Sequence of potentials lying between the U(5) and X(5) symmetries

Starting from the original collective Hamiltonian of Bohr and separating the $\ensuremath{\beta}$ and $\ensuremath{\gamma}$ variables as in the X(5) model of Iachello, an exactly soluble model corresponding to a harmonic oscillator potential in the $\ensuremath{\beta}$ variable [to be called $\mathrm{X}(5)\text{\ensuremath{-}}{\ensuremath{\beta}}^{2}$] is constructed. Furthermore, it is proved that the potentials of the form ${\ensuremath{\beta}}^{2n}$ (with $n$ being an integer) provide a ``bridge'' between this new $\mathrm{X}(5)\text{\ensuremath{-}}{\ensuremath{\beta}}^{2}$ model (occurring for $n=1$) and the X(5) model (corresponding to an infinite well potential in the $\ensuremath{\beta}$ variable, materialized for $n\ensuremath{\rightarrow}\ensuremath{\infty}$). Parameter-free (up to overall scale factors) predictions for spectra and $B(E2)$ transition rates are given for the potentials ${\ensuremath{\beta}}^{2}$, ${\ensuremath{\beta}}^{4}$, ${\ensuremath{\beta}}^{6}$, ${\ensuremath{\beta}}^{8}$, corresponding to ${R}_{4}=E(4)∕E(2)$ ratios of 2.646, 2.769, 2.824, and 2.852, respectively, compared to the ${R}_{4}$ ratios of 2.000 for U(5) and 2.904 for X(5). Hints about nuclei showing this behavior, as well as about potentials ``bridging'' the X(5) symmetry with SU(3) are briefly discussed.

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