Evolution of E2 transition strength in deformed hafnium isotopes from new measurements on $^{172}$Hf, $^{174}$Hf, and $^{176}$Hf

Background: The available data for E2 transition strengths in the region between neutron-deficient hafnium and platinum isotopes are far from complete. More and precise data are needed to enhance the picture of structure evolution in this region and to test state-of-the-art nuclear models. In a simple model, the maximum collectivity is expected at the middle of the major shell. However, for actual nuclei, particularly in heavy-mass regions, which should be highly complex, this picture may no longer be the case, and one should use a more realistic nuclear-structure model. We address this point by studying the spectroscopy of Hf as a representative case. Purpose: We remeasure the 2(1)(+) half-lives of Hf-172,Hf-174,Hf-176, for which there is some disagreement in the literature. The main goal is to measure, for the first time, the half-lives of higher-lying states of the rotational band. The new results are compared to a theoretical calculation for absolute transition strengths. Method: The half-lives were measured using gamma-gamma and conversion-electron-gamma delayed coincidences with the fast timing method. For the determination of half-lives in the picosecond region, the generalized centroid difference method was applied. For the theoretical calculation of the spectroscopic properties, the interacting boson model is employed, whose Hamiltonian is determined based on microscopic energy-density functional calculations. Results: The measured 2(1)(+) half-lives disagree with results from earlier gamma-gamma fast timing measurements, but are in agreement with data from Coulomb excitation experiments and other methods. Half-lives of the 4(1)(+) and 6(1)(+) states were measured, as well as a lower limit for the 8(1)(+) states. Conclusions: This work shows the importance of a mass-dependent effective boson charge in the interacting boson model for the description of E2 transition rates in chains of nuclei. It encourages further studies of the microscopic origin of this mass dependence. New experimental values on transition rates in nuclei from neighboring isotopic chains could support these studies.