Shell model study of isobaric analog states for $T_z= \pm 2$ nuclei using isospin non-conserving interactions

TL;DR

This work uses sd-shell shell-model calculations with isospin-non-conserving interactions USDC and USDCm to investigate mirror energy differences in $T_z=\pm 2$ nuclei across $A=20$–$36$, aiming to understand ISB effects and the Thomas–Ehrman shift. It analyzes spectra, MEDs, B(E2) transitions, and isoscalar/isovector E2 matrix elements ($M_0$, $M_1$), along with magnetic and quadrupole moments and single-particle occupancies, emphasizing the role of the weakly bound $s_{1/2}$ orbital and possible continuum coupling. The results generally reproduce experimental levels and MEDs across multiple mirror pairs, though notable TE-related deviations persist (e.g., $^{22}$Al, $^{24}$Si), and some cases require continuum coupling considerations. The study demonstrates that incorporating INC and Coulomb effects in USD-type interactions, together with continuum considerations near the proton drip line, is essential for accurately describing mirror nuclei and has implications for nuclear structure theory and related astrophysical rates.

Abstract

In order to comprehend the process underlying mirror energy differences in mirror pairs, we have performed shell-model calculations for $T_z= \pm 2$ $sd$-shell nuclei in the mass range $A$= 20 to 36 and neutron number varying from $N$= 8 to 20. Isospin-symmetry breaking (ISB) is responsible for the mirror energy difference of excited states. We have investigated the {\color{black}isospin non-conserving} interactions: USDC and USDCm to explore the low-lying energy spectra, mirror energy differences, isoscalar ($M_0$), isovector ($M_1$) matrix elements, \textit{E2} transition probability, magnetic ($μ$), and quadrupole moments ($Q$) of mirror-pair and compared them with their available experimental data. The impact of single-particle states on weakly bound and unbound nuclear states are investigated, especially those of the $s$-wave. We have also analyzed single proton/neutron separation energies and proton/neutron occupancy for ($T_z$=-2)/($T_z$=+2) $sd$-shell nuclei.

Figures (13)

• Figure 1: Comparison between the calculated and experimental NNDC energy levels (solid lines) and the proton and neutron thresholds (dashed lines)(left), mirror energy differences for low-lying states and proton and neutron occupancies of single-particle orbits for isobaric analog states of $^{20}$Mg and $^{20}$O (right).
• Figure 2: $B(E2)\downarrow$ values for (top) $T_z=-{2}$ and (bottom) $T_z=+{2}$ nuclei. We have taken result of VS-IMSRG interaction from the Ref. E2_sd.
• Figure 3: Comparison between the calculated and experimental NNDC energy levels (solid lines) and the proton and neutron thresholds (dashed lines)(left), mirror energy differences for low-lying states and proton and neutron occupancies of single-particle orbits for isobaric analog states of $^{22}$Al and $^{22}$F (right).
• Figure 4: Comparison between the calculated and experimental NNDC energy levels (solid lines) and the proton and neutron thresholds (dashed lines)(left), mirror energy differences for low-lying states and proton and neutron occupancies of single-particle orbits for isobaric analog states of $^{24}$Si and $^{24}$Ne (right).
• Figure 5: Comparison between the calculated and experimental NNDC energy levels (solid lines) and the proton and neutron thresholds (dashed lines)(left), mirror energy differences for low-lying states and proton and neutron occupancies of single-particle orbits for isobaric analog states of $^{26}$P and $^{26}$Na (right).