Mixed-symmetry octupole and hexadecapole excitations in the $N=52$ isotones

Background: Excitations with mixed proton-neutron symmetry have been previously observed in the $N=52$ isotones. Besides the well-established quadrupole mixed-symmetry states (MSS), octupole and hexadecapole MSS have been recently proposed for the nuclei ${}^{92}\mathrm{Zr}$ and ${}^{94}\mathrm{Mo}$.

Purpose: The heaviest stable $N=52$ isotone ${}^{96}\mathrm{Ru}$ was investigated to study the evolution of octupole and hexadecapole MSS with increasing proton number.

Methods: Two inelastic proton-scattering experiments on ${}^{96}\mathrm{Ru}$ were performed to extract branching ratios, multipole mixing ratios, and level lifetimes. From the combined data, absolute transition strengths were calculated.

Results: Strong $M1$ transitions between the lowest-lying $3^{-}$ and $4^{+}$ states were observed, providing evidence for a one-phonon mixed-symmetry character of the $3_2^{(-)}$ and $4_2^{+}$ states.

Conclusions: $sdg$-IBM-2 calculations were performed for ${}^{96}\mathrm{Ru}$. The results are in excellent agreement with the experimental data, pointing out a one-phonon hexadecapole mixed-symmetry character of the $4_2^{+}$ state. The $\langle 3_1^{-}\left|\right|M1\left|\right|3_2^{(-)}\rangle$ matrix element is found to scale with the $\langle 2_{\mathrm{s}}^{+}\left|\right|M1\left|\right|2_{\mathrm{ms}}^{+}\rangle$ matrix element.

Figure 1

Centroid shift of the $E_{\gamma }^0=944$ keV $4_2^{+}\to 4_1^{+}$ $\gamma$ transition of ${}^{96}\mathrm{Ru}$ as a function of $cos\left(\theta \right)$. $\theta$ is the angle between the recoil direction of motion and the direction in which the $\gamma$ ray is emitted. From the slope, the Doppler-shift attenuation factor is calculated.

Figure 2

Comparison of experimental (upper panel) and calculated (lower panel) level schemes for positive-parity low-spin states of ${}^{96}\mathrm{Ru}$. $M1$ and $E2$ transitions are indicated by gray and black arrows, respectively. The widths of the arrows are proportional to the transition strengths. States for which the IBM predicts an $F$-spin quantum number of $F_{\max }-1$ are marked with dashed lines.

Figure 3

Calculated $g$- and ($d+g$)-boson contents in the low-lying positive-parity states of ${}^{96}\mathrm{Ru}$, indicated with black and gray bars, respectively. The remaining fraction is related to $s$-boson components. Enhanced $g$-boson contents are predicted for the $1_1^{+}$, $3_1^{+}$, $4_1^{+}$, and $4_2^{+}$ states.

Figure 4

$M1$ strengths of the one-phonon MSS ($J_{II}^{\pi }$) to FSS ($J_I^{\pi }$) transitions in the even-even $N=52$ isotones of quadrupole (full squares), octupole (full circles), and hexadecapole (full diamonds) character. The values obtained in $sdg$-IBM-2 calculations (open triangles) are shown as well. Data for ${}^{92}\mathrm{Zr}$ and ${}^{94}\mathrm{Mo}$ are taken from [15] and [14], respectively. The $sdg$-IBM-2 results for ${}^{94}\mathrm{Mo}$ are adopted from [20].