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

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.

Figure 1

HFB energy of oxygen ${}^{20}\mathrm{O}$ with respect to mesh size $s$. The broken line is the HFB energy calculated with the 3DHO. Crosses ($\times$) denote the mesh sizes $s=0.67$, 0.71, 0.77, 0.83, 0.91, and 1.0 fm at which the HFB ground states are calculated.

Figure 2

Variations in the contributions of the components in the total energy of ${}^{20}\mathrm{O}$ with respect to the mesh size $s$. The proton (neutron) kinetic energy $E_{\mathrm{kin}}^{\left(p\right)}\left(E_{\mathrm{kin}}^{\left(n\right)}\right)$, proton (neutron) Gaussian mean-field energy $E_{\mathrm{G}}^{\left(p\right)}\left(E_{\mathrm{G}}^{\left(n\right)}\right)$, density-dependent part energy $E_{\mathrm{dd}}$, $LS$ part energy $E_{LS}$, Coulomb mean-field energy $E_{\mathrm{Cl}}$, and neutron pairing energy $E_{\mathrm{pair}}^{\left(n\right)}$ are displayed. Every quantity is measured from the value at $s=1.0\mathrm{fm}$.

Figure 3

HFB energy of magnesium ${}^{34}\mathrm{Mg}$ with respect to mesh size $s$. The broken line is the HFB energy calculated with the 3DHO. Crosses ($\times$) denote the mesh sizes $s=0.63$, 0.67, 0.71, 0.77, 0.83, 0.91, and 1.0 fm at which the HFB ground states are calculated.

Figure 4

Isoscalar quadrupole ($K=0$) vibrations of (a) ${}^{20}\mathrm{O}$ and (b) ${}^{34}\mathrm{Mg}$ by using the TDHFB with the LMHO. In each of the panels, the values of $Q_{20}\left(t\right)$ are measured from the initial value $Q_{20}(t=0)$.

Figure 5

(a) Deviation $\tilde{E}=E_{\mathrm{TDHFB}}-E_0$ of the TDHFB energy $E_{\mathrm{TDHFB}}$ from the initial value $E_0$ and (b) the neutron number ${\tilde{N}}_{\mathrm{neut}}=|N_{\mathrm{neut}}-12|$ from the accurate value 12 with respect to the time in the isoscalar quadrupole vibrations of ${}^{20}\mathrm{O}$ in Fig. 4.

Figure 6

Strength functions $S\left(E\right)$ of the isoscalar quadrupole ($K=0$) vibrations of (a) ${}^{20}\mathrm{O}$ and (b) ${}^{34}\mathrm{Mg}$ in Fig. 4 with respect to the excitation energy $E$. The solid (broken) curves are the results with the LMHO (3DHO), respectively. Artificial widths are 1.0 MeV in both panels (a) and (b).