Evolution of the “<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>β</mml:mi></mml:math>excitation” in axially symmetric transitional nuclei

The evolution of first excited ${0}^{+}$ states in transitional nuclei of the $N\ensuremath{\approx}90$ region is correlated with the structure of the ground band. Iachello's $X(5)$ solution for the Bohr-Hamiltonian for axially symmetric prolate nuclei is generalized to the transition path between $X(5)$ and the rigid-rotor limit using infinite square-well potentials in the quadrupole deformation parameter $\ensuremath{\beta}$ with boundaries at ${\ensuremath{\beta}}_{M}&gt;{\ensuremath{\beta}}_{m}\ensuremath{\geqslant}0$. Analytical solutions in terms of Bessel functions of first and second kind are derived and describe well the evolutionary trajectory of the studied ${0}_{2}^{+}$ states in this mass region.

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