Over the past few decades, the use of online Penning trap mass spectrometry (PTMS) has enabled precise, accurate mass measurements of rare isotopes as a probe of nuclear structure far from the valley of beta stability. As the first and only Penning trap coupled to a projectile fragmentation facility, the Low Energy Beam and Ion Trap (LEBIT) facility at the National Superconducting Cyclotron Laboratory allows precision mass measurements of exotic nuclei that are not available at other PTMS facilities.The nuclear shell model provides a robust framework for understanding nuclear structure effects. While nuclear shell structure is well understood for stable isotopes, the evolution of nuclear structure away from stability remains an active area of rare isotope research. The region near Z=28 and N=40 is a subject of great interest for nuclear structure studies due to spectroscopic signatures in 68Ni suggesting a subshell closure at N=40. Trends in nuclear masses do not appear to support this conclusion, however a complete picture of the mass surfaces in this region has so far been limited by the large uncertainty remaining for nuclei with N>40 along the iron (Z=26) and cobalt (Z=27) chains because these species are not available at traditional isotope separator online (ISOL) facilities. Recent Penning trap mass measurements of 68,69Co at LEBIT provide the first precise examination of nuclear masses beyond N=40 in the Co chain. The motivation, procedure, and results of these measurements are presented in this dissertation. Recent theoretical calculations for these isotopes are also presented, and the importance of these measurements and calculations for understanding the evolution of nuclear structure near 68Ni is discussed.In order to expand the reach of LEBIT to isotopes very far away from stability, a new Single Ion Penning Trap (SIPT) has been developed. Many rare isotopes far from stability can only be produced at very low rates incompatible with the current destructive measurement technique used for online PTMS, which requires ~100 ions or more to complete a mass measurement. SIPT employs a non-destructive measurement technique which enables complete mass determinations with a single ion. This technique has been used successfully at other facilities for stable particle measurements but has never before been extended to measurements of exotic radioisotopes. The design and offline commissioning tests of SIPT are presented in this work, demonstrating promising outlook for single ion rare isotope measurements in the near future.
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