Hartree Fock Bogoliubov Approach Research Articles

Cluster mean-field description of α emission

We show that the Hartree-Fock-Bogoliubov (HFB) method is able to describe experimental values of alpha decay widths by including a residual nucleon-nucleon Surface Gaussian Interaction (SGI) within the standard procedure used to calculate the nuclear mean field. We call this method the Cluster HFB (CHFB) approach. In this way we correct the deficient asymptotic behaviour of the corresponding single-particle (sp) wave functions generated by the standard mean field. The corrected mean field becomes a sum between the standard mean Woods-Saxon-like field and a cluster Gaussian component centered at the same radius as the SGI. Thus, we give a confirmation of the mean field plus cluster potential structure, which was assumed in our previous work on alpha-decay widths. Systematic calculations evidence the linear correlation between the SGI strength and fragmentation potential, allowing for reliable predictions concerning the half lives of superheavy emitters.

A Hartree-Fock-Bogoliubov Study on the Pairing Correlations of the Isotopes of Cobalt

Background: The phenomena of nucleon pairing could be outlined from the Bethe-Weizäcker semi-empirical formula, from which the nuclear properties, viz. the binding energy, stability, shape etc. could be clearly sketched. Though the pairing correlation seems to be a small correction to the binding energy term, it plays a determinative role in defining the structure of nuclear systems. The addition to the binding energy in turn affects the position of the isotope on the dripline and hence increases the stability.
 Purpose: To study the effects of pairing on the ground state properties of the isotopes of Cobalt.
 Methods: We use Hartree-Fock-Bogoliubov (HFB) theory for the study. The general wave functions for the HFB approach are determined from variational principle. The eigen functions for the Hamiltonian are connected with the particle operators through the Bogoliubov transformations. The Hartree-Fock energy is obtained through the minimization of the variational parameter and the HFB equation is solved by iterative diagonalization by restoring the particle number symmetry.
 Results: The HFB analysis substantiates the effect of pairing correlation on binding energies, neutron and proton pairing energies, neutron and proton pairing gaps and one- and two-neutron separation energies of the Cobalt isotopes. The binding energies and one and two-neutron separation energies match with the experimental values and for pairing energies and pairing gaps, the regions where pairing is significant and the effects of shell closure at the vicinity of magic configuration of neutrons could be recognized.
 Conclusion: The Hartree-Fock-Bogoliubov calculations of the effects of pairing could be used as an efficient tool to study the nuclear structure. It can be ascertained that pairing plays an important role in determining the ground state properties of atomic nuclei.

Study of Ground State Properties in Atomic Nuclei using Hartree Fock Bogoliubov Approach

Study of Ground State Properties in Atomic Nuclei using Hartree Fock Bogoliubov Approach

Microscopic description of quadrupole-octupole coupling in actinides with the Gogny-D1M energy density functional

The interplay between quadrupole and octupole degrees of freedom is discussed in a series of U, Pu, Cm and Cf isotopes both at the mean-field level and beyond. In addition to the static Hartree–Fock–Bogoliubov approach, dynamical beyond-mean-field correlations are taken into account via both parity restoration and symmetry-conserving generator coordinate method calculations based on the parametrization D1M of the Gogny energy density functional. Physical properties such as correlation energies, negative-parity excitation energies as well as reduced transition probabilities B(E1) and B(E3) are discussed in detail and compared with the available experimental data. It is shown that, for the studied nuclei, the quadrupole-octupole coupling is weak and to a large extent the properties of negative parity states can be reasonably well described in terms of the octupole degree of freedom alone.

Comparison between the Thomas–Fermi and Hartree–Fock–Bogoliubov Methods in the Inner Crust of a Neutron Star: The Role of Pairing Correlations

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.

Pairing Effects on Bubble Nuclei**Supported by the National Natural Science Foundation of China under Grant Nos U1832120 and 11675265, the State Scholarship Fund of China Scholarship Council under Grant No 201708130035, and the Natural Science Foundation for Outstanding Young Scholars of Hebei Province of China under Grant No A2018210146.

In the framework of the Skyrme–Hartree–Fock–Bogoliubov approach with the SkT interaction, the pairing effects on the proton bubble structures of 46Ar and 206Hg are discussed. In calculations, three kinds of pairing forces (volume, surface and mixed pairing interactions) are used. For 46Ar, it is shown that the bubble structure with the volume pairing is almost the same as that with the mixed pairing. The bubble with the surface pairing is less pronounced than those with the volume and mixed pairings. Analyzing the density distributions and occupation probabilities of the proton s states and the quasi-degeneracy between the proton 2s1/2 and 1d3/2 orbitals, we explain the difference between the bubble structure with the surface pairing and those with the volume and mixed pairings. For 206Hg, it is seen that the proton density distribution with the surface pairing is different from those with the volume and mixed pairings in the whole region of the radial distance. In addition, it is found that the bubbles with the three pairing forces are different from each other and the least pronounced bubble is obtained with the surface pairing. Thus the selection of the pairing force is important for the study of the nuclear bubble structure.

Structural investigation of neutron-deficient Pt isotopes: the case of 178Pt

Lifetime measurements with the recoil distance Doppler-shift technique have been performed to determine yrast E2 transition strengths in 178Pt. The experimental data are related to those on neighboring Pt isotopes, especially recent data on 180Pt, and compared to calculations within the interacting boson model and a Hartree-Fock Bogoliubov approach. These models predict prolate deformed ground states in Pt isotopes close to neutron midshell consistent with the experimental findings. Further, evidence was found that the prolate intruder structure observed in neutron-deficient Hg isotopes that is minimum in energy in 182Hg becomes the ground state configuration in 178Pt and neighboring 180Pt with nearly identical transition quadrupole moments. The new data on 178Pt are further discussed in the context of the systematics along the Pt isotopic chain with respect to a possible sharp shape transition towards a weakly deformed or a quasi-vibrational ground state whereas the prolate structure increases in energy in 174,176Pt and becomes an intruder configuration similar to, for example, 180,182Hg.

Shape transitions in odd-mass γ -soft nuclei within the interacting boson-fermion model based on the Gogny energy density functional

The interacting boson-fermion model (IBFM), with parameters determined from the microscopic Hartree-Fock-Bogoliubov (HFB) approximation, based on the parametrization D1M of the Gogny energy density functional (EDF), is employed to study the structural evolution in odd-mass $\gamma$-soft nuclei. The deformation energy surfaces of even-even nuclei, single-particle energies and occupation probabilities of the corresponding odd-mass systems have been obtained within the constrained HFB approach. Those building blocks are then used as a microscopic input to build the IBFM Hamiltonian. The coupling constants of the boson-fermion interaction terms are taken as free parameters, fitted to reproduce experimental low-lying spectra. The diagonalization of the IBFM Hamiltonian provides the spectroscopic properties for the studied odd-mass nuclei. The procedure has been applied to compute low-energy excitation spectra and electromagnetic transition rates, in the case of the $\gamma$-soft odd-mass systems $^{129-137}$Ba, $^{127-135}$Xe, $^{129-137}$La and $^{127-135}$Cs. The calculations provide a reasonable agreement with the available experimental data and agree well with previous results based on the relativistic mean-field approximation.

Collective excitations of a dilute Bose gas at finite temperature: time dependent Hartree–Fock–Bogoliubov theory

Using the time-dependent Hartree–Fock–Bogoliubov approach, where the condensate is coupled with the thermal cloud and the anomalous density, we study the equilibrium and the dynamical properties of three-dimensional quantum-degenerate Bose gas at finite temperature. Effects of the anomalous correlations on the condensed fraction and the critical temperature are discussed. In uniform Bose gas, useful expressions for the Bogoliubov excitations spectrum, the first and second sound, the condensate depletion and the superfluid fraction are derived. Our results are tested by comparing the findings computed by quantum Monte Carlo simulations. We present also a systematic investigation of the collective modes of a Bose condensate confined in an external trap. Our predictions are in qualitative agreement with previous experimental and theoretical results. We show in particular that our theory is capable of explaining the so-called anomalous behavior of the mode.

Hyperbolic Bloch equations: Atom-cluster kinetics of an interacting Bose gas

Experiments with ultracold Bose gases can already produce so strong atom–atom interactions that one can observe intriguing many-body dynamics between the Bose–Einstein condensate (BEC) and the non-condensed atoms. This dynamics is thoroughly analyzed with the cluster-expansion approach to uniquely identify atom-cluster dynamics within the many-body system. These clusters assign those atoms that are genuinely connected with one another. The excitation picture is applied to express the many-body state in terms of correlated atom clusters among the non-condensed atoms alone. Implicit notation formalism is developed to explicitly derive the quantum kinetics of all atom clusters. The clusters are shown to build up sequentially, from smaller to larger ones, which is utilized to nonperturbatively describe the interacting BEC with as few clusters as possible. This yields the hyperbolic Bloch equations (HBEs) that not only generalize the Hartree–Fock Bogoliubov approach but are also analogous to the semiconductor Bloch equations (SBEs). This connection is utilized to apply sophisticated many-body techniques of semiconductor quantum optics to BEC investigations. Here, the HBEs are implemented to determine how a strongly interacting Bose gas reacts to a fast switching from weak to strong interactions, often referred to as unitarity. The computations for 85Rb demonstrate that molecular states (dimers) depend on atom density, and that the many-body interactions create coherent transients on a 100 μs time scale converting BEC into non-condensed atoms via quantum depletion.

Multidimensional Skyrme-density-functional Study of the Spontaneous Fission of $^{238}$U

We determined the spontaneous fission lifetime of 238U by a minimization of the action integral in a three-dimensional space of collective variables. Apart from the mass-distribution multipole moments Q20 (elongation) and Q30 (left–right asymmetry), we also considered the pairing-fluctuation parameter λ2 as a collective coordinate. The collective potential was obtained self-consistently using the Skyrme energy density functional SkM*. The inertia tensor was obtained within the nonperturbative cranking approximation to the adiabatic time-dependent Hartree–Fock–Bogoliubov approach. As a result, the pairing-fluctuation parameter λ2 allowed us to control the pairing gap along the fission path, which significantly changed the spontaneous fission lifetime.

Tensor Force Effect on Shape Coexistence of N = 28 Neutron-Rich Isotones

The tensor force effect on potential energy surfaces of the N = 28 neutron-rich isotones is investigated by using the deformed Skyrme—Hartree—Fock—Bogoliubov approach with the T22 interaction. It is found that, without the tensor force, 40Mg and 46Ar have prolate and spherical ground states, respectively. The ground states of 42Si and 44S are oblate. The shape coexistence in 40Mg, 42Si and 44S is evident. However, the ground state deformations of these isotones are not changed and the shape coexistence in 42Si and 44S vanishes when the tensor force is switched on. Taking 42Si as an example, the disappearance of the shape coexistence is understood by analyzing the tensor force effect on the shell correction energies.

Excitation picture of an interacting Bose gas

Atomic Bose–Einstein condensates (BECs) can be viewed as macroscopic objects where atoms form correlated atom clusters to all orders. Therefore, the presence of a BEC makes the direct use of the cluster-expansion approach–lucrative e.g. in semiconductor quantum optics–inefficient when solving the many-body kinetics of a strongly interacting Bose. An excitation picture is introduced with a nonunitary transformation that describes the system in terms of atom clusters within the normal component alone. The nontrivial properties of this transformation are systematically studied, which yields a cluster-expansion friendly formalism for a strongly interacting Bose gas. Its connections and corrections to the standard Hartree–Fock–Bogoliubov approach are discussed and the role of the order parameter and the Bogoliubov excitations are identified. The resulting interaction effects are shown to visibly modify number fluctuations of the BEC. Even when the BEC has a nearly perfect second-order coherence, the BEC number fluctuations can still resolve interaction-generated non-Poissonian fluctuations.

Superheavy magic structures in the relativistic Hartree–Fock–Bogoliubov approach

We have explored the occurrence of the spherical shell closures for superheavy nuclei in the framework of the relativistic Hartree–Fock–Bogoliubov (RHFB) theory. Shell effects are characterized in terms of two-nucleon gaps δ2n(p). Although the results depend slightly on the effective Lagrangians used, the general set of magic numbers beyond 208Pb are predicted to be Z=120, 138 for protons and N=172, 184, 228 and 258 for neutrons, respectively. Specifically the RHFB calculations favor the nuclide 120304 as the next spherical doubly magic one beyond 208Pb. Shell effects are sensitive to various terms of the mean-field, such as the spin-orbit coupling, the scalar and effective masses.

α-decay calculations of heavy nuclei using an effective Skyrme interaction

Background: For nuclei heavier than ${}^{208}$Pb $\ensuremath{\alpha}$ decay is a dominating decay mode, and in the search of new superheavy elements one often observes chains of $\ensuremath{\alpha}$ decays.Purpose: Explore and test microscopic descriptions of $\ensuremath{\alpha}$ decay based on theories with effective nuclear interactions.Methods: The nuclear ground states are calculated with the Hartree-Fock-Bogoliubov (HFB) method using the Skyrme interaction. Microscopic $\ensuremath{\alpha}$-decay formation amplitudes are calculated from the HFB wave functions, and the $R$-matrix formalism is utilized to obtain decay probabilities.Results: Using a large harmonic-oscillator basis we obtain converged $\ensuremath{\alpha}$-decay widths. A comparison with experiment including all spherical even-even $\ensuremath{\alpha}$ emitting nuclei shows that the model consistently predicts too small formation amplitudes while relative values are in good agreement with experiment.Conclusions: The method was found to be numerically practical even with a large basis size. The comparison of formation amplitudes suggests that the pairing type correlations included in the HFB approach cannot produce sufficient $\ensuremath{\alpha}$-particle clustering.

Systematic study of tensor force effect on pseudospin orbital splittings in Sn isotopes

The tensor force effect on the proton pseudospin orbital splittings of the 1 and 1 states in Sn isotopes is systematically analyzed with 36 sets of the TIJ interactions in the framework of the Hartree–Fock–Bogoliubov approach. It is shown that not all tensor parametrizations preserve the pseudospin symmetry.

Study of spontaneous fission lifetimes using nuclear density functional theory

The spontaneous fission lifetimes have been studied microscopically by minimizing the collective action integral in a two-dimensional collective space of quadrupole moments (Q20 ,Q 22) representing elongation and triaxiality. The microscopic collective potential and inertia tensor are obtained by solving the self-consistent Hartree- Fock-Bogoliubov (HFB) equations with the Skyrme energy density functional and mixed pairing interaction. The mass tensor is computed within the perturbative Adiabatic Time- Dependent HFB (ATDHFB) approach in the cranking approximation. The dynamic fission trajectories have been obtained by minimizing the collective action using two different numerical techniques. The values of spontaneous fission lifetimes obtained in this way are compared with the static results.

Matrix Elements for Neutrinoless Double Beta Decay

The status of calculation of the neutrinoless double beta decay (0νββ-decay) nuclear matrix elements (NMEs) is reviewed. The 0νββ-decay NMEs obtained in the framework of the Quasiparticle Random Phase Approximation (QRPA), Large Scale Shell Model (LSSM), Interacting Boson Model (IBM), Projected Hartree-Fock Bogoliubov approach (P-HFB) and Energy Density Functional Method (EDF) are presented and discussed. The problem of reliable determination of the 0νββ-decay NMEs is addressed. It is manifested that the uncertainty associated with the calculation of the 0νββ-decay NMEs can be diminished by suitable chosen nuclear probes. Further, the recent development in the field of double beta decay showed that the search for the neutrinoless double electron capture process (0νECEC) may establish the Majorana nature of neutrinos. This possibility should be considered as alternative and complementary to searches for the 0νββ-decay.

Role of superfluidity in nuclear incompressibilities

Nuclei are propitious tools to investigate the role of the superfluidity in the compressibility of a Fermionic system. The centroid of the Giant Monopole Resonance (GMR) in Tin isotopes is predicted using a constrained Hartree-Fock Bogoliubov approach, ensuring a full self-consistent treatment. Superfluidity is found to favour the compressibitily of nuclei. Pairing correlations explain why doubly magic nuclei such as {sup 208}Pb are stiffer compared to open-shell nuclei. Fully self-consistent predictions of the GMR on an isotopic chain should be the way to microscopically extract both the incompressibility and the density dependence of a given energy functional. The macroscopic extraction of K{sub sym}, the asymmetry incompressibility, is questioned. Investigations of the GMR in unstable nuclei are called for. Pairing gap dependence of the nuclear matter incompressibility should also be investigated.

Physical origin of density dependent forces of Skyrme type within the quark meson coupling model

A density dependent, effective nucleon–nucleon force of the Skyrme type is derived from the quark–meson coupling model—a self-consistent, relativistic quark level description of nuclear matter. This new formulation requires no assumption that the mean scalar field is small and hence constitutes a significant advance over earlier work. The similarity of the effective interaction to the widely used SkM ∗ force encourages us to apply it to a wide range of nuclear problems, beginning with the binding energies and charge distributions of doubly magic nuclei. Finding acceptable results in this conventional arena, we apply the same effective interaction, within the Hartree–Fock–Bogoliubov approach, to the properties of nuclei far from stability. The resulting two neutron drip lines and shell quenching are quite satisfactory. Finally, we apply the relativistic formulation to the properties of dense nuclear matter in anticipation of future application to the properties of neutron stars.