Indirect study of the stellar ${}^{34}\mathrm{Ar}(\alpha ,p){}^{37}\mathrm{K}$ reaction rate through ${}^{40}\mathrm{Ca}(p,t){}^{38}\mathrm{Ca}$ reaction measurements

The ${}^{34}\mathrm{Ar}(\alpha ,p){}^{37}\mathrm{K}$ reaction is believed to be one of the last in a sequence of ($\alpha ,p$) and ($p,\gamma$) reactions within the $T_z=-1, sd$-shell nuclei, known as the $\alpha p$-process. This process is expected to influence the shape and rise times of luminosity curves coming from type I x-ray bursts (XRBs). With very little experimental information known on many of the reactions within the $\alpha p$-process, stellar rates are calculated using a statistical model, such as Hauser-Feshbach. Questions on the applicability of a Hauser-Feshbach model for the ${}^{34}\mathrm{Ar}(\alpha ,p){}^{37}\mathrm{K}$ reaction arise due to level density considerations in the compound nucleus, ${}^{38}\mathrm{Ca}$. We have performed high energy-resolution forward-angle ${}^{40}\mathrm{Ca}(p,t){}^{38}\mathrm{Ca}$ measurements with the $K=600$ spectrograph at iThemba LABS in order to identify levels above the $\alpha$-threshold in ${}^{38}\mathrm{Ca}$. States identified in this work were then used to determine the ${}^{34}\mathrm{Ar}(\alpha ,p){}^{37}\mathrm{K}$ reaction rate based on a narrow-resonance formalism. Comparisons are made to two standard Hauser-Feshbach model predicted rates at XRB temperatures.

Figure 1

${}^{38}\mathrm{Ca}$ spectra in the focal plane of the $K=600$ spectrograph shown at ${\theta }_{\text{lab}}=8^{\circ }$ with two different field settings [panels (a) and (b)] to cover a full energy region from the ground state to 13 MeV in excitation energy in ${}^{38}\mathrm{Ca}$. The spectra shown here have been background subtracted using two particle identification gates, namely energy loss and time of flight. Peaks with dark dots (blue online) are states that have been observed in previous experiments investigating ${}^{38}\mathrm{Ca}$, while peaks with lighter dots (orange online) represent states observed for the first time in this work.

Figure 2

Calculated proton particle-widths as a function of center-of-mass energy given a range of orbital angular momenta, $\ell$ = 0–4. A proton center-of-mass energy range of 1.5–5.5 MeV approximately corresponds to an excitation energy range of 6–10 MeV in ${}^{38}\mathrm{Ca}$.

Figure 3

Spin distributions for selected excitation energies in ${}^{38}\mathrm{Ca}$ based on Eq. (5) used in the framework of the BSFG model.

Figure 4

(a) $\alpha$-spectroscopic factors for states in ${}^{38}\mathrm{Ca}$ calculated using the shell model as described in Ref. [39] (shown in gray), along with the mapped values to states observed in this work (overlaid). (b) Level density of observed states in ${}^{38}\mathrm{Ca}$ from this work along with previous works.

Figure 5

(a) ${}^{34}\mathrm{Ar}(\alpha ,p){}^{37}\mathrm{K}$ reaction rates as a function of stellar temperature for statistical model predictions, non-smoker${}^{\mathrm{WEB}}$ v5.0w and talys 1.8, along with the two median rates calculated in this work, median rate 1 (without $\alpha$-clustering) and median rate 2 (with $\alpha$-clustering). (b) All rates are normalized to the non-smoker rate.