An approach to the equation of state for the inner crust of neutron stars based on Skyrme-type forces is presented. Working within the Wigner-Seitz picture, the energy is calculated by the TETF (temperature-dependent extended Thomas-Fermi) method, with proton shell corrections added self-consistently by the Strutinsky-integral method. Using a Skyrme force that has been fitted to both neutron matter and to essentially all the nuclear mass data, we find strong proton shell effects: proton numbers $Z$ = 50, 40 and 20 are the only values possible in the inner crust, assuming that nuclear equilibrium is maintained in the cooling neutron star right down to the ambient temperature.
Convergence problems with the TETF expansion for the entropy, and our way of handling them, are discussed. Full TETF expressions for the specific heat of inhomogeneous nuclear matter are presented. Our treatment of the electron gas, including its specific heat, is essentially exact, and is described in detail.
We analyse the temperature dependence of pairing correlations in the inner crust matter of neutron stars. The study is done in a finite-temperature HFB approach and by using a zero range pairing force adjusted to the pairing properties of infinite neutron matter. Within the same approach we investigate how the specific heat of the inner crust depends on temperature, matter inhomogeneity, and the assumption used for the pairing force. It is shown that in a physical relevant range of densities the pairing properties of inner crust matter depend significantly on temperature. The finite-temperatur…
The boundary between the solid crust, assumed to be in complete thermodynamic equilibrium (cold catalyzed matter), and the liquid core of a neutron star is studied using Skyrme SLy effective N-N interactions. An approximate value of the density at the inner edge of the crust is obtained from the threshold for the instability of homogeneous $npe$ matter with respect to small periodic density perturbations. Calculations of the bottom layer of the crust are performed within the Compressible Liquid Drop Model. Spherical nuclei are energetically preferred over exotic ones (cylinders, plates, tubes,…
We study, within a quantum mechanical framework based on self-consistent mean field theory, the interaction between a vortex and a nucleus immersed in a sea of free neutrons, a scenario representative of the inner crust of neutron stars. Quantal finite size effects force the vortex core outside the nucleus, influencing vortex pinning in an important way.
The structure of a vortex in the inner crust of neutron stars is calculated within the framework of quantum mean field theory taking into account the interaction with the nuclei composing the Coulomb lattice. Making use of the results obtained with different nuclear interactions, the pinning energy, relevant in the study of glitches, is worked out. Quantal size and density dependent effects are found to be important.
We investigate the nuclear collective excitations of Wigner-Seitz cells containing nuclear clusters immersed in a gas of neutrons. This baryonic non-uniform system is specific to the structure of inner crust matter of neutron stars. The collective excitations are studied in the framework of a spherical Hartree-Fock-Bogoliubov + Quasiparticle Random Phase Approximation, formulated in coordinate representation. The calculations are done for two representative Wigner-Seitz cells with baryonic density equal to 0.02 fm$^{-3}$ and 0.08 fm$^{-3}$. It is shown that the excitations with low multipolari…
We investigate the inner crust structure of neutron stars using the Skyrme-Hartree-Fock approach with the Coulomb interaction treated beyond the Wigner-Seitz approximation. Our results suggest that the shell effects associated with unbound neutrons play an important role and, in particular, lead to complicated phase transition pattern between various nuclear phases (as a function of the density). Namely, we show that the relative energies of different phases are rapidly oscillating functions of the neutron density. In the semiclassical approach this behavior is explained as an interference eff…
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.
In the inner crust of a neutron star, due to the high density and pressure, nuclei which are still present, are immersed in a neutron superfluid. One then expects that the dynamical properties of nuclei are significantly affected. In order to estimate the magnitude of the effect associated with the presence of a superfluid medium, we formulate the hydrodynamical approach to the nuclear dynamics in the inner crust of neutron stars. We calculate the renormalized nuclear mass and the strength of the medium-induced interaction between nuclei. We argue that these effects noticeably modify the prope…
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.
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…
The neutron-star inner crust is assumed to be superfluid at relevant temperatures. The contribution of neutron quasiparticles to thermodynamic and transport properties of the crust is therefore strongly suppressed by the pairing gap. Nevertheless, the neutron gas still has low-energy excitations, namely long-wavelength collective modes. We summarize different approaches to describe the collective modes in the crystalline phases of the inner crust and present an improved model for the description of the collective modes in the pasta phases within superfluid hydrodynamics.
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 investigate the pairing properties at finite temperature of the Wigner-Seitz cells in the inner crust of a neutron star obtained with the recent Brussels-Montreal Skyrme functional BSk21. In particular we analyze the phenomena of persistence and reentrance of pairing correlations and their impact on the specific heat in the low-density region of the inner crust.
We investigate the superfluid properties of the inner crust of neutron stars at finite temperature for different pairing functionals. We generalize the formalism adopted in previous article to include the effect of the temperature to calculate the specific heat of each given Wigner-Seitz cell. The calculations are done for two pairing forces, Gogny D1 and $V_{\text{low-k}}$, with finite range and a density dependent contact interaction. We compare in such a way the effect of the pairing strength and of the range on the thermal properties of the inner crust.
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 …
The inner crust of neutron stars formed of nuclear clusters immersed in a neutron sea has been widely studied in the framework of the Wigner-Seitz approximation since the seminal work of Negele and Vautherin. In this article, the validity of this approximation is discussed in the framework of the band theory of solids. For a typical cell of $^{200}$Zr, present in the external layers of the inner crust, it is shown that the ground state properties of the neutron gas are rather well reproduced by the Wigner-Seitz approximation, while its dynamical properties depend on the energy scale of the pro…
A fully self-consistent model of the neutron star inner crust based upon models of the nucleonic equation of state at zero temperature is constructed. The results nearly match those of previous calculations of the inner crust given the same input equation of state. The extent to which the uncertainties in the symmetry energy, the compressibility, and the equation of state of low-density neutron matter affect the composition of the crust are examined. The composition and pressure of the crust is sensitive to the description of low-density neutron matter and the nuclear symmetry energy, and the …
Superfluid properties of the inner crust matter of neutron stars, formed by nuclear clusters immersed in a dilute neutron gas, are analysed in a self- consistent HFB approach. The calculations are performed with two pairing forces, fixed so as to obtain in infinite nuclear matter the pairing gaps provided by the Gogny force or by induced interactions. It is shown that the nuclear clusters can either suppress or enhance the pairing correlations inside the inner crust matter, depending on the density of the surrounding neutrons. The profile of the pairing field in the inner crust is rather simil…
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…
We present calculations of the hydrodynamics of the inner crust of neutron stars, where a superfluid neutron liquid coexists with a lattice of neutron-rich nuclei. The long-wavelength collective oscillations are combinations of phonons in the lattice and phonons in the superfluid neutrons. Velocities of collective modes are calculated from information about effective nucleon-nucleon interactions derived from Lattimer and Swesty's microscopic calculations based on a compressible liquid drop picture of the atomic nuclei and the surrounding neutrons.
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 $10^{18…
We study the linear response of the inner crust of neutron stars within the Random Phase Approximation, employing a Skyrme-type interaction as effective interaction. We adopt the Wigner-Seitz approximation, and consider a single unit cell of the Coulomb lattice which constitutes the inner crust, with a nucleus at its center, surrounded by a sea of free neutrons. With the use of an appropriate operator, it is possible to analyze in detail the properties of the vibrations of the surface of the nucleus and their interaction with the modes of the sea of free neutrons, and to investigate the role o…
Within both dynamical and thermodynamical approaches using the equation of state for neutron-rich nuclear matter constrained by the recent isospin diffusion data from heavy-ion reactions in the same sub-saturation density range as the neutron star crust, the density and pressure at the inner edge separating the liquid core from the solid crust of neutron stars are determined to be 0.040 fm$^{-3}$ $\leq ρ_{t}\leq 0.065$ fm$^{-3}$ and 0.01 MeV/fm$^{3}$ $\leq P_{t}\leq 0.26$ MeV/fm$^{3}$, respectively. These together with the observed minimum crustal fraction of the total moment of inertia allow …
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 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 inner layers of a neutron star crust, composed of a Coulomb lattice of neutron rich nuclear clusters immersed in a sea of ``free'' superfluid neutrons, are closely analogous to periodic condensed matter systems such as electronic, photonic or phononic crystals. Applying methods from solid state physics to the neutron star context, we study the transport properties of those ``free'' neutrons for the outermost layers of the inner crust, near the drip point $ρ_{\rm drip} \sim 4\times 10^{11}$ g.cm$^{-3}$. In particular, we evaluate the effective neutron mass resulting from Bragg scattering by…
A semi-microscopic self-consistent quantum approach developed recently to describe the inner crust structure of neutron stars within the Wigner-Seitz method with the explicit inclusion of neutron and proton pairing correlations is used for finding the neutron drip point which separates the outer and inner crusts. The equilibrium configurations of the crust are examined in vicinity of the drip point and in the upper part of the inner crust, for the density region corresponding to average Fermi momenta $k_{\rm F}{=}0.2 ÷0.5 $fm$^{-1}$.
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.
Background: Low-lying collective excitations of inner crust matter in neutron stars are expected to have impact on observables such as the quasi-periodic oscillation in giant flares and the cooling of inner crust in transient phenomena. Coupling between Anderson-Bogoliubov superfluid phonon (AB phonon) in superfluid neutron gas and collective excitations of nuclear clusters is a key issue. Purpose: We intend to predict the nature of low-lying excitation modes of superfluid inner crust matter with focus on quadrupole excitations around a spherical nuclear cluster. We study how the AB phonon of …
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).
The mass, radius and crustal fraction of moment of inertia in neutron stars are calculated using $β$-equilibrated nuclear matter obtained from Skyrme effective interaction. The transition density, pressure and proton fraction at the inner edge separating the liquid core from the solid crust of the neutron stars are determined from the thermodynamic stability conditions using the KDE0v1 set. The neutron star masses obtained by solving the Tolman-Oppenheimer-Volkoff equations using neutron star matter obtained from this set is able to describe highly massive compact stars $\sim$2$M_{\odot}$. The…
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.
In this chapter we discuss some possible physical pictures that describe the constitution of the inner crust of compact objects. Different relativistic models both with constant couplings and density dependent ones are used. We calculate the liquid-gas phase transition in asymmetric nuclear matter from the thermodynamic and dynamic instabilities. The equations of state used to describe the crust are related to the crust-core transition properties. Cold and warm pasta phases with and without alpha particles are constructed. The influence of the pasta phase and its internal structure on the diff…
We study the effect of strong magnetic fields, of the order of $10^{15}-10^{17}$ G, on the extension of the crust of magnetized neutron stars. The dynamical instability region of neutron-proton-electron ($npe$) matter at subsaturation densities and the mode with the largest growth rate are determined within a relativistic mean field model. It is shown that a strong magnetic field has a large effect on the instability region, defining the crust-core transition as a wide density range.
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.
We analyse the effect of neutron superfluidity on the cooling time of inner crust matter in neutron stars, in the case of a rapid cooling of the core. The specific heat of the inner crust, which determines the thermal response of the crust, is calculated in the framework of HFB approach at finite temperature. The calculations are performed with two paring forces chosen to simulate the pairing properties of uniform neutron matter corresponding respectively to Gogny-BCS approximation and to many-body techniques including polarisation effects. Using a simple model for the heat transport across th…
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.
We compare two approaches to describe the inner crust of neutron stars: on the one hand, the simple coexistence of a liquid (clusters) and a gas phase, and on the other hand, the energy minimization with respect to the density profile, including Coulomb and surface effects. We find that the phase-coexistence model gives a reasonable description of the densities in the clusters and in the gas, but the precision is not high enough to obtain the correct proton fraction at low baryon densities. We also discuss the surface tension and neutron skin obtained within the energy minimization.
Even though the "free" neutrons in the inner crust of a neutron star are superfluid, they are still strongly coupled to nuclei due to non-dissipative entrainment effects. These effects have been systematically studied in all regions of the inner crust in the framework of the band theory of solids. Using concepts from solid-state physics, it is shown that the density of conduction neutrons, i.e. neutrons that are effectively "free", can be much smaller than the density of unbound neutrons (by an order of magnitude in some layers) due to Bragg scattering. These results suggest that a revision of…
In this paper, we present some relativistic mean-field inner crust equations of state that have recently been uploaded in the CompOSE online repository. These equations of state fulfill experimental and microscopic constraints, and are also able to reproduce two solar-mass stars. We integrate the TOV equations to obtain the mass-radius relation, and we also calculate the tidal deformability, compactness, and effective tidal deformability to compare with the latest astrophysical data from NICER and LIGO and Virgo.
The detailed knowledge of the inner crust properties of neutron stars might be important to explain different phenomena such as pulsar glitches or the possibility of an {\it r-process} site in neutron star mergers. It has been shown in the literature that quantal effects like shell correction or pairing may play a relevant role to determine the composition of the inner crust of the neutron star. In this paper we construct the equation of state of the inner crust using the finite-range Gogny interactions, where the mean field and the pairing field are calculated with same interaction. We have u…
The influence of the boundary conditions used in the Wigner-Seitz approximation applied to the neutron star inner crust is examined. The generalized energy functional method which includes neutron and proton pairing correlations is used. Predictions of two versions of the boundary conditions are compared with each other. The uncertainties in the equilibrium configuration (Z,R_c) of the crust, where Z is the proton charge and R_c the radius of the Wigner-Seitz cell, correspond to variation of Z by 2 -- 6 units and of R_c, by 1 -- 2 fm. The effect of the boundary conditions is enhanced at increa…
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 review the nuclear and condensed matter physics underlying the thermal and transport properties of the neutron star inner crust. These properties play a key role in interpreting transient phenomena such as thermal relaxation in accreting neutron stars, superbursts, and magnetar flares. We emphasize simplifications that occur at low temperature where the inner crust can be described in terms of electrons and collective excitations. The heat conductivity and heat capacity of the solid and superfluid phase of matter is discussed in detail and we emphasize its role in interpreting observations …
The neutron star maximum mass and the radius are investigated within the framework of the relativistic mean-field (RMF) model. The variation in the radius at the canonical mass, $R_{1.4}$, using different inner crust equation of state (EoS) with different symmetry energy slope parameter is studied. It is found that although the NS maximum mass and the corresponding radius do not vary much with different inner crust EoSs, the radius and the tidal deformability at 1.4$M_{\odot}$ vary with the different choice of crust EoS and variation of about 1-2 km is seen in the radius at the canonical mass.…
Monte Carlo simulations of neutron-rich matter of relevance to the inner neutron-star crust are performed for a system of A=5,000 nucleons. To determine the proton fraction in the inner crust, numerical simulations are carried out for a variety of densities and proton fractions. We conclude---as others have before us using different techniques---that the proton fraction in the inner stellar crust is very small. Given that the purported "nuclear pasta" phase in stellar crusts develops as a consequence of the long-range Coulomb interaction among protons, we question whether pasta formation is po…
We study proximity effect of pair correlation in the inner crust of neutron stars by means of the Skyrme-Hartree-Fock-Bogoliubov theory formulated in the coordinate space. We describe a system composed of a nuclear cluster immersed in neutron superfluid, which is confined in a spherical box. Using a density-dependent effective pairing interaction which 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 neutron superfluid is affected by the presence of the nuclear cluster. It is found that the proxim…
Superfluidity in the crust is a key ingredient for the cooling properties of proto-neutron stars. Present theoretical calculations employ the quasi-particle mean-field Hartree-Fock-Bogoliubov theory with temperature dependent occupation numbers for the quasi-particle states. Finite temperature stellar matter is characterized by a whole distribution of different nuclear species. We want to assess the importance of this distribution on the calculation of heat capacity in the inner crust. Following a recent work, the Wigner-Seitz cell is mapped into a model with cluster degrees of freedom. The fi…
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 symmetry energy effects on the location of the inner edge of neutron star crusts are studied. Three phenomenological models are employed in order to check the accuracy of the well known parabolic approximation of the equation of state for asymmetric nuclear matter in the determination of the transition density $n_t$ and transition pressure $P_t$. The results corroborate the statement that the error due to the assumption that a priori the equation of state is parabolic may introduce a large error in the determination of related properties of a neutron star as the crustal fraction of the mom…
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…
The Vlasov equation is used to determine the dispersion relation for the eigenmodes of magnetized nuclear and neutral stellar matter, taking into account the anomalous magnetic moment of nucleons. The formalism is applied to the determination of the dynamical spinodal section, a quantity that gives a good estimation of the crust-core transition in neutron stars. We study the effect of strong magnetic fields, of the order of $10^{15}-10^{17}$ G, on the extension of the crust of magnetized neutron stars. The dynamical instability region of neutron-proton-electron ($npe$) matter at subsaturation …
Context. In the cooling process of a non-accreting neutron star, the composition and properties of the crust are thought to be fixed at the finite temperature where nuclear reactions fall out of equilibrium. A lower estimation for this temperature is given by the crystallization temperature, which can be as high as $\approx 7\times 10^9$ K in the inner crust, potentially leading to sizeable differences with respect to the simplifying cold-catalyzed matter hypothesis. Aims. We extend the recent work by Fantina et al. (2019) on the outer crust, to the study of the crystallization of the inner cr…
Despite the absence of viscous drag, the neutron superfluid permeating the inner crust of a neutron star can still be strongly coupled to nuclei due to non-dissipative entrainment effects. Neutron superfluidity and entrainment have been systematically studied in all regions of the inner crust of a cold non-accreting neutron star in the framework of the band theory of solids. It is shown that in the intermediate layers of the inner crust a large fraction of "free" neutrons are actually entrained by the crust. The results suggest that a revision of the interpretation of many observable astrophys…
A semi-microscopic self-consistent quantum approach developed recently to describe the inner crust structure of neutron stars within the Wigner-Seitz (WS) method with the explicit inclusion of neutron and proton pairing correlations is further developed. In this approach, the generalized energy functional is used which contains the anomalous term describing the pairing. It is constructed by matching the realistic phenomenological functional by Fayans et al. for describing the nuclear-type cluster in the center of the WS cell with the one calculated microscopically for neutron matter. Previousl…
