Shape evolution of neutron-rich ${}^{106,108,110}\mathrm{Mo}$ isotopes in the triaxial degree of freedom

Background: Neutron-rich nuclei with mass number between 100 and 110 attract much attention, since several kinds of shapes, such as spherical, prolate, oblate, and triaxial shapes, are predicted. In particular, for neutron-rich Mo isotopes, different models predict different magnitudes and rigidity of triaxial deformation. Previous interpretations of experimental results based solely on low-lying $2_2^{+}$ states are insufficient to distinguish between the rigid triaxial shape, $\gamma$ vibration, or $\gamma$-soft rotor.

Purpose: The shape evolution of ${}^{106}\mathrm{Mo}$, ${}^{108}\mathrm{Mo}$, and ${}^{110}\mathrm{Mo}$ is investigated through their $2_1^{+}$-state lifetimes, decay-branching ratios of the $2_2^{+}$ state, and energies of the low-lying collective excited states with $K^{\pi }=0^{+},2^{+}$, and $4^{+}$.

Method: $\beta$-delayed $\gamma$-ray spectroscopy was employed for neutron-rich Nb and Zr isotopes produced at the RIKEN RI Beam Factory to populate excited states in ${}^{106}\mathrm{Mo}$, ${}^{108}\mathrm{Mo}$, and ${}^{110}\mathrm{Mo}$. The EUroball-RIKEN Cluster Array was used for high-resolution $\gamma$-ray detection and lifetimes of the $2_1^{+}$ states were determined using the UK fast-timing array of ${\mathrm{LaBr}}_3\left(\mathrm{Ce}\right)$ detectors.

Results: New $\gamma$-ray transitions and levels are reported, including newly assigned $0_2^{+}$ states in ${}^{108,110}\mathrm{Mo}$. Quadrupole deformations were obtained for ${}^{106,108,110}\mathrm{Mo}$ from their $2_1^{+}$ energies and lifetimes. The $\beta$-delayed neutron-emission probabilities of ${}^{108}\mathrm{Nb}$ and ${}^{110}\mathrm{Nb}$ were determined by examining the $\gamma$ rays of their respective daughter decays.

Conclusions: The even-odd energy staggering in the $2_2^{+}$ band was compared with typical patterns of the $\gamma$-vibrational band, rigid triaxial rotor, and $\gamma$-soft rotor. The very small even-odd staggering of ${}^{106}\mathrm{Mo}$, ${}^{108}\mathrm{Mo}$, and ${}^{110}\mathrm{Mo}$ favors a $\gamma$-vibrational band assignment. The kinematic moment of inertia for the $2_2^{+}$ band showed a trend similar to the ground-state band, which is as expected for the $\gamma$-vibrational band. Beyond-mean-field calculations employing the constrained Hartree-Fock-Bogoliubov and local quasiparticle-random-phase approximation method using the $\mathrm{SLy}5+\mathrm{T}$ interaction reproduced the ground and $2_2^{+}$ bands in ${}^{106}\mathrm{Mo}$ and ${}^{108}\mathrm{Mo}$. The collective wave functions are consistent with the interpretation of the $2_2^{+}$ band as the $\gamma$-vibrational band of the prolate shape. However, the staggering pattern observed in ${}^{110}\mathrm{Mo}$ differs from the one suggested in the calculations which predict a $\gamma$-soft rotor. There was no experimental indication of the oblate shape or the $\gamma$-soft rotor predicted in these Mo isotopes.

Figure 1

[(a), (b)] The $\beta$-delayed $\gamma$-ray spectrum of ${}^{106}\mathrm{Nb}$ obtained from the $\beta$-decay chain ${}^{106}\mathrm{Zr}\to {}^{106}\mathrm{Nb}\to {}^{106}\mathrm{Mo}$. The range of the time window was set to be 180 ms $<t_{\mathrm{ion}}-t_{\beta }<$ 2200 ms. The labeled peaks belong to ${}^{106}\mathrm{Mo}$. The identified background peaks are marked with asterisks. Other unknown peaks are mainly associated with parent ${}^{106}\mathrm{Zr}$ decays. [(c), (d)] The coincidence spectra gated on 171.4 and 784.6 keV.

Figure 2

The proposed level scheme of ${}^{106}\mathrm{Mo}$ obtained from the $\beta$-decay chain ${}^{106}\mathrm{Zr}\to {}^{106}\mathrm{Nb}\to {}^{106}\mathrm{Mo}$. The $Q({\beta }^{-}$) of ${}^{106}\mathrm{Nb}$ is taken from the atomic mass evaluation (AME2016 [33]). The arrow width is proportional to the relative intensity $I_{\gamma }$ ($\mathrm{Zr}\to \mathrm{Mo}$, given in Table 2). Red lines are the new levels and transitions. Spin parities of the known states are taken from ENSDF [34].

Figure 3

The time spectra of $\beta$-delayed $\gamma$ rays in the Mo isotopes. Dashed lines indicate the fitting region of the decay curve to determine the $\beta$-decay half-life, $T_{1/2}$. Orange lines are the constant background, which was determined by fitting to the negative-time region. The $\beta$-delayed $\gamma$ rays with 531.5, 462.6, 563.3, and 563.4 keV from the implanted ${}^{110}\mathrm{Nb}$ were selected as the $\beta$ decays of the high-spin state in ${}^{110}\mathrm{Nb}$.

Figure 4

[(a), (b)] The $\beta$-delayed $\gamma$-ray spectrum of ${}^{108}\mathrm{Nb}$ obtained from the $\beta$-decay chain ${}^{108}\mathrm{Zr}\to {}^{108}\mathrm{Nb}\to {}^{108}\mathrm{Mo}$. The range of the time window was set to be 80 ms $<t_{\mathrm{ion}}-t_{\beta }<$ 280 ms. The labeled peaks belong to ${}^{108}\mathrm{Mo}$. The identified background peaks are marked with asterisks. Other unknown peaks are mainly associated with parent ${}^{108}\mathrm{Zr}$ decays. [(c), (d)] The coincidence spectra gated on 192.8 and 700.7 keV.

Figure 5

The proposed level scheme of ${}^{108}\mathrm{Mo}$ obtained from the $\beta$-decay chain ${}^{108}\mathrm{Zr}\to {}^{108}\mathrm{Nb}\to {}^{108}\mathrm{Mo}$. Red lines are the new levels and transitions.

Figure 6

[(a), (b)] The $\beta$-delayed $\gamma$-ray spectrum of the implanted ${}^{110}\mathrm{Nb}$. The time window after the implantation of ${}^{110}\mathrm{Nb}$ was set to be less than 400 ms. The labeled peaks belong to ${}^{110}\mathrm{Mo}$. The identified background peaks are marked with asterisks. [(c), (d)] The coincidence spectra gated on 213.4 and 828.8 keV. (e) The $\beta$-delayed $\gamma$-ray spectrum obtained from the $\beta$-decay chain ${}^{110}\mathrm{Zr}\to {}^{110}\mathrm{Nb}\to {}^{110}\mathrm{Mo}$, where $\mathrm{\Delta }{t}_{\beta -\mathrm{ion}}$ from 30 to 250 ms was selected.

Figure 7

The proposed level scheme of ${}^{110}\mathrm{Mo}$ obtained from the $\beta$-decay of ${}^{110}\mathrm{Nb}$ isotopes implanted into WAS3ABi. Red lines are the new levels and transitions.

Figure 8

The time spectra of $2_1^{+}\to 0_1^{+}\gamma$-ray transition in ${}^{106}\mathrm{Mo}$, ${}^{108}\mathrm{Mo}$, and ${}^{110}\mathrm{Mo}$. $\mathrm{\Delta }T$ is the time from $\beta$-particle detection by the plastic scintillator to $\gamma$-ray detection by the ${\mathrm{LaBr}}_3$(Ce) array. The solid red lines are the best-fit curves using an exponential function and fixed constant background to the region indicated by the dashed red lines. The constant backgrounds, shown by the orange lines, were determined by fitting the region of 15 $<\mathrm{\Delta }T<$ 25 ns, 10 $<\mathrm{\Delta }T<$ 25 ns, and 8 $<\mathrm{\Delta }T<$ 25 ns for ${}^{106}\mathrm{Mo}$, ${}^{108}\mathrm{Mo}$, and ${}^{110}\mathrm{Mo}$, respectively.

Figure 9

The $\gamma$-ray energy spectra of the ${\mathrm{LaBr}}_3$(Ce) array. The energy region used to make the time spectra of Fig. 8 are highlighted with gray. The prompt, $\left|\mathrm{\Delta }T\right|<$ 1 ns, and delayed, $\mathrm{\Delta }T>1$ ns, components are shown by the red and blue dotted lines, respectively.

Figure 10

Experimental and theoretical $B(E2;2_1^{+}\to 0_1^{+}$) values of the even-even Mo isotopes. The experimental values were calculated by the use of the relation in Ref. [52]. The open circles are taken from Ref. [48]. The theoretical values were calculated using the five-dimensional collective Hamiltonian with the pairing-plus-quadrupole interaction parameters determined from the two kinds of the Skyrme-interaction parameters ($\mathrm{SLy}5+\mathrm{T}$ and SLy4).

Figure 11

Quadrupole deformation parameter $\beta$ for Zr (square) and Mo (circle) isotopes. Filled circles are the present results for the Mo isotopes. Filled squares for the Zr isotopes are the results from the same data set [53], but the values were recalculated from $B(E2;2_1^{+}\to 0_1^{+}$) by using the formula given in the review paper [48]. Open circles and squares are taken from the review paper [48] and a later work [54].

Figure 12

The $E_s\left(4\right)/E\left(2_1^{+}\right)$ ratio around neutron-rich $A=110$. The black-dashed lines represent the ideal values of three models; rigid-triaxial rotor, $\gamma$-unstable rotor, and $\gamma$-vibrational band. Filled square, circles, triangles, and inverted triangles represent Zr, Mo, Ru, and Pd isotopes, respectively.

Figure 13

The staggering pattern of $E_s\left(J\right)/E\left(2_1^{+}\right)$. The flat pattern indicates the $\gamma$-vibrational band, while the staggering pattern with low values at even and odd $J$ indicates the $\gamma$-soft and rigid triaxial rotor, respectively [4].

Figure 14

The kinematic moment of inertia for the ground band (black line with filled squares), $K^{\pi }=2^{+}$ band (blue line with filled triangles for even $J$ and green line for odd $J$), and $K^{\pi }=4^{+}$ band (red line with open circles for even $J$ and orange line for odd $J$) in ($\mathrm{a}){}^{106}\mathrm{Mo}$, ($\mathrm{b}){}^{108}\mathrm{Mo}$, and ($\mathrm{c}){}^{110}\mathrm{Mo}$.

Figure 15

The potential-energy surface and the collective-wave functions squared (with a factor of ${\beta }^4$) of low-lying states in ${}^{106}\mathrm{Mo}$, ${}^{108}\mathrm{Mo}$, and ${}^{110}\mathrm{Mo}$. The pairing-plus-quadrupole interaction and spherical single-particle energies used in the $\mathrm{CHFB}+\mathrm{LQRPA}$ calculations were fitted to the mean-field results obtained with the $\mathrm{SLy}5+\mathrm{T}$ interaction.

Figure 16

Same as Fig. 15, but with the SLy4 interaction.

Figure 17

The experimental and theoretical energies of the low-lying excited states in ${}^{106}\mathrm{Mo}$, ${}^{108}\mathrm{Mo}$, and ${}^{110}\mathrm{Mo}$. Black lines present the experimental results, and red and blue lines present the results from the theoretical calculations using $\mathrm{SLy}5+\mathrm{T}$ and SLy4, respectively.