Homogeneous Nuclear Matter Research Articles

Popcorn Transitions and Approach to Conformality in Homogeneous Holographic Nuclear Matter

We study cold and dense nuclear matter by using the gauge/gravity duality. To this end, we use the Witten–Sakai–Sugimoto model and the V-QCD models with an approach where the nuclear matter is taken to be spatially homogeneous. We focus on the “popcorn” transitions, which are phase transitions in the nuclear matter phases induced by changes in the layer structure of the configuration on the gravity side. We demonstrate that the equation of state for the homogeneous nuclear matter becomes approximately conformal at high densities, and compare our results to other approaches.

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.

Properties of the neutron star crust: Quantifying and correlating uncertainties with improved nuclear physics

A compressible liquid-drop model (CLDM) is used to correlate uncertainties associated with the properties of the neutron star (NS) crust with theoretical estimates of the uncertainties associated with the equation of state (EOS) of homogeneous neutron and nuclear matter. For the latter, we employ recent calculations based on Hamiltonians constructed using chiral effective-field theory ($\ensuremath{\chi}\mathrm{EFT}$). Fits to experimental nuclear masses are employed to constrain the CLDM further, and we find that they disfavor some of the $\ensuremath{\chi}\mathrm{EFT}$ Hamiltonians. The CLDM allows us to study the complex interplay between bulk, surface, curvature, and Coulomb contributions, and their impact on the NS crust. It also reveals how the curvature energy alters the correlation between the surface energy and the bulk symmetry energy. Our analysis quantifies how the uncertainties associated with the EOS of homogeneous matter implies significant uncertainties for the composition of the crust, its proton fraction, and the volume fraction occupied by nuclei. We find that the finite-size effects impact the crust composition but have a negligible effect on the net isospin asymmetry of matter. The isospin asymmetry is largely determined by the bulk properties and the isospin dependence of the surface energy. The most significant uncertainties associated with matter properties in the densest regions of the crust, the precise location of the crust-core transition, are found to be strongly correlated with uncertainties associated with the Hamiltonians. By adopting a unified model to describe the crust and the core of NSs, we tighten the correlation between their global properties such as their mass-radius relationship, moment of inertia, crust thickness, and tidal deformability with uncertainties associated with the nuclear Hamiltonians.

Neutron stars and phase diagram from a double hard-wall

Description of nuclear matter in the core of neutron stars eludes the main tools of investigation of QCD, such as perturbation theory and the lattice formulation of the theory. Recently, the application of the holographic paradigm (both via top-down and bottom-up models) to this task has led to many encouraging results, both qualitatively and quantitatively. We present our approach to the description of neutron star cores, relying on a simple model of the (double) hard-wall type: we discuss results concerning the nature of homogeneous nuclear matter at high density emerging from the model including a quarkyonic phase, the mass-radius relation for neutron stars, as well as the rather stiff equation of state we have found. We show how, despite the very simple model employed, for an appropriate calibration we are able to obtain neutron stars that only slightly fall short of the observational bounds on radius and tidal deformability.

Non-spherical nucleon clusters in the mantle of a neutron star: CLDM based on Skyrme-type forces

Neutron stars are superdense compact astrophysical objects. The central region of the neuron star (the core) consists of locally homogeneous nuclear matter, while in the outer region (the crust) nucleons are clustered. In the outer crust these nuclear clusters represent neutron-rich atomic nuclei and all nucleons are bound within them. Whereas in the inner crust some neutrons are unbound, but nuclear clusters still keeps generally spherical shape. Here we consider the region between the crust and the core of the star, so-called mantle, where non-spherical nuclear clusters may exist. We apply compressible liquid drop model to calculate the energy density for several shape types of nuclear clusters. It allows us to identify the most energetically favorable configuration as function of baryon number density. Employing four Skyrme-type forces (SLy4 and BSk24, BSk25, BSk26), which are widely used in the neutron star physics, we faced with strong model dependence of the ground state composition. In particular, in agreement with previous works within liquid drop model, mantle is absent for SLy4 (nuclear spheres directly transit into homogeneous nuclear matter; exotic nuclear shapes do not appear).

Thermal effects in hot and dilute homogeneous asymmetric nuclear matter

We present a comprehensive analysis of hot and dilute isospin-asymmetric nuclear matter employing the temperature-dependent effective-relativistic mean-field theory (E-RMF). The E-RMF is applied to study the effect of $\delta$ and $\omega-\rho$ meson cross-coupling on the thermal properties of asymmetric nuclear matter using two recently developed IOPB-I and G3 parameter sets. These sets are known to reproduce the nuclear matter properties in agreement with various experimental and observational constraints. We consider the nuclear matter to be homogeneous and study the equation of state (EoS) for densities, temperature and asymmetry which are relevant for astrophysical simulations such as supernovae explosion. The effect of temperature is investigated in reference to the density-dependent free symmetry energy and its higher-order derivatives using the well known parabolic approximation. The larger value of $\lambda_\omega$ cross-coupling in G3 in addition to the $\delta$ meson coupling in G3 smoothen the free symmetry energy. Thermal effects on various state variables are examined at fixed temperature and isospin asymmetry by separating their T=0 and the finite-T expressions. The thermal effects are mainly governed by effective mass with larger effective mass estimating larger thermal contribution. The effect of temperature on isothermal and isentropic incompressibility is discussed which is in harmony with various available microscopic calculations. The liquid-gas phase transition properties are examined in asymmetric matter with two conserved charges in the context of different slope parameter and comparable symmetry energy in IOPB-I and G3 set. The spinodal instability, binodal curve and critical properties are found to be influenced by the slope parameter $L_{sym}$.

Time-evolution of magnetic field in hot nuclear matter with fluctuating topological charge

The time-evolution of the magnetic field in hot homogeneous nuclear matter has two qualitatively different stages separated by the sphaleron transition time ${\ensuremath{\tau}}_{c}$. At early times, when the chiral conductivity ${\ensuremath{\sigma}}_{\ensuremath{\chi}}$ is a slow function of time, the soft chiral modes $k<{\ensuremath{\sigma}}_{\ensuremath{\chi}}$ of the magnetic field grow exponentially with time, which is known as the chiral instability. At later times ${\ensuremath{\sigma}}_{\ensuremath{\chi}}$ fluctuates due to the sphaleron transitions and can be regarded as a random process. It is argued that the average magnetic field is exponentially damped at later times. The time-evolution of the average field energy is more complicated and depends on the electrical conductivity of the chiral matter but does not depend on chirality. It exhibits instability only if the matter is a poor electrical conductor, such as the quark-gluon plasma near the critical temperature. The precise conditions for the instability and the growth rate of the unstable modes are derived.

Thermodynamically Consistent Equation of State for an Accreted Neutron Star Crust.

We study the equation of state (EOS) of an accreting neutron star crust. Usually, such an EOS is obtained by assuming (implicitly) that the free (unbound) neutrons and nuclei in the inner crust move together. We argue that this assumption violates the condition μ_{n}^{∞}=const, required for hydrostatic (and diffusion) equilibrium of unbound neutrons (μ_{n}^{∞} is the redshifted neutron chemical potential). We construct a new EOS respecting this condition, working in the compressible liquid-drop approximation. We demonstrate that it is close to the catalyzed EOS in most of the inner crust, being very different from EOSs of accreted crust discussed in the literature. In particular, the pressure at the outer-inner crust interface does not coincide with the neutron drip pressure, usually calculated in the literature, and is determined by hydrostatic (and diffusion) equilibrium conditions within the star. We also find an instability at the bottom of the fully accreted crust that transforms nuclei into homogeneous nuclear matter. It guarantees that the structure of the fully accreted crust remains self-similar during accretion.

Nuclear statistical equilibrium equation of state with a parametrized Dirac–Brückner Hartree–Fock calculation

Abstract We calculate new equations of state (EOSs) for astrophysical simulations in the framework of the extended nuclear statistical equilibrium, in which we minimize the free energy density for the full ensemble of nuclei in a hot and dense stellar environment. To evaluate bulk and surface energies of heavy nuclei and free energies of uniformly distributed nucleons, we use fitting formulae for the interaction energies and single-nucleon potentials at zero temperature of a Dirac–Brückner Hartree–Fock (DBHF) theory, one of the modern approaches to describe homogeneous nuclear matter. We find that the DBHF EOS exhibits larger mass fractions for medium-mass nuclei and smaller mass fractions for the other nuclei than the EOS obtained using the variational method (VM), another modern model for homogeneous nuclear matter. This effect is due to the more deeply bound energy for symmetric nuclear matter and the larger symmetry energy encoded in the DBHF EOS. At supra-nuclear densities, the DBHF EOS exhibits characteristics of a larger free energy, a higher pressure, and a larger neutron chemical potential of neutron-rich matter, which lead to a larger radius of cold neutron stars than that obtained by the VM EOS.

Cold dilute nuclear matter with α -particle condensation in a generalized nonlinear relativistic mean-field model

We explore the thermodynamic properties of homogeneous cold (zero-temperature) nuclear matter including nucleons and $\alpha$-particle condensation at low densities by using a generalized nonlinear relativistic mean-field (gNL-RMF) model. In the gNL-RMF model, the $\alpha$-particle is included as explicit degree of freedom and treated as point-like particle with its interaction described by meson exchanges and the in-medium effects on the $\alpha$ binding energy is described by density- and temperature-dependent energy shift with the parameters obtained by fitting the experimental Mott density. We find that below the dropping density $n_{\rm{drop}}$ ($\sim 3\times 10^{-3}$ fm$^{-3}$), the zero-temperature symmetric nuclear matter is in the state of pure Bose-Einstein condensate (BEC) of $\alpha$ particles while the neutron-rich nuclear matter is composed of $\alpha$-BEC and neutrons. Above the $n_{\rm{drop}}$, the fraction of $\alpha$-BEC decreases with density and vanishes at the transition density $n_t$ ($\sim 8\times 10^{-3}$ fm$^{-3}$). Above the $n_t$, the nuclear matter becomes pure nucleonic matter. Our results indicate that the empirical parabolic law for the isospin asymmetry dependence of nuclear matter equation of state is heavily violated by the $\alpha$-particle condensation in the zero-temperature dilute nuclear matter, making the conventional definition of the symmetry energy meaningless. We investigate the symmetry energy defined under parabolic approximation for the zero-temperature dilute nuclear matter with $\alpha$-particle condensation, and find it is significantly enhanced compared to the case without clusters and becomes saturated at about $7$ MeV at very low densities ($\lesssim 10^{-3}$ fm$^{-3}$). The critical temperature for $\alpha$-condensation in homogeneous dilute nuclear matter is also discussed.

Analysis of nuclear structure in a converging power expansion scheme

In the framework of the KIDS generalized energy density functional (EDF), the nuclear equation of state (EoS) is expressed as an expansion in powers of the Fermi momentum or the cubic root of the density ($\rho^{1/3}$). Although an optimal number of converging terms was obtained in specific cases of fits to empirical data and pseudodata, the degree of convergence remains to be examined not only for homogeneous matter but also for finite nuclei. One goal of the present work is to validate the minimal and optimal number of EoS parameters required for the description of homogeneous nuclear matter over a wide range of densities relevant for astrophysical applications. The major goal is to examine the validity of the adopted expansion scheme for an accurate description of finite nuclei. To this end we vary the values of the high-order derivatives of the EoS, namely the skewness of the energy of symmetric nuclear matter and the kurtosis of the symmetry energy, at saturation and examine the relative importance of each term in $\rho^{1/3}$ expansion for homogeneous matter. For given sets of EoS parameters determined in this way, we define equivalent Skyrme-type functionals and examine the convergence in the description of finite nuclei focusing on the masses and charge radii of closed-shell nuclei. The EoS of symmetric nuclear matter is found to be efficiently parameterized with only 3 parameters and the symmetry energy (or the energy of pure neutron matter) with 4 parameters when the EoS is expanded in the power series of the Fermi momentum. Higher-order EoS parameters do not produce any improvement, in practice, in the description of nuclear ground-state energies and charge radii, which means that they cannot be constrained by bulk properties of nuclei.

From homogeneous matter to finite nuclei: Role of the effective mass

Recent astronomical observations, nuclear-reaction experiments, and microscopic calculations have placed new constraints on the nuclear equation of state (EoS) and revealed that most nuclear structure models fail to satisfy those constraints upon extrapolation to infinite matter. A reverse procedure for imposing EoS constraints on nuclear structure has been elusive. Here we present for the first time a method to generate a microscopic energy density functional (EDF) for nuclei from a given immutable EoS. The method takes advantage of a natural Ansatz for homogeneous nuclear matter, the Kohn-Sham framework, and the Skyrme formalism. We apply it to the realistic nuclear EoS of Akmal-Pandharipande-Ravenhall and describe successfully closed-(sub)shell nuclei. In the process, we provide predictions for the neutron skin thickness of nuclei based directly on the given EoS. Crucially, bulk and static nuclear properties are found practically independent of the assumed effective mass value - a unique result in bridging EDF of finite and homogeneous systems in general.

Dense Matter and Neutron Stars: Some Basic Notions

A number of properties of dense matter can be understood semiquantitatively in terms of simple physical arguments. We begin with the outer parts of neutron stars, and consider the density at which pressure ionization occurs, the density at which electrons become relativistic, the density at which neutrons drip out of nuclei, and the size of the equilibrium nucleus in dense matter. Subsequently, we treat the so-called "pasta" phases expected to occur at densities just below the density at which the transition from the crust to the liquid core of a neutron star occurs. We then consider aspects of superfluidity in dense matter. Estimates of pairing gaps in homogeneous nuclear matter are given, and the effect of the dense medium on the interaction between nucleons is described. Finally, we turn to superfluidity in the crust of neutron stars and especially the neutron superfluid density, an important quantity in the theory of sudden speedups of the rotation rate of some pulsars.

Minimal nuclear energy density functional

We present a minimal nuclear energy density functional (NEDF) called "SeaLL1" that has the smallest number of possible phenomenological parameters to date. SeaLL1 is defined by 7 significant phenomenological parameters, each related to a specific nuclear property. It describes the nuclear masses of even-even nuclei with a mean energy error of 0.97 MeV and a standard deviation 1.46 MeV, two-neutron and two-proton separation energies with r.m.s. errors of 0.69 MeV and 0.59 MeV respectively, and the charge radii of 345 even-even nuclei with a mean error $\epsilon_r=$0.022 fm and a standard deviation $\sigma_r=$0.025 fm. SeaLL1 incorporates constraints on the EoS of pure neutron matter from quantum Monte Carlo calculations with chiral effective field theory two-body (NN) interactions at N3LO level and three-body (NNN) interactions at the N2LO level. Two of the seven parameters are related to the saturation density and the energy per particle of the homogeneous symmetric nuclear matter, one is related to the nuclear surface tension, two are related to the symmetry energy and its density dependence, one is related to the strength of the spin-orbit interaction, and one is the coupling constant of the pairing interaction. We identify additional phenomenological parameters that have little effect on ground-state properties, but can be used to fine-tune features such as the Thomas-Reiche-Kuhn sum rule, the excitation energy of the giant dipole and Gamow-Teller resonances, the static dipole electric polarizability, and the neutron skin thickness.

A Unified Equation of State on a Microscopic Basis : Implications for Neutron Stars Structure and Cooling

We discuss the structure of Neutron Stars by modelling the homogeneous nuclear matter of the core by a suitable microscopic Equation of State, based on the Brueckner-Hartree-Fock many-body theory, and the crust, including the pasta phase, by the BCPM energy density functional which is based on the same Equation of State. This allows for a uni ed description of the Neutron Star matter over a wide density range. A comparison with other uni ed approaches is discussed. With the same Equation of State, which features strong direct Urca processes and using consistent nuclear pairing gaps as well as effective masses, we model neutron star cooling, in particular the current rapid cooldown of the neutron star Cas A. We nd that several scenarios are possible to explain the features of Cas A, but only large and extended proton 1S0 gaps and small neutron 3PF2 gaps can accommodate also the major part of the complete current cooling data.

Low density nuclear matter with light clusters in a generalized nonlinear relativistic mean-field model

We systematically investigate the thermodynamic properties of homogeneous nuclear matter with light clusters at low densities and finite temperatures using a generalized nonlinear relativistic mean-field (gNL-RMF) model, in which light clusters up to $\alpha $ ($1 \le A \le 4$) are included as explicit degrees of freedom and treated as point-like particles with their interactions described by meson exchanges and the medium effects on the cluster binding energies are described by density- and temperature-dependent energy shifts with the parameters obtained by fitting the experimental cluster Mott densities. We find that the composition of low density nuclear matter with light clusters is essentially determined by the density- and temperature-dependence of the cluster binding energy shifts. Compared with the values of the conventional (second-order) symmetry energy, symmetry free energy and symmetry entropy, their fourth-order values are found to be significant at low densities ($\rho \sim 10^{-3}$ fm$^{-3}$) and low temperatures ($T \lesssim 3$ MeV), indicating the invalidity of the empirical parabolic law for the isospin asymmetry dependence of these nuclear matter properties. Our results indicate that in the density region of $\rho \gtrsim 0.02$ fm$^{-3}$, the clustering effects become insignificant and the nuclear matter is dominated by nucleon degree of freedom. In addition, we compare the gNL-RMF model predictions with the corresponding experimental data on the symmetry energy and symmetry free energy at low densities and finite temperatures extracted from heavy-ion collisions, and reasonable agreement is found.

Limiting symmetry energy elements from empirical evidence

In the framework of an equation of state (EoS) constructed from a momentum and density-dependent finite-range two-body effective interaction, the quantitative magnitudes of the different symmetry elements of infinite nuclear matter are explored. The parameters of this interaction are determined from well-accepted characteristic constants associated with homogeneous nuclear matter. The symmetry energy coefficient [Formula: see text], its density slope [Formula: see text], the symmetry incompressibility [Formula: see text] as well as the density-dependent incompressibility [Formula: see text] evaluated with this EoS are seen to be in good harmony with those obtained from other diverse perspectives. The higher order symmetry energy coefficients [Formula: see text], etc., are seen to be not very significant in the domain of densities relevant to finite nuclei, but gradually build up at supra-normal densities. The analysis carried out with a Skyrme-inspired energy density functional (EDF) obtained with the same input values for the empirical bulk data associated with nuclear matter yields nearly the same results.

Di-nucleon structures in homogeneous nuclear matter based on two- and three-nucleon interactions

We investigate homogeneous nuclear matter within the Brueckner-Hartree-Fock (BHF) approach in the limits of isospin-symmetric nuclear matter (SNM) as well as pure neutron matter at zero temperature. The study is based on realistic representations of the internucleon interaction as given by Argonne v18, Paris, Nijmegen I and II potentials, in addition to chiral N$^{3}$LO interactions, including three-nucleon forces up to N$^{2}$LO. Particular attention is paid to the presence of di-nucleon bound states structures in $^1\textrm{S}_0$ and $^3\textrm{SD}_1$ channels, whose explicit account becomes crucial for the stability of self-consistent solutions at low densities. A characterization of these solutions and associated bound states is discussed. We confirm that coexisting BHF single-particle solutions in SNM, at Fermi momenta in the range $0.13-0.3$~fm$^{-1}$, is a robust feature under the choice of realistic internucleon potentials.

Divergence of the isospin-asymmetry expansion of the nuclear equation of state in many-body perturbation theory

The isospin-asymmetry dependence of the nuclear matter equation of state obtained from microscopic chiral two- and three-body interactions in second-order many-body perturbation theory is examined in detail. The quadratic, quartic and sextic coefficients in the Maclaurin expansion of the free energy per particle of infinite homogeneous nuclear matter with respect to the isospin asymmetry are extracted numerically using finite differences, and the resulting polynomial isospin-asymmetry parametrizations are compared to the full isospin-asymmetry dependence of the free energy. It is found that in the low-temperature and high-density regime where the radius of convergence of the expansion is generically zero, the inclusion of higher-order terms beyond the leading quadratic approximation leads overall to a significantly poorer description of the isospin-asymmetry dependence. In contrast, at high temperatures and densities well below nuclear saturation density, the interaction contributions to the higher-order coefficients are negligible and the deviations from the quadratic approximation are predominantly from the noninteracting term in the many-body perturbation series. Furthermore, we extract the leading logarithmic term in the isospin-asymmetry expansion of the equation of state at zero temperature from the analysis of linear combinations of finite differences. It is shown that the logarithmic term leads to a considerably improved description of the isospin-asymmetry dependence at zero temperature.

Further explorations of Skyrme-Hartree-Fock-Bogoliubov mass formulas. XVI. Inclusion of self-energy effects in pairing

Extending our earlier work, a new family of three Hartree-Fock-Bogoliubov (HFB) mass models, labeled HFB-30, HFB-31, and HFB-32, is presented, along with their underlying interactions, BSk30, BSk31, and BSk32, respectively. The principle new feature is a purely phenomenological pairing term that depends on the density gradient. This enables us to have a bulk pairing term that is fitted to realistic nuclear-matter calculations in which for the first time the self-energy corrections are included, while the behavior of the nucleon effective masses in asymmetric homogeneous nuclear matter is significantly improved. Furthermore, in the particle-hole channel all the highly realistic constraints of our earlier work are retained. In particular, the unconventional Skyrme forces containing ${t}_{4}$ and ${t}_{5}$ terms are still constrained to fit realistic equations of state of neutron matter stiff enough to support the massive neutron stars PSR J1614--2230 and PSR J0348+0432. All unphysical long-wavelength spin and spin-isospin instabilities of nuclear matter, including the unphysical transition to a polarized state in neutron-star matter, are eliminated. Our three interactions are characterized by values of the symmetry coefficient $J$ of 30, 31, and 32 MeV, respectively. The best fit to the database of 2353 nuclear masses is found for model HFB-31 $(J=31\phantom{\rule{0.16em}{0ex}}\mathrm{MeV})$ with a model error of 0.561 MeV. This model also fits the charge-radius data with an root-mean-square error of 0.027 fm.