3.1.2. From *N* = 28 to *N* = 32 and Beyond

As the neutron number increases from *N* = 28, the Fermi surface moves to *p*3/2, which causes a different proton shell evolution from that for 20 ≤ *N* ≤ 28. Figure 2a indicates that the most prominent is that *s*1/2 goes down relative to *d*3/2. This is caused by a positive <sup>Δ</sup>*πp*3/2 (*εν<sup>d</sup>*3/2 − *εν<sup>s</sup>*1/2 ) = <sup>4</sup>{*V*<sup>m</sup> *pn*(*d*3/2, *<sup>p</sup>*3/2) − *<sup>V</sup>*<sup>m</sup> *pn*(*s*1/2, *p*3/2)} because the *d*3/2-*p*3/2 and *s*1/2-*p*3/2 pairs are labeled "{+(−)}" and "{−0}". Although the tensor force causes attraction for the former pair, the central force that favors the latter surpasses this effect due to a larger spatial overlap. It is thus predicted that the *d*3/2 orbital becomes the highest in the *sd* shell again at *N* = 32, leading to the reinversion of the *s*1/2-*d*3/2 level ordering.

Similar to that of Section 3.1.1, K isotopes play a key role in probing this level ordering from experiment. The observed hyperfine structure ruled out a 1/2<sup>+</sup> ground state for the *N* = 32 isotope 51K [32], and its measured *g*-factor of +0.3420(15) [32] is very close to that of the single-proton hole in *d*3/2. From these data, it is concluded in Ref. [32] that the ground state of 51K must be a 3/2<sup>+</sup> state that is dominated by the *π*(*d*3/2)−<sup>1</sup> configuration. The predicted reinversion has thus been confirmed by experiment.

A deeper understanding of shell evolution can be obtained from excitation energies. In Figure 4, the energies of the 1/2<sup>+</sup> <sup>1</sup> levels, measured from the 3/2<sup>+</sup> <sup>1</sup> levels in neutron-rich K isotopes, are compared to theory. Very recently, the first excited levels in 51,53K (*N* = 32, 34) were measured to be 0.74 and 0.84 MeV, respectively [33]. These states are assigned to be 1/2<sup>+</sup> from the observed parallel momentum distributions of the 51,53K residues after (*p*, 2*p*) reactions.

As shown in Figure 4, the measured values are lower than the shell-model results with the SDPF-MU interaction, 1.40 and 1.74 MeV, respectively. Although the calculated levels are located lower than those estimated from the ESPE, 1.71 and 2.50 MeV, respectively; the deviation from the experimental data may indicate the need of refining the monopole matrix elements, related to the shell evolution under discussion.

In Ref. [33], a modified SDPF-MU interaction was introduced (named SDPF-MUs) in which *V*<sup>m</sup> *<sup>T</sup>*=0(*s*, *p*) is shifted by +0.4 MeV, equivalent to a +0.2 MeV shift for the proton– neutron channel. The resulting 1/2<sup>+</sup> <sup>1</sup> levels in 51,53K are improved to be 0.85 and 0.79 MeV, respectively. These SDPF-MUs levels are also somewhat lower than those estimated from its ESPE, 0.95 and 1.38 MeV, respectively. This difference is caused by a many-body correlation, which makes single-hole strengths fragmented. Experimentally, three more levels are observed from the 52Ca(*p*, 2*p*)51K reaction [33], which may indicate some deviation from the single-hole nature for 1/2<sup>+</sup> <sup>1</sup> or proton *d*5/2 hole states fragmented.

As shown in Figure 4, the *E*(1/2<sup>+</sup> <sup>1</sup> ) <sup>−</sup> *<sup>E</sup>*(3/2<sup>+</sup> <sup>1</sup> ) value evolves in a non-monotonic way; that is, it decreases until *N* = 28 and then turns to increase. This evolution, following the ESPE, is caused by that of the ESPE of *πd*3/2 measured from *πs*1/2. The reinversion of the 1/2<sup>+</sup> <sup>1</sup> -3/2<sup>+</sup> <sup>1</sup> level ordering is a consequence of the non-monotonic evolution of single-particle level spacings.

**Figure 4.** Comparison of the evolution of the energy difference *E*(1/2<sup>+</sup> <sup>1</sup> ) <sup>−</sup> *<sup>E</sup>*(3/2<sup>+</sup> <sup>1</sup> ) in neutron-rich K isotopes between experiment and theory. The red circles represent experimental data, and the blue diamonds and the green triangles stand for the results of large-scale shell-model calculations with the SDPF-MU and the SDPF-MUs interactions, respectively. The dashed lines in blue and green are the corresponding values evaluated from the ESPE (i.e., *επ*(*s*1/2)−<sup>1</sup> − *επ*(*d*3/2)−<sup>1</sup> = *επ<sup>d</sup>*3/2 − *επs*1/2 ) for the SDPF-MU and the SDPF-MUs interaction, respectively.

Let us point out that such a non-monotonic evolution constitutes a strong evidence for the dominance of the effective interaction in shell evolution because simple one-body potential models like the Woods–Saxon ones always produce monotonic evolution of level spacings with changing mass number. Furthermore, in this particular case, the non-monotonic evolution is caused by the central force. To account for this, let us first remind one that the changes of *επ<sup>d</sup>*3/2 − *επ<sup>s</sup>*1/2 for *N* = 20–28 and for *N* = 28–32 amounts, respectively, to <sup>Δ</sup>*E*<sup>1</sup> = <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, *<sup>f</sup>*7/2)} and <sup>Δ</sup>*E*<sup>2</sup> = <sup>4</sup>{*V*<sup>m</sup> *pn*(*d*3/2, *p*3/2) − *V*<sup>m</sup> *pn*(*s*1/2, *p*3/2)}.

For the tensor force, *V*<sup>m</sup> *pn*(*s*1/2, *f*7/2) = *V*<sup>m</sup> *pn*(*s*1/2, *f*7/2) = 0 holds, and only the first terms contribute to Δ*E*<sup>1</sup> and Δ*E*2. As shown in Table 1, both of them are negative, and the *επ<sup>d</sup>*3/2 − *επ<sup>s</sup>*1/2 value keeps decreasing. On the other hand, the central-force contributions to Δ*E*<sup>1</sup> and Δ*E*<sup>2</sup> are negative and positive, respectively, thus producing a kink in *E*(1/2<sup>+</sup> <sup>1</sup> ) <sup>−</sup> *<sup>E</sup>*(3/2<sup>+</sup> <sup>1</sup> ) and *επ<sup>d</sup>*3/2 − *επ<sup>s</sup>*1/2 . Since this non-monotonic evolution is dominated by the central force, any microscopic model, with a reasonable two-body force, is able to describe that. In fact, both nonrelativistic and relativistic mean-field models produce similar effects [34,35].

Here, let us comment on the idea behind the empirical shift of monopole matrix elements employed in the SDPF-MUs interaction. As presented in Section 2.1, the crossshell part of the SDPF-MU interaction consists of the central, two-body spin–orbit, and tensor terms. Among them, the tensor term is the most strongly supported by microscopic theories in terms of the "renormalization persistency", named in Ref. [16]. On the other hand, the central term is constructed in a fully phenomenological way. The two-body spin–orbit term is too small to tune.

On the basis of this general consideration, it seems that the most reasonable method of monopole tuning is for the central term alone, with the other terms untouched. The SDPF-MUs interaction is made to follow this policy. With respect to the cross-shell interaction, the difference between SDPF-MU and SDPF-MUs is the shift of *V*<sup>m</sup> *<sup>T</sup>*=0(*s*, *p*). The shift, Δ*V*<sup>m</sup> *<sup>T</sup>*=0(*s*, *p*)=+0.4 MeV, is applied not only to the *p*3/2 orbital but also to the *p*1/2 orbital. The latter change is needed to keep the tensor term unchanged after carrying out the spin-tensor decomposition [36].

Finally, let us mention that the *V*<sup>m</sup> *pn*(*s*, *p*) monopole matrix elements contain non-negligible contributions from the two-body spin–orbit force. This feature is discussed in Section 3.2.2.
