Lifetime measurement for the $2_1^{+}$ state of ${}^{170}\mathrm{Hf}$

The lifetime of the $J^{\pi }=2_1^{+}$ state of ${}^{170}\mathrm{Hf}$ at 100.8 keV was measured. Excited states of ${}^{170}\mathrm{Hf}$ were populated with the ${}^{158}\mathrm{Gd}$(${}^{16}\mathrm{O}$,4nγ) reaction at the TANDEM-LINAC facility of the State University of New York (SUNY) at Stony Brook. A lifetime of $\tau =1.74\pm 0.06$ ns was found using the delayed γ-radiofrequency coincidence method with respect to the pulsed LINAC beam. It corresponds to an $E2$ transition strength of $181\pm 6$ W.u. to the $J^{\pi }=0_1^{+}$ ground state. With its increased precision, by one order of magnitude with respect to previous literature, this value serves as a normalization parameter for collective models for this nucleus.

Figure 1

Part of the γ-ray spectrum obtained in the experiment. The x-ray transitions of interest in ${}^{232}\mathrm{Th}$ and the delayed 100.8-keV $2_1^{+}\to 0_1^{+}$ transition in ${}^{170}\mathrm{Hf}$ are labeled.

Figure 2

Time difference histograms corresponding to the delayed 100.8-keV $2_1^{+}\to 0_1^{+}$ transition in ${}^{170}\mathrm{Hf}$ (black) and to the prompt 93.3-keV (online red) and 105.6-keV (online green) x-ray transitions in ${}^{232}\mathrm{Th}$. The histograms were normalized with respect to the one with maximum integral. Their shape is related to the time structure of the beam bunches.

Figure 3

Centroid diagram obtained in the ${}^{158}\mathrm{Gd}$(${}^{16}\mathrm{O}$,4nγ)${}^{170}\mathrm{Hf}$ fusion-evaporation reaction for one of the Ge detectors. Crosses represent the centroids of the time distributions of the prompt x-ray transitions. The solid circle represents the data point for the $2_1^{+}\to 0_1^{+}$ transition in ${}^{170}\mathrm{Hf}$. The radii of the symbols correspond to the statistical errors.

Figure 4

$2_1^{+}\to 0_1^{+}$ transition energies (top) and corresponding $B(E2;2_1^{+}\to 0_1^{+})$ values (bottom) across the Hf isotopic chain. Two conflicting values of $B(E2)$ are plotted for ${}^{174}\mathrm{Hf}$ corresponding to Coulomb excitation (top value) and delayed coincidence (bottom value) measurements. Data on Hf isotopes other than ${}^{170}\mathrm{Hf}$ were taken from Ref. [11].

Figure 5

$2_1^{+}\to 0_1^{+}$ transition energies (top) and corresponding $B(E2;2_1^{+}\to 0_1^{+})$ values (bottom) for nuclei with a $P$ parameter of $6.2\pm 0.3$ and situated in the p-p ($Z=50\text{−}66,N=82\text{−}104$) and p-h ($Z=68\text{−}82,N=82\text{−}104$) regions. Data other than the $B(E2;2_1^{+}\to 0_1^{+})$ value of ${}^{170}\mathrm{Hf}$ are taken from Ref. [11]. No $2_1^{+}\to 0_1^{+}$ transition energies or $B(E2;2_1^{+}\to 0_1^{+})$ values for ${}^{158,160}\mathrm{Nd}$ are known. The $B(E2;2_1^{+}\to 0_1^{+})$ values for ${}^{156,158}\mathrm{Sm}$ are also unknown.

Figure 6

Theoretical and experimental transition quadrupole moments, $Q_t$, as a function of spin, J, in ${}^{170}\mathrm{Hf}$. The dotted line represents the rigid rotor prediction and the solid curve is the CBS rotor model prediction with the $E2$ operator in the lowest order in the quadrupole deformation parameter, β ($\chi =0$). $Q_t$ values including a second-order correction ($\chi =-0.535$ [3]) in the $E2$ operator are represented by the dashed curve. Our experimental $Q_t(2_1^{+}\to 0_1^{+})$ (online red) is superimposed on the previously known experimental value.