Experimental study of the ${}^{24}\mathrm{Na}^m(d,p){}^{25}\mathrm{Na}$ reaction and implications for the influence of the ${}^{24}\mathrm{Al}^m$ isomer on $rp$-process nucleosynthesis

A radioactive beam of ${}^{24}\mathrm{Na}$ with $90\%$ of its content in its $1^{+}$ isomeric state (${\mathrm{E}}_{ex}$ = 0.472 MeV, $t_{1/2}$ = 20.18 ms) has been developed and used to perform a measurement of the ${}^{24}\mathrm{Na}^m(d,p){}^{25}\mathrm{Na}$ reaction at the John D. Fox Accelerator Laboratory at Florida State University. This reaction selectively populated $\ell$ = 0 transfers, allowing the study of low-spin states in ${}^{25}\mathrm{Na}$. Mirror symmetry arguments were then used to investigate the effects of the isomeric state of ${}^{24}\mathrm{Al}$ (${\mathrm{E}}_{ex}$ = 0.426 MeV, $t_{1/2}$ = 130 ms) on the astrophysical rate of the ${}^{24}\mathrm{Al}^m(p,\gamma ){}^{25}\mathrm{Si}$ reaction. Experimental parameters were extracted to provide, for the first time, an experimental reaction rate for the destruction of ${}^{24}\mathrm{Al}$ via proton captures in its isomeric state relevant to $rp$-process nucleosynthesis.

Figure 1

Schematics of the experimental setup used during the present ${}^{24}\mathrm{Na}$($d,p){}^{25}\mathrm{Na}$ experiment. The beam enters the reaction chamber from the left and is incident on a ${\mathrm{CD}}_2$ target. Backward scattered protons from the interaction of the beam with the target are measured in a silicon S2 detector. The heavy products as well as the unreacted beam are measured downstream in an ionization chamber. Two NaI detectors were placed outside the reaction chamber at $\sim {90}^{\circ }$ from the target position and are used to monitor the isomeric content of the beam via detection of the 472-keV $\gamma$ rays.

Figure 2

(a) Ionization chamber spectrum for a no-target 2-min run. The ${}^{24}\mathrm{Na}$ beam as well as its main contaminant, the primary ${}^{23}\mathrm{Na}$ beam, are indicated. The ratio of ${}^{23}\mathrm{Na}$ to ${}^{24}\mathrm{Na}$ was about 1:1. (b) Typical spectrum of one of the NaI detectors taken during a gold-target 2-min run. The peak indicated in red correspond to the 472-keV $\gamma$ ray in ${}^{24}\mathrm{Na}$ which was used to determine and monitor the isomeric content of the beam.

Figure 3

DWBA calculations for the production of a ${}^{24}\mathrm{Na}$ beam via the ${}^{23}\mathrm{Na}$($d,p){}^{24}\mathrm{Na}$ reaction. The production cross section as a function of the beam energy for the ground state (short dashed line), isomeric state (long dashed line), and total production of ${}^{24}\mathrm{Na}$ (solid line) are shown. In the inset, the DWBA calculated isomer ratio, or isomer to total ratio, is shown by the solid black line. The experimentally determined isomeric ratio is shown by the single point. Good agreement is observed between DWBA calculations and the experimental data point.

Figure 4

${}^{25}\mathrm{Na}$ apparent excitation energy spectrum from the ${}^{24}\mathrm{Na}$($d,p){}^{25}\mathrm{Na}$ reaction using the $Q$ value from the isomeric state, in the angular range of ${\theta }_{\mathrm{lab}}$ = $161.6^{\circ }\text{–}173.7^{\circ }$, measured in the present work. (a) The spectrum for states in ${}^{25}\mathrm{Na}$ populated in the present ${}^{24}\mathrm{Na}$(d,p) reaction. The contribution of $\beta$-decay background from the decay of the ${}^{24}\mathrm{Na}^{gs}$ to ${}^{24}\mathrm{Mg}$, scaled to fit the data (red thick line), is observed at energies above 5 MeV. (b) ${}^{25}\mathrm{Na}$ apparent excitation energy spectrum after the subtraction of the $\beta$-decay background contribution. The contribution from the ground state component of the beam is also shown (blue thick line). The ground state contribution was estimated from the work of Ref. [15]. Given the kinematics used ($Q$ value of the isomeric state), states populated by the ground state appear shifted up in energy by 472 keV. (c) The ${}^{25}\mathrm{Na}$ excitation energy spectrum populated by the ${}^{24}\mathrm{Na}^m$($d,p){}^{25}\mathrm{Na}$ reaction populated with only the isomeric component of the beam.

Figure 5

Angular distributions for the states populated in the present ${}^{24}\mathrm{Na}^m$($d,p){}^{25}\mathrm{Na}$ reaction at (a) ${\mathrm{E}}_{ex}$ = 1.069 MeV ($1/2^{+}$), (b) ${\mathrm{E}}_{ex}$ = 3.687 MeV ($3/2^{+}$), and (c) ${\mathrm{E}}_{ex}$ = 4.289 MeV ($1/2^{+}$). All three states show $\ell$ = 0 neutron transfers from the $1^{+}$ isomeric state in ${}^{24}\mathrm{Na}$. DWBA calculations were used to fit the experimental data. A chi-square minimization was used to determine the best value for the spectroscopic factors ($\mathrm{C}{}^2\mathrm{S}$).

Figure 6

${}^{25}\mathrm{Na}$ apparent excitation energy. The states in ${}^{25}\mathrm{Na}$ populated in the present ${}^{24}\mathrm{Na}^m$($d,p){}^{25}\mathrm{Na}$ reaction in the energy range ${\mathrm{E}}_{cm}$ = 0–5 MeV are compared with shell model predictions using the USDB interaction. Spectroscopic factors (right $y$ axis) extracted from fits to the experimental data (yellow bars) and predicted by the shell model (dark red bars for $\ell$ = 0 transfers, orange for $\ell$ = 2) are overlaid.

Figure 7

Level schemes for ${}^{25}\mathrm{Si}$ and ${}^{25}\mathrm{Na}$, and USDB shell model calculations for the A = 25 system. Level information for ${}^{25}\mathrm{Si}$ and ${}^{25}\mathrm{Na}$ taken from Refs. [6, 18, 27]. USDB level information from [25]. The dotted lines indicate mirror levels determined by Ref. [13]. The dashed lines indicate proposed mirror levels. The energies of the $5/2^{+}$ state at 4.087 MeV in ${}^{25}\mathrm{Si}$ and 4.429 MeV in ${}^{25}\mathrm{Na}$ are based on the USDB shell model calculations with Thomas-Ehrman shifts of $-0.479\mathrm{MeV}$ and 0.224 MeV, respectively [28, 29].

Figure 8

Ratio of the isomeric rates extracted in this work to the current ground state REACLIB rate [34]. (a) Individual isomeric rate contributions. The $1/2^{+}$ subthreshold resonance at ${\mathrm{E}}_r^m$ = $-0.038$ MeV (red line) and the $5/2^{+}$ state at ${\mathrm{E}}_r^m$ = 0.247 MeV (black lines) as a function of the temperature. The resonance energy of the $5/2^{+}$ resonance was varied by $\pm$ 100 keV (shaded area). (b) Total isomeric contribution ($1/2^{+}$ $+$ $5/2^{+}$) determined in this work. The shaded area indicates the uncertainty in the energy of the $5/2^{+}$ resonance.

Figure 9

The rate of the ${}^{24}\mathrm{Al}(p,\gamma ){}^{25}\mathrm{Si}$ reaction. The red lines show the contributions from the $1^{+}$ isomeric state in ${}^{24}\mathrm{Al}$ presented in this work, while the blue line shows the current recommended REACLIB rate [34] with the state from the work of Ref. [6] added in. The black line denotes the total rate of the ${}^{24}\mathrm{Al}(p,\gamma ){}^{25}\mathrm{Si}$.