3.2.3. From *Z* = 14 to *Z* = 8

Finally, protons in *d*5/2 are removed from *Z* = 14 to *Z* = 8. As shown in Figure 2b, the *N* = 28 shell gap sharply decreases again. Note that Figure 2b presents neutron ESPEs for the *N* = 28 cores. When a similar figure is drawn for the *N* = 20 cores, the ESPE of *p*3/2 shifts downward by ∼2 MeV , and the neutron *f*7/2 and *p*3/2 orbitals cross at around *Z* = 11. In this Section, the evolution of the *N* = 28 shell gap is examined; other gaps are difficult to access with the current experimental capability.

The change of the *N* = 28 shell gap with *Z* decreasing from 14 to 8 is estimated to be −Δ*πd*5/2 (*εν<sup>p</sup>*3/2 − *εν <sup>f</sup>*7/2 ) ≈ <sup>6</sup>{*V*<sup>m</sup> *pn*(*d*5/2, *<sup>f</sup>*7/2) − *<sup>V</sup>*<sup>m</sup> *pn*(*d*5/2, *p*3/2)}. The *d*5/2-*f*7/2 and *d*5/2-*p*3/2 pairs are {−+} and {+(+)}, respectively. Although the tensor force produces a slightly larger positive value for the former pair, the central attraction overrides this effect, thus, causing a negative value in total.

For such proton deficient isotopes, one cannot obtain sufficient experimental information from the nuclei around *N* = 28. Moreover, *N* = 20 isotones do not provide direct data for the present purpose because some isotopes in the "island of inversion" are strongly deformed. Hence, one relies on single-particle levels on top of the *N* = 16 cores, although *N* = 16 does not form a good closed shell except for with oxygen.

In Figure 6, the 3/2− <sup>1</sup> energy levels relative to 7/2<sup>−</sup> <sup>1</sup> are compared for experiment vs. theory. The data for 27Ne, 29Mg, and 31Si indicate a nearly linear change of these energies. Since the relevant one-neutron adding spectroscopic factors are not large, i.e., typically ∼0.5, as measured [50–52], these energy differences cannot be identified with the *N* = 28 shell gap. However, the linear evolution reminds one of the famous "Talmi plot" [2], which successfully predicted the 1/2<sup>+</sup> <sup>1</sup> level in 11Be from the linearity. Thus, this behavior is worthy of particular attention.

One can see from Figure 6 that the measurements are in a good agreement with the calculations based on the SDPF-MU and SDPF-MUs interactions. The SDPF-MUs interaction achieves better agreement because its *N* = 28 shell gap on for the 34Si core is improved (see Section 3.2.2). These two interactions are quite successful in reproducing the slope of *Ex*(3/2<sup>−</sup> <sup>1</sup> ) − *Ex*(7/2<sup>−</sup> <sup>1</sup> ).

**Figure 6.** Evolution of the *N* = 28 shell gap in going from *Z* = 8 to 14 estimated from the *Ex*(3/2− <sup>1</sup> ) − *Ex*(7/2<sup>−</sup> <sup>1</sup> ) values in the *N* = 17 isotones (solid lines) and from the ESPE calculations (dashed lines). Data are from Refs. [50–52].

As one can also see from Figure 6, the slope is quite similar to what the ESPE predicts. Since the *V*<sup>m</sup> *pn*(*d*5/2, *f*7/2) and *V*<sup>m</sup> *pn*(*d*5/2, *p*3/2) values are kept unchanged in making the SDPF-MUs interaction based on SDPF-MU, the *νp*3/2 ESPEs are parallel. On the other hand, these ESPEs are shifted downward in parallel from *Ex*(3/2<sup>−</sup> <sup>1</sup> ) − *Ex*(7/2<sup>−</sup> <sup>1</sup> ) by ∼1.5 MeV. This difference arises from the assumption of the *ν*(*d*5/2)6(*s*1/2)<sup>2</sup> closure taken here to evaluate the ESPE.

However, in reality, a significant number of neutron excitations to *<sup>d</sup>*3/2 occur in the 27Ne, 29Mg, and 31Si eigenstates. These neutron excitations attract a neutron in the *<sup>f</sup>*7/2 orbital more than a one in *p*3/2 because the *T* = 1 monopole matrix element of *d*3/2-*f*7/2 is more attractive than that of *d*3/2-*p*3/2, thus, shifting *Ex*(3/2<sup>−</sup> <sup>1</sup> ) − *Ex*(7/2<sup>−</sup> <sup>1</sup> ) upward. Such significant neutron excitation to *d*3/2 occurs similarly in the Ne, Mg, and Si isotopes. Hence, the evolution of *Ex*(3/2<sup>−</sup> <sup>1</sup> ) − *Ex*(7/2<sup>−</sup> <sup>1</sup> ) is predominantly changed by the ESPE, providing evidence for the narrowing *N* = 28 shell gap caused by the monopole matrix element *V*<sup>m</sup> *pn*(*d*5/2, *<sup>f</sup>*7/2) − *<sup>V</sup>*<sup>m</sup> *pn*(*d*5/2, *p*3/2).

It should be noted that the predicted *Ex*(3/2<sup>−</sup> <sup>1</sup> ) − *Ex*(7/2<sup>−</sup> <sup>1</sup> ) at *Z* = 8 is closer to the ESPE estimate than those of other isotopes. This is due to the fact that the assumed *N* = 16 closure works better at *Z* = 8 due to the occurrence of the *N* = 16 magic number.
