Understanding how nuclear shell structure and magic numbers emerge near stability but change as nuclides move away from stability is one of the main goals of modern nuclear structure studies. The N = 20 island of inversion is one of the oldest and best-studied examples of a magic number disappearing in exotic isotopes, but new observations are still emerging. Nuclides in this region exhibit collective behavior driven by the neutron particle-hole intruder configurations across the reduced gap between the sd and pf shells. The assignment of the 32Mg nuclide to the N = 20 island of inversion is supported from studies of the mass, excited states, and transition rate to the 2+1 state of 32Mg. However, a complete understanding of this classic island of inversion nuclide is still a challenge. The 0+2 state of 32Mg was observed at a lower energy than many models predicted, leading to a reinterpretation of the collectivity in 32Mg. Reduced transition rates are valuable observables to characterize collectivity. However, precise measurements of the reduced E2 transition rates in 32Mg are still needed to test the validity of recent theoretical calculations.This dissertation reports the results from an experimental study of the collectivity in 32Mg. The present work includes lifetime measurements for the lowest 0+, 2+, and 4+ excited states in 32Mg which are used to determine the reduced E2 transition rates. Lifetime measurements were performed using two different and complementary experimental approaches. First, the TRIPLEX device was used along with the Recoil-distance Method and Doppler-shift Attenuation Method to measure the lifetimes of the short-lived 2+1 and 4+1 states in 32Mg. The results reinforce the evidence from the energy systematics and earlier B(E2) measurements which suggests that the ground-state band in 32Mg is collective. Second, a novel in-flight technique referred to here as the Cascade Doppler-shift Method was used to observe decays from the 0+2 isomer in 32Mg. The 0+1 state was confirmed and the measured lifetime suggests a large collectivity in this state as well. The small partial cross section populating the 0+1 state in the 9Be(34Si,32Mg)X reaction provides experimental evidence for the dominance of intruder configurations in the 0+2 state.In summary, the lifetime measurements presented in this dissertation indicate the collectivity in 32Mg driven by intruder configurations that dominate the ground state as well as low-lying excited states. Experimental methods based on in-flight Doppler-shift techniques are demonstrated in this work and highlight how a broad range of lifetimes from 1 ps to tens of ns can be measured in rare-isotope beam experiments.
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