Lifetime measurements to investigate $\gamma$ softness and shape coexistence in ${}^{102}\mathrm{Mo}$

Lifetimes of low-spin excited states in ${}^{102}\mathrm{Mo}$ populated in a ${}^{100}\mathrm{Mo}({}^{18}\mathrm{O},{}^{16}\mathrm{O}){}^{102}\mathrm{Mo}$ two-neutron transfer reaction were measured using the recoil-distance Doppler-shift technique at the Cologne FN Tandem accelerator. Lifetimes of the $2_1^{+}$, $4_1^{+}$, $6_1^{+}$, $0_2^{+}$, $2_{\gamma }^{+}$, $3_{\gamma }^{+}$ states and one upper limit for the lifetime of the $4_{\gamma }^{+}$ state were obtained. The energy levels and deduced electromagnetic transition probabilities are compared with those obtained within the mapped interacting boson model framework with microscopic input from Gogny mean-field calculations. With the newly obtained signatures a more detailed insight in the $\gamma$ softness and shape coexistence in ${}^{102}\mathrm{Mo}$ is possible and discussed in the context of the $Z\approx 40$ and $N\approx 60$ region. The nucleus of ${}^{102}\mathrm{Mo}$ follows the $\gamma$ soft trend of the Mo isotopes. The properties of the $0_2^{+}$ state indicate, in contrast with the microscopic predictions, shape coexistence which also occurs in other $N=60$ isotones.

Figure 1

The energies of the first-excited $2_1^{+}$ states (filled symbols) for Sr ($Z=38$), Zr ($Z=40$), Mo ($Z=42$), and Ru ($Z=44$) isotopes with $N=52\text{–}64$. Data are taken from the Nuclear Data Sheets [9, 10, 11, 12, 13, 14, 15, 16, 17, 18]. Also the $B(E2;2_1^{+}\to 0_1^{+})$ for the Mo isotopes are shown (open symbols) where the values are taken from Refs. [13, 14, 15, 19, 20, 21, 22].

Figure 2

(a) Partial level scheme of the observed states in ${}^{102}\mathrm{Mo}$ using the ${}^{100}\mathrm{Mo}({}^{18}\mathrm{O},{}^{16}\mathrm{O}){}^{102}\mathrm{Mo}$ two-neutron transfer reaction. The width of the transition arrows corresponds to the intensities (see Table 1) and the dashed lines indicate known transitions not observed in this experiment. (b) The solar cell spectrum of the $15\mu \mathrm{m}$ distance. The rectangle shows the gate that has been used for the analysis of ${}^{102}\mathrm{Mo}$. (c) Particle gated singles $\gamma$-ray spectrum of the backward HPGe detector ring for the shortest distance of $15\mu \mathrm{m}$. The spectrum is shown for the energy range from $120\mathrm{keV}$ up to $720\mathrm{keV}$ in which the observed transitions of ${}^{102}\mathrm{Mo}$ are indicated and colored in red. The transitions marked with $\#$ belong to the Coulomb excitation of ${}^{100}\mathrm{Mo}$ and transitions marked with * stem from ${}^{104}\mathrm{Ru}$, populated by the $\alpha$-transfer reaction channel. Note that the $y$ scale is logarithmic. (d) Same for the energy range 680 up to $1450\mathrm{keV}$ with a linear $y$ scale.

Figure 3

The evolution of the shifted (blue) and unshifted (red) components in the backward ring for the $2_1^{+}\to 0_1^{+}$ (left panel), $0_2^{+}\to 2_1^{+}$ (middle panel), $4_1^{+}\to 2_1^{+}$ (middle panel), $2_{\gamma }^{+}\to 2_1^{+}$ (right panel) and $6_1^{+}\to 4_1^{+}$ (right panel) transitions for four distances, namely 15, 64, 214, and $1114\mu \mathrm{m}$. The solid line indicates the background level and different disturbing peaks were also fit (green). The disturbing transitions, i.e., at 536 and 600 keV with their shifted components belong to ${}^{100}\mathrm{Mo}$.

Figure 4

The decay curves for the lifetimes of the $6_1^{+}$, $3_{\gamma }^{+}$, $2_{\gamma }^{+}$, $4_1^{+}$, $0_2^{+}$, and $2_1^{+}$ states using the Bateman equations to fit the data of the backward ring at ${142}^{\circ }$. Note that the $x$ scale is logarithmic. The lifetimes are summarized in Table 2.

Figure 5

The DDC method for the $6_1^{+}$, $3_{\gamma }^{+}$, $2_{\gamma }^{+}$, $4_1^{+}$, $0_2^{+}$, and $2_1^{+}$ states using the program napatau [43] for the backward angle. The upper panel shows the individually obtained lifetimes. The lower panel the evolution of the shifted and unshifted component in addition with a fit which is used to obtain the derivative $\frac{d}{dx}{R}_i\left(x\right)$.

Figure 6

Contour plot of the deformation-energy surface in the ($\beta ,\gamma$) plane for ${}^{102}\mathrm{Mo}$ computed with the constrained HFB method by using the Gogny functional D1M (left) and with the mapped IBM (right). The red dot indicates the minimum of the energy surface plots and the difference between two neighboring contours is 100 keV.

Figure 7

The level energies of the ground-state band up to the $8_1^{+}$, the $\gamma$ band up to the $6_{\gamma }^{+}$ and the $0_2^{+}$ state which were observed in different experiments (a) and the same level energies for the IBM calculations. The numbers (in red) close to the arrows indicate the $B\left(E2\right)$ values in Weisskopf units. The $B(E2;3_{\gamma }^{+}\to 2_{\gamma }^{+})$ value indicated with an * is calculated using the limits of a pure $E2$ transition due to a lack of multipole mixing ratios.

Figure 8

The energies of the $0_2^{+}$ state (filled symbols) for ($Z=38$), Zr ($Z=40$), Mo ($Z=42$), and Ru ($Z=44$) isotopes with $N=56\text{–}62$. The data are taken from the Nuclear Data Sheets [11, 12, 13, 14, 15, 16, 17]. If available, the ${10}^3\times {\rho }^2\left(E0\right)$ are shown for the same isotopes (open symbols), where the data are taken from Refs. [34, 52] and for ${}^{102}\mathrm{Mo}$ from this work. Note that the values are slightly shifted along the $x$ axis to have a better separation of the values.