Investigation of $\gamma$ softness: Lifetime measurements in ${}^{104,106}\mathrm{Ru}$

Lifetimes of the $2_1^{+}, 4_1^{+}, 6_1^{+}, 2_{\gamma }^{+}$, and $3_{\gamma }^{+}$ states in ${}^{104,106}\mathrm{Ru}$ were measured by using the recoil-distance Doppler-shift technique and the Cologne Plunger device. Low-lying excited states in both nuclei were populated in a ${}^{104}\mathrm{Ru}({}^{18}\mathrm{O},{}^{18}\mathrm{O}){}^{104}\mathrm{Ru}$* inelastic scattering and in a ${}^{104}\mathrm{Ru}({}^{18}\mathrm{O},{}^{16}\mathrm{O}){}^{106}\mathrm{Ru}$ two-neutron transfer reaction using the Cologne FN Tandem accelerator. The experimental energy levels and deduced electromagnetic transition probabilities are compared in the context of $\gamma$ softness and the mapped interacting boson model with input from the microscopic self-consistent mean-field calculation using a Gogny interaction. The newly obtained results for the $\gamma$ band give a more detailed insight about the triaxial behavior of ${}^{104,106}\mathrm{Ru}$. The results will be discussed in the context of $\gamma$ soft and rigid triaxial behavior which is present in the neutron-rich Ru isotopes.

Figure 1

(a) The staggering parameter $S\left(J\right)$ for the ${}^{100–112}\mathrm{Ru}$ isotopes calculated using Eq. (1). The energy levels are taken from the Nuclear Data Sheets [22, 23, 24, 25, 26, 27, 28].

Figure 2

(a) The summed solar cell spectrum of all distances. The red rectangle shows the applied gate to select the recoiling ${}^{16,18}\mathrm{O}$ particles to obtain a $\gamma$ spectrum of ${}^{104,106}\mathrm{Ru}$. The gate on ${}^{16,18}\mathrm{O}$ has been applied asymmetric to avoid a possible contamination of the $\alpha$-transfer channel. (b), (c) The summed $\gamma$-ray spectrum for all distances after applying a particle gate which is indicated by the red rectangle in panel (a) for the energy range is from 170 up to 750 keV and 720 up to 1750 keV, respectively. The observed transitions of ${}^{104}\mathrm{Ru}$ are marked in blue and for ${}^{106}\mathrm{Ru}$ in red. The transitions marked with “*” belong to the inelastic scattering of ${}^{102}\mathrm{Ru}$, which are the $2_1^{+}\to 0_1^{+}$ and $4_1^{+}\to 2_1^{+}$ transitions. Note that the $y$ scale is logarithmic in panel (b) and a linear scale is used in panel (c).

Figure 3

Partial level scheme of the observed states in ${}^{104}\mathrm{Ru}$ and ${}^{106}\mathrm{Ru}$ populated in the inelastic scattering or in the two-neutron transfer reaction. The width of the transition arrows corresponds to the observed intensities (see Table 1) and the dashed lines indicate known transitions not observed in this experiment.

Figure 4

(a), (b) Particle-gated $\gamma$-ray spectrum detected by the backward detectors of the $6_1^{+}\to 4_1^{+}$ (581.1 keV), $2_{\gamma }^{+}\to 0_1^{+}$ (792.3 keV), and $3_{\gamma }^{+}\to 2_1^{+}$ (821.5 keV) decay transitions in ${}^{106}\mathrm{Ru}$ with the corresponding shifted peaks. A peak at 585 keV has included in the fit procedure to account for its contribution [see panel (a)], marked in green. However, the origin of the $\gamma$ ray is unclear. The $\gamma$-ray spectra of all distances are summed up due to low statistics and the fits of the shifted (blue) and unshifted (red) peaks are shown. (c)–(e) The result of the Monte Carlo simulation using Eq. (2), where no feeding is assumed (green) and the case where feeding is taken into account (red). Also the final lifetimes with the error bars are shown. See text in Sec. 3b2 for more details.

Figure 5

The evolution of the shifted (blue) and unshifted (red) components in the backward ring for the $2_1^{+}\to 0_1^{+}$ (left panel) and $4_1^{+}\to 2_1^{+}$ (right panel) transitions for four selected distances, namely 53, 143, 843, and $2843\mu \mathrm{m}$. Furthermore, the decay curves for the lifetimes of the $2_1^{+}$ and $4_1^{+}$ states using the Bateman equations to fit the data are shown. The DDC method for the $2_1^{+}$ and $4_1^{+}$ states using the program napatau [43] for the backward angle. The individual obtained lifetimes are shown in the upper panel, and the lower panel shows the evolution of the shifted (red) and unshifted (blue) component. In addition the fit of the data is shown, which is used to obtain the derivative $\frac{d}{dx}{R}_i\left(x\right)$. Note that the $x$ scale is logarithmic for the decay curves.

Figure 6

Contour plot of the deformation-energy surface in the $(\beta ,\gamma )$ plane for ${}^{104}\mathrm{Ru}$ (top) and ${}^{106}\mathrm{Ru}$ (bottom) computed with the constrained SCMF method by using the Gogny functional D1M (left) and with the mapped IBM (right). The red dot indicates the minima of the energy surface plots and the difference between two neighboring contours is 100 keV

Figure 7

The experimental (Expt.) and calculated ($\gamma$ soft and mapped IBM) low energy level scheme for ${}^{104}\mathrm{Ru}$ (top) and ${}^{106}\mathrm{Ru}$ (bottom). The deduced $B\left(E2\right)$ transition probabilities values are placed on the corresponding arrow and are given in Weisskopf units. Due to a lack of multipole mixing ratios, some $B\left(E2\right)$ values are calculated in the limits of a pure $E2$ transitions, which are marked with $\#$. The transition probability of the $0_2^{+}\to 0_1^{+}$ is given in ${10}^3\times \rho \left(E0\right)$.