Lifetime measurements in the tungsten isotopes ${}^{176,178,180}\mathrm{W}$

Lifetimes of yrast states in the rare-earth midshell isotopes ${}^{176,178,180}\mathrm{W}$ have been measured with fast-timing methods using the Cologne iron-free Orange spectrometer and the Cologne HORUS spectrometer and with the recoil distance Doppler shift method using the Cologne coincidence plunger setup. Different fusion evaporation reactions have been used to populated excited states in the investigated nuclei. Lifetimes of the $2_1^{+}$, $4_1^{+}$, $6_1^{+}$, and $8_1^{+}$ yrast states were measured in all three tungsten isotopes and in addition the lifetime of the ${10}_1^{+}$ state was measured in ${}^{176}\mathrm{W}$. Quadrupole deformation parameters, reduced transition probabilities, and ${\text{B}}_{4/2}$ ratios are extracted and discussed in the context of interacting boson model 1 calculations. The results with newly determined signatures largely confirm the investigated tungsten isotopes to be deformed prolate rotors.

Figure 1

(a) The elements of the Orange spectrometer, shown in a section through an $(r,z)$ plane [30]. (b) Schematic three-dimensional model of the Orange spectrometer. The copper coils are drawn in orange. The magnetic field is indicated by the green circular arrows inside the coil volume. The circular electron shields are shown by the transparent light gray forms. The dark gray cone represents the $\gamma$-ray shield to shield the electron detector from unwanted $\gamma$ rays. Possible flight trajectories of detected electrons are displayed by the black dotted curves. The symmetry axis ($z$) is displayed in black and the beam axis ($r$) in red.

Figure 2

Conversion electron momentum scans of ${}^{178,180}\mathrm{W}$ between 125 and 325 A corresponding to an electron residual energy range of 60 to 310 keV. The scan is performed in steps of 0.3 A with 10-s measurements and the electron counts were normalized to the beam current. The L and M peaks of the $2_1^{+}\to 0_1^{+}$ transition and the K, L, and M peaks of the $4_1^{+}\to 2_1^{+}$ of (a) the ${}^{178}\mathrm{W}$ reaction and (b) the ${}^{180}\mathrm{W}$ reaction are marked. The peaks at 137 and 144 A, marked with asterisks in (a), are the L and M peaks of the $2_1^{+}\to 0_1^{+}$ transition in ${}^{172}\mathrm{Yb}$ from the Coulomb excitation of the target.

Figure 3

LaBr spectra and HPGe spectra in coincidence with the $2_1^{+}\to 0_1^{+}$ transition (observed with HPGe detectors in HORUS in ${}^{176}\mathrm{W}$ and with the Orange spectrometer in ${}^{178,180}\mathrm{W}$) for (a) ${}^{176}\mathrm{W}$, (b) ${}^{178}\mathrm{W}$, and (c) ${}^{180}\mathrm{W}$ (c). In (a) and (b) the yrast cascade $\gamma$-ray lines are marked. In (c) the yrast $\gamma$-ray lines as well as some of the $K^{\pi }=2^{-}$ band $\gamma$-ray lines and the $8_1^{-}\to 8_1^{+}$ transition from the $K^{\pi }=8^{-}$ band to the ground state band are marked. The small overlay box shows the 902.8 keV $\gamma$ ray of the $2_1^{-}\to 2_1^{+}$ transition from the $K^{\pi }=2^{-}$ band to the ground state band.

Figure 4

Relevant level schemes of ${}^{176,178,180}\mathrm{W}$. All transitions depicted with solid lines were used for the lifetime analysis in this work, and are labeled with their respective transition energy. The level and transition energies were adopted from the Nuclear Data Sheets [15, 16, 17].

Figure 5

Schematic drawing of the HORUS setup. The red tubes are the BGO Compton shields, the blue cylinders are the HPGe detectors. The yellow shapes are the LaBr fast-timing scintillators attached to their photo-multiplier tubes, depicted in gray. The beam direction is depicted by a green arrow.

Figure 6

The PRDs of (a) the Orange setup and (b) the HORUS setup. The upper panels in (a) and (b) show the prompt response difference curve, determined with coincident $\gamma$ rays from a ${}^{152}\mathrm{Eu}$ source at target position. The fitted calibration functions are displayed by the red lines. The uncertainties are displayed by the green uncertainty bands. The zero reference is 344 keV in both cases. The residuals of the PRD fits with uncertainties and $1\sigma$ interval, displayed by the green lines, are depicted in the lower panels.

Figure 7

Time distributions for the $2_1^{+}$ states in (a) ${}^{176}\mathrm{W}$, (b) ${}^{178}\mathrm{W}$, and (c) ${}^{180}\mathrm{W}$. The fits to the exponential decay are shown as green lines, the constant background contribution is indicated by blue constants, and the fit with all components by red curves. The fit ranges are indicated by vertical lines. Each panel is labeled with the respective conversion electron and LaBr gates and resulting lifetimes. The yellow centroids are the prompt response functions of the HORUS respective Orange timing systems. The time resolution of the LaBr-Orange timing system is around 1000 ps [22], indicated by the widened PRF, shown in (b) and (c), compared to the LaBr-LaBr timing system of the HORUS spectrometer with a time resolution of about 340 ps.

Figure 8

Centroid difference of the $6_1^{+}\to 4_1^{+}\to 2_1^{+}\gamma$ cascade in ${}^{178}\mathrm{W}$, generated using a conversion electron gate on the $2_1^{+}\to 0_1^{+}$ transition. Displayed are the time distributions for delayed (blue) and antidelayed (red) gate sequence.

Figure 9

Analysis of the $4_1^{+}$ state of ${}^{178}\mathrm{W}$. In red and black, double gated HPGe and LaBr energy spectra with an Orange trigger gate on the $2_1^{+}\to 0_1^{+}$ transition and a LaBr gate on the (a) $4_1^{+}\to 2_1^{+}$ (237 keV), which results in an antidelayed time spectrum, and the (b) $6_1^{+}\to 4_1^{+}$ (351 keV), which results in a delayed time spectrum. The peaks of interest are well separated from contaminating transitions. In the lower parts of the plots, the measured centroid difference (green) and the measured background time response and corresponding interpolations (blue) are shown.

Figure 10

Projections of the $\gamma \text{−}\gamma$ coincidences with a gate on the flight component of the ${12}_1^{+}\to {10}_1^{+}$ transition in ${}^{176}\mathrm{W}$ are shown for different target-to-stopper distances under backward angle. Yrast transitions of interest are marked with dashed lines, where the left lines mark the Doppler-shifted and the right ones the unshifted components.

Figure 11

Example of the DDCM analysis for the $6_1^{+}$ state using the software napatau [48]. The upper panels show the corresponding obtained lifetimes. The lower panels show the evolution of the shifted component intensities (green data points and red curves) and derivates (light red data points and blue curves). The $\tau$ plots are depicted with their corresponding gate configurations and angles, where “fw” means forward angle and “bw” means backward angle.

Figure 12

The evolution of (a) ${\text{R}}_{4/2}$ ratios, (b) $B\left(E2\right)$ values, (c) ${\text{B}}_{4/2}$ ratios, and (d) experimental quadrupole deformation parameters of the tungsten isotopes around $N=104$ compared with the IBM-1 calculations from this work. The $\beta$ values obtained from the IBM-1 calculations are drawn with solid lines in the same color as the experimental values. Experimental values for the energies and the literature values of the $B\left(E2\right)$ values are taken from Refs. [15, 16, 17, 19, 50, 51, 52, 53, 54]. Rigid rotor and X(5) limits are shown as horizontal lines. The $\widehat{L}\widehat{L}$ criterion as mentioned in Sec. 4b is displayed as a dashed line in (a).

Figure 13

Excitation energies (top), reduced transition probabilities (middle), and experimental quadrupole deformation parameters ${\beta }_{I,I-2}$ (bottom) as a function of yrast spin for ${}^{172-182}\mathrm{W}$. The upper and the middle panels compare the experimental values to the IBM-1 calculations from Sec. 4b and the SU(3) and X(5) limits. The bottom plot compares the experimental values to the IBM-1. The literature values are taken from Refs. [50, 52, 64, 65].

Figure 14

(a) Experimental and (b) theoretical (IBM-1) level energies of the low-lying yrast $2^{+}$, $4^{+}$, $6^{+}$, $8^{+}$, and ${10}^{+}$ states as a function of the neutron number.