Lifetime measurement of neutron-rich even-even molybdenum isotopes

Background: In the neutron-rich $A\approx 100$ mass region, rapid shape changes as a function of nucleon number as well as coexistence of prolate, oblate, and triaxial shapes are predicted by various theoretical models. Lifetime measurements of excited levels in the molybdenum isotopes allow the determination of transitional quadrupole moments, which in turn provides structural information regarding the predicted shape change.

Purpose: The present paper reports on the experimental setup, the method that allowed one to measure the lifetimes of excited states in even-even molybdenum isotopes from mass $A=100$ up to mass $A=108$, and the results that were obtained.

Method: The isotopes of interest were populated by secondary knock-out reaction of neutron-rich nuclei separated and identified by the GSI fragment separator at relativistic beam energies and detected by the sensitive PreSPEC-AGATA experimental setup. The latter included the Lund-York-Cologne calorimeter for identification, tracking, and velocity measurement of ejectiles, and AGATA, an array of position sensitive segmented HPGe detectors, used to determine the interaction positions of the $\gamma$ ray enabling a precise Doppler correction. The lifetimes were determined with a relativistic version of the Doppler-shift-attenuation method using the systematic shift of the energy after Doppler correction of a $\gamma$-ray transition with a known energy. This relativistic Doppler-shift-attenuation method allowed the determination of mean lifetimes from 2 to 250 ps.

Results: Even-even molybdenum isotopes from mass $A=100$ to $A=108$ were studied. The decays of the low-lying states in the ground-state band were observed. In particular, two mean lifetimes were measured for the first time: $\tau =29.7_{-9.1}^{+11.3}$ ps for the $4^{+}$ state of ${}^{108}\mathrm{Mo}$ and $\tau =3.2_{-0.7}^{+0.7}$ ps for the $6^{+}$ state of ${}^{102}\mathrm{Mo}$.

Conclusions: The reduced transition strengths $B\left(E2\right)$, calculated from lifetimes measured in this experiment, compared to beyond-mean-field calculations, indicate a gradual shape transition in the chain of molybdenum isotopes when going from $A=100$ to $A=108$ with a maximum reached at $N=64$. The transition probabilities decrease for ${}^{108}\mathrm{Mo}$ which may be related to its well-pronounced triaxial shape indicated by the calculations.

Figure 1

Photograph of the experimental setup positioned behind the last FRS plastic detector.

Figure 2

Identification plot of species selected by the FRS for the setting centered on ${}^{109}\mathrm{Tc}$. The color bar on the right-hand side indicates the number of counts per bin.

Figure 3

The $\mathrm{\Delta }E\text{−}E$ spectrum measured by the LYCCA telescopes for nuclei produced in the secondary reaction of ${}^{109}\mathrm{Tc}$ ions. It allowed the identification of the proton number of the products. The color bar on the right-hand side indicates the number of counts per bin.

Figure 4

Mass identification based on the reconstruction of the mass given by Eq. (3). In the upper panel, only a gate on the LYCCA $\mathrm{\Delta }E\text{−}E$ was applied. For each histogram of the lower panel, two gates were applied, one on the relevant $\gamma$-ray transition (see legend) and one on the molybdenum isotope in the LYCCA $\mathrm{\Delta }E\text{−}E$ identification histogram in Fig. 3. The yield is the ratio of the number of $\gamma$ rays observed for the energy of the transition over the total number of $\gamma$ rays recorded in an energy range from 200 to 1000 keV. The lines corresponding to ${}^{108}\mathrm{Mo}$ and ${}^{104}\mathrm{Mo}$ are similar because the energies of the $4^{+}\to 2^{+}$ transition are close.

Figure 5

Observed transitions in even-even molybdenum isotopes. The blue curves correspond to the model fitted on the data and used to determine the mean energy of the observed transitions. The green dashed lines indicate the mean energy of the observed transitions as given in the fourth column of Table 3.

Figure 6

The curves correspond to the simulated values of $R_{\mathrm{shift}}$ as a function of $\tau$, the mean lifetime. The data points for ${}^{100,102}\mathrm{Mo}$ correspond to previous measurements [45]. In the legend “standard” stands for assumed velocity and target thickness, while “optimized” corresponds to the set of parameters $(t_{\mathrm{thick}},f)=(556\left(13\right)\mathrm{mg}/{\mathrm{cm}}^2,0.916\left(19\right))$ for which the simulations were consistent with known lifetimes (see text for details). For clarity, the data points of ${}^{104}\mathrm{Mo}$ and feeding simulations are not shown in this plot, but they were taken into account in the determination of $(t_{\mathrm{thick}},f)$.

Figure 7

Absolute value of transitional quadrupole moments as a function of the mass of the neutron-rich molybdenum isotopes. Some of the values are slightly displaced on the $x$ axis for clarity. The $Q_0^t\left(4^{+}\right)$ shown in this plots are extracted from this work. Instead the $Q_0^t\left(2^{+}\right)$ used the literature values (see Table 3). The arrow (for the $2^{+}$ state of ${}^{108}\mathrm{Mo})$ indicates that the previous lifetime measurement [53] has an error bigger than displayed here.

Figure 8

Comparison of the energies of the ground-state band members (open symbols) calculated with beyond mean-field calculation [15] with the literature value [45] (filled symbols). Dashed lines are used to guide the eye. Some of the values are slightly displaced on the $x$ axis for clarity.

Figure 9

Comparison of reduced transition strengths measured in this work with the experimental literature values (see Table 3) and with those obtained from BMF calculations. Because low-spin states were preferably populated in the present experiment, only levels up to $6^{+}$ are displayed. The arrow (for the $2^{+}$ state of ${}^{108}\mathrm{Mo})$ indicates that the previous lifetime measurement [53] has an error bigger than displayed here.

Figure 10

(a)–(e) Particle-number projected potential energy surfaces—normalized to the minimum of each surface contour line, separated by 0.25 MeV (dashed) and 2 MeV (continuous)—and (f)–(j) SCCM collective wave functions—normalized to one, reddish means large and blueish small—– for ${}^{100,102,104,106,108}\mathrm{Mo}$ isotopes calculated with the Gogny D1S interaction.