Hartree-Fock-Bogoliubov Calculations Research Articles

Determination of excitation function for alpha induced reactions on 237Np and 238Pu

The excitation function of the alpha particle-induced reactions on 237Np and 238Pu has been determined for the incident alpha energies ranging from threshold to 50 MeV using TALYS1.95 nuclear reaction code. In this study, the excitation function of (α,xn) reactions on 237Np and 238Pu has been determined by invoking the Jeukenne–Lejeune–Mahaux–Bruyeres Optical model potential. The microscopic nuclear level density from Hilaire’s combinatorial table is used to calculate the nuclear level density. Also, the excitation function has been evaluated by using the alpha potential of Watanabe folding approach with Koning–Delaroche spherical optical model potential and McFadden–Satchler potential. Furthermore, all the calculated cross-sections have been compared with the experimental results taken from EXFOR and the available evaluated cross-sections from TENDL2019 nuclear data library. The calculated cross sections using Jeukenne–Lejeune–Mahaux–Bruyeres Optical model potential together with microscopic nuclear level density from Hilaire’s combinatorial table incorporating the Skyrme–Hartree–Fock–Bogoliubov calculation results in good correspondence with the experimental values from EXFOR.

Robust formulation of Wick's theorem for computing matrix elements between Hartree-Fock-Bogoliubov wavefunctions.

Numerical difficulties associated with computing matrix elements of operators between Hartree-Fock-Bogoliubov (HFB) wavefunctions have plagued the development of HFB-based many-body theories for decades. The problem arises from divisions by zero in the standard formulation of the nonorthogonal Wick's theorem in the limit of vanishing HFB overlap. In this Communication, we present a robust formulation of Wick's theorem that stays well-behaved regardless of whether the HFB states are orthogonal or not. This new formulation ensures cancellation between the zeros of the overlap and the poles of the Pfaffian, which appears naturally in fermionic systems. Our formula explicitly eliminates self-interaction, which otherwise causes additional numerical challenges. A computationally efficient version of our formalism enables robust symmetry-projected HFB calculations with the same computational cost as mean-field theories. Moreover, we avoid potentially diverging normalization factors by introducing a robust normalization procedure. The resulting formalism treats even and odd number of particles on equal footing and reduces to Hartree-Fock as a natural limit. As proof of concept, we present a numerically stable and accurate solution to a Jordan-Wigner-transformed Hamiltonian, whose singularities motivated the present work. Our robust formulation of Wick's theorem is a most promising development for methods using quasiparticle vacuum states.

Roles of tensor and isoscalar pairing interactions in β-decay calculations for possible r-process waiting point nuclei with N ~ 82 and 126* *Supported by the National Natural Science Foundation of China (11575120, 11822504), and Science Specialty Program of Sichuan University (2020SCUNL210)

In this study, we adopt the self-consistent Hartree-Fock-Bogoliubov (HFB) theory with the proton-neutron quasi-particle random phase approximation (pnQRPA) based on the Skyrme force for calculation of the β − decay half-lives for nuclei with N ~ 82 and 126 on possible r-process paths. In the calculations, the Skyrme interaction (e.g., SKO') is adopted, and the tensor interaction is added self-consistently in both HFB and QRPA calculations. We systematically study how the half-life is changed by varying the strength of the triplet-even (TE) and triplet-odd (TO) components as well as the IS pairing. We find that a variation in strength of the IS pairing of approximately 20% does not produce a substantial effect on β-decay rates with or without the tensor force, while a strength variation of the TO tensor force considerably affects the change in the β-decay half-lives for the very neutron rich N ~ 82 and 126 isotonic chains. In addition, with the inclusion of the tensor force, the GT decay becomes dominant for very neutron-rich nuclei.

Constraining the relativistic mean-field models from PREX-2 data: effective forces revisited * *Supported by SERB, Department of Science and Technology, Govt. of India, Project No. CRG/2019/002691, FOSTECT Project No. FOSTECT.2019B.04, and FAPESP Project No. 2017/05660-0

Based on the current measurement of the neutron distribution radius ( ) of 208Pb from the PREX-2 data, we revisited the recently developed G3 and IOPB-I force parameters by fine-tuning some specific couplings within the relativistic mean-field (RMF) model. The ω– ρ-mesons coupling and the ρ-meson coupling are constrained to the experimental neutron radius of 208Pb without compromising the bulk properties of finite nuclei and infinite nuclear matter observables. The modified parameter sets are applied to calculate the gross properties of finite nuclei such as binding energies, charge distributions, nuclear radii, pairing gaps, and single-particle energies. The root-mean-square deviations in binding energy and charge radius are estimated with respect to the available experimental data for 195 even–even nuclei, and the results compare favourably with the well-calibrated effective interactions of Skyrme, Gogny and other relativistic mean-field parametrizations. The pairing gap estimations for modified G3 and IOPB-I for Sn isotopes are also compared with the Hartree–Fock–Bogoliubov calculation with the Gogny (D1S) interaction. The isotopic shift and single-particle energy spacing are also calculated and compared with the experimental data for both original and modified versions of the G3 and IOPB-I parameter sets. Subsequently, both the modified parameter sets are used to obtain the various infinite nuclear matter observables at saturation. In addition to these, the force parameters are adopted to calculate the properties of a high isospin asymmetry dense system such as neutron star matter and tested for validation using the constraint from GW170817 binary neutron star merger events. The tuned forces predict relatively good results for finite and infinite nuclear matter systems and the current limitation on the neutron radius from PREX-2. A systematic analysis using these two refitted parameter sets over the nuclear chart will be communicated shortly.

Axially-deformed solution of the Skyrme-Hartree-Fock-Bogoliubov equations using the transformed harmonic oscillator basis (IV) hfbtho (v4.0): A new version of the program

We describe the new version 4.0 of the code hfbtho that solves the nuclear Hartree-Fock-Bogoliubov problem by using the deformed harmonic oscillator basis in cylindrical coordinates. In the new version, we have implemented the restoration of rotational, particle number, and reflection symmetry for even-even nuclei. The restoration of rotational symmetry does not require using bases closed under rotation. Furthermore, we added the SeaLL1 functional and improved the calculation of the Coulomb potential. Finally, we refactored the code to facilitate maintenance and future developments. New version program summaryProgram title:hfbtho v4.0CPC Library link to program files:https://doi.org/10.17632/c5g2f92by3.2Code Ocean capsule:https://codeocean.com/capsule/5389629Licensing provisions: GPLv3Programming language: Fortran 2003Journal reference of previous version: R.N. Pérez, N. Schunck, R.-D. Lasseri, C. Zhang and J. Sarich, Comput. Phys. Commun. 220 (2017) 363Does the new version supersede the previous version: YesReasons for the new version: This version adds new capabilities to restore broken symmetries and determine corresponding quantum numbers of even-even nucleiSummary of revisions:1.Angular momentum projection for even-even nuclei in a deformed basis;2.Particle number projection for even-even nuclei in the quasiparticle basis;3.Implementation of the SeaLL1 functional;4.Expansion of the Coulomb potential onto Gaussians;5.MPI-parallelization of a single hfbtho execution;6.Code refactoring.Nature of problem:hfbtho is a physics computer code that is used to model the structure of the nucleus. It is an implementation of the energy density functional (EDF) approach to atomic nuclei, where the energy of the nucleus is obtained by integration over space of some phenomenological energy density, which is itself a functional of the neutron and proton intrinsic densities. In the present version of hfbtho, the energy density is derived either from the zero-range Skyrme or the finite-range Gogny effective two-body interaction between nucleons. Nuclear superfluidity is treated at the Hartree-Fock-Bogoliubov (HFB) approximation. Constraints on the nuclear shape allow probing the potential energy surface of the nucleus as needed, e.g., for the description of shape isomers or fission. A local scale transformation of the single-particle basis in which the HFB solutions are expanded provides a tool to properly compute the structure of weakly-bound nuclei. Restoration of the rotational, particle number, and reflection symmetry for even-even nuclei enables recovering the quantum numbers that are lost at the HFB approximation.Solution method: The program uses the axial harmonic oscillator (HO) or the transformed harmonic oscillator (THO) single-particle basis to expand quasiparticle wave functions. It iteratively diagonalizes the HFB Hamiltonian based on generalized Skyrme-like energy densities and zero-range pairing interactions or the finite-range Gogny force until a self-consistent solution is found. Lagrange parameters are used to impose constraints on HFB solutions, and their value is updated at each iteration from an approximation of the quasiparticle random phase approximation (QRPA) matrix. Symmetry restoration is implemented through standard projection techniques. Previous versions of the program were presented in [1-3].Additional comments including restrictions and unusual features: Axial and time-reversal symmetries are assumed in HFB calculations; y-simplex symmetry and even particle numbers are assumed in angular momentum projection.

Skyrme–Hartree–Fock–Bogoliubov Calculations of Even and Odd Neutron-Rich Mg Isotopes

Using the Skyrme functional with SIII, SKM*, SLy4, and UNE0 sets of parameters and the Hartree–Fock–Bogoliubov mean-field method; the ground-state properties of even-even and even-odd neutron-rich Mg isotopes have been investigated. The results of calculations of the binding energy per nucleon (B/A), the one- and two-neutron separation energies (Sn and S2n), proton and neutron rms radii, neutron pairing gap, and quadrupole deformation parameter (B2) have been compared with the available experimental data, the results of Hartree–Fock–Bogoliubov calculations based on the D1S Gogny force, and predictions of some nuclear models such as the Finite Range Droplet Model (FRDM) and Relativistic Mean-Field (RMF) model. Our results show good agreements in comparison with the experimental data and the results of the mentioned models.

Unified Equation of State for Neutron Stars Based on the Gogny Interaction

The effective Gogny interactions of the D1 family were established by D. Gogny more than forty years ago with the aim to describe simultaneously the mean field and the pairing field corresponding to the nuclear interaction. The most popular Gogny parametrizations, namely D1S, D1N and D1M, describe accurately the ground-state properties of spherical and deformed finite nuclei all across the mass table obtained with Hartree–Fock–Bogoliubov (HFB) calculations. However, these forces produce a rather soft equation of state (EoS) in neutron matter, which leads to predict maximum masses of neutron stars well below the observed value of two solar masses. To remove this limitation, we built new Gogny parametrizations by modifying the density dependence of the symmetry energy predicted by the force in such a way that they can be applied to the neutron star domain and can also reproduce the properties of finite nuclei as good as their predecessors. These new parametrizations allow us to obtain stiffer EoS’s based on the Gogny interactions, which predict maximum masses of neutron stars around two solar masses. Moreover, other global properties of the star, such as the moment of inertia and the tidal deformability, are in harmony with those obtained with other well tested EoSs based on the SLy4 Skyrme force or the Barcelona–Catania–Paris–Madrid (BCPM) energy density functional. Properties of the core-crust transition predicted by these Gogny EoSs are also analyzed. Using these new Gogny forces, the EoS in the inner crust is obtained with the Wigner–Seitz approximation in the Variational Wigner–Kirkwood approach along with the Strutinsky integral method, which allows one to estimate in a perturbative way the proton shell and pairing corrections. For the outer crust, the EoS is determined basically by the nuclear masses, which are taken from the experiments, wherever they are available, or by HFB calculations performed with these new forces if the experimental masses are not known.

Ground-State Properties of Even–Even Nuclei from He (Z = 2) to Ds (Z = 110) in the Framework of Skyrme–Hartree–Fock–Bogoliubov Theory

The ground-state properties of 2094 even–even nuclei with $$2\le Z\le 110$$ over a wide range of isotopes have been calculated and studied in the framework of the Hartree–Fock–Bogoliubov (HFB) theory with the SKI3 Skyrme type effective nucleon–nucleon interaction. The calculated ground-state values of the binding energy per nucleon (B.E/A), quadrupole deformation parameter (β2), two-neutron separation energy (S2n), charge radii (RC), proton rms radii, neutron rms radii and neutron pairing gap have been compared with the available experimental data of the atomic mass evaluation (AME2016), the results of Hartree–Fock–Bogoliubov calculations based on the D1S Gogny effective nucleon–nucleon interaction and predictions of some nuclear models such as the finite range droplet model (FRDM2012) and relativistic mean-field model. Based on the HFBTHO results, the parameters of the Skyrme SKI3-HFB method can be successfully used for describing the ground-state properties of the investigated nuclei, in particularly the neutron-rich nuclei and the exotic nuclei near the neutron drip-line.

Isobaric analog state energy in deformed nuclei: A toy model

A formula to evaluate the effects of a general deformation on the Coulomb direct contribution to the energy of the Isobaric Analog State (IAS) is presented and studied via a simple yet physical model. The toy model gives a reasonable account of microscopic deformed Hartree-Fock-Bogolyubov (HFB) calculations in a test case, and provides a guidance when predicting unknown IAS energies. Thus, deformed HFB calculations, to predict the IAS energies, are performed for several neutron-deficient medium-mass and heavy nuclei which are now planned to be studied experimentally.

Phase diagram of an extended Agassi model

Background: The Agassi model is an extension of the Lipkin-Meshkov-Glick model that incorporates the pairing interaction. It is a schematic model that describes the interplay between particle-hole and pair correlations. It was proposed in the 1960's by D. Agassi as a model to simulate the properties of the quadrupole plus pairing model. Purpose: The aim of this work is to extend a previous study by Davis and Heiss generalizing the Agassi model and analyze in detail the phase diagram of the model as well as the different regions with coexistence of several phases. Method: We solve the model Hamiltonian through the Hartree-Fock-Bogoliubov (HFB) approximation, introducing two variational parameters that play the role of order parameters. We also compare the HFB calculations with the exact ones. Results: We obtain the phase diagram of the model and classify the order of the different quantum phase transitions appearing in the diagram. The phase diagram presents broad regions where several phases, up to three, coexist. Moreover, there is also a line and a point where four and five phases are degenerated, respectively. Conclusions: The phase diagram of the extended Agassi model presents a rich variety of phases. Phase coexistence is present in extended areas of the parameter space. The model could be an important tool for benchmarking novel many-body approximations.

Nuclear structure investigation of even–even and odd Pb isotopes by using the Hartree–Fock–Bogoliubov method

The nuclear structure of even–even and odd lead isotopes ([Formula: see text]Pb) is investigated within the Hartree–Fock–Bogoliubov (HFB) theory. Calculations are performed for a wide range of neutron numbers, starting from the proton-rich side up to the neutron-rich side, by using the SLy4 Skyrme interaction and a new proposed formula for the pairing strength which is more precise for this region of nuclei as we did in previous works in the regions of Neodymium (Nd, [Formula: see text]) [Y. El Bassem and M. Oulne, Int. J. Mod. Phys. E 24 (2015) 1550073] and Molybdenum (Mo, [Formula: see text]) [Y. El Bassem and M. Oulne, Nucl. Phys. A 957 (2017) 22]. Such a new pairing strength formula allows reaching exotic nuclei region where the experimental data are not available. Calculated values of various physical quantities such as binding energy, two-neutron separation energy, quadrupole deformation, and rms-radii for protons and neutrons are discussed and compared with experimental data and some estimates of other nuclear models like Finite Range Droplet Model (FRDM), Relativistic Mean Field (RMF) model with NL3 functional (NL3), Density-Dependent Meson-Exchange Relativistic Energy Functional (DD-ME2) and results of HFB calculations based on the D1S Gogny effective nucleon–nucleon interaction (Gogny D1S).

A new statistical method for the structure of the inner crust of neutron stars

We investigated the structure of the low density regions of the inner crust of neutron stars using the Hartree–Fock–Bogoliubov (HFB) model to predict the proton content Z of the nuclear clusters and, together with the lattice spacing, the proton content of the crust as a function of the total baryonic density . The exploration of the energy surface in the configuration space and the search for the local minima require thousands of calculations. Each of them implies a HFB calculation in a box with a large number of particles, thus making the whole process very demanding. In this work, we apply a statistical model called Gaussian process emulator that makes the exploration of the energy surface ten times faster. We also present a novel treatment of the HFB equations that leads to an uncertainty on the total energy of per particle. Such a high precision is necessary to distinguish neighbour configurations around the energy minima.

Toroidal Nuclear Matter Distributions of Superheavy Nuclei from Constrained Skyrme--HFB Calculations

Using the Hartree-Fock-Bogoliubov (HFB) self-consistent mean-field theory with the SkM* Skyrme energy-density functional, we study nuclear structure properties of even-even superheavy nuclei (SHN) of $Z=120$ isotopes and $N=184$ isotones. The shape of the nucleus along the lowest energy curve as a function of the quadrupole moment $Q_{20}$ makes a sudden transition from the oblate spheroids (biconcave discs) to the toroidal shapes, in the region of large oblate quadrupole moments.

Reactions of exotic nuclei with the quark-meson coupling model

The nucleon-nucleon interaction is an important requirement for investigations of nuclear structure and reactions, as well as for astrophysical models such as r-process nucleosynthesis and neutron stars. The traditional approach to low-energy nuclear physics is to treat nucleons as immutable objects interacting via phenomenological forces. The use of phenomenological interactions, rather than one derived from a microscopic theory, raises questions as to the reliability of predictions for exotic regions of the nuclear chart. The quark-meson coupling (QMC) model uses a relativistic mean-field approach to provide a microscopically derived nucleon-nucleon interaction, which takes into account the quark structure of the nucleon.The Skyrme energy density functional is a popular phenomenological tool in studies of nuclear structure and reactions. In this work, the QMC density functional was used to produce a set of Skyrme parameterisations, in the hopes that they will give more reliable predictions for exotic nuclei. In conjunction with Hartree-Fock-Bogoliubov (HFB) calculations, the Skyrme-QMC (SQMC) parameterisations have been used to model the ground-state properties of individual nuclei and nucleus-nucleus potentials for Ca + Sn reactions. The SQMC parameterisation performs with an accuracy comparable to modern phenomenological functionals. From this, one can investigate the importance of the isovector terms of the nucleon-nucleon interaction, which are particularly significant for exotic, neutron-rich regions of the nuclear chart.One of the notable successes of the QMC model is its derivation of nuclear spin-orbit coupling. The isovector dependence of the spin-orbit equation of state is remarkably similar to that of the modern UNEDF1 phenomenological density functional. HFB calculations along the Sn isotopic chain reveal that the isovector properties of the spin-orbit term impact binding energies to a level that will be significant for astrophysical r-process modelling.

Hartree–Fock–Bogoliubov calculation of ground state properties of even–even and odd Mo and Ru isotopes

In a previous work (El Bassem and Oulne (2015) [20]), hereafter referred to as paper I, we have investigated the ground-state properties of Nd, Ce and Sm isotopes within Hartree–Fock–Bogoliubov method with SLy5 Skyrme force in which the pairing strength has been generalized with a new proposed formula. However, that formula is more appropriate for the region of Nd. In this work, we have studied the ground-state properties of both even–even and odd Mo and Ru isotopes. For this, we have used Hartree–Fock–Bogoliubov method with SLy4 Skyrme force, and a new formula of the pairing strength which is more accurate for this region of nuclei. The results have been compared with available experimental data, the results of Hartree–Fock–Bogoliubov calculations based on the D1S Gogny effective nucleon–nucleon interaction and predictions of some nuclear models such as Finite Range Droplet Model (FRDM) and Relativistic Mean Field (RMF) theory.

Odd–even staggering in neutron drip line nuclei

We have done systematic Hartree–Fock–Bogoliubov calculations in coordinate space on the one-quasi-particle energies and binding energy odd–even staggering (OES) in semi-magic nuclei with the zero-range volume, mixed and surface pairing forces in order to explore the influence of their density dependence. The odd-N isotopes are calculated within the blocking scheme. The strengths for the pairing forces are determined in two schemes by fitting locally to reproduce pairing gap in 120Sn and globally to all available data on the OES of semi-magic nuclei with Z≥8. In the former calculations, there is a noticeable difference between the neutron mean gaps in neutron-rich O, Ca, Ni and Sn isotopes calculated with the surface pairing and those with the mixed and volume pairing. The difference gets much smaller if the globally optimized pairing strengths are employed. The heavier Pb isotopes show the opposite trend. Moreover, large differences between the mean gap and the OES may be expected in both calculations when one goes towards the neutron drip line.

Properties of nuclear matter from macroscopic–microscopic mass formulas

Based on the standard Skyrme energy density functionals together with the extended Thomas–Fermi approach, the properties of symmetric and asymmetric nuclear matter represented in two macroscopic–microscopic mass formulas: Lublin–Strasbourg nuclear drop energy (LSD) formula and Weizsäcker–Skyrme (WS*) formula, are extracted through matching the energy per particle of finite nuclei. For LSD and WS*, the obtained incompressibility coefficients of symmetric nuclear matter are K∞=230±11 MeV and 235±11 MeV, respectively. The slope parameter of symmetry energy at saturation density is L=41.6±7.6 MeV for LSD and 51.5±9.6 MeV for WS*, respectively, which is compatible with the liquid-drop analysis of Lattimer and Lim [4]. The density dependence of the mean-field isoscalar and isovector effective mass, and the neutron–proton effective masses splitting for neutron matter are simultaneously investigated. The results are generally consistent with those from the Skyrme Hartree–Fock–Bogoliubov calculations and nucleon optical potentials, and the standard deviations are large and increase rapidly with density. A better constraint for the effective mass is helpful to reduce uncertainties of the depth of the mean-field potential.

Proton–neutron pairing in N = Z nuclei: Quartetting versus pair condensation

The isoscalar proton–neutron pairing and isovector pairing, including both isovector proton–neutron pairing and like-particle pairing, are treated in a formalism which conserves exactly the particle number and the isospin. The formalism is designed for self-conjugate (N=Z) systems of nucleons moving in axially deformed mean fields and interacting through the most general isovector and isoscalar pairing interactions. The ground state of these systems is described by a superposition of two types of condensates, i.e., condensates of isovector quartets, built by two isovector pairs coupled to the total isospin T=0, and condensates of isoscalar proton–neutron pairs. The comparison with the exact solutions of realistic isovector–isoscalar pairing Hamiltonians shows that this ansatz for the ground state is able to describe with high precision the pairing correlation energies. It is also shown that, at variance with the majority of Hartree–Fock–Bogoliubov calculations, in the present formalism the isovector and isoscalar pairing correlations coexist for any pairing interactions. The competition between the isovector and isoscalar proton–neutron pairing correlations is studied for N=Z nuclei with the valence nucleons moving in the sd and pf shells and in the major shell above 100Sn. We find that in these nuclei the isovector pairing prevail over the isoscalar pairing, especially for heavier nuclei.

Ground state properties of even–even and odd Nd, Ce and Sm isotopes in Hartree–Fock–Bogoliubov method

In this work, we have studied the ground state properties of both even–even and odd Nd isotopes within Hartree–Fock–Bogoliubov method with SLy5 Skyrme force in which the pairing strength has been generalized with a new proposed formula. We calculated binding energies, two-neutron separation energies, quadrupole deformation, charge, neutron and proton radii. Similar calculations have been carried out for Ce and Sm in order to verify the validity of our pairing strength formula. The results have been compared with available experimental data, the results of Hartree–Fock–Bogoliubov calculations based on the D1S Gogny effective nucleon–nucleon interaction and predictions of some nuclear models such as finite range droplet model (FRDM) and relativistic mean field (RMF) theory.

A study of self-consistent Hartree–Fock plus Bardeen–Cooper–Schrieffer calculations with finite-range interactions

In this work we test the validity of a Hartree–Fock plus Bardeen–Cooper–Schrieffer model in which a finite-range interaction is used in the two steps of the calculation by comparing the results obtained to those found in fully self-consistent Hartree–Fock–Bogoliubov calculations using the same interaction. Specifically, we consider the Gogny-type D1S and D1M forces. We study a wide range of spherical nuclei, far from the stability line, in various regions of the nuclear chart, from oxygen to tin isotopes. We calculate various quantities related to the ground state properties of these nuclei, such as binding energies, radii, charge and density distributions, and elastic electron scattering cross sections. The pairing effects are studied by direct comparison with the Hartree–Fock results. Despite its relative simplicity, in most cases, our model provides results very close to those of the Hartree–Fock–Bogoliubov calculations, and it reproduces the empirical evidence of pairing effects rather well in the nuclei investigated.