Evolution of $E2$ strength in the rare-earth isotopes ${}^{174,176,178,180}\mathrm{Hf}$

Mean lifetimes of yrast states of the isotopes ${}^{174,176,178,180}\mathrm{Hf}$ have been measured using fast-electronic scintillation timing. Excited states of ${}^{174,176,178}\mathrm{Hf}$ were populated via $\beta$ decay, while ${}^{180}\mathrm{Hf}$ was populated via Coulomb excitation. The lifetimes of the $2_1^{+}$ and $4_1^{+}$ states of all isotopes and the lifetimes of the $6_1^{+}$ states of ${}^{174,178}\mathrm{Hf}$ were measured, using the slope and the centroid shift methods. The mean lifetime, $\tau \left(4_1^{+}\right)=85\left(13\right)\mathrm{ps}$, of ${}^{178}\mathrm{Hf}$ has been determined for the first time. In addition, the mean lifetimes of the $2_1^{-}$ and the $3_1^{-}$ states of ${}^{176}\mathrm{Hf}$ have been determined. Systematic uncertainties on the evolution of data as a function of neutron number were reduced by using the same setup for all the isotopes of interest. The data are in agreement with other recent lifetime measurements where available and shows a shift of the maximum of collectivity for the Hf isotopic chain from neutron midshell at $N=104$ to $N=100$.

Figure 1

(a) Systematics of $g$ factors of first excited $2_1^{+}$ states of even-even nuclei around neutron number $N=104$ (marked with gray dashed line) of Er, Yb, Hf, W, and Os isotopes. The $g$ factor values seem to saturate around (or slightly below) $N=104$ apart from the Hf isotopic chain, which only has three data points. Data have been taken from Ref. [19]. (b) Systematics of $B(E2;2_1^{+}\to 0_{\mathrm{gs}}^{+})$ values of even-even Er, Yb, Hf, W, and Os isotopes around neutron number $N=104$ (marked with vertical gray dashed line) from literature data [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 28]. The values of different isotopes are slightly shifted for a better visibility.

Figure 2

Section of the $E_{\gamma ,{\mathrm{LaBr}}_3}\text{−}{E}_{\gamma ,{\mathrm{LaBr}}_3}$ matrix obtained after projection of the $E_{\gamma ,{\mathrm{LaBr}}_3}\text{−}{E}_{\gamma ,{\mathrm{LaBr}}_3}\text{−}\mathrm{\Delta }t$ cube. Marked are the coincidence areas of the $\gamma$-ray pair 93 and 213 keV, corresponding to the transitions $2_1^{+}\to 0_{\mathrm{gs}}^{+}$ and $4_1^{+}\to 2_1^{+}$ of ${}^{178}\mathrm{Hf}$. They are not distinguishable from the $\gamma$-ray pair $8_1^{-}\to 8_1^{+}$ ($88\mathrm{keV}$) and $4_1^{+}\to 2_1^{+}$ ($213\mathrm{keV}$) as mentioned in the text.

Figure 3

Partial energy spectra of all ${\mathrm{LaBr}}_3$ (red) and HPGe detectors (black) for (a) ${}^{174}\mathrm{Hf}$, (b) ${}^{176}\mathrm{Hf}$, (c) ${}^{178}\mathrm{Hf}$, and (d) ${}^{180}\mathrm{Hf}$. Transitions used for the determination of the lifetimes are denoted by blue triangles.

Figure 4

Logarithmic plots of the time-difference spectra for the decays of $2_1^{+}$ states of the isotopes ${}^{174,176,178,180}\mathrm{Hf}$ from (a)–(d). Final linear fits are indicated by the dashed black lines.

Figure 5

Logarithmic plots of the time-difference spectra for the decays of short lived (range of ps) yrast states of the isotopes ${}^{180,178,176,174}\mathrm{Hf}$. The delayed time spectra are shown in black (solid lines) and the antidelayed spectra in orange (dashed lines). (a) The time-difference spectra for the $6_1^{+}\to 4_1^{+}\to 2_1^{+}$ cascade of ${}^{180}\mathrm{Hf}$, (b) for the $6_1^{+}\to 4_1^{+}\to 2_1^{+}$ cascade of ${}^{178}\mathrm{Hf}$, (c) for the $8_1^{+}\to 6_1^{+}\to 4_1^{+}$ cascade of ${}^{178}\mathrm{Hf}$, (d) for the $3_1^{-}\to 4_1^{+}\to 2_1^{+}$ cascade of ${}^{176}\mathrm{Hf}$, (e) for the $4_3^{+}\to 4_1^{+}\to 2_1^{+}$ cascade of ${}^{174}\mathrm{Hf}$, and (f) for the $4_2^{+}\to 6_1^{+}\to 4_1^{+}$ cascade of ${}^{174}\mathrm{Hf}$.

Figure 6

Partial level schemes of the investigated Hf isotopes, ${}^{174-180}\mathrm{Hf}$ (a)–(d). Shown are the states and transitions used for fast-timing lifetime measurement. The determined mean lifetimes are given in yellow. In the case of ${}^{180}\mathrm{Hf}$, the transition widths correspond to the observed $\gamma$-ray intensity.

Figure 7

(a) Logarithmic plots of the time-difference spectrum for the decay of $2_1^{-}$ state of the isotopes ${}^{176}\mathrm{Hf}$. Final linear fit is indicated by the dashed black line. (b) Delayed (black) and antidelayed (orange) time distributions of the cascade ${(2/3)}^{-}\to 3_1^{-}\to 4_1^{+}$.

Figure 8

(a) $B(E2;2_1^{+}\to 0_{\mathrm{gs}}^{+})$ strengths in W.u. plotted over the product proton valence particles (holes) and of neutron valence particles (holes) $N_p{N}_n$ of the Er, Yb, Hf, and W isotopic chains. The gray dashed lines are plotted to guide the eyes. Data are taken from Refs. [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 24] and this work. (b) “Quadrupole” $B\left(E2\right)$ values, $B{\left(E2\right)}_Q$, vs. $N_p{N}_n$ obtained by removing the ${\beta }_4$ hexadecapole deformations as described in Ref. [46]. ${\beta }_2$ values are taken from Ref. [28] and the ${\beta }_4$ values from Ref. [47]. (c) ${\beta }_4$ hexadecapole deformations vs. the neutron number. Data are taken from Ref. [47].

Figure 9

(a) Depicted is the double difference $\delta V_{pn}$ [20] in a $N\text{−}Z$ chart. $N=104$ is marked by the dotted dashed line. The areas marked in blue is shown in (b). The dotted dashed lines are fits with a constant to the Hf data before and after midshell, illustrating the sudden increase of $\delta V_{pn}$ at $N=104$ and the constant behavior before and after $N=104$. The vertical gray dashed line marks the neutron midshell $N=104$. Binding energies taken from Refs. [52, 53].

Figure 10

(a) Normalized $B(E2;J\to J-2)$ transition strengths are plotted over $J$ for the isotopes ${}^{174,176,178,180}\mathrm{Hf}$. ${}^{178,180}\mathrm{Hf}$ are close to the rigid rotor limit. The $B(E2;6_1^{+}\to 4_1^{+})$ strength is significantly lower than the $B(E2;4_1^{+}\to 2_1^{+})$ strength of ${}^{174}\mathrm{Hf}$. The isotopes are slightly shifted for a better visibility. (b) Ratio $B_{4/2}$ of the $B\left(E2\right)$-transition strengths $B(E2;4_1^{+}\to 2_1^{+})$ and $B(E2;2_1^{+}\to 0_{\mathrm{gs}}^{+})$. Shown in the red box are the averaged values from this work and Refs. [12, 13, 24] and outside the box the known literature data taken from Refs. [5, 6, 7, 8, 9]. The dotted line represents the value obtained in the rigid rotor limit and in the vibrator limit.