Self-consistent mean-field methods with Skyrme-type effective interactions and semiclassical approximations, such as the Thomas-Fermi approach and its extensions are particularly well-suited for describing in a thermodynamically consistent way the various phases of the dense matter present in the interior of neutron stars. These methods have been applied to predict the composition of the different regions, including the inner crust constituted by nuclear clusters coexisting with free neutrons and electrons. Because of the computational cost, the energy is typically calculated for a few selected average baryon number densities, and the results are interpolated to obtain the pressure numerically. However, this may introduce systematic errors in the calculations of the global structure of a neutron star and its dynamical evolution. In this paper, we show how the full equation of state can be consistently calculated within the same framework by deriving exact formulas for the chemical potentials and for the pressure that can be easily implemented in existing computer codes. These formulas are applicable to both catalyzed and accreted crusts. We discuss in each case the suitable conditions to impose to determine the composition. Numerical examples are also presented and discussed. Results from refined calculations of the BSk24 equation of state for the inner crust of nonaccreted neutron stars and the corresponding adiabatic index are provided.
The equation of state (EoS) of the outer crust of a cold non-accreting magnetar has been determined using the model of Lai and Shapiro (1991). For this purpose, we have made use of the latest experimental atomic mass data complemented with a Hartree-Fock-Bogoliubov (HFB) mass model. Magnetar crusts are found to be significantly different from the crusts of ordinary neutron stars.
The equilibrium properties of the outer crust of cold nonaccreting magnetars (i.e. neutron stars endowed with very strong magnetic fields) are studied using the latest experimental atomic mass data complemented with a microscopic atomic mass model based on the Hartree-Fock-Bogoliubov method. The Landau quantization of electron motion caused by the strong magnetic field is found to have a significant impact on the composition and the equation of state of crustal matter. It is also shown that the outer crust of magnetars could be much more massive than that of ordinary neutron stars.
We investigated the role of a pairing correlation in the chemical composition of the inner crust of a neutron star with the extended Thomas-Fermi method, using the Strutinsky integral correction. We compare our results with the fully self-consistent Hartree-Fock-Bogoliubov approach, showing that the resulting discrepancy, apart from the very low density region, is compatible with the typical accuracy we can achieve with standard mean-field methods.
We perform a systematic investigation of the chemical composition of the inner crust of a neutron star, using the extended Thomas-Fermi approximation, the Strutinsky integral correction for shell effects, and the BCS approximation for pairing. Fifteen Skyrme functionals were selected, which cover the range of values of important bulk properties of infinite nuclear matter, while also having pure neutron matter (PNM) equation of states (EoS) with varying degrees of stiffness. We find that a functional's low-density PNM EoS is correlated with the number of protons found in the inner crust's nucle…
We perform a systematic comparison between the results obtained by solving fully self-consistently the Hartree-Fock-Bogoliubov equations, and those obtained using the semi-classical Extended Thomas-Fermi method, for various Wigner-Seitz cells within the inner crust of a neutron star. The lack of pairing correlations in the semi-classical approach leads to a large discrepancy between the two approaches. This discrepancy is well beyond the error of the quantum-mechanical calculation, and is related to spurious shell effects in the neutron gas.
After a brief review of the Hartree-Fock-Bogoliubov method with Skyrme effective interactions, we show how it can be applied to the description of various nuclear systems, from finite nuclei to neutron-star crusts.
We study the effects of the symmetry energy on the neutron drip density and properties of nuclei in neutron star crusts. The nonuniform matter around the neutron drip point is calculated by using the Thomas--Fermi approximation with the relativistic mean-field model. The neutron drip density and the composition of the crust are found to be correlated with the symmetry energy and its slope. We compare the self-consistent Thomas--Fermi approximation with other treatments of surface and Coulomb energies and find that these finite-size effects play an essential role in determining the equilibrium …
The equation of state (EOS) for neutron star (NS) crusts is studied in the Thomas-Fermi (TF) approximation using the EOS for uniform nuclear matter obtained by the variational method with the realistic nuclear Hamiltonian. The parameters associated with the nuclear three-body force, which are introduced to describe the saturation properties, are finely adjusted so that the TF calculations for isolated atomic nuclei reproduce the experimental data on masses and charge distributions satisfactorily. The resulting root-mean-square deviation of the masses from the experimental data for mass-measure…
We calculate the specific heat of the inner crust of neutron stars within a local-density approximation to an improved relativistic Hartree-Bogoliubov theory. Non-uniformness of the system enhances the specific heat in particular at low temperatures. The degree of enhancement is similar to that in the spherical phase of Elgarøy et al. We examine a schematic interpolation between the results of Broglia et al. adopting the Gogny force and ours based on the Lagrangian of the relativistic mean field model.
The bottom part of the neutron star crust is investigated using the Skyrme-Hartree-Fock approach with the Coulomb interaction treated beyond the Wigner-Seitz approximation. A variety of nuclear phases is found to coexist in this region. Their stability and relative energies are governed by the Coulomb, surface and shell energies. We have also found that a substantial contribution is coming from the spin-orbit interaction.
We study the impact of the symmetry energy on properties of nuclear pasta phases and crust-core transition in neutron stars. We perform a self-consistent Thomas--Fermi calculation employing the relativistic mean-field model. The properties of pasta phases presented in the inner crust of neutron stars are investigated and the crust-core transition is examined. It is found that the slope of the symmetry energy plays an important role in determining the pasta phase structure and the crust-core transition. The correlation between the symmetry energy slope and the crust-core transition density obta…
We present a detailed analysis of three models predicting the properties of non-uniform matter in the crust of neutron stars: the compressible liquid-drop model, the fourth order Extended Thomas Fermi (ETF) method, and ETF plus Strutinsky integral (ETFSI) correction. The former treats the nuclear clusters as uniform hard spheres, the second takes into account the density distribution which can be different for neutrons and protons, and the last one includes the proton shell effects within the Strutinsky approach. The purpose of this work is to understand the importance of the improvements in t…
We have studied the various properties of magnetically deformed atoms, replaced by deformed Wigner-Seitz cells, at the outer crust region of strongly magnetized neutron stars (magnetars) using a relativistic version of Thomas-Fermi model in cylindrical coordinates.
Properties of the matter in the inner crust of a neutron star are investigated in a Hartree-Fock plus BCS approximation employing schematic effective forces of the type of the Skyrme forces. Special attention is paid to differences between a homogenous and inhomogeneous description of the matter distribution. For that purpose self-consistent Hartree Fock calculations are performed in a spherical Wigner-Seitz cell. The results are compared to predictions of corresponding Thomas Fermi calculations. The influence of the shell structure on the formation of pairing correlations in inhomogeneous mat…
In the present paper we explore the neutron-drip region of cold non-rotating isolated neutron stars. We have performed extended nuclear-structure calculations for nuclei embedded in the electron gas. For modeling the outer crust we use a set of Wigner-Seitz cells, where every cell contains one nucleus surrounded by a cloud of relativistic electrons. Above the drip point a non-relativistic neutron gas occurs in the cell. These calculations are carried out within the Hartree-Fock approach in combination with Skyrme effective interactions. For every baryon density we have determined the configura…
Fully self-consistent band calculation has been performed for slab phase in neutron-star inner crust, using the BCPM energy density functional. Optimized slab structure is calculated at given baryon density either with the fixed proton ratio or with the beta-equilibrium condition. Numerical results indicate the band gap of in order of keV to tens of keV, and the mobility of dripped neutrons are enhanced by the Bragg scattering, which leads to the macroscopic effective mass, $\bar{m}^*_z/m_n=0.65\sim 0.75$ near the bottom of the inner crust in neutron stars. We also compare the results of the b…
Using the relativistic mean-field theory, we adopt two different methods, namely, the coexisting phase method and the self-consistent Thomas-Fermi approximation, to study the impacts of the nuclear symmetry energy on properties of neutron star crusts within a wide range of densities. It is found that the nuclear symmetry energy and its density slope play an important role in determining the pasta phases and the crust-core transition.
We analyse how the structure of the inner curst is influenced by the pairing correlations. The inner-crust matter, formed by nuclear clusters immersed in a superfluid neutron gas and ultra-relativistic electrons, is treated in the Wigner-Seitz approximation. The properties of the Wigner-Seitz cells, i.e., their neutron to proton ratio and their radius at a given baryonic density, are obtained from the energy minimization at beta equilibrium. To obtain the binding energy of baryonic matter we perform Skyrme-HFB calculations with zero-range density-dependent pairing forces of various intensities…
Adopting the density dependent relativistic mean-field (RMF) and relativistic Hartree-Fock (RHF) approaches, the properties of the nuclear fourth-order symmetry energy $S_4$ are studied within the covariant density functional (CDF) theory. It is found that the fourth-order symmetry energies are suppressed in RHF at both saturation and supranuclear densities, where the extra contribution from the Fock terms is demonstrated, specifically via the isoscalar meson-nucleon coupling channels. The reservation of $S_4$ and higher-order symmetry energies in the nuclear equation of state then affects ess…
In order to study structure of the crust in neutron stars, we develop a finite-temperature Skyrme-Hartree-Fock method in the full three-dimensional coordinate space using the Fermion operator expansion method. It provides us with a possible order-N approach to non-uniform neutron-star matters.
We investigate the thermal properties of the inner crust of a neutron star using the Hartree-Fock-Bogoliubov (HFB) formalism at finite temperature. We compare our results with the ones obtained solving the same equations, but within the BCS approximation. We observe that for the outermost regions of the inner crust, the two methods can show important differences, in particular when we use them to calculate the neutron specific heat of the system.
We discuss outer and inner crusts of neutron stars in strong magnetic fields. Here, we demonstrate the effect of Landau quantization of electrons on the ground state properties of matter in outer and inner crusts in magnetars. This effect leads to the enhancement of the electron number density in strong magnetic fields with respect to the zero field case. For the outer crust, we adopt the magnetic Baym-Pethick-Sutherland model and obtain the sequence of nuclei and equation of state (EoS). The properties of nuclei in the inner crust in the presence of strong magnetic fields are investigated usi…
The properties and composition of the outer crust of nonaccreting cold neutron stars are studied by applying the model of Baym, Pethick, and Sutherland (BPS) and taking into account for the first time triaxial deformations of nuclei. Two theoretical nuclear models, Hartree-Fock plus pairing in the BCS approximation (HF-BCS) with Skyrme SLy6 parametrization and Hartree-Fock-Bogolyubov (HFB) with Gogny D1S force, are used to calculate the nuclear masses. The two theoretical calculations are compared concerning their neutron drip line, binding energies, magic neutron numbers, and the sequence of …
Following an extremely interesting idea \cite{R1}, published long ago, the work function at the outer crust region of a strongly magnetized neutron star is obtained using relativistic version of Thomas-Fermi type model. In the present scenario, the work function becomes anisotropic; the longitudinal part is an increasing function of magnetic field strength, whereas the transverse part diverges. An approximate estimate of the electron density in the magnetosphere due to field emission and photo emission current, from the polar cap region are obtained.
The structure of Wigner-Seitz cells in the inner crust of neutron stars is investigated using a microcospic Hartree-Fock-BCS approach with finite range D1S and M3Y-P4 interactions. Large effects on the densities are found compared to previous predictions using Skyrme interactions. Pairing effects are found to be small, and they are attenuated by the use of finite range interactions in the mean field.
In the present study we investigate the static properties of nuclei in the inner crust of neutron stars. Using the Hartree-Fock method in coordinate space, together with the semiclassical approximation, we examine the patterns of phase transitions.
In neutron star crust, non-uniform structure of nuclear matter is expected, which is called the "pasta" structure. From the recent studies of giant flares in magnetars, these structures might be related to some observables and physical quantities of the neutron star crust. To investigate the above quantities, we numerically explore the pasta structures with a fully threedimensional geometry and study the properties of low-density nuclear matter, based on the relativistic mean-field model and the Thomas-Fermi approximation. We observe typical pasta structures for fixed proton number-fraction an…
The cell structure of clusters in the inner crust of a cold β-equilibrium neutron star is studied within a Thomas Fermi approach and compared with other approaches which include shell effects. Relativistic nuclear models are considered. We conclude that the symmetry energy slope L may have quite dramatic effects on the cell structure if it is very large or small. Rod-like and slab-like pasta clusters have been obtained in all models except one with a large slope L.
In the present work we study the effects of strongly quantizing magnetic fields and finite temperature on the properties of inner crusts of hot neutron stars. The inner crust of a neutron star contains neutron-rich nuclei arranged in a lattice and embedded in gases of free neutrons and electrons. We describe the system within the Wigner-Seitz (WS) cell approximation. The nuclear energy is calculated using Skyrme model with SkM* interaction. To isolate the properties of nuclei we follow the subtraction procedure presented by Bonche, Levit and Vautherin, within the Thomas-Fermi approximation. We…
We investigated the structure of the low density regions of the inner crust of neutron stars using the Hartree-Fock-Bogoliubov (HFB) model to predict the proton content $Z$ of the nuclear clusters and, together with the lattice spacing, the proton content of the crust as a function of the total baryonic density $ρ_b$. The exploration of the energy surface in the $(Z,ρ_b)$ configuration space and the search for the local minima require thousands of calculations. Each of them implies an HFB calculation in a box with a large number of particles, thus making the whole process very demanding. In th…
Properties of inhomogeneous nuclear matter are evaluated within a relativistic mean field approximation using density dependent coupling constants. A parameterization for these coupling constants is presented, which reproduces the properties of the nucleon self-energy obtained in Dirac Brueckner Hartree Fock calculations of asymmetric nuclear matter but also provides a good description for bulk properties of finite nuclei. The inhomogeneous infinite matter is described in terms of cubic Wigner-Seitz cells, which allows for a microscopic description of the structures in the so-called ``pasta-ph…
Globally neutral neutron stars, obtained from the solution of the called Einstein-Maxwell-Thomas-Fermi equations that account for all the fundamental interactions, have been recently introduced. These configurations have a more general character than the ones obtained with the traditional Tolman-Oppenheimer-Volkoff, which impose the condition of local charge neutrality. The resulting configurations have a less massive and thinner crust, leading to a new mass-radius relation. Signatures of this new structure of the neutron star on the thermal evolution might be a potential test for this theory.…
We study the impact of strong magnetic fields on the pasta phases that are expected to exist in the inner crust of neutron stars. We employ the relativistic mean field model to describe the nucleon interaction and use the self-consistent Thomas-Fermi approximation to calculate the nonuniform matter in neutron star crust. The properties of pasta phases and crust-core transition are examined. It is found that as the magnetic field strength $B$ is less than $10^{17}$ G, the effects of magnetic field are not evident comparing with the results without magnetic field. As $B$ is stronger than $10^{18…
We use the effective relativistic mean-field (E-RMF) model to study the crustal properties of the neutron star. The unified equations of state (EoS) are constructed using recently developed E-RMF parameter sets, such as FSUGarnet, IOPB-I, and G3. The outer crust composition is determined using the atomic mass evaluation 2020 data [Chinese Physics C {\bf 45}, 030002 (2021)] along with the available Hartree-Fock-Bogoliubov mass models [Phys. Rev. C {\bf 88}, 024308 (2013)] for neutron-rich nuclei. The structure of the inner crust is estimated by performing the compressible liquid drop model calc…
The equation of state and composition of the inner crust of neutron stars at zero temperature are calculated, using the T = 0 version of the TETFSI (temperature-dependent extended Thomas-Fermi plus Strutinsky integral) method, for each of a family of three functionals based on Skyrme-type forces BSk19, BSk20 and BSk21, which are characterized by different degrees of symmetry-energy stiffness, and also for the SLy4 functional. We also solve the Tolman-Oppenheimer-Volkoff equations to calculate the distribution of mass within the inner crust. Qualitatively similar results are found for all four …
To make best use of multi-faceted astronomical and nuclear data-sets, probability distributions of neutron star models that can be used to propagate errors consistently from one domain to another are required. We take steps toward a consistent model for this purpose, highlight where model inconsistencies occur and assess the resulting model uncertainty. Using two distributions of nuclear symmetry energy parameters - one uniform, the other based on pure neutron matter theory, we prepare two ensembles of neutron star inner crust models. We use an extended Skyrme energy-density functional within …
Magnetars are neutron stars endowed with surface magnetic fields of the order of $10^{14}-10^{15}$~G, and with presumably much stronger fields in their interior. As a result of Landau quantization of electron motion, the neutron-drip transition in the crust of a magnetar is shifted to either higher or lower densities depending on the magnetic field strength. The impact of nuclear uncertainties is explored considering the recent series of Brussels-Montreal microscopic nuclear mass models. All these models are based on the Hartree-Fock-Bogoliubov method with generalized Skyrme functionals. They …
The cell structure of $β$-stable clusters in the inner crust of cold and warm neutron stars is studied within the Thomas-Fermi approach using relativistic mean field nuclear models. The relative size of the inner crust and the pasta phase of neutron stars is calculated, and the effect of the symmetry energy slope parameter, $L$, on the profile of the neutron star crust is discussed. It is shown that while the size of the total crust is mainly determined by the incompressibility modulus, the relative size of the inner crust depends on $L$. It is found that the inner crust represents a larger fr…
We construct three new Hartree-Fock-Bogoliubov (HFB) mass models, labeled HFB-19, HFB-20, and HFB-21, with unconventional Skyrme forces containing $t_4$ and $t_5$ terms, i.e., density-dependent generalizations of the usual $t_1$ and $t_2$ terms, respectively. The new forces underlying these models are fitted respectively to three different realistic equations of state of neutron matter for which the density dependence of the symmetry energy ranges from the very soft to the very stiff, reflecting thereby our present lack of complete knowledge of the high-density behavior of nuclear matter. All …
We have investigated some of the properties of dense sub-nuclear matter at the crustal region (both the outer crust and the inner crust region) of a magnetar. The relativistic version of Thomas-Fermi (TF) model is used in presence of strong quantizing magnetic field for the outer crust matter. The compressed matter in the outer crust, which is a crystal of metallic iron, is replaced by a regular array of spherically symmetric Wigner-Seitz (WS) cells. In the inner crust region, a mixture of iron and heavier neutron rich nuclei along with electrons and free neutrons has been considered. Conventi…
The possible presence of amorphous and heterogeneous phases in the inner crust of a neutron star is expected to reduce the electrical conductivity of the crust, with potentially important consequences on the magneto-thermal evolution of the star. In cooling simulations, the disorder is quantified by an impurity parameter which is often taken as a free parameter. We aim to give a quantitative prediction of the impurity parameter as a function of the density in the crust,performing microscopic calculations including up-to-date microphysics of the crust. A multi-component approach is developed at…
The equation of state for neutron stars in a wide-density range at zero temperature is constructed. The chiral quark-meson coupling model within relativistic Hartree-Fock approximation is adopted for uniform nuclear matter. The coupling constants are determined so as to reproduce the experimental data of atomic nuclei and hypernuclei. In the crust region, nuclei are taken into account within the Thomas-Fermi calculation. All octet baryons are considered in the core region, while only $Ξ^{-}$ appears in neutron stars. The resultant maximum mass of neutron stars is $1.95M_\odot$, which is consis…
In this paper, we calculate the stable Wigner-Seitz (W-S) cells in the inner crust of neutron stars and we discuss the nuclear shell effects. A distinction is done between the shell effects due to the bound states and those induced by the unbound states, which are shown to be spurious. We then estimate the effects of the spurious shells on the total energy and decompose it into a smooth and a residual part. We propose a correction to the Hartree-Fock binding energy in Wigner-Seitz cell (HF-WS).
In this review paper we discuss the effects of pairing correlations on inner crust matter in the density region where nuclear clusters are supposed to coexist with nonlocalised neutrons. The pairing correlations are treated in the framework of the finite temperature Hartree-Fock-Bogoliubov approach and using zero range nuclear forces. After a short introduction and presentation of the formalism we discuss how the pairing correlations affect the structure of the inner crust matter, i.e., the proton to neutron ratio and the size of Wigner-Seitz cells. Then we show how the pairing correlations in…
The properties of inhomogeneous nuclear matter are investigated considering the self-consistent Skyrme Hartree-Fock approach with inclusion of pairing correlations. For a comparison we also consider a relativistic mean field approach. The inhomogeneous infinite matter is described in terms of cubic Wigner-Seitz cells, which leads to a smooth transition to the limit of homogeneous nuclear matter. The possible existence of various structures in the so-called pasta phase is investigated within this self-consistent approach and a comparison is made to results obtained within the Thomas-Fermi appro…
We investigate the structure of neutron star crusts, including the crust-core boundary, based on new Skyrme mean field models constrained by the bulk-matter equation of state from chiral effective field theory and the ground-state energies of doubly-magic nuclei. Nuclear pasta phases are studied using both the liquid drop model as well as the Thomas-Fermi approximation. We compare the energy per nucleon for each geometry (spherical nuclei, cylindrical nuclei, nuclear slabs, cylindrical holes, and spherical holes) to obtain the ground state phase as a function of density. We find that the size …
We study the role of the pasta phases on the properties of rotating and magnetized neutron stars. In order to investigate such systems, we make use of two different relativistic mean-field unified inner-crust--core equations of state, with a different density dependence of the symmetry energy, and an inner-crust computed within a Thomas-Fermi calculation. Special attention is given to the crust-core transition density, and the pasta phases effects on the global properties of stars. The effects of strong magnetic fields and fast rotation are computed by solving the Einstein-Maxwell equations se…
We present a new semiclassical theory for describing pairing in finite Fermi systems. It is based on taking the $\hbar \to 0$, i.e. Thomas-Fermi, limit of the gap equation written in the basis of the mean field (weak coupling). In addition to the position dependence of the Fermi momentum, the size dependence of the pairing force is also taken into account in this theory. Along isotopic chains the Thomas-Fermi gaps average the well known arch structure shown by the quantal gaps. This structure can be almost recovered in our formalism if some shell fluctuations are included in the level density.…
We investigated the superfluid properties of the inner crust of neutron stars, solving the Hartree-Fock-Bogoliubov equations in spherical Wigner-Seitz cells. Using realistic two-body interactions in the pairing channel, we studied in detail the Cooper-pair and the pairing-field spatial properties, together with the effect of the proton clusters on the neutron pairing gap. Calculations with effective pairing interactions are also presented, showing significant discrepancies with the results obtained with realistic pairing forces. At variance with recent studies on finite nuclei, the neutron coh…
The mean free path of neutrino in charged and neutral current reactions is calculated for inhomogeneous nuclear matter which is expected to appear in the crust of neutron stars. The relevant cross section depends on Fermi and Gamow-Teller strength distributions, which are derived from the large-scale shell model calculations within the self-consistent Skyrme-Hartree-Fock approach and in a relativistic mean-field model. The inhomogeneous nuclear matter is described in terms of cubic Wigner-Seitz cells, which allows for a microscopic description of the structures in the so-called pasta phase of …
Using the $\hbar$-expansion of the Green's function of the Hartree-Fock-Bogoliubov equation, we extend the second-order Thomas-Fermi approximation to generalized superfluid Fermi systems by including the density-dependent effective mass and the spin-orbit potential. We first implement and examine the full correction terms over different energy intervals of the quasiparticle spectra in calculations of finite nuclei. Final applications of this generalized Thomas-Fermi method are intended for various inhomogeneous superfluid Fermi systems.
In order to study structure of proto-neutron stars and those in subsequent cooling stages, it is of great interest to calculate inhomogeneous hot and cold nuclear matter in a variety of phases. The finite-temperature Hartree-Fock-Bogoliubov (FT-HFB) theory is a primary choice for this purpose, however, its numerical calculation for superfluid (superconducting) many-fermion systems in three dimensions requires enormous computational costs. To study a variety of phases in the crust of hot and cold neutron stars, we propose an efficient method to perform the FT-HFB calculation with the three-dime…
