Shell-model calculation with density-dependent interaction for $pf$-shell nuclei

TL;DR

This work develops a hybrid framework that merges density-functional theory with shell-model configuration mixing to study pf-shell nuclei in a $0ħω$ space, determining density-dependent TBMEs self-consistently from the ground-state SM wave function. It tests three finite-range density-dependent interactions—Gogny-D1S, Gogny-GT2, and M3Y-P6—against experimental data and GXPF1A, finding that all provide reasonable overall descriptions, with M3Y-P6 best capturing the $N=28$ magicity in $^{56}$Ni and related isotopes. The results show that density-dependent TBMEs can reproduce ground-state energies, low-lying spectra, and $E2$ strengths well, though cross-shell effects beyond the pf-shell and the exact treatment of deformation remain limiting factors. The study highlights the potential of density-functional–based SM interactions for describing shell evolution and magicity across mid-mfp mass regions, guiding future refinements in model space and spin-orbit/isovector terms.

Abstract

Shell-model calculations with density-dependent interactions are performed to investigate $pf$-shell nuclei, examining the ground-state energies, low-lying spectra, and $E2$ transition probabilities. The density-dependent terms in the interaction are self-consistently determined using the shell-model wave function for the ground state. We test three density-dependent interactions adapted from density functionals of Gogny-D1S, Gogny-GT2, and M3Y-P6. The shell-model results satisfactorily agree with the experimental data. However, the Gogny-D1S and Gogny-GT2 fail to reproduce the magicity of $N=28$, while it is properly described by the M3Y-P6 functional.

Figures (12)

• Figure 1: Ground-state energy of $^{44}$Ti as a function of the HO energy, $\hbar\omega$. The blue, red, and green solid lines are the SM results obtained with the Gogny-D1S, GT2, and M3Y-P6 functionals. The horizontal and vertical dotted lines denote the experimental value and the empirical HO energy, respectively.
• Figure 2: Correlation of the all TBMEs of GXPF1A against (a) Gogny-D1S, (b) Gogny-GT2, and (c) M3Y-P6 in $^{44}$Ti. The red open squares denote the $J=0$ matrix elements. The insets show the monopole matrix elements.
• Figure 3: Ground-state spin, $J$, given by the shell-model results with (a) Gogny-D1S, (b) Gogny-GT2, and (c) M3Y-P6 functionals. The spin $J$ (even-mass nuclei) or $2J$ (odd-mass nuclei) is indicated by colors. The circles (triangles) denote the agreement (disagreement) with the experimental data. The even-even nuclei are omitted since their ground-state spin-parities are trivial, $0^+$ without exception.
• Figure 4: One-neutron separation energies of (a) Ti, (b) Cr, (c) Fe, and (d) Ni isotopes as a function of the neutron number. The blue, red, and green lines denote the theoretical results with the Gogny-D1S, Gogny-GT2, and M3Y-P6 interactions, respectively. The black circles denote the experimental values taken from Wang2021.
• Figure 5: Level schemes of even-mass Ni isotopes. The experimental values are compared with the SM results obtained using GXPF1A, M3Y-P6, Gogny-GT2, and Gogny-D1S interactions.