Coulomb Excitation and Cascade Decay of Rotational States in Odd-Mass Nuclei

We have observed the gamma radiation from the first two excited states, as well as the cascade transition between them, by bombarding the following odd-mass rare-earth nuclei with 6-Mev alpha particles: ${\mathrm{Eu}}^{151}$, ${\mathrm{Eu}}^{153}$, ${\mathrm{Gd}}^{155}$, ${\mathrm{Gd}}^{157}$, ${\mathrm{Tb}}^{159}$, ${\mathrm{Ho}}^{165}$, ${\mathrm{Lu}}^{175}$, ${\mathrm{Hf}}^{177}$, ${\mathrm{Hf}}^{179}$, and ${\mathrm{Ta}}^{181}$. All of these nuclei were available as enriched targets. We also detected x-x, x-$\gamma$, and $\gamma -\gamma$ coincidences from these nuclei, establishing their previously suggested level schemes. All of them show rotational spectra except ${\mathrm{Eu}}^{151}$ ($N=88\mathrm{}$), which was found to have an excitation spectrum very different from that of ${\mathrm{Eu}}^{153}$ ($N=90\mathrm{}$) in both the intensity and position of the excited states. From the measured branching ratios of the second rotational states we determined the $M1-E2\mathrm{}$ mixture ratios for the cascade radiation, using the theoretical value for the $E2\mathrm{}$ component. This is reasonable because within the experimental uncertainties, mainly caused by our lack of information on total internal conversion coefficients, the ratios of $E2\mathrm{}$ transition probabilities from ground state to the two rotational states are in agreement with the theory. Combining the mixture ratio with the values of the ground state magnetic moments, we computed the so-called intrinsic and collective $g$ factors from the strong-coupling approximation of the Bohr-Mottelson theory. Two pairs of values are obtained because of the ambiguity in the phase of the mixture ratio. Appreciable deviations from the irrotational value $\frac{Z}{A}$ are observed for the collective $g$ factors. The mixtures we determine also agree with those known from radioactive decay for first-excited state transitions as predicted by the unified model.