Renormalized interactions with a realistic single-particle basis

Neutron-rich isotopes in the $\mathit{sdpf}$ space with $Z⩽14$ require modifications to derived effective interactions to agree with experimental data away from stability. A quantitative justification is given for these modifications because of the weakly bound nature of model space orbits via a procedure using realistic radial wave functions and realistic $\mathit{NN}$ interactions. The long tail of the radial wave function for loosely bound single-particle orbits causes a reduction in the size of matrix elements involving those orbits, most notably for pairing matrix elements, resulting in a more condensed level spacing in shell-model calculations. Example calculations are shown for ${}^{36}\mathrm{Si}$ and ${}^{38}\mathrm{Si}$.

Figure 1

Comparison of $\mathit{pf}$ neutron-neutron matrix elements (in MeV) for the renormalization procedure in the HO basis for the two target nuclei. The solid line $y=x$ denotes where the matrix elements would be identical.

Figure 2

Comparison of $\mathit{pf}$ neutron-neutron matrix elements (in MeV) for the renormalization procedure in the SHF basis for the two target nuclei. The solid line $y=x$ denotes where the matrix elements would be identical. Black dots correspond to matrix elements with $J>0$, whereas the $J=0$ matrix elements are split into three groups: $\mathit{ff}$-$\mathit{ff}$ (crosses), $\mathit{ff}$-$\mathit{pp}$ (diamonds), and $\mathit{pp}$-$\mathit{pp}$ (plus signs).

Figure 3

Comparison of the single-particle radial wave functions for ${}^{34}\mathrm{Si}$ in the HO and SHF bases.

Figure 4

Calculations of the lowest energy states for $J=0,2,4,6$ in ${}^{36}\mathrm{Si}$ relative to ${}^{34}\mathrm{Si}$ from the renormalization procedure for ${}^{34}\mathrm{Si}$ and ${}^{40}\mathrm{Ca}$, in the HO, CP, and SHF bases for both target nuclei. Crosses are used for calculations with ${}^{40}\mathrm{Ca}$ as the target nucleus.

Figure 5

Calculations of the lowest energy states for $J=0,2,4,6$ in ${}^{36}\mathrm{Si}$ relative to ${}^{34}\mathrm{Si}$ using the empirical SDPF-U interaction and the renormalization procedure for both ${}^{34}\mathrm{Si}$ and ${}^{40}\mathrm{Ca}$ as target nuclei, using the SHF and HO bases. Experimental data are shown for comparison, with a new mass from [11]. Crosses are used for calculations with ${}^{40}\mathrm{Ca}$ as the target nucleus.

Figure 6

Calculations for the lowest energy states for $J=0$ and $J=2$ in ${}^{38}\mathrm{Si}$ relative to ${}^{34}\mathrm{Si}$ using neutron-neutron matrix elements from SDPF-U and the renormalization procedure for both ${}^{34}\mathrm{Si}$ and ${}^{40}\mathrm{Ca}$ as target nuclei, using the SHF and HO bases. Crosses are used for calculations with ${}^{40}\mathrm{Ca}$ as the target nucleus.