3.1.1. From *N* = 20 to *N* = 28

As shown in Figure 2a, the most distinct property by filling the *ν f*7/2 orbital is that the *Z* = 16 shell gap sharply diminishes and that the order of *πs*1/2 and *πd*3/2 is finally inverted at the 48Ca core. The change of this shell gap is expressed as <sup>Δ</sup>*<sup>ν</sup> <sup>f</sup>*7/2 (*επ<sup>d</sup>*3/2 − *επ<sup>s</sup>*1/2 ) ≈ <sup>8</sup>{*V*<sup>m</sup> *pn*(*d*3/2, *<sup>f</sup>*7/2) − *<sup>V</sup>*<sup>m</sup> *pn*(*s*1/2, *f*7/2)}. Since the *d*3/2-*f*7/2 and *s*1/2-*f*7/2 pairs are labeled {−−} and {+0}, respectively, according to the rule, introduced in Section 2.1, this value is a large negative value. The actual number calculated with the SDPF-MU interaction is −3.32 MeV. If the tensor force is omitted from the interaction, this value decreases to −1.71 MeV, pointing to almost equal contributions of the central and tensor forces.

Experimentally, the evolution of the *Z* = 16 shell gap is well examined by the first excitation energies of 39K and 47K, which can be regarded as a proton hole in the 40Ca and 48Ca cores, respectively, from very large spectroscopic factors for the lowest two levels. The measured values of *E*(1/2<sup>+</sup> <sup>1</sup> ) <sup>−</sup> *<sup>E</sup>*(3/2<sup>+</sup> <sup>1</sup> ) for 39K and 47K are 2.52 MeV and −0.36 MeV, respectively.

Hence, if one assumes the pure single-hole states for the 1/2<sup>+</sup> <sup>1</sup> and 3/2<sup>+</sup> <sup>1</sup> states in 39K and 47K, the <sup>Δ</sup>*<sup>ν</sup> <sup>f</sup>*7/2 (*επ<sup>d</sup>*3/2 <sup>−</sup> *επ<sup>s</sup>*1/2 ) value estimated from these experimental data is −2.88 MeV. The corresponding value obtained from large-scale shell-model calculations is −3.33 MeV, which is somewhat overestimated; however, the sharp decrease of *E*(1/2<sup>+</sup> <sup>1</sup> ) <sup>−</sup> *<sup>E</sup>*(3/2<sup>+</sup> <sup>1</sup> ) in going from 39K to 47K is well explained. Note that this number is very close to that evaluated from the ESPE (−3.32 MeV; see the first paragraph of this Subsection) because the first two levels of 47K are very close to single-proton-hole states.

Another important property in filling the *ν f*7/2 orbital is that the proton spin–orbit splitting for the *d* orbitals sharply decreases. This is caused almost solely by the tensor force (Figure 2a) because the central force gives similar monopole matrix elements between the *d*3/2-*f*7/2 and *d*5/2-*f*7/2 pairs: those are {−−} and {−+} pairs, respectively. Hence, quantifying the spin–orbit splitting is the key to extracting the tensor-force driven shell evolution. By using the SDPF-MU interaction, the proton spin–orbit splittings for the *d* orbital are obtained to be 7.42 and 5.05 MeV for the 40Ca and 48Ca cores, respectively, indicating a more than 2 MeV reduction.

Unlike the cases of *d*3/2 and *s*1/2, the *d*5/2 proton hole does not appear as a nearly pure single-hole state because the excitation energy is much higher than other low-lying levels, making the hole state fragmented over many levels. For the present purpose, the distribution of spectroscopic factors provides crucial information. The one-proton removal spectroscopic factors from 40Ca and 48Ca were measured with reactions, such as (*d*, 3He) and (*e*,*e p*). Although the (*e*,*e p*) reaction gives more reliable spectroscopic factors, those measured for 40Ca are concerning only a few low-lying states. Thus, the (*d*, 3He) data were used to estimate the spin–orbit splitting for Ca isotopes from the centroid of the measured spectroscopic factors, as discussed in Refs. [27,28].

The centroid of the spectroscopic factors, actually, provides the exact single-particle energy. However, there are many energy levels that cannot be detected by the actual experiment because their spectroscopic factors are too small to be measured. Although each of these undetected levels has a tiny contribution to the centroid, the total effect is not negligible because the number of such levels is very large. In this sense, the centroid of the spectroscopic factors that is obtained from experiment cannot be free from uncertainty associated with the limited experimental sensitivity. Hence, in order to validate theoretical single-particle energies, it is rather helpful to compare between experiment and theory regarding how major peaks are distributed. The results are shown in Figure 3, in which the spectroscopic factor *C*2*S*(*j*) for the orbital *j* is defined as

$$\mathbb{C}^2 S(j) = \frac{\left| \langle \Psi\_B || a\_j^\dagger || \Psi\_A \rangle \right|^2}{2J\_B + 1},\tag{8}$$

where Ψ*<sup>A</sup>* and Ψ*<sup>B</sup>* are the wave functions of the nuclei *A* and *B*, respectively (here, *A* and *B* correspond to Ca and K isotopes, respectively), and *JB* is the angular momentum of *B*.

**Figure 3.** Distribution of the one-proton removal spectroscopic strengths (see Equation (8)) from 48Ca (**left**) and 40Ca (**right**) comparing experimentalal results ("Expt.") with shell-model calculations ("Calc."). The spectroscopic factors shown are divided by 2*j* + 1 to normalize to unity for fully occupied orbitals. The bin widths are 0.25 MeV. Data are from Refs. [29] (48Ca) and [30] (40Ca). See text for details.

For 48Ca, the calculations were carried out with the SDPF-MU interaction in the 0*h*¯ *ω* model space [18]. The present calculation successfully captures the characteristics of the measured distribution. For *s*1/2 and *d*3/2, although the strengths are dominated by the lowest states, some strengths remain in the states slightly below 4 MeV due to the coupling to the 2<sup>+</sup> <sup>1</sup> state. Note that the sum of the experimental strengths for *d*3/2 exceeds the sum-rule limit [29], indicating non-negligible uncertainties due to the reaction model employed. For *d*5/2, the calculation well reproduces three major peaks located at 3–4, 5–6, and 7–8 MeV, although the calculated peaks are located a few hundred keV lower than those of the experiment. If the tensor force is omitted, the calculated weight of the *d*5/2 strengths is shifted higher and fails to reproduce the data as presented in [18].

For 40Ca, as seen in Figure 3, the *d*5/2 strengths are highly fragmented as in 48Ca. This property is impossible to reproduce with the same setup as 48Ca, since only one 5/2<sup>+</sup> state appears in the 0*h*¯ *ω* calculation. It is also found that the 2*h*¯ *ω* calculation was not sufficient to obtain enough fragmentation because of much smaller level densities compared with the data. To resolve this problem, the large-scale shell-model calculations were done to allow many-particle many-hole excitations across the *N* = *Z* = 20 core. Since it is still difficult to perform such calculations in the full *sd*-*p f* valence shell, the *p*1/2 and *f*5/2 orbitals are omitted from the valence shell, thus enabling 6*h*¯ *ω* calculations with the KSHELL code [26].

The effective interaction is taken from Ref. [31], a modified SDPF-M interaction whose single-particle energies are fine-tuned to reproduce the correct one-neutron separation energies of 40,41Ca. Note that the original SDPF-M interaction [11] was designed for the full *sd* + *f*7/2 + *p*3/2 model space. One expects that the 6*h*¯ *ω* truncation is sufficient to achieve convergent results. The resulting spin–orbit splitting of the *d* orbitals for the 40Ca core is close to that of SDPF-MU, 7.49 MeV, estimated from the ESPE.

Figure 3 presents the results of the calculations. Similar to 48Ca, the agreement with experiment is quite satisfactory. For *<sup>d</sup>*3/2 and *<sup>s</sup>*1/2, the strengths near the 2<sup>+</sup> <sup>1</sup> level of 40Ca (∼4 MeV) are much smaller than those for 48Ca, in good accordance with the measured distribution [30]. For *d*5/2, the calculated three major peaks at 5–6, ∼6, and 7–8 MeV well correspond to the measured peaks, although the highest peak is more fragmented in the experiment.

The above detailed comparisons of spectroscopic distributions confirm that a large reduction of the spin–orbit splitting, which amounts to ∼2 MeV, occurs in reality as a *π* + *ρ* meson exchange tensor force produces.
