3.2.1. From *Z* = 20 to *Z* = 16

As shown in Figure 2b, with protons removed from *d*3/2, the *N* = 28 shell gap changes by −Δ*πd*3/2 (*εν<sup>p</sup>*3/2 − *εν <sup>f</sup>*7/2 ) ≈ <sup>4</sup>{*V*<sup>m</sup> *pn*(*f*7/2, *<sup>d</sup>*3/2) − *<sup>V</sup>*<sup>m</sup> *pn*(*p*3/2, *d*3/2)}. Note that the negative sign in −Δ*πd*3/2 is needed because the shell evolution is considered with decreasing *Z*. Since *f*7/2-*d*3/2 and *p*3/2-*d*3/2 are {−−} and {+(−)} pairs, respectively, this quantity should be negative. As discussed in Section 3.1.1, the strongly attractive monopole matrix element of *V*<sup>m</sup> *pn*(*f*7/2, *d*3/2) causes the rapid decrease of the *Z* = 16 shell gap in going from *N* = 20 to *N* = 28.

The decrease of the *N* = 28 shell gap is difficult to evaluate from experimental data in the vicinity of *N* = 28 isotones because the corresponding sulfur isotopes are deformed. As emphasized in Section 2.2, however, this decrease can be probed from another isotone chain. In this case, *N* = 20 isotones provide useful information since both 36S and 40Ca are regarded as doubly-closed-shell nuclei with rather large first excitation energies (>3 MeV).

From the 36S(*d*, *p*)37S reaction data, the *ν f*7/2 and *νp*3/2 strengths are concentrated in the ground state and the 0.646 MeV state, respectively, while some strengths remain in the 3/2<sup>−</sup> state at 3.263 MeV with *<sup>C</sup>*2*<sup>S</sup>* ≈ 0.14 and in the 7/2<sup>−</sup> state at 3.443 MeV with *<sup>C</sup>*2*<sup>S</sup>* ≈ 0.06 [37]. The measured spectroscopic strengths, thus, indicate a small *<sup>N</sup>* = <sup>28</sup> shell gap that is less than 1 MeV on top of the 36S core. The shell-model calculation with the SDPF-MU interaction rather well reproduces this feature with *C*2*S*(*f*7/2) = 0.86 at *Ex* = 0 MeV, *C*2*S*(*p*3/2) = 0.77 at *Ex* = 0.56 MeV, *C*2*S*(*p*3/2) = 0.19 at *Ex* = 2.97 MeV, and *C*2*S*(*f*7/2) = 0.07 at *Ex* = 3.21 MeV. The calculated *N* = 28 shell gap for the 36S core is 0.32 MeV.

Similar data exist for the 40Ca core. The *p*3/2 strengths are fragmented into the states at 1.94, 2.46, and 4.60 MeV, which is impossible to reproduce with the 0*h*¯ *ω* calculations. The centroids of the spectroscopic factors measured with the 40Ca(*d*, *p*)41Ca reaction [38] suggest that the *N* = 28 shell gap for 40Ca is 2.5 MeV. The SDPF-MU interaction produces the *N* = 28 shell gap of 2.94 MeV, which is slightly larger than this value. Hence, a large decrease of the *N* = 28 shell gap is confirmed, although the SDPF-MU interaction may overestimate this decrease by a few hundred keV.

The reduction of the *N* = 28 shell gap should have a significant impact on the *N* = 28 closed-shell structure. The breaking of the *N* = 28 closure can be probed with one-neutron removal spectroscopic strengths from *p*3/2: if no *νp*3/2 strengths are observed, then no neutrons occupy the *p*3/2 orbital, implying a complete closure. Although summing up all the *p*3/2 strengths are desirable for a quantitative evaluation, excited states available in neutron-rich nuclei are limited. For this purpose, the strengths of the first 3/2− levels between experiment and theory are compared and the results are shown in Figure 5.

It is natural that the strength for 48Ca is very small. As the proton number is away from *Z* = 20, the strengths are naively expected to increase due to deformation caused by valence proton particles or holes. If deformation is controlled by the number of valence protons alone, those spectroscopic factors should be symmetric with respect to *Z* = 20. However, the observed spectroscopic factors are rather large for the *Z* < 20 isotones, whereas they remain small for the *Z* > 20 isotones.

**Figure 5.** One-neutron removal spectroscopic factors of the 3/2− <sup>1</sup> states from the ground states of *N* = 28 isotones. The crosses denote the calculated neutron occupation numbers in the ground states of the *N* = 28 isotones. Data are from Refs. [39] (*Z* = 20, 22, and 24), [40] (*Z* = 18), and [41] (*Z* = 16).

This behavior is well reproduced by the shell-model calculations with the SDPF-MU interaction. The same trend is seen in the *νp*3/2 occupation numbers, which are the upper limit of these spectroscopic factors. Hence, one concludes that the breaking of the *N* = 28 closure is much greater for *Z* < 20 than for *Z* > 20 and that the reduction of the *N* = 28 shell gap for lower *Z* works to enhance this property.

From Figure 5, it may look unexpected that the *C*2*S* value for *Z* = 16 is only half the neutron *p*3/2 occupation number of 44S unlike that for 46Ar. This is caused by a unique nuclear structure of 44S. As pointed out in Ref. [25], sulfur isotopes around 44S have two nearly degenerate deformed neutron orbitals on the Fermi surface with Ω*<sup>π</sup>* = 7/2<sup>−</sup> and 3/2−, which make the *K<sup>π</sup>* = 7/2<sup>−</sup> and 3/2<sup>−</sup> bands in 43S, respectively, by one-neutron occupation. Here, Ω and *K* are, respectively, single-particle and total angular-momentum projection onto the symmetry axis, and *π* is parity. The 3/2− <sup>1</sup> state in 43S is a *<sup>K</sup><sup>π</sup>* <sup>=</sup> 3/2<sup>−</sup> member. Due to the near degeneracy of the Ω*<sup>π</sup>* = 7/2<sup>−</sup> and 3/2<sup>−</sup> orbitals, the ground state of 44S has a strongly mixed configuration with two neutrons in Ω*<sup>π</sup>* = 7/2<sup>−</sup> and those in Ω*<sup>π</sup>* = 3/2−. As a result, about half the ground-state wave function of 44S, i.e., the part with two neutrons occupying the Ω*<sup>π</sup>* = 3/2−, is able to contribute to populating the *K<sup>π</sup>* = 3/2<sup>−</sup> band in 43S. The remaining fractions of *C*2*S* should be distributed to the excited 3/2<sup>−</sup> states, which was indeed observed [41].

Let us comment on other shell gaps. The discussions of the *N* = 32 shell gap is given in Section 3.2.2, and here, just a brief remark to be made about the *N* = 34 shell gap. A recent 54Ca(*p*, *pn*)53Ca measurement clarified that the *N* = 34 shell closure is rather good [42], while the *<sup>N</sup>* = 34 shell gap for the 54Ca core was estimated to be ∼ 2.5 MeV from the GXPF1Br interaction [43]. It was predicted that this shell gap enlarges with decreasing *Z* and that the fingerprint of the enlargement can be seen in the 2<sup>+</sup> <sup>1</sup> energies of the *N* = 34 isotones with *Z* < 20 [44,45].

This prediction was confirmed later by measuring the 2<sup>+</sup> <sup>1</sup> level in 52Ar that is located at 1.656(18) MeV [46]. Interestingly, this level is even higher than that of the *N* = 28 isotope, 46Ar. The change of the *N* = 34 shell gap from *Z* = 20 to 16 is expressed as −Δ*πd*3/2 (*εν <sup>f</sup>*5/2 − *εν<sup>p</sup>*1/2 ) ≈ <sup>4</sup>{*V*<sup>m</sup> *pn*(*d*3/2, *<sup>p</sup>*1/2) − *<sup>V</sup>*<sup>m</sup> *pn*(*d*3/2, *f*5/2)}. The *d*3/2-*p*1/2 and *d*3/2 *f*5/2 are {+(+)} and {−+} pairs, respectively. Since the former pair is the most unfavored combination in energy in terms of both the central and tensor forces, this value is positive leading to the enlargement of the *N* = 34 shell gap. Experimental evaluation of this enhancement is difficult for *N* = 34 cores, but it is, however, possible for *N* = 20 cores

through spectroscopic strengths. Although the measured *f*5/2 strengths are not complete for the 36S core, such an enlargement possibly occurs from the existing data (Figure 3 of [14]).
