different surface symmetry energies. The calculated half-lives show a downward tendency for different surface symmetry energies which can be caused by various neutron skin thicknesses.

Isospin dependent hybrid model for studying isoscaling in heavy ion collisions around the Fermi energy domain

Investigation of observables from nuclear multifragmentation reactions depending on isospin led to the development of a hybrid model. The mass and charge distribution as well as isotopic distribution was studied using this model for 112Sn+112Sn reaction as well as 124Sn+124Sn reactions at different energies. The agreement of the results obtained from the model with those from experimental data confirms the accuracy of the model. Isoscaling coefficients were extracted from these observables which can throw light on the symmetry energy coefficient. Another important facet of this model is that temperature of the studied reaction can be directly extracted using this model.

Nuclear Symmetry Energy Effects on the Bulk Properties of Neutron-rich Finite Nuclei

We systematically study the effect of the nuclear symmetry energy in the basic properties of finite, neutron-rich, heavy nuclei where symmetry energy plays a dominant role. We employ a variational method, in the framework of the Thomas-Fermi approximation, to study the effect of the symmetry energy on the neutron skin thickness and symmetry energy coefficients of various nuclei. The isospin asymmetry function a(r) is directly related to the symmetry energy as a consequence of the variational principle. In addition to this, the Coulomb interaction is included in a self-consistent way. The energy density of the asymmetric nuclear matter that is used, has its origins in a momentum-dependent interaction.

Density dependence of the symmetry energy constrained by β−-, 2β−-, and 4β−-decay energies

In this paper, the β−-, 2β−-, and 4β−-decay energies Q(β−), Q(2β−), and Q(4β−) given in the newly updated atomic mass evaluation (AME2016) are used to systematically constrain not only the quadratic symmetry energy coefficient and the surface-volume symmetry energy coefficient ratio , but also the quartic symmetry energy coefficient . Based on the modified Bethe–Weizsäcker mass formula, the decay energies Q(β−), Q(2β−), and Q(4β−) are all derived to be closely related to the quantities , κ, and , and then their values can be obtained by fitting the decay energies given in the AME2016. Besides, with the extracted values of and κ, by applying the relationship between the quadratic density-dependent symmetry energy S(ρ) and the mass-dependent symmetry energy coefficient of finite nuclei asym(A), the density slopes L(ρ0) for two different forms of S(ρ) at the saturation density ρ0 are constrained and the values are calculated to be L(ρ0) = 42.2 ± 8.0 MeV and L(ρ0) = 42.8 ± 7.6 MeV, respectively.

Density Dependence of Nuclear Matter Symmetry Energy: VMC Calculations

A Variational Monte-Carlo (VMC) method is employed to investigate the density dependence of the symmetry energy of isospin asymmetric nuclear matter. The realistic Urbana $$V$$ 14 nucleon–nucleon interaction potential of Lagaris and Pandharipande was used in the VMC calculations with addition of a phenomenological density-dependent term to simulate many-body interactions. The symmetry energy is obtained for different densities and compared with the data found in the literature, and it was observed that the results obtained in this study reasonably agree with the results found in the literature, and the symmetry energy is found to increase almost linearly with the density. We obtained the symmetry energy coefficient at saturation density $$\rho=0.16$$ fm $${}^{-3}$$ to be about 26.89 MeV, which is in agreement with the empirical value 30 $$\pm$$ 4 MeV. Also, the incompressibility factor of the nuclear matter and the equation of the state of pure neutron matter were reported. The results obtained are consistent with those obtained by various authors with different potentials and techniques.

Isoscaling of mass A ≃ 40 reconstructed quasiprojectiles from collisions in the Fermi energy regime

Isoscaling studies of fragments with Z=1-8 from reconstructed quasiprojectiles of mass A ≃40 from the 40Ca + 112,124Sn and 40Ar + 112,124Sn reactions at beam energy of 45 MeV/nucleon were performed. After initial efforts to obtain isoscaling using pairs of systems differing in their neutron-to-proton ratio N/Z, “intra-system” isoscaling for each of these four systems was obtained using fragment sources restricted in two narrow N/Z zones, one neutron rich and one neutron poor. The observed isoscaling behaviour was excellent and the isoscaling parameter α was found to decrease as the excitation energy increases. Corrections due to undetected neutrons were also taken into account in the source N/Z determination by using the theoretical models DIT (Deep Inelastic Transfer) and SMM (Statistical Multifragmentation Model) along with a software replica of the experimental setup. The reduced isoscaling parameter α/∆ was found to decrease as the excitation energy of the quasiprojectile source increases, in good agreement with recent work on reconstructed mass A ≃ 80 quasiprojectiles. This decrease of α/∆ may point to a decrease of the symmetry energy coefficient with increasing excitation energy.

Nuclear mass parabola and its applications * * Supported by National Natural Science Foundation of China, (11961131010, 11422548 and U1867212), and the Foundation of Guangxi innovative team (2017GXNSFGA198001, 2016GXNSFFA380001), and innovation fund of undergraduate at Anyang Normal University (ASCX/2019-Z055)

We propose a method for extracting the properties of the isobaric mass parabola based on the total double -decay energies of isobaric nuclei. Two important parameters of the mass parabola, the location of the most -stable nuclei and the curvature parameter , are obtained for 251 A values, based on the total double -decay energies of nuclei compiled in the AME2016 database. The advantage of this approach is that the pairing energy term caused by the odd-even variation can be removed in the process, as well as the mass excess of the most stable nuclide for the mass number A, which are employed in the mass parabolic fitting method. The Coulomb energy coefficient MeV is determined by the mass difference relation for mirror nuclei, and the symmetry energy coefficient is also studied by the relation .

Temperature-dependent symmetry energy of neutron-rich thermally fissile nuclei

Background: The density-dependent symmetry energy coefficient plays a crucial role in understanding a variety of issues in nuclear physics as well as nuclear astrophysics. It is quite interesting and crucial to determine the symmetry energy coefficient and its related observables for neutron-rich thermally fissile nuclei at finite temperature.Purpose: We evaluate the symmetry energy coefficient, neutron pressure, and symmetry energy curvature of a finite nucleus from the corresponding quantities of infinite nuclear matter. Moreover, we correlate an effective symmetry energy coefficient and its related observables with the neutron skin thickness of neutron-rich thermally fissile nuclei at a finite temperature.Methods: The temperature-dependent relativistic mean field model (TRMF) is used to obtain the ground and excited state bulk properties of finite nuclei and the energy density, pressure, and the symmetry energy for infinite nuclear matter. The TRMF model with FSUGarnet, IOPB-I, and NL3 parameter sets is used for the present analysis. The effective nuclear matter properties are used to estimate the corresponding quantities of finite nuclei by using the local density approximation.Results: Nuclear bulk properties such as binding energy, quadrupole deformation, root-mean-square charge radius of the nuclei, and the equation of state and symmetry energy for infinite symmetric nuclear matter are estimated within the TRMF model. The nuclear matter observables at the local density of the nuclei serve as an input to obtain the effective symmetry energy coefficient, neutron pressure, and the symmetry energy curvature of $^{234,236,250}\mathrm{U}$ and $^{240}\mathrm{Pu}$ nuclei. The influence of temperature and density on these properties for neutron-rich thermally fissile nuclei is observed. A correlation is established between the neutron skin thickness and the neutron pressure of the nuclei.Conclusions: The studied properties of nuclei such as effective symmetry energy coefficient, neutron pressure and symmetry energy curvature can be used in the synthesis of neutron-rich thermally fissile nuclei. The method presented here (fully microscopic) can be used further to study the properties of exotic and superheavy nuclei from the corresponding quantities of nuclear matter and vice versa.