Effect of superstrong magnetic fields on phase transition temperatures of neutron matter at supranuclear densities with unconventional Skyrme forces to superfluid states with anisotropic spin-triplet p -wave pairing

The nonlinear integral equations for the components of the order parameter of dense superfluid neutron matter (SNM) with spin-triplet anisotropic p-wave pairing (similar to pairing in 3He−A2 in magnetic fields, i.e., with spin S = 1 and orbital moment L = 1 of anisotropic Cooper pairs of neutrons) in superstrong magnetic fields (exceeding the 1017 G) are solved analytically in the limit of zero temperature. These solutions are derived for the family of so-called BSk-type generalized parameterizations of the effective Skyrme forces (with three terms dependent on density n) in neutron matter. The obtained general solutions for splitting of energy gap in SNM in superstrong magnetic fields are specified for the generalized BSk21 parameterization of the effective Skyrme forces at nuclear density n0 = 0.17 fm–3 and at two supranuclear densities n = 1.25n0 and n = 1.5n0 for magnetic fields H = Z⋅1017 G, where 1≤ Z ≤ 10. The main results are the splitting of energy gap and its asymmetry which increase nonlinearly with growing both superstrong magnetic field H and supranuclear density n > n0. Such effects in SNM might exist in liquid outer core (at densities n>∼⁡n0) in strongly magnetized neutron stars known as “magnetars”.

Magnetic properties of a dense superfluid neutron matter (relevant to the physics in cores of magnetars, namely the strongly magnetized neutron stars) at supranuclear densities n > n0 (where n0 = 0.17 fm–3 is the saturation nuclear density) with generalized Skyrme effective forces (with three density-dependent terms) and with spin-triplet anisotropic p-wave pairing (similar to 3He-A in magnetic fields, i.e. with spin S = 1 and orbital moment L = 1 of anisotropic Cooper pairs of neutrons) in the presence of a superstrong magnetic field (exceeding the 1017 G) are studied within the framework of the non-relativistic generalized Fermi-liquid theory at zero temperature. The upper limit for the density range of a neutron matter is restricted by the magnitude 3n0 in order to avoid the account of relativistic corrections growing with density. The approximate general formula (valid for any parameterization of the Skyrme forces) is derived here analytically for the magnetic susceptibility (which contains additional correction depending nonlinearly on superstrong magnetic field H and on the density n) of a superfluid neutron matter in the limit of zero temperature. The obtained general formula for magnetic susceptibility is specified for the generalized BSk21 parameterization of the Skyrme forces and figures for corresponding values are plotted on the interval 1.5n0 ≤ n ≤ 3n0 and for superstrong magnetic fields 2⋅1017 G ≤ H ≤ 2⋅1018 G. It is established that the high-density ferromagnetic instability is removed in neutron matter with the generalized Skyrme forces (in particular, with the generalized BSk21 parameterization) not only in normal, but also in superfluid neutron matter with spin-triplet anisotropic p-wave pairing at supranuclear densities and in superstrong magnetic fields.

The previously derived nonlinear integral equations for the components of the order parameter (OP) of dense superfluid neutron matter (SNM) with anisotropic spin-triplet p-wave pairing (similar to 3He-A), taking into account the effects of magnetic field and finite temperatures, are reduced here to the equations for the two components of OP at the limit of zero temperature. In this article, these equations (which are valid for arbitrary parametrization of the effective Skyrme interaction in neutron matter) are specified and solved numerically for the generalized BSk21 parametrization of the effective Skyrme forces (with additional terms dependent on density n) in neutron matter. The primary result is the splitting of the energy gap, calculated in the energy spectrum of neutrons in SNM (nonlinearly increasing under a moderately strong magnetic field H), which has a nonlinear dependency on density n in the limiting case of zero temperature. A small asymmetry (nonlinearly increasing with magnetic field) of the energy gap splitting has also been obtained in the range of moderately strong magnetic fields 1016 G ≤ H ≤ 1017 G. Neutron matter phase transitions to superfluid states of such a type and magnetic field strength might occur (and exist) at subnuclear and supranuclear densities, as in the liquid outer core of magnetars (strongly magnetized neutron stars).

Within a generalized non-relativistic Fermi-liquid approach we have found general analytical formulae for phase-transition temperatures T c,1(n, H) and T c,2(n, H) (which are nonlinear functions of density, n, and linear of magnetic field, H) for phase transitions in spatially uniform, dense, pure neutron matter from normal to superfluid states with spin-triplet p-wave pairing (similar to anisotropic superfluid phases 3He - A1 and 3He - A2) in steady and homogeneous sufficiently strong magnetic field (but |µn|H ≪ E c < ɛ F(n), where µn is the magnetic dipole moment of a neutron, E c is the cutoff energy and ɛ F(n)is the Fermi energy in neutron matter). General formulae for T c,1,2(n,H) are valid for arbitrary parameterization of the effective Skyrme forces in neutron matter. We have used for definiteness the so-called SLy2, Gs and RATP parameterizations of the Skyrme forces with different exponents in their power dependence on density n (at sub- and supranuclear densities) from the interval 0.7 n 0 ≲ n < n c(Skyrme)< 2 n 0, where n 0 =0.17 fm−3 is the nuclear density and n c(Skyrme)is the the critical density of the ferromagnetic instability in superfluid neutron matter. These phase transitions might exist in the liquid outer core of magnetized neutron stars.

Magnetic properties of a dense superfluid neutron matter (relevant to the physics of neutron star cores) at subnuclear and supranuclear densities (in the range 0.5 < n=n0 < 3.0, where n0 = 0.17 (fm^-3) is the saturation nuclear density) with the so-called generalized Skyrme effective forces BSk18, BSk19, BSk20, BSk21 (containing additional unconventional density-dependent terms) and with spin-triplet p-wave pairing (with spin S = 1 and orbital moment L = 1) in the presence of a strong magnetic field are studied within the framework of the non-relativistic generalized Fermi-liquid theory at zero temperature. The upper limit for the density range of a neutron matter is restricted by the magnitude 3n0 in order to avoid the account of relativistic corrections growing with density. The general formula obtained in [1] (valid for any parametrization of the Skyrme forces) for the magnetic susceptibility of a superfluid neutron matter at zero temperature is specified here for the new BSk18-BSk21 parametrizations of the Skyrme interaction. As is known, all previous conventional Skyrme interactions predict spin instabilities in a normal (nonsuperfluid) neutron matter beyond the saturation nuclear density. It is obtained in the present work that, for the model of superfluid neutron matter with new generalized BSk18-BSk21 parametrizations, such phase transition to the ferromagnetic state occurs neither at subnuclear nor at supranuclear densities. Thus, the high-density ferromagnetic instability is removed in the neutron matter with new generalized Skyrme forces BSk18-BSk21 not only in normal, but also in superfluid states with anisotropic spin-triplet pairing.

Analytical expressions for the specific heat of superfluid Fermi liquids (SFLs) with anisotropic spin-triplet p-wave pairing of the 3He-A type have been obtained based on a generalised Fermi-liquid approach both at the low temperatures of 0 < T ≪Tc0 (n) and near the temperature of the phase transition [Tc0 (n)] from a normal to a superfluid phase in the absence of a magnetic field. Apart from liquid 3He-A, we also study dense superfluid neutron matter (SNM) with anisotropic spin-triplet p-wave pairing (similar to 3He-A) at subnuclear (n < n0, where n0 = 0.17 fm–3 is the nuclear density) and supernuclear densities (n > n0) in view of generalised Skyrme forces (with additional terms depending on the density n). At 0 < T ≪Tc0 (n), we have obtained an asymptotic expansion for the specific heat CSNM(SFL)(T,n), in which, apart from the main term ∼ T 3 (known for 3He-A in the limit of T → 0) there is an additional correction term (∼ T5), which may reach several percent of the contribution from the main term to the decomposition of the specific heat of SNM (or SFLs). An analytical formula has also been derived for calculating the specific heat CSNM(SFL)(T,n) at temperatures near Tc0(n). The expressions obtained for the CSNM(T,n) functions (valid for an arbitrary parametrisation of the effective Skyrme interaction in neutron matter) have been defined for SNM with the generalised BSk21 parametrisation of Skyrme forces. In addition, dependency graphs were plotted for the specific heat CBSk21(t, y) in the reduced temperature range of 0 < t ≡ T/Tc0(n) ≪ 1 and at t ≲ 1 for dense SNM (at 0.1 ≤ y ≡ n/n0 ≤ 1.7). These results may be of interest for neutron star physics in connection with neutron star cooling (in the presence of neutron superfluidity with anisotropic spin-triplet p-wave Cooper pairing in the outer part of dense liquid neutron star cores).

A generalized non-relativistic Fermi-liquid approach was used to find analytical formulas for temperatures Tc1(n, H) and Tc2(n, H) (which are functions nonlinear of density n and linear of magnetic field H) of phase transitions in spatially uniform dense pure neutron matter from normal to superfluid states with spin-triplet p-wave pairing (similar to anisotropic superfluid phases 3He-A1 and 3He-A2) in steady and homogeneous strong magnetic field (but |μn| H ≪ Ec < ∊F(n), where μn is the magnetic dipole moment of a neutron, Ec is the cutoff energy and ∊F(n) is the Fermi energy in neutron matter). General formulas for Tc1, 2 (n, H) (valid for arbitrary parameterization of the effective Skyrme interaction in neutron matter) are specified here for generalized BSk18 parameterization of the Skyrme forces (with additional terms dependent on density n) on the interval 0.3 n0 < n < nc (BSk18) ≍ 2.7952 · n0, where n0 = 0.17 fm−3 is nuclear density and at critical density nc(BSk18) triplet superfluidity disappears, Tc0(n, cH = 0) = 0. Expressions for phase transition temperatures Tc0(n)<0.09MeV (at Ec = 10MeV) and Tc1, 2(n, H) are realistic non-monotone functions of density n for BSk18 parameterization of the Skyrme forces (contrary to their monotone increase for all previous BSk parameterizations). Phase transitions to superfluid states of such type might occur in liquid outer core of magnetars (strongly magnetized neutron stars).

In the framework of the generalized non-relativistic Fermi-liquid approach we study phase transitions in spatially uniform dense pure neutron matter from normal to superfluid states with a spin-triplet p-wave pairing (similar to anisotropic superfluid phases 3He-A1 and 3He-A2) in a steady and homogeneous strong magnetic field H (but , where is the magnetic dipole moment of a neutron, is the cutoff energy and is the Fermi energy in neutron matter with density of particles n). The previously derived general formulas (valid for the arbitrary parametrization of the effective Skyrme interaction in neutron matter) for phase transition (PT) temperatures (which are nonlinear functions of the density n and linear functions of the magnetic field H) are specified here for new generalized BSk20 and BSk21 parameterizations of the Skyrme forces (with additional terms dependent on the density n) in the interval , where is the nuclear density. Our main results are mathematical expressions and figures for PT temperatures in the absence of magnetic field, and (at ), and in strong magnetic fields (which may approach to or even more as in the liquid outer core of magnetars —strongly magnetized neutron stars). These are realistic non-monotone functions with a bell-shaped density profile.

The previously derived equations for the components of the order parameter (OP) of dense superfluid neutron matter (SNM) with anisotropic spin-triplet p-wave pairing and with taking into account the effects of magnetic field and finite temperatures are reduced to the single equation for the one-component OP in the limit of zero magnetic field. Here this equation is solved analytically for arbitrary parametrization of the effective Skyrme interaction in neutron matter and as the main results the energy gap (in the energy spectrum of neutrons in SNM) is obtained as nonlinear function of temperature T and density n in two limiting cases: for low temperatures near T = 0 and in the vicinity of phase transition temperature Tc0(n) for dense neutron matter from normal to superfluid state. These solutions for the energy gap are specified for generalized BSk21 and BSk24 parametrizations of the Skyrme forces (with additional terms dependent on density n) and figures are plotted on the interval 0.1n0 &amp;lt; n &amp;lt; 2.0n0, where n0 = 0.17 fm−3 is nuclear density.

The influences of electron screening (ES) and electron energy correction (EEC) are investigated by superstrong magnetic field (SMF). We also discuss in detail the discrepant factor between our results and those of Fushiki, Gudmundsson and Pethick (FGP) in SMF. The results show that SMF has only a slight effect on ES when B < 109 T on the surfaces of most neutron stars. Whereas for some magnetars, SMF influence ES greatly when B > 109 T. For instance, due to SMF the ES potential may be increased about 23.6% and the EEC may be increased about 4 orders of magnitude at ρ/μe = 1.0 × 106 mol/cm3 and T9 = 1. On the other hand, the discrepant factor shows that our results are in good agreement with FGP's when B < 109 T. But the difference will be increased with increasing SMF.

Based on the theory of the relativistic mean-field effective interactions, we explore the effect of superstrong magnetic fields (SMFs) on nuclear single-particle level structure, nuclear blinding energy, and the electron Fermi energy in strongly magnetized super-Chandrasekhar white dwarfs (SMSWDs). We also discuss the electron capture (EC) instability from the threshold mass density (TMD) and threshold pressure (TP) for some typical iron group nuclei in SMSWD due to SMFs. At relatively low densities (e.g., ρ 7 = 57, 97, 197), the EC rates increase and they then decrease by more than four orders of magnitude. However, at relatively higher density (e.g., ρ 7 = 4197), the rates increase by about two orders of magnitude when 2 × 103 < B 12 < 5 × 104, and then the SMFs have a minor effect on EC rates. The TMD (TP) can increase by about 23% ∼ 29%, due to an increase in the electron–ion interaction and nuclear binding energy in SMFs. As SMFs further increase, the TMD (TP) increase almost linearly for one Landau levels system. For a two and three Landau level system, the SMFs have a slight influence on the TMD (TP) (e.g., at B 12 = 103.5), and then TMD (TP) increases by about 13% ∼ 14%, and finally decreases by about 27.85% (27.34%). The shift of the onset of EC can make the maximum mass of the SMSWD increase by more than 2.28% for a one Landau level system. This influence from TMD (TP) can be larger than that of general relativity for SMSWD, which contains some iron group nuclei.

A dense superfluid pure neutron matter with an effective Skyrme interaction, which depends on the density n of the neutrons with spin‐triplet p‐wave pairing similar to those of 3He‐A1 and 3He‐A, is studied in a strong uniform static magnetic field H in the framework of a generalized non‐relativistic Fermi‐liquid theory. We present an analytic solution for a set of previously derived nonlinear integral equations for the components of the order parameter and an effective magnetic field Heff. The obtained general formulas (valid for arbitrary parameterization of Skyrme forces) for the phase transition temperatures Tc1,2 of the neutron matter (from normal to a superfluid state of 3He‐A1 type and then to a 3He‐A type state, respectively) and the expression for Heff at T=0 are linear functions of strong magnetic field H and nonlinear functions of n. The gap equation is also solved at T=0.

In the framework of the model of a degenerate relativistic ideal neutron–proton–electron gas (np+e− gas) in an external superstrong constant and homogeneous magnetic field, we study the effect of the magnetic field on the state of chemical equilibrium of the np+e− gas and on the processes of electronic (β−) and positronic (β+) nucleon decay taking the effects due to the interaction between the nucleon anomalous magnetic moments and the magnetic field into account. For sufficiently large values of the magnetic induction, the proton density in chemical equilibrium must exceed the neutron density. Including the interaction between the nucleon anomalous magnetic moments Mn,p and the magnetic field results in an insignificant reduction of the proton density, but, as in the case Mn,p = 0, the proton density in chemical equilibrium in the presence of the superstrong magnetic field exceeds the neutron density. We show that if the interaction between the nucleon anomalous magnetic moments and the superstrong magnetic field is taken into account, then the positronic decay of a free proton (i.e., a proton not entering the composition of an atomic nucleus) into a neutron, a positron, and a neutrino can become energetically allowed. We discuss the necessary conditions for realizing the phase transition from the nucleon phase to the quark phase of the substance in the central region of a strongly magnetized neutron star.

Effects of medium polarization are studied for $^1S_0$ pairing in neutron and nuclear matter. The screening potential is calculated in the RPA limit, suitably renormalized to cure the low density mechanical instability of nuclear matter. The selfenergy corrections are consistently included resulting in a strong depletion of the Fermi surface. All medium effects are calculated based on the Brueckner theory. The $^1S_0$ gap is determined from the generalized gap equation. The selfenergy corrections always lead to a quenching of the gap, which is enhanced by the screening effect of the pairing potential in neutron matter, whereas it is almost completely compensated by the antiscreening effect in nuclear matter.

We investigated the neutrino energy loss by electron capture of the nuclide 56Fe, 56Co, 56Ni, 56Mn and 56Cr in superstrong magnetic field at the crusts of neutron stars. The results showed that the superstrong magnetic field has only a slight effect on the neutrino energy loss rates when B13 G on surfaces of most neutron stars. Whereas for some magnetars, the range of the magnetic field is 1013—1015 G, the neutrino energy loss rates would be lowered greatly and may be even decreased more than five orders of magnitude by superstrong magnetic field.