*4.1. Shape Coexistence in* <sup>154</sup>*Sm*

Figure 13 shows low-lying energy levels of 154Sm. The present MCSM calculation can describe the four low-lying bands including the negative-parity one. The agreement between the experimental levels in Figure 13a and the theoretical levels in Figure 13b is rather good. Although the importance of the quadrupole interaction is evident for the formation of deformed rotational bands, one can investigate to what extent the monopole interaction is involved. The monopole interaction here was obtained from the shell–model interactions, comprising the central, tensor, and other components.

**Figure 13. Left**: (**a**) Experimental energy levels [42], (**b**) calculated original energy levels and (**c**,**d**) monopole-frozen energy levels of 154Sm. **Right**: ESPE (vertical position) and occupation number (horizontal width). Taken from Figures 2 and 3 of [71].

The monopole interaction is an operator, but we "freeze" it now: its ESPE expectation values ˆ *p*,*n <sup>j</sup>* are calculated for the state to be specified, and the obtained values are adopted as the SPEs,  *p*,*n* 0;*<sup>j</sup>* in Equation (4), with the monopole interaction removed. We then perform the shell-model calculation and draw the PES. This toy game is called the "monopolefrozen" analysis [71], as the monopole properties are included only through the specified state. Figure 13c exhibits the energy levels obtained by the monopole-frozen analysis referring to the ground state. The band built on the 0<sup>+</sup> <sup>2</sup> state (often called the *β* band) is lifted up by 0.5 MeV (∼50% of the original excitation energy), suggesting that the active monopole interaction produces a substantial lowering of this state. Figure 13d shows the monopole-frozen analysis referring to the spherical HF state: the ground state is no longer prolate, but triaxial, with the wave function close to the 0<sup>+</sup> <sup>2</sup> state of the original Hamiltonian. Thus, the crucial effect of the monopole interaction is verified.

Figure 13 right shows the actual values of ˆ *p*,*n <sup>j</sup>* for the 0<sup>+</sup> <sup>1</sup> and 0<sup>+</sup> <sup>2</sup> states. This figure demonstrates the significant differences between two sets of the ESPE values. The occupation numbers are also different: there are more half-filled orbits for the 0<sup>+</sup> <sup>2</sup> state, which is indicative of its triaxial nature. The smaller occupation numbers of unique-parity orbits are also consistent with the tendency away from the prolate shape.

We now introduce the deformation parameters *β*<sup>2</sup> and *γ* [13], and their meanings are sketched in Figure 14a. The parameter *β*<sup>2</sup> represents the magnitude of the ellipsoidal deformation from sphere. The ellipsoid has three axes: the longest, middle, and shortest. The parameter *γ* is an angle between 0◦ and 60◦ and represents mutual relations among the lengths of these axes: *γ* = 0◦ means that the middle and shortest axes have the same length (prolate); *γ* = 60◦ implies that the longest and the middle ones have the same length (oblate); and *γ* values in between stand for intermediate situations, called triaxial. Figures 11 and 12 include them. The *β*<sup>2</sup> and *γ* parameters can be obtained, in some approximation, from intrinsic quadrupole moments through the formulas [72],

$$\beta\_2 = \sqrt{5/16\pi} \left\{ (\varepsilon + \varepsilon\_p' + \varepsilon\_n')/\varepsilon \right\} \left( 4\pi/3R\_0^2 A^{5/3} \right) \sqrt{(Q\_0)^2 + 2(Q\_2)^2},\tag{18}$$

and

$$\gamma = \arctan\left(\sqrt{2}Q\mathbf{z}/Q\_0\right),\tag{19}$$

where *e* is the unit charge; *e <sup>p</sup>* (*e n*) denotes proton (neutron) effective charge induced by inmedium (or core-polarization) effects; and *R*<sup>0</sup> stands for the radius parameter of the droplet model (spherical background) (see [73] for some detailed explanation). The relations in Equations (18) and (19) worked very well in many works, for instance [64,69–71].

Figure 14c,d shows the T-plot for the original interaction, where the PES is shown by using *β*<sup>2</sup> and *γ* as coordinates (see Figure 14a). Figure 14e,f depicts the T-plot for the monopole-frozen interaction obtained with the spherical HF state. The T-plot patterns are consistent with the above features suggested by the shell–model diagonalization. The cut of the PES shown in Figure 14b suggests that the local minimum is raised by the monopolefrozen process referring to the ground state.

Figure 14c,d depict a valley of the PES with a local minimum around *γ* = 15◦. Similar valleys are seen in the PES obtained by the mean-field calculations [74,75], implying that this valley likely has a common origin. On the other hand, one can state that the present monopole effect results in not only the valley but also the local minimum, and the latter plays essential roles in the formation and stability of the side bands. It is of interest to refine the monopole interaction in mean-field models.

Regarding the *β* vibration picture of the 0<sup>+</sup> <sup>2</sup> state, the present view is opposed to such a conventional view. The triaxial deformation is shared by the members not only of the 0<sup>+</sup> <sup>2</sup> band but also of the 2<sup>+</sup> <sup>3</sup> band (usually called *γ* band), as can be verified by their T-plots. Namely, the 0<sup>+</sup> <sup>2</sup> state is the "ground" state of the triaxial states to which both the 0<sup>+</sup> <sup>2</sup> and 2<sup>+</sup> <sup>3</sup> bands belong. In short, this is a shape coexistence between the prolate and triaxial shapes assisted by the interplay between the monopole interaction and the quadrupole deformation. It is noted that the *β* vibration picture of the 0<sup>+</sup> <sup>2</sup> states has been investigated from experimental viewpoints [76,77].

**Figure 14.** Properties of the 0<sup>+</sup> 1,2 states of 154Sm. (**a**) Deformation parameters and shapes. (**b**) Lowest values of PES for a given *γ* value for the original case (red) as well as for the prolate (blue) and spherical (green) monopole-frozen cases. (**c**–**f**) Three-dimensional T-plot in the original and spherical monopole-frozen cases. Based on Figure 3 of [71].
