This dissertation details the initiation of a new experimental program to study $\beta$ decay that is now in use at the National Superconducting Cyclotron Laboratory and will be an integral part of the science conducted at the Facility for Rare Isotope Beams. This experimental program studies the $\beta$-decay properties of nuclides relevant to the astrophysical $r$ process with the total absorption spectroscopy technique. Descriptions of $r$-process nucleosynthesis, an overview of $\beta$ decay and $\gamma$ decay, the experimental setups, and analysis procedures are included in this dissertation.This dissertation contains the commissioning experiments of this experimental program. These commissioning experiments were performed at the Coupled Cyclotron Facility at the National Superconducting Cyclotron Laboratory and combined charged-particle detection using silicon detectors and $\gamma$-ray detection using a segmented total absorption spectrometer called the Summing NaI(Tl) (SuN) detector.The commissioning experiment with a thermalized beam examined the $\beta$ decay of $^{76}$Ga. The extracted $\beta$-decay half-life agrees with previously published values. However, the extracted $\beta$-decay feeding intensity distribution disagrees with the existing decay scheme at the National Nuclear Data Center. The extracted distribution provided experimental data in the A = 76 mass chain. This experimental data can constrain nuclear structure models that calculate nuclear matrix elements for neutrinoless double-$\beta$ decay.The commissioning experiment with a fast beam studied neutron-rich nuclides in the A = 100-110 mass region. This experiment was the first-ever application of the total absorption spectroscopy technique with a fast beam produced via projectile fragmentation. $\beta$-decay half-lives were extracted for $^{99}$Y, $^{101}$Zr, $^{102}$Zr, $^{102m}$Nb, $^{103}$Nb, $^{104m}$Nb, and $^{109}$Tc. Overall, the extracted half-lives agree with previously published values. Additionally, the $\beta$-decay feeding intensity distributions and $B$(GT) distributions were extracted for $^{101}$Zr, $^{102}$Zr, and $^{109}$Tc. The extracted distributions were compared to QRPA calculations, which are commonly used to provide $\beta$-decay properties in $r$-process reaction network calculations. In these comparisons, none of the QRPA calculations were able to reproduce the extracted distributions. The extracted distributions were compared to another set of QRPA calculations in an attempt to learn about the shape of the ground state of the parent nucleus. For $^{101}$Zr and $^{102}$Zr, calculations assuming a pure shape configuration (oblate or prolate) were not able to reproduce the extracted distributions. These results may indicate that some type of mixture between oblate and prolate is necessary to reproduce the extracted distributions. For $^{109}$Tc, a comparison of the extracted distribution with QRPA calculations suggests a dominant oblate configuration.
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