Introduction

Nuclei northeast of double magic nucleus ⁴⁰Ca are of significant interest in both experiments and shell model theory. The research in Ref. [1] highlighted the doubly magic nature of ⁵⁴Ca and provided direct experimental evidence for the onset of a sizable subshell closure at N = 34. Possible variations of magic numbers have attracted considerable interest in nuclear physics [2, 3]. For exotic nuclei, the universality of magic numbers does not extend to stable nuclei; some nuclei lose their magic properties [4-8], whereas others acquire new magic numbers [1, 8-12]. The significantly larger cross-section observed in ⁵³Ca provides direct evidence for the N = 34 shell closure. [13]. Interaction cross-sections for ^42-51Ca were measured for the first time on a carbon target at 280 MeV per nucleon [14]. A recent publication [15] showcased cutting-edge experimental and theoretical advancements related to 2p decay, including technological innovations for measuring nucleon-nucleon correlations and developments in models linking structural attributes and decay properties.

Charge resolution is crucial for particle identification in ionization measurement system (IMS) experiments involving ion pairs [16]. Additionally, Ref. [17] predicted the cross sections of ejectile residues (ERIs) in 140 MeV/u projectile fragmentation reactions with ^78,86Kr / ^58,64Ni / ^40,48Ca + ⁹Be using the proposed model. Another study [18] applied the continuum Skyrme Hartree-Fock-Bogoliubov (HFB) theory with Green’s function method to study neutron-rich Ca, Ni, Zr, and Sn isotopes, accurately reproducing the experimental double- and single-neutron separation energies. Lastly, Ref. [19] utilized a dinuclear system model to explore reactions of ^40,48Ca projectiles with ²³⁸U, ²⁴²Pu, and ²⁴³Am targets at various incident energies. This study analyzed the dependence of calculated synthesis cross sections on collision orientations, discussed the isospin effect of projectiles, and explored the influence of entrance channel effects on the synthesis cross-sections of superheavy nuclei.

In theoretical studies, determining the half-lives of β decay and β delayed neutron emission (βn) is crucial for advancing fundamental science, nuclear physics, and industrial applications [20]. By including contributions of the three-body force, the effective shell model Hamiltonian can reproduce the experimental shell evolution towards and beyond the closure at N = 28 [21].

Examinations of the excited states in Ti isotopes have been performed within a single j-shell framework [22]. Comparative research on pf shell nuclei has been conducted using both the cranked Nilsson–Strutinsky model and the spherical shell model [23]. Over the past few decades, various models have been designed to analyze the spectroscopic properties of atomic nuclei. Large-scale spherical shell model calculations, such as those for a set of interactions [24], show excellent agreement with observed data. Based on existing interactions, this Hamiltonian utilizes the pf shell as the valence shell. More recently, the EPQQM model, initially designed for $N\approx Z$ nuclei by Hassegawa and Kaneko [25], has proven effective in different mass regions, including proton-rich pf shells [26], $p{f}_{5/2}{g}_{9/2}$ shells [27], neutron-rich fpg shells [28], and sd-pf shells [29]. In the heavy neutron enrichment region near A=130, the EPQQM model successfully elucidated low-energy and high-core excitations [30-32], confirming the persistence of the N=82 shell closure. In addition, it provides a comprehensive description of the ground-state inversions driven by monopole interactions between protons and neutrons [33-35].

In nuclear physics, the intruder orbit g_9/2 plays a significant role in connecting the df and pf shells. Intruder states, which extend beyond valence configurations, significantly affect the structural properties of atomic nuclei. They can cause inversions in ground-state or low-lying excited-state, altering the energy level order predicted by the conventional shell model. Intruder states also influence various nuclear properties, including energy and structure. The Ref. [36] studied nuclei near a double magic nucleus ⁴⁰Ca using the intruder orbit g_9/2 and successfully reproduced the low level positive-parity states of ⁴²Ca, ⁴²Sc, and ^42-44Ti, while predicting the negative-parity level. This confirms that the intruder orbit g_9/2 is crucial for studying high-energy state near the double magic nucleus ⁴⁰Ca. Furthermore, the intruder orbit i_13/2 plays a crucial role [37] in the Sn isotope chain. In this study, the EPQQM model, including the intruder orbit g_9/2, was used to explore the nuclear energy levels in ⁴⁴Ca. The inclusion of the intruder orbit provides a comprehensive description of the positive-parity and high-energy states with negative parity in these nuclei.

The model’s calculations were compared with experimental results to validate its accuracy, emphasizing the importance of including intruder states in nuclear structure. Intruder states may be linked to shape coexistence, where different shapes coexist in the ground state of a nucleus due to the mixing of distinct configurations [38, 39]. This phenomenon is observed in numerous atomic nuclei, and integrating intruder states into the model space offers a more comprehensive depiction of shape coexistence. Such considerations are crucial for comprehending the evolution of nuclear shape in response to variations in neutron or proton numbers and for investigating the potential presence of novel magic numbers.

Notably, in the EPQQM model, particularly for unstable nuclei near ¹³²Sn, the inclusion of a monopole correction led to the development of a new Hamiltonian above ¹³²Sn, incorporating both core excitation and the intruder orbit i_13/2. This enhanced model successfully elucidates low-lying and high-spin excitations, and predicts and confirms the anti-aligned low-lying excitations in ¹²⁹Cd. By encompassing the cross-shell orbitals, the model overcomes limitations associated with closed-shell interactions, rendering it suitable for investigating high-spin spectra in unstable nuclei. Furthermore, careful selection of the model space, considering single-particle energies, is paramount for ensuring the accuracy of the shell model calculations.

This study utilized the EPQQM model, incorporating the g_9/2 intruder orbit within the pf shell model space to investigate the level spectra of ⁴⁴Ca. The EPQQM interaction effectively describes the positive-parity levels and intruder states in these isotopes, offering configurations for negative-parity levels. The calculations were performed using the NUSHELLX@MSU shell model code [40].

Hamiltonian

In the proton-neutron (pn) representation, the EPQQM Hamiltonian consists of pairing forces, multipole terms, and monopole corrections [36, 26, 32]: $\begin{array}{l}H=H_{\text{sp}}+H_{P_0}+H_{P_2}+H_{QQ}+H_{OO}+H_{HH}+H_{\text{mc}}\\ ={\displaystyle \sum _{\alpha \mathrm{,}i}{\epsilon }_a^i{c}_{\alpha \mathrm{,}i}^{†}{c}_{\alpha \mathrm{,}i}}-\frac{1}{2}{\displaystyle \sum _{J=\mathrm{0,2}}{\displaystyle \sum _{i{i}^{\prime }}{g}_{J\mathrm{,}i{i}^{\prime }}}}{\displaystyle \sum _M{P}_{JM\mathrm{,}i{i}^{\prime }}^{†}}{P}_{JM\mathrm{,}i{i}^{\prime }}\\ -\frac{1}{2}{\displaystyle \sum _{i{i}^{\prime }}\frac{{\chi }_{\mathrm{2,}i{i}^{\prime }}}{b^4}}{\displaystyle \sum _M:}{Q}_{2M\mathrm{,}i{i}^{\prime }}^{†}{Q}_{2M\mathrm{,}i{i}^{\prime }}:\\ -\frac{1}{2}{\displaystyle \sum _{i{i}^{\prime }}\frac{{\chi }_{\mathrm{3,}i{i}^{\prime }}}{b^6}}{\displaystyle \sum _M:}{O}_{3M\mathrm{,}i{i}^{\prime }}^{†}{O}_{3M\mathrm{,}i{i}^{\prime }}:\\ -\frac{1}{2}{\displaystyle \sum _{i{i}^{\prime }}\frac{{\chi }_{\mathrm{4,}i{i}^{\prime }}}{b^8}}{\displaystyle \sum _M:}{H}_{4M\mathrm{,}i{i}^{\prime }}^{†}{H}_{4M\mathrm{,}i{i}^{\prime }}:\\ +{\displaystyle \sum _{a\le c\mathrm{,}i{i}^{\prime }}{k}_{\text{mc}}}\left(ia\mathrm{,}{i}^{\prime }c\right){\displaystyle \sum _{JM}{A}_{JM}^{†}}\left(ia\mathrm{,}{i}^{\prime }c\right)A_{JM}\left(ia\mathrm{,}{i}^{\prime }c\right)\mathrm{.}\end{array}$ (1) The effective interaction (1) encompasses the single-particle Hamiltonian (H_sp), J=0 and J=2 pairing terms ($P_0^{†}{P}_0$ and $P_2^{†}{P}_2$), quadrupole-quadrupole interactions ($Q^{†}Q$), octupole-octupole interactions ($O^{†}O$), hexadecapole-hexadecapole interactions ($H^{†}H$), and monopole corrections (H_mc). In the pn representation, $P_{JM\mathrm{,}i{i}^{\prime }}^{†}$ and $A_{JM}^{†}\left(ia\mathrm{,}{i}^{\prime }c\right)$ serve as pair operators, while $Q_{2M\mathrm{,}i{i}^{\prime }}^{†}$, $O_{3M\mathrm{,}i{i}^{\prime }}^{†}$, and $H_{4M\mathrm{,}i{i}^{\prime }}^{†}$ represent the quadrupole, octupole, and hexadecapole operators. Here, i (i’) denotes the nucleon index, where i is a proton, and i’ is a neutron. The parameters $g_{J\mathrm{,}i{i}^{\prime }}$, χ_2,ii’, χ_3,ii’, χ_4,ii’, and $k_{\text{mc}}\left(ia\mathrm{,}{i}^{\prime }c\right)$ represent the respective strengths of forces, and b is the range parameter of the harmonic oscillator. The model space for protons (neutrons) includes the entire pf shell orbitals (1f_7/2, 1f_5/2, 2p_1/2, 2p_3/2) with a core nucleus of ⁴⁰Ca. In addition, the 1g_9/2 orbital is introduced as an intruder state to study high-energy levels.

The two-body force strengths of the EPQQM model are presented in Table 1 [36]. The parameters of the proton (neutron) J = 0 and J =2 pairing forces were confirmed using data from 2⁺, 4⁺, and 6⁺ in ⁴²Ti (⁴²Ca). Quadrupole–quadrupole and octupole-octupole forces significantly affect spectra with more than two valence particles, such as ^42-44Ti. The hexadecapole-hexadecapole force primarily affected the high-spin levels. The parameters between protons and neutrons were confirmed by the odd-odd nuclear ⁴²Sc. In Ref. [36], two monopole correction terms were added to modify the monopole force between orbits 1f_7/2 and 1g_9/2 as $Mc1\equiv k_{\text{mc}}\left(\nu f_{7/2}\mathrm{,}\nu g_{9/2}\right)=-1.10\text{ MeV}$, and $Mc2\equiv k_{\text{mc}}\left(\pi f_{7/2}\mathrm{,}\pi g_{9/2}\right)=-1.10\text{ MeV}$. In this study, we increased the monopole term Mc3(Mc4) to modify the energy gap between the neutron orbits 2p_3/2(1f_5/2) and 1f_7/2. $\begin{array}{l}Mc3\equiv k_{\text{mc}}\left(\nu f_{7/2}\mathrm{,}\nu p_{3/2}\right)=0.18\text{ MeV}\mathrm{,}\\ Mc4\equiv k_{\text{mc}}\left(\pi f_{7/2}\mathrm{,}\pi f_{5/2}\right)=0.65\text{ MeV}\mathrm{.}\end{array}$ (2) The strengths are determined by the 2⁺ states in ⁴⁸Ca and ⁵⁰Ca and their monopole effects are discussed in Sect. 3.2.

Results and discussions

3.1

Level spectra of ^44-58Ca

In this section, we investigate the level spectra of ^44-58Ca, including the g_9/2 orbitals in ⁴⁴Ca and ⁴⁶Ca. As shown in Fig. 1(a), shell model calculations revealed the shared major configurations of ⁴⁴Ca in the ground and excited states (2⁺, 4⁺, and 6⁺). However, differences emerged in minor configurations, notably in the ground-state composition. For the second 2⁺ state, the EPQQM and a set of interactions exhibited varying configurations. The EPQQM model predicted the second 4⁺ states with 79.44% $\nu f_{7/2}^4$ and 4.97% $\nu f_{7/2}^3{p}_{3/2}$, whereas the set of effective interactions had a higher predominance of approximately 96.15% $\nu f_{7/2}^4$. For negative-parity states, the EPQQM model accurately predicts 3^-, 4^-, and 5^- states with the $\nu f_{7/2}^3{g}_{9/2}$ configuration. The set of effective interactions failed to reproduce negative-parity states due to the absence of cross-shell and intruder orbits in the model space.

Fig. 1Full-screen View

Experimental states of ⁴⁴Ca and ⁴⁶Ca [41], and the shell model calculations by EPQQM (Th.1) and a set of effective interactions (Th.2)

Recent studies on Ca isotopes have explored the nuclear entropy, revealing an approximate particle-hole symmetry between ⁴²Ca and ⁴⁶Ca. In the current model space, calculations for states (0⁺,2⁺, 4⁺, and 6⁺) show shape-coexistence configurations. For the 4⁺ level coupled with the $\nu f_{7/2}^6$ configuration, the calculated energy closely aligns with the experimental value of 2.574 MeV. Figure 1(b) shows a primary $\nu f_{7/2}^6$ configuration for the ground state and excited states (2⁺, 4⁺, 6⁺), and another shared configuration for the excited states (1⁺, 3⁺, 5⁺). Calculated using EPQQM and a set of effective interactions, some differences exist in the minor configurations. In the EPQQM model, the ground state 0⁺ comprises roughly 79.21% of $\nu f_{7/2}^6$, 8.78% of $\nu f_{7/2}^4{p}_{3/2}^2$, and 4.34% of $\nu f_{7/2}^4{f}_{5/2}^2$. Conversely, the ground state 0⁺ of a set of effective interactions was dominated by approximately 92.79% of $\nu f_{7/2}^6$.

In ⁴⁸Ca, stable properties are exhibited, featuring an excitation spectrum of up to 13 MeV. Ground- and excited-state calculations revealed coexistence features for the 2⁺, 3⁺, 4⁺, and 5⁺ states. As shown in Fig. 2(a), the dominant configuration for the excited states is $\nu f_{7/2}^7{p}_{3/2}$, whereas the secondary configurations exhibit slight differences. The 1⁺ state in the EPQQM model exhibits a primary configuration of $\nu f_{7/2}^6{p}_{3/2}^2$, with secondary configurations comprising 7.36% of $\nu f_{7/2}^6{p}_{3/2}{p}_{1/2}$ and 1.95% of $\nu f_{7/2}^7{f}_{5/2}$. Understanding these energy levels and configuration distributions is crucial for advancing our knowledge of nuclear structures.

Fig. 2Full-screen View

Experimental states of ⁴⁸Ca and ⁵⁰Ca [41], and the shell model calculations by EPQQM (Th.1) and a set of effective interactions (Th.2)

In ⁵⁰Ca, the ground state has a short half-life of 13.45 s. As illustrated in Fig. 2(b). the excited states can reach 11 MeV with a maximum spin of 8⁺. The coexistence configurations are shown in the ground state and the 2⁺ excited state as $\nu f_{7/2}^8{p}_{3/2}^2$. Similarly, excited states 3⁺, 4⁺, and 5⁺ have the same preliminary configuration. Figure 2(b) illustrates that the ground state 0⁺ is primarily composed of approximately 74.61% of $\nu f_{7/2}^8{p}_{3/2}^2$ and 6.6% of $\nu f_{7/2}^6{p}_{3/2}^4$. The configuration of ground state 0⁺ is dominated by approximately 86.92% of $\nu f_{7/2}^8{p}_{3/2}^2$ and 6.11% of $\nu f_{7/2}^8{p}_{1/2}^2$ in a set of effective interactions. The 2⁺ excited state is primarily composed of 79.45% of $\nu f_{7/2}^8{p}_{3/2}^2$ whereas in a set of effective interactions, it predominantly consists of 85.72% of $\nu f_{7/2}^8{p}_{3/2}^2$. Notable differences in the secondary configurations were observed for the 4⁺ and 5⁺ energy levels. These findings enhance the understanding of the energy levels and configuration distributions of ⁵⁰Ca. Recent discoveries of neutron shell closures at N=32 and N=34 in the pf-shell region have been attributed to the filling of neutron orbits 2p_1/2 and 2p_3/2, respectively. These shell closures were determined based on observations related to the first 2⁺ excitation energies, transition probabilities, mass measurements, E(2⁺), and neutron stripping cross-sections.

For ⁵²Ca (Fig. 3(a)), the calculated results revealed shared primary features in the configurations of the 1⁺ and 2⁺ excited states. The EPQQM model exhibited increased secondary configurations for the ground state 0⁺ and the 2⁺ excited states. Specifically, the ground state 0⁺ is predominantly composed of approximately 78.06% of $\nu f_{7/2}^8{p}_{3/2}^4$ and 8.12% of $\nu f_{7/2}^8{p}_{3/2}^2{p}_{1/2}^2$, whereas in a set of effective interactions, it primarily consists of approximately 83.11% $\nu f_{7/2}^8{p}_{3/2}^4$ and 11% of $\nu f_{7/2}^8{p}_{3/2}^2{p}_{1/2}^2$. For the 2⁺ excited state, the EPQQM model is primarily composed of 84.83% of $\nu f_{7/2}^8{p}_{3/2}^3{p}_{1/2}$ and 4.94% of $\nu f_{7/2}^8{p}_{3/2}^2{p}_{1/2}^2$, while a set of effective interactions model predominantly consists of 89.51% of $\nu f_{7/2}^8{p}_{3/2}^3{p}_{1/2}$ and 4.12% of $\nu f_{7/2}^8{p}_{3/2}^2{p}_{1/2}^2$. Overall, the excited states 1⁺, 2⁺, 3⁺, and 4⁺ provide reliable predictions of the energy-level structure of ⁵²Ca within the model space calculations.

Fig. 3Full-screen View

Experimental states of ⁵²Ca and ⁵⁴Ca [41], and the shell model calculations by EPQQM (Th.1) and a set of effective interactions (Th.2)

In Fig. 3(b), the ground state 0⁺ of ⁵⁴Ca is 91.64% $\nu f_{7/2}^8{p}_{3/2}^4{p}_{1/2}^2$, whereas in a set of effective interactions, it is 95.09%. The 2⁺ state is 89.96% $\nu f_{7/2}^8{p}_{3/2}^4{f}_{5/2}{p}_{1/2}$, while in a set of effective interactions, the 2⁺ state is 94.54% $\nu f_{7/2}^8{p}_{3/2}^4{f}_{5/2}{p}_{1/2}$. Predictions for the 3⁺ and 4⁺ states are provided, and the absence of negative-parity levels is due to the exclusion of the g_9/2 orbital when the model space dimension overflows.

For ⁵⁶Ca (Figs. 4(a)), the ground state and the 2⁺ and 4⁺ excited states have coexisting configurations. The 1⁺ and 5⁺ excited states exhibit consistent primary configurations across both the EPQQM and a set of effective interactions. Notable differences arise in the secondary configurations of the ground state and the 2⁺ state between EPQQM and a set of effective interactions. At the 3⁺ level, there is a variation in the primary configuration between the two interactions. Overall, these calculations provide insight into the predicted energy levels and configurations of ⁵⁶Ca.

Fig. 4Full-screen View

Experimental states of ⁵⁶Ca and ⁵⁸Ca [41], and the shell model calculations by EPQQM (Th.1) and a set of effective interactions (Th.2)

Figure 4(b) shows the calculated results for the energy-level structure of ⁵⁸Ca. Due to the absence of experimental values for comparison, these results serve as predictions. In the EPQQM model, the ground state 0⁺ is mainly composed of $\nu f_{7/2}^8{p}_{3/2}^4{f}_{5/2}^4{p}_{1/2}^2$ and $\nu f_{7/2}^6{p}_{3/2}^4{f}_{5/2}^6{p}_{1/2}^2$, while in a set of effective interactions, it is primarily $\nu f_{7/2}^8{p}_{3/2}^4{f}_{5/2}^4{p}_{1/2}^2$. For the 2⁺ state, both models show the dominance of $\nu f_{7/2}^8{p}_{3/2}^4{f}_{5/2}^4{p}_{1/2}^2$. The 4⁺ and 3⁺ states are primarily $\nu f_{7/2}^8{p}_{3/2}^4{f}_{5/2}^4{p}_{1/2}^2$ and $\nu f_{7/2}^7{p}_{3/2}^4{f}_{5/2}^5{p}_{1/2}^2$. The EPQQM with the monopole effects of Mc3 and Mc4 accurately reproduces the level structure of Ca isotopes from N =22 to N = 30 and predicts some useful states in ^52-58Ca in the absence of experimental data.

3.2

The level spectra and B(E2) with Monopole effects

In this section, the analysis of E₂₊ and E₄₊ span isotopes from ⁴²Ca to ⁵⁸Ca (Fig. 5). As indicated by the gray points in ⁴²Ca,the interaction in Ref. [36] plays an important role in nuclei close to ⁴⁰Ca. After the neutron number increased, the 2⁺ levels increased to approximately 3.8  MeV in ⁴⁸Ca. This interaction provides only the 2⁺ state at 2.8MeV. Here, the neutron number is magic and the 2⁺ level is from one neutron excited across the N = 28 shell. The datum 3.8 MeV in level 2⁺ of ⁴⁸Ca marks an energy gap of N = 28 and determines the strength of the monopole terms Mc3 and Mc4. With their monopole effects, the calculations reproduced the data for the 2⁺ and 4⁺ states very well (Fig. 5(a,b)). As indicated by the black solid points, we predict the 2⁺ state of ⁵⁴Ca at approximately 5  MeV. This implies that the energy gap for N = 34 is higher than that for the magic number N = 28. which supports “a new nuclear magic number from the level structure of ⁵⁴Ca" [1].

Fig. 5Full-screen View

Experimental 2⁺,4⁺ states, B(E2) values from 2⁺ to 0⁺ [41], and the shell model calculations by EPQQM and a set of effective interactions. Label +Mc3,4 means including the monopole effects of Mc3 and Mc4

As shown in Fig. 5, the EPQQM with the monopole effects of Mc3 and Mc4 reproduced the level structure of the Ca isotopes from N =22 to N = 30. For configurations, the 2⁺ state of ⁴⁸Ca has 77.46% of $\nu f_{7/2}^7{p}_{3/2}$, and 6.98% of $\nu f_{7/2}^6{p}_{3/2}^2$. This state is almost entirely composed of one neutron excited across the N = 28 shell. The 2⁺ state of ⁴⁶Ca is 85% of $\nu f_{7/2}^6$ and 4.38% of $\nu f_{7/2}^5{p}_{3/2}$ and the component crossing N = 28 is very small. In ⁵⁴Ca, the 2⁺ state is 89.96 % of νf_5/2 p_1/2. Orbits νf_7/2 and νp_3/2 are completely occupied. The high energy of the 2⁺ state indicates an energy gap between orbits νf_5/2 and νp_1/2.

In the Ca isotopes from N = 22 to N = 38, the monopole effects of Mc3 and Mc4 have almost no impact on the B(E2) values from the 2⁺ to 0⁺ states. Both the set of effective interactions and the interactions in Ref. [36] yielded B(E2) values near zero (Fig. 5(d)). The experimental data exhibited a peak at ⁴⁴Ca, followed by a general decrease in the B(E2) values for isotopes with higher neutron numbers (beyond ⁴⁴Ca). Theoretical models generally predict lower and more stable B(E2) values near zero. The Mc3,4 monopole effects partially capture the overall trend observed in the experimental data, although they underestimate the values for ⁴⁴Ca and ⁴⁶Ca. This discrepancy suggests potential limitations in the ability of the model to fully capture the specific nuclear structure effects that influence B(E2) transitions in this isotopic chain.

Summary

The level spectra and monopole effects of the Ca isotopes were investigated using a model space that included the pf shell and the intruder orbital g_9/2. This study enhances the understanding of the energy spectrum by considering the effects of the intruder orbit g_9/2. The EPQQM with the monopole effects of Mc3 and Mc4 reproduced the level structure of Ca isotopes from N = 22 to N = 30, and the main conclusions are as follows:

(1) In ⁴⁴Ca, positive-parity states are effectively reproduced. The negative-parity states coupled with the intruder orbital g_9/2 are predicted to be approximately 4 MeV.

(2) In ^46-58Ca, we reproduced the existing data well; for example, 1⁺, 3⁺, and 5⁺ in ⁴⁶Ca, and 5⁺, 6⁺ in ⁴⁸Ca. The excited states 2⁺ and 4⁺ in ⁵⁰Ca were close to the experimental data, whereas the 2⁺ and 4⁺ states provided good predictions for ⁵⁴Ca.

(3) For the Ca isotopes from N = 22 to N = 38, both E₂₊ and E₄₊ reached peak values at the neutron magic number N = 28. The values of $E_{4^{+}}/E_{2^{+}}$ reproduce the experimental trend from ⁴²Ca to ⁵⁰Ca.

(4) With the monopole effects between νf_7/2 and νp_3/2 (νf_5/2) in data of ⁴⁸Ca and ⁵⁰Ca, the EPQQM interaction predicts a high-energy 2⁺ state in ⁵⁴Ca, which supports a new nuclear magic number from the level structure of ⁵⁴Ca.