Cluster and Halo Structures of Light Nuclei within the NUCLEI-PACK Framework

As part of the ongoing NUCLEI-PACK project, this study presents a semi-classical framework for exploring the microscopic geometry of light and exotic nuclei based on optimized sphere packing of nucleons and clusters. Starting from explicit nucleon coordinates generated by the packing algorithm, the model provides direct access to charge, matter, and core--valence radii, allowing quantitative analysis of clustering and halo formation. The study covers one-nucleon halo nuclei ($^{11}$Be, $^{15}$C, $^{19}$C, and $^{8}$B) and two-nucleon halo systems ($^{6}$He, $^{11}$Li, $^{19}$B, and $^{17}$Ne). For the halo systems, the fitted geometric offset parameter $\Delta$ exhibits an inverse correlation with the nucleon separation energy, reflecting the increasing spatial decoupling between the core and valence nucleons in weakly bound configurations. The framework also reproduces characteristic neutron--neutron separations and opening angles in Borromean nuclei and qualitatively captures the geometric arrangement of $\alpha$ clusters ($^{6}$Li, $^{7}$Li, and $^{12}$C). These results demonstrate that a simple geometric framework can effectively capture the essential features of both halo and cluster structures, providing an intuitive and computationally efficient link between nuclear geometry, binding, and experimentally observed radii.