**3. The Region of 42Si**

In this Section, results for two often used effective Hamiltonians for this model space, called SDPF-MU [82] and SDFP-U-SI [83], together with those based on the IM-SRG method [20] are compared. The MU and U-SI Hamiltonians are "universal" in the sense that a single Hamiltonian with a smooth mass-dependence is applied to a wide mass region. MU is used for all nuclei in this model space, while U-SI was designed for *Z* ≤ 14 (the SDPF-U version was designed for *Z* > 14 [83]).

The 2<sup>+</sup> energy in 42Si [*Z*, *N*]=[14, 28] (0.74 MeV) is low compared to those in 34Si [14,20] (3.33 MeV) and 48Ca [20,28] (3.83 MeV). 34Si and 48Ca are doubly magic due to the *LS* magic number 20. 28Si [14,14] has a known intrinsic oblate deformation [84].

The 2<sup>+</sup> energy in 20C [6,14] (1.62 MeV) is low compared to those in 14C [6,8] (7.01 MeV) and 22O [8,14] (3.20 MeV). 14C and 22O are doubly magic due to the *LS* magic number 8. Hartree–Fock calculations [85] as well as CI calculations for the *Q* moment within the *<sup>p</sup>* − *sd* model space [86] show that 12C and 20C have intrinsic oblate shapes.

The oblate shapes for 28Si and 42Si are shown by their *E*2 maps in Figures 10 and 11. The transition from spherical to oblate shapes for the *jj* doubly-magic numbers can be qualitatively understood in the Nilsson diagram as shown, for example, for 42Si in Figure 12. The highest filled Nilsson orbitals have rather flat energies between *β* = 0 and *β* = −0.3. The important aspect is the concave bend of the 2Ω*<sup>π</sup>* [N,n*z*,Λ]=1<sup>+</sup> [2,2,0] proton and 1<sup>−</sup> [3,3,0] neutron Nilsson orbitals for oblate shapes. For the heavier *jj* doubly-magic nuclei, increases and the *j* = - + 1/2 orbital decreases in energy, the bend will not be so large and the energy minima come closer to *β* = 0. This is illustrated in Figure 10. In Figure 10a and Figure 10c, the 0*d* spin–orbit gap is small enough to give an oblate rotational pattern. The oblate shape is manifest in the positive *Q* moments. In Figure 10a, the 0*d* spin–orbit gap is increased by 1 MeV and the rotational energy pattern is broken. The pattern in Figure 10a is similar to that obtained for 56Ni in the *f p* model space as shown in Figure 13. An interesting feature for 56Ni is the relatively strong 0<sup>+</sup> <sup>2</sup> to 2<sup>+</sup> <sup>1</sup> *B*(*E*2). I am not aware of a simple explanation for this.

**Figure 10.** Electric quadrupole (*E*2) maps for 28Si. The results shown are based on the universal *sd*-shell version-B (USDB) Hamiltonian with (**a**) the 0*d* spin–orbit energy gap increased by 1 MeV, (**b**) in the *sd* model space, and (**c**) with the 0*d* spin–orbit energy gap decreased by 1 MeV. For each *J* value, ten states were calculated. The widths of the lines are proportional to the reduced electric quadrupole transition strength, *B*(*E*2). Lines for *B*(*E*2) less than 5% of the largest value are not shown. The radius of the circles are proportional to spectroscopic quadrupole moment, *Qs* (2). To set the scale, for panel (**b**) the 2<sup>+</sup> <sup>1</sup> to 0<sup>+</sup> <sup>1</sup> *<sup>B</sup>*(*E*2) = 82 e<sup>2</sup> fm<sup>4</sup> and *Qs* (2<sup>+</sup> <sup>1</sup> ) = +19 e fm are used.

**Figure 11.** *E*2 maps for 42Si obtained with the SDPF-IMSRG [20] (**a**), SDPF-MU [82] (**b**), and SDPF-U-SI [83] (**c**) Hamiltonians. See Figure 10 and text for details.

**Figure 12.** Nilsson diagram for 42Si. At the deformation parameter *β* = 0, the orbitals are labeled by *nr*, -, 2*j* (see text for definitions), and, at larger deformation, the orbitals are labeled by the Nilsson quantum numbers 2Ω [N,n*z*,Λ]. The negative and positive parities is shown by the blue and red lines, respectively. The black dots show the highest Nilsson states occupied. This figure is made using the code WSBETA [87] with the potential choice ICHOIC = 3. The spin–orbit potential are reduced here for protons to make the spherical energies for the 0*d*3/2 and 1*s*1/2 orbitals approximately the same.

**Figure 13.** *E*2 map for 56Ni obtained with the GXFP1A Hamiltonian [88,89] in the full *f p* model space. See Figure 10 and text for details.

The oblate bands in 28Si and 42Si are linked to the 0*d*5/2 and 0 *f*7/2 orbitals. For completeness, the *E*2 maps for 12C and 20C obtained with the WBP Hamiltonian [90] are shown in Figure 14. For these nuclei, the oblate ground-state bands are linked with the 0*p*3/2 and 0*d*5/2 orbitals.

**Figure 14.** *E*2 maps for 12C (**bottom**) and 20C (**top**), obtained with the WBP Hamiltonian [90]. See Figure 10 and text for details.

For CI calculations, the *B*(*E*2) values depend on the effective charge parameters *ep* and *en*. In the harmonic-oscillator basis, the *E*2 operator connects states within a major shell as well as those that change *No* by two. The *E*2 strength function contains low-lying Δ*No* = 0 strength as well "giant-quadrupole" strength near an energy of 2*h*¯ *ω*. The effective charges account for the renormalization of the proton and neutron components of the *E*2 matrix elements within the CI basis of a major shell due to admixtures of the 1*p*–1*h*, Δ*No* = 2 proton configurations. For the calculations, shown here, effective charges, which depend on the model space, are used. The effective charges are chosen to best reproduce observed *B*(*E*2) values and quadrupole moments within that model space. These are the *sd* model space with *ep* = 0.45 and *en* = 0.36 [91], the *f p* model space with *ep* = *en* = 0.50 [88] and the neutron-rich *sd* − *p f* model space with *ep* = *en* = 0.35 [82]. Since low-lying excitations in nuclei are mostly isoscalar, only *ep* + *en* is well determined. It takes special situations such as a comparison of B(E2) in mirror nuclei [92] to obtain the isovector combination *ep* − *en*.

The isoscalar effective charge decreases for more neutron-rich nuclei (e.g., the drop from 0.5 in the *f p* model space to 0.35 in the *sd* model space). This can be understood by the macroscopic model of Mottelson [93], by the microscopic Hartree–Fock calculations of Sagawa et al. [85], and by the microscopic models, discussed in [94,95]. Microscopic models also give an orbital dependence to the effective charge. A recent example of this is for the relatively small B(E2) value for the the 1/2<sup>+</sup> to 5/2<sup>+</sup> transition in 21O [96]. This transition is dominated by the 1*s*1/2–0*d*5/2 *E*2 matrix element, and the relatively small neutron effective charge is due to the node in the 1*s*1/2 wavefunction.

The results for CI calculations for 42Si are shown in Figure 11 for three Hamiltonians. The IMSRG Hamiltonian is based on a VS-IMSRG calculation [20] similar to that used in [12]. The interpretation of the spectroscopic quadrupole moments, *Qs*, shown in Figure 11 in terms of an intrinsic shape, *Qo*, is given by the rotational formula [97],

$$Q\_s = \frac{2K^2 - J(J+1)}{(J+1)(2J+3)} Q\_o \,\text{e}\_{\prime} \tag{2}$$

with the Nilsson quantum number *K* = 0 for the ground-state bands in even–even nuclei. The MU [82] and IMSRG [20] calculations show an intrinsic oblate ground-state band, (*Qs* > 0 and *Qo* < 0), followed by a large energy gap to other more complex states. The U-SI Hamiltonian [83] also gives an oblate ground-state band, but there is also an intrinsic prolate band at relatively low energy. The presence of this low-lying prolate band dramatically increases the level density below 4 MeV [98,99].

The Nilsson diagram in Figure 12 shows a higher-energy prolate minimum related to a crossing of the 1− [3,2,1] and 7− [3,0,3] Nilsson orbitals near *β* = +0.3. At present, there is not enough experimental information to determine the energy of the prolate band in 42Si. The structure of 42Si is a touchstone for understanding all of the nuclei near the drip line in this mass region. More complete experimental results for the energy levels of 42Si are needed. The low-lying structure of 42Si depends on the details of the neutron ESPE that are affected by the continuum for the 0*p* orbitals. The deformed neutron ESPE need to be established by one-neutron transfer reactions on 42Si.

Deformation for *N* = 28 as a function of *Z* is determined by how the proton Nilsson orbitals are filled in Figure 12. When six protons are added to make 48Ca with *Z* = 20, there is a sharp energy minimum for protons at *β* = 0, and thus 48Ca is doubly magic. For 44S, the protons have a intrinsic prolate minimum near *β* = +0.2 where the neutrons are near the crossing of the 2Ω*<sup>π</sup>* = 1<sup>−</sup> and 7<sup>−</sup> orbitals [100]. In 44S, a *K* = 4<sup>+</sup> isomer at 2.27 MeV coming from the two quasi-particle state made from these two neutron orbitals was observed [101]. In 43S rotational bands associated with these, two Ω states have been observed [102]. All of these features are reproduced by CI calculations based on the SDPF-MU [82] and SDPF-U [83] Hamiltonians. At higher excitation energy, the CI energy spectra are more complex than anything that could be easily understood by the collective model.

The *E*2 map obtained with the SDPF-MU Hamiltonian for 40Mg is shown in Figure 15. In this case, the ground-state band has an intrinsic prolate shape. In the nuclear chart, prolate shapes are most common [103], in contrast to the oblate shapes obtained for *jj* magic numbers discussed above. The oblate shape for 40Mg can be understood in the Nilsson diagram of Figure 12. When two protons are removed, the energy minimum for protons shifts to positive *β* in the 3<sup>−</sup> [2,1,1] orbital. The experimental energy of the first 2<sup>+</sup> is 500(14) keV [9] compared to the result of 718 keV obtained with the SDPF-MU Hamiltonian. Models that explicitly include the -= 1 levels in the continuum are needed.

**Figure 15.** *E*2 map for 40Mg obtained with the SDFPF-MU Hamiltonian.
