A=42 nuclei: $^{42}$Sc and $^{42}$Ca

Within the conventional shell model, the $^{42}$Ca and $^{42}$Sc nuclei are treated as two-body systems above the core of $^{40}$Ca. Hence, they are often used by the shell-model community to adjust the isoscalar $T$=0;$I$=1, 3, 5, 7, and isovector $T$=1;$I$=0, 2, 4, 6 matrix elements within the $f_{7/2}$ shell. Here, we use these nuclei to test our NCCI model but, at least at this stage, without an intention of refitting the interaction. The aim of this exercise is to capture global trends and tendencies, which may allow us to identify systematic features of the NCCI model in describing these seemingly simple nuclei. From the perspective of our approach, such tests are by no means trivial, because these nuclei are here treated within the full core-polarization effects included, cf. discussion in Refs. [53,54].

The results of the NCCI calculations for the isovector and isoscalar multiplets in $A=42$ nuclei are depicted in Figs. 5 and 6, and collected in Table 5. The reference states used in the calculation for $^{42}$Sc are listed in Table 6. They cover all fully aligned ( $K_\nu = K_\pi$) states, which are almost purely isoscalar, all possible antialigned states ( $K_\nu = - K_\pi$), and two $K=1$ aligned states. The antialigned states manifestly violate the isospin symmetry and, as discussed in Ref. [55], are approximately fifty-fifty mixtures of the isoscalar and isovector components. The $K=1$ aligned states also violate the isospin symmetry.

Figure 5: Excitation energies of the isovector (circles) and isoscalar (squares) multiplets in $^{42}$Sc with respect to the $0^+$ state. Theoretical and experimental results are marked with open and filled symbols, respectively.

Figure 6: Same as in Fig. 5, but for the isovector multiplets in $^{42}$Sc (circles) and $^{42}$Ca (diamonds).

Table 5: Excitation energies (total energies) of low-lying states (ground states) in $^{42}$Sc and $^{42}$Ca. For $^{42}$Sc, we show calculated and experimental energies of the isovector ($I$=0$^+$, 2$^+$, 4$^+$, and 6$^+$) and isoscalar ($I$=1$^+$, 3$^+$, 5$^+$, and 7$^+$) multiplets. For $^{42}$Ca, we show the analogous energies of the isovector multiplet. All energies are in MeV.

	$^{42}$Sc	$^{42}$Sc	$^{42}$Ca	$^{42}$Ca
$I^{\rm\pi}$	$\Delta$E $_{\rm I}^{\rm (th)}$	$\Delta$E $_{\rm I}^{\rm (exp)}$	$\Delta$E $_{\rm I}^{\rm (th)}$	$\Delta$E $_{\rm I}^{\rm (exp)}$
0$^+$	$-$352.961	$-$354.687	$-$360.200	$-$361.895
2$^+$	1.012	1.586	1.357	1.525
4$^+$	1.590	2.815	2.005	2.752
6$^+$	1.696	(3.200)	2.154	3.189
1$^+$	1.785	0.611
3$^+$	1.656	1.490
5$^+$	1.336	1.510
7$^+$	$-$0.347	0.617

The following three general conclusions can be drown from the results presented in Fig. 5:

• The model lacks isoscalar pairing $T=0,I=1$ correlations. A similar deficiency was already observed in $^{6}$Li, Sec. 4.1.

• The model strongly prefers fully aligned isoscalar $T=0,I_{\rm max}=7$ states. Again, the conclusion is consistent with the one drown from the calculated spectrum of $^{6}$Li.

• The energy range spanned by the isovector states, $\Delta E_{T=1} = E_{T=1,I=6} - E_{T=1,I=0}$, is by a factor of two smaller in theory than in experiment. It is not obvious, however, whether this difference signalizes that the model underestimates the isovector pairing correlations, overbinds the stretched (aligned) configurations, or both.

Table 6: Similar as in Table 3, but for $^{42}$Sc. Here, the reference Slater determinants correspond to configurations $\nu f_{7/2}\otimes \pi f_{7/2}$, and are labeled by intrinsic $K$ quantum numbers of valence neutrons and protons as $\vert\nu; K_\nu\rangle \otimes \vert\pi ; K_\pi\rangle$. Reference states $i$=1-4 correspond to antialigned configurations, $K_\nu = - K_\pi$, thus carrying no net intrinsic alignment. Reference states $i$=5-8 represent aligned configurations, $K_\nu = K_\pi$, thus having the total alignments of 7, 5, 3, and 1, respectively. The remaining two configurations $i$=9-10 carry net alignments of 1. The Table also lists the HF energies $\Delta$E $_{\rm I=\vert K\vert}$ relative to the $\vert\nu ;\tfrac{1}{2}\rangle \otimes \vert\overline{ \pi ;\tfrac{1}{2}}\rangle$ solution. The last column shows excitation energy of the lowest $I=\vert K\vert$ state projected from a given Slater determinant.

$i$	$\vert^{42}$Sc$;i\rangle$	$\Delta$E$_{\rm HF}$	$\beta _2$	$\gamma$	$\Delta$E $_{\rm I=\vert K\vert}$
1	$\vert\nu ;\tfrac{1}{2}\rangle \otimes \vert\overline{ \pi ;\tfrac{1}{2}}\rangle$	0.000	0.063	0	0.000
2	$\vert\nu ;\tfrac{3}{2}\rangle \otimes \vert\overline{\pi ;\tfrac{3}{2}}\rangle$	0.802	0.031	0	0.561
3	$\vert\nu ;\tfrac{5}{2}\rangle \otimes \vert\overline{\pi ;\tfrac{5}{2}}\rangle$	0.986	0.008	60	0.551
4	$\vert\nu ;\tfrac{7}{2}\rangle \otimes \vert\overline{\pi ;\tfrac{7}{2}}\rangle$	0.759	0.062	60	0.085
5	$\vert\nu ;\tfrac{7}{2}\rangle \otimes \vert\pi ;\tfrac{7}{2}\rangle$	$-$0.929	0.061	60	-0.647
6	$\vert\nu ;\tfrac{5}{2}\rangle \otimes \vert\pi ;\tfrac{5}{2}\rangle$	0.082	0.007	60	1.160
7	$\vert\nu ;\tfrac{3}{2}\rangle \otimes \vert\pi ;\tfrac{3}{2}\rangle$	0.345	0.032	0	1.594
8	$\vert\nu ;\tfrac{1}{2}\rangle \otimes \vert\pi ;\tfrac{1}{2}\rangle$	0.340	0.060	0	1.719
9	$\vert\nu ;\tfrac{3}{2}\rangle \otimes \vert\pi ;-\tfrac{1}{2}\rangle$	0.716	0.043	0	2.164
10	$\vert\nu ;\tfrac{5}{2}\rangle \otimes \vert\pi ;-\tfrac{3}{2}\rangle$	0.986	0.011	0	2.338

In the case of $^{42}$Ca, we focused on calculating the excitation energies of the $0^+$ states, addressing, in particular, the question of structure and excitation energy of the intruder configuration. Experimentally, the intruder configuration is observed at very low excitation energy of 1.843MeV, see Ref. [56] and references cited therein. In the calculations presented below we assumed that the structure of intruder state is associated with (multi)particle-(multi)hole excitations across the $N=Z=20$ magic gap, which in $^{40}$Ca is of the order of 7.0MeV, see Ref. [53] and references cited therein. The mechanism bringing the intruder configuration down in energy is sketched in Fig. 7.

The energy needed to elevate particles from the $d_{3/2}$ subshell to $f_{7/2}$ is at (near)spherical shape reduced by the energy associated with the spontaneous breaking of spherical symmetry in the intruder configuration, and further, by a rotational correction energy associated with the symmetry restoration. Owing to the configuration interaction, an additional gain in energy is expected too. The rotational correction and configuration interaction are also expected to lower slightly the MF g.s. energy. As shown in Fig. 7, the final value of the intruder excitation energy is an effect of rather a delicate interplay of several factors. Therefore, it is not surprising that the intruder states pose a real challenge for both the state-of-the-art nuclear shell models and MF-rooted theories.

Figure 7: Schematic illustration of the interplay between the primary physical ingredients contributing to the excitation energy of the intruder state within our model. See text for details.

In the NCCI calculations presented below, we mix states projected from the antialigned configurations that are listed in Table 7. The reference states can be divided into two classes. The first four configurations do not involve any cross-shell excitations. They correspond to the $\vert K\rangle \otimes \vert\overline{K}\rangle$ 0p-0h $(\nu f_{7/2})^2$ configurations with magnetic quantum number of $K$=1/2, 3/2, 5/2, and 7/2, respectively. The three remaining configurations are the lowest MF configurations involving two $f_{7/2}^4 d_{3/2}^{-2}:\, (\nu f_{7/2})^2\otimes (\pi f_{7/2})^2 \otimes (\pi d_{3/2})^{-2}$, four $f_{7/2}^6 d_{3/2}^{-4}:\, (\nu f_{7/2})^4\otimes (\nu d_{3/2})^{-2} \otimes (\pi f_{7/2})^2 \otimes (\pi d_{3/2})^{-2}$, and six $f_{7/2}^8 d_{3/2}^{-6}:\, (\nu f_{7/2})^4\otimes (\nu d_{3/2})^{-2} \otimes (\pi f_{7/2})^4 \otimes (\pi d_{3/2})^{-4}$ holes in $d_{3/2}$ shell, respectively.

Table 7: Similar as in Table 3, but for $^{42}$Ca. Here, the Slater determinants are labeled by spherical quantum numbers pertaining to active neutron orbitals. The last column shows excitation energies of the lowest $0^+$ states projected from a given Slater determinant.

$i$	$\vert^{42}$Ca$;i\rangle$	$\Delta$E$_{\rm HF}$	$\beta _2$	$\gamma$	$\Delta$E$_{I=0}$
1	$\vert\tfrac{1}{2}\rangle \otimes \vert\overline{\tfrac{1}{2}}\rangle$	0.000	0.069	0$^\circ$	0.000
2	$\vert\tfrac{3}{2}\rangle \otimes \vert\overline{\tfrac{3}{2}}\rangle$	0.516	0.033	0$^\circ$	0.765
3	$\vert\tfrac{5}{2}\rangle \otimes \vert\overline{\tfrac{5}{2}}\rangle$	0.544	0.007	60$^\circ$	0.770
4	$\vert\tfrac{7}{2}\rangle \otimes \vert\overline{\tfrac{7}{2}}\rangle$	0.084	0.061	60$^\circ$	0.315
5	$f_{7/2}^4\, d_{3/2}^{-2}$	10.001	0.288	14$^\circ$	6.860
6	$f_{7/2}^6\, d_{3/2}^{-4}$	10.986	0.414	22$^\circ$	6.498
7	$f_{7/2}^8\, d_{3/2}^{-6}$	14.937	0.542	12$^\circ$	9.619

The results of the NCCI calculations in $^{42}$Ca are depicted in Figs. 6 and 8 and collected in Tables 5 and 7. Figure 6 shows the $I=0^+,2^+,4^+$, and 6$^+$ states - the isovector $T$=1 multiplet - obtained within the NCCI calculations involving only $(\nu f_{7/2})^2$ reference states. The results are qualitatively similar to those in $^{42}$Sc. In both cases, theoretical spectra are compressed as compared to data. Detailed quantitative comparison reveals, however, surprisingly large differences between the theoretical and experimental spectra.

First, the energy differences $\delta E_{\rm I} = \Delta E_I ( ^{42}{\rm Ca}) - \Delta E_I ( ^{42}{\rm Sc})$ for $I=2^+, 4^+, 6^+$ are positive (negative) in theory (experiment), respectively. Second, the absolute values of $\vert\delta E_{\rm I}\vert$ are a few times larger in theory as compared to the data. It means that the model tends to overestimate the ISB effects in clearly an unphysical manner. This influences the ISB correction to the $0^+\longrightarrow 0^+$ Fermi $\beta$-decay matrix element, which in the present NCCI calculation rises to $\delta_{\rm C} \approx 2.2$%. Most likely, the unphysical component in the ISB effect is related to the time-odd polarizations and matrix elements originating from these fields, which are essentially absent in even-even systems. One should also remember that in the Skyrme functionals, including, of course, the SV force used here, the time-odd terms are very purely constrained.

Figure 8: The lowest $0^+$ states projected from $(f_{7/2})^2$ (0p-0h), $(f_{7/2})^4 (d_{3/2})^{-2}$ (2p-2h), $(f_{7/2})^6 (d_{3/2})^{-4}$ (4p-4h), and $(f_{7/2})^8 (d_{3/2})^{-6}$ (6p-6h) reference states. Open (filled) diamonds refer to calculations performed using the SV and SV$_{\rm SO}$ functionals, respectively. These results do not include configuration mixing. The right part shows excitation energies of the intruder state obtained within the NCCI theory with SV and SV$_{\rm SO}$ interactions.

Figure 8 shows the $0^+$ states calculated using functionals SV. The left part of the figure depicts the lowest $0^+$ states projected from the reference states $(f_{7/2})^2$ (0p-0h), $(f_{7/2})^4 (d_{3/2})^{-2}$ (2p-2h), $(f_{7/2})^6 (d_{3/2})^{-4}$ (4p-4h), and $(f_{7/2})^8 (d_{3/2})^{-6}$ (6p-6h). These results do not include configuration mixing. Note, that symmetry restoration itself changes the optimal intruder configuration to $(\nu f_{7/2})^4\otimes (\nu d_{3/2})^{-2} \otimes (\pi f_{7/2})^2 \otimes (\pi d_{3/2})^{-2}$ as compared to MF, which favors $(\nu f_{7/2})^2\otimes (\pi f_{7/2})^2 \otimes (\pi d_{3/2})^{-2}$.

The right part of the figure shows excitation energies of the intruder states as obtained within the NCCI calculations. Here, all reference states listed in Table 7 were included. For the SV force, the excitation energy of the lowest intruder configuration equals 7.5MeV, and exceeds the data by 5.7MeV. The main reason of the disagreement is related to an unphysically large $N=Z=20$ shell gap: the bare $N=20$ gap deduced directly from the s.p. HF levels in $^{40}$Ca equals as much as 11.5MeV. Its value exceeds the experimental gap by almost 4.5MeV (for an overview of experimental data, see Ref. [53] and references cited therein). It is therefore not surprising that the combined effects of deformation and rotational correction are unable to compensate for the large energy needed to lift the particles from the $d_{3/2}$ to $f_{7/2}$ shell, see Fig. 7.

To investigate interplay between the s.p. and collective effects, we repeated the NCCI calculations using the functional SV$_{\rm SO}$, which differs from SV in a single aspect, namely, we increased its spin-orbit strength by a factor of 1.2. This readjustment allows to reduce a disagreement between theoretical and experimental binding energies in $N\approx Z$ $sd$ and lower-$pf$ shell nuclei to $\pm$1% level as shown in Ref. [57]. When applied to the heaviest $N=Z$ nucleus $^{100}$Sn and its neighbor $^{100}$In it gives 827.710 MeV and 833.067 MeV what is in an impressive agreement with the experimental binding energies equal 825.300 MeV (833.110 MeV) in $^{100}$Sn ($^{100}$In), respectively. Ability to reproduce masses is among the most important indicators of a quality of DFT-based models. Such a readjustment of the SO strength is also the simplest and most efficient mechanism allowing us to reduce the magic $Z=N=20$ gap [58]. For the SV$_{\rm SO}$ force, the bare gap equals 9.6MeV, which is by almost 1.9MeV smaller than the original SV gap, but still it is much larger, by circa 2.6MeV, than the experimental value. Results of the NCCI calculations obtained using functional SV$_{\rm SO}$ are shown in Fig. 8. Now the projected and NCCI calculations both favor the configuration $(\nu f_{7/2})^4\otimes (\nu d_{3/2})^{-2} \otimes (\pi f_{7/2})^2 \otimes (\pi d_{3/2})^{-2}$. We also note that for both the SV and SV$_{\rm SO}$ functionals, geometrical properties of the reference states (deformations) are very similar.

When discussing the influence of various effects on the final position of the intruder state, it is worth stressing the role of the symmetry restoration. The rotational correction lowers the intruder state by 4.9MeV, bringing its excitation energy to 2.3MeV, which is only 0.5MeV above the experiment. However, after the configuration mixing, the excitation energy of the intruder state increases to about 3.6MeV, that is, it becomes again 1.7MeV higher as compared to data. This is due to the configuration mixing in the ground state, which lowers its energy by almost 1MeV, whereas it leaves the position of the intruder state almost unaffected. The reason for that is the fact that the $(\nu f_{7/2})^2$ antialigned reference states (states 1-4 in Tables 7 and 8) are almost linearly dependent and thus mix relatively strongly. Conversely, at deformations corresponding to the intruder configuration $(\nu f_{7/2})^4\otimes (\nu d_{3/2})^{-2} \otimes (\pi f_{7/2})^2 \otimes (\pi d_{3/2})^{-2}$, the Nilsson scheme prevails. Therefore, the intruder configurations become almost linearly independent and appear to mix very weakly. The amount of the mixing was tested by performing additional calculations of matrix elements between the lowest $(\nu f_{7/2})^4\otimes (\nu d_{3/2})^{-2} \otimes (\pi f_{7/2})^2 \otimes (\pi d_{3/2})^{-2}$ configuration and the excited configurations involving the same number of $(d_{3/2})^{-4}$ holes. All these matrix elements turned out to be negligibly small.

Table 8: Same as in Table 7, but for the functional SV$_{\rm SO}$.

$i$	$\vert^{42}$Ca$;i\rangle$	$\Delta$E$_{\rm HF}$	$\beta _2$	$\gamma$	$\Delta$E$_{I=0}$
1	$\vert\tfrac{1}{2}\rangle \otimes \vert\overline{\tfrac{1}{2}}\rangle$	0.000	0.064	0$^\circ$	0.000
2	$\vert\tfrac{1}{2}\rangle \otimes \vert\overline{\tfrac{1}{2}}\rangle$	0.517	0.032	0$^\circ$	0.679
3	$\vert\tfrac{1}{2}\rangle \otimes \vert\overline{\tfrac{1}{2}}\rangle$	0.496	0.007	60$^\circ$	0.676
4	$\vert\tfrac{1}{2}\rangle \otimes \vert\overline{\tfrac{1}{2}}\rangle$	0.006	0.061	60$^\circ$	0.200
5	$f_{7/2}^4\, d_{3/2}^{-2}$	8.399	0.276	15$^\circ$	5.085
6	$f_{7/2}^6\, d_{3/2}^{-4}$	7.377	0.402	22$^\circ$	2.548
7	$f_{7/2}^8\, d_{3/2}^{-6}$	9.955	0.532	15$^\circ$	4.103