Hamiltonian

In the proton-neutron (pn) representation, the present Hamiltonian is composed of the pairing-plus-multipole force and the monopole corrections: $\begin{array}{l}H=H_{\text{sp}}+H_{P_0}+H_{P_2}+H_{QQ}+H_{OO}+H_{HH}+H_{\text{mc}}\\ ={\displaystyle \sum _{\alpha \mathrm{,}i}{\epsilon }_a^i{c}_{\alpha \mathrm{,}i}^{†}{c}_{\alpha \mathrm{,}i}}-\frac{1}{2}{\displaystyle \sum _{J=\mathrm{0,2}}{\displaystyle \sum _{i{i}^{\prime }}{g}_{J\mathrm{,}i{i}^{\prime }}}}{\displaystyle \sum _M{P}_{JM\mathrm{,}i{i}^{\prime }}^{†}}{P}_{JM\mathrm{,}i{i}^{\prime }}\\ -\frac{1}{2}{\displaystyle \sum _{i{i}^{\prime }}\frac{{\chi }_{\mathrm{2,}i{i}^{\prime }}}{b^4}}{\displaystyle \sum _M:}{Q}_{2M\mathrm{,}i{i}^{\prime }}^{†}{Q}_{2M\mathrm{,}i{i}^{\prime }}:\\ -\frac{1}{2}{\displaystyle \sum _{i{i}^{\prime }}\frac{{\chi }_{\mathrm{3,}i{i}^{\prime }}}{b^6}}{\displaystyle \sum _M:}{O}_{3M\mathrm{,}i{i}^{\prime }}^{†}{O}_{3M\mathrm{,}i{i}^{\prime }}:\\ -\frac{1}{2}{\displaystyle \sum _{i{i}^{\prime }}\frac{{\chi }_{\mathrm{4,}i{i}^{\prime }}}{b^8}}{\displaystyle \sum _M:}{H}_{4M\mathrm{,}i{i}^{\prime }}^{†}{H}_{4M\mathrm{,}i{i}^{\prime }}:\\ +{\displaystyle \sum _{a\le c\mathrm{,}i{i}^{\prime }}{k}_{\text{mc}}}\left(ia\mathrm{,}{i}^{\prime }c\right){\displaystyle \sum _{JM}{A}_{JM}^{†}}\left(ia\mathrm{,}{i}^{\prime }c\right)A_{JM}\left(ia\mathrm{,}{i}^{\prime }c\right)\mathrm{.}\end{array}$ (1) This effective interaction (1) includes the single-particle Hamiltonian (H_sp), the J=0 and J=2 pairing ($P_0^{†}{P}_0$ and $P_2^{†}{P}_2$), the quadrupole-quadrupole ($Q^{†}Q$), the octupole-octupole ($O^{†}O$), the hexadecapole-hexadecapole ($H^{†}H$) terms, and the monopole corrections (H_mc). In the pn representation, $P_{JM\mathrm{,}i{i}^{\prime }}^{†}$ and $A_{JM}^{†}(ia\mathrm{,}{i}^{\prime }c)$ are the pair operators, while $Q_{2M\mathrm{,}i{i}^{\prime }}^{†}$, $O_{3M\mathrm{,}i{i}^{\prime }}^{†}$, and $H_{4M\mathrm{,}i{i}^{\prime }}^{†}$ are the quadrupole, octupole, and hexadecapole operators, in which i (i’) are indices for protons (neutrons). The parameters gJ,ii’, χ_2,ii’, χ_3,ii’, χ_4,ii’, and k_mc(ia,i’c) are the corresponding force strengths, and b is the harmonic-oscillator range parameter. The two-body force strengths suited for the present model space are listed in Table 1.

Model space

In addition to the Hamiltonian, the choice of model spaces with single particle energies are crucial for shell model calculations. In this work, the model space for protons (neutrons) consists of all the pf shell orbits (1f_7/2, 1f_5/2, 2p_1/2, 2p_3/2) with frozen core ⁴⁰Ca. Further, an intruder state, the 1g_9/2 orbit is added to study high energy levels. The low-lying levels in ⁴¹Ca (⁴¹Sc) are selected as neutron (proton) single particle states in this model space to obtain single particle energies. Marking ${\epsilon }_{f7/2}^{\pi }$ as the single particle energy (SPE) of the ground state 7/2^- in ⁴¹Sc, π denoting proton (ν neutron), we obtain SPE as the difference in binding energies (BE) of ⁴¹Sc and ⁴⁰Ca, ${\epsilon }_{f7/2}^{\pi }=BE\left({}^{41}\text{Sc}\right)-BE\left({}^{40}\text{Ca}\right)=-1.089\text{M}e\text{V}\mathrm{.}$ (2) The other proton SPE in ⁴¹Sc can be obtained by adding this value ${ϵ}_{f7/2}^{\pi }=-1.089$ MeV to the excited energies. Thus, all SPEs are obtained as follows (MeV): $\begin{array}{l}{\epsilon }_{f7/2}^{\pi }=-1.089\mathrm{,}{\epsilon }_{f7/2}^{\nu }=-8.387\mathrm{,}\\ {\epsilon }_{p3/2}^{\pi }=0.627\mathrm{,}{\epsilon }_{p3/2}^{\nu }=-6.444\mathrm{,}\\ {\epsilon }_{f5/2}^{\pi }=1.499\mathrm{,}{\epsilon }_{f5/2}^{\nu }=-5.811\mathrm{,}\\ {\epsilon }_{p1/2}^{\pi }=2.376\mathrm{,}{\epsilon }_{p1/2}^{\nu }=-4.773\mathrm{,}\\ {\epsilon }_{g9/2}^{\pi }=3.425\mathrm{,}{\epsilon }_{g9/2}^{\nu }=-3.936\mathrm{.}\end{array}$ (3) For these neutron SPEs, the excited energies are taken as 0, 1.943, 2.576, 3.614 and 4.451 MeV from 7/2^-, 3/2^-, 5/2^-, 3/2^- and 9/2⁺ levels in ⁴¹Ca. For proton SPEs, they are 0, 1.716, 2.588, 3.465, and 4.514 MeV from 7/2^-, 3/2^-, 5/2^-, 1/2^-, and 9/2⁺ levels in ⁴¹Sc.

Monopole corrections

In the present Hamiltonian, the monopole corrections can be investigated as follows: $Mc=k_{\text{mc}}(ia\mathrm{,}{i}^{\prime }c){\displaystyle \sum _{JM}{A}_{JM}^{†}}\left(ia\mathrm{,}{i}^{\prime }c\right)A_{JM}\left(ia\mathrm{,}{i}^{\prime }c\right)\mathrm{.}$ (4) Here $A_{JM}^{†}(ia\mathrm{,}{i}^{\prime }c)$ and AJM represent the pair operators, and k_mc is the monopole force strength. Monopole correction terms should consider corresponding data. Note that the maximum multiplet is J = 8 coupled by the configuration 1f_7/2 1g_9/2. The datum 8^- at 6.408 MeV in ⁴²Ca is selected to determine the force strength of monopole correction between neutron orbits 1f_7/2 and 1g_9/2. There is no datum 8^- in ⁴²Ti; therefore, the force strength between proton orbits 1f_7/2 and 1g_9/2 refers to the value in ⁴²Ca. $\begin{array}{l}Mc1\equiv k_{\text{mc}}\left(\nu f_{7/2}\mathrm{,}\nu g_{9/2}\right)=-1.10\text{ MeV}\mathrm{,}\\ Mc2\equiv k_{\text{mc}}\left(\pi f_{7/2}\mathrm{,}\pi g_{9/2}\right)=-1.10\text{ MeV}\mathrm{.}\end{array}$ (5) Owing to missing three-body contributions in the Hamiltonians derived from realistic two-body potentials [32, 33], monopole corrections become necessary for two-body interactions [34]. The extended pairing plus multipole-multipole force (EPQQM model) is advantageous in employing monopole correction (Mc) terms [35]. Monopole-driven shell evolutions are discussed with different effects of tensor forces in Ref. [36]. Monopole interactions are critical in shifting the spherical-shell model to account for the nonconventional shell evolution observed in neutron-rich nuclei. Experimental studies have shown that the monopole interaction between the neutron orbits h_11/2 and d_3/2 provides an effective description of ground-state inversions observed in isotopes ranging from ¹³⁰In(¹²⁹Cd) to ¹²⁸In(¹²⁷Cd) [37]. The energy gap of the neutron N = 82 can be modified by using a monopole term between h_11/2 and f_7/2. Experimental confirmation has validated the inverse low-lying levels of ¹²⁹Cd predicted by the monopole effect between g_9/2 and h_11/2. Monopole corrections have recently helped in constructing a new Hamiltonian above ¹³²Sn with core excitations and an intruder orbit i_13/2.

⁴²Ca

The g_9/2 orbit plays a crucial role in the energy spectrum of ⁴²Ca. Being an intruder orbit, it lies outside the traditional valence shell configuration of ⁴⁰Ca, which is composed of the pf shell. By including the g_9/2 orbit in the model space, additional degrees of freedom for nucleons are introduced, leading to new configurations and the possibility of new quantum numbers.

⁴²Ca, as a stable nucleus, exhibits a rich spectrum of excited states up to 11 MeV and spins up to J = 12. Only positive parity levels can be coupled with two valence nucleons in the pf shell, with the highest spin J = 6, resulting from the configuration coupling. However, negative parity levels arise from two distinct sources: cross-shell excitations or coupling with the intruder orbit g_9/2. In this study, we employ the EPQQM interaction to discuss levels coupled by the intruder orbit g_9/2 and the SDPFMU interaction to explore cross-shell excitations.

The determination of pairing and multipole force strengths in the neutron model space is facilitated by studying the low-lying states of ⁴²Ca. Figure 1 depicts a comparison between the experimental data for ⁴²Ca and shell model calculations. The ground state and excited levels 2⁺, 4⁺, and 6⁺ exhibit the same main configuration $\nu f_{7/2}^2$ as suggested by the EPQQM, GXPF1A, and SDPFMU interactions. The second 2⁺ and 4⁺ levels predominantly feature the configuration $\nu f_{7/2}{p}_{3/2}$.

Fig. 1Full-screen View

The experimental energy levels in ⁴²Ca [30], and the theoretical levels calculated by interactions EPQQM, GXPF1A, and SDPFMU

Regarding negative parity levels, the 8^- configuration $\nu f_{7/2}{g}_{9/2}$ closely corresponds to the observed 8^- state at 6.408 MeV. This configuration successfully reproduces the (3^-), (5^-), and 6^- states. The 7^- state at 6.703 MeV is in proximity to the datum at 6.718 MeV, although its parity information remains unconfirmed. The levels 1^-, 2^-, and 4^- are well-predicted in this study. However, the GXPF1A interaction fails to reproduce the negative parity levels in ⁴²Ca due to the absence of cross-shell and intruder orbits in the model space. In contrast, the SDPFMU interaction, which considers the cross-shell proton (neutron) d_5/2 orbit under Z=20 (N=20), successfully predicts negative levels around 6 MeV with the primary configuration ν d^-15/2f³7/2.

The low-lying states of ⁴²Ca can be used to determine the pairing and multipole force strengths in the neutron model space. As shown in Fig. 1, the experimental data in ⁴²Ca are listed and compared with shell model calculations. The ground state and excited levels 2⁺, 4⁺, and 6⁺ are consistently described by the same main configuration $\nu f_{7/2}^2$ according to three different interactions: EPQQM, GXPF1A, and SDPFMU. There are also small differences in the mixed minor configurations. In the EPQQM model, the ground state 0⁺ has approximately 82% of $\nu f_{7/2}^2$ 6.3 % of $\nu p_{3/2}^2$, and 8.3% of $\nu f_{5/2}^2$. The configurations $\nu p_{1/2}^2$ and $\nu g_{9/2}^2$ occupy only about 2% respectively. In GXPF1A model, the ground state 0⁺ has 97 % of $\nu f_{7/2}^2$, and only 1.6 % of $\nu p_{3/2}^2$. In SDPFMU model, the ground state 0⁺ has 92 % of $\nu f_{7/2}^2$, and no other configuration more than 1.52%.

In EPQQM, the first 2⁺ state has 79.8 % of $\nu f_{7/2}^2$ and 10.4% of $\nu f_{7/2}{p}_{3/2}$, while it has 96.8% of $\nu f_{7/2}^2$ and 2.8 % of $\nu f_{7/2}{p}_{3/2}$ by GXPF1A model. The second 2⁺ level has 73.7% of configuration $\nu f_{7/2}{p}_{3/2}$ and 16.8% of $\nu f_{7/2}^2$ in this work, while it has 92.7% of $\nu f_{7/2}{p}_{3/2}$ in GXPF1A model. The difference increases in the minor configuration of this level, between EPQQM and GXPF1A models. The second 4⁺ has 97.1% of $\nu f_{7/2}{p}_{3/2}$ in EPQQM, while it has 93.2% of $\nu f_{7/2}{p}_{3/2}$ in GXPF1A. The difference decreases in the second 4⁺.

For negative parity level states, the 8^- coupled by $\nu f_{7/2}{g}_{9/2}$ is very close to the datum 8^- at 6.408 MeV. This configuration has 8 members from the 1^- to 8^- level, effectively reproducing the data of (3^-), (5^-), and 6^- states. The 7^- state at 6.703 MeV is close to the datum 6.718 MeV, which includes unconfirmed parity information. This study successfully predicts the levels 1^-, 2^-, and 4^-. The 8^- level at 6.468MeV has 100 % of $\nu f_{7/2}{g}_{9/2}$, because only this configuration can produce the J = 8 state. In addition to the main configuration $\nu f_{7/2}{g}_{9/2}$, the 7^- state has a minor configuration of $\nu f_{5/2}{g}_{9/2}$, only 0.5%. The 6^- (5^-)state has a minor configuration of $\nu p_{3/2}{g}_{9/2}$, 2.9 (3.93) %. The levels 4^- and 3^- also have a little part of $\nu p_{3/2}{g}_{9/2}$. The levels 2^- and 1^- are only reproduced by configuration $\nu f_{7/2}{g}_{9/2}$ in the present model space. For the other two interactions, the GXPF1A is not able to reproduce the negative parity levels in ⁴²Ca owing to the lack of cross-shell and intrusive orbits in the model space. While considering the cross-shell proton (neutron) d_5/2 orbit under Z=20 (N=20), the SDPFMU provides negative levels at approximately 6 MeV with the main configuration $\nu d_{5/2}^{-1}{f}_{7/2}^3$.

⁴²Ti

The ground state of ⁴²Ti, which is the mirror nucleus of ⁴²Ca, is unstable with a half-life of 211.7 ms (19), as determined from analysis of beta decay and correlated implantations. Levels of ⁴²Ti excit up around 7 MeV with the highest spin Jπ = 6⁺. The low-lying states in ⁴²Ti are used to determine the strengths of pairing and multipole forces in the proton model space. In the even-even nucleus of ⁴²Ti, the EPQQM model suggests that the main configuration of the 0⁺, 2⁺, 4⁺, and 6⁺ levels is $\pi f_{7/2}^2$ (Fig. 2). In this work, the ground state 0⁺ has 81 % of $\pi f_{7/2}^2$, 8.8 % of $\pi f_{5/2}^2$, and 6 % of $\pi p_{3/2}^2$. In GXPF1A model, the ground state has 97 % of $\pi f_{7/2}^2$ and 1.6 % of $\pi p_{3/2}^2$. In SDPFMU model, the ground state has 73.4 % of $\pi f_{7/2}^2$ and 16.3 % of $\pi p_{1/2}^2$. These three models have the same main configuration and only some difference in minor configurations. In the second 2⁺ and 4⁺ levels, these three interactions show a small difference in configurations. The EPQQM has 78.7 % (94.4 %) configuration $\pi f_{7/2}{p}_{3/2}$ in the second 2⁺ (4⁺) levels, while in GXPF1A is 92.7 % (93.2 %) The SDPFMU has 53 % (65 %) of $\pi f_{7/2}{p}_{3/2}$ in the second 2⁺ (4⁺) levels respectively.

Fig. 2Full-screen View

The calculations and experimental energy levels in ⁴²Ti[30].

In this study, the negative parity levels ranging from 1^- to 8^- coupled by the configuration $\pi f_{7/2}{g}_{9/2}$ are found to be good predictions, which appear at approximately 6 MeV. There is still no experimental data of negative parity levels in this energy range. The high-spin level 8^- at 6.602 MeV is only coupled by the configuration $\pi f_{7/2}{g}_{9/2}$. The level 7^- at 6.651 MeV has 99.7% of $\pi f_{7/2}{g}_{9/2}$ and 0.3% of $\pi f_{5/2}{g}_{9/2}$. The level 6^- at 5.593 MeV has 96.0% of $\pi f_{7/2}{g}_{9/2}$ and 4.0% of $\pi p_{3/2}{g}_{9/2}$. The level 5^- (4^-) at 5.789(6.567) MeV has 97.1(95.6)% of $\pi f_{7/2}{g}_{9/2}$ and 2.6(3.5)% of $\pi p_{3/2}{g}_{9/2}$. The levels 1^-, 2^-, and 3^- are also coupled by configuration $\pi f_{7/2}{g}_{9/2}$ completely. Another type of negative parity levels are coupled by neutron cross-shell excitations from the sd shell under N=20. The 1^- and 3^- levels, obtained from the SDPFMU interaction, exhibit a dominant configuration of $\pi f_{7/2}^2\nu d^{-1}5/2f7/2$.

When comparing the experimental data of ⁴²Ca and ⁴²Ti, some differences are observed in the low-lying energy levels. For example, the first 2⁺, 4⁺, and 6⁺ levels of ⁴²Ti are slightly higher than those of ⁴²Ca, while the GXPF1A interaction predicts the same energy values. The EPQQM interaction, by incorporating different force strengths for protons and neutrons, effectively captures the distinction between ⁴²Ca and ⁴²Ti. As shown in Table 1, the pairing force strength of g₀,pp (g_2,pp) is 0.450 (0.470), while g₀,nn (g_2,nn) is 0.422 (0.449). This small discrepancy in force strengths can accurately reproduce the energy difference between ⁴²Ca and ⁴²Ti as observed in experiments.

⁴²Sc

As an odd-odd nucleus, ⁴²Sc offers a unique opportunity to investigate the correlations between proton and neutron states. The excited states in ⁴²Sc extend up to energies of 13 MeV, with the highest observed spin being J = 15. However, at present, the parity information for these states remains unknown. Theoretical calculations suggest that the configurations of low-lying states in ⁴²Sc are formed by coupling the proton and neutron single particle states appearing in ⁴¹Sc and ⁴¹Ca (Fig. 3). Shell-model calculations predict that the main configuration of the ground state in ⁴²Sc is $\pi f_{7/2}\nu f_{7/2}$ that couples eight states from 0⁺ to 7⁺. Here, the 0⁺ state is taken as the ground state in ⁴²Sc and it matches the experimental data well. The interactions of GXPF1A (Th. 2) and SDPFMU (Th. 3) provide the 7⁺ state as the lowest state. The 7⁺ level at 0.616 MeV from the level systematics and shell model expectations is a long half-life isomer with an average of 61.7(4)s, which is significantly longer than the ground state 0⁺ at 680.79(28) ms [30].

Fig. 3Full-screen View

The calculations and experimental energy levels in ⁴²Sc [30].

In EPQQM model, the ground state 0⁺ has 91.4% of configuration $\pi f_{7/2}\nu f_{7/2}$ and 7.7% of configuration $\pi p_{3/2}\nu p_{3/2}$. The first excited state 1⁺ at 0.361MeV has 94.0% of configuration $\pi f_{7/2}\nu f_{7/2}$ and 4.6% of configuration $\pi p_{3/2}\nu p_{3/2}$. The level 7⁺ at 0.965 MeV is near to the datum 0.616 MeV, which is completely coupled by configuration $\pi f_{7/2}\nu f_{7/2}$. The level 5⁺ at 1.570 MeV is close to the datum 1.510 MeV, which has 77.3% of configuration $\pi f_{7/2}\nu f_{7/2}$, 12.6% of configuration $\pi p_{3/2}\nu f_{7/2}$, and 9.9% of configuration $\pi f_{7/2}\nu p_{3/2}$. The level 6⁺ is predicted at 1.763 MeV with a main configuration of $\pi f_{7/2}\nu f_{7/2}$. Theoretical calculations based on the three different interactions efficiently fit the experimental data of the observed 1⁺, 3⁺, and 5⁺ states.

As for negative parity levels coupled by the intruder orbit g_9/2, they lie above 5 MeV with a main configuration $\pi f_{7/2}\nu g_{9/2}$. There is no datum observed in this energy extent. The level 1^- at 5.269MeV is completely coupled by the configuration $\pi f_{7/2}\nu g_{9/2}$. The level 2^- at 5.488MeV has 99.6 % of configuration $\pi f_{7/2}\nu g_{9/2}$. The level 8^- at 5.569MeV is completely coupled by the configuration $\pi f_{7/2}\nu g_{9/2}$. The SDPFMU interaction provides negative parity levels as neutron core excitations around 5 MeV, with the main configuration $\pi f_{7/2}\nu d_{5/2}^{-1}{f}_{7/2}^2$.

⁴³Ti

In this study, the energy levels in ⁴³Ti are examined as an additional test of the current model (Fig. 4). Owing to the presence of an odd number of valence nucleons, the pf shell model space only couples negative parity levels. The positive parity levels can be coupled through intruder orbit g_9/2 or cross-shell excitations. The ground state 7/2^- is well reproduced by all three different models with a main configuration of $\pi f_{7/2}^2\nu f_{7/2}$, while some difference exists in percentages and minor configurations. In EPQQM, the ground state has 76.9% of configuration $\pi f_{7/2}^2\nu f_{7/2}$, 6.1% of configuration $\pi f_{5/2}^2\nu f_{7/2}$, and 5.0% of configuration $\pi p_{3/2}^2\nu f_{7/2}$. In GXPF1A, the ground state has 86.8% of configuration $\pi f_{7/2}^2\nu f_{7/2}$, 2.3% of configuration $\pi f_{5/2}^2\nu p_{1/2}$, and 1.8% of configuration $\pi f_{7/2}{p}_{3/2}\nu f_{7/2}$. In SDPFMU, the ground state has 79.2% of configuration $\pi f_{7/2}^2\nu f_{7/2}$, 4.7% of configuration $\pi p_{1/2}^2\nu f_{1/2}$, and 3.8% of configuration $\pi f_{7/2}{p}_{3/2}\nu f_{7/2}$.

Fig. 4Full-screen View

Calculations and experimental energy levels in ⁴³Ti [30].

The state 3/2^- has mixed configurations as follows: 27.9% (30.4%) of $\pi f_{7/2}^2\nu f_{7/2}$, 28.0% (23.9%) of $\pi f_{7/2}^2\nu p_{3/2}$, and 20.7% (24.1%) $\pi f_{7/2}{p}_{3/2}\nu f_{7/2}$ by EPQQM (GXPF1A) model. The SDPFMU model has some difference in this state’s configurations, which has a major part of 50.1% $\pi f_{7/2}{f}_{5/2}\nu f_{7/2}$, and a minor part of 24.6% of $\pi f_{7/2}^2\nu f_{7/2}$. In EPQQM, the level 11/2^- at 1.842 MeV is very close to the datum 1.858 MeV, with a difference factor 0.991. Its state has 70.7% of configuration $\pi f_{7/2}^2\nu f_{7/2}$ and 16.7% of configuration $\pi f_{7/2}{p}_{3/2}\nu f_{7/2}$. The levels 15/2^-, and 19/2^- also have a main configuration $\pi f_{7/2}^2\nu f_{7/2}$.

The positive parity states around 6 MeV are coupled by two main configurations $\pi f_{7/2}{g}_{9/2}\nu f_{7/2}$ and $\pi f_{7/2}^2\nu g_{9/2}$. The level 11/2⁺ at 5.708 MeV has 95.1% of $\pi f_{7/2}{g}_{9/2}\nu f_{7/2}$. The level 13/2⁺ at 5.990 MeV has 91.4% of $\pi f_{7/2}{g}_{9/2}\nu f_{7/2}$. The levels 15/2⁺, 17/2⁺, and 19/2⁺ have about 95 % of $\pi f_{7/2}{g}_{9/2}\nu f_{7/2}$. Except the level 9/2⁺, the states from 1/2⁺ to 21/2⁺ around 6 MeV are all coupled by this configuration. The level 9/2⁺ at 4.923 MeV belongs to configuration $\pi f_{7/2}^2\nu g_{9/2}$. The interaction SDPFMU produces positive parity states near 2 MeV, which are coupled by neutron core excitations of configuration $\pi f_{7/2}^2\nu d_{5/2}^{-1}{f}_{7/2}^2$.

⁴⁴Ti

As a self-conjugated nucleus, ⁴⁴Ti is a benchmark for shell model calculations in the northeast of the doubly magic nucleus ⁴⁰Ca. With two protons and two neutrons as valence nucleons, the nucleus ⁴⁴Ti has more complicated and high energy states. The excited states in ⁴⁴Ti reach up to 16 MeV, and the highest spin is 15 with negative parity information. For the positive parity levels shown in Fig. 5, the 0⁺, 2⁺, 4⁺, 6⁺, and 12⁺ levels have a main configuration $\pi f_{7/2}^2\nu f_{7/2}^2$ according to the three different interactions, while some difference is found in percentages and minor configurations.

Fig. 5Full-screen View

The calculations and experimental energy levels in ⁴⁴Ti[30].

In the EPQQM model, the ground state 0⁺ of ⁴²Ca is composed of 56.7% $\pi f_{7/2}^2\nu f_{7/2}^2$ configuration. On the other hand, in the GXPF1A and SDPFMU models, the ground state 0⁺ consists of 70.4% and 78.5% $\pi f_{7/2}^2\nu f_{7/2}^2$ configurations, respectively.

Similarly, for the 2⁺ state, the EPQQM model predicts a 50.0% $\pi f_{7/2}^2\nu f_{7/2}^2$ configuration, while the GXPF1A and SDPFMU models assign 64.1% and 79.1% $\pi f_{7/2}^2\nu f_{7/2}^2$ configurations, respectively.

In the case of the 4⁺ state, the EPQQM model predicts a 54.8% $\pi f_{7/2}^2\nu f_{7/2}^2$ configuration, whereas the GXPF1A and SDPFMU models assign 56.7% and 74.6% $\pi f_{7/2}^2\nu f_{7/2}^2$ configurations, respectively.

Overall, the configurations in the GXPF1A and SDPFMU models tend to be more concentrated in the states of ⁴²Ca compared to the EPQQM model, indicating a stronger dominance of the $\pi f_{7/2}^2\nu f_{7/2}^2$ configuration in these models.

In this study, the calculated energy levels from 0⁺ to 6⁺ are found to be in good agreement with the experimental data, indicating a successful reproduction of these states by the model used. Furthermore, when considering the high spin levels 8⁺, 10⁺, and 12⁺, the EPQQM model yields better results compared to the GXPF1A and SDPFMU models.

With even-valence nucleons, only positive parity levels can be coupled in the model space of the pf shell. The negative parity levels are from cross-shell excitations or are coupled by the intruder orbit g_9/2. As neutron core excitations, the SDPFMU interactions provide negative parity states from 1^- to 12^- in the energy range from 4 to 9 MeV. Their main configuration is $\pi f_{7/2}^2\nu d_{5/2}^{-1}{f}_{7/2}^3$, while the 5^- to 9^- states fit the corresponding experimental data well.

For the negative parity states coupled by the intruder orbit g_9/2, the EPQQM model reproduces very well the data (1^-), (8^-), (10^-), (11^-), and (12^-), which has a main configuration $\pi f_{7/2}^2\nu f_{7/2}{g}_{9/2}$. This good fit, coupled with intruder orbit g_9/2 in nuclei ⁴²Sc, ⁴²Ti, and ⁴³Ti confirms our predictions. The level 1^- at 6.309 MeV has 77.5 % of configuration $\pi f_{7/2}^2\nu f_{7/2}{g}_{9/2}$, which is very close to datum 1^- at 6.220 MeV. The level 8^- at 7.012 MeV has 79.5 % of configuration $\pi f_{7/2}^2\nu f_{7/2}{g}_{9/2}$, with a difference factor 1.013 to the datum (8^-) at 6.924 MeV. The level 10^- at 8.968 MeV has a difference factor 1.012 by comparing with datum (10^-) at 8.861 MeV. The level 11^- at 9.815 MeV has 86.8 % of configuration $\pi f_{7/2}^2\nu f_{7/2}{g}_{9/2}$, with a difference factor 1.009 to its corresponding datum (11^-) at 9.723 MeV. The level 12^- at 10.455 MeV has 86.8 % of configuration $\pi f_{7/2}^2\nu f_{7/2}{g}_{9/2}$, with a difference factor 0.9992 to its corresponding datum (12^-) at 10.463 MeV. The shell model with EPQQM interaction works very accurately even at about 10 MeV.