394 pages, 17 figures, 2 tables, published in Atomic Data and Nuclear Data Tables, data file in the TXT form is available for download under "Ancillary files"
发表信息：
Peng Guo, et. al. (DRHBc Mass Table Collaboration), Atomic Data and Nuclear Data Tables 158 (2024) 101661
The mass table in the deformed relativistic Hartree-Bogoliubov theory in continuum (DRHBc) with the PC-PK1 density functional has been established for even-$Z$ nuclei with $8\le Z\le120$, extended from the previous work for even-even nuclei [Zhang $\it{et.~al.}$ (DRHBc Mass Table Collaboration), At. Data Nucl. Data Tables 144, 101488 (2022)]. The calculated binding energies, two-nucleon and one-neutron separation energies, root-mean-square (rms) radii of neutron, proton, matter, and charge distributions, quadrupole deformations, and neutron and proton Fermi surfaces are tabulated and compared with available experimental data. A total of 4829 even-$Z$ nuclei are predicted to be bound, with an rms deviation of 1.477 MeV from the 1244 mass data. Good agreement with the available experimental odd-even mass differences, $α$ decay energies, and charge radii is also achieved. The description accuracy for nuclear masses and nucleon separation energies as well as the prediction for drip lines is compared with the results obtained from other relativistic and nonrelativistic density functional. The comparison shows that the DRHBc theory with PC-PK1 provides an excellent microscopic description for the masses of even-$Z$ nuclei. The systematics of the nucleon separation energies, odd-even mass differences, pairing energies, two-nucleon gaps, $α$ decay energies, rms radii, quadrupole deformations, potential energy curves, neutron density distributions, and neutron mean-field potentials are discussed.
The deformed relativistic Hartree-Bogoliubov theory in continuum with the density-dependent meson-nucleon couplings is developed. The formulism is briefly presented with the emphasis on handling the density-dependent couplings, meson fields, and potentials in axially deformed system with partial wave method. Taking the neutron-rich nucleus $^{38}$Mg as an example, the newly developed code is verified by the spherical relativistic continuum Hartree-Bogoliubov calculations, where only the spherical components of the densities are considered. When the deformation is included self-consistently, it…
The Thermal Hartree-Fock-Bogoliubov approximation with reprojection to good quantum numbers is analyzed in a canonical ensemble. Simple recipes are given for the evaluation of traces, gradients and expectation values in the canonical ensemble.
Halo phenomenon in deformed nuclei is studied by using a fully self-consistent deformed relativistic Hartree-Bogoliubov model in a spherical Woods-Saxon basis with the proper asymptotic behavior at large distance from the nuclear center. Taking a deformed neutron-rich and weakly bound nucleus $^{44}$Mg as an example and by examining contributions of the halo, deformation effects, and large spatial extensions, we show a decoupling of the halo orbitals from the deformation of the core.
After a brief review of the Hartree-Fock-Bogoliubov method with Skyrme effective interactions, we show how it can be applied to the description of various nuclear systems, from finite nuclei to neutron-star crusts.
We solve the Hartree-Fock-Bogoliubov (HFB) equations for a spherical mean field and a pairing potential with the inverse Hamiltonian method, which we have developed for the solution of the Dirac equation. This method is based on the variational principle for the inverse Hamiltonian, and is applicable to Hamiltonians that are bound neither from above nor below. We demonstrate that the method works well not only for the Dirac but also for the HFB equations.
A formula to calculate a norm overlap between Hartree-Fock-Bogoliubov (HFB) states with the odd number parity (one quasi-particle excited states) is derived with help of the Grassmann numbers and the Fermion coherent states. The final form of the formula is expressed in terms of a product of the Pfaffian for a neighboring even-even system (the zero quasi-particle state), and an extra factor consisting of the Bogoliubov transformation matrix and the anti-symmetric matrix in Thouless' HFB ansatz for the even-even system.
In order to describe the exotic nuclear structure in unstable odd-$A$ or odd-odd nuclei, the deformed relativistic Hartree Bogoliubov theory in continuum has been extended to incorporate the blocking effect due to the odd nucleon. For a microscopic and self-consistent description of pairing correlations, continuum, deformation, blocking effects, and the extended spatial density distribution in exotic nuclei, the deformed relativistic Hartree Bogoliubov equations are solved in a Woods-Saxon basis in which the radial wave functions have a proper asymptotic behavior at large $r$. The formalism an…
We have studied odd-even nuclear mass staggering with the Skyrme-Hartree-Fock-Bogoliubov theory by employing isoscalar and isovector contact pairing interactions. By reproducing the empirical odd-even mass differences of the Sn isotopic chain, the strengths of pairing interactions are determined. The optimal strengths adjusted in this work can give better description of odd-even mass differences than that fitted by reproducing the experimental neutron pairing gap of $^{120}$Sn.
To describe the yrast states in weakly bound nuclei, I directly solve the coordinate-space cranked Skyrme-Hartree-Fock-Bogoliubov equation on a three-dimensional lattice with the continuum states discretized in a box. After the numerical demonstration for the ground-state band in a medium-mass nucleus, I apply the newly-developed method to neutron-rich even-$N$ Mg isotopes. I find that the appearance of the significantly low $I^π=2^+$ state in $^{40}$Mg is mainly due to the suppression of pairing. The calculation predicts that the $2^+$ state in $^{42}$Mg appears as high in energy as in $^{34\…
The accuracy of three different sets of Hartree-Fock-Bogoliubov calculations of nuclear binding energies is systematically evaluated. To emphasize minor fluctuations, a second order, four-point mass relation, which almost completely eliminates smooth aspects of the binding energy, is introduced. Applying this mass relation yields more scattered results for the calculated binding energies. By examining the Gaussian distributions of the non-smooth aspects which remain, structural differences can be detected between measured and calculated binding energies. Substructures in regions of rapidly cha…
We present a pfaffian formula for projection and symmetry restoration for wave functions of the general Bogoliubov form, including quasiparticle excited states and linear combinations of them. This solves a long-standing problem in calculating states of good symmetry, arising from the sign ambiguity of the commonly used determinant formula. A simple example is given of projecting good particle number and angular momentum from a Bogoliubov wave function in the Fock space of a single j-shell.
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.
An extension of the time-dependent Hartree-Fock-Bogoliubov theory (ETDHFB) which includes higher-order effects such as screening of the pairing correlation is proposed. ETDHFB is applied to a fermion system trapped in a harmonic potential to test its feasability by comparison with the exact solution. With the use of perturbative expressions for the pairing tensor and the two-body density matrix derived from ETDHFB, the screening effect is investigated for atomic fermion systems and isotopes of tin nuclei. It is found that the screening effect on the pairing correlation is not significant.
The $N = 28$ shell gap in sulfur, argon, calcium and titanium isotopes is investigated in the framework of relativistic continuum Hartree-Bogoliubov (RCHB) theory. The evolutions of neutron shell gap, separation energy, single particle energy and pairing energy are analyzed, and it is found that $N = 28$ shell gap is quenched in sulfur isotopes but persists in argon, calcium and titanium isotopes. The evolution of $N = 28$ shell gap in $N = 28$ isotonic chain is discussed, and the erosion of $N = 28$ shell gap is understood with the evolution of potential with proton number.
Symmetry-projected Hartree-Fock-Bogoliubov (HFB) equations are derived using the variational ansatz for the generalized one-body density-matrix in the Valatin form. It is shown that the projected-energy functional can be completely expressed in terms of the HFB density-matrix and the pairing-tensor. The variation of this projected-energy is shown to result in HFB equations with modified expressions for the pairing-potential and the Hartree-Fock field. The expressions for these quantities are explicitly derived for the case of particle number-projection. The numerical applicability of this proj…
Background: The Density-constraint Time-dependent Hartree-Fock method is currently the tool of choice to predict fusion cross-sections. However, it does not include pairing correlations, which have been found recently to play an important role. Purpose: To describe the fusion cross-section with a method that includes the superfluidity and to understand the impact of pairing on both the fusion barrier and cross-section. Method: The density-constraint method is tested first on the following reactions without pairing, $^{16}$O+$^{16}$O and $^{40}$Ca+$^{40}$Ca. A new method is developed, the Densi…
We present an overview of the Hartree-Fock-Bogoliubov (HFB) theory of nucleonic superfluidity for finite nuclei. After introducing basic concepts related to pairing correlations, we show how the correlated pairs are incorporated into the HFB wave function. Thereafter, we present derivation and structure of the HFB equations within the superfluid nuclear density functional formalism and discuss several aspects of the theory, including the unitarity of the Bogoliubov transformation in truncated single-particle and quasiparticle spaces, form of the pairing functional, structure of the HFB continu…
We perform particle-number projected mean-field study using the recently developed symmetry-projected Hartree-Fock-Bogoliubov (HFB) equations. Realistic calculations have been performed in sd- and fp-shell nuclei using the shell model empirical intearctions, USD and GXPFIA. It is demonstrated that the mean-field results for energy surfaces, obtained with these shell model interactions, are quite similar to those obtained using the density functional approaches. Further, it is shown that particle-number projected results, for neutron rich isotopes, can lead to different ground-state shapes in c…
Ground state properties of carbon isotopes, including root-mean-square radii, neutron separation energies, single particle spectra, and shapes are systematically studied with the deformed relativistic Hartree-Bogoliubov theory in continuum. The calculations with the effective interaction PK1 reproduce the available data reasonably well. The shell evolution in this isotopic chain is investigated by examining the single particle spectra. The inversion of neutron orbitals $ν2s_{1/2}$ and $ν1d_{5/2}$ compared with the order of neutron orbitals in stable nuclei is revealed in $^{15-20,22}\mathrm{C}…
The Relativistic Hartree-Fock-Bogoliubov model for axially deformed nuclei (RHFBz) is introduced. The model is based on an effective Lagrangian with density-dependent meson-nucleon couplings in the particle-hole channel, and the pairing part of the Gogny force is used in the pairing channel. The RHFBz quasiparticle equations are solved by expansion in the basis of a deformed harmonic oscillator. Illustrative RHFBz calculations are performed for Carbon, Neon and Magnesium isotopes. The effect of the explicitly including the pion field is investigated for binding energies, deformation parameters…
We provide an efficient and accurate method for converting Hartree-Fock-Bogoliubov wave functions into matrix product states (MPSs). These wave functions, also known as Bogoliubov vacua, exhibit a peculiar entanglement structure that the eigenvectors of the reduced density matrix are also Bogoliubov vacua. We exploit this important feature to obtain their optimal MPS approximation and derive an explicit formula for corresponding MPS matrices. The performance of our method is benchmarked with the Kitaev chain and the Majorana-Hubbard model on the honeycomb lattice. The approach facilitates the …
We formulate a new scheme of the Hartree-Fock-Bogoliubov mean-field theory applicable to weakly bound and pair correlated deformed nuclei using the coordinate-space Green's function technique. On the basis of a coupled-channel representation of the quasiparticle wave function expanded in terms of the partial waves, we impose the correct boundary condition of the asymptotically out-going waves on the continuum quasiparticle states. We perform numerical analysis for $^{38}$Mg to illustrate properties of the continuum quasiparticle states and the pair correlation in deformed nuclei near the neutr…
We show that the lowest-energy solution of the Hartree-Fock-Bogoliubov (HFB) equation has the even particle-number parity as long as the time-reversal symmetry is conserved in the HFB Hamiltonian without null eigenvalues. Based on this finding, we give a rigorous foundation of a method for solving the HFB equation to describe the ground state of odd-mass nuclei by employing a time-reversal anti-symmetric constraint operator to the Hamiltonian, where one obtains directly the ground state as a self-consistent solution of the cranked-HFB-type equation. Numerical analysis is done for the neutron-r…
The transfer reaction between two nuclei in the superfluid phase is studied with the Time-dependent Hartree-Fock-Bogoliubov (TDHFB) theory. In order to restore the symmetry of the relative gauge angle a set of independent TDHFB evolutions is done. Then the transfer probability is computed using a triple projection method. This method is first tested to determine the transfer probabilities on a toy model and compared to the exact solution. It is then applied to the reactions $^{20}$O+$^{20}$O and $^{14}$O+$^{20}$O in a realistic framework with a Gogny interaction.
Weakly-bound deformed nuclei have been studied by the Skyrme Hartree-Fock-Bogoliubov (HFB) approach in large coordinate-space boxes. In particular, the box-size dependence of the HFB calculations of weakly-bound deformed nuclei are investigated, including the particle density and pairing density distributions at nuclear surfaces, the near-threshold resonant and continuum quasiparticle spectra, and energetic properties. The box size may have larger influences in pairing properties than in other bulk properties. We demonstrate that large-box calculations of weakly-bound nuclei are important to p…
In a previous work [Int. J. Mod. Phys. E 24, 1550073 (2015)], hereafter referred 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 wh…
We present a new formula to calculate matrix elements of a general unitary operator with respect to Hartree-Fock-Bogoliubov states allowing multiple quasi-particle excitations. The Balian-Brézin decomposition of the unitary operator (Il Nuovo Cimento B 64, 37 (1969)) is employed in the derivation. We found that this decomposition is extremely suitable for an application of Fermion coherent state and Grassmann integrals in the quasi-particle basis. The resultant formula is compactly expressed in terms of the Pfaffian, and shows the similar bipartite structure to the formula that we have previou…
A numerical method to integrate the time-dependent Hartree-Fock Bogoliubov (TDHFB) equations with Gogny interaction is proposed. The feasibility of the TDHFB code is illustrated by the conservation of the energy, particle numbers, and center-of-mass in the small amplitude vibrations of oxygen 20. The TDHFB code is applied to the isoscalar quadrupole and/or isovector dipole vibrations in the linear (small amplitude) region in oxygen isotopes (masses A = 18,20,22 and 24), titanium isotopes (A = 44,50,52 and 54), neon isotope (A = 26), and magnesium isotopes (A = 24 and 34). The isoscalar quadrup…
The coordinate-space representation of the Hartree-Fock-Bogoliubov theory is the method of choice to study weakly bound nuclei whose properties are affected by the quasiparticle continuum space. To describe such systems, we developed a three-dimensional Skyrme-Hartree-Fock-Bogoliubov solver HFBFFT based on the existing, highly optimized and parallelized Skyrme-Hartree-Fock code Sky3D. The code does not impose any self-consistent spatial symmetries such as mirror inversions or parity. The underlying equations are solved in HFBFFT directly in the canonical basis using the fast Fourier transform.…
The quasi-particle energy spectrum of the Hartree-Fock-Bogoliubov (HFB) equations contains discrete bound states, resonances, and non-resonant continuum states. We study the structure of the unbound quasi-particle spectrum of weakly bound nuclei within several methods that do not rely on imposing scattering or outgoing boundary conditions. Various approximations are examined to estimate resonance widths. It is shown that the stabilization method works well for all HFB resonances except for very narrow ones. The Thomas-Fermi approximation to the non-resonant continuum has been shown to be very …
In this work, we have studied 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 bind- ing 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 bas…
The equation of state (EoS) of the outer crust of a cold non-accreting magnetar has been determined using the model of Lai and Shapiro (1991). For this purpose, we have made use of the latest experimental atomic mass data complemented with a Hartree-Fock-Bogoliubov (HFB) mass model. Magnetar crusts are found to be significantly different from the crusts of ordinary neutron stars.
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. The restoration of rotational symmetry does not require using bases closed under rotation. Furthermore, the SeaLL1 functional was added and the calculation of the Coulomb potential was improved. Finally, the code has been refactored to facilitate maintenance and future developments.
