*4.2. Collective Bands and γ Vibration in* <sup>166</sup>*Er*

The features of the collective motion in 166Er have been studied by the MCSM similarly well (see Figure 15a). Among rotational nuclei, 166Er is characterized by particularly lowlying 2<sup>+</sup> <sup>2</sup> state and the *<sup>γ</sup>* band built on it. Aage Bohr stressed that this 2<sup>+</sup> <sup>2</sup> state was a *γ* vibration from the prolate ground state [9–11,13]. The relatively strong 2<sup>+</sup> <sup>2</sup> <sup>→</sup> <sup>0</sup><sup>+</sup> <sup>1</sup> E2 transition (B(E2)∼5 W.u., see Figure 15a, was ascribed to the annihilation of one *γ* phonon in the 2<sup>+</sup> <sup>2</sup> state. This was one of the major points of the Nobel lecture by Aage Bohr and has been a common sense as stated in many textbooks of nuclear physics. We now challenge

this traditional belief, by utilizing the recent MCSM calculation. It is reminded that no firm experimental evidence to uniquely pin down the *γ*-vibration nature of 166Er has been reported and also that in a systematic calculation of many heavy nuclei [78], the excitation energies of the 2<sup>+</sup> <sup>2</sup> states in the *γ* band appeared to be about twice higher than the observed values, despite much better description of those of the 2<sup>+</sup> <sup>1</sup> state in the ground band.

**Figure 15.** Experimental and calculated properties of the lowest states of 166Er. (**a**) Energy levels and electromagnetic transitions (W.u.) [42] as well as spectroscopic electric quadrupole moments (eb) [79]. (**b**) Three-dimensional PES and its cut surface for *β*<sup>2</sup> = 0.3. (**c**–**e**) T-plots for the 0<sup>+</sup> <sup>1</sup> and 2<sup>+</sup> 2 states and for the monopole-frozen 0<sup>+</sup> <sup>1</sup> state at spherical shape. Based on Figure 4 of [71].

Figure 15b shows the calculated PES, which shows the minimum not at *γ* = 0◦ but around *γ* = 9◦ (see also [80]). The T-plot is shown for the 0<sup>+</sup> <sup>1</sup> and 2<sup>+</sup> <sup>2</sup> states in Figure 15c and Figure 15d, respectively. The patterns of the T-plot circles are nearly identical between these two panels. This is consistent with a (rigid) triaxial interpretation, and indeed E2 transition strengths follow the predictions of the Davydov triaxial model [81,82] with *γ* = 9◦. Certainly, a pure rigid triaxiality is not the correct picture, and there are quantum fluctuations, as evident from Figure 15c,d [80]. After all, the displacement from the *γ* = 0◦ is obvious. The triaxiality of 166Er is also suggested by the triaxial projected shell model, although the rigid-triaxiality is not an outcome but an assumption [83,84].

The experimentally known *J<sup>π</sup>* = 4<sup>+</sup> state around 2 MeV excitation energy provides a long-standing puzzle [85,86]: the observed relatively strong E2 transition from this state to the 2<sup>+</sup> <sup>2</sup> state looks like a sign that the 2<sup>+</sup> <sup>2</sup> state and this *<sup>J</sup><sup>π</sup>* = 4<sup>+</sup> states are the singleand double-phonon states in the *γ* vibration picture (à la A. Bohr [9,10]), respectively, but the excitation energy of this *J<sup>π</sup>* = 4<sup>+</sup> state is too high for a double-phonon excitation. The present calculation, on the other hand, reproduces both the excitation energy and the E2 transition strength, and this *J<sup>π</sup>* = 4<sup>+</sup> state appears as the *K<sup>π</sup>* = 4<sup>+</sup> member of the triaxial states including the 0<sup>+</sup> <sup>1</sup> and 2<sup>+</sup> 1,2 states (see Figure 15c,d) [71,80]. Thus, the triaxiality is shown to be one of the key aspects for understanding/predicting the shapes of heavy nuclei.

The monopole-frozen analysis referring to the spherical CHF state shows that the ground state moves to *γ* = 0◦, confirming the important role of the monopole interaction activated. The triaxial ground states are now shown to appear in a large number of nuclei in the nuclear chart, besides the known triaxial domain [87].
