Analytical and Mathematical Framework of Collective Dynamics in Medium-Mass Nuclei

Abstract

The theoretical treatment of atomic nuclei as strongly coupled quantum many-body systems requires a unified mathematical description that connects collective macroscopic surface deformations with microscopic single-particle shell states, pairing correlations, and experimental target validation. Building upon classical liquid drop dynamics, Mayer-Jensen spin orbit shell models, and BCS pairing theory, this paper presents a rigorous analytical formulation of quadrupole surface oscillations coupled to single-particle Woods-Saxon orbitals. We derive explicit differential forms for the collective Bohr Hamiltonian, quantify quadrupole deformed potential splitting via perturbation theory, formulate quasiparticle gap equations, and evaluate transition probability matrices B(E2) across rotational-vibrational states. Furthermore, we integrate experimental diagnostics using Scanning Electron Microscopy coupled with Energy-Dispersive X-ray Spectroscopy (SEM-EDX) for validating enriched isotopic target purity, surface morphology, and thickness uniformity. Synthetic functional graphs, vector SEM target schematics, EDX spectral profiles, and pairing gap curves are provided to demonstrate shell-gap evolution and shape-phase transitions in medium-mass nuclei.