Octupole collectivity in $^{220}$Rn and $^{224}$Ra

Collective properties of the radioactive nuclei $^{220}$Rn and $^{224}$Ra have been studied via Coulomb excitation of a 2.8$\,$A.MeV radioactive ion beam (RIB) incident upon $^{60}$Ni, $^{112,114}$Cd and $^{120}$Sn targets. The experiments took place at the REX-ISOLDE RIB facility, CERN. De-excitation $\gamma$-ray yields following multiple-step Coulomb excitation were detected in coincidence with recoiling target nuclei in the Miniball spectrometer. For the first time, B(E3;3$^+ \rightarrow 0^+$) values have been directly measured with a radioactive ion beam. In the process, $^{224}$Ra becomes the heaviest post-accelerated RIB to date at ISOLDE (with the possible exception of the quasi-stable $^{238}$U). The measurements presented in this thesis represent a tripling of the number of nuclei around Z$\simeq88$ and N$\simeq134$, for which direct measurements of the octupole collectivity have been performed. The only previous measurements being for the relatively long-lived $^{226}$Ra. The $\gamma$-ray yields, in conjunction with previously measured spectroscopic data, were used to determine electromagnetic matrix elements using the least-squares search code, extsc{Gosia}. In total, nine E1, E2 and E3 matrix elements were measured in $^{220}$Rn (plus six upper limits) and 12 (plus four upper limits) in $^{224}$Ra. The measured B(E3;3$^+ \rightarrow 0^+$) values in $^{220}$Rn and $^{224}$Ra are 32$\pm4 \,$W.u. and 42$\pm3\,$W.u., respectively. A new state has been observed at 937.8(8)$\,$keV in $^{220}$Rn and is proposed to be the 2$^+$ member of the K=2, $\gamma$-vibrational band. The results are interpreted in terms of the collectivity and deformation, and are compared with the predictions of self-consistent mean-field theory and cluster model calculations.