The main topic of this thesis concerns the unusual features of the nuclear structure of $^{78}$Ge. The discovery of a sequence of levels separated by one unit of angular momentum for which the expected crossover transitions carrying two units of angular momentum are not observed. This result stands in contrast to the neighboring even-even Ge and Se nuclei, as well as the results of most model calculations. The level structures of adjacent $^{82,80}$Ge are also studied to place the results for $^{78}$Ge in context. Likewise, shell-model calculations are performed as comparisons with the structures for all three nuclei.

The data come from experiments at Argonne National Laboratory using the Gammasphere detector array. These experiments are important because of the broad interest in the structure of $^{76}$Ge and neighboring nuclei, owing to a world-wide effort in the search for neutrinoless double $\beta$ decay. Owing to the unusual features of this level sequence, it is labelled as a $\kappa$ band, taken from the Greek word καινουργιος, meaning new. The results pose a challenge to theorists to find ways to develop models that can fit both these features, as well as the other aspects of the structure of $^{78}$Ge. In addition, this study is important determining why no sequence like this has been found in any of the adjacent nuclei.