Boundary Conditions of Wigner-Seitz Cell in Inner Crust of Neutron Stars with Relativistic Mean Field Approach

The microscopic structure of the Wigner-Seitz （W-S） cell in the inner crust of neutron stars is investigated with the relativistic mean field （RMF） approach. The W-S cell is composed of a cluster of neutrons and protons localized in a region around the centre and surrounded by a neutron gas of approximately uniform density. In order to generate the density of the W-S cell, appropriate boundary conditions in the calculation of the single-particle wavefunctions are necessary. We emphasize on the choice of the boundary conditions in the RMF approach. Three kinds of boundary conditions are suggested. The properties of the W-S cell with the three kinds of boundary conditions are investigated. The neutron density distributions in the RMF and Hartree-Fock-Bogoliubov （HFB） models are compared. It is found that the neutron gas densities of the W-S cell in the RMF model is higher than those obtained in the HFB model.     

Hartree-Fock and Hartree-Bogolyubov calculations inp shell nuclei

Hartree-Fock (HF) and Hartree-Fock-Bogolyubov (HFB) calculations have been performed for the 1p shell nuclei. Nuclear deformations are assumed to be at most axially symmetric. The HFB transformation is restricted to allow forp-p andn-n pairing only.Volkov's force, a soft-core, two-body interaction of semi-realistic nature, is used which does not produce any single-particle spin-orbit splittings. Coulomb force and the usual correction for centre-of-mass motion are taken into account. The calculations are carried out in a single-particle basis including all states up to principle oscillator quantum numberN=3 (in some cases,N=4). Binding energies, rms radii, density distributions, and quadrupole moments are calculated and found to be in reasonable agreement with experiment. Large Hartree-Fock energy gaps are obtained. They prevent the pairing correlations considered from becoming effective in an HFB approach and from changing the HF ground-state properties appreciably. In non-selfconjugate nuclei, the Pauli principle, rather than the Coulomb interaction, yields large differences between the charge and mass distributions. A theorem on selfconsistent symmetries is proved. The coefficients of the HFB transformation turn out to be real, if time-reversal and angular momentum projection flip are selfconsistent symmetries.     

Towards a Skyrme–Hartree–Fock–Bogolyubov Mass Formula

In this work, we explain our astrophysical motivations for deriving a mass formula based on HFB calculations with a Skyrme interaction. We give an overview of existing mass formulae and present briefly the last HF+BCS mass formula [1]. The Skyrme force MSk7 [1] is considered in the study of shell effects at N=82, in the neutron-rich region far from stability, within the HFB and HF+BCS theories, and compared with results obtained using the forces SkP^δ and SkP^δρ [2]. This revised version was published online in July 2006 with corrections to the Cover Date.     

Non-local mean field effect on nuclei near sub-shell

Evolutions of single-particle energies and Z=64 sub-shell along the isotonic chain of N=82 are investigated in the density dependent relativistic Hartree–Fock (DDRHF) theory in comparison with other commonly used mean field models such as Skyrme HF, Gogny HFB and density dependent relativistic Hartree model (DDRMF). The pairing is treated in the BCS scheme, except for Gogny HFB. It is pointed out that DDRHF reproduces well characteristic features of experimental Z-dependence of both spin–orbit and pseudo-spin–orbit splittings around the sub-shell closure Z=64. Non-local exchange terms of the isoscalar σ and ω couplings play dominant roles in the enhancements of the spin–orbit splitting of proton 2d states, which is the key ingredient to give the Z=64 sub-shell closure properly. On the other hand, the π and ρ tensor contributions for the spin–orbit splitting cancel each other and the net effect becomes rather small. The enhancement of the sub-shell gaps towards Z=64 is studied by the DDRHF, for which the local terms of the scalar and vector meson couplings are found to be important.     

Hartree-Fock-Bogolyubov description of nuclei near the neutron-drip line

We consider the Hartree-Fock-Bogolyubov theory of nuclei in the coordinate representation and derive and solve the HFB equation for the Skyrme effective interaction. Ground-state wave functions and energies of the tin isotopes with 100 ? A ? 176 have been determined and the results have been compared with the predictions of the HF+BCS and macroscopic-microscopic models. The lightest tin isotope which is unstable with respect to a neutron emission is predicted by the HFB method to be ¹⁵³Sn. In the region of nuclei where experimental data are not available the macroscopic-microscopic and self-consistent approximations give substantially different results.     

Theory of the cranked temperature-dependent Hartree-Fock-Bogoliubov approximation and parity projected statistics

Projected statistics is proposed to select states of specified space parity and number parity from the Fock space. In the framework of the self-consistent mean field method the temperature-dependent (or thermal) HFB (THFB) equation with the constraints of angular momentum as well as particle number can be derived from the variation after projections of space parity and number parity. By considering the behavior of the quasiparticle distribution function in the limit of zero temperature, it is shown that the self-consistently blocked HFB equation follows from the THFB equation in this limit. An improved level density formula which is free from the problem of enhancement at low temperatures is derived.     

Consistent empirical physical formulas for potential energy curves of 38–66Ti isotopes by using neural networks

Nuclear shape transition has been actively studied in the past decade. In particular, the understanding of this phenomenon from a microscopic point of view is of great importance. Because of this reason, many works have been employed to investigate shape phase transition in nuclei within the relativistic and nonrelativistic mean field models by examining potential energy curves (PECs). In this paper, by using layered feed-forward neural networks (LFNNs), we have constructed consistent empirical physical formulas (EPFs) for the PECs of ^38–66Ti calculated by the Hartree-Fock-Bogoliubov (HFB) method with SLy4 Skyrme forces. It has been seen that the PECs obtained by neural network method are compatible with those of HFB calculations.     

Linear response theory in nuclei

A new method is proposed for the treatment of a perturbing field as well as the perturbation due to the addition of a particle. The variational equations obtained from a Thouless-type trial function are linearized. The higher order equations are similar to the first order equations, only the inhomogenous terms are different. These equations have been derived for the Hartree-Fock-Bogolyubov (HFB) ground state of the N-particle system. An application of the method to the exactly soluble Lipkin model is also presented. The proposed method meets a high degree of success in the model case. We further argue that the present method is superior to the blocked HFB method and also numerically simpler, for the addition of a particle.     

Translationally noninvariant path integral for Bose-Einstein condensate

The Bogolyubov [Hartree-Fock-Bogolyubov (HFB)] method performs the one-particle (mean-field) approximation in the theory of Bose-Einstein condensation (BEC). Various generalizations of this method are possible. Apart from a nonlinear theory, taking the correlation effects into consideration, the HFB approximation for translationally noninvariant systems describes an instructive phenomenon. This paper is devoted to the treatment of two cases: superfluid ⁴He in porous media and atomic BEC in traps subjected to the gravitational field. Both these systems show the dependence of a critical BEC temperature T _c on their nonuniform properties in space.     

On microscopic theory of radiative nuclear reaction characteristics

A survey of some results in the modern microscopic theory of properties of nuclear reactions with gamma rays is given. First of all, we discuss the impact of Phonon Coupling (PC) on the Photon Strength Function (PSF) because it represents the most natural physical source of additional strength found for Sn isotopes in recent experiments that could not be explained within the standard HFB + QRPA approach. The self-consistent version of the Extended Theory of Finite Fermi Systems in the Quasiparticle Time Blocking Approximation is applied. It uses the HFB mean field and includes both the QRPA and PC effects on the basis of the SLy4 Skyrme force. With our microscopic E1 PSFs, the following properties have been calculated for many stable and unstable even–even semi-magic Sn and Ni isotopes as well as for double-magic ¹³²Sn and ²⁰⁸Pb using the reaction codes EMPIRE and TALYS with several Nuclear Level Density (NLD) models: (1) the neutron capture cross sections; (2) the corresponding neutron capture gamma spectra; (3) the average radiative widths of neutron resonances. In all the properties considered, the PC contribution turned out to be significant, as compared with the standard QRPA one, and necessary to explain the available experimental data. The results with the phenomenological so-called generalized superfluid NLD model turned out to be worse, on the whole, than those obtained with the microscopic HFB + combinatorial NLD model. The very topical question about the M1 resonance contribution to PSFs is also discussed.     

Crossing of the πd<Subscript>5/2</Subscript> and πg<Subscript>7/2</Subscript> orbitals in the <Subscript>51</Subscript>Sb isotopes

Self-consistent calculations using the D1S Gogny force have been performed in order to study the mechanism involved in the crossing of the πd _5/2 and πg _7/2 orbitals in the Sb isotopes. This inversion is well predicted by the HFB + blocking calculations with spherical symmetry performed for the odd-A Sb isotopes. In addition, several HFB and HF calculations have been performed for even-even nuclei of the five neighbouring isotopic chains (Z = 46 to 54, from the proton dripline to N = 82). The results obtained for the binding energies of the two proton orbitals indicate that the radii of the systems play an important role in the crossing, even though some particular πν interactions also give a contribution. The spin-orbit interaction, which is known to be concentrated mainly at the nuclear surface, is proposed to be the main responsible of the crossing.     

Symmetry conserving generalisation of Hartree Fock Bogoliubov theory

We derive a generalisation of the HFB equations which conserve particle number. This is achieved in using the equation of motion method or alternatively the Green's function technique. The price we have to pay is that there is not only one mean field for the particle numberN but a set of coupled mean field equations for the whole bandN, N±2,N±4... Nevertheless we think that our theory is a quite interesting variant in comparison with the conventional projection technique. We apply our theory to simple models and find that the results are excellent.     

Restoration of the broken D2-symmetry in the mean field description of rotating nuclei

Signature effects observed in rotational bands are a consequence of an inherent D2-symmetry. This symmetry is naturally broken by the mean field cranking approximation when a tilted (non-principal) axis orientation of the nuclear spin becomes stable. The possible tunneling forth and back between the two symmetry-related minima in the double-humped potential-energy surface appears as a typical bifurcation of the rotational band. We describe this many-body process in which all nucleons participate by diagonalizing the nuclear Hamiltonian within a selected set of tilted and non-tilted cranking quasiparticle states. This microscopic approach is able to restore the broken D2 symmetry and reproduce the quantum fluctuations between symmetry-related HFB states which emerge as splitting of the band energies and in parallel staggering in intraband M1 transitions.     

On the solution of the number-projected Hartree-Fock-Bogolyubov equations

The numerical solution of the recently formulated number-projected Hartree-Fock-Bogolyubov (HFB) equations is studied in an exactly solvable cranked-deformed shell-model Hamiltonian. It is found that the solution of these number-projected equations involves similar numerical effort as that of bare HFB. We consider that this is significant progress in the mean-field studies of quantum many-body systems. The results of the projected calculations are shown to be in almost complete agreement with the exact solutions of the model Hamiltonian. The phase transition obtained in the HFB theory as a function of the rotational frequency is shown to be smeared out with the projection.     

Simple applications of the pairing-plus-quadrupole-plus-octupole model

The octupole-octupole interaction has been added to the old pairing-plus-quadrupole model. Symmetries of the HF mean field are discussed. An overview is presented of the ground-state deformations for a singlej orbital and two opposite-parity,j, j – 3 orbitals. Breaking of various symmetries can be studied within this model. It is shown that the pairing and the quadrupole-quadrupole interaction stabilize the mean field against octupole deformation. For the singlej orbital, except for one particularj value, no non-axial quadrupole deformations were found. For two opposite-parityj, j –3 orbitals, non-axial octupole deformations occur in most cases. Also axial and non-axial quadrupole deformations were found to be induced by octupole deformations.Communicated by: X. Campi     

Test of three-body contact Skyrme forces with spin excitations in deformed nuclei

Experimental data on spin M1 strength distributions in even-even rare-earth nuclei are compared to theoretical results of selfconsistent HF+RPA calculations with separable spin-spin residual interactions derived from the two alternative versions (two-body density-dependent and three-body contact terms) of the Skyrme force Sk3. It is shown that the two versions produce quite different spin M1 strength distributions, though they generate the same HF mean field. The experimental data favour the two-body over the three-body version of the Skyrme interactions Sk1–Sk6.     

THE SELF-CONSISTENT FIELD EQUATIONS AT FINITE NUCLEAR TEMPERATURE——THE GENERALIZED HFB EQUATIONS

In order to study the variation of shell effects as well as pairing effects with nuclear temperature, we have extended the work of Sano and Yamasaki to generalize the HFB equations to finite nuclear temperature with the help of variational principle. When nullear temperature θ→0, sulh generalized HFB equations reduce to the conventional HFB equations. While the fermi gas model results can be obtained in the high temperature limit.     

Spontaneous symmetry breaking approach to La2CuO4 properties: Hints for matching the Mott and Slater pictures

Special solutions of the Hartree-Fock (HF) problem for Coulomb interacting electrons described by a simple model of the Cu-O planes in La₂CuO₄ are presented. One of the mean field states obtained, is able to predict some of the most interesting properties of this material, such as its insulator character and the antiferromagnetic order. The natural appearance of pseudogaps in some states of this material is also indicated by another of the HF states obtained. These surprising results follow after eliminating spin and crystal symmetry restrictions usually imposed on the single particle HF orbitals, by employing the rotational invariant formulation of the HF scheme originally introduced by Dirac. Therefore, it is exemplified here, how up to know considered strong correlation effects, can be described by improving the HF solution of the considered system. In other words, it has been argued, that defining correlation effects as the ones shown by the system and not predicted by the HF best (lowest energy) solution, allows to explain important, up to know considered as strong correlation properties, as simple mean field ones. The discussion also helps to clarify the role of the antiferromagnetism and pseudogaps in the physical properties of the HTSC materials and indicates a promising way to start conciliating the Mott and Slater pictures in the physics of the transition metal oxides and other strongly correlated electron systems.     

The effect of HF field on diffusion and convection in a plasma column

A cylindrical plasma model with both axial and azimuthal steady magnetic field is used. Nonlinear excitation of time averaged electric currents by the HF field is analyzed. Ponderomotive forces of the HF field also lead to translational and rotational plasma motions. Radial plasma diffusion can be modified by the presence of the HF field at sufficiently low frequencies. The analysis can be applied to problems of anomalous transport.     

Correlation effects on the energy spectra of quantum dot electrons with harmonic model interactions

The low lying excitation energy spectra of two, three and five quantum dot electrons with harmonic model interactions in a large magnetic field are calculated by the Hartree Fock(HF) methods. Correlation effects on the energy level structures are investigated by comparing the HF results with the exact ones. It is found that the pure collective excitations(center of mass mode quanta) existing in the exact energy spectra do not appear in the HF energy spectra. The degeneracies of energy levels are also re...