$\gamma$-ray spectroscopy of the $A=23$, $T=1/2$ nuclei ${}^{23}$Na and ${}^{23}$Mg: High-spin states, mirror symmetry, and applications to nuclear astrophysical reaction rates

Background: Obtaining reaction rates for nuclear astrophysics applications is often limited by the availability of radioactive beams. Indirect techniques to establish reaction rates often rely heavily on the properties of excited states inferred from mirror symmetry arguments. Mirror energy differences can depend sensitively on nuclear structure effects.

Purpose: The present work sets out to establish a detailed comparison of mirror symmetry in the $A=23$, $T=1/2$ mirror nuclei ${}^{23}$Na and ${}^{23}$Mg both to high spin, and high excitation energy, including beyond the proton threshold. These data can be used to benchmark state-of-the-art shell-model calculations of these nuclei.

Methods: Excited states in ${}^{23}$Na and ${}^{23}$Mg were populated using the ${}^{12}$C(${}^{12}$C,$p$) and ${}^{12}$C(${}^{12}$C,$n$) reactions at beam energies of 16 and 22 MeV, and their resulting $\gamma$ decay was measured with Gammasphere.

Results: Level schemes for ${}^{23}$Na and ${}^{23}$Mg have been considerably extended; highly excited structures have been found in ${}^{23}$Na, as well as their counterparts in ${}^{23}$Mg for previously known rotational structures in ${}^{23}$Na. Mirror symmetry has been investigated up to an excitation energy of 8 MeV and spin-parity of 13/2${}^{+}$. Excited states in the region above the proton threshold have been studied in both nuclei.

Conclusions: A detailed exploration of mirror symmetry has been performed which heavily constrains expectations as to how mirror energy differences should evolve for different structures. Agreement with shell-model calculations provides confidence in using such estimations where real data are absent.

Figure 1

Angular correlation ratios for $\gamma$ rays of well established multipolarity in ${}^{23}$Na. The open circles are known stretched-$E1$ transitions while the filled squares are stretched-$E2$ ones.

Figure 2

Fractional Doppler shifts as a function of the trial lifetime calculated using the model described in the text. The solid curve is for ${}^{23}$Mg residues, while the dashed curve is for ${}^{23}$Na ones.

Figure 3

Sum of double gates on the 450-, 1600-, 663- and 2739-keV transitions in the $\gamma$-$\gamma$-$\gamma$ cube. Strong transitions in ${}^{23}$Mg are labeled with their energy in keV.

Figure 4

Sum of double gates on the 440-, 1636-, 627- and 2830- and 701-keV transitions in the $\gamma$-$\gamma$-$\gamma$ cube. Strong transitions in ${}^{23}$Na are labeled with their energy in keV.

Figure 5

Positive-parity rotational bands in ${}^{23}$Na. The $\gamma$-ray and level energies are given in keV, while the width of the arrows reflects the intensity of the transitions. Many other positive-parity states were identified in the present work, but are suppressed in this figure for reasons of clarity.

Figure 6

Comparison of the positive-parity rotational bands in (a) ${}^{23}$Na and (b) ${}^{23}$Mg. The $\gamma$-ray and level energies are given in keV, while the width of the arrows reflects the intensity of the transitions. The intensity has been scaled arbitrarily to 100$\%$ for the 1600- and 1636-keV transitions in ${}^{23}$Mg and ${}^{23}$Na, respectively.

Figure 7

Comparison of (a) the positive-parity states in ${}^{23}$Na with (b) the predictions of full shell-model calculations described in the text. The decay intensity from each state is normalized to 100$\%$ with the width of the arrows corresponding to the fraction of the decay intensity. Weaker transitions are suppressed for the purpose of clarity.

Figure 8

Plot of the difference in excitation energy predicted by the $sd$ shell model compared with the experimentally observed states in ${}^{23}$Na. The red circles are for the ground-state band, the blue diamonds are for the excited $K=1/2^{+}$ band, and the green squares are for the excited $K=11/2^{+}$ band.

Figure 9

Mirror energy differences (MED) between positive-parity states in ${}^{23}$Na and ${}^{23}$Mg plotted as a function of excitation energy in ${}^{23}$Mg for (a) $J=1/2$; (b) $J=3/2$; (c) $J=5/2$; (d) $J=7/2$; (e) $J=9/2$; (f) $J=11/2$; (g) $J=13/2$. The blue diamonds are the experimental data, while the red circles are from the $sd$-shell-model calculations described in the text.

Figure 10

Comparison of the negative-parity bands in (a) ${}^{23}$Na and (b) ${}^{23}$Mg; level and transition energies are given in keV. The width of the arrows denotes the relative strength of decay branches.

Figure 11

States of presumed negative parity in ${}^{23}$Na; level and transition energies are given in keV. The width of the arrows denotes the relative strength of the decay branches.

Figure 12

Comparison of the yrast negative-parity states in ${}^{23}$Na [the K = (1/2${}^{-}$)${}_1$ band] with the predictions of the PSPDF shell model (not to scale).

Figure 13

Mirror energy differences (MED) between negative-parity states in ${}^{23}$Na and ${}^{23}$Mg plotted as a function of excitation energy in ${}^{23}$Mg. The red circles are for the $K={(1/2^{-})}_1$ band, while the blue diamonds are for the second negative-parity band. The green squares are calculated values for the MED for the first negative-parity band, obtained using PSPDF shell-model calculations (see text).

Figure 14

PSPDF calculations of negative-parity states in ${}^{23}$Na.

Figure 15

Spectrum gated by the 1821-keV $\gamma$ ray, which depopulates the 7/2${}^{-}$ state at 8945 keV. The 1087-keV $\gamma$ ray feeding this level is clearly visible as are the transitions below the 1821-keV $\gamma$ ray in the decay scheme.

Figure 16

Spectrum gated by the 3914-keV $\gamma$ ray, which depopulates the 5/2${}^{+}$ state at 3914 keV. The 5030- and 5292-keV $\gamma$ rays which depopulate the astrophysically relevant states at 8944 and 9211 keV in ${}^{23}$Na, are marked.