Nuclear Matter Incompressibility Research Articles

Isoscalar giant quadrupole resonance of even-even 112-124Sn isotopes using BCS-QRPA

Using self-consistent Bardeen - Cooper - Schriffer + Hartree - Fock and quasiparticle random phase approximation, the isoscalar giant quadrupole resonance in the isotopes of 112,114,116,118,120,122,124Sn has been studied in this work. Five sets of Skyrme-type interactions of different values of the nuclear matter incompressibility coefficient KNM and effective mass m*/m are used in the calculations. Additionally, the impact of different types of pairing forces (i.e., volume, surface, and mixed) is examined. Comparisons are made between the computed strength distributions, centroid energies Ecen, scaled energies Es, and constrained energies Econ of the isoscalar giant quadrupole resonance and the available experimental data. Analysis is done on the relationships between KNM and m*/m, and the estimated properties.

Isoscalar monopole response in the neutron-rich molybdenum isotopes using self-consistent QRPA

The isoscalar giant monopole resonance (ISGMR) of even molybdenum isotopes 92,94,96,98,100Mo has been studied within the Skyrme self-consistent Hartree - Fock - Bardeen, Cooper, and Schrieffer and quasi-particle random phase approximation. Ten sets of Skyrme-type interactions of different values of the nuclear matter incompressibility coefficient KNM are used in the calculations. The calculated strength distributions, centroid energies Ecen, scaled energies Es and constrained energies Econ of ISGMR are compared with available experimental data. Due to the appropriate value of the nuclear matter incompressibility KNM, several types of Skyrme interactions were successful in describing the ISGMR strength distribution in the 92,94,96,98,100Mo isotopes. As a result, high correlations between Ecen and KNM were found.

Excitation of the isoscalar giant monopole resonance and incompressibility of nuclear matter: resolution of a long-standing puzzle

Excitation of the isoscalar giant monopole resonance and incompressibility of nuclear matter: resolution of a long-standing puzzle

Density-dependent relativistic mean field approach and its application to single-Λ hypernuclei in oxygen hyperisotopes* *Supported by the Fundamental Research Funds for the Central Universities, Lanzhou University (lzujbky-2022-sp02, lzujbky-2023-stlt01), the National Natural Science Foundation of China (11875152, 12275111), and the Strategic Priority Research Program of Chinese Academy of Sciences (XDB34000000)

The in-medium feature of nuclear force, which includes both nucleon-nucleon () and hyperon-nucleon () interactions, impacts the description of single-Λ hypernuclei. With the alternated mass number or isospin of hypernuclei, such effects may be unveiled by analyzing the systematic evolution of the bulk and single-particle properties. From a density-dependent meson-nucleon/hyperon coupling perspective, a new effective interaction in the covariant density functional (CDF) theory, namely, DD-LZ1-, is obtained by fitting the experimental data of Λ separation energies for several single-Λ hypernuclei. It is then used to study the structure and transition properties of single-Λ hypernuclei in oxygen hyperisotopes, in comparison with those determined using several selected CDF Lagrangians. A discrepancy is explicitly observed in the isospin evolution of spin-orbit splitting with various effective interactions, which is attributed to the divergence of the meson-hyperon coupling strengths with increasing density. In particular, the density-dependent CDFs introduce an extra contribution to reduce the value but enhance the isospin dependence of the splitting, which originates from the rearrangement terms of Λ self-energies. In addition, the characteristics of hypernuclear radii are studied along the isotopic chain. Owing to the impurity effect of the Λ hyperon, a size shrinkage is observed in the matter radii of hypernuclei compared with the cores of normal nuclei, and its magnitude is further elucidated to correlate with the incompressibility of nuclear matter. Moreover, there is a sizable model-dependent trend in which the Λ hyperon radii evolve with neutron number, which is decided partly by the in-medium interactions and core polarization effects.

ISOSCALAR DIPOLE RESPONSE IN 92Mo AND 100Mo ISOTOPES

In this work, the Isoscalar (IS) Giant Dipole Resonance (GDR) and Pygmy Dipole Resonance (PDR) of 92Mo and 100Mo Isotopes were calculated in the farmwork of the self-consistent quasi-particle random phase approximation (QRPA) based on the results of Hartree-Fock-Bardeen, Cooper and Schrieffer (HF-BCS) using 10 Skyrme-type interactions: KDE0v1, eMSL08, SKX, SGOI, v080, SKP, SIV, SIII, SKIII, and SGI. The strength distributions of isoscalar dipole response were compared with the available experimental data. Also, we discussed the statistical relation between the centroid energy and the nuclear matter incompressibility KNM.

Isoscalar giant monopole strength in Ni58, Zr90, Sn120 , and Pb208

Background: Inelastic $\ensuremath{\alpha}$-particle scattering at energies of a few hundred MeV and very-forward scattering angles including 0\ifmmode^\circ\else\textdegree\fi{} has been established as a tool for the study of the isoscalar giant monopole (IS0) strength distributions in nuclei. This compressional mode of nuclear excitation can be used to derive the incompressibility of nuclear matter.Purpose: An independent investigation of the IS0 strength in nuclei across a wide mass range was performed using the ${0}^{\ensuremath{\circ}}$ facility at iThemba Laboratory for Accelerator Based Sciences (iThemba LABS), South Africa, to understand differences observed between IS0 strength distributions in previous experiments performed at the Texas A University (TAMU) Cyclotron Institute, USA and the Research Center for Nuclear Physics (RCNP), Japan.Methods: The isoscalar giant monopole resonance (ISGMR) was excited in $^{58}\mathrm{Ni}, ^{90}\mathrm{Zr}, ^{120}\mathrm{Sn}$, and $^{208}\mathrm{Pb}$ using $\ensuremath{\alpha}$-particle inelastic scattering with a 196 MeV $\ensuremath{\alpha}$ beam and scattering angles ${\ensuremath{\theta}}_{\text{Lab}}={0}^{\ensuremath{\circ}}$ and ${4}^{\ensuremath{\circ}}$. The K600 magnetic spectrometer at iThemba LABS was used to detect and momentum analyze the inelastically scattered $\ensuremath{\alpha}$ particles. The IS0 strength distributions in the nuclei studied were deduced with the difference-of-spectra (DoS) technique including a correction factor for the ${4}^{\ensuremath{\circ}}$ data based on the decomposition of $L0$ cross sections in previous experiments.Results: IS0 strength distributions for $^{58}\mathrm{Ni}, ^{90}\mathrm{Zr}, ^{120}\mathrm{Sn}$, and $^{208}\mathrm{Pb}$ are extracted in the excitation-energy region ${E}_{\mathrm{x}}=9--25$ MeV. Using correction factors extracted from the RCNP experiments, there is a fair agreement with their published IS0 results. Good agreement for IS0 strength in $^{58}\mathrm{Ni}$ is also obtained with correction factors deduced from the TAMU results, while marked differences are found for $^{90}\mathrm{Zr}$ and $^{208}\mathrm{Pb}$.Conclusions: Previous measurements show significant differences in the IS0 strength distributions of $^{90}\mathrm{Zr}$ and $^{208}\mathrm{Pb}$. This work demonstrates clear structural differences in the energy region of the main resonance peaks with possible impact on the determination of the nuclear matter incompressibility presently based on the IS0 centroid energies of these two nuclei. The results also suggest that, for an improved determination of the incompressibility, theoretical approaches should aim at a description of the full strength distributions rather than the centroid energy only.

Isoscalar Giant Monopole Resonance in Spherical Nuclei as a Nuclear Matter Incompressibility Indicator

The incompressibility of both nuclear matter and finite nuclei is estimated by the monopole compression modes in nuclei in the framework of a nonrelativistic Hartree–Fock–Bogoliyubov method and the coherent density fluctuation model. The monopole states originate from vibrations of the nuclear density. The calculations in the model for the incompressibility in finite nuclei are based on the Brueckner energy–density functional for nuclear matter. Results for the energies of the breathing vibrational states and finite nuclei incompressibilities are obtained for various nuclei and their values are compared with recent experimental data. The evolution of the isoscalar giant monopole resonance (ISGMR) along Ni, Sn, and Pb isotopic chains is discussed. This approach can be applied to analyses of neutron stars properties, such as incompressibility, symmetry energy, slope parameter, and other astrophysical quantities, as well as for modelling dynamical behaviors within stellar environments.

Extracting nuclear matter properties from the neutron star matter equation of state using deep neural networks

The extraction of the nuclear matter properties from neutron star (NS) observations is nowadays an important issue, in particular, the properties that characterize the symmetry energy which are essential to describe correctly asymmetric nuclear matter. We use deep neural networks (DNNs) to map the relation between cold $\beta$-equilibrium NS matter and the nuclear matter properties. Assuming a quadratic dependence on the isospin asymmetry for the energy per particle of homogeneous nuclear matter and using a Taylor expansion up to fourth order in the iso-scalar and iso-vector contributions, we generate a dataset of different realizations of $\beta$-equilibrium NS matter and the corresponding nuclear matter properties. The DNN model was successfully trained, attaining great accuracy in the test set. Finally, a real case scenario was used to test the DNN model, where a set of 33 nuclear models, obtained within a relativistic mean field approach or a Skyrme force description, were fed into the DNN model and the corresponding nuclear matter parameters recovered with considerable accuracy, in particular, the standard deviations $\sigma(L_{\text{sym}})= 12.85$ MeV and $\sigma(K_{\text{sat}})= 41.02$ MeV were obtained, respectively, for the slope of the symmetry energy and the nuclear matter incompressibility at saturation.

Finite-range simple effective interaction including tensor terms

The prediction of single particle level crossing phenomenon between $2p_{3/2}$ and $1f_{5/2}$ orbitals in $Ni$- and $Cu$-isotopic chains by the finite range simple effective interaction without requiring the tensor part is discussed. In this case the experimentally observed crossing could be studied as a function of nuclear matter incompressibility, $K(\rho_0)$. The estimated crossing for the neutron number $N$=46 could be reproduced by the equation of state corresponding to $K(\rho_0)$=240 MeV. However, the observed proton gaps between the $1h_{11/2}$ and $1g_{7/2}$ shells in $Sn$ and $Sb$ isotopic chain, and the neutron gaps between the $1i_{13/2}$ and $1h_{9/2}$ shells in $N$=82 isotones, as well as the shell closure properties at $N$=28 require explicit consideration of a tensor part as the central contribution is not enough to initiate the required level splittings.

Bayesian inference of finite-nuclei observables based on the KIDS model

Bayesian analyses on both isoscalar and isovector nuclear interaction parameters are carried out based on the Korea-IBS-Daegu-SKKU (KIDS) model under the constraints of nuclear structure data of $^{208}$Pb and $^{120}$Sn. Under the constraint of the neutron-skin thickness, it is found that incorporating the curvature parameter $K_{sym}$ of nuclear symmetry energy as an independent variable significantly broadens the posterior probability distribution function (PDF) of the slope parameter $L$, and affects the related correlations. Typically, the anticorrelation between $L$ and the symmetry energy at saturation density disappears, while a positive correlation between $L$ and $K_{sym}$ is observed. Under the constraint of the isoscalar giant monopole resonance (ISGMR), incorporating the skewness parameter as an independent variable also significantly broadens the posterior PDF of the nuclear matter incompressibility $K_0$. Even with the broad uncertainties of higher-order parameters of the equation of state (EOS), robust constraints of $L<90$ MeV and $K_0<270$ MeV are obtained. Our study quantifies the consistency between the constraints on $L$ from the neutron-skin data of PREXII and isovector giant dipole resonance (IVGDR) data, and the constraints on $K_0$ from the data of ISGMR in $^{208}$Pb and $^{120}$Sn.

The influence of the nuclear matter incompressibility on heavy-ion fusion reactions at near- and above-barrier energies

Based on simulating the saturation property of cold nuclear matter for complete fusion channel of two approaching nuclei, we systematically investigate the role of nuclear incompressibility in the nucleus-nucleus potential and fusion-evaporation cross section of 69 medium-heavy and heavy mass systems with 208≤Z1Z2≤1520. Our observations based on the uncoupled single-barrier potential model and the M3Y double-folding potential which is corrected with a repulsive core term reveal that the incompressibility of cold nuclear matter can be responsible for the discrepancy between the measured and calculated values of the fusion cross sections at energies around and above the Coulomb barrier. In other words, one can find that the high energy fusion cross sections can not be reproduced using the standard form of the M3Y-DF potential model without considering the repulsive core effects. We show that the strength of the repulsive contact term and also the incompressibility of the compound nuclei exhibit systematic behaviors with respect to the atomic and mass numbers of reacting nuclei. It is shown that the hardness level of nuclear matter reduces when one passes from lighter to heavier systems.

Bayesian uncertainty quantification for nuclear matter incompressibility

Within a Bayesian statistical framework using the standard Skyrme-Hartree-Fock model, the maximum {\it a posteriori} (MAP) values and uncertainties of nuclear matter incompressibility and isovector interaction parameters are inferred from the experimental data of giant resonances and neutron-skin thicknesses of typical heavy nuclei. With the uncertainties of the isovector interaction parameters constrained by the data of the isovector giant dipole resonance and the neutron-skin thickness, we have obtained $K_0 = 223_{-8}^{+7}$ MeV at 68% confidence level using the data of the isoscalar giant monopole resonance in $^{208}$Pb measured at the Research Center for Nuclear Physics (RCNP), Japan, and at the Texas A&M University (TAMU), USA. Although the corresponding $^{120}$Sn data gives a MAP value for $K_0$ about 5 MeV smaller than the $^{208}$Pb data, there are significant overlaps in their posterior probability distribution functions.

Heavy-ion fusion cross sections in the barrier and high-energy regions: Signature of energy dependence in the repulsive core potential

The effect of incompressibility of cold nuclear matter (CNM) on the high-energy fusion data of the 26Mg+248Cm, 36S+238U, 48Ca+208Pb, 50Ti+209Bi and 40,48Ca+238U colliding systems with 1152≤Z1Z2≤1840 has been analyzed using coupled-channels (CC) calculations. To apply the effect of saturation property of nuclear matter, we use the M3Y double-folding model which is corrected with a repulsive term. The repulsive parameters are adjusted to provide an optimum fit to the available experimental fusion cross sections. For each of the 26Mg+248Cm, 36S+238U, 48Ca+208Pb colliding systems, the best fit to the measured fusion cross sections at high energies is achieved using the CC calculations based on the M3Y+Repulsion potential with a constant value for the strength of the repulsive contact term. While focusing on the heavy-ion fusion reactions existing in the mass range Z1Z2>1800 reveals that it is necessary to consider an energy-dependent behavior for the repulsive parameters. When these conditions are imposed on entrance channel potential, it is possible to obtain an excellent description of the fusion data for 50Ti+209Bi, 40Ca+238U, and 48Ca+238U reactions. Our findings indicate that the extracted values of the strength Vrep of the repulsive contact term increase with increasing incident energy Ec.m.. In addition, we conclude that the repulsive part of the density distribution becomes more diffused with an increase in bombarding energy.

Equation of state of asymmetric nuclear matter and the tidal deformability of neutron star

Neutron star (NS) is a unique astronomical compact object where the four fundamental interactions have been revealed from the observation and studied in different ways. While the macroscopic properties of NS like mass and radius can be determined within the General Relativity using a realistic equation of state (EOS) of NS matter, such an EOS is usually generated by a nuclear structure model like, e.g., the nuclear mean-field approach to asymmetric nuclear matter. Given the radius of NS extended to above 10 km and its mass up to twice the solar mass, NS is expected to be tidally deformed when it is embedded in a strong tidal field. Such a tidal effect was confirmed unambiguously in the gravitation wave signals detected recently by the LIGO and Virgo laser interferometers from GW170817, the first ever direct observation of a binary NS merger. A nonrelativistic mean-field study is carried out in the present work within the Hartree-Fock formalism to construct the EOS of NS matter, which is then used to determine the tidal deformability, gravitational mass, and radius of NS. The mean-field results are compared with the constraints imposed for these quantities by the global analysis of the observed GW170817 data, and a strong impact by the incompressibility of nuclear matter on the hydrostatic configuration of NS is shown.

Bayesian inference of the incompressibility, skewness and kurtosis of nuclear matter from empirical pressures in relativistic heavy-ion collisions

Within the Bayesian statistical framework we infer the incompressibility K 0, skewness J 0 and kurtosis Z 0 parameters of symmetric nuclear matter (SNM) at its saturation density ρ 0 using the constraining bands on the pressure in cold SNM in the density range of 1.3ρ 0 to 4.5ρ 0 from transport model analyses of kaon production and nuclear collective flow in relativistic heavy-ion collisions. As the default option assuming the K 0, J 0 and Z 0 have Gaussian prior probability distribution functions (PDFs) with the means and variances of 235 ± 30, −200 ± 200 and −146 ± 1728 MeV, their posterior most probable values are narrowed down to MeV, − MeV and at 68% confidence level, respectively. The results are largely independent of the prior PDFs of J 0 and Z 0 used. However, if one adopts the strong belief that the incompressibility K 0 has a uniform prior PDF within its absolute boundary of 220–260 MeV as one can find easily in the literature, the posterior most probable values of K 0, J 0 and Z 0 shift to MeV, MeV and MeV, respectively. While the posterior PDFs of the SNM EOS parameters depend somewhat on the prior PDF of K 0 used, the results from using different prior PDFs are qualitatively consistent. The uncertainties of all three parameters are significantly reduced especially for the J 0 and Z 0 parameters compared to their current values.

Non-monotonic potential description of the cross-section, vector and tensor analyzing powers of the 6Li+12C elastic scattering at 30 and 50 MeV†

This work illustrates, for the first time, the analysis of tensor analyzing powers (T 20, T 21, T 22) along with the differential cross-section (CS) and the vector analyzing power iT 11 for the 6Li+12C elastic scattering at 30 and 50 MeVwithin the framework of an optical model (OM) using microscopic shallow non-monotonic (NM) potentials. The NM potential is generated from the energy density functional formalism (EDF) [Brueckner et al., Phys. Rev., 168 (1968) 1184] using a realistic two-nucleon interaction incorporating Pauli Exclusion principle. The shallow NM potential can describe the experimental angular distributions of CS and analyzing powers of the elastic scattering data. The OM analysis of the data at this energy does not indicate their sensitivity on the nuclear matter incompressibility K.

Non-monotonic nucleus-nucleus potential and incompressibility of infinite cold nuclear matter

A novel method for the determination of the yet not well-known quantity of nuclear incompressibility, K is presented. Non-monotonic (NM) nucleus-nucleus potentials from the energy-density functional (EDF) theory including the Pauli principle have been considered for K in the range 188-266 MeV. The experimental cross sections of 16O+16O elastic scattering over the 31-350 MeV incident energies have been analyzed in the optical model using the NM potentials. Sensitivity of K on the elastic scattering data is studied and its value for infinite cold nuclear matter deduced to be 222 ± 5 MeV

Quantifying uncertainties and correlations in the nuclear-matter equation of state

We perform statistically rigorous uncertainty quantification (UQ) for chiral effective field theory ($\chi$EFT) applied to infinite nuclear matter up to twice nuclear saturation density. The equation of state (EOS) is based on high-order many-body perturbation theory calculations with nucleon-nucleon and three-nucleon interactions up to fourth order in the $\chi$EFT expansion. From these calculations our newly developed Bayesian machine-learning approach extracts the size and smoothness properties of the correlated EFT truncation error. We then propose a novel extension that uses multitask machine learning to reveal correlations between the EOS at different proton fractions. The inferred in-medium $\chi$EFT breakdown scale in pure neutron matter and symmetric nuclear matter is consistent with that from free-space nucleon-nucleon scattering. These significant advances allow us to provide posterior distributions for the nuclear saturation point and propagate theoretical uncertainties to derived quantities: the pressure and incompressibility of symmetric nuclear matter, the nuclear symmetry energy, and its derivative. Our results, which are validated by statistical diagnostics, demonstrate that an understanding of truncation-error correlations between different densities and different observables is crucial for reliable UQ. The methods developed here are publicly available as annotated Jupyter notebooks.

Study of the surface energy coefficient used in nuclear proximity potential from density-dependent nucleon–nucleon interactions

The nuclear surface tension coefficient, γ, of the proximity potential is systematically investigated from the microscopic double-folding (DF) model accompanied by the realistic density-dependent nucleon–nucleon (NN) interactions. In the present work, we select six versions of density-dependent M3Y–Paris effective NN forces, including DDM3Y1 (with K=176MeV), CDM3Y1 (with K=188MeV), CDM3Y3 (with K=217MeV), CDM3Y5 (with K=241 MeV), CDM3Y6 (with K=252MeV) and BDM3Y1 (with K=270MeV). A total of 70 heavy-ion fusion reactions with condition 48≤Z1Z2≤2460 for the charge product of their reacting nuclei have been analyzed. By fitting the calculated values against the asymmetry parameter As, we present a new analytical formula for the parameter γ which is directly dependent on the nuclear matter (NM) incompressibility constant K. It is shown that the presently obtained formula in proximity potential model can reproduce the fusion data well. In this study, a discussion is also presented about the influence of the incompressibility constant K on the interaction potentials and also on the fusion cross sections resulting from the proximity approach for different heavy-ion reactions.

Cross-section, vector and tensor analyzing powers of the 6Li + 12C elastic scattering at 30 and 50 MeV: a non-monotonic potential description

This work reports, for the first time, the analysis of tensor analyzing powers (T20, T21, T22) along with the differential cross-section (CS) and the vector analyzing power iT11 for the 6Li + 12C elastic scattering at 30 and 50 MeV within the framework of an optical model (OM) using microscopic shallow non-monotonic (NM) potentials. The NM potential is generated from the energy density functional formalism (EDF) (Brueckner et al (1968) Phys. Rev. 168 1184) using a realistic two-nucleon interaction incorporating the Pauli exclusion principle. The shallow NM potential can describe the experimental angular distributions of CS and analyzing powers of the elastic scattering data. The OM analysis of the data at 30 and 50 MeV does not indicate their sensitivity on the nuclear matter incompressibility K.