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The $\ensuremath{\beta}$ decays of $^{103,104m}\mathrm{Nb}$ were studied with the Summing NaI(Tl) (SuN) detector at the National Superconducting Cyclotron Laboratory. The $\ensuremath{\beta}$-decay feeding intensity distribution ${I}_{\ensuremath{\beta}}(E)$ for each isotope was extracted by measuring $\ensuremath{\gamma}$ rays in coincidence with an emitted electron. The ${I}_{\ensuremath{\beta}}(E)$ was extracted via the total absorption spectroscopy technique. The ${I}_{\ensuremath{\beta}}(E)$ for each nucleus was compared to predictions made by the quasiparticle random-phase approximation (QRPA) model which is commonly used to calculate $\ensuremath{\beta}$-decay properties for astrophysical applications. The main goal was to provide experimental data for neutron-rich nuclei, relevant to the astrophysical $r$ process. In addition, the extracted $\ensuremath{\beta}$-decay feeding intensity distributions can lead to a better understanding of nuclear structure in a region of rapid structure changes around $A=100$. Finally, experimental data for $^{104m}\mathrm{Nb}$ are also of interest to antineutrino studies of nuclear reactors.

The $\ensuremath{\beta}$ decay of $^{101,102}\mathrm{Zr}$ and $^{109}\mathrm{Tc}$ was studied using the technique of total absorption spectroscopy. The experiment was performed at the National Superconducting Cyclotron Laboratory using the Summing NaI(Tl) (SuN) detector in the first-ever application of total absorption spectroscopy with a fast beam produced via projectile fragmentation. The $\ensuremath{\beta}$-decay feeding intensity and Gamow-Teller transition strength distributions were extracted for these three decays. The extracted distributions were compared to three different quasiparticle random-phase approximation (QRPA) models based on different mean-field potentials. A comparison with calculations from one of the QRPA models was performed to learn about the ground-state shape of the parent nucleus. For $^{101}\mathrm{Zr}$ and $^{102}\mathrm{Zr}$, 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 shapes is necessary to reproduce the extracted distributions. For $^{109}\mathrm{Tc}$, a comparison of the extracted distributions with QRPA calculations suggests a dominant oblate configuration. The other two QRPA models are commonly used to provide $\ensuremath{\beta}$-decay properties in $r$-process network calculations. This work shows the importance of making comparisons between the experimental and theoretical $\ensuremath{\beta}$-decay distributions, rather than just half-lives and $\ensuremath{\beta}$-delayed neutron emission probabilities, as close to the $r$-process path as possible.

$\ensuremath{\beta}$-decay half-lives of neutron-rich nuclides in the $A=100--110$ mass region have been measured using an implantation station installed inside of the Summing NaI(Tl) (SuN) detector at the National Superconducting Cyclotron Laboratory. Accurate half-lives for these nuclides are important for nuclear astrophysics, nuclear structure, and nuclear technology. The half-lives from the present work are compared with previous measurements, showing overall good agreement.