Collective excitations of ${}^{96}\mathrm{Ru}$ by means of $(p,p^{\prime }\gamma )$ experiments

Background: One-phonon mixed-symmetry quadrupole excitations are a well-known feature of near-spherical, vibrational nuclei. Their interpretation as a fundamental building block of vibrational structures is supported by the identification of multiphonon states resulting from a coupling of fully-symmetric and mixed-symmetric quadrupole phonons. In addition, the observation of strong $M1$ transitions between low-lying $3^{-}$ and $4^{+}$ states has been interpreted as an evidence for one-phonon mixed-symmetry excitations of octupole and hexadecapole character.

Purpose: The aim of the present study is to identify collective one- and two-phonon excitations in the heaviest stable $N=52$ isotone ${}^{96}\mathrm{Ru}$ based on a measurement of absolute $M1$, $E1$, and $E2$ transition strengths.

Methods: Inelastic proton-scattering experiments have been performed at the Wright Nuclear Structure Laboratory (WNSL), Yale University, and the Institute for Nuclear Physics (IKP), University of Cologne. From the acquired proton-$\gamma$ and $\gamma \gamma$ coincidence data we deduced spins of excited states, $\gamma$-decay branching ratios, and multipole mixing ratios, as well as lifetimes of excited states via the Doppler-shift attenuation method (DSAM).

Results: Based on the new experimental data on absolute transition strengths, we identified the $2^{+}$ and $3^{+}$ members of the two-phonon mixed-symmetry quintuplet $(2_{1,\mathrm{ms}}^{+}\otimes 2_{1,\mathrm{s}}^{+})$. Furthermore, we observed strong $M1$ transitions between low-lying $3^{-}$ and $4^{+}$ states suggesting one-phonon symmetric and mixed-symmetric octupole and hexadecapole components in their wave functions, respectively. The experimental results are compared to $sdg$-IBM-2 and shell-model calculations.

Conclusions: Both the $sdg$-IBM-2 and the shell-model calculations are able to describe key features of mixed-symmetry excitations of ${}^{96}\mathrm{Ru}$. Moreover, they support the one-phonon mixed-symmetry hexadecapole assignment of the experimental $4_2^{+}$ state.

Figure 1

(a) Proton-$\gamma$ coincidence matrix obtained in the ${}^{96}\mathrm{Ru}$ experiment at WNSL, Yale. Transitions stemming from ${}^{96}\mathrm{Ru}$ are visible as thin horizontal lines due to the better energy resolution of the HPGe detectors compared to the silicon particle detectors. (b) The summed HPGe singles spectrum is already dominated by transitions of ${}^{96}\mathrm{Ru}$, marked with asterisks. (c) By applying a proton gate on the excitation energy of the $4_2^{+}$ state of ${}^{96}\mathrm{Ru}$ at $E_x=2462\mathrm{keV}$, the peak-to-background ratio is significantly improved. (d) The proton singles spectrum is dominated by elastically scattered protons and reactions in the backing material. (e) Applying a $\gamma$ gate on the $4_2^{+}\to 4_1^{+}$ transition at $E_{\gamma }=944\mathrm{keV}$ reveals peaks in the excitation spectrum corresponding to the excitation of the $4_2^{+}$ level as well as states feeding the $4_2^{+}$ state.

Figure 2

Results of the $\gamma \gamma$ angular correlation analysis for the $1518\mathrm{keV}\to 832\mathrm{keV}\to \mathrm{g}.\mathrm{s}.$ (a), the $2283\mathrm{keV}\to 832\mathrm{keV}\to \mathrm{g}.\mathrm{s}.$ (b), the $2462\mathrm{keV}\to 1518\mathrm{keV}\to 832\mathrm{keV}$ (c), and the $3077\mathrm{keV}\to 1818\mathrm{keV}\to 832\mathrm{keV}$ (d) cascades. The obtained multipole mixing ratios ${\delta }_1$ of the first transitions are quoted in the insets. A dominant $M1$ character is obtained for the $2462\mathrm{keV}\to 1518\mathrm{keV}$ (c) and $3077\mathrm{keV}\to 1818\mathrm{keV}$ (d) transitions. In contrast, an assumed pure $E2$ character cannot describe the experimental data for those transitions.

Figure 3

Centroid shift of the $\gamma$-ray peaks obtained for the $2_3^{+}\to 2_1^{+}$ transition (a), the $4_2^{+}\to 4_1^{+}$ transition (b), the $3\to 2_1^{+}$ transition with the $J=3$ state at $E_x=2851\mathrm{keV}$ (c), and the $2_5^{+}\to 2_2^{+}$ transition (d) as a function of $cos\mathrm{\Theta }$. The linear trend giving the Doppler-shift attenuation factor is evident.

Figure 4

The upper panel shows the measured $M1$ strengths distribution for the decay of low-lying non-yrast $2^{+}$ states to the $2_1^{+}$ state. The corresponding $E2$ transition strengths to the ground state are shown in the lower panel. Arrows mark transitions with strengths smaller than 0.05 W.u. Because of its characteristic decay properties, the $2_3^{+}$ state represents a good realization of the one-phonon mixed-symmetry quadrupole excitation.

Figure 5

Decay properties of candidates for the $1^{+}$, $2^{+}$, and $3^{+}$ members of two-phonon mixed-symmetry quintuplet. The thickness of the arrows corresponds to the measured $\gamma$-decay intensities. If available, the measured $M1$ and $E2$ transition strengths are quoted as well. $M1$ transitions are indicated with the unit ${\mu }_N^2$, while $E2$ transitions are quoted in W.u. The dashed horizontal line indicates the sum of the excitation energies of the $2_1^{+}$ and $2_3^{+}$ states.

Figure 6

Ratio of the $\langle 3_1^{-}\left|\right|M1\left|\right|3_i^{-}\rangle$ and $\langle 2_{1,\mathrm{s}}^{+}\left|\right|M1\left|\right|2_{1,\mathrm{ms}}^{+}\rangle$ matrix elements for nuclei in the $A\approx 100$ mass region and for ${}^{144}\mathrm{Nd}$. The ratio for ${}^{96}\mathrm{Ru}$ is a factor of 2 lower compared to the other $N=52$ isotones ${}^{92}\mathrm{Zr}$ and ${}^{94}\mathrm{Mo}$, but close to the value for ${}^{96}\mathrm{Mo}$. For ${}^{114}\mathrm{Cd}$, only an upper limit has been reported up to now. The data, except for ${}^{96}\mathrm{Ru}$, are adopted from [25].

Figure 7

Model space used for the present SM calculations. The single-particle energies for the proton (left) and neutron orbits (right) relative to the ${}^{88}\mathrm{Sr}$ core are adopted from Ref. [63].

Figure 8

Comparison of the experimental level scheme (middle) with the results of the $sdg$-IBM-2 (left) and shell-model calculations (right). In the cases where a one-to-one correspondence of the experimental and calculated states on the basis of similar decay properties was obtained, they are connected with dashed lines.

Figure 9

Calculated $B\left(E2\right)$ strength of the ground-state transitions of the $2_1^{+}$ (upper panel) and the $2_3^{+}$ (lower panel) states as a function of the isoscalar (left) and isovector (right) effective charges. The respective other effective charge was fixed to zero. As suggested by Eq. (6), the calculated transition strengths are perfectly described by a fit with a second order polynomial (solid line). The correlation of the $2_1^{+}\to 0_1^{+}$ and $2_3^{+}\to 0_1^{+}$ transition strengths with the isoscalar and isovector effective charges, respectively, were used to fix the proton and neutron effective charges $e_{\pi }$ and $e_{\nu }$.

Figure 10

Results of the shell-model calculations for the five lowest-lying $2^{+}$ states. (a) The calculated $E2$ strengths to the ground state and (b) $M1$ strengths for the decay to the $2_1^{+}$ state are compared to the experimental data. Arrows indicate upper limits for the corresponding transition strengths. Based on their $\gamma$-decay properties, the $2_1^{+}$ and $2_3^{+}$ states are identified to be the one-phonon symmetric and mixed-symmetry excitations, respectively. (c) The calculated values of $R_{\mathrm{E}2}$, that are based on SM $B\left(E2\right)$ values, support the different proton-neutron symmetry of the wave functions of the $2_1^{+}$ and $2_3^{+}$ states. The dashed line separates the regions of dominant isovector and dominant isoscalar excitations from the ground state.

Figure 11

Same as Fig. 10, but for the five lowest-lying $4^{+}$ states. While the $4_1^{+}$ state shows signatures of a symmetric one-phonon hexadecapole configuration, the $\gamma$-decay properties and the ratio $R_{\mathrm{E}4}$ hint toward a one-phonon hexadecapole configuration with mixed proton-neutron symmetry for the $4_3^{+}$ shell-model state, which we assign to the experimental $4_2^{+}$ state.

Figure 12

Compilation of experimental data on one- and two-phonon mixed-symmetry states of the $N=52$ isotones ${}^{92}\mathrm{Zr}$, ${}^{94}\mathrm{Mo}$, and ${}^{96}\mathrm{Ru}$. The left panel shows the excitation energies while the $M1$, $E1$, and $E2$ transition strengths are compiled in the right panel. Included are data on one-phonon quadrupole ($2_{1,\mathrm{ms}}^{+}$) and two-phonon quadrupole ($2_{2,\mathrm{ms}}^{+}$, $3_{1,\mathrm{ms}}^{+}$) mixed-symmetry states in the first and second rows as well as data on candidates for one-phonon octupole ($3_2^{-}$) and hexadecapole mixed-symmetry states ($4_2^{+}$) in rows three and four. Data for ${}^{92}\mathrm{Zr}$ and ${}^{94}\mathrm{Mo}$ are taken from Refs. [14, 15].