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Physical Review C

Time-dependent Hartree-Fock-Bogoliubov calculations using a Lagrange mesh with the Gogny interaction

Yukio Hashimoto

Published Sep 6, 2013• 25 citations
Publisher

Abstract

A new numerical method of calculating the Hartree-Fock-Bogoliubov (HFB) and time-dependent HFB (TDHFB) with the Gogny interaction is proposed. The three-dimensional harmonic-oscillator (3DHO) basis functions are replaced by one-dimensional spatial grid points of Lagrange mesh plus two-dimensional harmonic-oscillator basis functions in the perpendicular plane to the direction of the spatial grid points [Lagrange mesh and harmonic oscillator (LMHO)]. By using the LMHO, the calculations of the HFB and TDHFB are carried out in the typical nuclei of ${}^{20}\mathrm{O}$ and ${}^{34}\mathrm{Mg}$ as an illustration of the feasibility of the TDHFB formulation with the LMHO. The strength functions of the quadrupole vibrations ($K=0$) are obtained and are compared with the ones calculated with the 3DHO.

DOI
10.1103/physrevc.88.034307

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A new numerical method of calculating the Hartree-Fock-Bogoliubov (HFB) and time-dependent HFB (TDHFB) with the Gogny interaction is proposed. The three-dimensional harmonic-oscillator (3DHO) basis functions are replaced by one-dimensional spatial grid points of Lagrange mesh plus two-dimensional harmonic-oscillator basis functions in the perpendicular plane to the direction of the spatial grid points [Lagrange mesh and harmonic oscillator (LMHO)]. By using the LMHO, the calculations of the HFB and TDHFB are carried out in the typical nuclei of 20O{}^{20}\mathrm{O}20O and 34Mg{}^{34}\mathrm{Mg}34Mg as an illustration of the feasibility of the TDHFB formulation with the LMHO. The strength functions of the quadrupole vibrations (K=0K=0K=0) are obtained and are compared with the ones calculated with the 3DHO.
We present simple equations for a canonical-basis (Cb) formulation of the time-dependent Hartree-Fock-Bogoliubov (TDHFB) theory. The equations are obtained from the TDHFB theory with an approximation that the pair potential is assumed to be diagonal in the Cb. The Cb formulation significantly reduces the computational cost. We apply the method to linear-response calculations for even-even light nuclei and demonstrate its capability and accuracy by comparing our results with recent calculations of the quasiparticle random-phase approximation with Skyrme functionals. We show systematic studies of E1E1E1 strength distributions for Ne and Mg isotopes. The evolution of the low-lying pygmy strength seems to be determined by the interplay of several factors, which include the neutron excess, the separation energy, the neutron-shell effects, the deformation, and the pairing.
We have developed a new theory to study excited states and dynamics of atomic nuclei, which we call the Canonical-basis time-dependent Hartree-Fock-Bogoliubov (Cb-TDHFB). The CbTDHFB is a simplified theory of the full TDHFB theory, treating pairing energy functional with a BCS-like approximation which is assumed to be diagonal in the canonical basis. Implementing the theory with a real-space and real-time method, we can describe nuclear excitations and dynamics taking account of deformation effects and pairing correlations, even for heavy nuclei with a reasonable computational costs. In this thesis, we first present a derivation of the CbTDHFB theory and we apply the method to linear-response calculations for even-even light to heavy nuclei and demonstrate its capability and accuracy by comparing our results with recent calculations of the quasi-particle random-phase approximation (QRPA) which is equivalent to the small amplitude limit of TDHFB, with Skyrme functionals. We derived the Cb-TDHFB equations starting with the full TDHFB equations, introducing a simplified form for the pairing energy functional. In the derivation, we start with the ordinary TDHFB equation written in terms of the time-dependent generalized density matrix R(t) which is composed of the time-dependent normal density matrix ρ(t) and the pairing tensor κ(t). We then rewrite the equation in the canonical basis, which diagonalize ρ(t). The derived TDHFB equations in the canonical basis has a very simple form. However, they are not useful practically for a general case since it costs much to obtain the canonical basis themselves at each time. Solving the time-dependent equation for the canonical basis is as difficult as solving the full TDHFB equation. This difficulty disappears if we employ a simplified pairing energy density functional, which is equivalent to that employed in the BCS approximation for ground state calculations. For linear response calculations, we present results of the Cb-TDHFB calculations for excited states of some spherical and deformed nuclei to show performance of the method. We show strength functions of isovector dipole and isoscalar quadrupole excitations for some spherical and deformed nuclei. First, we show the comparison results of Cb-TDHFB with the QRPA in order to confirm the justification of canonical-basis formulation and the reduction of computational cost with this new framework. We show isoscalar quadrupole strength functions of neutron-rich nucleus 34Mg for confirmation of our new method, and also discuss full self-consistent calculations of 24Mg as an example of light nucleus and 208Pb and 154Sm as an example of heavy nuclei, in order to be clear, especially, the effects of spin-orbit and Coulomb residual interaction. Next, in order to show typical strength functions for spherical and deformed nuclei, we discuss isovector dipole and isoscalar quadrupole mode for 34Si as an example of spherical nucleus, for 34Mg as an example of prolate deformed nucleus and for 24Ne as an example of oblate deformed nucleus. For a confirmation of computational cost of the calculation using Cb-TDHFB, we isovector dipole strength functions of deformed heavy nuclei 172Yb and 236−240U. We show then the comparison our results with experimental data for photo-nuclear reaction cross sections. And we also show systematic calculations low-energy E1 strength which is often called Pygmy dipole resonance and which is well-known as a important strength to understand nucleosynthesis. In this thesis, we discuss the appearance of low-energy E1 strength functions of C, O, Ne, Mg, Si, S, Ar and Ca isotopes which are relatively light nuclei.
A canonical-basis formulation of the time-dependent Hartree-Fock-Bogoliubov (TDHFB) theory is obtained with an approximation that the pair potential is assumed to be diagonal in the time-dependent canonical basis. The canonical-basis formulation significantly reduces the computational cost. We apply the method to linear-response calculations for even-even nuclei. E 1 strength distributions for proton-rich Mg isotopes are systematically calculated. The calculation suggests strong Landau damping of giant dipole resonance for drip-line nuclei.