Shell-model studies of the $rp$ reaction ${}^{35}$Ar($p$,$\gamma$)${}^{36}$K

We present results for levels in ${}^{36}$K (the mirror of nucleus ${}^{36}$Ar) that are used in $rp$ reaction rate calculations. The levels are also determined from the isobaric mass multiplet equation and the binding energies of the $T=1$ analog states as a check on the assignment of spins and parity. Where the analog states are not known, the levels are calculated with two-body interactions that use the $sd$-shell interactions USDA and USDB as the charge-independent parts, with a Coulomb, charge-dependent, and charge-asymmetric Hamiltonian added. The $\gamma$-decay lifetimes and ${}^{35}$Al to ${}^{36}$K spectroscopic factors are calculated with the same interactions, and together with experimental information on the levels of excited states, are used to determine the ${}^{35}$Ar($p,\gamma$)${}^{36}$K reaction rates.

Figure 1

$c$ coefficients from the IMME ($B=a+b{T}_z+c{T}_z^2$) vs state number (in order of increasing energy) in ${}^{36}$K based on experimental energies from Ref. [6] (closed circles) and energies calculated from usdb-cdpn (open circles).

Figure 2

Experimental energies of the isobaric $T=1$ triplets for $A=36$. The energies of ${}^{36}$Ar are relative to the lowest 2${}^{+}$ $T=1$ state at 6.611 MeV. Negative parity states are connected by dashed lines. The solid lines connect positive parity states considered to be analogs on the basis of our IMME predictions. The proton separation energy in ${}^{36}$K is shown by the horizontal line on the left-hand side. The data are from Endt [6] except for those above the proton separation energy in ${}^{36}$K for which we use the newer values from Wrede et al. [7]. The cross on the 2.282 MeV 5${}^{-}$ state in ${}^{36}$K indicates what this level was associated with the 2${}_3^{+}$ state by Wrede et al. Our reasons for associating the 2${}_3^{+}$ level with the higher state at the 2.446 MeV state are discussed in the text. The levels labeled ${}^{36}$K IMME are based on Eq. (4) with experimental binding energies of ${}^{36}$Cl and ${}^{36}$Ar and with the theoretical $c$ coefficient [Eq. (3)]. The crosses correspond to predicted energies without experimental counterparts.

Figure 3

Total $rp$ reaction rate vs temperature $T9$ (gigaK) (top panel) and the contribution of each of the final states (lower panel) with usdb-cdpn. ${\Gamma }_{\gamma }$ was calculated for ${}^{36}$K levels.

Figure 4

Total $rp$ reaction rates of usda-cdpn and usd-cdpn vs usdb-cdpn compared.

Figure 5

The usdb-cdpn present rate divided by the rate given in the 2010 evaluation (Table B.58 of Ref. [8]); solid line for the median rate and the dashed lines for the low and high rates.