Comment on top-on-top mechanism in triaxial strongly deformed even mass nuclei

We have derived the algebraic solution to the particle-rotor model with high $j$ nucleon coupled to a triaxially deformed core, $H=H_{\rm rot} +H_{\rm sp}$. The rotating core top with ${\vec R}={\vec I}-{\vec j}$ and the single-particle top with ${\vec j}$, are strongly correlating each other. We call this mechanism as top-on-top mechanism, where the Coriolis term, ${\vec I}\cdot{\vec j}$ in $H_{\rm rot}$, is explicitly taken into account, giving a big difference from the wobbling model. The algebraic solution to the top-on-top mechanism clarifies not only the energy level scheme, but also gives the approximate selection rules in the strength of transitions among bands. If the single-particle angular momentum $j$ is assumed to be the sum of two angular momenta as $j=j_1 +j_2$ and the value of integer $j$ keeps constant over some range of $I$, then the algebraic solution is easily extended to the even-even nucleus with alignment of integer $j$. Although several candidates of TSD bands are observed in Hf isotopes, no linking transitions between (0,0) and (1,0) are found. The rough estimation of the transition rates give a factor of $(\frac{I-j}{I})^3$ both in $B(E2)$ and $B(M1)$ values for the transitions with $\Delta I=1$ among the favored (0,0) and the unfavored (1,0) bands. The value of $I-j$ is smaller for even-$A$ case than odd-$A$ case, which makes the observation of the other partner band difficult.