Evaluation of the ${}^{35}\mathrm{K}(p,\gamma ){}^{36}\mathrm{Ca}$ reaction rate using the ${}^{37}\mathrm{Ca}(p,d){}^{36}\mathrm{Ca}$ transfer reaction

Background: A recent sensitivity study has shown that the ${}^{35}\mathrm{K}(p,\gamma ){}^{36}\mathrm{Ca}$ reaction is one of the ten $(p,\gamma )$ reaction rates that could significantly impact the shape of the calculated x-ray burst light curve. Its reaction rate used up to now in type I x-ray burst calculations was estimated using an old measurement for the mass of ${}^{36}\mathrm{Ca}$ and theoretical predictions for the partial decay widths of the first $2^{+}$ resonance with arbitrary uncertainties.

Purpose: In this work, we propose to reinvestigate the ${}^{35}\mathrm{K}(p,\gamma ){}^{36}\mathrm{Ca}$ reaction rate, as well as related uncertainties, by determining the energies and decay branching ratios of ${}^{36}\mathrm{Ca}$ levels, within the Gamow window of x-ray bursts, in the 0.5 to 2 GK temperature range.

Method: These properties were studied by means of the one-neutron pickup transfer reaction ${}^{37}\mathrm{Ca}(p,d){}^{36}\mathrm{Ca}$ in inverse kinematics using a radioactive beam of ${}^{37}\mathrm{Ca}$ at 48 MeV ${\mathrm{nucleon}}^{-1}$. The experiment was performed at the GANIL facility using the liquid hydrogen target CRYPTA, the MUST2 charged particle detector array for the detection of the light charged particles, and a zero degree detection system for the outgoing heavy recoil nuclei.

Results: The atomic mass of ${}^{36}\mathrm{Ca}$ is confirmed and new resonances have been proposed together with their proton decay branching ratios. This spectroscopic information, used in combination with very recent theoretical predictions for the $\gamma$-decay width, were used to calculate the ${}^{35}\mathrm{K}(p,\gamma ){}^{36}\mathrm{Ca}$ reaction rate. The recommended rate of the present work was obtained within a uncertainty factor of 2 at $1\sigma$. This is consistent with the previous estimate in the x-ray burst temperature range. A large increase of the reaction rate was found at higher temperatures due to two newly discovered resonances.

Conclusions: The ${}^{35}\mathrm{K}(p,\gamma ){}^{36}\mathrm{Ca}$ thermonuclear reaction rate is now well constrained by the present work in a broad range of temperatures covering those relevant to type I x-ray bursts. Our results show that the ${}^{35}\mathrm{K}(p,\gamma ){}^{36}\mathrm{Ca}$ reaction does not affect the shape of the x-ray burst light curve, and that it can be removed from the list of the few influential proton radiative captures reactions having a strong impact on the light curve.

Figure 1

Schematic layout (not to scale) of the experimental setup. The MUST2 telescopes are represented together with the two CATS beam tracker detectors, the CRYPTA liquid hydrogen target, the ionization chamber (IC), the drift chambers (DC), and the plastic scintillator.

Figure 2

Left: Energy loss vs TOF identification of the nuclei produced along with ${}^{37}\mathrm{Ca}$. Right: Identification of heavy transfer residues from their energy loss, measured in the ZDD ionization chamber, and their time of flight measured between the first CATS detector and the plastic scintillator located at the end of the ZDD (see text for details). The blue, purple and green contours show the regions corresponding to outgoing Ca, K, and Ar nuclei, respectively.

Figure 3

$E_{\text{CsI}}^{\text{ref}}$ vs $E_{\text{CsI}}^{\mathrm{\Delta }E}$ scatter plot for one CsI of one MUST2 telescope. The data points from the ${}^{38}\mathrm{Ca}(p,d){{}^{37}\mathrm{Ca}}_{\text{g.s.}}$ (blue), ${}^{35}\mathrm{Ar}(p,d){{}^{34}\mathrm{Ar}}_{\text{g.s.}}$ (green), ${}^{37}\mathrm{Ca}(p,d){{}^{36}\mathrm{Ca}}_{\text{g.s.}}$ (cyan), and ${}^{38}\mathrm{Ca}(p,d){{}^{37}\mathrm{Ca}}_{1.6\text{MeV}}$ (magenta) reactions are shown. A linear fit (red curve) was done using the ${}^{38}\mathrm{Ca}(p,d){{}^{37}\mathrm{Ca}}_{\text{g.s.}}$ and ${}^{35}\mathrm{Ar}(p,d){{}^{34}\mathrm{Ar}}_{\text{g.s.}}$ reactions. The linear fit describes well the data points from 0 to 6 MeV in excitation energy (written in red) in ${}^{36}\mathrm{Ca}$ ($R^2=0.992$).

Figure 4

Excitation energy spectrum of ${}^{36}\mathrm{Ca}$ reconstructed from the measurement of energy and angle of the deuteron and gated on outgoing Ca (a) or K (b) nuclei in the ZDD system. The center-of-mass energy of the protons emitted from unbound states in ${}^{36}\mathrm{Ca}$ has been added to the one-proton separation energy in the excitation energy spectrum $E_p^{\mathrm{c}.\mathrm{m}.}$ of (c). In all spectra, the red lines display the best fits ($p$ value = 0.67 for $E_x$ and 0.82 for $E_p^{\mathrm{c}.\mathrm{m}.}$) and colored dashed lines represent the different resonances in ${}^{36}\mathrm{Ca}$ used for the fit. Those especially relevant for x-ray bursts are shown in the level scheme on the left with the same color codes. Energy of exited states are taken from Ref. [14] for the $2_1^{+}$ state and from our work otherwise (highlighted in red).

Figure 5

Proton-deuteron angular correlation corresponding to the $2_1^{+}$ state as a function of the center-of-mass angle of the proton in the ${}^{36}\mathrm{Ca}$ frame. The red line shows the best fit obtained with $k_{\max }=2$ ($p$ value = 0.52).

Figure 6

The ${}^{35}\mathrm{K}(p,\gamma ){}^{36}\mathrm{Ca}$ reaction rate calculated (in units of ${\mathrm{cm}}^3{\mathrm{mol}}^{-1}{\mathrm{s}}^{-1}$) in this work (top) and in Ref. [16] (bottom). The red curve represent the total reaction rate, which includes the contributions of resonant (RC) and direct (DC) captures. The thickness of the curves represents a coverage probability of $68\%$. The indicated range of 0.5–2 GK is typical of x-ray burst temperatures.

Figure 7

Ratio of rates normalized to our recommended reaction rate. The areas delimited by the thick and thin black lines and centered around 1 represent the recommended values at the 68% and 95% confidence levels, respectively. The thick and dashed blue lines correspond to the reaction rate given by Iliadis et al. [16] and at the 68% confidence level, normalized to our recommended value.