Exploring the magical tenacity of doubly closed-core nuclei with gallium and francium isotopes

This thesis presents nuclear spins, electromagnetic moments, and changes in the mean-square charge radii for radioactive gallium isotopes, and the latter two for the francium isotope $^{214}$Fr with additional $\alpha$-decay spectroscopy. Measurements were made using the CRIS technique at the ISOLDE facility in CERN, facilitating some of the most sensitive nuclear structure studies on exotic nuclei to date. Where applicable, experimental results are compared to empirical, and LSSM calculations. The amalgamated data set has presented the opportunity of probing the sturdiness of three doubly-magic nuclei with respect to their local vicinities: $^{56}$Ni, $^{78}$Ni, and $^{208}$Pb. A magical $^{56}$Ni core is confirmed with an electromagnetic-moment analysis of $^{65,67,69}$Ga, albeit of a soft nature. Comparisons with shell model calculations not only confirm increased mixtures of $\pi2p_{3/2}/\pi1f_{5/2}$ configurations and the presence of neutron correlations towards $N = 28$, but they also highlight systematic trends with respect to the neighbouring odd-$Z$ isotope chains. Analyses of neutron-rich species up to $^{82}$Ga reveal a rich collection of nuclear-structure effects: a gradual proton-occupation migration from $\pi2p_{3/2} \rightarrow \pi1f_{5/2}$ for the odd-$A$ species which culminates in a ground-state spin reversal involving those spins at $^{81}$Ga, isomeric presence in $^{80}$Ga at $N = 49$, and a kink in the change in the mean-square charge radii for $^{82}$Ga at $N = 51$. Overall, this adds further evidence for a robust doubly magic $^{78}$Ni core. Measurements of $^{214}$Fr with t$_{1/2} = 5$ ms represents the shortest-lived isotope to have been measured with laser spectroscopy techniques at an online facility, in which the ISOL production mechanism is the limiting factor. With its $g$ factor suggesting a relatively pure ($\pi1h_{9/2} \otimes \pi2g_{9/2}$) configuration for $I^{\pi} = 1^{−}$ and a kink in its charge-radii value at $N = 127$, the magical effects of $^{208}$Pb can be observed five protons away.