Restoration of pseudo-spin symmetry in N = 32 and N = 34 isotones described by relativistic Hartree-Fock theory

The restoration of pseudo-spin symmetry(PSS) along the N = 32 and N = 34 isotonic chains and the physics behind are studied by applying the relativistic Hartree-Fock theory with the effective Lagrangian PKA1. Taking the proton pseudo-spin partners(π2s_(1/2), π1d_(3/2)) as candidates, the systematic restoration of PSS along both isotonic chains is found from sulphur(S) to nickel(Ni), while an obvious PSS violation from silicon(Si) to sulphur is discovered near the drip lines. The effects of the tensor force components are investigated, introduced naturally by the Fock terms, which can only partially interpret the systematics from calcium to nickel, whereas they fail for the overall trends. Further analysis following the Schr?dinger-like equation of the lower component of Dirac spinor shows that contributions from the Hartree terms dominate the overall systematics of the PSS restoration. Such effects can be self-consistently interpreted by the evolution of the proton central density profiles along both isotonic chains. In particular, the PSS violation is found to tightly relate to the dramatic changes from the bubble-like density profiles in silicon to the central-bumped ones in sulphur.     

Renormalized Hartree-Fock equations in a nuclear relativistic quantum field theory

Renormalized Hartree-Fock equations are derived for an infinite system of mesons and baryons in the framework of a relativistic quantum field theory. Direct and exchange diagrams in the baryon propagator are summed self-consistently to all orders, and the effects of occupied negative-energy states in the Dirac sea are included. The required counterterm subtractions are defined by conventional renormalization conditions, but they need not be evaluated explicitly. The result is a set of finite nonlinear integral equations for the baryon self-energy that includes vacuum fluctuation effects from virtual N$\text{N}$ pairs in the many-body wavefunction at finite density.     

Properties of nuclear and neutron matter in a relativistic Hartree-Fock theory

Relativistic Hartree-Fock (HF) equations are derived for an infinite system of mesons and baryons in the framework of a renormalizable relativistic quantum field theory. The derivation is based on a diagrammatic approach and Dyson's equation for the baryon propagator. The result is a set of coupled, nonlinear integral equations for the baryon self-energy with a self-consistency condition on the single-particle spectrum. The HF equations are solved for nuclear and neutron matter in the Walecka model, which contains neutral scalar and vector mesons. After renormalizing model parameters to reproduce nuclear matter saturation properties, HF results at low to moderate densities are similar to those in the mean-field (Hartree) approximation. Self-consistent exchange corrections to the Hartree equation of state become negligible at high densities. Rho- and pi-meson exchanges are incorporated using a renormalizable gauge-theory model. A chiral transformation of the lagrangian is used to replace the pseudoscalar πN coupling with a pseudovector coupling, for which one-pion exchange is a reasonable first approximation. This transformation maintains the model's renormalizability so that corrections may be evaluated. Pion exchange has a small effect on the HF results of the Walecka model and brings HF results in closer agreement with the mean-field theory. The diagrammatic techniques used here retain the mesonic degrees of freedom and are simple enough to be extended to more refined self-consistent approximations.     

Neutron star equation of state in density dependent relativistic Hartree-Fock theory

The equation of state of neutron stars is studied in the newly developed density dependent relativistic Hartree-Fock (DDRHF) theory with the effective interaction PKO1 and applied to describe the properties of neutron stars. The results are compared with the recent observational data of compact stars and those calculated with the relativistic mean field (RMF) effective interactions. The maximum mass of neutron stars calculated with PKO1 is about 2.45 M_⊙, which consists with high pulsar mass from PSR B1516+02B recently reported. The influence of Fock terms on the cooling of neutron stars is discussed as well.     

Mapping exchange in relativistic Hartree-Fock

We show that formally for the standard ansatz relativistic point-coupling mean-field (RMF-PC) model a Lagrangian density is not equivalent in Hartree and Hartree-Fock approximations. The equivalency can be achieved only if we use a “complete” ansatz at the cost of introducing new parameters in the model. An approximate treatment of the exchange terms from standard RMF-PC indicates that these effects cannot be easily, if at all, absorbed by a Dirac-Hartree approximation.     

Restoration of rotational symmetry in deformed relativistic mean-field theory

We report on a very recently developed three-dimensional angular momentum projected relativistic mean-field theory with point-coupling interaction (3DAMP+RMF-PC). Using this approach the same effective nucleon-nucleon interaction is adopted to describe both the single-particle and collective motions in nuclei. Collective states with good quantum angular momentum are built projecting out the intrinsic deformed mean-field states. Results for ²⁴Mg are shown as an illustrative application.     

Restoration of rotational symmetry in deformed relativistic mean-field theory

We report on a very recently developed three-dimensional angular momentum projected relativistic mean-field theory with point-coupling interaction (3DAMP+RMF-PC). Using this approach the same effective nucleon-nucleon interaction is adopted to describe both the single-particle and collective motions in nuclei. Collective states with good quantum angular momentum are built projecting out the intrinsic deformed meanfield states. Results for 24Mg are shown as an illustrative application.     

Neutron star equation of state in density dependent relativistic Hartree-Fock theory*

The equation of state of neutron stars is studied in the newly developed density dependent relativistic Hartree-Fock (DDRHF) theory with the effective interaction PKO1 and applied to describe the properties of neutron stars. The results are compared with the recent observational data of compact stars and those calculated with the relativistic mean field (RMF) effective interactions. The maximum mass of neutron stars calculated with PKO1 is about 2.45 M☉, which consists with high pulsar mass from PSR B1516+02B recently reported. The influence of Fock terms on the cooling of neutron stars is discussed as well.     

Spin symmetry in the relativistic modified Rosen-Morse potential with the approximate centrifugal term

By applying a Pekeris-type approximation to the centrifugal term, we study the spin symmetry of a Dirac nucleon subjected to scalar and vector modified Rosen-Morse potentials. A complicated energy equation and associated two-component spinors with arbitrary spin-orbit coupling quantum number k are presented. The positive-energy bound states are checked numerically in the case of spin symmetry. The relativistic modified Rosen-Morse potential cannot trap a Dirac nucleon in the limiting case α→ 0.     

The minimax technique in relativistic Hartree-Fock calculations

The online version of the original article can be found at     

The minimax technique in relativistic Hartree-Fock calculations

Using the set of trial spinors and the Dirac-Coulomb Hamiltonian (H _DC) we discuss the role of the minimax theorem in relativistic Hartree-Fock calculations. In principle, the minimax theorem guarantees the occurrence of an upper bound. We also consider a scaling of the functionsu _i and discuss the condition to derive the relativistic hypervirial theorem; the variational procedure represented by the condition serves as an example of the minimax technique. Single zeta calculations onH ₂ ⁺ ,H ₂ and He are analysed. The effect of enlarging the basis is investigated for the He atom. The “upper bound” obtained by usingcoherent basis spinors differs from the result of the (random) linear variation using the kinetically balanced basis set by an amount which is at most of orderc ⁻⁴. Use of thecoherent basis set is advocated. An erratum to this article is available at .     

Pseudospin symmetry in the Dirac phenomenology

In the phenomenological relativistic framework of the Dirac equation for spherical nuclei, we use different kinds of single-particle central potentials ( Σ_S + Σ₀ to investigate certain aspects of the spin and pseudospin (PS) symmetries. Neither the splitting of PS doublets (PSDs) nor the similarity of the radial parts of the small components (F/r of the corresponding Dirac spinors have been found related with the magnitude of Σ_S + Σ₀ , in the sense predicted by several authors in the last decade. This conclusion is shown to be valid, in particular, for a potential of Coulomb type. We give a simple explanation for the strong correlation established in the relativistic calculations between the similarity of the radial parts of the big (small) components of the Dirac spinors of two spin (pseudospin) partners and the number of their nodes. The direct effects of the so-called PS symmetry-breaking term (and its singularity point) on the F functions of the PSDs are also analysed.     

Classical limit of real Dirac theory: Quantization of relativistic central field orbits

The classical limit of real Dirac theory is derived as the lowest-order contribution in of a new, exact polar decomposition. The resulting classical spinor equation is completely integrated for stationary solutions to arbitrary central fields. Imposing single-valuedness on the covering space of a bivector-valued extension to these classical solutions, orbital angular momentum, energy, and spin directions are quantized. The quantization of energy turns out to yield the WKB formula of Bessey, Uhlenbeck, and Good. It is demonstrated that the success of Sommerfeld's old quantization is due to a complete mutual cancellation between wave mechanical half-integers and spin in the particular case of the relativistic Kepler problem.     

Multi-configuration Hartree-Fock theory in nuclei

A variational method for the self-consistent solution of the nuclear many body problem with the inclusion of correlations is formulated. The trial function in this multiconfiguration-Hartree-Fock (MCHF) theory is a linear combination of unrestricted Slater determinants. The MCHF equations are given and a simple procedure for solving them is outlined. A great advantage of this method is that it also yields the excited states. It is shown that the trial function is stable against particle-hole excitations. Therefore the Slater determinants differ from each other at least by two particle — two hole excitations. This method is applied to the Lipkin model. In the MCHF method the difference to the exact solution is reduced by a factor three to ten compared with the corresponding value in the HF approach.     

Finding Dirac Spin Effect in NUT Spacetime

In the present study, we are interested in finding the spin precession of a Dirac particle in expanding and rotating NUT spaeetime. A tetrad with two functions to be determined is applied to the field equation of the teleparallel theory of gravity via a coordinate transformation. The vector, the axial-vector and the tensor parts of the torsion tensor are obtained. We found that the vector parts are in the radial and Ф-directions. The axial-vector torsion is along r-direction while its other components along θ and oh-directions vanish everywhere. The vector connected with Dirac spin has been evaluated as well.     

Single-particle potential in a relativistic Hartree-Fock mean field approximation

A relativistic Hartree-Fock mean field approximation is investigated in a model in which the nucléon field interacts with scalar and vector meson fields. The Hartree-Fock potential felt by individual nucléons enters in a relativistic Dirac single-particle equation. It is shown that in the case of symmetric nuclear matter one can always find a potential which is fully equivalent to the most general mean field and which is only the sum of a Lorentz scalar, of one component of a Lorentz tensor and of the fourth component of a Lorentz vector. A non-relativistic potential is derived which yields exactly the same single-particle energies and elastic scattering phase shifts as the relativistic Hartree-Fock potential. Analytical results are presented in the case of nuclear matter. A local density approximation is constructed which enables one to consider finite nuclei. The input parameters of the model can be chosen in such a way that the empirical saturation properties of nuclear matter are well reproduced. Good agreement is obtained between the calculated non-relativistic potential and the empirical value of the real part of the optical-model potential at low and at intermediate energy. At intermediate energy, the wine-bottle bottom shape which had previously been found for the potential in the framework of the relativistic Hartree approximation is maintained when the Fock contribution is included.     

Pseudospin symmetry as an accidental symmetry in the relativistic framework

We analyse the arguments used in the relativistic context to base the quasi-degeneracy of pseudospin doublets (PSDs) observed in atomic nuclei on the smallness of the single-particle central potential (Σ _S + Σ₀), discussing, especially, the implications of the results obtained in the limit (Σ _S + Σ₀ = 0. We study also the transition from a relativistic model, where Σ _S + Σ₀ is a harmonic-oscillator potential and exhibits degenerate PSDs, to a more realistic one with broken pseudospin symmetry. We examine, in particular, the effect of the corresponding pseudospin symmetry-breaking term on the Dirac spinors of the PSDs. An extension of the Nilsson model to the relativistic case is also considered. Communicated by V. Vento     

Spin precession in the relativistic two-body problem

Synge's approximation procedure is applied to calculate the spin precession of either body in a binary system consisting of two rotating, spherical, rigid bodies of comparable mass and radius. The calculations are valid for the case in which the mass-radius ratio of each body, as well as the ratio of the radius of either body to the distance between their centers, is small. The results agree with those of earlier authors, who use different techniques, except for a term that arises from the effect of the rotation on the stress within the bodies. This term is similar in form to the quadrupole term of Barker and O'Connell, which they obtain when they allow the bodies to become distorted under the influence of the rotation.