E2 Rotational Invariants of $0^+_1$ and $2^+_1$ states for $^{106}$Cd: the Emergence of Collective Rotation

The collective structure of $^{106}$Cd is elucidated by multi-step Coulomb excitation of a 3.849 MeV/$A$ beam of $^{106}$Cd on a 1.1 mg/cm$^2$ $^{208}$Pb target using GRETINA-CHICO2 at ATLAS. Fourteen $E2$ matrix elements were obtained. The nucleus $^{106}$Cd is a prime example of emergent collectivity that possesses a simple structure: it is free of complexity caused by shape coexistence and has a small, but collectively active number of valence nucleons. This work follows in a long and currently active quest to answer the fundamental question of the origin of nuclear collectivity and deformation, notably in the cadmium isotopes. The results are discussed in terms of phenomenological models, the shell model, and Kumar-Cline sums of $E2$ matrix elements. The ${\langle 0_2^+ ||E2||2_1^+ \rangle}$ matrix element is determined for the first time, providing a total, converged measure of the electric quadrupole strength, $\langle Q^2 \rangle$, of the first-excited $2_1^+$ level relative to the $0_1^+$ ground state, which does not show an increase as expected of harmonic and anharmonic vibrations. Strong evidence for triaxial shapes in weakly collective nuclei is indicated; collective vibrations are excluded. This is contrary to the only other cadmium result of this kind in $^{114}$Cd by C. Fahlander et al., Nucl. Phys. A485, 327 (1988), which is complicated by low-lying shape coexistence near midshell.