Empirically determined limits on absolute boson numbers for well deformed nuclei

In the context of the consistent $Q$ formalism of the interacting boson approximation, which contains only one essential parameter, denoted $\chi$, it is possible to investigate the limits of allowable boson numbers $N$ for well deformed nuclei. This is done by examining the agreement with experiment of certain $E2$ branching ratios as a function of $N$. The results show that, in general, the best agreement is obtained for $N\approx N_{\mathrm{IBA}}$, where $N_{\mathrm{IBA}}$ is the usual interacting boson approximation value defined as the sum of half the number of valence protons and neutrons; moreover, significant discrepancies with experiment are obtained if $N<\left(\frac{3}{4}\right)N_{\mathrm{IBA}}$. The effect of additional terms in the interacting boson approximation Hamiltonian is also investigated.