This study investigates the nuclear structure of Te and Xe isotopes using algebraic collective models. These mid-mass nuclei exhibit collective excitations ranging from spherical vibrations to varying degrees of deformation. Te isotopes are treated as near-spherical or weakly deformed systems and analyzed using the interacting boson model (IBM). The \({}^{118}\) Te isotope is described within the U(5) symmetry of the IBM, which represents spherical vibrational behavior. In heavier Te isotopes (A = 120–130), features intermediate between U(5) and O(6) symmetries appear, thereby indicating mixed-mode structures. These are modeled by introducing perturbative contributions from the O(6) Casimir operator into the U(5)-based Hamiltonian. The study also applies Iachello’s critical-point symmetry E(5), which characterizes the phase transition between spherical and \(\gamma\) -soft shapes, to the \({}^{128}\) Xe and \({}^{130}\) Xe isotopes. Calculations of low-lying energy spectra and electromagnetic transition probabilities are used to assess consistency with E(5) symmetry. These results offer new insights into shape evolution in mid-mass nuclei and highlight the effectiveness of algebraic models in describing nuclear phase transitions.
