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

Starting from the original collective Hamiltonian of Bohr and separating the β and γ variables as in the X(5) model of Iachello, an exactly soluble model corresponding to a harmonic oscillator potential in the β variable [to be called X(5)-β²] is constructed. Furthermore, it is proved that the potentials of the form β²ⁿ (with n being an integer) provide a ``bridge'' between this new X(5)-β² model (occurring for $n=1$) and the X(5) model (corresponding to an infinite well potential in the β variable, materialized for n→∞). Parameter-free (up to overall scale factors) predictions for spectra and $B(E2)$ transition rates are given for the potentials β², β⁴, β⁶, β⁸, corresponding to R₄=E(4)∕E(2) ratios of 2.646, 2.769, 2.824, and 2.852, respectively, compared to the R₄ 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.