Comparative Study of the Liquid Drop and Collective Models for Nuclear Binding and Low-Energy Structural Properties of Selected Nuclei
This study comparatively evaluates the applicability of the liquid drop and collective nuclear models in describing the energy and structural properties of four selected even–even nuclei, ¹⁵²Sm, ¹⁶⁸Er, ¹³²Sn, and ²⁰⁸Pb. The nuclei were deliberately selected to represent two contrasting structural regimes: deformed, collective systems and nearly spherical, closed-shell systems. The study is a review-based comparative analysis supported by numerical evaluation and validation against published nuclear data. For the liquid-drop description, nuclear binding is considered through the semi-empirical mass-formula framework, including volume, surface, Coulomb, asymmetry, and pairing contributions. The collective description is assessed through low-lying excitation energies, rotational-band systematics, the ratio R₄/₂ = E(4₁⁺)/E(2₁⁺), and the quadrupole deformation parameter β₂. Binding-energy information was taken from the AME2020 evaluation, while level-structure information was drawn from ENSDF/NNDC and IAEA nuclear-data resources. The analyzed values give B/A of 8.24, 8.13, 8.36, and 7.87 MeV for ¹⁵²Sm, ¹⁶⁸Er, ¹³²Sn, and ²⁰⁸Pb, respectively. The corresponding E(2₁⁺) values are 121.8, 79.8, 4041, and 4085 keV, while β₂ is 0.30, 0.34, approximately zero, and approximately zero. The results show that the liquid-drop framework is effective for global binding-energy trends, whereas the collective model provides a more appropriate description of low-energy rotational behavior and deformation in ¹⁵²Sm and ¹⁶⁸Er. Neither framework alone describes all observed features; closed-shell effects require explicit microscopic treatment.