This thesis describes the use of the \((d,^2\mathrm{He})\) reaction in inverse kinematics with rare-isotope beams to constrain electron-capture rates, which are important for accurately simulating the heating and cooling processes in neutron star crusts. Electron-capture rates are calculated from Gamow-Teller (GT) strengths, which can be extracted experimentally in a model-independent way using charge-exchange reactions at intermediate energies. Following a successful pilot study with the \(^{14}\mathrm{O}(d,^2\mathrm{He})\) and \(^{13}\mathrm{N}(d,^2\mathrm{He})\) reactions in inverse kinematics using the Active-Target Time Projection Chamber with the S800 spectrometer at the Facility of Rare Isotope Beam, a second experiment was performed to investigate the \(^{33}\mathrm{Al}(d,^2\mathrm{He})\) and \(^{32}\mathrm{Mg}(d,^2\mathrm{He})\) reactions using an identical setup.The nuclei \(^{33}\mathrm{Al}\) and \(^{32}\mathrm{Mg}\) are located in or near the \(N=20\) "island of inversion," a region where shell evolution leads to the inversion of the \(sd\) and \(pf\) shells. These nuclei play an important role in understanding the Urca mechanism in neutron star crusts, where rapid cycles between electron capture and \(\beta^-\) decay result in significant cooling due to strong neutrino emission. This thesis presents the experimental setup, data analysis, and results for the extracted GT strengths from these reactions. GT strengths were extracted for one state in the \(^{33}\mathrm{Al}(d,^2\mathrm{He})\) reaction and two states in the \(^{32}\mathrm{Mg}(d,^2\mathrm{He})\) reaction. The results provide evidence that the assumption of a closed-shell structure is invalid for \(^{33}\mathrm{Al}\) and \(^{32}\mathrm{Mg}\). Although the uncertainties in the extracted GT strengths are large, there is reasonable agreement with shell-model calculations that include contributions from both the \(sd\)- and \(pf\)-shell configurations.In addition to the experimental work, weak interaction rates and their impact on neutron star crusts were examined for nuclei in the \(sd\) and \(pf\) shells, for which precise shell-model calculations are already established. Neutron star crust simulations utilized three different sets of weak interaction rates: (1) rates derived solely from Quasiparticle Random Phase Approximation (QRPA) calculations, (2) shell-model rates for \(sd\) and \(pf\) shell nuclei, combined with QRPA rates for all other nuclei, and (3) experimental data for \(sd\) and \(pf\) shells when available, supplemented by shell-model calculations, with QRPA rates used for all other nuclei. The inclusion of experimental data for weak interaction rates leads to a suppression of both heating and cooling effects, and the results align more closely with the rates from shell-model calculations than with those from QRPA calculations.The results from this work underscore the importance of precise weak interaction rates for realistic neutron star modeling and demonstrate the effectiveness of the \((d,^2\mathrm{He})\) reaction in inverse kinematics as a tool for extracting essential nuclear data.
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