Triaxiality in the mid-shell nucleus ${}^{112}\mathrm{Pd}$

Lifetimes of low-spin excited states in ${}^{112}\mathrm{Pd}$ were measured using the recoil-distance Doppler-shift technique. The nucleus of interest was populated in a ${}^{110}\mathrm{Pd}({}^{18}\mathrm{O},{}^{16}\mathrm{O}){}^{112}\mathrm{Pd}$ reaction using the Cologne FN Tandem accelerator. Three lifetimes of ground-state band members and one lifetime of the $\gamma$ band were measured. From these lifetimes reduced transition probabilities were extracted and compared to interacting boson model, $\gamma$-soft calculations, and Davydov calculations. The lifetime of the $2_{\gamma }^{+}$ gives some insights on the nuclear shape and structure of the $\gamma$ band. The deduced transition rates show an indicator for a rigid triaxial nucleus as well as more indicators for a $\gamma$-soft nucleus.

Figure 1

The staggering parameter $S\left(J\right)$ for the ${}^{108\text{–}116}\mathrm{Pd}$ isotopes calculated using Eq. (1), where ${}^{112}\mathrm{Pd}$ is indicated with the dashed line at $N=66$ (adapted from Ref. [15]). The energy values are taken from ${}^{108}\mathrm{Pd}$ ($N=62$) [16], ${}^{110}\mathrm{Pd}$ ($N=64$) [17], ${}^{112}\mathrm{Pd}$ ($N=66$) [18, 19], ${}^{114}\mathrm{Pd}$ ($N=68$) [18, 20], and ${}^{116}\mathrm{Pd}$ ($N=70$) [18, 21]. Note that some energy levels are in parentheses, which means that the spins are not finally assigned. This holds especially for the states with spin higher than $J=7$.

Figure 2

(a) The level scheme of the observed states in ${}^{112}\mathrm{Pd}$, where the width of the transition arrows indicates the intensity (see Table 1). Dashed lines are used to indicate possible feeding transitions, which have not been observed in this experiment. (b, c) The $\gamma$-ray spectrum with the sum over all distances using a particle gate on the backscattered ${}^{16,18}\mathrm{O}$ fragments for the backward detectors. All transitions which were observed in the current work are colored blue for the unshifted and red for the shifted components. Additionally, a dashed line is indicating the unshifted peak. The transitions marked with a $\#$ belong to ${}^{181}\mathrm{Ta}$ which was the stopper and transitions marked with a * belong to the Coulomb excitation channel, i.e., ${}^{110}\mathrm{Pd}$. Note the logarithmic $y$ scale due to statistical reasons.

Figure 3

(a) Fit of the decaying $\gamma$ ray of the $6_1^{+}$ state with the unshifted peak at $667.5\mathrm{keV}$ and the corresponding shifted peak at $658\mathrm{keV}$, where the summed particle-gated $\gamma$-ray spectra of each distance were used due to the low statistics. The ratio ${\mathrm{R}}_{\mathrm{sum}}$ was obtained with a fit and was used to determine the lifetime according to Eq. (2). The peaks marked with an asterisk are the shifted and unshifted components of the $6_1^{+}\to 4_1^{+}$ transition in ${}^{110}\mathrm{Pd}$. A fit to those components (green) is included to eliminate the contributions on the investigated case where the shifted component is colored in red and the unshifted in blue. (b) The simulated lifetimes of the $6_1^{+}$ state with (red) and without (blue) feeding. See text for details of the simulation. The upper and lower limits of the results are indicated with dashed lines.

Figure 4

Particle-gated spectrum of the backward and forward ring for all distances and the evolution of the $2_1^{+}\to 0_1^{+}$ ($348.6\mathrm{keV}$), $2_2^{+}\to 2_1^{+}$ ($388.0\mathrm{keV}$), and $4_1^{+}\to 2_1^{+}$ ($534.3\mathrm{keV}$) transitions. The ring and the transition are indicated on the upper part of the figure. Furthermore, the distance is placed on the right hand side of the figure. The spectra indicate the background level, the shifted peak (red), the unshifted peak (blue), and also different disturbing transitions which were also fit (green). The disturbing transitions belong either to ${}^{110}\mathrm{Pd}$ or to ${}^{181}\mathrm{Ta}$. Note that a logarithmic scale is used for the backward and forward spectra for the region 320 keV up to 410 keV.

Figure 5

The decay curves for the lifetime of the $4_1^{+}$, $2_2^{+}$, and $2_1^{+}$ states using the Bateman equations to fit the data. The upper panel shows the data points of the forward ring at ${45}^{\circ }$ and the lower panel of the backward ring at ${142}^{\circ }$. The resulting lifetimes are summarized in Table 2.

Figure 6

The DDCM method for the $4_1^{+}$, $2_2^{+}$, and $2_1^{+}$ using the program napatau [45] for the backward angle. The upper panel shows the individually obtained lifetime. The middle panel shows the evolution of the shifted component and the fit which is used to obtain the derivative $\frac{d}{dx}{R}_i\left(x\right)$, whereas the unshifted component and the corresponding fit are shown in the lower panel.

Figure 7

The level energies of the ground state up to the ${10}_1^{+}$ and the $\gamma$ band up to the $8_2^{+}$ state for the (a) experimental and (b) IBM-1 calculations. (c) The $\gamma$-soft calculations using the model explained in Refs. [48, 49] and using the code from Ref. [50], where the parameters $B=146$, $a=0.12$, $C_2=47$, $G=3.42$, and $C_8=f=0$ are used. Last, the Davydov-Filippov calculations with (d) $\gamma =27.5^{\circ }$ and (e) $\gamma ={30}^{\circ }$, where the $8_2^{+}$ state is not shown which is positioned at about $5\mathrm{MeV}$. All excitation energies are given in keV and a further description of the models and calculations are given in Sec. 4.

Figure 8

The experimentally deduced staggering parameter compared to the staggering parameter of the IBM-1 calculations, $\gamma$-soft calculations, and the Davydov-Filippov calculations with $27.5^{\circ }$ and ${30}^{\circ }$. For further explanations see text.

Figure 9

The potential energy surface (PES) of the IBM-1 calculation using the parameters described in Sec. 4a.