Superfluidity in neutron matter

The question of superfluidity in neutron matter is investigated in the framework of BCS-Bogoljubov-Theory. Solving the gap-equation for a semirealistic hard-core and a soft-core potential, a rapidly converging numerical method is developed. The results are applied to neutron star models.     

Pion condensation in neutron star matter: (II). Nuclear forces and stability

This paper presents several stable models of charged-pion condensed neutron star matter. The non-relativistic limit of the chirally symmetric Weinberg Lagrangian is used to describe interactions of the condensed pion field with the nucleons, as well as the pi-pi interactions of the condensed field. In the absence of nucleon-nucleon interactions, matter in this model is unstable, tending to ever-increasing baryon density and condensate wave vector. The connection of this model of condensation with the σ-model is shown.A general framework for including nuclear forces is then laid out. Results are given for a simple model in which the nuclear forces are assumed to produce an interaction energy V(ρ) dependent only on the total baryon density, independent of the degree of pion condensation, and also to produce a constant G-matrix element g in the nucleon-nucleon charge exchange channel. In the absence of condensation the equation of state reduces to that of interacting normal matter. We also consider effects of beta equilibrium and form factors in the p-wave pion-nucleon interaction. The condensed models are stable. Depending on the choice of parameters the models exhibit first- or second-order pion condensation phase transitions, or both.     

Triplet cooper pairing in nuclear matter and rotational states of superdeformed nuclei

One hundred and sixty-one rotational bands of superdeformed states in nuclei are considered on the basis of a model that admits triplet Cooper pairing in superfluid nuclear matter. The behavior of the dynamical moment of inertia for such states is investigated within this model, which is shown to comply well with available experimental data and to describe successfully the rotational spectra of superdeformed states.     

Superfluidity in neutron matter and symmetric nuclear matter and the effect of a microscopic three-body force

The neutron ³PF₂ pairing gap in pure neutron matter, neutron ³PF₂ gap and neutron-proton ³SD₁ gap in symmetric nuclear matter have been studied by using the Brueckner-Hartree-Fock(BHF) approach and the BCS theory. We have concentrated on investigating and discussing the three-body force effect on the nucleon superfluidity. The calculated results indicate that the three-body force enhances remarkably the ³PF₂ superfluidity in neutron matter. It also enhances the ³PF₂ superfluidity in symmetric nuclear matter and its effect increases monotonically as the Fermi-momentum k_F increases, whereas the three-body force is shown to influence only weakly the neutron-proton ³SD₁ gap in symmetric nuclear matter.     

New method in bardeen-cooper-schrieffer theory: Tiplet pairing in superfluid dense neutron matter

On the basis of the method outlined in the first part of this review, the properties of superfluid dense neutron matter are analyzed in the density region where the spin of a Cooper pair and its total angular momentum are S=1 and J=2, respectively. An analytic solution to the problem of ³ P ₂ pairing in neutron matter is presented. Basic features of the structure and of the energy spectrum of superfluid phases are discussed. Degeneracy that is absolutely dissimilar to that which is associated with the phase transformation of the order parameter in the S-pairing problem is a distinct feature of the structure of the aforementioned phases. It appears that one or even a few numbers characterizing the weight of components associated with different values of the projection M of the total angular momentum J=2 of a Cooper pair can be chosen arbitrarily, while the others adjust to them in accordance with universal laws. As a result, the structure of any phase depends neither on the density, nor on the temperature, nor on any other input parameter. The phases found here form two groups degenerate in energy. One of these groups comprises phases for which the sign of the order parameter remains unchanged over the entire Fermi surface, while the other consists of phases whose order parameter has a zero. The energy splitting between the phases from the different groups is calculated analytically as a function of temperature. The relative magnitude of this splitting changes from approximately 3% at T=0 to zero in the vicinity of the critical point T _c. The role of tensor forces in dense neutron matter is analyzed. It is shown that the mixing of the orbital angular momenta L=1 and L=3 of Cooper pairs that is induced by tensor forces completely removes degeneracy peculiar to the ³ P ₂-pairing problem—the number of phases and their structure at a given temperature are tightly fixed, while the energy spectrum of the phases splits completely.     

Two quasiparticle scattering and pairing in neutron matter with Skyrme interactions

Pairing matrix elements in neutron matter are computed employing Skyrme interactions under different approaches. We compare the pairing strengths calculated from the exact scattering amplitudes to those obtained as one neglects the integral kernel of the Bethe-Salpeter equation, and to the matrix element of the bare particle-particle interaction with and without rearrangement contribution. The effect upon the gap is analyzed in the frame of the simple weak coupling approximation, in order to indicate possible outcomes of more refined treatments of the pairing problem.     

Superfluidity in the inner crust of neutron stars

We present a mean-field quantum calculation of the specific heat in the inner crust of neutron stars, taking into account the inhomogeneous character of the system, in which a lattice of neutron-rich nuclei coexists with a gas of unbound neutrons.Received: 10 December 2002, Published online: 24 February 2004PACS: 21.60.-n Nuclear structure models and methods - 26.60. + c Nuclear matter aspects of neutron stars - 97.60.Jd Neutron stars     

OBEP and eikonal form factor: (II). Nuclear matter results

Nuclear matter properties are calculated in a first-order Brueckner-Bethe calculation using a one-boson exchange potential recently proposed by the authors, in which the phenomenological cutoff of dipole type used so far has been replaced by a form factor obtained from an eikonal approximation to multiple vector meson exchange processes. We find ?23.5 MeV saturation energy at a Fermi momentum k_F = 1.77 fm^?1, i.e. about 12 MeV more binding than realistic OBEP using dipole-type cutoffs and about 8 MeV overbinding compared to the empirical value of 16 MeV. On the other hand, estimates suggest that, compared to the Reid soft-core potential, this new OBEP predicts about 1.5 MeV more binding in the case of the triton and about 4 MeV more binding in the case of ¹⁶O, i.e. gives nearly the correct empirical result. The additional binding is traced back to the small deuteron D-state probability of 4.32% predicted by this OBEP, which is a consequence of the special structure of the eikonal form factor. However, taking the effect of the Δ-resonance into account recently given by Green and Niskanen, one arrives at ?14 MeV saturation energy for nuclear matter at k_F = 1.36 fm^?1, whereas the results for the triton and ¹⁶O are changed to a negligible extent only.     

Perturbation theory for projected states in the pairing force model: (II). The problem of convergence

In a previous paper second-order calculations were carried out on the two-level pairing force model using perturbation theory for projected states and the results were compared with those of BCS or ordinary perturbation theory. In the present paper criteria for convergence are applied to two different forms of the perturbation series for both the projected and ordinary perturbation theories. It is found that the superior results obtained for the convergence rates in the projected theory tally neatly with the results of the second-order calculations, and give further support to the use of the projected perturbation theory.     

Some nuclear matter calculations: (II). Insertions in hole and particle lines

Brueckner-Goldstone diagrams that represent insertions in the particle and hole lines of the basic two-body ladder diagram are calculated with the Reid soft-core potential. The bubble insertions in particle lines are calculated exactly, off the energy shell, for small excitations (≦ 2k_F), and contribute 1.5 MeV to the binding energy of nuclear matter. Comparisons are made with the approximations used by Brueckner and Gammel (BG) and by Bethe, Brandow and Petschek (BBP). The BG method is found to be fairly accurate for the treatment of small excitations. The BBP method, used in Dahlblom's three-body work, underestimates the binding by 0.7 MeV. The third-order insertion in hole lines contributes 1.8 MeV to the binding energy, when the reaction matrix is calculated with the bubble insertions in particle lines. Without these bubble insertions the contribution is 1.0 MeV. The difference is due to the slower healing when bubble insertions in particle lines are included as the gap between hole and particle energies then decreases. Inclusion of other higher-order diagrams reduces the contribution from the third-order insertion from 1.8 to 1.2 MeV. Day estimated this contribution to be 0.6 MeV. Bethe, using Dahlblom's and Day's results concludes that the Reid soft core gives 15.4 MeV binding. Our calculations suggest that this could be increased to 16.7–17.3 MeV.     

Nuclear matter approach to the heavy-ion optical potential: (II). Imaginary part

The heavy-ion optical potentials are constructed in a nuclear matter approach, for the ¹⁶O + ¹⁶O, ⁴⁰Ca + ¹⁶O and ⁴⁰Ca + ⁴⁰Ca elastic scattering at the incident energies per nucleon E^lab/A ? 45 MeV. The energy density formalism is employed assuming that the complex energy density of colliding heavy ions is a functional of the nucleon density ?(r), the intrinsic kinetic energy density τ⁽²⁾(r) and the average momentum of relative motion per nucleon K_r(≦ 1.5 fm^?1). The complex energy density is numerically evaluated for the two units of colliding nuclear matter with the same values of ρ, τ⁽²⁾ and K_r. The Bethe-Goldstone equation is solved for the corresponding Fermi distribution in momentum space using the Reid soft-core interaction. The “self-consistent” single-particle potential for unoccupied states which is continuous at the Fermi surface plays a crucial role to produce the imaginary part. It is found that the calculated optical potentials become more attractive and absorptive with increasing incident energy. The elastic scattering and the reaction cross sections are in fair agreement with the experimental data.     

Properties of quark matter governed by quantum chromodynamics (II). Renormalization schemes in quark matter

Renormalization schemes are examined (in the Coulomb gauge) for quantum chromodynamics in the presence of quark matter. We demand that the effective coupling constant for all schemes become congruent with the vacuum QCD running coupling constant as the matter chemical potential, μ, goes to zero. Also, to enable us to standardize with the vacuum QCD running coupling constant at some asymptotic momentum transfer, $\text{|}\text{p}{}_0\text{|,}\text{we keep}\text{μ ? ¦}\text{p}{}_0\text{¦}$, to ensure that the matter contribution is negligible at this point. This means all schemes merge with vacuum QCD at $\text{|}\text{p}{}_0\text{|}$ and beyond. Two renormalization group invariants are shown to emerge: (i) the effective or invariant charge, g_inv², which is, however, scheme dependent and (ii) g²(M)/S(M), where S(M)^?1 is the Coulomb propagator, which is scheme independent. The only scheme in which g_inv² is scheme independent and identical to g²(M)/S(M) is the screened charged scheme (previous paper) characterised by the normalization of the entire Green function, S^?1, to unity. We conclude that this is the scheme to be used if one wants to identify with the experimental effective coupling in perturbation theory. However, if we do not restrict to perturbation theory all schemes should be allowed. Although we discuss matter QCD in the Coulomb gauge, the above considerations are quite general to gauge theories in the presence of matter.     

Equation of state of superfluid neutron matter and the calculation of the 1S0 pairing gap

We present a quantum Monte Carlo study of the zero-temperature equation of state of neutron matter and the computation of the 1S0 pairing gap in the low-density regime with rho < 0.04 fm(-3). The system is described by a nonrelativistic nuclear Hamiltonian including both two- and three-nucleon interactions of the Argonne and Urbana type. This model interaction provides very accurate results in the calculation of the binding energy of light nuclei. A suppression of the gap with respect to the pure BCS theory is found, but sensibly weaker than in other works that attempt to include polarization effects in an approximate way.     

Effects of pairing correlation on neutron drop

In this work, the effects of the pairing correlation on the properties of neutron drops N=6-50 trapped in a harmonic oscillator potential with ω = 10 Me V are investigated by comparing the results given by the Skyrme Hartree-Fock and Hartree-FockBogoliubov theories with the density-dependent delta interaction(DDDI) pairing force. The results showed that the pairing correlation slightly made the neutron drops more bound, and increased the central neutron density, the spin-orbit and pseudo spin-orbit splittings. Thus, the pairing correlation must be accounted for to improve the Skyrme functional compared with the ab initio calculations. Furthermore, although the single-particle energy gaps with or without pairing were similar, the shell closures varied due to pair scattering. Here, the shell closures in neutron drops using the Sk M* parameter set and DDDI pairing force were found at N=8, 16, and 32.     

Superfluidity of a perfect quantum crystal II

This paper continues the work begun in a previous paper [Eur. Phys. J. B 71, 85 (2009)]. To treat the equations that describe a crystal with condensate that can be superfluid, a method termed the Kirkwood approximation is used. Earlier, the method was found to be rather seminal when applied to a classical crystal. In the case of a simple cubic lattice, solutions to the equations under study can be expressed in terms of the well-known Mathieu functions. A more realistic case of the face centered cubic lattice is also considered although in this case the three-dimensional equations cannot be reduced to one-dimensional ones. Condensate crystals without superfluidity are studied first and then the same crystals in a superfluid state. It is shown in particular that a crystal in which the condensate is formed is energetically preferable with respect to the same quantum crystal without condensate at absolute zero of temperature. Therefore, on lowering the temperature there must somewhere occur Bose-Einstein condensation in the crystal. In the concluding section, we discuss various physical aspects of the problem.     

Triplet superfluidity in neutron matter with Skyrme forces at subnuclear and supranuclear densities in a strong magnetic field

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 |μ _n |H ≪ E _c < ɛ _F (n), where μ _n 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.