Comparative Study of the Liquid Drop and Collective Models for Nuclear Binding and Low-Energy Structural Properties of Selected Nuclei

Abstract : 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.