Spectroscopic criteria for identification of nuclear tetrahedral and octahedral symmetries: Illustration on a rare earth nucleus

We formulate criteria for identification of the nuclear tetrahedral and octahedral symmetries and illustrate for the first time their possible realization in a rare earth nucleus ${}^{152}\mathrm{Sm}$. We use realistic nuclear mean-field theory calculations with the phenomenological macroscopic-microscopic method, the Gogny-Hartree-Fock-Bogoliubov approach, and general point-group theory considerations to guide the experimental identification method as illustrated on published experimental data. Following group theory the examined symmetries imply the existence of exotic rotational bands on whose properties the spectroscopic identification criteria are based. These bands may contain simultaneously states of even and odd spins, of both parities and parity doublets at well-defined spins. In the exact-symmetry limit those bands involve no $E2$ transitions. We show that coexistence of tetrahedral and octahedral deformations is essential when calculating the corresponding energy minima and surrounding barriers, and that it has a characteristic impact on the rotational bands. The symmetries in question imply the existence of long-lived shape isomers and, possibly, new waiting point nuclei—impacting the nucleosynthesis processes in astrophysics—and an existence of 16-fold degenerate particle-hole excitations. Specifically designed experiments which aim at strengthening the identification arguments are briefly discussed.

Figure 1

Total nuclear energy for the nucleus ${}^{152}\mathrm{Sm}$, calculated using the standard macroscopic-microscopic method with the realistic phenomenological mean field in the form of the so-called Woods-Saxon universal realization of Ref. [19]. The vertical axis corresponds to the ${\alpha }_{32}$-multipole deformation which represents the tetrahedral-symmetry nuclear shape. At each $({\alpha }_{32},{\alpha }_{20})$ point the energy has been minimized over the octahedral deformations $o_1$ and $o_2$. Here and in other figures the energies are normalized assuming that the liquid-drop contribution is zero at zero deformation; $E_o$ gives then the shell and pairing energy at the spherical shape, whereas ${\mathrm{E}}_{\min }$ gives the energy at the absolute minimum.

Figure 2

Total energy surface of the ${}^{152}\mathrm{Sm}$ nucleus projected on the $(o_1,{\alpha }_{32})$ plane, $o_1$ defined in Eq. (5). Here all other deformations are set to zero.

Figure 3

Similar to Fig. 1, but for the axial-hexadecapole deformation ${\alpha }_{40}$ replacing the tetrahedral one, ${\alpha }_{32}$, at the vertical axis. For increasing quadrupole deformation an increase in the axial hexadecapole deformation causes a decrease in the energy of the ground-state minimum stronger as compared to the octahedral one, $E_{\min }=-3.71$ MeV compared to $-2.60$ MeV, cf. Figs. 1 and 4 (for explanation see text).

Figure 4

Similar to Fig. 3, but illustrating the impact of $o_1$. The absolute minimum lies higher, $E_{\min }=-2.60$ MeV compared to $E_{\min }=-3.71$ MeV in Fig. 3.

Figure 5

Graphical representation of the experimental data from Table 2. Curves represent the fit according to Eq. (11) and are not meant “to guide the eye.” Point $\left[0^{+}\right]$, marked with square brackets, represents a prediction by extrapolation; it is not an experimental datum.