Effective interactions in the $sd$ shell

We perform a quantitative study of the microscopic effective shell-model interactions in the valence $sd$ shell, obtained from modern nucleon-nucleon potentials, chiral N3LO, JISP16, and Daejeon16, using no-core shell-model wave functions and the Okubo-Lee-Suzuki transformation. We investigate the monopole properties of those interactions in comparison with the phenomenological universal $sd$-shell interaction, USDB. Theoretical binding energies and low-energy spectra of O isotopes and of selected $sd$-shell nuclei, are presented. We conclude that there is a noticeable improvement in the quality of the effective interaction when it is derived from the Daejeon16 potential. We show that its proton-neutron centroids are consistent with those from USDB. We then propose monopole modifications of the Daejeon16 centroids in order to provide an adjusted interaction yielding significantly improved agreement with the experiment. A spin-tensor decomposition of two-body effective interactions is applied in order to extract more information on the structure of the centroids and to understand the reason for deficiencies arising from our current theoretical approximations. The issue of the possible role of the three-nucleon forces is addressed.

Figure 1

Low-energy spectra (eight lowest states) of ${}^{18}\mathrm{F}$ obtained from USDB and from the microscopic effective interactions in comparison with experiment (eight lowest positive-parity states). The USDB single-particle energies have been used in all calculations, except for DJ16th, where theoretical values from Table 2 are used with DJ16 TBMEs. DJ16A is a monopole-modified version of DJ16 (see Sec. 4d).

Figure 2

Experimental ground state energies of O isotopes relative to the ground state energy of ${}^{16}\mathrm{O}$ in comparison with theoretical results, calculated using microscopic effective interactions. Calculations using JISP16 and DJ16 are performed with the USDB single-particle energies and with the ${(A/A_0)}^{-0.3}$ mass dependence of TBMEs. Results denoted as JISP16th (empty cyan squares) and DJ16th (empty black circles) are obtained from the same valence-space TBMEs (without any mass dependence) and with the theoretical single-particle energies. The experimental data (extrapolations included) are from AME2012 [56].

Figure 3

Experimental low-energy spectrum of ${}^{21,23}\mathrm{O}$ in comparison with theoretical results, calculated using the phenomenological USDB interaction and the microscopic effective interactions. Calculations with JISP16 and DJ16 employ the USDB single-particle energies and the ${(A/A_0)}^{-0.3}$ mass dependence of TBMEs, while spectra denoted as JISP16th and DJ16th are obtained from the same valence-space TBMEs (without any mass dependence) and with the theoretical single-particle energies.

Figure 4

Variation of ESPEs in O isotopes with neutron number $N$ calculated using the empirical (USDB) and the microscopic effective interactions (which employ the scaling with $A$ described in the text). DJ16A is a monopole-modified version of DJ16 (see text for details). The results within IMSRG have been obtained using the original Hamiltonians from Ref. [26] for O isotopes.

Figure 5

Variation of the neutron ESPEs in $N=14$ isotones calculated using the phenomenological (USDB) and the microscopic effective interactions.

Figure 6

Experimental low-energy spectra of ${}^{21-24}\mathrm{O}$ in comparison with theoretical results, obtained from USDB and from the microscopic effective interactions. The experimental data are from Ref. [61].

Figure 7

Experimental ground state energies of O isotopes relative to the ground state energy of ${}^{16}\mathrm{O}$, in comparison with theoretical results, obtained from USDB and from the microscopic effective interactions. The experimental data (extrapolations included) are from AME2012 [56].

Figure 8

Low-energy spectra of odd-$A{}^{19-27}\mathrm{F}$ and ${}^{39}\mathrm{K}$, obtained from USDB and microscopic effective interactions, in comparison with the experimental data on positive-parity states from Refs. [61, 62, 63]. For ${}^{39}\mathrm{K}$, we show experimentally deduced centroids from Ref. [64].

Figure 9

Low-energy spectrum of ${}^{26}\mathrm{F}$ obtained from USDB and microscopic effective interactions, in comparison with the experimental data from Ref. [62]. The spectrum entitled “IMSRG” is quoted also from Ref. [62].

Figure 10

Low-energy spectrum of ${}^{22}\mathrm{Na}$ obtained from USDB and microscopic effective interactions, in comparison with the experimental data on positive-parity states from Ref. [61]. $T=0$ states are shown in black, while $T=1$ states are plotted in red.

Figure 11

Low-energy spectra of ${}^{28,29}\mathrm{Si}$ obtained from USDB and microscopic effective interactions, in comparison with the experimental data on positive-parity states from Ref. [61].

Figure 12

Low-energy spectrum of ${}^{32}\mathrm{S}$ obtained from USDB and microscopic effective interactions, in comparison with the experimental data on positive-parity states from Ref. [61].

Figure 13

Low-energy spectrum of ${}^{24}\mathrm{Mg}$ obtained from USDB and microscopic effective interactions, in comparison with the experimental data on positive-parity states from Ref. [61]. Different rotation-vibration bands are distinguished by color.