Inner Crust Of Neutron Stars Research Articles

Role of neutron pairing with density-gradient dependence in the semimicroscopic treatment of the inner crust of neutron stars

Role of neutron pairing with density-gradient dependence in the semimicroscopic treatment of the inner crust of neutron stars

Effect of the equation of state for dilute neutron matter on the composition of the inner crust of neutron stars

Effect of the equation of state for dilute neutron matter on the composition of the inner crust of neutron stars

Superfluid fraction in the slab phase of the inner crust of neutron stars

Superfluid fraction in the slab phase of the inner crust of neutron stars

Orbital-free density functional theory: Differences and similarities between electronic and nuclear systems

Abstract Orbital-free density functional theory (OF-DFT) has been used when studying atoms, molecules, and solids. In nuclear physics, applications of OF-DFT have been quite scarce so far, as DFT has been widely applied to the study of many nuclear properties mostly within the Kohn–Sham (KS) scheme. There are many realizations of nuclear KS-DFT, but computations become very demanding for heavy systems, such as superheavy nuclei and the inner crust of neutron stars, and it is hard to describe exotic nuclear shapes using a finite basis made with a limited number of orbitals. These bottlenecks could, in principle, be overcome by an orbital-free formulation of DFT. This work is a first step towards the realistic application of OF-DFT to nuclei. In particular, we have implemented possible choices for an orbital-free kinetic energy and solved the associated Schrödinger equation either with simple potentials or with simplified nuclear density functionals. While the former choice sheds light on the differences between electronic and nuclear systems, the latter choice allows us to discuss the practical applications to nuclei as well as open questions.

Equation of state of superfluid neutron matter with low-momentum interactions

The behavior of the unbound neutrons in the inner crust of neutron stars is linked to various neutron-star characteristics such as glitches in the repetition rate of pulsars. This paper presents a novel calculation of low-density superfluid neutron matter using techniques and interactions previously applied to the study of finite nuclei. The starting point is a two-body interaction softened using renormalization-group techniques. The resulting interaction is used in a Hartree-Fock-Bogoliubov calculation that includes pairing. This provides a good starting point for the many-body perturbation expansion, and for including more sophisticated nuclear interactions in future work.

Effects of pairing gap and band gap on superfluid density in the inner crust of neutron stars

Calculations of the superfluid density in the inner crust of neutron stars using different approaches are in strong disagreement, which causes a debate on the accountability of pulsar glitches based on superfluidity. Using a simple unified model, we study the dependence on approximation of the superfluid density in a periodic potential. In comparison with the Hartree-Fock-Bogoliubov theory, which treats the effects of the band gap and the pairing gap on equal footing, we examine a Hartree-Fock-BCS-type (HF-BCS-type) approximation in which the former is preferentially incorporated and another approximation in which the latter is preferentially incorporated. We find that, when the pairing gap and the band gap are comparable as in the inner crust of neutron stars, they need to be treated on equal footing, and the HF-BCS approximation can considerably underestimate the superfluid density even if the pairing gap is much smaller than the Fermi energy. Our result suggests that the validity of the HF-BCS approximation for evaluating the superfluid density in neutron star crusts is questionable.

Low Density Neutron Star Matter with Quantum Molecular Dynamics: The Role of Isovector Interactions

The effect of isospin-dependent nuclear forces on the inner crust of neutron stars is modeled within the framework of Quantum Molecular Dynamics (QMD). To successfully control the density dependence of the symmetry energy of neutron-star matter below nuclear saturation density, a mixed vector-isovector potential is introduced. This approach is inspired by the baryon density and isospin density-dependent repulsive Skyrme force of asymmetric nuclear matter. In isospin-asymmetric nuclear matter, the system shows nucleation, as nucleons are arranged into shapes resembling nuclear pasta. The dependence of clusterization in the system on the isospin properties is also explored by calculating two-point correlation functions. We show that, as compared to previous results that did not involve such mixed interaction terms, the energy symmetry slope L is successfully controlled by varying the corresponding coupling strength. Nevertheless, the effect of changing the slope of the nuclear symmetry energy L on the crust-core transition density does not seem significant. To the knowledge of the authors, this is the first implementation of such a coupling in a QMD model for isospin asymmetric matter, which is relevant to the inner crust of neutron and proto-neutron stars.

Stability of Spherical Nuclei in the Inner Crust of Neutron Stars

Neutron stars are the densest objects in the Universe. In this paper, we consider the so-called inner crust—the layer where neutron-excess nuclei are immersed in the degenerate gas of electrons and a sea of quasi-free neutrons. It was generally believed that spherical nuclei become unstable with respect to quadrupole deformations at high densities, and here, we consider this instability. Within the perturbative approach, we show that spherical nuclei with equilibrium number density are, in fact, stable with respect to infinitesimal quadrupole deformation. This is due to the background of degenerate electrons and associated electrostatic potential, which maintain stability of spherical nuclei. However, if the number of atomic nuclei per unit volume is much less than the equilibrium value, instability can arise. To avoid confusion, we stress that our results are limited to infinitesimal deformations and do not guarantee strict thermodynamic stability of spherical nuclei. In particular, they do not exclude that substantially non-spherical nuclei (so-called pasta phase) represent a thermodynamic equilibrium state of the densest layers of the neutron star crust. Rather, our results point out that spherical nuclei can be metastable even if they are not energetically favourable, and the timescale of transformation of spherical nuclei to the pasta phases should be estimated subsequently.

Superfluid density in disordered pasta phases in neutron star crusts

We calculate the superfluid density of nucleons in disordered pasta phases in the inner crust of neutron stars using an effective medium approach which parallels that previously used for calculating the electrical conductivity of terrestrial matter. We allow for the effect of entrainment, the fact that the current density of one species of nucleon depends on the gradient of the phase not only of the same species but also of the other species. The superfluid density tensors for the perfectly ordered pasta phases are very anisotropic, and we derive expression for the effective superfluid densities of the disordered phases.

Time-dependent extension of the self-consistent band theory for neutron star matter: Anti-entrainment effects in the slab phase

Background: In the solid crust of neutron stars, a variety of crystalline structure may exist. Recently the band theory of solids has been applied to the inner crust of neutron stars and significance of the entrainment between dripped neutrons and the solid crust was advocated. Since it influences interpretations of various phenomena of neutron stars, it has been desired to develop deeper understanding of the microphysics behind. Purpose: The purpose of the present article is to propose a fully self-consistent microscopic framework for describing time-dependent dynamics of neutron star matter, which allows us to explore diverse properties of nuclear matter, including the entrainment effect. Results: As the first application of the time-dependent self-consistent band theory for nuclear systems, we investigate the slab phase of nuclear matter with various proton fractions. From a dynamic response of the system to an external force, we extract the collective mass of a slab, associated with entrained neutrons as well as bound nucleons. We find that the extracted collective mass is smaller than a naive estimation based on a potential profile and single-particle energies. We show that the reduction is mainly caused by "counterflow" of dripped neutrons towards the direction opposite to the motion of the slabs. We interpret it as an "anti-entrainment" effect. As a result, the number of effectively bound neutrons is reduced, indicating an enhancement of the number density of conduction neutrons. We demonstrate that those findings are consistent with a static treatment in the band theory of solids. *shortened due to the arXiv's word limit.

Unified equations of state for cold nonaccreting neutron stars with Brussels-Montreal functionals. III. Inclusion of microscopic corrections to pasta phases

Earlier fourth-order extended Thomas-Fermi calculations with functional BSk24 of pasta phases in the inner crust of neutron stars are here repeated with Strutinsky-integral and pairing corrections included for protons. The main consequence of these corrections is to eliminate the spaghetti phase at zero temperature and significantly reduce the range of densities over which the lasagna phase is found. On the other hand, the microscopic corrections have a negligible impact on the equation of state and on the proton fraction.

The effect of the energy functional on the pasta-phase properties of catalysed neutron stars

Nuclear pasta, that is an inhomogeneous distribution of nuclear matter characterised by non-spherical clustered structures, is expected to occur in a narrow spatial region at the bottom of the inner crust of neutron stars, but the width of the pasta layer is strongly model dependent. In the framework of a compressible liquid-drop model, we use Bayesian inference to analyse the constraints on the sub-saturation energy functional and surface tension imposed by both ab-initio chiral perturbation theory calculations and experimental measurements of nuclear masses. The posterior models are used to obtain general predictions for the crust-pasta and pasta-core transition with controlled uncertainties. A correlation study allows us to extract the most influential parameters for the calculation of the pasta phases. The important role of high-order empirical parameters and of the surface tension is underlined.

Effect of the Landau quantization of electrons on the inner crusts of hot neutron stars

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 obtain the equilibrium properties of inner crust for various density, temperatures and magnetic fields by minimizing the free energy of the WS cell satisfying the charge neutrality and $\beta-$equilibrium conditions. We infer that at a fixed baryon density and temperature, strong quantizing magnetic field reduces the cell radii, neutron and proton numbers in the cell compared with the field free case. However, the nucleon number in the nucleus increases in presence of magnetic field. The free energy per nucleon also decreases in magnetized inner crust. On the other hand, we find that finite temperature tends to smear out the effects of magnetic field. Our results can be important in the context of $r-$process nucleosynthesis in the binary neutron star mergers.

Nuclear pasta and symmetry energy in the relativistic point-coupling model

Nonuniform structure of low-density nuclear matter, known as nuclear pasta, is expected to appear not only in the inner crust of neutron stars but also in core-collapse supernova explosions and neutron-star mergers. We perform fully three-dimensional calculations of inhomogeneous nuclear matter and neutron-star matter in the low-density region using the Thomas-Fermi approximation. The nuclear interaction is described in the relativistic mean-field approach with the point-coupling interaction, where the meson exchange in each channel is replaced by the contact interaction between nucleons. We investigate the influence of nuclear symmetry energy and its density dependence on pasta structures by introducing a coupling term between the isoscalar-vector and isovector-vector interactions. It is found that the properties of pasta phases in the neutron-rich matter are strongly dependent on the symmetry energy and its slope. In addition to typical shapes like droplets, rods, slabs, tubes, and bubbles, some intermediate pasta structures are also observed in cold stellar matter with a relatively large proton fraction. We find that nonspherical shapes are unlikely to be formed in neutron-star crusts, since the proton fraction obtained in $\beta$ equilibrium is rather small. The inner crust properties may lead to a visible difference in the neutron-star radius.

Tasting nuclear pasta made with classical molecular dynamics simulations

Nuclear clusters or voids in the inner crust of neutron stars were predicted to have various shapes collectively nicknamed nuclear pasta. The recent review in Ref. \cite{Lopez1} by L\'opez, Dorso and Frank summarized their systematic investigations into properties especially the morphological and thermodynamical phase transitions of the nuclear pasta within a Classical Molecular Dynamics model, providing further stimuli to find more observational evidences of the predicted nuclear pasta in neutron stars.

Impact of strong magnetic fields on the inner crust of neutron stars

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 ${10}^{18}$ G, the onset densities of pasta phases and crust-core transition density decrease significantly, and the density distributions of nucleons and electrons are also changed obviously.

From odd-even staggering to the pairing gap in neutron matter

The properties of neutron matter are integral to the correct description of neutron stars as well as the description of neutron-rich nuclei. One key property of neutron matter is its superfluid behavior in a range of densities relevant to the inner crust of neutron stars. This work investigates the finite-size effects in the pairing gap of a pure neutron matter superfluid system at densities found in the inner crust of cold neutron stars. The BCS (Bardeen-Cooper-Schrieffer) treatment of superfluidity gives rise to the mean-field pairing gap, while a projection after variation (PBCS theory) can provide a beyond-mean-field pairing gap through an odd-even staggering formula. While these two pairing gap results should agree in the thermodynamic limit, in this work we demonstrate that this is the case for systems far from the thermodynamic limit as well. These results can prove valuable to the study of neutron matter since they can connect ab initio approaches to other approaches found in the literature. This is a first step towards a model-independent extraction of the pairing gap in neutron matter.

Proximity effect of pair correlation in the inner crust of neutron stars

Abstract We study the proximity effect of pair correlation in the inner crust of neutron stars by means of the Skyrme–Hartree–Fock–Bogoliubov theory formulated in coordinate space. We describe a system composed of a nuclear cluster immersed in neutron superfluid confined in a spherical box. Using a density-dependent effective pairing interaction that reproduces both the pair gap of neutron matter obtained in ab initio calculations and that of finite nuclei, we analyze how the pair condensate in a neutron superfluid is affected by the presence of the nuclear cluster. It is found that the proximity effect is characterized by the coherence length of the neutron superfluid measured from the edge position of the nuclear cluster. The calculation predicts that the proximity effect has a strong density dependence. In the middle layers of the inner crust with baryon density $5 \times 10^{-4}$ fm$^{-3} \mathop < \limits_ \sim \ \rho_b \mathop < \limits_ \sim \ 2\times 10^{-2}$ fm$^{-3}$, the proximity effect is strongly limited in the vicinity of the nuclear cluster, i.e., in a sufficiently smaller area than the Wigner–Seitz cell. In contrast, the proximity effect is predicted to extend to the whole volume of the Wigner–Seitz cell in shallow layers of the inner crust with $\rho_b \mathop < \limits_ \sim \ 2 \times 10^{-4}$ fm$^{-3}$, and in deep layers with $\rho_b \mathop > \limits_ \sim \ 5 \times 10^{-2}$ fm$^{-3}$.

Neutron star cooling and GW170817 constraint within quark-meson coupling models * *This work is a part of the project INCT-FNA Proc. (464898/2014-5), was partially supported by CNPq (Brazil) (301155.2017-8)(D.P.M.), 310242/2017-7, 406958/2018-1 (O.L.), 308486/2015-3 (T.F.), 433369/2018-3 (M.D.), by Capes-PNPD program (C.V.F.), and by Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) under the thematic projects 2013/26258-4 (O.L, T.F.)

In the present work, we used five different versions of the quark-meson coupling (QMC) model to compute astrophysical quantities related to the GW170817 event and the neutron star cooling process. Two of the models are based on the original bag potential structure and three versions consider a harmonic oscillator potential to confine quarks. The bag-like models also incorporate the pasta phase used to describe the inner crust of neutron stars. With a simple method studied in the present work, we show that the pasta phase does not play a significant role. Moreover, the QMC model that satisfies the GW170817 constraints with the lowest slope of the symmetry energy exhibits a cooling profile compatible with observational data.

Nucleon localization function in rotating nuclei

Background: An electron localization function was originally introduced to visualize bond structures in molecules. It became a useful tool to describe electron configurations in atoms, molecules and solids. In nuclear physics, a nucleon localization function (NLF) has been used to characterize clusters in light nuclei, fragment formation in fission and pasta phases in the inner crust of neutron stars. Purpose: We use the NLF to study the nuclear response to fast rotation. Methods: We generalize the NLF to the case of nuclear rotation. The extended expressions involve both time-even and time-odd local densities. Since current density and density gradient contribute to the NLF primarily at the surface, we propose a simpler spatial measure given by the kinetic-energy density. Illustrative calculations for the superdeformed yrast band of $^{152}$Dy were carried out by using the cranked Skyrme-Hartree-Fock method. We also employed the cranked harmonic-oscillator model to gain insights into patterns revealed by the NLF at high angular momentum. Results: In a deformed rotating nucleus, several NLFs can be introduced, depending on the definition of the spin-quantization axis and self-consistent symmetries of the system. The oscillating pattern of the NLF can be explained by a constructive interference between the kinetic-energy and particle densities. The nodal pattern seen in the NLF along the major axis of a rotating nucleus comes from single-particle orbits with large aligned angular momentum. The variation of the NLF along the minor axis is traced back to deformation-aligned orbits. Conclusions: The NLF allows a simple interpretation of the shell structure evolution in the rotating nucleus in terms of the angular-momentum alignment of individual nucleons. We expect that the NLF will be useful for the characterization of other collective modes and time-dependent processes.