Extraction of Gamow-Teller strengths in the ${\beta }^{+}$ direction with the ($d,{}_2\mathrm{He}$) reaction in inverse kinematics

The ($d,{}_2\mathrm{He}$) reaction in inverse kinematics has been developed for experiments with rare-isotope beams to constrain electron-capture rates needed for astrophysical simulations of processes in dense nuclear environments such as supernovae and neutron star crusts. The first experiment focused on the measurement of the ${}_{14}O(d,{}_2\mathrm{He})$ and ${}_{13}N(d,{}_2\mathrm{He})$ reactions in inverse kinematics, utilizing the active-target time-projection chamber placed in front of the S800 magnetic spectrograph. This work focuses on the experimental and analysis details, and presents the results for the ${}_{13}N(d,{}_2\mathrm{He})$ reaction, which is important for constraining electron captures rates on ${}_{13}N$ in the preexplosion convective phase of Type Ia supernova. The extracted Gamow-Teller transition strengths associated with electron capture on ${}_{13}N$ are consistent with those previously obtained from the analog transitions from ${}_{13}C$. The successful development of the ($d,{}_2\mathrm{He}$) reaction in inverse kinematics presents a novel opportunity for performing experiments aimed at constraining electron-capture rates in nuclei far from stability.