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Register a new userApplication of the inverse Hamiltonian method to Hartree-Fock-Bogoliubov calculations
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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.
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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 Density-constraint Time-dependent Hartree-Fock-Bogoliubov method. Using the Gogny-TDHFB code, it is applied to the reactions $^{20}$O+$^{20}$O and $^{44}$Ca+$^{44}$Ca. Results: The Gogny approach for systems without pairing reproduces the experimental data well. The DC-TDHFB method is coherent with the TDHFB fusion threshold. The effect of the phase-lock mechanism is shown for those reactions. Conclusions: The DC-TDHFB method is a useful new tool to determine the fusion potential between superfluid systems and to deduce their fusion cross-sections.
The variational Hartree-Fock-Bogoliubov (HFB) mean-field theory is the starting point of various (ab initio) many-body methods dedicated to superfluid systems. While taking the zero-pairing limit of HFB equations constitutes a text-book problem when the system is of closed-(sub)shell character, it is typically, although wrongly, thought to be ill-defined whenever the naive filling of single-particle levels corresponds to an open-shell system. The present work demonstrates that the zero-pairing limit of an HFB state is mathematically well-defined, independently of the closed- or open-shell character of the system in the limit. Still, the nature of the limit state strongly depends on the underlying shell structure and on the associated naive filling reached in the zero-pairing limit for the particle number A of interest. All the analytical findings are confirmed and illustrated numerically. While HFB theory has been intensively scrutinized formally and numerically over the last decades, it still uncovers unknown and somewhat unexpected features. From this general perspective, the present analysis demonstrates that HFB theory does not reduce to Hartree-Fock theory even when the pairing field is driven to zero in the HFB Hamiltonian matrix.
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 $delta_{2n(p)}$. Although the results depend slightly on the effective Lagrangians used, the general set of magic numbers beyond $^{208}$Pb 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 $^{304}$120 as the next spherical doubly magic one beyond $^{208}$Pb. Shell effects are sensitive to various terms of the mean-field, such as the spin-orbit coupling, the scalar and effective masses.
In order to study structure of proto-neutron stars and those in subsequent cooling stages, it is of great interest to calculate inhomogeneous hot and cold nuclear matter in a variety of phases. The finite-temperature Hartree-Fock-Bogoliubov (FT-HFB) theory is a primary choice for this purpose, however, its numerical calculation for superfluid (superconducting) many-fermion systems in three dimensions requires enormous computational costs. To study a variety of phases in the crust of hot and cold neutron stars, we propose an efficient method to perform the FT-HFB calculation with the three-dimensional (3D) coordinate-space representation. Recently, an efficient method based on the contour integral of Greens function with the shifted conjugate-orthogonal conjugate-gradient method has been proposed [Phys. Rev. C 95, 044302 (2017)]. We extend the method to the finite temperature, using the shifted conjugate-orthogonal conjugate-residual method. We benchmark the 3D coordinate-space solver of the FT-HFB calculation for hot isolated nuclei and fcc phase in the inner crust of neutron stars at finite temperature. The computational performance of the present method is demonstrated. Different critical temperatures of the quadrupole and the octupole deformations are confirmed for $^{146}$Ba. The robustness of the shape coexistence feature in $^{184}$Hg is examined. For the neutron-star crust, the deformed neutron-rich Se nuclei embedded in the sea of superfluid low-density neutrons appear in the fcc phase at the nucleon density of 0.045 fm$^{-3}$ and the temperature of $k_B T=200$ keV. The efficiency of the developed solver is demonstrated for nuclei and inhomogeneous nuclear matter at finite temperature. It may provide a standard tool for nuclear physics, especially for the structure of the hot and cold neutron-star matters.
To study the exotic odd nuclear systems, the self-consistent continuum Skyrme-Hartree-Fock-Bogoliubov theory formulated with Greens function technique is extended to include blocking effects with the equal filling approximation. Detailed formula are presented.To perform the integrals of the Greens function properly, the contour paths $C_{rm b}^{-}$ and $C_{rm b}^{+}$ introduced for the blocking effects should include the blocked quasi-particle state but can not intrude into the continuum area. By comparing with the box-discretized calculations, the great advantages of the Greens function method in describing the extended density distributions, resonant states, and the couplings with the continuum in exotic nuclei are shown. Finally, taking the neutron-rich odd nucleus $^{159}$Sn as an example, the halo structure is investigated by blocking the quasi-particle state $1p_{1/2}$. It is found that it is mainly the weakly bound states near the Fermi surface that contribute a lot for the extended density distributions at large coordinate space.
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