Predictions of the interacting boson approximation in a consistentQframework

The predictions of the interacting boson approximation are studied in the consistent $Q$ formalism in which the same parametrization of the boson quadrupole operator is used in both the Hamiltonian and in the $E2\mathrm{}$ operator. In this scheme, wave functions, relative energies of states of the same spin, and all relative $B\left(E2\mathrm{}\right)\mathrm{}$ values depend on only a single parameter, ${\chi }_Q$, which appears in the internal structure of the operator $Q$. This feature allows a number of simple results to be obtained, principally through the construction of contour plots of various observables in terms of ${\chi }_Q$ and the boson number $N$. The entire SU(3)—O(6) region, including both limiting symmetries, can be treated by allowing ${\chi }_Q$ to vary between its respective limiting values for those two symmetries. For deformed nuclei, a number of characteristic features are obtained, involving the predicted decay of the $\gamma$ band and the energy and decay of the first $0^{+}$ excitation. It is shown that the dominance of the $\beta \to \gamma$ over $\beta \to g$ matrix elements and the near equality of $\beta \to \gamma$ and $\gamma \to g$ $E2\mathrm{}$ matrix elements are inherent features of the model. The automatic inclusion of band mixing in the interacting boson approximation is discussed in terms of the mixing parameter $Z_{\gamma }$ and it is shown that the interacting boson approximation reproduces the empirical systematics in $Z_{\gamma }$. The concepts of the intrinsic state formalism are reviewed in the context of the consistent $Q$ framework and shown to imply vanishing $\beta \to g$ transitions, for any boson number, in the absence of $K$ mixing effects. The O(6) limit obtained with the consistent $Q$ formalism is shown to be a special case of the general limit. Finally, transition regions are discussed, particularly the SU(3)—O(6) case, in terms of "trajectories" in ${\chi }_Q$ between its limiting values. A number of qualitative parameter-free predictions for the evolution of energy or $E2\mathrm{}$ branching ratios are thus obtained.