Systematic study of infrared energy corrections in truncated oscillator spaces with Gogny energy density functionals

We study the convergence properties of nuclear binding energies and two-neutron separation energies obtained with self-consistent mean-field calculations based on the Hartree-Fock-Bogolyubov (HFB) method with Gogny-type effective interactions. Owing to lack of convergence in a truncated working basis, we employ and benchmark one of the recently proposed infrared energy correction techniques to extrapolate our results to the limit of an infinite model space. We also discuss its applicability to global calculations of nuclear masses.

Figure 1

Spherical harmonic oscillator levels for two different values of the oscillator length (a) $b=1.65$ fm and (b) $b=2.45$ fm.

Figure 2

HFB energies calculated in different bases $N_{\mathrm{OS}}=11,\cdots ,21$ (see labels at each curve) as a function of oscillator length $b$ for (a) ${}^{16}\mathrm{O}$ and (b) ${}^{120}\mathrm{Cd}$.

Figure 3

Convergence patters of HFB energies with enlargement of the basis dimension, defined as $E^{\prime }\left(N_{\mathrm{OS}}\right)=E_{\min }\left(N_{\mathrm{OS}}\right)-E_{\min }\left(11\right)$ for (a) cadmium and (b) oxygen nuclei.

Figure 4

Obtained HFB energy gains by including two more major oscillator shells in the working basis, $\mathrm{\Delta }E\left(N_{\mathrm{OS}}\right)=E_{\min }(N_{\mathrm{OS}}-2)-E_{\min }\left(N_{\mathrm{OS}}\right)$, for (a) oxygen and (b) cadmium isotopes.

Figure 5

(a) calculated HFB energies for the nucleus ${}^{16}\mathrm{O}$ using SHO bases with $N_{\mathrm{OS}}=11,\cdots ,17$ as a function of $L_{\mathrm{eff}}$. Filled symbols indicate g.s. energies having ${\mathrm{\Lambda }}_{\mathrm{UV}}>750$ MeV/$c$, the red line shows the converged result, and other continuous and dot-dashed lines are the corresponding fits to Eq. (13); (b) the inset shows $E_{\infty }$ values for different extrapolations; (c) HFB energies calculated in a basis having $N_{\mathrm{OS}}=13$ and filtered out by conditions ${\mathrm{\Lambda }}_{\mathrm{thr}}=850$, 950, and 1050 MeV/$c$. The corresponding fits are shown with dashed lines; (d) residuals defined as $dE=E\left(L_{\mathrm{eff}}\right)-E_L\left(L_{\mathrm{eff}}\right)$; and (e) the corresponding extrapolations. See figure labels.

Figure 6

Difference of the calculated $E_{\min }\left(21\right)$ and the energy obtained with IR extrapolations from different combinations of basis dimension $N_{\mathrm{OS}}$ and ${\mathrm{\Lambda }}_{\mathrm{thr}}$ values.

Figure 7

Similar to Figs. 5 and 5, but for ${}^{120}\mathrm{Cd}$ nucleus for $N_{\mathrm{OS}}=15$ and 17 bases. The $E_{\min }$ values is indicated by the dashed red line.

Figure 8

Similar to Fig. 6 but for ${}^{120}\mathrm{Cd}$.

Figure 9

$\mathrm{\Delta }{E}_{\mathrm{thr}}, \mathrm{\Delta }{E}_{\mathrm{OS}}$, and $\mathrm{\Delta }{E}_{\mathrm{gain}}$ for ${}^{90–152}\mathrm{Cd}$ isotopes as a function of the neutron number.

Figure 10

Similar to Fig. 9 but as a function of the $S_{2n}$, which were obtained directly from HFB calculations in the $N_{\mathrm{OS}}=21$ basis without extrapolations.

Figure 11

$S_{2n}$ energies of cadmium isotopes calculated in different basis dimensions (shaded bins). Color code and symbols are same as in Fig. 3.

Figure 12

The plot shows the results of IR extrapolation for nuclei in cadmium, tin, and tellurium isotopic chains as a function of neutron number in form of the differences $E_{\min }\left(21\right)-E_{\infty }\left(N_{\mathrm{OS}}\right)$ with momentum threshold ${\mathrm{\Lambda }}_{\mathrm{thr}}=750$ MeV/$c$. Two dimensions of model space are considered: $N_{\mathrm{OS}}=17$ (hollow symbols) and $N_{\mathrm{OS}}=19$ (filled symbols), separately. The associated error bars represent the spread in corresponding $E_{\infty }$ values from the variation of the threshold parameter in the interval $700<{\mathrm{\Lambda }}_{\mathrm{thr}}<900$ MeV/$c$.