$\beta$-delayed $\gamma$ decay of $^{20}$Mg and the $^{19}$Ne(p,$\gamma$)$^{20}$Na breakout reaction in Type I X-ray bursts

Certain astrophysical environments such as thermonuclear outbursts on accreting neutron stars (Type-I X-ray bursts) are hot enough to allow for breakout from the Hot CNO hydrogen burning cycles to the rapid proton capture (rp) process. An important breakout reaction sequence is $^{15}$O($\alpha$,$\gamma$)$^{19}$Ne(p,$\gamma$)$^{20}$Na and the $^{19}$Ne(p,$\gamma$)$^{20}$Na reaction rate is expected to be dominated by a single resonance at 457 keV above the proton threshold in $^{20}$Na. The reaction rate depends strongly on whether this $^{20}$Na state at excitation energy 2647 keV has spin and parity of 1$^+$ or 3$^+$. Previous $^{20}$Mg ($J^\pi$=0$^+$) $\beta^+$ decay experiments have relied almost entirely on searches for $\beta$-delayed proton emission from this resonance in $^{20}$Na to limit the log $ft$ value. However there is a non-negligible $\gamma$-ray branch expected that must also be limited experimentally to determine the log $ft$ value and constrain $J^\pi$. We have measured the $\beta$-delayed $\gamma$ decay of $^{20}$Mg to complement previous $\beta$-delayed proton decay work and provide the first complete limit based on all energetically allowed decay channels through the 2647 keV state. Our limit confirms a 1$^+$ assignment for this state is highly unlikely.