Predictions of the interacting boson approximation in a consistent<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>Q</mml:mi></mml:math>framework

Abstract

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 $E2$ operator. In this scheme, wave functions, relative energies of states of the same spin, and all relative $B(E2)$ values depend on only a single parameter, ${\ensuremath{\chi}}_{Q}$, 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 ${\ensuremath{\chi}}_{Q}$ and the boson number $N$. The entire SU(3)---O(6) region, including both limiting symmetries, can be treated by allowing ${\ensuremath{\chi}}_{Q}$ 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 $\ensuremath{\gamma}$ band and the energy and decay of the first ${0}^{+}$ excitation. It is shown that the dominance of the $\ensuremath{\beta}\ensuremath{\rightarrow}\ensuremath{\gamma}$ over $\ensuremath{\beta}\ensuremath{\rightarrow}g$ matrix elements and the near equality of $\ensuremath{\beta}\ensuremath{\rightarrow}\ensuremath{\gamma}$ and $\ensuremath{\gamma}\ensuremath{\rightarrow}g$ $E2$ 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 ${Z}_{\ensuremath{\gamma}}$ and it is shown that the interacting boson approximation reproduces the empirical systematics in ${Z}_{\ensuremath{\gamma}}$. The concepts of the intrinsic state formalism are reviewed in the context of the consistent $Q$ framework and shown to imply vanishing $\ensuremath{\beta}\ensuremath{\rightarrow}g$ 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 ${\ensuremath{\chi}}_{Q}$ between its limiting values. A number of qualitative parameter-free predictions for the evolution of energy or $E2$ branching ratios are thus obtained.NUCLEAR STRUCTURE Consistent $Q$ formalism of the IBA. Energy and $B(E2)$ predictions, contour plots, ${\ensuremath{\chi}}_{Q}$ trajectories, intrinsic state formalism.