Properties of ${}^{26}\mathrm{Mg}$ and ${}^{26}\mathrm{Si}$ in the $\mathit{sd}$ shell model and the determination of the ${}^{25}\mathrm{Al}$($p$,$\gamma$)${}^{26}\mathrm{Si}$ reaction rate

We present theoretical results for the ${}^{25}\mathrm{Al}$($p,\gamma$)${}^{26}\mathrm{Si}$ resonance-capture rate. The isobaric mass multiplet equation is used to determine the energies and $J^{\pi }$ values of states in ${}^{26}\mathrm{Si}$ based upon those observed in ${}^{26}\mathrm{Mg}$ and ${}^{26}\mathrm{Al}$ together with $\mathit{sd}$ shell calculations for the $c$ coefficients. Three Hamiltonians for the $\mathit{sd}$ shell, USD, USDA and USDB, are used to estimate the theoretical uncertainties in the $\gamma$-decay and proton-decay widths that go into the resonance-capture rate.

Figure 1

$c$ coefficients from the IMME vs state number (in order of increasing energy) in ${}^{26}\mathrm{Si}$ based on experimental energies (closed circles) and energies calculated from USDB (open circles).

Figure 2

Experimental excitation energies in ${}^{26}\mathrm{Si}$ and ${}^{26}\mathrm{Mg}$ [2]. Solid lines are for the states in ${}^{26}\mathrm{Si}$ with firm $J^{\pi }$ values. Dashed lines are for states in ${}^{26}\mathrm{Si}$ with uncertain $J^{\pi }$ values with the most likely mirror associations in ${}^{26}\mathrm{Mg}$ with known $J^{\pi }$ values. The three states indicated by lines to the right do not have known counterparts in ${}^{26}\mathrm{Si}$.

Figure 3

Experimental excitation energies in ${}^{26}\mathrm{Si}$ [2] vs predicted energies $E_{\text{th}}$ based on Eq. (7). Dashed lines are for states in ${}^{26}\mathrm{Si}$ with uncertain $J^{\pi }$ values with the most likely mirror associations in ${}^{26}\mathrm{Mg}$ with known $J^{\pi }$ values. The three states indicated by lines to the right do not have known counterparts in ${}^{26}\mathrm{Mg}$. The energies of two of these (2${}^{+}$ and 4${}^{+}$) are based on Eq. (7). The energy of the 3${}^{+}$ is shifted down by 250 keV from its position in ${}^{26}\mathrm{Mg}$. Level number 20 in Table is not shown in this figure (see the discussion in the text).

Figure 4

Total resonance-capture reaction rate vs temperature $T_9$ in GK (top panel) and the contribution of each of the final states (lower panel) with USDB. In the lower panel, the dominant contribution below log$T_9$ $=$ $-$0.8 is from state number 15, the 1${}^{+}$ state at 5.675 MeV. Between log$T_9$ $=$ $-$0.7 and 0.5, the dominant contribution is from state number 16, the 3${}^{+}$ state at 5.915 MeV.

Figure 5

Rates calculated with different assumptions divided by the USDB results given in Fig. 4.

Figure 6

Ratio given by the rate calculated with $Q$ $=$ 5.5123 MeV divided by the rate obtained with $Q$ $=$ 5.5173 MeV.

Figure 7

Relative contributions to the rate for log$T_9=1$. The top and bottom curves are binned over 0.5 MeV to show the integrated contribution for each bin. The middle curve shows the contribution from individual states.

Figure 8

(a) Present rate divided by the rate given in Table VII of [2]. (b) Present rate divided by the rate given in the 2010 Evaluation (Table B.37 of [17]); solid line for the median rate and dashed lines for the low and high rates.