Conclusions and Discussion

The vanishing of intra-band E2 transitions, supporting the tetrahedral symmetry interpretation at the bottom of the odd-spin negative-parity band, has been confirmed along with the two-orders-of-magnitude differences in the $B$(E2)/$B$(E1) branching ratios of two negative-parity bands. Thanks to the $\gamma\gamma\gamma$ coincidences, we have established new inter-band transitions. As it is known from general considerations, cf. Ref. [9], a $K^\pi=0^-$ band must not have even spins and it follows that the even-spin negative-parity band discussed here must not be interpreted as $0^-$, in contrast to some first claims in the past. We were not able to establish the $4^-\to2^-$ transition which most likely signifies that it is very weak or non-existent. Therefore, it will be even more important to find-out whether the $\Delta I=1$ transitions connecting the even- and odd-spin negative-parity bands have the $M1$-character, which could suggest the presence of high-$K$ components in the underlying band-heads.

Let us emphasize at this point that the tetrahedral configurations, as predicted by theory, are markedly non-axial and, therefore, are expected to strongly mix components of wave-functions with various quantum numbers $K$: the strongest component associated with the geometry of shapes based on the $Y_{3+2}+Y_{3-2}$ spherical harmonics should be $K=2$. Theoretical calculations based on the generalised collective rotor Hamiltonian that includes terms of the third order in angular momentum² indicate that the structure of the wave-function of the $1^-$ state is exceptional since, in contrast to states with $I\geq2$, it must not manifest the tetrahedral symmetry. In other words, for $I\leq1$ the tetrahedral symmetry is excluded; actually $1^-$ state wave-function manifest an axial symmetry. Consequently, the role of the $1^-$ state, often treated as a member of the (expected to be) the tetrahedral band, is special in that even if connected to the $3^-$ state via an E2 transition, in principle possible due to an expected to be strong a K-mixing, its underlying symmetry must not be tetrahedral. Our experiement, similarly to the preceding ones, gives no sign of the $3^-\to1^-$ transition either what signifies that the corresponding E2 transition, if exists, must be very weak.