Proton decays from $\alpha$-unbound states in ${}^{22}\mathrm{Mg}$ and the ${}^{18}\mathrm{Ne}{(\alpha ,p_0)}^{21}\mathrm{Na}$ cross section

Background: Type I x-ray bursts provide an opportunity to constrain the equation of state of nuclear matter. Observations of the light curves from these bursts allow the compactness of neutron stars to be constrained. However, the behavior of these light curves also depends on a number of important thermonuclear reaction rates. One of these reactions, ${}^{18}\mathrm{Ne}(\alpha ,p){}^{21}\mathrm{Na}$, has been extensively studied but there is some tension between the rate calculated from spectroscopic information of states above the $\alpha$-particle threshold in ${}^{22}\mathrm{Mg}$ and the rate determined from time-reversed measurements of the cross section.

Purpose: The time-reversed measurement of the cross section is only sensitive to the ground state-to-ground state contribution. Therefore, corrections must be made to this reaction rate to account for the contribution of branches to excited states in ${}^{21}\mathrm{Na}$. At present this is done with statistical models which may not be applicable in such light nuclei. Basing the correction of the time-reversed cross section on experimental data is much more robust.

Method: The ${}^{24}\mathrm{Mg}(p,t){}^{22}\mathrm{Mg}$ reaction was used to populate states in ${}^{22}\mathrm{Mg}$. The reaction products from the reaction were analysed by the K600 magnetic spectrometer at iThemba Laboratories, South Africa. Protons decaying from excited states of ${}^{22}\mathrm{Mg}$ ($S_p=5502$ keV) were detected in an array of five double-sided silicon strip detectors placed at backward angles. The branching ratio for proton decays to the ground state of ${}^{21}\mathrm{Na}, B_{p_0}$, was determined by comparing the inclusive (triton-only focal-plane) and exclusive (focal-plane gated on a specific proton decay) spectra.

Results: The experimental proton decay branching ratio to the ground state of ${}^{21}\mathrm{Na}$ from excited states in ${}^{22}\mathrm{Mg}$ were found to be a factor of about two smaller than the ratios predicted by Hauser-Feshbach models. Using the experimental branchings for a recalculation of the ${}^{18}\mathrm{Ne}(\alpha ,p_0){}^{21}\mathrm{Na}$ cross section leads to a considerably improved agreement with previous reaction data. Updated information on the disputed number of levels around $E_x\approx 9$ MeV and on the possible ${}^{18}\mathrm{Ne}(\alpha ,2p){}^{20}\mathrm{Ne}$ cross section at astrophysical energies is also reported.

Conclusions: The proton decay branching of excited states in ${}^{22}\mathrm{Mg}$ to the ground state of ${}^{21}\mathrm{Na}$ have been measured using the K600 Q2D spectrometer at iThemba Laboratories coupled to the double-sided silicon-strip detector array CAKE. Using these experimental data, the modeling of the ${}^{18}\mathrm{Ne}(\alpha ,p_0){}^{21}\mathrm{Na}$ cross section has been improved. The result is not only in better agreement with previous cross section data but also consistent with a recent direct measurement of ${}^{18}\mathrm{Ne}(\alpha ,p){}^{21}\mathrm{Na}$. This strengthens the case for the application of statistical models for these reactions.

Figure 1

Particle identification for the tritons in the K600 magnetic spectrometer. The triton locus is encircled by a black border which is representative of the software gate used. Most of the other events are caused by beamstop-induced background or by deuterons caused by reactions in the target.

Figure 2

Inclusive excitation-energy spectrum of levels in ${}^{22}\mathrm{Mg}$. The proton-decay threshold ($S_p=5.502$ MeV) and the $\alpha$-particle threshold ($S_{\alpha }=8.142$ MeV) is indicated.

Figure 3

Excitation energy vs. silicon energy coincidence spectrum. (Top) Combined coincidence spectrum (for reference), the broadening of the coincidence loci is due to the effective target thickness change experienced by the decaying proton. (Bottom) Coincidence spectrum selected on a small number of angular rings (ring 1 (${\theta }_{\mathrm{lab}}=161.4^{\text{o}}$) to ring 4 (${\theta }_{\mathrm{lab}}=151.4^{\text{o}}$)) on the silicon detectors; the coincidence loci are much narrower due to the smaller range of effective target thicknesses. Each decay locus from $p_0$ to $p_5$ is indicated. The contaminants seen in the kinematically-inaccessible region, to the left of the $p_0$ decay locus, are from oxygen.

Figure 4

Exclusive (gated on specific proton decays to different final ${}^{21}\mathrm{Na}$ states) excitation-energy spectra. These spectra are gated on the $p_0$ (top) to $p_{4+5}$ (bottom) decay channels. The $p_{4+5}$ spectrum includes the $p_4$ and $p_5$ decay locus as they are 30 keV apart and not resolved. The $p_0$ and $p_1$ spectra have a bin size of 2.5 keV, the other spectra have 10-keV binning. The states that correspond to RCNP data from Matic et al. [19] are labeled in the top panel. Table 2 lists the details of each state that corresponds to each label in the top panel of this figure.

Figure 5

(Top) Center-value $B_{p_0}$ as a function of $E_x$ (blue) with quadratic fit to the data (black). The uncertainties are not indicated on individual $B_{p_0}$ data points in the upper panel but are present in the lower panel. Quadratic fits to the upper and lower limits of the $B_{p_0}$ branching ratios are shown as black-dotted lines. $B_{p_0}$ for the ${}^{18}\mathrm{Ne}(\alpha ,p){}^{21}\mathrm{Na}$ reaction calculated with talys is shown in green. The non-smoker $B_{p_0}$ using the SMARAGD model is shown by the red line. (Bottom) $B_{p_i}$ and the sum of the proton branching ratios as a function as excitation energy. The bin size is 100 keV.

Figure 6

Cross sections for ${}^{18}\mathrm{Ne}(\alpha ,p_0){}^{21}\mathrm{Na}$ and ${}^{18}\mathrm{Ne}(\alpha ,p_0){}^{21}\mathrm{Na}$ calculated with the smaragd code [40, 41] with and without modified $p_0$ width. The shaded band shows the experimental uncertainty on the average $p_0$ branching.