Multi-quasiparticle excitations and the impact of high-j intruder orbital in the N = 51 nucleus 93Mo

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Hao Wang, Ke-Yan Ma, Yi-Heng Wu, Yi-Feng Lv, Hao-Nan Pan and Jing-Bin Lu. Multi-quasiparticle excitations and the impact of high-j intruder orbital in the N = 51 nucleus 93Mo[J]. Chinese Physics C.
Hao Wang, Ke-Yan Ma, Yi-Heng Wu, Yi-Feng Lv, Hao-Nan Pan and Jing-Bin Lu. Multi-quasiparticle excitations and the impact of high-j intruder orbital in the N = 51 nucleus 93Mo[J]. Chinese Physics C. shu
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Multi-quasiparticle excitations and the impact of high-j intruder orbital in the N = 51 nucleus 93Mo

    Corresponding author: Ke-Yan Ma, mky@jlu.edu.cn
    Corresponding author: Jing-Bin Lu, ljb@jlu.edu.cn
  • 1. College of Physics, Jilin University, Changchun 130012, China
  • 2. School of Physics and Electronic Engineering, An Qing Normal University, Anqing 246133, China

Abstract: The level structures of $^{93}$Mo are investigated using the Large Scale Shell Model calculations. A reasonable agreement between the experimental and calculated values is obtained. The calculated results show that the lower-lying states are mainly dominated by the proton excitations from the $1f_{5/2}$, $2p_{3/2}$ and $2p_{1/2}$ orbitals into the higher orbitals across the Z = 38 or Z = 40 subshell closure. For the higher-spin states, the multi-particle excitations including a $2d_{5/2}$ neutron across the N = 56 subshell closure into the high-j intruder $1h_{11/2}$ orbital are essential. Moreover, the previous unknown spin-parity assignments of the six higher excited states in $^{93}$Mo are reasonably inferred from shell model calculations.

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    I.   INTRODUCTION
    • The level structures of nuclei near the Z = 38 semimagic and N = 50 magic shell have been the focus of experimental and theoretical researches in recent years [1-9]. A number of interesting phenomena have been reported in this mass region, such as the single-particle excitation [3-8], isomeric states [8-12], collective rotation [13, 14], and core breaking [15-18]. The level structures of Zr, Nb, and Tc isotones around $ N = 50 $ have been extended to higher spins and well described by the shell model [16-19]. These studies show that the low-lying states are dominated by the particle-hole excitations from $ \pi $$ p_{1/2} $ orbital into the $ \pi $$ g_{9/2} $ orbital, whereas the higher level structures could be interpreted by the multi-particle excitations in a larger configuration space, and even the excitation of the core has to be taken into account. However, there are relatively few studies in the neighboring Mo isotopes around N = 50 magic shell closure, especially N = 51 nucleus $ ^{93} $Mo.

      For the nucleus $ ^{93} $Mo, the previous theoretical studies mainly focused on several low-lying levels using small truncation space, i.e., ignored certain orbitals, proton 1$ f_{5/2} $ [8] and neutron 1$ g_{9/2} $ [8, 20], 1$ h_{11/2} $ [20] orbitals, etc. However, these orbitals play an important role in neighboring N = 51 isotones, where the excited states can be principally generated via three different mechanisms: (a) the (1$ f_{5/2} $, 2$ p_{3/2} $, 2$ p_{1/2} $) $ \rightarrow $ 1$ g_{9/2} $ proton excitations; (b) 1$ g_{9/2} $ $ \rightarrow $ 2$ d_{5/2} $ neutron-core excitation; or (c) 2$ d_{5/2} $ $ \rightarrow $ 1$ h_{11/2} $ neutron excitation [15-23]. Therefore, to describe the level structures of $ ^{93} $Mo more reasonably, the proton 1$ f_{5/2} $ and neutron (1$ g_{9/2} $, 1$ h_{11/2} $) orbitals are necessary to be taken into consideration in the calculations. For this purpose, the large scale shell model calculations considering the orbitals mentioned above are performed to investigate the origin and components of the excited states in $ ^{93} $Mo. Besides, $ ^{93} $Mo has been observed up to higher excitation energy of about 10 MeV [9]. However, due to the limitation of the experimental conditions, the spin-parity assignments of the higher-lying states are undefinite. So, it is necessary to theoretically predict the multipolarities of the higher-spin states in $ ^{93} $Mo.

    II.   SHELL MODEL CALCULATIONS
    • The level structures of $ ^{93} $Mo are investigated by the large scale shell model calculations with the code NUSHELLX [24]. The SNE model space and SNET interaction are adopted in the code, and the model space includes eight proton orbitals (1$ f_{5/2} $, 2$ p_{3/2} $, 2$ p_{1/2} $, 1$ g_{9/2} $, 1$ g_{7/2} $, 2$ d_{5/2} $, 2$ d_{3/2} $, 3$ s_{1/2} $) and nine neutron orbitals (1$ f_{5/2} $, 2$ p_{3/2} $, 2$ p_{1/2} $, 1$ g_{9/2} $, 1$ g_{7/2} $, 2$ d_{5/2} $, 2$ d_{3/2} $, 3$ s_{1/2} $, 1$ h_{11/2} $) relative to an inert $ ^{56} $Ni (Z = 28, N = 28) core. The single-particle energies relative to the $ ^{56} $Ni core were set as $ \varepsilon^{\pi}_{1f_{5/2}} $ = 0.525 MeV, $ \varepsilon^{\pi}_{2p_{3/2}} $ = 1.228 MeV, $ \varepsilon^{\pi}_{2p_{1/2}} $ = 5.106 MeV, $ \varepsilon^{\pi}_{1g_{9/2}} $ = 5.518 MeV, $ \varepsilon^{\pi}_{1g_{7/2}} $ = 20.656 MeV, $ \varepsilon^{\pi}_{2d_{5/2}} $ = 18.893 MeV, $ \varepsilon^{\pi}_{2d_{3/2}} $ = 20.016 MeV, $ \varepsilon^{\pi}_{3s_{1/2}} $ = 16.895 MeV, $ \varepsilon^{\nu}_{1f_{5/2}} $ = 0 MeV, $ \varepsilon^{\nu}_{2p_{3/2}} $ = 0 MeV, $ \varepsilon^{\nu}_{2p_{1/2}} $ = 0 MeV, $ \varepsilon^{\nu}_{1g_{9/2}} $ = 0 MeV, $ \varepsilon^{\nu}_{1g_{7/2}} $ = 4.352 MeV, $ \varepsilon^{\nu}_{2d_{5/2}} $ = 2.313 MeV, $ \varepsilon^{\nu}_{2d_{3/2}} $ = 3.440 MeV, $ \varepsilon^{\nu}_{3s_{1/2}} $ = 1.532 MeV, and $ \varepsilon^{\nu}_{1h_{11/2}} $ = -0.589 MeV. These single-particle energies and the corresponding values of the strengths of the residual interactions were used to calculated level energies [17, 24].

    III.   RESULTS AND DISCUSSION
    • The nucleus $ ^{93} $Mo has 14 valence protons and 23 valence neutrons in the configuration space. Due to the large dimensionality of the matrices involved, truncations were employed to make the calculations feasible. In order to employ an appropriate truncation, we examine the contribution from proton excitation across the Z = 50 shell closure by two sets of shell model calculations SM1 and SM1*. For SM1, the valence proton space is restricted to $ \pi $($ 1f_{5/2}^{4-6} $, $ 2p_{3/2}^{3-4} $, $ 2p_{1/2}^{0-2} $, $ 1g_{9/2}^{2-5} $, $ 1g_{7/2}^{0-0} $, $ 2d_{5/2}^{0-0} $, $ 2d_{3/2}^{0-0} $, $ 3s_{1/2}^{0-0} $). Simultaneously, only a single $ 1g_{9/2} $ neutron is allowed to be excited across the N = 50 closed shell. Considering that the $ 2d_{3/2} $ and $ 3s_{1/2} $ orbitals are too far from the Fermi surface, no neutrons are allowed to be excited to these orbitals. On the basis of SM1 truncation, an additional proton across the Z = 50 core into the 2$ d_{5/2} $ orbital is considered in SM1*. Excitation energies for positive and negative parity states of $ ^{93} $Mo obtained from the SM1 and SM1* calculations in comparison with the experimental values taken from the Refs. [9, 25, 26], are shown in Fig. 1 and Fig. 2, respectively. As shown in Figs. 1 and 2, the predicted energies within SM1 and SM1* are very close to each other, and the calculated occupation number of protons in the 2$ d_{5/2} $ orbital above the Z = 50 core is small with a value of 0.0 $ \sim $ 0.1, thereby indicating a quite small contribution from proton excitation beyond Z = 50 for the states presented in Figs. 1 and 2. Meanwhile, the calculated energies within SM1, not including the proton excitation beyond Z = 50 core, are closer to the experiment ones than those within SM1*. Hence, the proton excitation across the Z = 50 shell closure is not taken into account in the subsequent SM2 calculations.

      Figure 1.  Comparison of experimental and calculated energy levels of positive-parity states in $^{93}$Mo within SM1 and SM2 configuration spaces.

      Figure 2.  Comparison of experimental and calculated energy levels of negative-parity states in $^{93}$Mo within SM1 and SM2 configuration spaces.

      The SM2 has the same proton configuration space as SM1. Besides, on the basis of the neutron configuration space of SM1, SM2 allows the valence neutron across the N = 56 subshell closure into the higher-lying $ 1h_{11/2} $ orbital. Excitation energies for positive and negative parity states of $ ^{93} $Mo obtained from the SM2 calculations are also presented in Fig. 1 and Fig. 2, respectively. The main partitions of the wave functions for each state within SM1 and SM2 are presented in Table 1, characterized by two main configurations with larger contributions.

      $I^{\pi}\;(\hbar)$$E_{(exp)}$ (keV)SM1SM2
      $E_{(cal)}$ (keV)Wave function $\pi\otimes\nu$Partitions (%)$E_{(cal)}$ (keV)Wave function $\pi \otimes\nu $Partitions (%)
      $1/2_{1}^{+}$94313426 4 2 2 $\otimes$ 2 10 1 064.5113346 4 2 2 $\otimes$ 2 10 1 064.21
      6 4 0 4 $\otimes$ 2 10 1 025.306 4 0 4 $\otimes$ 2 10 1 025.44
      $1/2_{2}^{+}$243727536 4 0 4 $\otimes$ 2 10 1 041.8027526 4 0 4 $\otimes$ 2 10 1 041.73
      6 4 2 2 $\otimes$ 2 10 1 027.706 4 2 2 $\otimes$ 2 10 1 027.90
      $3/2_{1}^{+}$149217216 4 2 2 $\otimes$ 2 10 1 059.7917186 4 2 2 $\otimes$ 2 10 1 059.55
      6 4 0 4 $\otimes$ 2 10 1 026.506 4 0 4 $\otimes$ 2 10 1 026.53
      $3/2_{2}^{+}$218125236 4 2 2 $\otimes$ 2 10 1 067.8425236 4 2 2 $\otimes$ 2 10 1 067.63
      6 4 0 4 $\otimes$ 2 10 1 023.356 4 0 4 $\otimes$ 2 10 1 023.38
      $5/2_{1}^{+}$006 4 2 2 $\otimes$ 2 10 1 059.0106 4 2 2 $\otimes$ 2 10 1 058.92
      6 4 0 4 $\otimes$ 2 10 1 030.266 4 0 4 $\otimes$ 2 10 1 030.25
      $5/2_{2}^{+}$169514436 4 2 2 $\otimes$ 2 10 1 055.4014436 4 2 2 $\otimes$ 2 10 1 055.33
      6 4 0 4 $\otimes$ 2 10 1 024.876 4 0 4 $\otimes$ 2 10 1 024.95
      $5/2_{3}^{+}$214216436 4 2 2 $\otimes$ 2 10 1 043.2116476 4 2 2 $\otimes$ 2 10 1 042.84
      6 4 0 4 $\otimes$ 2 10 1 024.436 4 0 4 $\otimes$ 2 10 1 025.23
      $5/2_{4}^{+}$239824396 4 2 2 $\otimes$ 2 10 1 066.7524376 4 2 2 $\otimes$ 2 10 1 066.23
      6 4 0 4 $\otimes$ 2 10 1 022.226 4 0 4 $\otimes$ 2 10 1 022.46
      $7/2_{1}^{+}$136312306 4 2 2 $\otimes$ 2 10 1 058.1312296 4 2 2 $\otimes$ 2 10 1 058.14
      6 4 0 4 $\otimes$ 2 10 1 025.416 4 0 4 $\otimes$ 2 10 1 024.48
      $7/2_{2}^{+}$152020666 4 2 2 $\otimes$ 2 10 1 034.7620706 4 2 2 $\otimes$ 2 10 1 034.45
      6 4 0 4 $\otimes$ 2 10 1 023.886 4 0 4 $\otimes$ 2 10 1 024.00
      $7/2_{3}^{+}$247925366 4 2 2 $\otimes$ 2 10 1 068.2225366 4 2 2 $\otimes$ 2 10 1 067.93
      6 4 0 4 $\otimes$ 2 10 1 020.726 4 0 4 $\otimes$ 2 10 1 020.82
      $9/2_{1}^{+}$147714106 4 2 2 $\otimes$ 2 10 1 061.4314106 4 2 2 $\otimes$ 2 10 1 061.37
      6 4 0 4 $\otimes$ 2 10 1 025.116 4 0 4 $\otimes$ 2 10 1 025.09
      $9/2_{2}^{+}$240924786 4 2 2 $\otimes$ 2 10 1 061.4124796 4 2 2 $\otimes$ 2 10 1 060.86
      6 4 0 4 $\otimes$ 2 10 1 022.196 4 0 4 $\otimes$ 2 10 1 022.32
      $11/2_{1}^{+}$224721376 4 2 2 $\otimes$ 2 10 1 073.0621356 4 2 2 $\otimes$ 2 10 1 072.95
      6 4 0 4 $\otimes$ 2 10 1 017.416 4 0 4 $\otimes$ 2 10 1 017.40
      $13/2_{1}^{+}$216221646 4 2 2 $\otimes$ 2 10 1 072.8521616 4 2 2 $\otimes$ 2 10 1 072.54
      6 4 0 4 $\otimes$ 2 10 1 018.626 4 0 4 $\otimes$ 2 10 1 018.70
      $13/2_{2}^{+}$266829026 4 2 2 $\otimes$ 2 10 1 060.3829016 4 2 2 $\otimes$ 2 10 1 059.55
      6 4 0 4 $\otimes$ 2 10 1 027.876 4 0 4 $\otimes$ 2 10 1 028.23
      $15/2_{1}^{+}$264228616 4 2 2 $\otimes$ 2 10 1 059.2128596 4 2 2 $\otimes$ 2 10 1 059.06
      6 4 0 4 $\otimes$ 2 10 1 029.046 4 0 4 $\otimes$ 2 10 1 029.08
      $17/2_{1}^{+}$243028306 4 2 2 $\otimes$ 2 10 1 030.9528306 4 2 2 $\otimes$ 2 10 1 030.95
      6 4 0 4 $\otimes$ 2 10 1 025.516 4 0 4 $\otimes$ 2 10 1 025.28
      $21/2_{1}^{+}$242523126 4 2 2 $\otimes$ 2 10 1 073.3223096 4 2 2 $\otimes$ 2 10 1 073.09
      6 4 0 4 $\otimes$ 2 10 1 019.516 4 0 4 $\otimes$ 2 10 1 019.52
      Continued on next page

      Table 1.  Main partitions of the wave functions for $^{93}$Mo with SM1 and SM2 configuration spaces. The wave function for a particular angular momentum state would be composed of several partitions. Each partition is of the form $P = \pi[p(1), p(2), p(3), p(4)]$ $\otimes$ $\nu[n(1), n(2), n(3), n(4)]$, where $p(i)$ represents the number of valence protons occupying the $1f_{5/2}$, $2p_{3/2}$, $2p_{1/2}$, and $1g_{9/2}$ orbits, and $n(j)$ represents the number of valence neutrons in the $2p_{1/2}$, $1g_{9/2}$, $2d_{5/2}$, and $1h_{11/2}$ orbits.

      The $ I^{\pi} $ = 5/2$ ^{+} $ ground state is predominantly generated by the coupling of a 1$ g_{9/2} $ proton pair and one unpaired neutron in the 2$ d_{5/2} $ orbital. The state with energy of 1363 keV may correspond to the predicted 7/2$ _{1}^{+} $ state, which is consistent with the observed result of Ref. [25]. As presented in Table 1, the dominating contributions to the 1/2$ ^{+} $, 3/2$ ^{+} $, 7/2$ ^{+} $, 9/2$ ^{+} $, 11/2$ ^{+} $, 13/2$ ^{+} $, 15/2$ ^{+} $, 17/2$ ^{+} $, and 21/2$ ^{+} $ states obtained in shell model arise from the configurations $ \pi(1g_{9/2})^{2} $ $ \otimes $ $ \nu(2d_{5/2}) $, mixing with the $ \pi(1g_{9/2})^{4} $ $ \otimes $ $ \nu(2d_{5/2}) $, i.e., a proton pair is excited from the completely filled 2$ p_{1/2} $ orbital across the Z = 40 subshell closure into the 1$ g_{9/2} $ orbital. The calculations predict that the yrast 25/2$ ^{+} $ and 29/2$ ^{+} $ states, involve the same neutron configurations as the ground state 5/2$ _{1}^{+} $, and have the $ \pi(1g_{9/2})^{4} $ proton configurations. For the observed I = (35/2) state at 7268 keV, its parity was not assigned arising from the weak experimental statistics in the previous work [9]. The present large scale shell model calculations show that the predicted energy is much closer to the experimental value with the positive parity rather than negative parity. Thus, the I = (35/2) state is tentatively suggested as a positive parity state, i.e., 35/2$ _{1}^{+} $. The 35/2$ _{1}^{+} $ state could be interpreted as $ \pi[(1f_{5/2})^{-1}(2p_{1/2})^{1}(1g_{9/2})^{4}] $ $ \otimes $ $ \nu(2d_{5/2}) $, involving excitations of a 2$ p_{1/2} $ proton pair across the Z = 40 closed subshell into the 1$ g_{9/2} $ orbital and one unpaired 1$ f_{5/2} $ proton across the Z = 38 subshell into the 2$ p_{1/2} $ orbital. Besides, the 35/2$ _{1}^{+} $ state also includes proton excitation from the completely filled 2$ p_{3/2} $ orbital to the 2$ p_{1/2} $ orbital, leading to the configuration $ \pi[(1f_{5/2})^{-1}(2p_{3/2})^{-1}(1g_{9/2})^{4}] $ $ \otimes $ $ \nu(2d_{5/2}) $. As shown in Fig. 1, the predicted excitation energies from the ground state 5/2$ ^{+} $ to 35/2$ _{1}^{+} $ state within SM1 and SM2 are close to each other and in reasonable agreement with the experimental values. However, there are significant discrepancies (about 2 MeV) between the SM1 and SM2 for the higher-spin states at the energies of 8335 and 8820 keV (see Fig. 1). Considering the electromagnetic transition multipolarities and the discrepancies between calculated and experimental values, we tend to assign the states with the energies of 8335 and 8820 keV as the yrast and the yrare 37/2$ ^{+} $ states, respectively. One can see from Fig. 1 that the predicted energies of the 37/2$ _{1}^{+} $ and 37/2$ _{2}^{+} $ states within SM2 are more reasonable than SM1. However, the large discrepancies between calculations of SM1 and experimental values may be a consequence of the truncation space. As listed in Table 1, the 37/2$ _{1}^{+} $ and 37/2$ _{2}^{+} $ states within SM2 are mainly generated by the excitation of one neutron across the N = 56 subshell closure into the 1$ h_{11/2} $ orbital. It indicates that the high-j intruder 1$ h_{11/2} $ orbital may play an important role in the higher-spin states.

      Similar situation also appears in the negative parity states of $ ^{93} $Mo. As can be seen from Fig. 2, the calculations within SM2 containing the 1$ h_{11/2} $ orbital provide an improved description for higher-spin states in comparison with the SM1. In SM2, the energy difference of the predicted 37/2$ _{1}^{-} $ and 39/2$ _{1}^{-} $ states is 477 keV, which is close to the energy of the observed 573 keV $ \gamma $ ray [9], and the calculated 41/2$ _{1}^{-} $ state is in reasonable agreement with the experimental one. Thus, these states at energies of 8597, 9170, and 9646 keV may correspond to the predicted states with energies of 8604, 9081, and 9387 keV, respectively, namely, the 37/2$ _{1}^{-} $, 39/2$ _{1}^{-} $, and 41/2$ _{1}^{-} $ states. However, for the levels mentioned above, the predicted energies within SM1 are much larger than experimental values, along with a relatively large differences of 1.2 $ \sim $ 2.2 MeV, which indicates that the contribution of excitations from the neutron 2$ d_{5/2} $ orbital to the high-j intruder 1$ h_{11/2} $ orbital across the N = 56 subshell cannot be ignored. In addition, the yrast 33/2$ ^{-} $ state is also well reproduced in SM2 with an energy difference of only 81 keV, while the calculated energy in SM1 is 502 keV higher than experimental value. Therefore, there is better agreement between the experiment and theory for higher-spin states, using the extended truncation space of SM2.

      Based on two sets of shell model calculations, the 41/2$ _{1}^{-} $ state involves the neutron-core excitation $ \nu[(1g_{9/2})^{-1}(2d_{5/2})^{2}] $ in SM1, however, this state obtained in SM2 mainly arises from (2$ d_{5/2} \rightarrow 1h_{11/2} $) neutron excitation without neutron-core excitation. It indicates that the neutron-core excitation may not be involved below the 41/2$ _{1}^{-} $ state in $ ^{93} $Mo. For the 37/2$ _{1}^{-} $, 39/2$ _{1}^{-} $, and 41/2$ _{1}^{-} $ states, the calculations within SM2 predict that they involve mainly one neutron excitation from the 2$ d_{5/2} $ orbital to the 1$ h_{11/2} $ orbital, i.e., a configuration of $ \pi[(1f_{5/2})^{-1}(2p_{1/2})^{1}(1g_{9/2})^{4}] $ $ \otimes $ $ \nu(1h_{11/2}) $. Besides, the mentioned above states also include the excitation of one proton from the completely filled 2$ p_{3/2} $ orbital to the 2$ p_{1/2} $ orbital. And the 33/2$ ^{-} $ state also has the $ \nu(1h_{11/2}) $ as the main component, involving a proton pair excitations from the 2$ p_{1/2} $ orbital into 1$ g_{9/2} $ orbital. Based on above arguments, the contribution of the 1$ h_{11/2} $ orbital to the higher-spin states cannot be ignored whether for positive parity or negative parity states in $ ^{93} $Mo.

      The lower negative parity states with spin $ I^{\pi} $ = 5/2$ ^{-} $, 7/2$ ^{-} $, 9/2$ ^{-} $, 11/2$ ^{-} $, 13/2$ ^{-} $, 15/2$ ^{-} $, 23/2$ ^{-} $, and 27/2$ ^{-} $, based on SM1 and SM2, are reasonably reproduced as shown in Table 1. Meanwhile, these states obtained in shell model show the characteristics of multiplet, with $ \pi[(2p_{1/2})^{1}(1g_{9/2})^{3}] $ $ \otimes $ $ \nu(2d_{5/2}) $ configurations contributing maximally.

      In order to understand the contribution of different orbitals to each state more intuitively, we have also calculated occupation number for protons and neutrons based on the configuration space of SM2. The calculated occupation number for protons and neutrons are shown in Figs. 3 and 4, respectively. It can be seen from Fig. 3 that a large contribution in both kinds of parity states (I $ \geqslant $ 25/2) is coming from the proton $ 1g_{9/2} $ orbital. It can also be seen from Table 1 that the wave functions of these states involve proton excitations from the $ 1f_{5/2} $, 2$ p_{3/2} $, or 2$ p_{1/2} $ orbitals into the 1$ g_{9/2} $ orbital. As can be seen from Fig. 4, in the higher-spin states, major contribution from the intruder 1$ h_{11/2} $ orbital to both kinds of parity states is clearly evident, especially for negative parity states. Thus, the level structures of $ ^{93} $Mo in higher-spin states should include the component of the neutron 1$ h_{11/2} $ orbital.

      Figure 3.  Calculated occupation number of the 1$ f_{5/2} $, 2$ p_{3/2} $, 2$ p_{1/2} $, and 1$ g_{9/2} $ orbits for the protons in $ ^{93} $Mo based on SM2

      Figure 4.  Calculated occupation number of the 1$ g_{9/2} $, 2$ d_{5/2} $, and 1$ h_{11/2} $ orbits for the neutrons in $ ^{93} $Mo based on SM2

    IV.   SUMMARY
    • Large Scale Shell Model calculations have been performed in $ ^{93} $Mo based on two different configuration spaces. The calculations within SM1 well reproduced the low-lying states, but cannot reasonably describe the higher-spin states. An improved description for higher-spin states was obtained using extended space including the $ \nu(1h_{11/2}) $ orbital in SM2. It predicts that the higher-spin states of $ ^{93} $Mo could be principally interpreted by the configurations dominated by (i) proton excitations from the completely filled 1$ f_{5/2} $, 2$ p_{3/2} $, 2$ p_{1/2} $ subshells into the higher orbitals, and (ii) neutron excitation from the interior of N = 56 subshell closure into the high-j intruder 1$ h_{11/2} $ orbital. This indicates that the neutron 1$ h_{11/2} $ orbital is essential to obtain a more appropriate description for the higher-spin levels.

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