Introduction

The existence of multi-quasiparticle configurations is consistently predicted in the low-energy-level structures of the nuclei near the N=50 closed shell. For N=47 even-odd nuclei, three-quasiparticle excitations are expected to dominate the level schemes, and the simplest excitations available will come from the three-neutron-hole $\nu g_{9/2}^{-3}$ configuration. Under the Pauli principle, the coupling of three νg_9/2 neutron holes generates spins from Jπ = 3/2⁺ to 21/2⁺ except for 19/2⁺. Indeed, the 7/2⁺ and 11/2⁺ states identified in the low-energy level schemes of the N=47 even-odd isotones ⁸⁵Sr (Z=38) [1-3], ⁸⁷Zr (Z=40) [4, 5], and ⁸⁹Mo (Z=42) [6-8] have been widely suggested to arise from the configuration of $\nu g_{9/2}^{-3}$.

By contrast, in the N=47 isotones heavier than ⁸⁵Sr, the $\pi g_{9/2}$ protons become more active as the proton number Z increases and play an important role in building low-lying positive-parity levels. For example, a study on high-spin states in ⁸⁹Mo [6-8] proposed that strongly populated positive-parity states of up to 25/2⁺ are mainly built from the seniority-three $\pi g_{9/2}^2\nu g_{9/2}^{-1}$ configuration. Therefore, it is of great interest and challenge to search for the low-lying 7/2⁺ and 11/2⁺ levels in ⁹¹Ru (Z=44), which will extend the systematic trend of the seniority-three $\nu g_{9/2}^{-3}$ configuration in the N=47 nuclei to larger proton numbers and provide an ideal case for studying the competition between proton-particle and neutron-hole excitations from the g_9/2 orbitals.

The excited states in ⁹¹Ru were investigated by Arnell et al. [9] via (α, xn) reactions. Subsequently, the level scheme was extended to Jπ=41/2^- at an excitation energy of 8 MeV by Heese et al. [10] using the reaction ⁵⁸Ni(³⁶Ar, 2p1n)⁹¹Ru at a beam energy of 149 MeV.

Fusion-evaporation reactions are crucial for populating excited states in nuclei and synthesizing superheavy nuclei [11-13]. This reaction mechanism results in a large excitation energy and angular momentum in the nucleus. However, this reaction populates a considerable number of exit channels, complicating the selection of a specific channel for this study. As a typical solution, data analysis often involves using auxiliary measurements to detect charged particles and neutrons simultaneously [14]. In this paper, we report the results of a low-energy fusion-evaporation reaction designed to populate low- to medium-spin states in nuclei in the A~90 mass region. The experiment and data analysis will be briefly described in the next section, followed by the details of the semiempirical shell model calculation we performed to interpret our data together with the level systematics along the N=47 even-odd isotonic chain.

Experimental details and methods

The experiment was performed using the fusion-evaporation reactions ³⁶Ar + ⁵⁸Ni at a bombarding energy of E(³⁶Ar) = 111 MeV. The beam was delivered using the GANIL CIME cyclotron and focused on a 99.83% isotopically enriched 6.0 mg/cm² thick ⁵⁸Ni target. The low-spin states in ⁹¹Ru were populated in the 2p1n exit channel of the ³⁶Ar + ⁵⁸Ni fusion-evaporation reactions. Charged-particle emission following the decay of the ⁹⁴Pd compound nucleus was detected using a DIAMANT detector system consisting of 80 CsI scintillators [15, 16]. The Neutron Wall [17], comprising 50 liquid scintillator detectors covering a solid angle of $1\pi$ in the forward direction, was used to detect the evaporated neutrons. γ rays emitted from the reaction products were detected using an EXOGAM Ge clover detector array [18]. In the experiment, seven segmented clover detectors were placed in an angle of 90° and four detectors at an angle of 135° relative to the beam direction, leaving room for the Neutron Wall at forward angles. EXOGAM was used in a close-packed configuration, with the front part of the BGO Compton suppression shield removed from the clover detector (Further details can be found in Ref. [19]). Events were collected when at least one γ-ray was detected in the Ge clover detectors, coinciding with at least one neutron in the Neutron Wall. Under these conditions, 4×10⁹ events were recorded.

In the offline analysis, the reaction channel leading to ⁹¹Ru was selected by sorting events containing two protons fired in the DIAMANT CsI detectors together with one detected neutron. By using this filter, approximately 2.6×10⁷ γ–γ coincidence events were selected. The event data were mainly from the 2p1n reaction channel, with some contamination from other channels, such as 3p and 3p1n which were the main reaction channels in this experiment. The efficiency and energy calibrations of the Ge clover detectors were performed using a standard radioactive γ-ray source ¹⁵²Eu. After the gain matching of all the Ge clover detectors, the coincidence data were sorted into symmetric and asymmetric (angle-dependent) matrices for subsequent analysis.

The spin and parity of the levels were deduced from information on both the directional correlations of the γ rays from the oriented states (DCO ratios) [20] and γ-ray polarization asymmetries [21]. In the γ-ray decay process, the emitted photon has an integer intrinsic spin L, which is called the multipolarity of γ radiation (for example, $L=\mathrm{1,2}$ refer to dipole and quadrupole radiation, respectively). When a nucleus decays from an excited state, the angular momentum L carried by the γ transition is determined by the selection law $L⩾\text{Δ}J=\left|J_i-J_f\right|$ where Ji and Jf are the spins of the initial and final nuclear states, respectively. In practice, only the two lowest multipolarities allowed by the selection law are observed; among them, the $L=\text{Δ}J$ transition is called stretched radiation. The DCO ratio extracted directly in this experiment($R_{\text{DCO}}^{\text{exp}}$) has a value of ~1 for known pure stretched (ΔJ=2) quadrupole transitions and of ~0.6 for known stretched (ΔJ=1) dipole transitions when gating on quadrupole transitions. However, if the gate is set on a pure stretched dipole transition, the values will become ~1.6 and ~1, respectively. Normalization should be performed for the DCO ratio such that the effect of the gating transition type can be canceled. The normalized DCO ratio ($R_{\text{DCO}}^{\text{norm}}$) is defined as: $R_{\text{DCO}}^{\text{norm}}=a\cdot R_{\text{DCO}}^{\text{exp}}$ (1) where a is a factor with a value of 1 when gating on the quadrupole transitions or 0.6 if the gate is set on the dipole transitions. The multipolarity of the emitted γ rays has been determined in this work based on the $R_{\text{DCO}}^{\text{norm}}$ value, which is ~1 for known pure stretched quadrupole transitions and is ~0.6 for known stretched dipole transitions.

The linear polarization of γ-ray transitions, which indicates their electromagnetic nature, is extracted from the scattering asymmetry between the planes perpendicular and parallel to the reaction plane. The measurement was facilitated using an EXOGAM clover detector with four crystals in one cryostat, wherein each crystal can operate as a scatterer and the adjacent crystals as absorbers. This asymmetry is represented as: $A=\frac{\left[b{N}_{\perp }\right]-N_{\parallel }}{\left[b{N}_{\perp }\right]+N_{\parallel }}\mathrm{,}$ (2) where $N_{\perp }$ and $N_{\parallel }$ are the numbers of scattered photons of a given γ ray, which are scattered perpendicular and parallel to the reaction plane, respectively. These were extracted from their respective asymmetric matrices. Parameter b is the inherent geometrical asymmetry of the detection system and is estimated from the measure of asymmetry between the parallel and perpendicular scattering of γ-rays from an unpolarized radioactive source, for which, ideally, $N_{\perp }=N_{\parallel }$, but is actually $b{N}_{\perp }=N_{\parallel }$. The value of b was obtained from measurements using a standard ¹⁵²Eu radioactive source of 1.05(3).

The polarization asymmetry A is negative for stretched purely magnetic transitions and positive for stretched purely electric transitions. Note that for mixed M1+E2 transitions, the $R_{\text{DCO}}^{\text{norm}}$ ratios can vary between 0.6 and 1.0, depending on the δ multipole mixing ratio of the γ-ray. Further ambiguity arises for nonstretched (ΔI=0) pure E1 (or M1) transitions, where $R_{\text{DCO}}^{\text{norm}}$ for a nonstretched dipole transition with $\delta \approx 0$ mixing ratio is approximately the same as that for a stretched quadrupole transition [22, 23]. These ambiguities can be resolved by simultaneously measuring the linear polarizations of the γ-ray transitions. For example, stretched E1, E2 or unstretched M1 transitions and stretched M1 or unstretched E1 transitions have opposite linear polarization values [23]. Therefore, to determine the multipolarity and electromagnetic nature of a transition, both the DCO ratio and the polarization asymmetry should be measured.

Results

The partial-level scheme for ⁹¹Ru shown in Fig. 1 was established based on an analysis of γ-ray coincidence relationships. Spin parity assignments were derived from an analysis of the normalized DCO ratios and Compton asymmetries.

Fig. 1Full-screen View

(Color online) Partial low-energy level scheme of ⁹¹Ru established by this work. The newly observed transitions are indicated by red arrows. The widths of the arrows represent the relative intensities of the γ rays

Note that stretched quadrupole transitions cannot be distinguished from ΔI=0 dipole transitions or certain mixed ΔI=1 transitions based only on DCO ratios. In these cases, simultaneously measuring the linear polarizations of the transitions can provide supporting arguments for spin assignments. Figure 2 illustrates a two-dimensional plot of the asymmetries as a function of the normalized DCO ratios. As shown in the plot, the polarization and multipolarity measurements together provided reasonable spin parity assignments for the levels.

Fig. 2Full-screen View

(Color online) Two-dimensional plot of the asymmetries as a function of the normalized DCO ratios ($R_{\text{DCO}}^{\text{norm}}$ s) of the γ rays shown in the partial decay scheme of ⁹¹Ru (see Fig. 1). Stretched E1, E2, and M1 transitions and non-stretched E1 transitions are indicated in the plot. The dashed lines parallel to the y-axis correspond to the values obtained for known pure stretched dipole and quadrupole transitions, respectively. These lines have been drawn to guide the eye. For the definition of $R_{\text{DCO}}^{\text{norm}}$, see the text

New transitions were assigned to ⁹¹Ru based on coincidences with γ rays already known in this nucleus. Typical prompt γ-γ coincidence spectra for ⁹¹Ru are shown in Fig. 3. The γ-ray transitions of ⁹¹Ru are listed in Table 1. Their relative intensities were obtained from the total projection of the Eγ-Eγ matrix. If the peak-to-background ratio in the total projection was too low or if there was contamination in the peak from other γ-ray transitions, the relevant transition energies were selected in the coincidence matrix, and the obtained projected spectra were used to fit the relative intensities. The transition energies were also measured from the total projection of the matrix. The energy uncertainties presented in the table are the sums of the statistical and calibration errors.

Fig. 3Full-screen View

(Color online) Typical coincidence spectra for ⁹¹Ru gated on the 361, 1003, and 234 keV γ rays, corresponding to the transitions that depopulate the 15/2^- and 13/2^- states in ⁹¹Ru, respectively. The 296, 300, 455, 549, 612, 755, 825, and 871 keV lines shown in the spectra belong to the transitions decaying the states of ⁹¹Ru at the excitation energies above 2.5 MeV, thus these transitions did not present in the low-energy level scheme as shown in Fig. 1. Contaminant lines indicated with filled and open circles in the figure come from the 3p1n channel (⁹⁰Tc) and the 3p channel (⁹¹Tc), respectively

Below the previously known 13/2^- level at 1893 keV, a transition sequence consisting of 1003 keV and 890 keV γ rays fed into the ground state is observed. The ordering of the two new transitions is fixed by a newly observed 844 keV transition that crosses the transition sequence, as shown in Fig. 1. The normalized DCO ratio and asymmetry measured for the 890 keV transition are 0.63(5) and -0.03(1), respectively, indicating a stretched M1 multipolarity, which leads to the spin parity assignment of 11/2⁺ for the new state at 890 keV (denoted ${11/2}_1^{+}$ in the partial level scheme, as shown in Fig. 1). A stretched E1 character for the 1003 keV transition was obtained from the results of the normalized DCO ratio (0.59±0.04) and linear polarization (0.12±0.05) measurements. Thus, this γ-ray was assigned to the 13/2^-→ ${11/2}_1^{+}$ transition, which provides a supplementary argument for the previous assignment of the 13/2^- state at 1893 keV.

The newly observed γ-rays of 234 keV, 1660 keV, and 1614 keV were assigned to transitions between the 13/2^- level and the 9/2⁺ ground state. These new transitions are shown in Fig. 3. Based on the obtained normalized DCO ratios and Compton asymmetries, we assigned M1 and E2 multipolarities to 1660 keV and 1614 keV γ rays, respectively. The DCO ratio analysis shows that the 234 keV transition has dipole multipolarity (see Table 1). These assignments imply the existence of a 7/2⁺ state at 46 keV excitation energy and a second 11/2⁺ level at 1660 keV excitation energy (as shown in Fig. 1, marked ${11/2}_2^{+}$). The 46 keV γ-ray to the ground state cannot be observed in EXOGAM Clovers because it is fully absorbed by the material located around the target, including the DIAMANT array. In addition, a new transition of 686 keV was observed and assigned to connect the ${11/2}_2^{+}$ and 13/2⁺ states. The combination of the normalized DCO and linear polarization data shows that the 686 keV γ ray has M1 multipolarity, which confirms that the 1660 keV level has Jπ=11/2⁺.

Calculation and discussion

The low-energy level schemes of the N=47 nuclei with $Z\ge 38$ display a complex structure in the presence of high-energy γ rays, irregular level spacings, and parallel decay branches, indicating the dominance of single-particle excitations or configurations associated with the weak coupling of a g_9/2 neutron hole to core excitations. The 7/2⁺, $11/2_1^{+}$, and $11/2_2^{+}$ states newly identified below the last 13/2^- level in ⁹¹Ru were grouped into a positive-parity structure. Very similar structures have also been found previously in lighter N=47 isotones, down to ⁸⁵Sr [3, 5-8], allowing a detailed study of level systematics with varying proton numbers.

Figure 4 compares the positive-parity structures and 13/2^- states in the N=47 even-odd isotones from ⁸⁵Sr to ⁹¹Ru. The $2_1^{+}$ and $2_2^{+}$ levels in the N=48 even-even nuclei from ⁸⁶Sr to ⁹²Ru (except $2_2^{+}$ in ⁹²Ru) are also shown [24-27]. Figure 4 shows that the energies of the two 11/2⁺ states and low-lying 7/2⁺ level in ⁹¹Ru fit well with the level systematics of N=47 isotones. Note that the systematic occurrence of low-lying 7/2⁺ states in the N=47 nuclei is an important feature of the three-neutron-hole $\nu g_{9/2}^{-3}$ configuration [4, 28]. The energies of the $11/2_1^{+}$ levels in the N=47 isotones from ⁸⁵Sr to ⁹¹Ru agree well with those of the $2_1^{+}$ states in the adjacent N=48 even-even cores. The latter are known to be excitations dominated by a two-neutron–hole configuration ($\nu g_{9/2}^{-2}$)2⁺ [24, 27, 29, 30]. Based on energy systematics, it can be expected that the $11/2_1^{+}$ states arise from the coupling of a g_9/2 neutron hole to the 2⁺ neutron core excitations, resulting in a three-neutron-hole configuration of ($\nu g_{9/2}^{-3}$)11/2⁺. In particular, in the semi-magic nucleus ⁸⁵Sr with only three neutron holes outside the doubly closed-shell nucleus ⁸⁸Sr, the 7/2⁺ and $11/2_1^{+}$ states have been suggested as nearly pure ν $g_{9/2}^{-3}$ excitations [1].

Fig. 4Full-screen View

(Color online) Level systematics of positive-parity structures below 13/2^- states in N=47 even-odd isotones ⁸⁵Sr, ⁸⁷Zr, ⁸⁹Mo, and ⁹¹Ru (present work). The $2_1^{+}$ and $2_2^{+}$ levels in the N=48 even-even nuclei from ⁸⁶Sr to ⁹²Ru (except $2_1^{+}$ in ⁹²Ru) are also shown (red)

Figure 4 shows that the $11/2_2^{+}$ states remain almost constant as Z. This feature is also observed for the $2_2^{+}$ levels in the N=48 nuclei ⁸⁶Sr (1854 keV), ⁸⁸Zr (1818 keV), and ⁹⁰Mo (1897 keV), which lie at almost the same energy as the first-excited 2⁺ state (1836 keV) in the ⁸⁸Sr core [31], varying by only 61 keV. The $2_1^{+}$ level in ⁸⁸Sr is a proton core excited state, in which the main component is a proton 1p-1h excitation from πp_3/2 into the πp_1/2 orbital [30, 31]. Therefore, it is reasonable to assume that the $11/2_2^{+}$ levels in the N=47 isotones are associated with weak coupling of νg_9/2 neutron holes to the proton-core 2⁺ excitation.

To confirm the validity of the above interpretation of the low-energy structure of N=47 isotones, we performed a semiempirical shell model calculation, which is discussed in the next two subsections.

4.2

Results

The calculated results for the 7/2⁺, 11/2 $11/2_1^{+}$, and $11/2_2^{+}$ states of the N=47 isotones are shown in Fig. 5 (red lines). Input data for the calculation are taken from the neighboring nuclei ^85-87Sr [1, 24, 37], ^88-90Zr [4, 25, 38, 39], ^90-92Mo [26, 40, 41], and ^92-94Ru [27, 42, 43]. The ground-state masses required for the calculations were obtained from Ref. [44]). The calculation details are as follows:

Fig. 5Full-screen View

Calculated level energies, using the semiempirical shell model, of the 7/2⁺, $11/2_1^{+}$, and $11/2_2^{+}$ states in the even-odd N=47 isotones ⁸⁷Zr, ⁸⁹Mo, and ⁹¹Ru (red lines) are compared to the experimental data (black bars)

The energy of the 7/2⁺ state of ⁹¹Ru from the (${\pi }_{0^{+}}^6\nu g_{9/2}^{-3}$)7/2⁺ configuration was calculated as $\begin{array}{c}{E}_{\left({\pi }_{0^{+}}^6\nu g_{9/2}^{-3}\right){7/2}^{+}}^{{}^{\text{91}}\text{Ru}}=E_{\left(\nu g_{9/2}^{-3}\right){7/2}^{+}}^{{}^{\text{85}}\text{Sr}}+2E_{\left({\pi }_{0^{+}}^6\nu g_{9/2}^{-1}\right){9/2}^{+}}^{{}^{\text{93}}\text{Ru}}+E_{\left(\nu g_{9/2}^{-2}\right)0^{+}}^{{}^{\text{86}}\text{Sr}}\\ -E_{\left({\pi }_{0^{+}}^6\right)0^{+}}^{{}^{\text{94}}\text{Ru}}-2E_{\left(\nu g_{9/2}^{-1}\right){9/2}^{+}}^{{}^{\text{87}}\text{Sr}}\\ -E_{\left(\nu g_{9/2}^{-3}\right){9/2}^{+}}^{{}^{\text{85}}\text{Sr}}-E_{\left({\pi }_{0^{+}}^6\right)0^{+}\left(\nu g_{9/2}^{-2}\right)0^{+}}^{{}^{\text{92}}\text{Ru}}+S\\ =232\text{keV}+2\times 0\text{keV}+0\text{keV}-0\text{keV}\\ -2\times 0\text{keV}-0\text{keV}-0\text{keV}-233.2\text{keV}\\ =-1.2\text{keV}\mathrm{.}\end{array}$ (8) With two protons less than ⁹¹Ru, the nucleus ⁸⁹Mo will have seven quasiparticles relative to the ⁸⁸Sr core. The energy of the (${\pi }_{0^{+}}^4\nu g_{9/2}^{-3}$)11/2⁺ state was calculated as $\begin{array}{c}{E}_{\left({\pi }_{0^{+}}^4\nu g_{9/2}^{-3}\right){11/2}^{+}}^{{}^{\text{89}}\text{Mo}}=E_{\left(\nu g_{9/2}^{-3}\right){11/2}^{+}}^{{}^{\text{85}}\text{Sr}}+2E_{\left({\pi }_{0^{+}}^4\nu g_{9/2}^{-1}\right){9/2}^{+}}^{{}^{\text{91}}\text{Mo}}\\ +E_{\left(\nu g_{9/2}^{-2}\right)0^{+}}^{{}^{\text{86}}\text{Sr}}-E_{\left({\pi }_{0^{+}}^4\right)0^{+}}^{{}^{\text{92}}\text{Mo}}-2E_{\left(\nu g_{9/2}^{-1}\right){9/2}^{+}}^{{}^{\text{87}}\text{Sr}}\\ -E_{\left(\nu g_{9/2}^{-3}\right){9/2}^{+}}^{{}^{\text{85}}\text{Sr}}-E_{\left({\pi }_{0^{+}}^4\right)0^{+}\left(\nu g_{9/2}^{-2}\right)0^{+}}^{{}^{\text{90}}\text{Mo}}+S\\ =1221\text{keV}+2\times 0\text{keV}+0\text{keV}-0\text{keV}\\ -2\times 0\text{keV}-0\text{keV}-0\text{keV}+S\\ =1099.9\text{keV}\mathrm{,}\end{array}$ (9) In this reduction, the mass term S is $\begin{array}{ll}S\hfill & =2M_{{}^{\text{91}}\text{Mo}}+M_{{}^{\text{88}}\text{Sr}}+M_{{}^{\text{86}}\text{Sr}}-2M_{{}^{\text{87}}\text{Sr}}\hfill \\ \hfill & -M_{{}^{\text{92}}\text{Mo}}-M_{{}^{\text{90}}\text{Mo}}=-121.1\text{keV}\mathrm{.}\hfill \end{array}$ (10) Using a similar calculation, the excitation energy of the (${\pi }_{0^{+}}^4\nu g_{9/2}^{-3}$)7/2⁺ state was deduced as 110.9 keV. Based on these considerations, the excitation energies of the five quasiparticle seniority-three (${\pi }_{0^{+}}^2\nu g_{9/2}^{-3}$), 11/2⁺ and 7/2⁺ configurations in ⁸⁷Zr were calculated as 1181.9 keV and 192.9 keV.

As argued above, the $11/2_2^{+}$ state in the N=47 isotones arises from the weak coupling of nucleons to the $2_1^{+}$ level of ⁸⁸Sr, which is interpreted as a proton p-h core excitation. In ⁸⁵Sr, the three-quasiparticle weak-coupling state of $\left[{\left(\pi p_{3/2}^{-1}\pi p_{1/2}\right)}^{2+}{\left(\nu g_{9/2}^{-3}\right)}_{\upsilon =1}^{9/2+}\right]11/2^{+}$ was observed at 1658 keV, where it is suggested that the three neutron holes are in a seniority υ=1 configuration. The appropriate configuration for this state in ⁹¹Ru is $\left[{\left({\pi }^6\right)}_{\upsilon =0}^{0+}{\left(\pi p_{3/2}^{-1}\pi p_{1/2}\right)}^{2+}{\left(\nu g_{9/2}^{-3}\right)}_{\upsilon =1}^{9/2+}\right]11/2^{+}$ in which the p-h excitation is coupled to a six-proton-particle configuration with seniority υ=0. The calculation of the excitation energy of this nine-quasiparticle state can start from the corresponding quantity in ⁸⁵Sr and yields $\begin{array}{l}{E}_{\left[{\left({\pi }^6\right)}_{\upsilon =0}^{0+}{\left(\pi p_{3/2}^{-1}\pi p_{1/2}\right)}^{2+}{\left(\nu g_{9/2}^{-3}\right)}_{\upsilon =1}^{9/2+}\right]{11/2}^{+}}^{{}^{\text{91}}\text{Ru}}\\ =E_{\left[{\left(\pi p_{3/2}^{-1}\pi p_{1/2}\right)}^{2+}{\left(\nu g_{9/2}^{-3}\right)}_{\upsilon =1}^{9/2+}\right]{11/2}^{+}}^{{}^{\text{85}}\text{Sr}}+E_{{\left({\pi }^6\right)}_{\upsilon =0}^{0+}{\left(\nu g_{9/2}^{-2}\right)}_{\upsilon =0}^{0+}}^{{}^{\text{92}}\text{Ru}}\\ +E_{{\left({\pi }^6\right)}_{\upsilon =0}^{0+}\left(\nu g_{9/2}^{-1}\right)}^{{}^{\text{93}}\text{Ru}}-E_{{\left({\pi }^6\right)}_{\upsilon =0}^{0+}}^{{}^{\text{94}}\text{Ru}}-E_{{\left(\nu g_{9/2}^{-2}\right)}_{\upsilon =0}^{0^{+}}}^{{}^{\text{86}}\text{Sr}}-E_{\nu g_{9/2}^{-1}}^{{}^{\text{87}}\text{Sr}}+S\\ =1658\text{keV}+0\text{keV}+0\text{keV}-0\text{keV}\\ -0\text{keV}-0\text{keV}+S\\ =1341.8\text{keV}\mathrm{.}\end{array}$ (11) In this case, the mass term S is $\begin{array}{ll}S\hfill & =M_{{}^{\text{85}}\text{Sr}}+M_{{}^{\text{92}}\text{Ru}}+M_{{}^{\text{93}}\text{Ru}}+M_{{}^{\text{88}}\text{Sr}}-M_{{}^{\text{94}}\text{Ru}}\hfill \\ \hfill & -M_{{}^{\text{86}}\text{Sr}}-M_{{}^{\text{87}}\text{Sr}}-M_{{}^{\text{91}}\text{Ru}}=-316.2\text{keV}\mathrm{.}\hfill \end{array}$ (12) The excitation energy of the seven-quasiparticle $\left[{\left({\pi }^4\right)}_{\upsilon =0}^{0+}{\left(\pi p_{3/2}^{-1}\pi p_{1/2}\right)}^{2+}{\left(\nu g_{9/2}^{-3}\right)}_{\upsilon =1}^{9/2+}\right]11/2^{+}$ state in ⁸⁹Mo is calculated in a similar approach as above to be $\begin{array}{l}{E}_{\left[{\left({\pi }^4\right)}_{\upsilon =0}^{0+}{\left(\pi p_{3/2}^{-1}\pi p_{1/2}\right)}^{2+}{\left(\nu g_{9/2}^{-3}\right)}_{\upsilon =1}^{9/2+}\right]{11/2}^{+}}^{{}^{\text{89}}\text{Mo}}\\ =E_{\left[{\left(\pi p_{3/2}^{-1}\pi p_{1/2}\right)}^{2+}{\left(\nu g_{9/2}^{-3}\right)}_{\upsilon =1}^{9/2+}\right]{11/2}^{+}}^{{}^{\text{85}}\text{Sr}}+E_{{\left({\pi }^4\right)}_{\upsilon =0}^{0+}{\left(\nu g_{9/2}^{-2}\right)}_{\upsilon =0}^{0+}}^{{}^{\text{90}}\text{Mo}}\\ +E_{{\left({\pi }^4\right)}_{\upsilon =0}^{0+}\left(\nu g_{9/2}^{-1}\right)}^{{}^{\text{91}}\text{Mo}}-E_{{\left({\pi }^4\right)}_{\upsilon =0}^{0+}}^{{}^{\text{92}}\text{Mo}}-E_{{\left(\nu g_{9/2}^{-2}\right)}_{\upsilon =0}^{0^{+}}}^{{}^{\text{86}}\text{Sr}}-E_{\nu g_{9/2}^{-1}}^{{}^{\text{87}}\text{Sr}}+S\\ =1658\text{keV}+0\text{keV}+0\text{keV}-0\text{keV}-0\text{keV}-0\text{keV}+S\\ =1478.4\text{keV}\mathrm{.}\end{array}$ (13) The mass term S is $\begin{array}{ll}S\hfill & =M_{{}^{\text{85}}\text{Sr}}+M_{{}^{\text{90}}\text{Mo}}+M_{{}^{\text{91}}\text{Mo}}+M_{{}^{\text{88}}\text{Sr}}-M_{{}^{\text{92}}\text{Mo}}\hfill \\ \hfill & -M_{{}^{\text{86}}\text{Sr}}-M_{{}^{\text{87}}\text{Sr}}-M_{{}^{\text{89}}\text{Mo}}=-179.6\text{keV}\mathrm{.}\hfill \end{array}$ (14) A similar calculation gives a value of 1649.1 keV for the excitation energy of the five-quasiparticle configuration $\left[{\left({\pi }^2\right)}_{\upsilon =0}^{0+}{\left(\pi p_{3/2}^{-1}\pi p_{1/2}\right)}^{2+}{\left(\nu g_{9/2}^{-3}\right)}_{\upsilon =1}^{9/2+}\right]11/2^{+}$ in ⁸⁷Zr.

A comparison between the semiempirical shell model calculation and the experimental observations is shown in Fig. 5. The figure shows that the estimate based on the three-neutron-hole $\nu g_{9/2}^{-3}$ structure provides an excellent description of the decreasing excitation energies of the 7/2⁺ and $11/2_1^{+}$ levels along the N=47 isotonic chain. This trend reflects the behavior of the seniority scheme, in which the $\nu g_{9/2}^{-3}$ configuration gradually falls in energy as the g_9/2 proton shell is filled until midshell. It can be seen in Fig. 5 that the difference between the experimental energy and the theoretical value for the 7/2⁺ level is only 8 keV for both ⁸⁷Zr and ⁸⁹Mo, and 47 keV for ⁹¹Ru. However, the deviation becomes higher in the case of $11/2_1^{+}$. The value was 56 keV in ⁸⁷Zr and then increased to almost 100 keV in both ⁸⁹Mo and ⁹¹Ru. This indicates that the low-lying 7/2⁺ levels of the N=47 isotones originate from the configuration $\nu g_{9/2}^{-3}$ and remain quite pure with increasing Z until ⁹¹Ru. However, the $11/2_1^{+}$ state would have smoothly increasing configuration mixing from ⁸⁷Zr to ⁹¹Ru, whereas this state might also be dominated by the $\nu g_{9/2}^{-3}$ configuration. It is shown in Fig. 5 that the calculated weak-coupling state of $\left(\pi 2^{+}\nu g_{9/2}^{-1}\right)11/2^{+}$ reproduces the experimentally observed $11/2_2^{+}$ level for ⁸⁷Zr with a deviation of only 9 keV. Note, however, that the deviation amounts to 168 keV and 318 keV as the calculation goes to the heavier isotones ⁸⁹Mo and ⁹¹Ru, respectively. These larger deviations were possibly due to significant mixing from the other configurations.

Note that the $11/2_1^{+}$ and $11/2_2^{+}$ states were observed in the same low-spin-level schemes, lying close in energy. Owing to the energy proximity and spin-parity identity, it is most likely that the two 11/2⁺ states are admixed with each other. This might account for the above finding that, compared to the 7/2⁺ case, the $11/2_1^{+}$ and $11/2_2^{+}$-level energies deviate more strongly from the experiment for both ⁸⁹Mo and ⁹¹Ru, with the mixing likely being more significant in magnitude. Under the assumption of two-state mixing between ($\nu g_{9/2}^{-3}$)11/2⁺ and ($\pi 2^{+}\nu g_{9/2}^{-1}$)11/2⁺, we have estimated the percentages of the $\nu g_{9/2}^{-3}$ components of the $11/2_1^{+}$ states in each even-odd N=47 isotone by using the semiquantitative two-state mixing calculation (cf. Ref. [45]). The calculated results show that the $\nu g_{9/2}^{-3}$ configuration contributes 94%, 79%, and 73% of the wave functions to the $11/2_1^{+}$ states of ⁸⁷Zr, ⁸⁹Mo, and ⁹¹Ru, respectively. These data clearly demonstrate that the two 11/2⁺ states in the N=47 isotones are correlated, and that the mixing between the two states gradually increases with increasing proton number. This is a general property of the np interaction that strongly breaks seniority and further enhances the correlation as each proton is added to the ⁸⁸Sr core, resulting in a strong seniority violation in ⁹¹Ru.

The final remark concerns the $11/2_2^{+}$ state. Its energy remains almost constant with increasing proton number, which is quite different from the behaviors of the 7/2⁺ and $11/2_1^{+}$ levels, where the energy level is systematically lower. The main reason behind this $11/2_2^{+}$ evolution is the basic feature of a quantum system arising from the mixing of two states: the unperturbed states repel each other, thereby increasing the energy of the upper state, which further increases the energy of the upper state and cancels the energy dropping effect of the unperturbed $11/2_2^{+}$ state with increasing Z, resulting in the observed constancy of its energy.

summary

The excited states in ⁹¹Ru were populated via the fusion-evaporation reaction ⁵⁸Ni(³⁶Ar, 2p1nγ)⁹¹Ru at a beam energy of 111 MeV. Spins and parities were assigned by measuring the DCO ratios and the linear polarization of γ rays. Three low-energy states with spins 7/2⁺, $11/2_1^{+}$, and $11/2_2^{+}$ were identified in ⁹¹Ru for the first time and fitted the level systematics of N=47 even-odd isotones. A semiempirical shell model was used to explain the systematics of these positive-parity states in the N=47 isotones. The calculations reproduced the energy spectra of N=47 isotones up to ⁹¹Ru. Based on a semiquantitative calculation of two-state mixing, it was found that seniority is increasingly broken in the first excited 11/2⁺ state as protons are added to the ⁸⁸Sr core, resulting in a gradually stronger violation from ⁸⁷Zr to ⁹¹Ru.