Finite Nuclei Research Articles

Implications of PREX-2 on the electric dipole polarizability of neutron-rich nuclei

The recent announcement by the PREX collaboration of an unanticipated thick neutron skin in 208Pb (Rskin208) has challenged our understanding of neutron rich matter in the vicinity of nuclear saturation density. Whereas earlier constraints indicate that the symmetry energy is relatively soft, the PREX-2 result seems to suggest the opposite. We confront constraints on the symmetry energy obtained from measurements of the electric dipole polarizability against those informed by the PREX-2 measurement of Rskin208 and by the correlations that it entails. Covariant energy density functionals informed by the properties of finite nuclei are used to compute the electric dipole response of 48Ca, 68Ni, 132Sn, and 208Pb. The set of functionals used in this work are consistent with experimental data, yet are flexible enough in that they span a wide range of values of Rskin208. It is found that theoretical predictions of the electric dipole polarizability that are consistent with the PREX-2 measurement systematically overestimate the corresponding values extracted from the direct measurements of the distribution of electric dipole strength. The neutron skin thickness of 208Pb extracted from parity violating electron scattering and the electric dipole polarizability measured in photoabsorption experiments are two of the cleanest experimental tools used to constrain the symmetry energy around nuclear saturation density. However, the recent value of Rskin208 that suggests a fairly stiff symmetry energy stands in stark contrast to the conclusions derived from the electric dipole polarizability. At present, we offer no solution to this dilemma.

Crust-core interface and bulk neutron star properties

The nuclear symmetry energy plays an important role in the description of the properties of finite nuclei as well as neutron stars. Especially, for low values of baryon density, the accurate description of the crust-core interface strongly depends on the symmetry energy. Usually, the well known parabolic approximation is employed for the definition of the symmetry energy without avoiding some drawbacks. In the present paper, a class of nuclear models, suitable for the description of the inner and outer core of neutron stars, is applied in studying the effect of higher orders of the expansion of the energy on the location of the crust-core transition. The thermodynamical and dynamical methods are used for the determination of the transition density ${n}_{\mathrm{t}}$ and pressure ${P}_{\mathrm{t}}$. The corresponding energy density functional is applied for the study of some relevant properties of both nonrotating and slowly rotating neutron stars. We found that the larger the value of the slope parameter $L$, the slower the convergence of the expansion. In addition, a universal relation is presented between ${n}_{\mathrm{t}}$ and $L$, by employing the full expression and dynamical approach. The crustal moment of inertia is very sensitive to the location of the transition while the effects are moderated concerning the critical angular velocity of the $r$-mode instability and minimum mass configuration. The effect on the tidal deformability is less but not negligible. In any case, the use of the parabolic approximation leads to the overestimation of ${n}_{\mathrm{t}}$ and ${P}_{\mathrm{t}}$ and, consequently, on inaccurate predictions. Moreover, in some cases, even the matching process at the interface may affect considerably the predictions, introducing errors of the same order with the one due to the employed method.

From noise to information: The transfer function formalism for uncertainty quantification in reconstructing the nuclear density

The neutron distribution of neutron-rich nuclei provides critical information on the structure of finite nuclei and neutron stars. Parity violating experiments -- such as PREX and CREX -- provide a clean and largely model-independent determination of neutron densities. Such experiments, however, are challenging and expensive which is why sound statistical arguments are required to maximize the information gained. For this goal we introduce a new framework, "the transfer function formalism", aimed at uncertainty quantification, model selection, and experimental design in the context of neutron densities. The transfer functions (TFs) are built analytically by expressing the linear response of the objective function to small perturbations of the data. Using the TF formalism, we are able to analyze the expected overall uncertainty -- quantified in terms of bias and variance -- of the mean square radius and interior density of $^{48}$Ca and $^{208}$Pb. Using relativistic mean field models as a proxy for the weak-charge density -- and assuming that a total of five measurements could be performed on the weak form factor of $^{48}$Ca and $^{208}$Pb -- we identify the optimal models and experimental locations that minimize the combined radius and interior uncertainty for both nuclei. We also explore the use of the TF formalism to understand the influence of prior distributions for the model parameters, as well as the optimization of model hyperparameters not constrained by the data.

Chiral quark-meson coupling models for finite nuclei and their (e,e′p) reactions

The chiral quark-meson coupling (CQMC) models are applied to revisiting static properties of spherical nuclei in comparison with the results from the conventional quantum hadrodynamics (QHD)-like models as well as the available experimental data. They are also used to describe the nuclei off which the electrons scatter to understand their dynamic properties. After calibrating the model parameters at equilibrium nuclear matter density, binding energies, charge radii, single-particle spectra, and density distributions of the nuclei are analyzed and compared. The nonlinear scalar self-interaction in each model is also discussed in consideration of the reproduction of nuclear saturation properties.

Tensor and Coulomb-exchange terms in the relativistic mean-field model with δ -meson and isoscalar-isovector coupling

Tensor couplings and Coulomb-exchange terms in the relativistic mean-field (RMF) model impact the neutron skin thickness of nuclei [N. Liliani, A. M. Nugraha, J. P. Diningrum, and A. Sulaksono, Phys. Rev. C 93, 054322 (2016)], and a significantly thicker skin of $^{208}\mathrm{Pb}$ predicted by PREX-II leads to the existence of a phase transition in the interior of neutron stars (NSs) [F. J. Fattoyev, J. Piekarewicz, and C. J. Horowitz, Phys. Rev. Lett. 120, 172702 (2018)]. In this work, we revisit the effects of tensors and Coulomb-exchange terms within the RMF model with the $\ensuremath{\delta}$-meson and isovector-isoscalar coupling on finite nuclei. We also perform a systematic investigation of the implicit effect of both terms on the nuclear matter and NS properties. We confirmed in our previous work [N. Liliani, A. M. Nugraha, J. P. Diningrum, and A. Sulaksono, Phys. Rev. C 93, 054322 (2016)] that the roles of tensor couplings, Coulomb exchange, and isovector-isoscalar coupling on the bulk properties of finite nuclei are crucial. Conversely, the $\ensuremath{\delta}$-meson term's inclusion yields no significant effect on the bulk properties of finite nuclei. The tensor couplings and Coulomb-exchange terms implicitly influence the isoscalar and isovector parts of nuclear matter. By contrast, the $\ensuremath{\delta}$-meson and isovector-isoscalar coupling significantly impact the isovector parts of nuclear matter predictions. Except for the binding energy and the equation of state of pure neutron matter in low-density cases, the nuclear matter property predictions of all parameter sets used are relatively compatible with experimental data and the chiral effective field theory result. Moreover, the tensor couplings, Coulomb-exchange, $\ensuremath{\delta}$-meson, and isovector-isoscalar coupling terms play a significant role in predicting the canonical NS radius but not in the maximum mass of NS. For relatively large ${\ensuremath{\eta}}_{2\ensuremath{\rho}}$, the compatibility of the canonical NS radius predicted by the model with all of the observation results used in this work is better than that with the tensor couplings and Coulomb exchange excluded. Moreover, the presence of tensor couplings, Coulomb exchange, and various isovector-isoscalar nonlinear coupling terms influence the relation between NS tidal deformation and neutron skin thickness of $^{208}\mathrm{Pb}$. However, tensor couplings and Coulomb-exchange terms play a crucial role in making the NS tidal deformability prediction compatible with the constraints from observations and experimental data. In conclusion, tensor couplings and Coulomb-exchange RMF models have an essential role in finite nuclei predictions and implicitly influence the nuclear matter and NS property predictions.

Clustering in nuclear systems: from finite nuclei to neutron stars

Clustering in nuclear systems: from finite nuclei to neutron stars

NMR Indirect Spin–Spin Coupling Constants in a Modern Quasi-Relativistic Density Functional Framework

A quasi-relativistic implementation of NMR indirect spin-spin coupling constants is presented. The exact two-component (X2C) Hamiltonian and its diagonal local approximation to the unitary decoupling transformation (DLU) are utilized together with the (modified) screened nuclear spin-orbit approach. In a restricted kinetic balance, the finite nucleus model is available for both the scalar and vector potentials. The implementation supports density functionals up to the fourth rung of Jacob's ladder, i.e., (range-separated) hybrid and local hybrid functionals based on a seminumerical ansatz. We assess the quality of our quasi-relativistic X2C approach by comparison with "fully" relativistic four-component results for small main-group molecules and alkynyl compounds. The mean absolute error introduced by the DLU scheme is less than 0.05 × 1019 T J-2 of the reduced coupling constant for the small main-group molecules and 0.5 Hz for the alkynyl compounds. Thus, the error is significantly smaller than finite nucleus size effects for heavy elements. The basis set convergence and the impact of different density functional approximations are further studied. We propose a simple scheme to develop segmented-contracted relativistic all-electron basis sets for NMR spin-spin couplings. Our implementation allows us to perform calculations of extended molecules with reasonable computational effort, which is illustrated for the 1J(119Sn, 31P) coupling constant of a low-valent tin phosphinidenide complex. The corresponding results are in good agreement with the experimental findings.

Theoretical uncertainties on the extraction of in-medium NN cross sections by different Pauli blocking algorithms * *This work was partly inspired by the transport code comparison project, and it was supported by the National Natural Science Foundation of China (11875323, 11705163, 11790320, 11790323, 11961141003), the National Key R&D Program of China (2018YFA0404404), the Continuous Basic Scientific Research Project (WDJC-2019-13, BJ20002501)

Three typical Pauli blocking algorithms in quantum molecular dynamics type models are investigated in the nuclear matter, the nucleus, and heavy ion collisions. In nuclear matter, the blocking ratios obtained with the three algorithms are underestimated by 13%-25% compared to the corresponding analytical values. For a finite nucleus, spurious collisions occur around the surface of the nucleus owing to the defects of the Pauli blocking algorithms. In the simulations of heavy ion collisions, the uncertainty of stopping power arising from the different Pauli blocking algorithms is less than 5%. Furthermore, the in-medium effects of nucleon-nucleon (NN) cross sections on the nuclear stopping power are discussed. Our results show that the transport model calculations with free NN cross sections result in the stopping power decreasing with beam energy when the beam energy is less than 300 MeV/u. To increase or decrease the values of the stopping power, the transport model calculations need enhanced or suppressed model dependent in-medium NN cross sections that are expected to be smaller than the true in-medium NN cross sections.

Finite-size instabilities in finite-range forces

It has been recently shown that some Gogny finite-range interactions suffer from finite-size instabilities in coordinate-space calculations [Eur. Phys. J. A 55, 150 (2019)]. We confirm this finding by using the Hartree-Fock (HF) method in the quasilocal approximation to finite-range forces. The use of the quasilocal approximation substantially simplifies the calculations as compared with those including the exact exchange contribution to the energy and HF fields. The quantity most affected by the finite-size instabilities in the coordinate-space calculations is the spatial density at the origin that wildly oscillates as the HF iterative process proceeds. In addition to the recent ${\mathrm{D1M}}^{*}$ parametrization of the Gogny force, we find that the D1M parametrization also shows this deficiency in several nuclei. We find that the harmonic-oscillator basis with its ultraviolet cutoff provides converged results in a wide and realistic range of basis sizes. This result serves as a justification of the numerous calculations with D1M and ${\mathrm{D1M}}^{*}$ in finite nuclei that show no trace of instability.

Neutron drop trapped in axially deformed external fields

The neutron drop is firstly investigated in an axially symmetric harmonic oscillator (ASHO) field, whose potential strengths of different directions can be controlled artificially. The shape of the neutron drop will change from spherical to oblate or prolate according to the anisotropy of the external field. With the potential strength increasing in the axial direction, the neutron prefers to occupy the orbital perpendicular to the symmetry axis. On the contrary, the neutron likes to stay in the orbital parallel to the symmetry axis when the potential strength increases in the radial direction. Meanwhile, when the potential strength of one direction disappears, the neutron drop cannot bind together. These investigations are not only helpful to simulate the properties of neutrons in finite nuclei but also provide the theoretical predictions to the future artificial operations on the nuclei like the ultracold atom system, for a deeper realization of quantum many-body systems.

Discerning nuclear pairing properties from magnetic dipole excitation

Pairing correlation of Cooper pair is a fundamental property of multi-fermion interacting systems. For nucleons, two modes of the Cooper-pair coupling may exist, namely of $$S_{12}=0$$ with $$L_{12}=0$$ (spin-singlet s-wave) and $$S_{12}=1$$ with $$L_{12}=1$$ (spin-triplet p-wave). In nuclear physics, it has been an open question whether the spin-singlet or spin-triplet coupling is dominant, as well as how to measure their role. We investigate a relation between the magnetic-dipole (M1) excitation of nuclei and the pairing modes within the framework of relativistic nuclear energy-density functional (RNEDF). The pairing correlations are taken into account by the relativistic Hartree-Bogoliubov (RHB) model in the ground state, and the relativistic quasi-particle random-phase approximation (RQRPA) is employed to describe M1 transitions. We have shown that M1 excitation properties display a sensitivity on the pairing model involved in the calculations. The systematic evaluation of M1 transitions together with the accurate experimental data enables us to discern the pairing properties in finite nuclei.

Evolution of magnetic dipole strength in Sn100−140 isotope chain and the quenching of nucleon g factors

The evolution of electromagnetic transitions along isotope chains is of particular importance for the nuclear structure and dynamics, as well as for the r-process nucleosynthesis. Recent measurement of inelastic proton scattering on even-even $^{112-124}$Sn isotopes provides a novel insight into the isotopic dependence of E1 and M1 strength distributions. We investigate M1 transitions in even-even $^{100-140}$Sn isotopes from a theoretical perspective, based on relativistic nuclear energy density functional. The M1 transition strength distribution is characterized by an interplay between single and double-peak structures, that can be understood from the evolution of single-particle states, their occupations governed by the pairing correlations, and two-quasiparticle transitions involved. It is shown that discrepancy between model calculations and experiment on B(M1) transition strength is considerably reduced than previously known, and the quenching of the g-factors for the free nucleons needed to reproduce the experimental data on M1 transition strength amounts $g_{eff}/g_{free}$=0.80-0.93. Since some of the B(M1) strength above the neutron threshold may be missing in the inelastic proton scattering measurement, further experimental studies are required to confirm if only small modifications of the bare g-factors are actually needed when applied in finite nuclei.

Role of the unitary correlation operator on high-momentum antisymmetrized molecular dynamics using the bare NN interaction for 3H and 4He

Abstract We extend high-momentum antisymmetrized molecular dynamics (HMAMD) by incorporating the short-range part of the unitary correlation operator method (UCOM) as the variational method of finite nuclei. In this HMAMD+UCOM calculation of light nuclei, HMAMD is mainly in charge of the tensor correlation with up to four-body correlation, while the short-range correlation is further improved by using UCOM. The binding energies of the $^3$H and $^4$He nuclei are calculated with HMAMD+UCOM using the AV8$'$ bare nucleon–nucleon ($NN$) interaction. The different roles of the short-range and tensor correlations from HMAMD and UCOM are analyzed in the numerical results. Compared with previous calculations based on different variational methods, this newly extended HMAMD+UCOM method can almost provide consistent results with ab initio results.

From Finite Nuclei to Neutron Stars: The Essential Role of High-Order Density Dependence in Effective ForcesSupported by the National Key R&D Program of China (Grant No. 2018YFA0404403), the National Natural Science Foundation of China (Grant Nos. 11975032, 11835001, 11790325, and 11961141003).

A unified description of finite nuclei and equation of state of neutron stars presents both a major challenge and also opportunities for understanding nuclear interactions. Inspired by the Lee–Huang–Yang formula of hard-sphere gases, we develop effective nuclear interactions with an additional high-order density dependent term. While the original Skyrme force SLy4 is widely used in studies of neutron stars, there are not satisfactory global descriptions of finite nuclei. The refitted SLy4’ force can improve descriptions of finite nuclei but slightly reduces the radius of neutron star of 1.4M ⊙ with M ⊙ being the solar mass. We find that the extended SLy4 force with a higher-order density dependence can properly describe properties of both finite nuclei and GW170817 binary neutron stars, including the mass-radius relation and the tidal deformability. This demonstrates the essential role of high-order density dependence at ultrahigh densities. Our work provides a unified and predictive model for neutron stars, as well as new insights for the future development of effective interactions.

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.

Impacts of nucleon-nucleon short-range correlations on neutron stars

Nucleon-nucleon short-range correlations (SRCs) are inevitable consequences of realistic nucleon-nucleon interactions. The corresponding SRC pairs, characterized by high relative momenta, have been verified experimentally to be important in finite nuclei. In this work, we study the impacts of nucleon-nucleon SRCs on the physical properties of neutron stars. Considering the small center-of-mass momenta of SRC pairs, new nucleon momentum distributions with high-momentum tails (HMTs) are proposed by extrapolating well-established experimental results of SRCs in finite nuclei to neutron star matter. HMTs with different shapes are adopted to estimate theoretical uncertainties. It is found that nucleon-nucleon SRCs generally stiffen the equation of state of neutron star matter and increase the maximum mass of neutron stars.

Constraining Isovector Nuclear Interactions with Giant Dipole Resonance and Neutron Skin in 208Pb from a Bayesian ApproachSupported by the National Natural Science Foundation of China (Grant No. 11922514).

The remaining uncertainties in relation to isovector nuclear interactions call for reliable experimental measurements of isovector probes in finite nuclei. Based on the Bayesian analysis, although neutron-skin thickness data or isovector giant dipole resonance data in 208Pb can constrain only one isovector interaction parameter, correlations among other parameters can also be built. Using combined data for both the neutron-skin thickness and the isovector giant dipole resonance helps to significantly constrain all isovector interaction parameters; as such, it serves as a useful methodology for future research.

Nuclear Equation of State in the Relativistic Point-Coupling Model Constrained by Excitations in Finite Nuclei

Nuclear equation of state is often described in the framework of energy density functional. However, the isovector channel in most functionals has been poorly constrained, mainly due to rather limited available experimental data to probe it. Only recently, the relativistic nuclear energy density functional with an effective point-coupling interaction was constrained by supplementing the ground-state properties of nuclei with the experimental data on dipole polarizability and isoscalar monopole resonance energy in 208Pb, resulting in DD-PCX parameterization. In this work, we pursue a complementary approach by introducing a family of 8 relativistic point-coupling functionals that reproduce the same nuclear ground-state properties, including binding energies and charge radii, but in addition have a constrained value of symmetry energy at saturation density in the range J = 29, 30, …, 36 MeV. In the next step, this family of functionals is employed in studies of excitation properties such as dipole polarizability and magnetic dipole transitions, and the respective experimental data are used to validate the optimal choice of functional as well as to assess reliable values of the symmetry energy and slope of the symmetry energy at saturation.

Manifestation of hidden symmetries in baryonic matter: From finite nuclei to neutron stars

When hadron-quark continuity is formulated in terms of a topology change at a density higher than twice the nuclear matter density [Formula: see text], the core of massive compact stars can be described in terms of quasiparticles of fractional baryon charges, behaving neither like pure baryons nor like deconfined quarks. Hidden symmetries, both local gauge and pseudo-conformal (or broken scale), emerge and give rise both to the long-standing “effective [Formula: see text]” in nuclear Gamow–Teller (GT) transitions at [Formula: see text] and to the pseudo-conformal sound velocity [Formula: see text] at [Formula: see text]. It is suggested that what has been referred to, since a long time, as “quenched [Formula: see text]” in light nuclei reflects what leads to the dilaton-limit [Formula: see text] at near the (putative) infrared fixed point of scale invariance. These properties are confronted with the recent observations in GT transitions and in astrophysical observations.

Application of the coherent density fluctuation model to study the nuclear matter properties of finite nuclei within the relativistic mean-field formalism

We obtained a density-dependent analytical expression of binding energy per nucleon for different neutron-proton asymmetry of the nuclear matter (NM) with a polynomial fitting, which manifests the results of effective field theory motivated relativistic mean-field (E-RMF) model. This expression has the edge over the Br$\ddot{u}$ckner energy density functional [Phys. Rev. {\bf 171}, 1188 (1968)] since it resolves the Coster-Band problem. The NM parameters like incompressibility, neutron pressure, symmetry energy, and its derivatives are calculated using the acquired expression of energy per nucleon. Further, the weight function calculated by E-RMF densities are folded with calculated NM parameters within coherent density fluctuation model to find the properties of closed/semi-closed-shell even-even $^{16}$O, $^{40}$Ca, $^{48}$Ca, $^{56}$Ni, $^{90}$Zr, $^{116}$Sn, and $^{208}$Pb nuclei. The values obtained for the neutron pressure $P^{A}$, symmetry energy $S^{A}$ and its derivative $L_{sym}^A$ known as slope parameter, lie within a narrow domain whereas there is a large variation in isoscalar incompressibility $K^{A}$ and surface incompressibility $K_{sym}^{A}$ while moving from light to heavy nuclei. The sizable variation in $K^{A}$ and $K_{sym}^{A}$ for light and heavy nuclei depicts their structural dependence due to the peculiar density distribution of each nucleus. A comparison of surface quantities calculated in the present work has also been made with ones obtained via Br$\ddot{u}$ckner energy density functional.