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

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 ${}^{172,174,176}\mathrm{Hf}$, 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 $\gamma \text{−}\gamma$ 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 $\gamma \text{−}\gamma$ 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.

Figure 1

Conversion-electron momentum spectra of the three investigated nuclei ($I$ is the current applied to the magnetic spectrometer; see text for more details). Different conversion lines from the same nuclear transition are marked accordingly. The step intervals during the scans were 0.3 A (${}^{172}\mathrm{Hf}$ and ${}^{174}\mathrm{Hf}$) and 0.5 A $\left({}^{176}\mathrm{Hf}\right)$.

Figure 2

Photograph of the $\gamma$ detector array in maintenance position. The position of the four BGO shielded LaBr detectors (red circles) and the two unshielded LaBr detectors (blue crosses) can be seen. Only the BGO shield is mounted where the germanium detector was positioned during the experiment (green star). The view is from the target position; the beam direction is from right to left.

Figure 3

$\gamma$-ray spectra of ${}^{174}\mathrm{Hf}$ from one experimental run with different coincidence conditions. (a) A spectrum measured with the Ge detector in coincidence with electrons corresponding to the L conversion of the $2_1^{+}\to 0_1^{+}$ transition. The Ge detector was shielded with lead and copper to suppress x rays. (b) LaBr $\gamma \text{−}\gamma$ projection without electron coincidence condition. (c) LaBr $\gamma \text{−}\gamma$ projection with the same electron coincidence condition as in (a).

Figure 4

ce-$\gamma$ (a),(b) and $\gamma \text{−}\gamma$ (c) delayed coincidence time spectra of the cascade $4_1^{+}\to 2_1^{+}\to 0_1^{+}$. Compton background has been subtracted to yield spectrum (c). The half-life was determined by fitting the slope. The best fit of an exponential function is shown as a solid line. The random background, shown as a dashed line, was determined separately.

Figure 5

Examples of gated LaBr $\gamma \gamma$ coincidence spectra for ${}^{174}\mathrm{Hf}$. The transition which was gated on is indicated in the top-right corner of each panel. No coincident electron was demanded.

Figure 6

PRD calibration points measured with a ${}^{152}\mathrm{Eu}$ source for ${}^{172}\mathrm{Hf},{}^{174}\mathrm{Hf}$ (a), and ${}^{176}\mathrm{Hf}$ (b). The reference energy is $E_{\gamma }=344$ keV. The fitted calibration function is shown as a solid line. The uncertainty of $\pm 10$ ps is indicated as shifted dashed lines. For better comparison the data point at $E_{\gamma }=40$ keV is shown as an inlay. See text for more details.

Figure 7

$\gamma \text{−}\gamma$ time difference spectra for direct decay-feeder cascades in the nuclei ${}^{172}\mathrm{Hf}$ (left), ${}^{174}\mathrm{Hf}$ (center), and ${}^{176}\mathrm{Hf}$ (right). The delayed spectra, with the decay gated on the stop branch, are shown in gray (green). The antidelayed spectra, where the decay is gated on the start branch, are shown in black (red). The top row shows the time spectra for the $6_1^{+}\to 4_1^{+}\to 2_1^{+}$ cascade (with ce gate), the center row those for the $8_1^{+}\to 6_1^{+}\to 4_1^{+}$ cascade, and the bottom those for the ${10}_1^{+}\to 8_1^{+}\to 6_1^{+}$ cascade (both without ce coincidence condition).

Figure 8

Comparison of measured $B(E2,2_1^{+}\to 0_1^{+})$ values of ${}^{172}\mathrm{Hf},{}^{174}\mathrm{Hf}$, and ${}^{176}\mathrm{Hf}$. The respective adopted values found in the nuclear data sheets [40, 42, 48] are shown as a solid line with dashed lines indicating the uncertainty. Other data are from Hansen et al. [49], Bjerrefard et al. [50], Fossan and Herskind [51], Abou-Leila [52], Charvet et al. [53], Ejiri and Hagemann [54], Ronningen et al. [55], Hammer et al. [56], Tanaka et al. [57], and Régis [45]. The experimental method is indicated by ce-ce, $\gamma \text{−}\gamma ,\beta$-ce (delayed ce-ce, $\gamma \text{−}\gamma$, and $\beta$-ce coincidence fast timing), Clx (Coulomb excitation), and $\mu$-atom (muonic atom spectroscopy).

Figure 9

Experimental $B(E2;J\to J-2)$ values in hafnium isotopes $\left(Z=72\right)$ (a) and the ratio $B_{42}=B(E2,4_1^{+}\to 2_1^{+})/B(E2,2_1^{+}\to 0_1^{+})$ (b). Values measured in this work are marked with a box. Other values are taken from the Nuclear Data Sheets [58, 59, 60, 61, 62].

Figure 10

Contour plots of the deformation energy surfaces in terms of the quadrupole deformations ${\beta }_2$ and $\gamma$ for the nuclei ${}^{168\text{–}178}\mathrm{Hf}$, obtained from the Gogny HFB calculations using the D1M interaction. The color scale varies in steps of 0.25 MeV and the contour lines are drawn in steps of 0.5 MeV. The range of the plot is $0.0\le \beta \le 0.5$ and $0^{\circ }\le \gamma \le {60}^{\circ }$. The absolute minimum is identified by an open circle.

Figure 11

The ${\beta }_{2,\text{min}}$ values, where the HFB energy has its absolute minimum are plotted as a function of neutron number for the ${}^{166\text{–}180}\mathrm{Hf}$ isotopes. Solid and dotted curves represent the results from D1S and D1M interactions, respectively.

Figure 12

Experimental (a) and the theoretical (D1M parametrization) (b) level energies of the low-lying yrast $2^{+},4^{+},6^{+},8^{+}$, and ${10}^{+}$ states as functions of neutron number.

Figure 13

The moments of inertia (MOIs) of the ground-state rotational band of the ${}^{166\text{–}180}\mathrm{Hf}$ nuclei obtained in different approaches are plotted as a function of neutron number. The MOIs are computed using the cranking calculations of the IBM in the coherent state “IBM-cranking” and the self-consistent HFB cranking method that leads to the Thouless-Valatin MOI with both the D1M “Gogny-HFB TV (D1M)” and the D1S “Gogny-HFB TV (D1S)” parametrizations. Also shown are the MOIs extracted from the $2_1^{+}$ excitation energies of the IBM-2 “$E\left(2_1^{+}\right)$ IBM” and the experiment “$E\left(2_1^{+}\right)$ Exp.” using the rigid rotor formula.

Figure 14

The $B\left(E2\right)$ values for hafnium $\left(Z=72\right)$ isotopes calculated with two different choices of effective boson charge $e_{\mathrm{B}}$: (a) $e_{\mathrm{B}}=0.123e\text{b}$ fixed for all isotopes; (b) $e_B$ different from nucleus to nucleus, each determined by adjusting the transition quadrupole moment $Q_t(2_1^{+}\to 0_1^{+})$ from the IBM-2 calculation to fit ${\beta }_{2,\min }$ of the Gogny-HFB D1M calculation. The boson effective charges $e_B$ used for the latter calculation are tabulated in Table 7.

Figure 15

Comparison of experimental and theoretical reduced transition strengths in hafnium isotopes $\left(Z=72\right)$. (a) Calculation from Fig. 14 $\left(e_{\mathrm{B}}=0.123e\mathrm{b}\right)$. (b) Calculation from Fig. 14 ($e_{\mathrm{B}}$ mass dependent).

Figure 16

Comparison of experimental and theoretical reduced transition strengths in hafnium isotopes $\left(Z=72\right)$. (a) Calculation from Fig. 14 ($e_{\mathrm{B}}$ mass dependent). (b) $B_{42}$ as defined in the text. The IBM SU(3) limit of $B_{42}=10/7$ is indicated as a gray line.