Configurations and decay hindrances of high-$K$ states in ${}^{180}\mathrm{Hf}$

Multi-quasiparticle high-$K$ states, several of which are isomeric, were observed in ${}^{180}\mathrm{Hf}$ with the Gammasphere array. Lifetimes in the ns-$\mu \mathrm{s}$ range were determined using centroid-shift and decay measurements within a $\mu \mathrm{s}$ coincidence time window. The configurations of high-$K$ states involve two and four quasiparticles, with states up to $K^{\pi }=\left({18}^{-}\right)$ established. High-$K$ excitations are found to be progressively more favored with increasing excitation energy. The $K$ quantum number is quite robust up to the highest spins observed, as evidenced by the large values of the reduced hindrance for isomeric decays. Rotational bands built on three high-$K$ states are identified, and the measured branching ratios in these sequences enable the assignment of underlying configurations. Multi-quasiparticle calculations using the Lipkin-Nogami approach for pairing, with blocking included, reproduce the observed high-$K$ energies quite well.

Figure 1

Decay scheme illustrating high-$K$ structures in ${}^{180}\mathrm{Hf}$. The half-lives of the $K$ isomers or the established limits are also indicated.

Figure 2

Time difference spectrum between transitions feeding (238 and 325 keV) and deexciting (1066 keV) the $K^{\pi }=4^{-}$ isomeric state. A half-life of 0.52(8) $\mu \mathrm{s}$ is deduced. The time distribution for prompt transitions is presented for comparison.

Figure 3

Centroids in the time distribution of the 1394-keV transition deexciting the $K^{\pi }=4^{-}$ state. The centroid of the 1315-keV transition from a prompt state is given for comparison. A distinct shift of 4 ns is evident for the 1394-keV transition. See text for details.

Figure 4

The variation of the centroid of the time distributions as a function of energy for prompt transitions. In the case of transitions from isomeric states in ${}^{179,180}\mathrm{Hf}$ (colored points), considerable shifts in the centroids are clearly visible.

Figure 5

Gated spectra illustrating transitions in rotational bands built on the $K^{\pi }={10}^{+},{11}^{-}$ and ${14}^{+}$ states. $\mathrm{\Delta }\mathit{I}=1$ transitions are below 400 keV, and $\mathrm{\Delta }\mathit{I}=2$ $\gamma$ rays above 500 keV. Transitions from the ground-state band in ${}^{180}\mathrm{Hf}$, present due to random coincidence events, are marked with asterisks. Additional transitions in the spectra are visible through coincidences with $\gamma$ rays from other structures in ${}^{180}\mathrm{Hf}$. The 569-keV transition in ${}^{207}\mathrm{Pb}$ (beam) is visible through cross coincidence. The spectrum for the $K^{\pi }={10}^{+}$ sequence is obtained by means of a double coincidence gate on the 243- and 1040-keV transitions, while that for the $K^{\pi }={11}^{-}$ band is obtained by summing double gates on 1118-keV with 243- and 268-keV transitions, respectively. The $K^{\pi }={14}^{+}$ spectrum is obtained by summing double coincidence gates on in-band transitions and ones deexciting the band-head state.

Figure 6

Time difference spectrum between transitions feeding (184 and 321 keV) and deexciting (539 keV) the $K^{\pi }={12}^{+}$ isomeric state. A half-life of 0.94(11) $\mu \mathrm{s}$ is deduced.

Figure 7

Comparison of relative intensities of 137-, 184-, and 270-keV $\gamma$ rays with gates on transitions: (a) above the $K^{\pi }={14}^{+}$ level, and (b) below the $K^{\pi }={12}^{+}$ state. The similarity of the ratio of relative intensities of the 270-keV $\gamma$ ray with the 137- and 184-keV transitions in the two spectra preclude the long-lived $({\mathit{T}}_{1/2}\gg 10$ $\mu \mathrm{s})$ isomeric character reported earlier [8] for the level at 2535 keV, and suggest $T_{1/2}\ll 1$ $\mu \mathrm{s}$.

Figure 8

Variation with spin of the difference between excitation energies of high-$K$ states and yrast collective rotational levels in ${}^{180}\mathrm{Hf}$.

Figure 9

Reduced hindrances $\left(f_{\nu }\right)$ for decays from $K^{\pi }=6^{+}$ isomers to the $4^{+}$ state in the ground-state band of even Hf isotopes. The hindrance is most pronounced in the case of ${}^{176}\mathrm{Hf}$, where the configuration is lowest in energy, and decreases towards the neutron- and proton-rich sides. In all cases, the uncertainties in reduced hindrances are quite small, and therefore not visible outside the symbols.

Figure 10

Values of $\left|(g_K-g_R)/Q_0\right|$ as a function of spin for $K^{\pi }={10}^{+},{11}^{-}$, and ${14}^{+}$ configurations in ${}^{180}\mathrm{Hf}$. Measured values, obtained from branching ratios as described in the text, are displayed along with expected values (dotted lines).

Figure 11

Excitation energies, with a rigid rotor reference subtracted as indicated, as a function of spin for the yrast positive-parity states, and bands built on the ${10}^{+}$ states (both even and odd spin) in Hf, W, and Os isotones with $N=108$.

Figure 12

Excitation energies, with a rigid rotor reference subtracted, as a function of spin for the yrast positive-parity states, and bands built on the $8^{+}$ states (both even and odd spin) in W and Os isotones with $N=106$.