3.2.2. From *Z* = 16 to *Z* = 14

Two protons are removed from the *s*1/2 orbital as moving from *Z* = 16 to *Z* = 14. Although the *N* = 28 Si isotope, 42Si, is strongly deformed, the *N* = 20 isotope, 34Si, can be regarded as a doubly closed-shell nucleus: its first excited state is 0<sup>+</sup> (not 2+) and is located as high as 2.719(3) MeV [47]. In addition, a proton knockout experiment from 34Si [48] indicated small spectroscopic strengths of *s*1/2 below *Ex* ≈ 4 MeV, thus, suggesting a good *π*(*d*5/2)<sup>6</sup> closure in 34Si. For this reason, it is a good approximation to substitute the yrast levels in 35Si for the neutron effective single-particle energies on top of the 34Si core.

As shown in Figure 2b, the *N* = 28 shell gap changes by −Δ*πs*1/2 (*εν<sup>p</sup>*3/2 − *εν <sup>f</sup>*7/2 ) ≈ <sup>2</sup>{*V*<sup>m</sup> *pn*(*s*1/2, *<sup>f</sup>*7/2) − *<sup>V</sup>*<sup>m</sup> *pn*(*s*1/2, *p*3/2)} from *Z* = 16 to *Z* = 14. The *s*1/2-*f*7/2 and *s*1/2-*p*3/2 pairs are {+0} and {−0}, respectively, since the tensor force does not contribute to the monopole matrix elements for *s*1/2 . As discussed next, the spin–orbit force also adds a negative value for the *s*1/2-*p*3/2, and therefore the *N* = 28 shell gap should enlarge. This enlargement is estimated from the yrast levels in 35Si and 37S to be +0.264 MeV. The shellmodel calculations with the SDPF-MU interaction lead to +0.667 MeV, which is somewhat too large.

On the other hand, when one uses the SDPF-MUs interaction [33]—the one introduced in Section 3.1.2—to reproduce the 1/2<sup>+</sup> levels in 51,53K, this value is modified to be +0.317 MeV. Note that the −Δ*πs*1/2 (*εν<sup>p</sup>*3/2 − *εν <sup>f</sup>*7/2 ) values estimated from the ESPEs of SDPF-MU and SDPF-MUs are +0.78 and +0.37 MeV, respectively. These two independent experimental data—K isotopes and *N* = 21 isotones—consistently require about a +0.2 MeV modification of *V*<sup>m</sup> *pn*(*s*, *p*) matrix elements for the SDPF-MU interaction. This looks like due to the uncertainty of the central force that is determined empirically with a simple potential.

Next, the evolution of the *N* = 32 shell gap is discussed. The 34Si(*d*, *p*) reaction experiment in inverse kinematics found two prominent *l* = 1 peaks at 0.910 and 2.044 MeV, the former and the latter of which should be the 3/2− and 1/2− levels, respectively [14]. The interval of these two levels, 1.134 MeV, is much smaller than the corresponding value of 37S, 1.911 MeV. If these values are identical with the spin–orbit splitting between the *p* orbitals, the data point to a sharp reduction of 0.857 MeV. Since the matrix elements for the *s*1/2-*p*3/2 and *s*1/2-*p*1/2 pairs have no tensor contributions and the same central strengths (see Table 1), only the spin–orbit force can change this shell gap in terms of the shell model.

As pointed out in Section 2.1, the two-body spin orbit force produces particularly large monopole matrix elements between the *s* and *p* orbitals. The reduction of the *p* orbital splitting is evaluated from the ESPEs of the SDPF-MU interaction to be 0.54 MeV, while the actual shell-model calculation produces a 0.758 MeV reduction of the 3/2− <sup>1</sup> -1/2<sup>−</sup> <sup>1</sup> level splitting in going from 37S to 35Si. Hence, although the two-body spin–orbit force is the dominant source of the observed reduction, correlation energy may account for the energy of a hundred keV order.

The origin of the observed reduction is still controversial. It is claimed [49] that Woods– Saxon potentials well account for the observed reduction of the spin–orbit splitting in going from the 40Ca to 34Si and that this occurs due to weak binding for lower *Z* isotopes. This effect causes a gradual reduction with decreasing *Z*, whereas the two-body spin–orbit force affects the *p* orbital splitting primarily with *s*1/2 filled. Hence, one of the key issues to discriminate these effects is to establish how sharp this reduction occurs from the 36S to 34Si cores compared to that occurring from the 40Ca to 36S cores. Although one-neutron adding spectroscopic factors are measured for the 36S and 40Ca cores, the experimental uncertainty does not converge within the required accuracy (see the Supplemental Material of Ref. [1]).
