## Yale Graduate School of Arts and Sciences Dissertations

Fall 10-1-2021

Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

#### Department

Physics

The statistical model of compound-nucleus reactions has important applications in fundamental nuclear science, nuclear astrophysics, and nuclear technology. This model relies on two theoretical areas: (i) statistical reaction theory, which describes the compound nucleus with the Gaussian orthogonal ensemble (GOE) of random-matrix theory; and (ii) statistical properties of nuclei, i.e., nuclear structure observables that determine statistical-model predictions of reaction rates. The GOE statistical theory predicts that the partial widths of compound-nucleus resonances follow the Porter-Thomas distribution (PTD) and that total $\gamma$-decay widths have a narrow distribution. However, recent experiments measured width distributions that were broader than statistical-model predictions. We study these results with resonance-reaction models based on the GOE. Nuclear level densities are important statistical properties of nuclei and inputs to the statistical model. Mean-field methods are widely used to calculate level densities microscopically but neglect important correlations. We introduce two novel methods for symmetry projection after variation in the finite-temperature mean-field approximation and calculate nuclear state densities with exact particle-number projection. Moreover, we calculate state densities in the configuration-interaction shell model framework using the static-path plus random-phase approximation (SPA+RPA). The SPA+RPA includes static fluctuations and small-amplitude time-dependent quantal fluctuations beyond the mean field. We find that the SPA+RPA state densities agree with exact shell model Monte Carlo (SMMC) state densities and improve significantly over mean-field state densities in heavy lanthanide nuclei.