Nuclear mass measurement has maintained an important position in the field of nuclear physics for a little over a century. Nuclear masses provide key evidence of the structural transformation of nuclei away from the valley of beta-stability and are essential input for many simulations of extreme astrophysical environments. However, obtaining these masses is often a challenging endeavor due to the low production cross sections and short half-lives of the exotic nuclei which are of particular interest. To this end, the time-of-flight mass measurement technique has been developed to obtain the masses of several nuclei at once to precisions of 1 part in 10^5 with virtually no half-life limitation. This dissertation contains a description of the experiment, analysis, and results of the second implementation of the time-of-flight nuclear mass measurement technique at the National Superconducting Cyclotron Laboratory. 18 masses were obtained for neutron-rich isotopes of argon through iron, where the masses of 48Ar, 49Ar, 56Sc, 57Sc, 64Cr, 67Mn, and 69Fe were measured for the first time. These newly obtained masses were applied to outstanding problems in nuclear structure and nuclear astrophysics, resulting in significant scientific advances. The measurement results for 48Ar and 49Ar, which were found to have atomic mass excesses of -22.28(31) MeV and -17.8(1.1) MeV, respectively, provide strong evidence for the closed shell nature of neutron number N = 28 in argon. It follows that argon is therefore the lowest even-Z element exhibiting the N = 28 closed shell. The masses of 64Cr, 67Mn, and 69Fe, which were found to have atomic mass excesses of -33.48(44) MeV, -34.09(62) MeV, and -39.35(60) MeV, respectively, show signs of nuclear deformation occurring around the N = 40 subshell. In addition, we found 64Cr is substantially less bound than predicted by global mass models that are commonly used in nuclear astrophysics simulations, resulting in a significant reduction in the predicted strength and depth of electron capture heating in the accreted neutron star crust due to the rather abundant A = 64 mass-chain. The reported value for the atomic mass excess of 56Sc, -24.85(59)(+0,-54) MeV, which contains an asymmetric systematic uncertainty due to potential isomeric contamination, results in a smaller than expected odd-even mass staggering in the A = 56 mass chain. Depending on the choice of theoretical models for electron capture transition strengths and energies, this could lead to strong Urca cooling in accreted neutron star crusts, due to the large amount of A = 56 material predicted to be present on the surface of accreted neutron stars.
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